International
Tables for Crystallography Volume A1 Symmetry relations between space groups Edited by Hans Wondratschek and Ulrich Müller © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. A1, ch. 2.1, pp. 4258
doi: 10.1107/97809553602060000542 Chapter 2.1. Guide to the subgroup tables and graphs^{a}Institut für Kristallographie, Universität, D76128 Karlsruhe, Germany, and ^{b}Departamento de Física de la Materia Condensada, Facultad de Ciencias, Universidad del País Vasco, Apartado 644, E48080 Bilbao, Spain This chapter is the users' guide to the subgroup tables and graphs of Part 2 of this volume. Sections 2.1.1 and 2.1.2 refer to the contents, arrangement and structure of the subgroup tables of Chapters 2.2 and 2.3 . In the next three sections, the data listed for the maximal translationengleiche subgroups (Section 2.1.3), maximal klassengleiche subgroups (Section 2.1.4) and for the series of the (infinitely many) maximal isomorphic subgroups (Section 2.1.5) are described in detail with many examples. Section 2.1.6 describes the data listed for minimal supergroups. The last section, Section 2.1.7, familiarizes the user with the subgroup graphs of the space groups listed in Chapters 2.4 and 2.5 . The chapter is supplemented by the subgroup table of the space group R3 , which is printed inside the front cover of the volume with some explanatory text. 
In this chapter, the subgroup tables, the subgroup graphs and their general organization are discussed. In the following sections, the different types of data are explained in detail. For every plane group and every space group there is a separate table of maximal subgroups and minimal supergroups. These items are listed either individually, or as members of (infinite) series, or both. In addition, there are graphs of translationengleiche and klassengleiche subgroups which contain for each space group all kinds of subgroups, not just the maximal ones.
The presentation of the planegroup and spacegroup data in the tables of Chapters 2.2 and 2.3 follows the style of the tables of Parts 6 (plane groups) and 7 (space groups) in Vol. A of International Tables for Crystallography (2005), henceforth abbreviated as IT A. The data comprise:
For the majority of groups, the data can be listed completely on one page. Sometimes two pages are needed. If the data extend less than half a page over one full page and data for a neighbouring spacegroup table `overflow' to a similar extent, then the two overflows are displayed on the same page. Such deviations from the standard sequence are indicated on the relevant pages by a remark Continued on… . The two overflows are separated by a rule and are designated by their headlines.
The sequence of the plane groups and space groups in this volume follows exactly that of the tables of Part 6 (plane groups) and Part 7 (space groups) in IT A. The format of the subgroup tables has also been chosen to resemble that of the tables of IT A as far as possible. Examples of graphs of subgroups can also be found in Section 10.1.4.3 of IT A, but only for subgroups of point groups. The graphs for the space groups are described in Section 2.1.7.
Some basic data in these tables have been repeated from the tables of IT A in order to allow the use of the subgroup tables independently of IT A. These data and the main features of the tables are described in this section. More detailed descriptions are given in the following sections.
The headline contains the specification of the space group for which the maximal subgroups are considered. The headline lists from the outside margin inwards:
As in IT A, for each plane group and space group a set of symmetry operations is listed under the heading `Generators selected'. From these group elements, can be generated conveniently. The generators in this volume are the same as those in IT A. They are explained in Section 2.2.10 of IT A and the choice of the generators is explained in Section 8.3.5 of IT A.
The generators are listed again in this present volume because many of the subgroups are characterized by their generators. These (often nonconventional) generators of the subgroups can thus be compared with the conventional ones without reference to IT A.
Like the generators, the general position has also been copied from IT A, where an explanation can be found in Section 2.2.11 . The general position in IT A is the first block under the heading `Positions', characterized by its site symmetry of 1. The elements of the general position have the following meanings:
Many of the subgroups in these tables are characterized by the elements of their general position. These elements are specified by numbers which refer to the corresponding numbers in the general position of . Other subgroups are listed by the numbers of their generators, which again refer to the corresponding numbers in the general position of . Therefore, the listing of the general position of as well as the listing of the generators of is essential for the structure of these tables. For examples, see Sections 2.1.3 and 2.1.4.
All 17 planegroup types ^{1} and 230 spacegroup types are listed and described in IT A. However, whereas each planegroup type is represented exactly once, 44 spacegroup types, i.e. nearly 20%, are represented twice. This means that the conventional setting of these 44 spacegroup types is not uniquely determined and must be specified. The same settings underlie the data of this volume, which follows IT A as much as possible.
There are three reasons for listing a spacegroup type twice:
If there is a choice of setting for the space group , the chosen setting is indicated under the HM symbol in the headline. If a subgroup belongs to one of these 44 spacegroup types, its `conventional setting' must be defined. The rules that are followed in this volume are explained in Section 2.1.2.5.
As in the subgroup data of IT A, the sequence of the maximal subgroups is as follows: subgroups of the same kind are collected in a block. Each block has a heading. Compared with IT A, the blocks have been partly reorganized because in this volume all maximal isomorphic subgroups are listed, whereas in IT A only a few of them are described. In addition, the subgroups are described here in more detail.
The sequence of the subgroups within each block follows the value of the index; subgroups of lowest index are listed first. Subgroups having the same index are listed according to their lattice relations to the lattice of the original group , cf. Section 2.1.4.3. Subgroups with the same lattice relations are listed in decreasing order of spacegroup number.
Conjugate subgroups have the same index and the same spacegroup number. They are grouped together and connected by a brace on the lefthand side. Conjugate classes of maximal subgroups and their lengths are therefore easily recognized. In the series of maximal isomorphic subgroups, braces are inapplicable so here the conjugacy classes are stated explicitly.
The block designations are:
The multiple listing of 44 spacegroup types has implications for the subgroup tables. If a subgroup belongs to one of these types, its `conventional setting' must be defined. In many cases there is a natural choice; sometimes, however, such a choice does not exist, and the appropriate conventions have to be stated.
The three reasons for listing a space group twice will be discussed in this section, cf. Section 2.1.2.3.

