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. 5458
Section 2.1.7. The subgroup 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 
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.
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.)
Internationale Tabellen zur Bestimmung von Kristallstrukturen (1935). 1. Bd. Edited by C. Hermann. Berlin: Borntraeger. (In German, English and French.) (Abbreviated IT 35.)
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.