International
Tables for Crystallography Volume A Spacegroup symmetry Edited by Th. Hahn © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. A, ch. 2.2, pp. 3538
Section 2.2.15. Maximal subgroups and minimal supergroups^{a}Institut für Kristallographie, RheinischWestfälische Technische Hochschule, Aachen, Germany, and ^{b}Laboratorium voor Chemische Fysica, Rijksuniversiteit Groningen, The Netherlands 
The present section gives a brief summary, without theoretical explanations, of the sub and supergroup data in the spacegroup tables. The theoretical background is provided in Section 8.3.3 and Part 13 . Detailed sub and supergroup data are given in International Tables for Crystallography Volume A1 (2004).
2.2.15.1. Maximal nonisomorphic subgroups^{6}
The maximal nonisomorphic subgroups of a space group are divided into two types: For practical reasons, type II is subdivided again into two blocks:
IIa the conventional cells of and are the same
IIb the conventional cell of is larger than that of . ^{7}
Block IIa has no entries for space groups with a primitive cell. For space groups with a centred cell, it contains those maximal subgroups that have lost some or all centring translations of but none of the integral translations (`decentring' of a centred cell).
Within each block, the subgroups are listed in order of increasing index [i] and in order of decreasing spacegroup number for each value of i.

Another set of klassengleiche subgroups are the isomorphic subgroups listed under IIc, i.e. the subgroups which are of the same or of the enantiomorphic spacegroup type as . The kind of listing is the same as for block IIb. Again, one entry may correspond to more than one isomorphic subgroup.
As the number of maximal isomorphic subgroups of a space group is always infinite, the data in block IIc are restricted to the subgroups of lowest index. Different kinds of cell enlargements are presented. For monoclinic, tetragonal, trigonal and hexagonal space groups, cell enlargements both parallel and perpendicular to the main rotation axis are listed; for orthorhombic space groups, this is the case for all three directions, a, b and c. Two isomorphic subgroups and of equal index but with cell enlargements in different directions may, nevertheless, play an analogous role with respect to . In terms of group theory, and then are conjugate subgroups in the affine normalizer of , i.e. they are mapped onto each other by automorphisms of .^{10} Such subgroups are collected into one entry, with the different vector relationships separated by `or' and placed within one pair of parentheses; cf. example (4).
Examples

If is a maximal subgroup of a group , then is called a minimal supergroup of . Minimal nonisomorphic supergroups are again subdivided into two types, the translationengleiche or t supergroups I and the klassengleiche or k supergroups II. For the minimal t supergroups I of , the listing contains the index [i] of in , the conventional Hermann–Mauguin symbol of and its spacegroup number in parentheses.
There are two types of minimal k supergroups II: supergroups with additional centring translations (which would correspond to the IIa type) and supergroups with smaller conventional unit cells than that of (type IIb). Although the subdivision between IIa and IIb supergroups is not indicated in the tables, the list of minimal supergroups with additional centring translations (IIa) always precedes the list of IIb supergroups. The information given is similar to that for the nonisomorphic subgroups IIb, i.e., where applicable, the relations between the basis vectors of group and supergroup are given, in addition to the Hermann–Mauguin symbols of and its spacegroup number. The supergroups are listed in order of increasing index and increasing spacegroup number.
The block of supergroups contains only the types of the nonisomorphic minimal supergroups of , i.e. each entry may correspond to more than one supergroup . In fact, the list of minimal supergroups of should be considered as a backwards reference to those space groups for which appears as a maximal subgroup. Thus, the relation between and can be found in the subgroup entries of .
Example:
Minimal nonisomorphic supergroupsBlock I lists, among others, the entry [2] Pnma (62). Looking up the subgroup data of Pnma (62), one finds in block I the entry [2] . This shows that the setting of Pnma does not correspond to that of but rather to that of . To obtain the supergroup referred to the basis of , the basis vectors b and c must be interchanged. This changes Pnma to Pnam, which is the correct symbol of the supergroup of .
Note on R supergroups of trigonal P space groups: The trigonal P space groups Nos. 143–145, 147, 150, 152, 154, 156, 158, 164 and 165 each have two rhombohedral supergroups of type II. They are distinguished by different additional centring translations which correspond to the `obverse' and `reverse' settings of a triple hexagonal R cell; cf. Chapter 1.2 . In the supergroup tables of Part 7 , these cases are described as [3] R3 (obverse) (146); [3] R3 (reverse) (146) etc.
No data are listed for isomorphic supergroups IIc because they can be derived directly from the corresponding data of subgroups IIc (cf. Part 13 ).
In the subgroup data, a′, b′, c′ are the basis vectors of the subgroup of the space group . The latter has the basis vectors a, b, c. In the supergroup data, a′, b′, c′ are the basis vectors of the supergroup and a, b, c are again the basis vectors of . Thus, a, b, c and a′, b′, c′ exchange their roles if one considers the same group–subgroup relation in the subgroup and the supergroup tables.
Examples

References
International Tables for Crystallography (2004). Vol. A1, edited by H. Wondratschek & U. Müller. Dordrecht: Kluwer Academic Publishers.