Finite Group Databasebeta

← C23×C32×D12 #13181712

Subgroups — C23×C32×D12 #13181712

Order 1728.

9116 conjugacy classes of subgroups; only the 3258 normal subgroups are shown.

normal subgroups only

This group has 9116 conjugacy classes of subgroups — too many to draw. Only its 3258 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C24×C32×D3C212yesmaximal, complemented
2C23×C33×C4C211yesmaximal, characteristic, complemented
3C22×C32×D12C2128yesmaximal, direct product factor
4C23×C32×D4—31nomaximal
5C23×C3×D12C314yesmaximal, direct product factor
6C24×C33C2211yescharacteristic
7C23×C32×D3—216no—
8C23×C32×D3C22128yescomplemented
9C22×C33×C4C22114yescomplemented
10C2×C32×D12C221112yesdirect product factor
11C25×C32—32no—
12C24×C3×D3C2×C318yescomplemented
13C22×C32×D4—328no—
14C23×C32×C4—24no—
15C23×C32×C4D311yescharacteristic, complemented
16C23×C32×C4C2×C314yescomplemented
17C22×C3×D12C2×C31112yesdirect product factor
18C23×C33D418yescomplemented
19C23×C33C2317yes—
20C22×C32×D3—2112no—
21C22×C32×D3C23156yescomplemented
22C2×C33×C4C23128yescomplemented
23C32×D12C23164yesdirect product factor
24C23×C3×D4—34no—
25C23×D12C3211yescharacteristic, direct product factor
26C24×C32—24no—
27C24×C32—328no—
28C24×C32—616no—
29C24×C32C2×D311yescharacteristic
30C24×C32C22×C314yes—
31C23×C3×D3—264no—
32C23×C3×D3C22×C31112yescomplemented
33C2×C32×D4—3112no—
34C22×C32×C4—256no—
35C22×C32×C4C2×D3114yescomplemented
36C22×C32×C4C22×C3156yescomplemented
37C2×C3×D12C22×C31448yesdirect product factor
38C22×C33C2×D4128yescomplemented
39C22×C33C2417yes—
40C2×C32×D3—2112no—
41C2×C32×D3C24116yescomplemented
42C33×C4C2418yescomplemented
43C25×C3—38no—
44C24×D3C2×C3212yescomplemented
45C22×C3×D4—3112no—
46C23×C3×C4—24no—
47C23×C3×C4C3×D314yescomplemented
48C23×C3×C4C2×C3211yescharacteristic, complemented
49C22×D12C2×C32128yesdirect product factor
50C23×C32—232no—
51C23×C32—228no—
52C23×C32—356no—
53C23×C32—6112no—
54C23×C32D1218yesdirect product factor
55C23×C32C3×D4132yescomplemented
56C23×C32C22×D317yes—
57C23×C32C23×C3128yes—
58C22×C3×D3—2448no—
59C22×C3×D3C23×C31224yescomplemented
60C32×D4—364no—
61C2×C32×C4—2112no—
62C2×C32×C4C22×D3128yescomplemented
63C2×C32×C4C23×C31112yescomplemented
64C3×D12C23×C31256yesdirect product factor
65C23×D4—31noSylow
66C2×C33C22×D4114yescomplemented
67C2×C33C2511yescharacteristic
68C32×D3—216no—
69C24×C3—24no—
70C24×C3—3112no—
71C24×C3—664no—
72C24×C3C2×C3×D314yes—
73C24×C3C22×C3211yescharacteristic
74C23×D3—216no—
75C23×D3C22×C32128yescomplemented
76C2×C3×D4—3448no—
77C22×C3×C4—256no—
78C22×C3×C4C2×C3×D3156yescomplemented
79C22×C3×C4C22×C32114yescomplemented
80C2×D12C22×C321112yesdirect product factor
81C22×C32—228no—
82C22×C32—2112no—
83C22×C32—316no—
84C22×C32—6112no—
85C22×C32C2×D12128yesdirect product factor
86C22×C32C2×C3×D41112yescomplemented
87C22×C32C23×D317yes—
88C22×C32C24×C3128yes—
89C2×C3×D3—2448no—
90C2×C3×D3C24×C3164yescomplemented
91C32×C4—232no—
92C32×C4C23×D318yescomplemented
93C32×C4C24×C3132yescomplemented
94C25—32no—
95C22×D4—328no—
96C23×C4C32×D311yescharacteristic, complemented
97C33C23×D411yescharacteristic, complemented, Sylow
98C23×C3—232no—
99C23×C3—228no—
100C23×C3—3224no—
101C23×C3—6448no—
102C23×C3C3×D12132yesdirect product factor
103C23×C3C32×D418yescomplemented
104C23×C3C22×C3×D3128yes—
105C23×C3C23×C3217yes—
106C22×D3—2112no—
107C22×D3C23×C32156yescomplemented
108C3×D4—3256no—
109C2×C3×C4—2112no—
110C2×C3×C4C22×C3×D31112yescomplemented
111C2×C3×C4C23×C32128yescomplemented
112D12C23×C32164yesdirect product factor
113C2×C32—256no—
114C2×C32—24no—
115C2×C32—616no—
116C2×C32C22×D12114yesdirect product factor
117C2×C32C22×C3×D4156yescomplemented
118C2×C32C24×D311yescharacteristic
119C2×C32C25×C314yes—
