Finite Group Databasebeta

← C22×#6291 #11724605

Subgroups — C22×#6291 #11724605

Order 864.

1952 conjugacy classes of subgroups; only the 320 normal subgroups are shown.

normal subgroups only

This group has 1952 conjugacy classes of subgroups — too many to draw. Only its 320 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
1C23×C3⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
2C23×C32×D3C211yesmaximal, characteristic, complemented
3C22×C3×C3⋊Dic3C211yesmaximal, characteristic, complemented
4C2×#6291C2112yesmaximal, direct product factor
5C22×#638—31nomaximal
6C22×#626—34nomaximal
7C23×C33C2211yescharacteristic
8C22×C3⋊(C3⋊D3)—24no—
9C22×C3⋊(C3⋊D3)C2216yescomplemented
10C22×C32×D3—24no—
11C22×C32×D3C2216yescomplemented
12C2×C3×C3⋊Dic3C2216yescomplemented
13#6291C22116yesdirect product factor
14C24×C32—31no—
15C23×C3⋊D3—31no—
16C23×C3⋊D3—34no—
17C23×C3⋊D3—64no—
18C23×C3×D3D314yescomplemented
19C2×#638—312no—
20C22×C3⋊Dic3D311yescharacteristic, complemented
21C22×C3×Dic3—34no—
22C2×#626—348no—
23C22×C33D414yescomplemented
24C22×C33C2313yes—
25C2×C3⋊(C3⋊D3)—212no—
26C2×C3⋊(C3⋊D3)C2314yescomplemented
27C2×C32×D3—212no—
28C2×C32×D3C2314yescomplemented
29C3×C3⋊Dic3C2314yescomplemented
30C22×#67—94no—
31C22×D12—91no—
32C23×C32—24no—
33C23×C32—36no—
34C23×C32—64no—
35C23×C32C2×D314yes—
36C23×C32C2×D311yescharacteristic
37C22×C3⋊D3—36no—
38C22×C3⋊D3—324no—
39C22×C3⋊D3—624no—
40C22×C3⋊D3—632no—
41C22×C3⋊D3—64no—
42C22×C3⋊D3—616no—
43C22×C3×D3—216no—
44C22×C3×D3C2×D3124yescomplemented
45#638—316no—
46C2×C3⋊Dic3C2×D316yescomplemented
47C2×C3×Dic3—324no—
48#626—364no—
49C2×C33C2×D416yescomplemented
50C2×C33C2411yescharacteristic
51C3⋊(C3⋊D3)—24no—
52C32×D3—24no—
53C24×C3—34no—
54C23×D3—94no—
55C23×D3—91no—
56C23×D3—184no—
57C23×D3C3⋊D311yescharacteristic, complemented
58C22×C3×C4—91no—
59C2×#67—948no—
60C22×Dic3—34no—
61C2×D12—912no—
62C22×C32—216no—
63C22×C32—212no—
64C22×C32—34no—
65C22×C32—612no—
66C22×C32D1214yescomplemented
67C22×C32#67116yescomplemented
68C22×C32C22×D3112yes—
69C22×C32C22×D313yes—
70C2×C3⋊D3—34no—
71C2×C3⋊D3—316no—
72C2×C3⋊D3—616no—
73C2×C3⋊D3—696no—
74C2×C3⋊D3—612no—
75C2×C3⋊D3—648no—
76C2×C3×D3—248no—
77C2×C3×D3C22×D3116yescomplemented
78C3⋊Dic3C22×D314yescomplemented
79C3×Dic3—316no—
80C22×D4—271noSylow
81C33C22×D411yescharacteristic, complemented, Sylow
82C23×C3—24no—
83C23×C3—324no—
84C23×C3—616no—
85C23×C3D3214yes—
86C23×C3C2×C3⋊D311yescharacteristic
87C22×D3—24no—
88C22×D3—924no—
89C22×D3—96no—
90C22×D3—1824no—
91C22×D3—1832no—
92C22×D3—1816no—
93C22×D3—184no—
94C22×D3C2×C3⋊D316yescomplemented
95C2×C3×C4—96no—
96#67—964no—
97C2×Dic3—324no—
98D12—916no—
99C2×C32—24no—
100C2×C32—224no—
101C2×C32—64no—
