Finite Group Databasebeta

← C24×#626 #11919169

Subgroups — C24×#626 #11919169

Order 1152.

23929 conjugacy classes of subgroups; only the 4868 normal subgroups are shown.

normal subgroups only

This group has 23929 conjugacy classes of subgroups — too many to draw. Only its 4868 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
1C25×C3⋊D3C211yesmaximal, characteristic, complemented
2C25×C3×D3C211yesmaximal, characteristic, complemented
3C24×C3×Dic3C211yesmaximal, characteristic, complemented
4C23×#626C2160yesmaximal, direct product factor
5C24×#67—31nomaximal
6C24×D12—31nomaximal
7C25×C32C2211yescharacteristic
8C24×C3⋊D3—216no—
9C24×C3⋊D3C22130yescomplemented
10C24×C3×D3—216no—
11C24×C3×D3C22130yescomplemented
12C23×C3×Dic3C22130yescomplemented
13C22×#626C221560yesdirect product factor
14C26×C3—31no—
15C25×D3—31no—
16C25×D3—31no—
17C25×D3—61no—
18C25×D3D311yescharacteristic, complemented
19C24×C3×C4—31no—
20C23×#67—360no—
21C24×Dic3D311yescharacteristic, complemented
22C23×D12—360no—
23C24×C32D4116yescomplemented
24C24×C32C23115yes—
25C23×C3⋊D3—2240no—
26C23×C3⋊D3C231140yescomplemented
27C23×C3×D3—2240no—
28C23×C3×D3C231140yescomplemented
29C22×C3×Dic3C231140yescomplemented
30C2×#626C231960yesdirect product factor
31C24×D4—91noSylow
32C25×C3—21no—
33C25×C3—330no—
34C25×C3—616no—
35C25×C3C2×D311yescharacteristic
36C25×C3C2×D311yescharacteristic
37C24×D3—216no—
38C24×D3—330no—
39C24×D3—330no—
40C24×D3—630no—
41C24×D3—632no—
42C24×D3—616no—
43C24×D3—616no—
44C24×D3C2×D3130yescomplemented
45C23×C3×C4—330no—
46C22×#67—3560no—
47C23×Dic3C2×D3130yescomplemented
48C22×D12—3560no—
49C23×C32C2×D41120yescomplemented
50C23×C32C24135yes—
51C22×C3⋊D3—2560no—
52C22×C3⋊D3C241120yescomplemented
53C22×C3×D3—2560no—
54C22×C3×D3C241120yescomplemented
55C2×C3×Dic3C241120yescomplemented
56#626C241256yesdirect product factor
57C26—31no—
58C26—91no—
59C23×D4—960no—
60C24×C4—31no—
61C24×C3—216no—
62C24×C3—215no—
63C24×C3—3140no—
64C24×C3—6240no—
65C24×C3D12116yescomplemented
66C24×C3#67116yescomplemented
67C24×C3C22×D3115yes—
68C24×C3C22×D3115yes—
69C23×D3—2240no—
70C23×D3—3140no—
71C23×D3—3140no—
72C23×D3—6140no—
73C23×D3—6480no—
74C23×D3—6240no—
75C23×D3—6240no—
76C23×D3C22×D31140yescomplemented
77C22×C3×C4—3140no—
78C2×#67—3960no—
79C22×Dic3C22×D31140yescomplemented
80C2×D12—3960no—
81C22×C32C22×D41140yescomplemented
82C22×C32C25115yes—
83C2×C3⋊D3—2240no—
84C2×C3⋊D3C25116yescomplemented
85C2×C3×D3—2240no—
86C2×C3×D3C25116yescomplemented
87C3×Dic3C25116yescomplemented
88C25—330no—
89C25—616no—
90C25—930no—
91C25—1816no—
92C25D3211yescharacteristic
93C22×D4—9560no—
94C23×C4—330no—
95C23×C3—235no—
96C23×C3—2120no—
97C23×C3—3120no—
98C23×C3—6560no—
99C23×C3C2×D121120yescomplemented
100C23×C3C2×#671120yescomplemented
101C23×C3C23×D3135yes—
102C23×C3C23×D3135yes—
103C22×D3—2560no—
104C22×D3—3120no—
105C22×D3—3120no—
106C22×D3—6120no—
107C22×D3—61120no—
108C22×D3—6560no—
109C22×D3—6560no—
110C22×D3C23×D31120yescomplemented
111C2×C3×C4—3120no—
112#67—3256no—
113C2×Dic3C23×D31120yescomplemented
114D12—3256no—
115C2×C32C23×D4130yescomplemented
116C2×C32C2611yescharacteristic
117C3⋊D3—216no—
118C3×D3—216no—
119C24—3140no—
120C24—6240no—
121C24—9140no—
122C24—18240no—
123C24#626116yesdirect product factor
124C24C2×D32115yes—
125C2×D4—9960no—
126C22×C4—3140no—
127C22×C3—2140no—
128C22×C3—215no—
129C22×C3—316no—
130C22×C3—6240no—
131C22×C3C22×D121140yescomplemented
132C22×C3C22×#671140yescomplemented
133C22×C3C24×D3115yes—
134C22×C3C24×D3115yes—
135C2×D3—2240no—
136C2×D3—316no—
137C2×D3—316no—
138C2×D3—616no—
139C2×D3—6480no—
140C2×D3—6240no—
141C2×D3—6240no—
142C2×D3C24×D3116yescomplemented
143C3×C4—316no—
144Dic3C24×D3116yescomplemented
145C32C24×D411yescharacteristic, complemented, Sylow
146C23—3120no—
147C23—6560no—
148C23—9120no—
149C23—18560no—
150C23C2×#6261120yesdirect product factor
151C23C22×D32135yes—
152D4—9256no—
153C2×C4—3120no—
154C2×C3—230no—
155C2×C3—21no—
156C2×C3—616no—
157C2×C3C23×D12130yescomplemented
158C2×C3C23×#67130yescomplemented
159C2×C3C25×D311yescharacteristic
160C2×C3C25×D311yescharacteristic
161D3—216no—
162D3—632no—
163D3—616no—
164D3—616no—
165C22—316no—
166C22—6240no—
167C22—916no—
168C22—18240no—
169C22C22×#6261140yesdirect product factor
170C22C23×D32115yes—
171C4—316no—
172C3—21no—
173C3C24×D1211yescharacteristic, complemented
174C3C24×#6711yescharacteristic, complemented
175C2—616no—
176C2—1816no—
177C2C23×#626130yesdirect product factor
178C2C24×D3211yescharacteristic
subgroupG/H
center, hypercenterC25D32
commutator, cosocleC2×C32C26
FrattiniC2C24×D32
Fitting, socleC25×C32C22
hypocenterC32C24×D4
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C25

