Finite Group Databasebeta

← C22×#6290 #11724604

Subgroups — C22×#6290 #11724604

Order 864.

1632 conjugacy classes of subgroups; only the 320 normal subgroups are shown.

normal subgroups only

This group has 1632 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⋊D3C211yesmaximal, characteristic, complemented
2C23×C32×D3C211yesmaximal, characteristic, complemented
3C22×C3⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
4C2×#6290C2112yesmaximal, direct product factor
5C22×#638—31nomaximal
6C22×#625—34nomaximal
7C23×C33C2211yescharacteristic
8C22×C3×C3⋊D3—24no—
9C22×C3×C3⋊D3C2216yescomplemented
10C22×C32×D3—24no—
11C22×C32×D3C2216yescomplemented
12C2×C3⋊(C3⋊Dic3)C2216yescomplemented
13#6290C22116yesdirect product factor
14C24×C32—31no—
15C23×C3⋊D3D311yescharacteristic, complemented
16C23×C3×D3—34no—
17C23×C3×D3D314yescomplemented
18C2×#638—312no—
19C22×C3⋊Dic3—31no—
20C22×C3⋊Dic3—34no—
21C22×C3⋊Dic3—64no—
22C2×#625—348no—
23C22×C33D414yescomplemented
24C22×C33C2313yes—
25C2×C3×C3⋊D3—212no—
26C2×C3×C3⋊D3C2314yescomplemented
27C2×C32×D3—212no—
28C2×C32×D3C2314yescomplemented
29C3⋊(C3⋊Dic3)C2314yescomplemented
30C22×#67—94no—
31C22×#67—91no—
32C23×C32—24no—
33C23×C32—36no—
34C23×C32—64no—
35C23×C32C2×D314yes—
36C23×C32C2×D311yescharacteristic
37C22×C3⋊D3—24no—
38C22×C3⋊D3C2×D316yescomplemented
39C22×C3×D3—216no—
40C22×C3×D3—324no—
41C22×C3×D3—616no—
42C22×C3×D3C2×D3124yescomplemented
43#638—316no—
44C2×C3⋊Dic3—36no—
45C2×C3⋊Dic3—324no—
46C2×C3⋊Dic3—624no—
47#625—364no—
48C2×C33C2×D416yescomplemented
49C2×C33C2411yescharacteristic
50C3×C3⋊D3—24no—
51C32×D3—24no—
52C24×C3—34no—
53C24×C3—91no—
54C23×D3—34no—
55C23×D3C3⋊D311yescharacteristic, complemented
56C2×#67—948no—
57C2×#67—912no—
58C22×Dic3—94no—
59C22×Dic3—91no—
60C22×Dic3—184no—
61C22×C32—216no—
62C22×C32—212no—
63C22×C32—34no—
64C22×C32—612no—
65C22×C32#67116yescomplemented
66C22×C32#6714yescomplemented
67C22×C32C22×D3112yes—
68C22×C32C22×D313yes—
69C2×C3⋊D3—212no—
70C2×C3⋊D3C22×D314yescomplemented
71C2×C3×D3—248no—
72C2×C3×D3—316no—
73C2×C3×D3—648no—
74C2×C3×D3C22×D3116yescomplemented
75C3⋊Dic3—34no—
76C3⋊Dic3—316no—
77C3⋊Dic3—616no—
78C22×D4—271noSylow
79C33C22×D411yescharacteristic, complemented, Sylow
80C23×C3—24no—
81C23×C3—324no—
82C23×C3—616no—
83C23×C3—96no—
84C23×C3—184no—
85C23×C3D3214yes—
86C23×C3C2×C3⋊D311yescharacteristic
87C22×D3—24no—
88C22×D3—324no—
89C22×D3—616no—
90C22×D3C2×C3⋊D316yescomplemented
91#67—964no—
92#67—916no—
93C2×Dic3—924no—
94C2×Dic3—96no—
95C2×Dic3—1824no—
