Chapter 2.1 Guide to the subgroup tables and graphs

Symmetry relations between space groups

First Online Edition (2006)

Part 2. Maximal subgroups of the plane groups and space groups

  1. Hans Wondratschek1,
  2. Mois I. Aroyo2

Published Online: 1 JAN 2006

DOI: 10.1107/97809553602060000542

International Tables for Crystallography

International Tables for Crystallography

How to Cite

Wondratschek, H. and Aroyo, M. I. 2006. Guide to the subgroup tables and graphs. International Tables for Crystallography. A1:2:2.1:42–58.

Author Information

  1. 1

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

  2. 2

    Departamento de Física de la Materia Condensada, Facultad de Ciencias, Universidad del País Vasco, Apartado 644, E-48080 Bilbao, Spain

Publication History

  1. Published Online: 1 JAN 2006


This chapter is the users’ guide to the subgroup tables and graphs of Part 2 of this volume. Sections 2.1.1 and 2.1.2 refer to the contents, arrangement and structure of the subgroup tables of Chapters 2.2 and 2.3. In the next three sections, the data listed for the maximal translationengleiche subgroups (Section 2.1.3), maximal klassengleiche subgroups (Section 2.1.4) and for the series of the (infinitely many) maximal isomorphic subgroups (Section 2.1.5) are described in detail with many examples. Section 2.1.6 describes the data listed for minimal supergroups. The last section, Section 2.1.7, familiarizes the user with the subgroup graphs of the space groups listed in Chapters 2.4 and 2.5. The chapter is supplemented by the subgroup table of the space group R3, which is printed inside the front cover of the volume with some explanatory text.


  • Bärnighausen trees;
  • Hermann–Mauguin symbols;
  • Schoenflies symbols;
  • basis transformations;
  • conjugate subgroups;
  • generators;
  • graphs of group–subgroup relations;
  • international symbols;
  • isomorphic subgroups;
  • klassengleiche subgroups;
  • minimal supergroups;
  • maximal subgroups;
  • space groups;
  • subgroup graphs;
  • isomorphic supergroups;
  • translationengleiche supergroups;
  • translationengleiche subgroups