Remarks (see also the following examples):

The necessary adjustment is performed through a coordinate transformation, i.e. by a change of the basis and by an origin shift, see Section 2.1.3.3.
Example 2.1.2.5.1
, No. 10; unique axis b.
II Maximal klassengleiche subgroups, Enlarged unit cell

Example 2.1.2.5.2
, No. 10; unique axis c.
II Maximal klassengleiche subgroups, Enlarged unit cell

Example 2.1.2.5.3
, No. 50; origin choice 1.
I Maximal (monoclinic) translationengleiche subgroups

Altogether, 24 orthorhombic, tetragonal and cubic space groups with inversions are listed twice in IT A. There are three kinds of possible ambiguities for such groups with two origin choices:

The seven trigonal space groups with a rhombohedral lattice are often called rhombohedral space groups. Their HM symbols begin with the lattice letter R and they are listed with both hexagonal axes and rhombohedral axes.
Rules:

Remarks:

Example 2.1.2.5.7
, No. 146. Maximal klassengleiche subgroups of index 2 and 3. Comparison of the subgroup data for the two settings of shows that the subgroups (145), (144) and (143) of index 3 appear in the block `Loss of centring translations' for the hexagonal setting and in the block `Enlarged unit cell' for the rhombohedral setting.
The sequence of the blocks has priority over the classification by increasing index. Therefore, in the setting `hexagonal axes', the subgroups of index 3 precede the subgroup of index 2. The complete general position is listed for the maximal ksubgroups of index 3 in the setting `hexagonal axes'; only the generator is listed for rhombohedral axes.
In this block, all maximal tsubgroups of the plane groups and the space groups are listed individually. Maximal tsubgroups are always nonisomorphic.
For the sequence of the subgroups, see Section 2.1.2.4. There are no lattice relations for tsubgroups because the lattice is retained. Therefore, the sequence is determined only by the rising value of the index and by the decreasing spacegroup number.
The listing is similar to that of IT A and presents on one line the following information for each subgroup : Conjugate subgroups are listed together and are connected by a left brace.
The symbols have the following meaning:

Remarks:

The description of the subgroups can be explained by the following four examples.
Example 2.1.3.2.3
, No. 137, ORIGIN CHOICE 2
I Maximal translationengleiche subgroups
…
(59, ) 1; 2; 5; 6; 9; 10; 13; 14
Comments:

Example 2.1.3.2.4
: , No. 151
I Maximal translationengleiche subgroups
Comments:

Each tsubgroup is defined by its representatives, listed under `sequence' by numbers each of which designates an element of . These elements form the general position of . They are taken from the general position of and, therefore, are referred to the coordinate system of . In the general position of , however, its elements are referred to the coordinate system of . In order to allow the transfer of the data from the coordinate system of to that of , the tools for this transformation are provided in the columns `matrix' and `shift' of the subgroup tables. The designation of the quantities is that of IT A Part 5 and is repeated here for convenience.
In the following, columns and rows are designated by boldface italic lowercase letters. Point coordinates , translation parts of the symmetry operations and shifts are represented by columns. The sets of basis vectors and are represented by rows [indicated by , which means `transposed']. The quantities with unprimed symbols are referred to the coordinate system of , those with primes are referred to the coordinate system of .
The following columns will be used ( is analogous to w):
The matrices W and of the symmetry operations, as well as the matrix P for a change of basis and its inverse , are designated by boldface italic uppercase letters ( is analogous to W): Let be the row of basis vectors of and the basis of , then the basis is expressed in the basis by the system of equations ^{2}or
The column p of coordinates of the origin of is referred to the coordinate system of and is called the origin shift. The matrix–column pair (P, p) describes the transformation from the coordinate system of to that of , for details, cf. IT A, Part 5 . Therefore, P and p are chosen in the subgroup tables in the columns `matrix' and `shift', cf. Section 2.1.3.2. The column `matrix' is empty if there is no change of basis, i.e. if P is the unit matrix I. The column `shift' is empty if there is no origin shift, i.e. if p is the column o consisting of zeroes only.
A change of the coordinate system, described by the matrix–column pair (P, p), changes the point coordinates from the column x to the column . The formulae for this change do not contain the pair (P, p) itself, but the related pair :
Not only the point coordinates but also the matrix–column pairs for the symmetry operations are changed by a change of the coordinate system. A symmetry operation is described in the coordinate system of by the system of equations i.e. by the matrix–column pair (W, w). The symmetry operation will be described in the coordinate system of the subgroup by the equation and thus by the pair . This pair can be calculated from the pair by solving the equations and
Example 2.1.3.3.1
Consider the data listed for the tsubgroups of , No. 31:This means that the matrices and origin shifts are