96C2×C32—24no—
97C2×C32—224no—
98C2×C32—64no—
99C2×C32C2×#67124yescomplemented
100C2×C32C2×#6716yescomplemented
101C2×C32C23×D314yes—
102C2×C32C23×D311yescharacteristic
103C3⋊D3—24no—
104C3×D3—216no—
105C3×D3—616no—
106C24—31no—
107C24—91no—
108C2×D4—2712no—
109C22×C4—271no—
110C22×C3—216no—
111C22×C3—212no—
112C22×C3—316no—
113C22×C3—648no—
114C22×C3—94no—
115C22×C3—1812no—
116C22×C3#625116yescomplemented
117C22×C3#63814yescomplemented
118C22×C3C2×D32112yes—
119C22×C3C22×C3⋊D313yes—
120C2×D3—212no—
121C2×D3—316no—
122C2×D3—648no—
123C2×D3C22×C3⋊D314yescomplemented
124Dic3—916no—
125Dic3—94no—
126Dic3—1816no—
127C32—24no—
128C32C22×#6714yescomplemented
129C32C22×#6711yescharacteristic, complemented
130C23—36no—
131C23—64no—
132C23—96no—
133C23—184no—
134C23D3×C3⋊D311yescharacteristic
135D4—2716no—
136C2×C4—276no—
137C2×C3—24no—
138C2×C3—224no—
139C2×C3—616no—
140C2×C3—184no—
141C2×C3C2×#625124yescomplemented
142C2×C3C2×#63816yescomplemented
143C2×C3C22×D3214yes—
144C2×C3C23×C3⋊D311yescharacteristic
145D3—24no—
146D3—616no—
147C22—34no—
148C22—612no—
149C22—94no—
150C22—1812no—
151C22#629014yesdirect product factor
152C22C2×D3×C3⋊D313yes—
153C4—274no—
154C3—24no—
155C3C22×#62514yescomplemented
156C3C22×#63811yescharacteristic, complemented
157C2—64no—
158C2—184no—
159C2C2×#629016yesdirect product factor
160C2C22×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⋊D3C2semidirect
2. C23×C32×D3C2semidirect
3. C22×C3⋊(C3⋊Dic3)C2semidirect
4. C2×#6290C2direct (N × H)
9. C22×C3×C3⋊D3C22semidirect
11. C22×C32×D3C22semidirect
12. C2×C3⋊(C3⋊Dic3)C22semidirect
13. #6290C22direct (N × H)
15. C23×C3⋊D3D3semidirect
17. C23×C3×D3D3semidirect
23. C22×C33D4semidirect
26. C2×C3×C3⋊D3C23semidirect
28. C2×C32×D3C23semidirect
29. C3⋊(C3⋊Dic3)C23semidirect
38. C22×C3⋊D3C2×D3semidirect
42. C22×C3×D3C2×D3semidirect
48. C2×C33C2×D4semidirect
55. C23×D3C3⋊D3semidirect
65. C22×C32#67semidirect
66. C22×C32#67semidirect
70. C2×C3⋊D3C22×D3semidirect
74. C2×C3×D3C22×D3semidirect
79. C33C22×D4semidirect
90. C22×D3C2×C3⋊D3semidirect
99. C2×C32C2×#67semidirect
100. C2×C32C2×#67semidirect
116. C22×C3#625semidirect
117. C22×C3#638semidirect
123. C2×D3C22×C3⋊D3semidirect
128. C32C22×#67semidirect
129. C32C22×#67semidirect
141. C2×C3C2×#625semidirect
142. C2×C3C2×#638semidirect
151. C22#6290direct (N × H)
155. C3C22×#625semidirect
156. C3C22×#638semidirect
159. C2C2×#6290direct (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⋊D32
C23×C32×D32
C22×C3⋊(C3⋊Dic3)2
C2×#62902
C22×#6383
C22×#6253