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Introduction

This page presents the hierarchical structure (graph) of t-subgroups.t-subgroup hierarchical structure issubgroup relationships of crystallographic point groupssimilar to. In other words, for a given space group \(G\) with crystal class (corresponding point group) \(G’\) , \(G\) the crystal classes of t-subgroups are necessarily \(G’\) subgroups of

However, independently describing all 230 types of space groups one by one would be extensive and difficult.For this reason, we first present large hierarchical structure graphs with high-symmetry space groups as vertices, and do not re-present space groups appearing in intermediate levels.Additionally, we consolidate space groups with identical hierarchical structures as much as possible, and distinguish them by the background color of the space group symbol.With these considerations, the relationships of t-subgroups can be summarized in 19 graphs.The table below shows the first graph number where a given space group appears.All space groups appear in one of the 19 graphs.

How to Read the Graphs

• Lines connecting space groups indicatedownward t-maximal subgrouprelationships.
  • Double or triple lines indicate the existence of two or three normal maximal subgroups.
• Within each graph,space groups aligned at the same level have equal crystal class order.Higher levels have larger orders.(subgroup relationships of crystallographic point groupsSee)。
• Space group symbolwith multiple background colors represent consolidated subgroup relationships with identical hierarchical structures.
  • At the top level, independent background colors are assigned to multiple space groups.
  • At levels other than the top, multiple background colors may be assigned to a single space group (for example in 1.1. graph,\(P23\)) . Such space groups arecommon subgroupsmeaning.
• The bottom level of all graphs contains space groups belonging to crystal classes \(3\), \(m\), \(2\), or \(\bar{1}\).Actually, these space groups are not the true bottom level, but as t-maximal subgroups \(P1\) have.Originally, \(P1\) should be connected with downward lines to, butis omitted to avoid the enlargement of the graph.。
• The number shown at the bottom right of each graph (2.4.1.1.and others) corresponds to International Tables for Crystallography A1 (1^st edition) in the table number.

1. Cubic crystal system

1.1. \(Pm\bar{3}m \,_{(221)}\), \(Pn\bar{3}n\,_{(222)}\), \(Pm\bar{3}n\,_{(223)}\), \(Pn\bar{3}m\,_{(224)}\)

2.4.1.1 + 2.4.1.2 + 2.4.1.3 + 2.4.1.4

1.2. \(Fm\bar{3}m\), \(Fd\bar{3}c\)

2.4.1.5 + 2.4.1.8

1.3. \(Fm\bar{3}c\), \(Fd\bar{3}m\)

2.4.1.6 + 2.4.1.7

1.4. \(Im\bar{3}m\)

2.4.1.9

1.5. \(Ia\bar{3}d\)

2.4.1.10

1.6. \(P4_132\), \(P4_332\), \(Pa\bar{3}\)

2.4.1.11

2. Tetragonal crystal system

2.1. \(P4/mcc\), \(P4_2/mcm\)

2.4.2.1 + 2.4.1.7

2.2. \(P4/ncc\), \(P4_2/ncm\)

2.4.2.6 + 2.4.2.12

2.3. \(P4/nbm\), \(P4_2/nbc\)

2.4.2.2 + 2.4.2.8

2.4. \(P4/mbm\), \(P4/nmm\), \(P4_2/mnm\)

2.4.2.3 + 2.4.2.5 + 2.4.2.10

2.5. \(P4_2/mbc\), \(P4_2/nmc\), \(P4/mnc\)

2.4.2.9 + 2.4.2.4 + 2.4.2.11

2.6. \(P4_122\), \(P4_322\)

2.4.2.13

3. Hexagonal crystal system

3.1. \(P6/mmm\), \(P6/mcc\)

2.4.3.1 + 2.4.3.2

3.2. \(P6_3/mcm\), \(P6_3/mmc\)

2.4.3.3 + 2.4.3.4

3.3. \(P6_122\), \(P6_422\), \(P6_522\), \(P6_222\)

2.4.3.5

4. Orthorhombic crystal system

4.1. \(Pmma\), \(Pmna\), \(Pbcm\)

2.4.4.1 + 2.4.4.3 + 2.4.4.5

4.2. \(Pnna\), \(Pcca\), \(Pbcn\)

2.4.4.2 + 2.4.4.4 + 2.4.4.6

4.3. \(Pnma\)

2.4.4.7

4.4. \(Cmce\)

2.4.4.8