Example 2.1.3.3.2
Evaluation of the tsubgroup data of the space group , No. 151, started in Example 2.1.3.2.4. The evaluation is now continued with the columns `sequence', `matrix' and `shift'. They are used for the transformation of the elements of to their conventional form. Only the monoclinic tsubgroups are of interest here because the trigonal subgroup is already in the standard setting.
One takes from the tables of subgroups in Chapter 2.3 Designating the three matrices by , , , one obtains with the corresponding inverse matrices and the origin shifts For the three new bases this means All these bases span orthohexagonal cells with twice the volume of the original hexagonal cell because for the matrices holds.
In the general position of , No.151, one findsThese entries represent the matrix–column pairs : Application of equations (2.1.3.8) on the matrices and (2.1.3.9) on the columns of the matrix–column pairs results in All translation vectors of are retained in the subgroups but the volume of the cells is doubled. Therefore, there must be centringtranslation vectors in the new cells. For example, the application of equation (2.1.3.9) with to the translation of with the vector , i.e. , results in the column , i.e. the centring translation of the subgroup. Either by calculation or, more easily, from a small sketch one sees that the vectors for , for (and for ) correspond to the cellcentring translation vectors of the subgroup cells.
Comments:
This example reveals that the conjugation of conjugate subgroups does not necessarily imply the conjugation of the representatives of these subgroups in the general positions of IT A. The three monoclinic subgroups in this example are conjugate in the group by the screw rotation. Conjugation of the representative (4) by the screw rotation of results in the representative (5) with the column , which is not exactly the representative (5) but one of its translationally equivalent elements of retained in .
The listing of the maximal klassengleiche subgroups (maximal ksubgroups) of the space group is divided into the following three blocks for practical reasons:
Loss of centring translations. Maximal subgroups of this block have the same conventional unit cell as the original space group . They are always nonisomorphic and have index 2 for plane groups and index 2, 3 or 4 for space groups.
Enlarged unit cell. Under this heading, maximal subgroups of index 2, 3 and 4 are listed for which the conventional unit cell has been enlarged. The block contains isomorphic and nonisomorphic subgroups with this property.
Series of maximal isomorphic subgroups. In this block all maximal isomorphic subgroups of a space group are listed in a small number of infinite series of subgroups with no restriction on the index, cf. Sections 2.1.2.4 and 2.1.5.
The description of the subgroups is the same within the same block but differs between the blocks. The partition into these blocks differs from the partition in IT A, where the three blocks are called `maximal nonisomorphic subgroups IIa', `maximal nonisomorphic subgroups IIb' and `maximal isomorphic subgroups of lowest index IIc'.
The kind of listing in the three blocks of this volume is discussed in Sections 2.1.4.2, 2.1.4.3 and 2.1.5 below.
Consider a space group with a centred lattice, a space group whose HM symbol does not start with the lattice letter P but with one of the letters A, B, C, F, I or R. The block contains those maximal subgroups of which have fully or partly lost their centring translations and thus are not tsubgroups. The conventional unit cell is not changed.
Only in space groups with an Fcentred lattice can the centring be partially lost, as is seen in the list of the space group Fmmm, No. 69. On the other hand, for , No. 196, the maximal subgroups , No. 195, or , No. 198, have lost all their centring translations.
For the block `Loss of centring translations', the listing in this volume is the same as that for tsubgroups, cf. Section 2.1.3. The centring translations are listed explicitly where applicable, e.g. for space group , No. 5, unique axis b
In this line, the representatives of the general position are .
The listing differs from that in IT A in only two points:

The sequence of the subgroups in this block is one of decreasing spacegroup number of the subgroups.
Under the heading `Enlarged unit cell', those maximal ksubgroups are listed for which the conventional unit cell is enlarged relative to the unit cell of the original space group . All maximal ksubgroups with enlarged unit cell of index 2, 3 or 4 of the plane groups and of the space groups are listed individually. The listing is restricted to these indices because 4 is the highest index of a maximal nonisomorphic subgroup, and the number of these subgroups is finite. Maximal subgroups of higher indices are always isomorphic to and their number is infinite.
The description of the subgroups with enlarged unit cell is more detailed than in IT A. In the block IIb of IT A, different maximal subgroups of the same spacegroup type with the same lattice relations are represented by the same entry. For example, the eight maximal subgroups of the type , No. 69, of space group , No. 47, are represented by one entry in IT A.
In the present volume, the description of the maximal subgroups in the block `Enlarged unit cell' refers to each subgroup individually and contains for each of them a set of spacegroup generators and the transformation from the setting of the space group to the conventional setting of the subgroup .
Some of the isomorphic subgroups listed in this block may also be found in IT A in the block `Maximal isomorphic subgroups of lowest index IIc'.
Subgroups with the same lattice are collected in small blocks. The heading of each such block consists of the index of the subgroup and the lattice relations of the sublattice relative to the original lattice. Basis vectors that are not mentioned are not changed.
Example 2.1.4.3.1
This example is taken from the table of space group , No. 20.
Enlarged unit cell
The entries mean:
Columns 1 and 2: HM symbol and spacegroup number of the subgroup; cf. Section 2.1.3.2.
Column 3: generators, here the pairsfor the six lines listed in the same order.
Column 4: basis vectors of referred the basis vectors of . means ; means .
Column 5: origin shifts, referred to the coordinate system of . These origin shifts by o, a and 2a for the first triplet of subgroups and o, b and 2b for the second triplet of subgroups are translations of . The subgroups of each triplet are conjugate, indicated by the left braces.
Often the lattice relations above the data describing the subgroups are the same as the basis vectors in column 4, as in this example. They differ in particular if the sublattice of is nonconventionally centred. Examples are the Hcentred subgroups of trigonal and hexagonal space groups.
The sequence of the subgroups is determined
For sublattices with twice the volume of the unit cell, the sequence of the different cell enlargements is as follows:

