(i) Blocks I and IIa
In blocks I and IIa, every maximal subgroup of a space group is listed with the following information:
[i] HMS1 (HMS2, No.) Sequence of numbers.
The symbols have the following meaning:
[i]: index of in (cf. the footnote
to Section 8.1.6);
HMS1: Hermann–Mauguin symbol of , referred to the coordinate system and setting of ; this symbol may be unconventional;
(HMS2, No.): conventional short Hermann–Mauguin symbol of , given only if HMS1 is not in conventional short form, and the spacegroup number of .
Sequence of numbers: coordinate triplets of retained in . The numbers refer to the numbering scheme of the coordinate triplets of the general position of (cf. Section 2.2.9). The following abbreviations are used:
Block I (all translations retained): 
Number 
Coordinate triplet given by Number, plus those obtained by adding all centring translations of . 
(Numbers) 
The same, but applied to all Numbers between parentheses. 
Block IIa (not all translations retained): 

Coordinate triplet obtained by adding the translation to the triplet given by Number. 

The same, but applied to all Numbers between parentheses. 

In blocks I and IIa, sets of conjugate subgroups are linked by lefthand braces. For an example, see space group (148) below.
(ii) Block IIb
Whereas in blocks I and IIa every maximal subgroup of is listed, this is no longer the case for the entries of block IIb. The information given in this block is:
[i] HMS1 (Vectors) (HMS2, No.)
The symbols have the following meaning:
[i]: index of in ;
HMS1: Hermann–Mauguin symbol of , referred to the coordinate system and setting of ; this symbol may be unconventional;^{8}
(Vectors): basis vectors a′, b′, c′ of in terms of the basis vectors a, b, c of . No relations are given for unchanged axes, e.g. is not stated;
(HMS2, No.): conventional short Hermann–Mauguin symbol, given only if HMS1 is not in conventional short form, and the spacegroup number of .
In addition to the general rule of increasing index [i] and decreasing spacegroup number (No.), the sequence of the IIb subgroups also depends on the type of cell enlargement. Subgroups with the same index and the same kind of cell enlargement are listed together in decreasing order of spacegroup number (see example 1 below).
In contradistinction to blocks I and IIa, for block IIb the coordinate triplets retained in are not given. This means that the entry is the same for all subgroups that have the same Hermann–Mauguin symbol and the same basisvector relations to , but contain different sets of coordinate triplets. Thus, in block IIb, one entry may correspond to more than one subgroup,^{9} as illustrated by the following examples.
Examples
(1)
Each of the subgroups is referred to its own distinct basis a′, b′, c′, which is different in each case. Apart from the translations of the enlarged cell, the generators of the subgroups, referred to a′, b′, c′, are as follows:
There are thus 2, 1 or 4 actual subgroups that obey the same basisvector relations. The difference between the several subgroups represented by one entry is due to the different sets of symmetry operations of that are retained in . This can also be expressed as different conventional origins of with respect to .
(2)
The nine subgroups of type P31m may be described in two ways:
(i) By partial `decentring' of ninetuple cells (, , ) with the same orientations as the cell of the group in such a way that the centring points 0, 0, 0; ; (referred to a′, b′, c′) are retained. The conventional spacegroup symbol P31m of these nine subgroups is referred to the same basis vectors , , , but to different origins; cf. Section 2.2.15.5. This kind of description is used in the spacegroup tables of this volume.
(ii) Alternatively, one can describe the group with an unconventional Hcentred cell (, , ) referred to which the spacegroup symbol is H31m. `Decentring' of this cell results in the conventional spacegroup symbol P31m for the subgroups, referred to the basis vectors a′, b′, c′. This description is used in Section 4.3.5
.


(iii) Subdivision of k subgroups into blocks IIa and IIb
The subdivision of k subgroups into blocks IIa and IIb has no grouptheoretical background and depends on the coordinate system chosen. The conventional coordinate system of the space group (cf. Section 2.1.3
) is taken as the basis for the subdivision. This results in a uniquely defined subdivision, except for the seven rhombohedral space groups for which in the spacegroup tables both `rhombohedral axes' (primitive cell) and `hexagonal axes' (triple cell) are given (cf. Section 2.2.2). Thus, some k subgroups of a rhombohedral space group are found under IIa (klassengleich, centring translations lost) in the hexagonal description, and under IIb (klassengleich, conventional cell enlarged) in the rhombohedral description.
Example:
Hexagonal axes
Rhombohedral axes
Apart from the change from IIa to IIb, the above example demonstrates again the restricted character of the IIb listing, discussed above. The three conjugate subgroups of index [3] are listed under IIb by one entry only, because for all three subgroups the basisvector relations between and are the same. Note the brace for the IIa subgroups, which unites conjugate subgroups into classes.
