International Tables for Crystallography is the definitive resource and reference work for crystallography and structural science. Volume A1, an extension of and supplement to Volume A, presents a systematic treatment of the maximal subgroups and minimal supergroups of the crystallographic plane groups and space groups. The volume is divided into 3 parts: Part 1: An introduction to the theory of space groups at various levels and with many examples. Part 2: A complete listing of all maximal subgroups and minimal supergroups for each plane group and space group. Graphs that illustrate the group-subgroup relations are also presented. Part 3: Lists the relations between the Wyckoff positions of every maximal subgroup of every space group, including the cell transformations and coordinate transformations. New to the second edition of Volume A1:A new chapter on building trees of group-subgroup relations for crystal structures that can be derived from a high-symmetry structure type (aristotype). A new chapter on the Bilbao Crystallographic Server, describing the databases and computer programs that are related to the subjects of the volume. A detailed discussion of the listed supergroup data and a description of a procedure for the complete derivation of the minimal supergroups from the listed (complete) data on maximal subgroups. This volume is invaluable for the study of group-subgroup relations, and is therefore essential for those interested in phase transitions, the systematic compilation of crystal structures, twinning phenomena and related fields of crystallographic research. More information on the series can be found at: http://it.iucr.org