With a few exceptions for trigonal, hexagonal and cubic space groups, ksubgroups with enlarged unit cells and index 3 or 4 are isomorphic. To each of the listed sublattices belong either one or several conjugacy classes with three or four conjugate subgroups or one or several normal subgroups. Only the sublattices with the numbers (5)(a)(v), (5)(b)(i), (5)(c)(ii), (6)(iii) and (7)(i) have index 4, all others have index 3. The different cell enlargements are listed in the following sequence:
(1) Triclinic space groups:
(2) Monoclinic space groups:
(a) Space groups , , , , (unique axis b):
(b) Space groups , , , , (unique axis c):
(c) Space groups , , (unique axis b):
(d) Space groups , , (unique axis c):
(e) All space groups with C lattice (unique axis b):
(f) All space groups with A lattice (unique axis c):
(3) Orthorhombic space groups:
(4) Tetragonal space groups:
(5) Trigonal space groups:
(a) Trigonal space groups with hexagonal P lattice:
(b) Trigonal space groups with rhombohedral R lattice and hexagonal axes:
(c) Trigonal space groups with rhombohedral R lattice and rhombohedral axes:
(6) Hexagonal space groups:
(7) Cubic space groups with P lattice:
Maximal subgroups of index higher than 4 have index p, or , where p is prime, are necessarily isomorphic subgroups and are infinite in number. Only a few of them are listed in IT A in the block `Maximal isomorphic subgroups of lowest index IIc'. Because of their infinite number, they cannot be listed individually, but are listed in this volume as members of series under the heading `Series of maximal isomorphic subgroups'. In most of the series, the HM symbol for each isomorphic subgroup will be the same as that of . However, if is an enantiomorphic space group, the HM symbol of will be either that of or that of its enantiomorphic partner.
Example 2.1.5.1.1
Two of the four series of isomorphic subgroups of the space group , No. 76, are (the data on the generators are omitted):
On the other hand, the corresponding data for , No. 78, are
Note that in both tables the subgroups of the type , No. 78, are listed first because of the rules on the sequence of the subgroups.
If an isomorphic maximal subgroup of index is a member of a series, then it is listed twice: as a member of its series and individually under the heading `Enlarged unit cell'.
Most isomorphic subgroups of index 3 are the first members of series but those of index 2 or 4 are rarely so. An example is the space group , No. 77, with isomorphic subgroups of index 2 (not in any series) and 3 (in a series); an exception is found in space group , No. 75, where the isomorphic subgroup for is the first member of the series .
The conventional basis of the unit cell of each isomorphic subgroup in the series has to be defined relative to the basis of the original space group. For this definition the prime p is frequently sufficient as a parameter.
Example 2.1.5.2.1
The isomorphic subgroups of the space group , No. 93, can be described by two series with the bases of their members:
In other cases, one or two positive integers, here called q and r, define the series and often the value of the prime p.
Example 2.1.5.2.2
In space group , No. 174, the series is listed. The values of q and r have to be chosen such that while and p is prime.
Example 2.1.5.2.3
In the space group , No. 11, unique axis c, the series is listed. Here p and q are independent and q may take the p values for each value of p.
Each of the sublattices discussed in Section 2.1.4.3.2 is common to a conjugacy class or belongs to a normal subgroup of a given series. The subgroups in a conjugacy class differ by the positions of their conventional origins relative to the origin of the space group . To define the origin of the conventional unit cell of each subgroup in a conjugacy class, one, two or three integers, called u, v or w in these tables, are necessary. For a series of subgroups of index p, or there are p, or conjugate subgroups, respectively. The positions of their origins are defined by the p or or permitted values of u or u, v or u, v, w, respectively.
Example 2.1.5.3.1
The space group , , No. 112, has two series of maximal isomorphic subgroups . For one of them the lattice relations are , listed as for the transformation matrix. The index is . For each value of p there exist exactly conjugate subgroups with origins in the points , where the parameters u and v run independently: and .
In another type of series there is exactly one (normal) subgroup for each index p; the location of its origin is always chosen at the origin of and is thus not indicated as an origin shift.
Example 2.1.5.3.2
Consider the space group , No. 29. Only one subgroup exists for each value of p in the series . This is indicated in the tables by the statement `no conjugate subgroups'.
The generators of the p (or or ) conjugate isomorphic subgroups are obtained from those of by adding translational components. These components are determined by the parameters p (or q and r, if relevant) and u (and v and w, if relevant).
Example 2.1.5.4.1
Space group , No. 198.
In the series defined by the lattice relations and the origin shift there exist exactly conjugate subgroups for each value of p. The generators of each subgroup are defined by the parameter p and the triplet in combination with the generators (2), (3) and (5) of . Consider the subgroup characterized by the basis and by the origin shift . One obtains from the generator (2) of the corresponding generator of by adding the translation vector to the translation vector of the generator (2) of and obtains , so that this generator of is written .
For most space groups, there is only one description of their series of the isomorphic subgroups. However, if a space group is described twice in IT A, then there are also two different descriptions of these series. This happens for monoclinic space groups with the settings unique axis b and unique axis c, for some orthorhombic, tetragonal and cubic space groups with origin choice 1 and origin choice 2 and for trigonal space groups with rhombohedral lattices with hexagonal axes and rhombohedral axes.
In the monoclinic space groups, the series in the listings `unique axis b' and `unique axis c' are closely related by a simple cyclic permutation of the axes a, b and c, see IT A, Section 2.2.16 .
In trigonal space groups with rhombohedral lattices, the series with hexagonal axes and with rhombohedral axes appear to be rather different. However, the `rhombohedral' series are the exact transcript of the `hexagonal' series by the same transformation formulae as are used for the different monoclinic settings. However, the transformation matrices P and in Part 5 of IT A are more complicated in this case.
Example 2.1.5.5.1
Space group , No. 148. The second series is described with hexagonal axes by the basis transformation , i.e. and the origin shift . We discuss the basis transformation first. It can be written in analogy to Part 5 , IT A. Here is the row of basis vectors of the conventional hexagonal basis. The matrix X is defined byWith rhombohedral axes, equation (2.1.5.1) would be written with the matrix Y to be determined.
The transformation from hexagonal to rhombohedral axes is described by where the matricesare listed in IT A, Table 5.1.3.1 , see also Figs. 5.1.3.6 (a) and (c) in IT A.
Applying equations (2.1.5.3), (2.1.5.1) and (2.1.5.2), one gets From equation (2.1.5.4) it follows that One obtains Y from equation (2.1.5.5) by matrix multiplication, and from Y for the bases of the subgroups with rhombohedral axesThe column of the origin shift in hexagonal axes must be transformed by . The result is the column in rhombohedral axes.
Space groups with two origin choices are always described in the same basis, but origin 1 is shifted relative to origin 2 by the shift vector s. For most space groups with two origins, the appearance of the two series related by the origin shift is similar; there are only differences in the generators.
Example 2.1.5.5.2
Consider the space group , No. 48, in both origin choices and the corresponding series defined by and . In origin choice 1, the generator (5) of is described by the `coordinates' . The translation part of the third generator of stems from the term in the first `coordinate' of the generator (5) of . Because must be a translation vector of , p is odd. Such a translation part is not found in the generators (2) and (3) of because the term does not appear in the `coordinates' of the corresponding generators of .
The situation is inverted in the description for origin choice 2.
The translation term appears in the first and second generator of and not in the third one because the term occurs in the first `coordinate' of the generators (2) and (3) of but not in the generator (5).
The term appears in both descriptions. It is introduced in order to adapt the generators to the origin shift .
In other space groups described in two origin choices, surprisingly, the number of series is different for origin choice 1 and origin choice 2.
Example 2.1.5.5.3
In the tetragonal space group , No. 141, for origin choice 1 there is one series of maximal isomorphic subgroups of index , p prime, with the bases and origin shifts . For origin choice 2, there are two series with the same bases but with the different origin shifts and . What are the reasons for these results?
For origin choice 1, the term appears in the first and second `coordinates' of all generators (2), (3), (5) and (9) of . This term is the cause of the translation vectors and in the generators of .
For origin choice 2, fractions and appear in all `coordinates' of the generator (3) of . As a consequence, translational parts with vectors and appear if . On the other hand, translational parts with vectors are introduced in the generators of if holds.
Another consequence of the fractions and occurring in the generator (3) of is the difference in the origin shifts. They are for and for (mod 4). Thus, the one series in origin choice 1 for odd p is split into two series in origin choice 2 for and . ^{3}
In the previous sections, the relation was seen from the viewpoint of the group . In this case, was a subgroup of . However, the same relation may be viewed from the group . In this case, is a supergroup of . As for the subgroups of , cf. Section 1.2.6 , different kinds of supergroups of may be distinguished. The following definitions are obvious.
Definition 2.1.6.1.1. Let be a maximal subgroup of . Then is called a minimal supergroup of . If is a translationengleiche subgroup of then is a translationengleiche supergroup (tsupergroup) of . If is a klassengleiche subgroup of , then is a klassengleiche supergroup (ksupergroup) of . If is an isomorphic subgroup of , then is an isomorphic supergroup of . If is a general subgroup of , then is a general supergroup of .
The search for supergroups of space groups is much more difficult than the search for subgroups. One of the reasons for this difficulty is that the search for subgroups is restricted to the elements of the space group itself, whereas the search for supergroups has to take into account the whole (continuous) group of all isometries. For example, there are only a finite number of subgroups of any space group for any given index i. On the other hand, there may not only be an infinite number of supergroups of a space group for a finite index i but even an uncountably infinite number of supergroups of .
Example 2.1.6.1.2
Let . Then there is an infinite number of tsupergroups of index 2 because there is no restriction for the sites of the centres of inversion and thus of the conventional origin of .
In the tables of this volume, a supergroup of a space group is listed by its type if is listed as a subgroup of . The entry contains at least the index of in , the conventional HM symbol of and its spacegroup number. Additional data may be given for klassengleiche supergroups. More details, e.g. the representatives of the general position or the generators as well as the transformation matrix and the origin shift, would only duplicate the subgroup data. The number of supergroups belonging to one entry can neither be concluded from the subgroup data nor is it listed among the supergroup data.
Like the subgroup data, the supergroup data are also partitioned into blocks.
For each space group , under this heading are listed those spacegroup types for which appears as an entry under the heading I Maximal translationengleiche subgroups. The listing consists of the index in brackets […], the conventional HM symbol and (in parentheses) the spacegroup number (…). The space groups are ordered by ascending spacegroup number. If this line is empty, the heading is printed nevertheless and the content is announced by `none', as in , No. 191.
The supergroups listed on the line I Minimal translationengleiche supergroups are realized only if the lattice conditions of fulfil the lattice conditions for . For example, if , No. 89, is a supergroup of , No. 16, two of the three independent lattice parameters a, b, c of must be equal (or in crystallographic practice, approximately equal). These must be a and b if c is the tetragonal axis, b and c if a is the tetragonal axis or c and a if b is the tetragonal axis. In the latter two cases, the setting of has to be adapted to the conventional caxis setting of . For the cubic supergroup , No. 195, all three lattice parameters of must be (approximately) equal. Such conditions are always to be taken into consideration if the tsupergroup belongs to a different crystal family^{4} to the original group. Therefore, for there is no lattice condition for the supergroup because and belong to the same crystal family.
Klassengleiche supergroups always belong to the crystal family of . Therefore, there are no restrictions for the lattice parameters of .
The block II Minimal nonisomorphic klassengleiche supergroups is divided into two subblocks with the headings Additional centring translations and Decreased unit cell. If both subblocks are empty, only the heading of the block is listed, stating `none' for the content of the block, as in , No. 191.
If at least one of the subblocks is nonempty, then the heading of the block and the headings of both subblocks are listed. An empty subblock is then designated by `none'; in the other subblock the supergroups are listed. The kind of listing depends on the subblock. Examples may be found in the tables of , No. 16, and , No. 228.
Under the heading `Additional centring translations', the supergroups are listed by their indices and either by their nonconventional HM symbols, with the spacegroup numbers and the standard HM symbols in parentheses, or by their conventional HM symbols and only their spacegroup numbers in parentheses. Examples are provided by space group , No. 61, with both subblocks nonempty and by space group , No. 16, with supergroups only under the heading `Additional centring translations'.
Under the heading `Decreased unit cell' each supergroup is listed by its index and by its lattice relations, where the basis vectors , and refer to the supergroup and the basis vectors a, b and c to the original group . After these data are listed either the nonconventional HM symbol, followed by the spacegroup number and the conventional HM symbol in parentheses, or the conventional HM symbol with the spacegroup number in parentheses. Examples are provided again by space group , No. 61, with both subblocks occupied and space group , No. 216, with an empty subblock `Additional centring translations' but data under the heading `Decreased unit cell'.
Each space group has an infinite number of isomorphic subgroups because the number of primes is infinite. For the same reason, each space group has an infinite number of isomorphic supergroups . They are not listed in the tables of this volume because they are implicitly listed among the subgroup data.
The group–subgroup relations between the space groups may also be described by graphs. This way is chosen in Chapters 2.4 and 2.5 . Graphs for the group–subgroup relations between crystallographic point groups have been published, for example, in Internationale Tabellen zur Bestimmung von Kristallstrukturen (1935) and in IT A (2005), Fig. 10.1.4.3 . Three kinds of graphs for subgroups of space groups have been constructed and can be found in the literature:

