The database of maximal subgroups on the Bilbao Crystallographic Server is based on the data of maximal subgroups of space groups as listed in International Tables for Crystallography, Vol. A1: Symmetry relations between space groups (hereafter referred to as IT A1).

The Bilbao Crystallographic Server includes also programs treating different problems related to group-subgroup relations of space groups. These programs use the retrieval tools for accessing the necessary space-group information and apply group-theoretical algorithms in order to obtain specific results which are not available in the subgroup databases. Examples are the determination of different subgroups of a space group and their distribution into conjugacy classes, or the calculation of possible supergroups of a given space-group type and a fixed index. There one can find also a program for calculating the splitting rules of the Wyckoff positions for a group-subgroup pair.

The practical includes exercises with the subgroup databases and computer tools:

1. Introduction and use of the databases of the Bilbao Crystallographic server of maximal subgroups, series of isomorphic subgroups and minimal supergroups.
2. Group-subgroup relations between space groups: computer programs on the Bilbao Crystallographic Server:
   1. basic programs: SUBGROUPGRAPH, HERMANN, SUPERGROUPS.
   2. Graph of maximal subgroups for a general group-subgroup pair; Hermann theorem: twins and antiphase domains; coset decomposition
   3. Symmetry relations between Wyckoff positions for group-subgroup pairs (the program WYCKSPLIT).
   4. Computer programs for determination of structure relationships: CELLSUB and COMMONSUB.
3. Supergroups of space groups: computer tools for their calculation (the programs MINSUP, SUPERGROUPS, CELLSUP and COMMONSUP).
4. Structure utilities
   1. equivalent structure descriptions: EQUIVSTRU.
   2. Structure descriptions compatible with symmetry reduction: TRANSTRU.

Supplementary Material