120C3×D3—264no—
121C24—328no—
122C24—616no—
123C24C2×C32×D311yescharacteristic
124C2×D4—3112no—
125C22×C4C2×C32×D3114yescomplemented
126C22×C3—228no—
127C22×C3—2112no—
128C22×C3—364no—
129C22×C3—6448no—
130C22×C3C2×C3×D121112yesdirect product factor
131C22×C3C2×C32×D4128yescomplemented
132C22×C3C23×C3×D3128yes—
133C22×C3C24×C3217yes—
134C2×D3—2112no—
135C2×D3C24×C32116yescomplemented
136C3×C4—232no—
137C3×C4C23×C3×D3132yescomplemented
138C3×C4C24×C3218yescomplemented
139C32—24no—
140C32C23×D1211yescharacteristic, direct product factor
141C32C23×C3×D414yescomplemented
142C23—356no—
143C23—6112no—
144C23C32×D1218yesdirect product factor
145C23C22×C32×D317yes—
146D4—364no—
147C2×C4C22×C32×D3128yescomplemented
148C2×C3—256no—
149C2×C3—24no—
150C2×C3—664no—
151C2×C3C22×C3×D12156yesdirect product factor
152C2×C3C22×C32×D4114yescomplemented
153C2×C3C24×C3×D314yes—
154C2×C3C25×C3211yescharacteristic
155D3—216no—
156C22—316no—
157C22—6112no—
158C22C2×C32×D12128yesdirect product factor
159C22C23×C32×D317yes—
160C4C23×C32×D318yescomplemented
161C3—24no—
162C3C23×C3×D1214yesdirect product factor
163C3C23×C32×D411yescharacteristic, complemented
164C2—616no—
165C2C22×C32×D12114yesdirect product factor
166C2C24×C32×D311yescharacteristic
subgroupG/H
centerC24×C32C2×D3
commutator, cosocleC2×C3C25×C32
FrattiniC2C24×C32×D3
FittingC23×C33×C4C2
socleC24×C33C22
hypercenterC23×C32×C4D3
hypocenterC3C23×C32×D4
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C24×C32 ◁ C23×C32×C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C24×C32×D3C2semidirect
2. C23×C33×C4C2semidirect
3. C22×C32×D12C2direct (N × H)
5. C23×C3×D12C3direct (N × H)
8. C23×C32×D3C22semidirect
9. C22×C33×C4C22semidirect
10. C2×C32×D12C22direct (N × H)
12. C24×C3×D3C2×C3semidirect
15. C23×C32×C4D3semidirect
16. C23×C32×C4C2×C3semidirect
17. C22×C3×D12C2×C3direct (N × H)
18. C23×C33D4semidirect
21. C22×C32×D3C23semidirect
22. C2×C33×C4C23semidirect
23. C32×D12C23direct (N × H)
25. C23×D12C32direct (N × H)
32. C23×C3×D3C22×C3semidirect
35. C22×C32×C4C2×D3semidirect
36. C22×C32×C4C22×C3semidirect
37. C2×C3×D12C22×C3direct (N × H)
38. C22×C33C2×D4semidirect
41. C2×C32×D3C24semidirect
42. C33×C4C24semidirect
44. C24×D3C2×C32semidirect
47. C23×C3×C4C3×D3semidirect
48. C23×C3×C4C2×C32semidirect
49. C22×D12C2×C32direct (N × H)
54. C23×C32D12direct (N × H)
55. C23×C32C3×D4semidirect
59. C22×C3×D3C23×C3semidirect
62. C2×C32×C4C22×D3semidirect
63. C2×C32×C4C23×C3semidirect
64. C3×D12C23×C3direct (N × H)
66. C2×C33C22×D4semidirect
75. C23×D3C22×C32semidirect
78. C22×C3×C4C2×C3×D3semidirect
79. C22×C3×C4C22×C32semidirect
80. C2×D12C22×C32direct (N × H)
85. C22×C32C2×D12direct (N × H)
86. C22×C32C2×C3×D4semidirect
90. C2×C3×D3C24×C3semidirect
92. C32×C4C23×D3semidirect
93. C32×C4C24×C3semidirect
96. C23×C4C32×D3semidirect
97. C33C23×D4semidirect
102. C23×C3C3×D12direct (N × H)
103. C23×C3C32×D4semidirect
107. C22×D3C23×C32semidirect
110. C2×C3×C4C22×C3×D3semidirect
111. C2×C3×C4C23×C32semidirect
112. D12C23×C32direct (N × H)
116. C2×C32C22×D12direct (N × H)
117. C2×C32C22×C3×D4semidirect
125. C22×C4C2×C32×D3semidirect
130. C22×C3C2×C3×D12direct (N × H)
131. C22×C3C2×C32×D4semidirect
135. C2×D3C24×C32semidirect
137. C3×C4C23×C3×D3semidirect
138. C3×C4C24×C32semidirect
140. C32C23×D12direct (N × H)
141. C32C23×C3×D4semidirect
144. C23C32×D12direct (N × H)
147. C2×C4C22×C32×D3semidirect
151. C2×C3C22×C3×D12direct (N × H)
152. C2×C3C22×C32×D4semidirect
158. C22C2×C32×D12direct (N × H)
160. C4C23×C32×D3semidirect
162. C3C23×C3×D12direct (N × H)
163. C3C23×C32×D4semidirect
165. C2C22×C32×D12direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C24×C32×D32
C23×C33×C42
C22×C32×D122
C23×C32×D43
C23×C3×D123