A complete collection of graphs of the first two kinds is presented in this volume: in Chapter 2.4 those displaying the translationengleiche or tsubgroup relations and in Chapter 2.5 those for the klassengleiche or ksubgroup relations. Neither type of graph is restricted to maximal subgroups but both contain t or ksubgroups of higher indices, with the exception of isomorphic subgroups, cf. Section 2.1.7.3 below.
The group–subgroup relations are direct relations between the space groups themselves, not between their types. However, each such relation is valid for a pair of space groups, one from each of the types, and for each space group of a given type there exists a corresponding relation. In this sense, one can speak of a relation between the spacegroup types, keeping in mind the difference between space groups and spacegroup types, cf. Section 1.2.5.3 .
The space groups in the graphs are denoted by the standard HM symbols and the spacegroup numbers. In each graph, each spacegroup type is displayed at most once. Such graphs are called contracted graphs here. Without this contraction, the more complex graphs would be much too large for the page size of this volume.
The symbol of a space group is connected by uninterrupted straight lines with the symbols of all maximal nonisomorphic subgroups or minimal nonisomorphic supergroups of . In general, the maximal subgroups of are drawn on a lower level than ; in the same way, the minimal supergroups of are mostly drawn on a higher level than . For exceptions see Section 2.1.7.3. Multiple lines may occur in the graphs for tsubgroups. They are explained in Section 2.1.7.2. No indices are attached to the lines. They can be taken from the corresponding subgroup tables of Chapter 2.3 , and are also provided by the general formulae of Section 1.2.8 . For the ksubgroup graphs, they are further specified at the end of Section 2.1.7.3.
Let be a space group and () the normal subgroup of all its translations. Owing to the isomorphism between the factor group and the point group , see Section 1.2.5.4 , according to the first isomorphism theorem, Ledermann (1976), tsubgroup graphs are the same (up to the symbols) as the corresponding graphs between point groups. However, in this volume, the graphs are not complete but are contracted by displaying each spacegroup type at most once. This contraction may cause the graphs to look different from the pointgroup graphs and also different for different space groups of the same point group, cf. Example 2.1.7.2.1.
One can indicate the connections between a space group and its maximal subgroups in different ways. In the contracted tsubgroup graphs one line is drawn for each conjugacy class of maximal subgroups of . Thus, a line represents the connection to an individual subgroup only if this is a normal maximal subgroup of , otherwise it represents the connection to more than one subgroup. The conjugacy relations are not necessarily transferable to nonmaximal subgroups, cf. Example 2.1.7.2.2. On the other hand, multiple lines are possible, see the examples. Although it is not in general possible to reconstruct the complete graph from the contracted one, the content of information of such a graph is higher than that of a graph which is drawn with simple lines only.
The graph for the space group at its top also contains the contracted graphs for all subgroups which occur in it, see the remark below Example 2.1.7.2.2.
Owing to lack of space for the large graphs, in all graphs of tsubgroups the group , No. 1, and its connections have been omitted. Therefore, to obtain the full graph one has to supplement the graphs by at the bottom and to connect by one line to each of the symbols that have no connection downwards.
Within the same graph, symbols on the same level indicate subgroups of the same index relative to the group at the top. The distance between the levels indicates the size of the index. For a more detailed discussion, see Example 2.1.7.2.2. For the sequence and the numbers of the graphs, see the paragraph below Example 2.1.7.2.2.
Example 2.1.7.2.1
Compare the tsubgroup graphs in Figs. 2.4.4.2 , 2.4.4.3 and 2.4.4.8 of , No. 52, , No. 53, and , No. 64, respectively. The complete (uncontracted) graphs would have the shape of the graph of the point group with at the top (first level), seven point groups^{5} (, , , , , and ) in the second level, seven point groups (, , , , , and ) in the third level and the point group at the bottom (fourth level). The group is connected to each of the seven subgroups at the second level by one line. Each of the groups of the second level is connected with three groups of the third level by one line. All seven groups of the third level are connected by one line each with the point group 1 at the bottom.
The contracted graph of the point group would have at the top, three pointgroup types (, and ) at the second level and three pointgroup types (, m and ) at the third level. The point group 1 at the bottom would not be displayed (no fourth level). Single lines would connect with 222, with 2, with 2, with m and with ; a double line would connect with m; triple lines would connect with , with and 222 with 2.
The number of fields in a contracted tsubgroup graph is between the numbers of fields in the full and in the contracted pointgroup graphs. The graph in Fig. 2.4.4.2 of , No. 52, has six spacegroup types at the second level and four spacegroup types at the third level. For the graph in Fig. 2.4.4.3 of , No. 53, these numbers are seven and five and for the graph in Fig. 2.4.4.8 of , No. 64 (formerly ), the numbers are seven and six. However, in all these graphs the number of connections is always seven from top to the second level and three from each field of the second level downwards to the ground level, independent of the amount of contraction and of the local multiplicity of lines.
Example 2.1.7.2.2
Compare the tsubgroup graphs shown in Fig. 2.4.1.1 for , No. 221, and Fig. 2.4.1.5 , , No. 225. These graphs are contracted from the pointgroup graph . There are altogether nine levels (without the lowest level of ). The indices relative to the top space groups and are 1, 2, 3, 4, 6, 8, 12, 16 and 24, corresponding to the pointgroup orders 48, 24, 16, 12, 8, 6, 4, 3 and 2, respectively. The height of the levels in the graphs reflects the index; the distances between the levels are slightly distorted in order to adapt to the density of the lines. From the top spacegroup symbol there are five lines to the symbols of maximal subgroups: The three symbols at the level of index 2 are those of cubic normal subgroups, the one (tetragonal) symbol at the level of index 3 represents a conjugacy class of three, the symbol , No. 166, at the level of index 4 represents a conjugacy class of four subgroups.
The graphs differ in the levels of the indices 12 and 24 (orthorhombic, monoclinic and triclinic subgroups) by the number of symbols (nine and seven for index 12, five and three for index 24). The number of lines between neighbouring connected levels depends only on the number and kind of symbols in the upper level. This property makes such graphs particularly useful.
However, for nonmaximal subgroups the conjugacy relations may not hold. For example, in Fig. 2.4.1.1 , the space group has three normal maximal subgroups of type and is thus connected to its symbol by a triple line, although these subgroups are conjugate subgroups of the nonminimal supergroup .
The tsubgroup graphs in Figs. 2.4.1.1 and 2.4.1.5 contain the tsubgroups of (221) and (225) and their relations. In addition, the tsubgroup graph of includes the tsubgroup graphs of , , , , , , , , , etc., that of includes those of , , , , also etc. Thus, many other graphs can be extracted from the two basic graphs. The same holds for the other graphs displayed in Figs. 2.4.1.2 to 2.4.4.8 : each of them includes the contracted graphs of all its subgroups. For this reason one does not need 229 or 218 different graphs to cover all tsubgroup graphs of the 229 spacegroup types but only 37 ( can be excluded as trivial).
The preceding Example 2.1.7.2.2 suggests that one should choose the graphs in such a way that their number can be kept small. It is natural to display the `big' graphs first and later those smaller graphs that are still missing. This procedure is behind the sequence of the tsubgroup graphs in this volume.