102C2×C32C2×D1216yescomplemented
103C2×C32C2×#67124yescomplemented
104C2×C32C23×D314yes—
105C2×C32C23×D311yescharacteristic
106C3⋊D3—632no—
107C3⋊D3—64no—
108C3⋊D3—616no—
109C3×D3—216no—
110C24—31no—
111C24—271no—
112C2×D4—2712no—
113C22×C4—91no—
114C22×C3—216no—
115C22×C3—212no—
116C22×C3—316no—
117C22×C3—648no—
118C22×C3#626116yescomplemented
119C22×C3#63814yescomplemented
120C22×C3C2×D32112yes—
121C22×C3C22×C3⋊D313yes—
122C2×D3—212no—
123C2×D3—916no—
124C2×D3—94no—
125C2×D3—1816no—
126C2×D3—1896no—
127C2×D3—1848no—
128C2×D3—1812no—
129C2×D3C22×C3⋊D314yescomplemented
130C3×C4—94no—
131Dic3—316no—
132C32—24no—
133C32C22×D1211yescharacteristic, complemented
134C32C22×#6714yescomplemented
135C23—36no—
136C23—64no—
137C23—276no—
138C23—544no—
139C23D3×C3⋊D311yescharacteristic
140D4—2716no—
141C2×C4—96no—
142C2×C3—24no—
143C2×C3—224no—
144C2×C3—616no—
145C2×C3C2×#626124yescomplemented
146C2×C3C2×#63816yescomplemented
147C2×C3C22×D3214yes—
148C2×C3C23×C3⋊D311yescharacteristic
149D3—24no—
150D3—1832no—
151D3—1816no—
152D3—184no—
153C22—34no—
154C22—612no—
155C22—274no—
156C22—5412no—
157C22#629114yesdirect product factor
158C22C2×D3×C3⋊D313yes—
159C4—94no—
160C3—24no—
161C3C22×#62614yescomplemented
162C3C22×#63811yescharacteristic, complemented
163C2—64no—
164C2—544no—
165C2C2×#629116yesdirect product factor
166C2C22×D3×C3⋊D311yescharacteristic
subgroupG/H
center, hypercenterC23D3×C3⋊D3
commutator, cosocleC2×C33C24
FrattiniC2C22×D3×C3⋊D3
Fitting, socleC23×C33C22
hypocenterC33C22×D4
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C23×C3⋊(C3⋊D3)C2semidirect
2. C23×C32×D3C2semidirect
3. C22×C3×C3⋊Dic3C2semidirect
4. C2×#6291C2direct (N × H)
9. C22×C3⋊(C3⋊D3)C22semidirect
11. C22×C32×D3C22semidirect
12. C2×C3×C3⋊Dic3C22semidirect
13. #6291C22direct (N × H)
18. C23×C3×D3D3semidirect
20. C22×C3⋊Dic3D3semidirect
23. C22×C33D4semidirect
26. C2×C3⋊(C3⋊D3)C23semidirect
28. C2×C32×D3C23semidirect
29. C3×C3⋊Dic3C23semidirect
44. C22×C3×D3C2×D3semidirect
46. C2×C3⋊Dic3C2×D3semidirect
49. C2×C33C2×D4semidirect
57. C23×D3C3⋊D3semidirect
66. C22×C32D12semidirect
67. C22×C32#67semidirect
77. C2×C3×D3C22×D3semidirect
78. C3⋊Dic3C22×D3semidirect
81. C33C22×D4semidirect
94. C22×D3C2×C3⋊D3semidirect
102. C2×C32C2×D12semidirect
103. C2×C32C2×#67semidirect
118. C22×C3#626semidirect
119. C22×C3#638semidirect
129. C2×D3C22×C3⋊D3semidirect
133. C32C22×D12semidirect
134. C32C22×#67semidirect
145. C2×C3C2×#626semidirect
146. C2×C3C2×#638semidirect
157. C22#6291direct (N × H)
161. C3C22×#626semidirect
162. C3C22×#638semidirect
165. C2C2×#6291direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3⋊(C3⋊D3)2
C23×C32×D32
C22×C3×C3⋊Dic32
C2×#62912
C22×#6383
C22×#6263