Complements

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

NH ≅ G/Ntype
1. C25×C3⋊D3C2semidirect
2. C25×C3×D3C2semidirect
3. C24×C3×Dic3C2semidirect
4. C23×#626C2direct (N × H)
9. C24×C3⋊D3C22semidirect
11. C24×C3×D3C22semidirect
12. C23×C3×Dic3C22semidirect
13. C22×#626C22direct (N × H)
18. C25×D3D3semidirect
21. C24×Dic3D3semidirect
23. C24×C32D4semidirect
26. C23×C3⋊D3C23semidirect
28. C23×C3×D3C23semidirect
29. C22×C3×Dic3C23semidirect
30. C2×#626C23direct (N × H)
44. C24×D3C2×D3semidirect
47. C23×Dic3C2×D3semidirect
49. C23×C32C2×D4semidirect
52. C22×C3⋊D3C24semidirect
54. C22×C3×D3C24semidirect
55. C2×C3×Dic3C24semidirect
56. #626C24direct (N × H)
65. C24×C3D12semidirect
66. C24×C3#67semidirect
76. C23×D3C22×D3semidirect
79. C22×Dic3C22×D3semidirect
81. C22×C32C22×D4semidirect
84. C2×C3⋊D3C25semidirect
86. C2×C3×D3C25semidirect
87. C3×Dic3C25semidirect
99. C23×C3C2×D12semidirect
100. C23×C3C2×#67semidirect
110. C22×D3C23×D3semidirect
113. C2×Dic3C23×D3semidirect
115. C2×C32C23×D4semidirect
123. C24#626direct (N × H)
131. C22×C3C22×D12semidirect
132. C22×C3C22×#67semidirect
142. C2×D3C24×D3semidirect
144. Dic3C24×D3semidirect
145. C32C24×D4semidirect
150. C23C2×#626direct (N × H)
157. C2×C3C23×D12semidirect
158. C2×C3C23×#67semidirect
169. C22C22×#626direct (N × H)
173. C3C24×D12semidirect
174. C3C24×#67semidirect
177. C2C23×#626direct (N × H)

Maximal subgroups

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

subgroupindex
C25×C3⋊D32
C25×C3×D32
C24×C3×Dic32
C23×#6262
C24×#673
C24×D123