For the index of a maximal tsubgroup, Lemma 1.2.8.2.3 is repeated: the index of a maximal nonisomorphic subgroup is always 2 for oblique, rectangular and square plane groups and for triclinic, monoclinic, orthorhombic and tetragonal space groups . The index is 2 or 3 for hexagonal plane groups and for trigonal and hexagonal space groups . The index is 2, 3 or 4 for cubic space groups .
There are 29 graphs for klassengleiche or ksubgroups, one for each crystal class with the exception of the crystal classes 1, and with only one spacegroup type each: , No. 1, , No. 2, and , No. 174, respectively. The sequence of the graphs is determined by the sequence of the point groups in IT A, Table 2.1.2.1 , fourth column. The graphs of , and are nearly trivial, because to these crystal classes only two spacegroup types belong. The graphs of with 22, of with 28 and of with 20 spacegroup types are the most complicated ones.
Isomorphic subgroups are special cases of ksubgroups. With the exception of both partners of the enantiomorphic spacegroup types, isomorphic subgroups are not displayed in the graphs. The explicit display of the isomorphic subgroups would add an infinite number of lines from each field for a space group back to this field, or at least one line (e.g. a circle) implicitly representing the infinite number of isomorphic subgroups, see the tables of maximal subgroups of Chapter 2.3 .^{6} Such a line would have to be attached to every spacegroup symbol. Thus, there would be no more information.
Nevertheless, connections between isomorphic space groups are included indirectly if the group–subgroup chain encloses a space group of another type. In this case, a space group may be a subgroup of a space group and a subgroup of , where and belong to the same spacegroup type. The subgroup chain is then – – . The two space groups and are not identical but isomorphic. Whereas in general the label for the subgroup is positioned at a lower level than that for the original space group, for such relations the symbols for and can only be drawn on the same level, connected by horizontal lines. If this happens at the top of a graph, the top level is occupied by more than one symbol (the number of symbols in the top level is the same as the number of symmorphic spacegroup types of the crystal class).
Horizontal lines are drawn as left or right arrows depending on the kind of relation. The arrow is always directed from the supergroup to the subgroup. If the relation is twosided, as is always the case for enantiomorphic spacegroup types, then the relation is displayed by a pair of horizontal lines, one of them formed by a right and the other by a left arrow. In the graph in Fig. 2.5.1.5 for crystal class , the connections of with and with are displayed by doubleheaded arrows instead. Furthermore, some arrows in Fig. 2.5.1.5 , crystal class , and Fig. 2.5.1.6 , , are dashed or dotted in order to better distinguish the different lines and to increase clarity.
The different kinds of relations are demonstrated in the following examples.
Example 2.1.7.3.1
In the graph in Fig. 2.5.1.1 , crystal class 2, a space group may be a subgroup of index 2 of a space group by `Loss of centring translations'. On the other hand, subgroups of in the block `Enlarged unit cell', belong to the type , see the tables of maximal subgroups in Chapter 2.3 . Therefore, both symbols are drawn at the same level and are connected by a pair of arrows pointing in opposite directions. Thus, the top level is occupied twice. In the graph in Fig. 2.5.1.2 of crystal class m, both the top level and the bottom level are occupied twice.
Example 2.1.7.3.2
There are four symbols at the top level of the graph in Fig. 2.5.1.4 , crystal class 222. Their relations are rather complicated. Whereas one can go (by index 2) from directly to a subgroup of type and vice versa, the connection from directly to is oneway. One always has to pass on the way from to a subgroup of the types or . Thus, the only maximal subgroup of among these groups is . One can go directly from to but not to etc.
Because of the horizontal connecting arrows, it is clear that there cannot be much correspondence between the level in the graphs and the subgroup index. However, in no graph is a subgroup positioned at a higher level than the supergroup.
Example 2.1.7.3.3
Consider the graph in Fig. 2.5.1.6 for crystal class . To the space group (65) belong maximal nonisomorphic subgroups of the 11 spacegroup types (from left to right) (72), (63), (74), (59), (55), (50), (51), (53), (66), (47) and (71). Although all of them have index 2, their symbols are positioned at very different levels of the graph.
The table for the subgroups of in Chapter 2.3 lists 22 nonisomorphic ksubgroups of index 2, because some of the spacegroup types mentioned above are represented by two or four different subgroups. This multiplicity cannot be displayed by multiple lines because the density of the lines in some of the ksubgroup graphs does not permit this kind of presentation, e.g. for . The multiplicity may be taken from the subgroup tables in Chapter 2.3 , where each nonisomorphic subgroup is listed individually.
Consider the connections from (65) to (55). There are among others: the direct connection of index 2, the connection of index 4 over (72), the connection of index 8 over (74) and (51). Thus, starting from the same space group of type one arrives at different space groups of the type with different unit cells but all belonging to the same spacegroup type and thus represented by the same field of the graph.
The index of a ksubgroup is restricted by Lemma 1.2.8.2.3 and by additional conditions. For the following statements one has to note that enantiomorphic space groups are isomorphic.

