Chapter 1.2 General introduction to the subgroups of space groups

Symmetry relations between space groups

Second Online Edition (2011)

Part 1. Space groups and their subgroups

  1. Hans Wondratschek

Published Online: 15 DEC 2011

DOI: 10.1107/97809553602060000791

International Tables for Crystallography

International Tables for Crystallography

How to Cite

Wondratschek, H. 2011. General introduction to the subgroups of space groups. International Tables for Crystallography. A1:1:1.2:7–24.

Author Information

  1. Institut für Kristallographie, Universität, D-76128 Karlsruhe, Germany

Publication History

  1. Published Online: 15 DEC 2011


This general introduction deals with the theory of the main subjects of this volume: space groups and their subgroups. After some general remarks, the definitions and the corresponding lemmata of mappings (and their description by matrices) and groups (in particular symmetry groups, their classifications and their subgroups) are given. The different types of subgroups are defined and explained using several examples. The practical use of the abstract group–subgroup relations of space groups is explained and demonstrated using examples of their application to domain structures in Section 1.2.7. The chapter closes with a list of the most important lemmata on group–subgroup relations between space groups (without proofs).


  • isomorphic subgroups;
  • space groups;
  • maximal subgroups;
  • minimal supergroups;
  • non-isomorphic subgroups;
  • translationengleiche subgroups;
  • klassengleiche subgroups;
  • symmetry operations;
  • Hermann’s theorem;
  • translation groups;
  • coordinate systems;
  • Euclidean normalizers;
  • Lagrange’s theorem;
  • crystallographic bases;
  • conjugacy classes;
  • cosets;
  • domains;
  • enantiomorphic space groups;
  • group–subgroup relations;
  • groups;
  • homomorphism;
  • isomorphism;
  • mappings;
  • affine normalizers;
  • phase transitions;
  • point groups;
  • crystal classes;
  • crystal systems;
  • crystal families;
  • motions;
  • isometries