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. 4245
Section 2.1.2. Structure of the subgroup tables^{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 
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.
References
International Tables for Xray Crystallography (1952, 1965, 1969). Vol. I, Symmetry groups, edited by N. F. M. Henry & K. Lonsdale. Birmingham: Kynoch Press.Google Scholar