There are no graphs for plane groups in this volume. The four graphs for tsubgroups of plane groups are apart from the symbols the same as those for the corresponding space groups: –, –, – and –, where the graphs for the space groups are included in the tsubgroup graphs in Figs. 2.4.1.1 , 2.4.3.1 , 2.4.2.1 and 2.4.2.3 , respectively.
The ksubgroup graphs are trivial for the plane groups , , , , and because there is only one plane group in its crystal class. The graphs for the crystal classes and consist of two plane groups each: and , and . Nevertheless, the graphs are different: the relation is onesided for the tetragonal planegroup pair as it is in the spacegroup pair and it is twosided for the hexagonal planegroup pair as it is in the spacegroup pair (81)– (82). The graph for the three plane groups of the crystal class m corresponds to the spacegroup graph for the crystal class 2.
Finally, the graph for the four plane groups of crystal class has no direct analogue among the ksubgroup graphs of the space groups. It can be obtained, however, from the graph in Fig. 2.5.1.3 of crystal class by removing the fields of (15) and (11) with all their connections to the remaining fields. The replacements are then: (12) by (9), (10) by (6), (13) by (7) and (14) by (8).
If a subgroup is not maximal then there must be a group–subgroup chain – of maximal subgroups with more than two members which connects with . There are three possibilities: may be a tsubgroup or a ksubgroup or a general subgroup of . In the first two cases, the application of the graphs is straightforward because at least one of the graphs will permit one to find the possible chains directly. If is a ksubgroup of , isomorphic subgroups have to be included if necessary. If is a general subgroup of one has to combine t and ksubgroup graphs, but the problem is only slightly more complicated. This is because for a general subgroup , Hermann's theorem 1.2.8.1.2 states the existence of an intermediate group with and the properties is a ksubgroup of and is a tsubgroup of .
Thus, however long and complicated the real chain may be, there is also always a chain for which only two graphs are needed: a tsubgroup graph for the relation between and and a ksubgroup graph for the relation between and .
There is, however, a severe shortcoming to using contracted graphs for the analysis of group–subgroup relations, and great care has to be taken in such investigations. All subgroups with the same spacegroup type are represented by the same field of the graph, but these different nonmaximal subgroups may permit different routes to a common original (super)group.
Example 2.1.7.5.1
An example for translationengleiche subgroups is provided by the group–subgroup chain of index 12. The contracted graph may be drawn by the program Subgroupgraph from the Bilbao Crystallographic Server, http://www.cryst.ehu.es/ . It is shown in Fig. 2.1.7.1; each field represents all occurring subgroups of a spacegroup type: (139) represents three subgroups, (166) represents four subgroups, and (12) represents nine subgroups belonging to two conjugacy classes. Fig. 2.1.7.1 is part of the contracted total graph of the translationengleiche subgroups of the space group , which is displayed in Fig. 2.4.1.5 . With Subgroupgraph one can also obtain the complete graph between and the set of all nine subgroups of the type . It is too large to be reproduced here.

