International
Tables for Crystallography Volume E Subperiodic groups Edited by V. Kopský and D. B. Litvin © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. E, ch. 1.2, pp. 1722
Section 1.2.15. Maximal subgroups and minimal supergroups^{a}Department of Physics, University of the South Pacific, Suva, Fiji, and Institute of Physics, The Academy of Sciences of the Czech Republic, Na Slovance 2, PO Box 24, 180 40 Prague 8, Czech Republic, and ^{b}Department of Physics, Penn State Berks Campus, The Pennsylvania State University, PO Box 7009, Reading, PA 196106009, USA 
In IT A (2005), for the representative space group of each spacegroup type the following information is given:
However, Bieberbach's theorem for space groups, i.e. the classification into isomorphism classes is identical with the classification into affine equivalence classes, is not valid for subperiodic groups. Consequently, to obtain analogous tables for the subperiodic groups, we provide the following information for each representative subperiodic group:
where isotypic means `belonging to the same subperiodic group type'. The cases of maximal enantiomorphic subgroups of lowest index and minimal enantiomorphic supergroups of lowest index arise only in the case of rod groups.
The maximal nonisotypic nonenantiomorphic subgroups S of a subperiodic group G are divided into two types:
Type II is subdivided again into two blocks:
Block IIa has no entries for subperiodic groups with a primitive cell. Only in the case of the nine centred layer groups are there entries, when it contains those maximal subgroups S which have lost all the centring translations of G but none of the integral translations.
In blocks I and IIa, every maximal subgroup S of a subperiodic group G is listed with the following information:The symbols have the following meaning:
Examples
The HMS1 symbol in each of the three subgroups S is given in the tetragonal coordinate system of the group G. In the first case, is not the conventional short Hermann–Mauguin symbol and a second conventional symbol is given. In the latter two cases, since the subgroups are orthorhombic rod groups, a second conventional symbol of the subgroup in an orthorhombic coordinate system is given.
Whereas in blocks I and IIa every maximal subgroup S of G is listed, this is no longer the case for the entries of block IIb. The information given in this block is
The symbols have the following meaning:

Examples

Another set of klassengleiche subgroups is that listed under IIc, i.e. the subgroups S which are of the same or of the enantiomorphic subperiodic group type as G. Again, one entry may correspond to more than one isotypic subgroup:

Examples

If G is a maximal subgroup of a group H, then H is called a minimal supergroup of G. Minimal supergroups are again subdivided into two types, the translationengleiche or t supergroups I and the klassengleiche or k supergroups II. For the t supergroups I of G, the listing contains the index [i] of G in H and the conventional Hermann–Mauguin symbol of H. For the k supergroups II, the subdivision between IIa and IIb is not made. The information given is similar to that for the subgroups IIb, i.e. the relations between the basis vectors of group and supergroup are given, in addition to the Hermann–Mauguin symbols of H. Note that either the conventional cell of the k supergroup H is smaller than that of the subperiodic group G, or H contains additional centring translations.
Example: G: Layer group (L15)
Minimal nonisotypic nonenantiomorphic supergroups:
Block I lists [2] pmam, [2] pmma and [2] pmmn. Looking up the subgroup data of these three groups one finds [2] p2_{1}/m11. Block I also lists [2] pbma. Looking up the subgroup data of this group one finds [2] p12_{1}/m1 (p2_{1}/m11). This shows that the setting of pbma does not correspond to that of p2_{1}/m11 but rather to p12_{1}/m1. To obtain the supergroup H referred to the basis of p2_{1}/m11, the basis vectors a and b must be interchanged. This changes pbma to pmba, which is the correct symbol of the supergroup of p2_{1}/m11.
Block II contains two entries: the first where the conventional cells are the same with the supergroup having additional centring translations, and the second where the conventional cell of the supergroup is smaller than that of the original subperiodic group.
No data are listed for supergroups IIc, because they can be derived directly from the corresponding data of subgroups IIc.
Example: G: Rod group (R29)
The maximal isotypic subgroup of lowest index of is found in block IIc: [3] (c′ = 3c). By interchanging c′ and c, one obtains the minimal isotypic supergroup of lowest index, i.e. [3] (3c′ = c).
References
International Tables for Crystallography (2005). Vol. A. Spacegroup symmetry, edited by Th. Hahn. Heidelberg: Springer. [Previous editions: 1983, 1987, 1992, 1995 and 2002. Abbreviated as IT A (2005).]