Finite Group Databasebeta

← C24×#213 #11708961

Subgroups — C24×#213 #11708961

Order 768.

15454 conjugacy classes of subgroups; only the 4427 normal subgroups are shown.

normal subgroups only

This group has 15454 conjugacy classes of subgroups — too many to draw. Only its 4427 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×C3×D4C211yesmaximal, characteristic, complemented
2C24×D12C211yesmaximal, characteristic, complemented
3C23×#213C2160yesmaximal, direct product factor
4C24×C3⋊C8C211yesmaximal, characteristic, complemented
5C24×D8—31nomaximal, Sylow
6C26×C3—21no—
7C25×D3—21no—
8C23×C3×D4—215no—
9C23×C3×D4C22130yescomplemented
10C24×C3×C4C2211yescharacteristic
11C23×D12—215no—
12C23×D12C22130yescomplemented
13C22×#213C221560yesdirect product factor
14C23×C3⋊C8C22130yescomplemented
15C24×D4—31no—
16C24×D4D311yescharacteristic, complemented
17C23×D8—360no—
18C24×C8—31no—
19C25×C3—230no—
20C25×C3—416no—
21C25×C3D411yescharacteristic
22C24×D3—230no—
23C24×D3—416no—
24C22×C3×D4—2210no—
25C22×C3×D4C231140yescomplemented
26C23×C3×C4D4115yes—
27C23×C3×C4C23115yes—
28C22×D12—2210no—
29C22×D12C231140yescomplemented
30C2×#213C231960yesdirect product factor
31C22×C3⋊C8C231140yescomplemented
32C26—21no—
33C26—61no—
34C23×D4—215no—
35C23×D4—330no—
36C23×D4—615no—
37C23×D4C2×D3130yescomplemented
38C24×C4C2×D311yescharacteristic
39C22×D8—3560no—
40C23×C8—330no—
41C24×C3—2140no—
42C24×C3—4240no—
43C24×C3D8116yescomplemented
44C24×C3C2×D4115yes—
45C23×D3—2140no—
46C23×D3—4240no—
47C2×C3×D4—2420no—
48C2×C3×D4C241120yescomplemented
49C22×C3×C4C2×D41105yes—
50C22×C3×C4C24135yes—
51C2×D12—2420no—
52C2×D12C241120yescomplemented
53#213C241256yesdirect product factor
54C2×C3⋊C8C241120yescomplemented
55C25—230no—
56C25—416no—
57C25—630no—
58C25—1216no—
59C25#6711yescharacteristic
60C22×D4—2210no—
61C22×D4—3140no—
62C22×D4—6210no—
63C22×D4C22×D31140yescomplemented
64C23×C4#67115yes—
65C23×C4C22×D3115yes—
66C2×D8—3960no—
67C22×C8—3140no—
68C23×C3—2120no—
69C23×C3—4560no—
70C23×C3C2×D81120yescomplemented
71C23×C3C22×D4135yes—
72C22×D3—2120no—
73C22×D3—4560no—
74C3×D4—2120no—
75C3×D4C25116yescomplemented
76C2×C3×C4C22×D41105yes—
77C2×C3×C4C25115yes—
78D12—2120no—
79D12C25116yescomplemented
80C3⋊C8C25116yescomplemented
81C24—2140no—
82C24—4240no—
83C24—6140no—
84C24—12240no—
85C24#213116yesdirect product factor
86C24C2×#67115yes—
87C2×D4—2420no—
88C2×D4—3120no—
89C2×D4—6420no—
90C2×D4C23×D31120yescomplemented
91C22×C4C2×#671105yes—
92C22×C4C23×D3135yes—
93D8—3256no—
94C2×C8—3120no—
95C22×C3—216no—
96C22×C3—4240no—
97C22×C3C22×D81140yescomplemented
98C22×C3C23×D4115yes—
99C2×D3—216no—
100C2×D3—4240no—
101C3×C4C23×D4115yes—
102C3×C4C2611yescharacteristic
103C23—2120no—
104C23—4560no—
105C23—6120no—
106C23—12560no—
107C23C2×#2131120yesdirect product factor
108C23C22×#67135yes—
109D4—2120no—
110D4—316no—
111D4—6120no—
112D4C24×D3116yescomplemented
113C2×C4C22×#671105yes—
114C2×C4C24×D3115yes—
115C8—316no—
116C2×C3—416no—
117C2×C3C23×D8130yescomplemented
118C2×C3C24×D411yescharacteristic
119D3—416no—
120C22—216no—
121C22—4240no—
122C22—616no—
123C22—12240no—
124C22C22×#2131140yesdirect product factor
125C22C23×#67115yes—
126C4C23×#67115yes—
127C4C25×D311yescharacteristic
128C3C24×D811yescharacteristic, complemented, Sylow
129C2—416no—
130C2—1216no—
131C2C23×#213130yesdirect product factor
132C2C24×#6711yescharacteristic
subgroupG/H
centerC25#67
commutator, cosocleC3×C4C26
FrattiniC4C25×D3
FittingC24×C3×D4C2
socleC25×C3D4
hypercenterC24×D4D3
hypocenterC3C24×D8
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C25 ◁ C24×C4 ◁ C24×D4

Complements

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

NH ≅ G/Ntype
1. C24×C3×D4C2semidirect
2. C24×D12C2semidirect
3. C23×#213C2direct (N × H)
4. C24×C3⋊C8C2semidirect
9. C23×C3×D4C22semidirect
12. C23×D12C22semidirect
13. C22×#213C22direct (N × H)
14. C23×C3⋊C8C22semidirect
16. C24×D4D3semidirect
25. C22×C3×D4C23semidirect
29. C22×D12C23semidirect
30. C2×#213C23direct (N × H)
31. C22×C3⋊C8C23semidirect
37. C23×D4C2×D3semidirect
43. C24×C3D8semidirect
48. C2×C3×D4C24semidirect
52. C2×D12C24semidirect
53. #213C24direct (N × H)
54. C2×C3⋊C8C24semidirect
63. C22×D4C22×D3semidirect
70. C23×C3C2×D8semidirect
75. C3×D4C25semidirect
79. D12C25semidirect
80. C3⋊C8C25semidirect
85. C24#213direct (N × H)
90. C2×D4C23×D3semidirect
97. C22×C3C22×D8semidirect
107. C23C2×#213direct (N × H)
112. D4C24×D3semidirect
117. C2×C3C23×D8semidirect
124. C22C22×#213direct (N × H)
128. C3C24×D8semidirect
131. C2C23×#213direct (N × H)

Maximal subgroups

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

subgroupindex
C24×C3×D42
C24×D122
C23×#2132
C24×C3⋊C82
C24×D83