Contracted graph of the group–subgroup chains from (225) to those subgroups with index 12 which belong to the spacegroup type (12). The graph forms part of the total contracted graph of tsubgroups of (Fig.2.4.1.5 ). 
More instructive are the complete graphs for different single subgroups of the type of . They can be obtained with the program Symmodes from the same server as above. In Fig. 2.1.7.2 such a graph is displayed for one of the six subgroups of type of index 12 whose monoclinic axes point in the directions of . Similarly, in Fig. 2.1.7.3 the complete graph is drawn for one of the three subgroups of of index 12 whose monoclinic axes point in the directions of . It differs markedly from the contracted graph and from the first complete graph. It is easily seen that it may be very misleading to use the contracted graph or the wrong individual complete graph instead of the right individual complete graph.

Complete graph of the group–subgroup chains from (225) to one representative of those six (12) subgroups with index 12 whose monoclinic axes are along the directions of . 
The following example deals with general subgroups.
Example 2.1.7.5.2
The crystal structures of SrTiO_{3} and KCuF_{3} both belong to the spacegroup type , No. 140. They can be derived by different distortions from the same ideal perovskite ABX_{3} structure with the space group , No. 221. Common to both chains is Hermann's group of space group , No. 123, and the unit cell of ABX_{3}. This is the only intermediate group for SrTiO_{3}. For the pair ABX_{3}–KCuF_{3}, on the other hand, there exists another chain with an intermediate space group of type , No. 226. The combination of the graph in Fig. 2.4.1.1 for tsubgroups with the graph in Fig. 2.5.2.7 for ksubgroups is thus possible for both crystal–chemical relations. The combination of the graph in Fig. 2.5.5.5 for ksubgroups with the graph in Fig. 2.4.1.6 for tsubgroups is, however, only meaningful for the chain ABX_{3}–KCuF_{3}. This cannot be concluded from the contracted graphs but can be seen from the complete graph, as displayed in Fig. 1 of Wondratschek & Aroyo (2001).
References
Ascher, E. (1968). Lattices of equitranslation subgroups of the space groups. Geneva: Battelle Institute.Bärnighausen, H. (1980). Group–subgroup relations between space groups: a useful tool in crystal chemistry. MATCH Comm. Math. Chem. 9, 139–175.
International Tables for Crystallography (2005). Vol. A, Spacegroup symmetry, edited by Th. Hahn, 5th ed. Heidelberg: Springer. (Abbreviated IT A.)
International Tables for Xray Crystallography (1952, 1965, 1969). Vol. I, Symmetry groups, edited by N. F. M. Henry & K. Lonsdale. Birmingham: Kynoch Press.
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Ledermann, W. (1976). Introduction to group theory. London: Longman. (German: Einführung in die Gruppentheorie, Braunschweig: Vieweg, 1977.)
Neubüser, J. & Wondratschek, H. (1966). Untergruppen der Raumgruppen. Krist. Tech. 1, 529–543.
Wondratschek, H. & Aroyo, M. I. (2001). The application of Hermann's group in group–subgroup relations between space groups. Acta Cryst. A57, 311–320.