Finite Group Databasebeta

← D4×D12 #5549

Subgroups — D4×D12 #5549

Order 192.

428 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 123 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
1C2×D3×D4C212yesmaximal, complemented
2C22×D12C212yesmaximal, complemented
3C3×C4×D4C211yesmaximal, characteristic, complemented
4#930C212yesmaximal, complemented
5#895C212yesmaximal, complemented
6#882C214yesmaximal, complemented
7#874C211yesmaximal, characteristic, complemented
8C4×D12C211yesmaximal, characteristic, complemented
9D42—31nomaximal, Sylow
10C23×D3C2214yescomplemented
11C2×C3×D4C2211yescharacteristic
12C22×C3×C4C2212yescomplemented
13C2×#67C2214yescomplemented
14D3×D4—28no—
15C2×D12—28no—
16C2×D12C2211yescharacteristic
17C2×D12C2212yescomplemented
18C2×D12C2212yescomplemented
19C2×D12C2214yescomplemented
20C2×D12C2212yescomplemented
21C2×C4×D3C2212yescomplemented
22C3×C4⋊C4C2211yescharacteristic, complemented
23C3×C22⋊C4C2212yescomplemented
24C3×C42C2211yescharacteristic, complemented
25(C2×D3)⋊C4C2212yescomplemented
26(C2×D3)⋊C4C2214yescomplemented
27C4⋊Dic3C2211yescharacteristic, complemented
28C22×D4—32no—
29C22×D4—32no—
30#127—31no—
31#121—32no—
32#121—32no—
33#120—34no—
34C4×D4—31no—
35C4×D4D311yescharacteristic, complemented
36C23×C3C2312yes—
37C22×D3—28no—
38C22×D3—28no—
39C22×D3—44no—
40C22×D3C2314yes—
41C22×D3C2312yes—
42C3×D4D414yescomplemented
43C2×C3×C4—24no—
44C2×C3×C4C2311yescharacteristic
45C2×C3×C4C2312yes—
46C2×C3×C4C2311yescharacteristic
47C2×C3×C4C2311yescharacteristic
48#67—28no—
49C2×Dic3C2312yes—
50D12—28no—
51D12—24no—
52D12—24no—
53D12—42no—
54D12D414yesdirect product factor
55C4×D3—24no—
56C24—34no—
57C2×D4—34no—
58C2×D4—31no—
59C2×D4—34no—
60C2×D4—32no—
61C2×D4—32no—
62C2×D4—32no—
63C2×D4—68no—
64C2×D4—68no—
65C2×D4C2×D311yescharacteristic
66C22×C4—32no—
67C22×C4C2×D312yescomplemented
68C4⋊C4—31no—
69C4⋊C4C2×D311yescharacteristic, complemented
70C22⋊C4—32no—
71C22⋊C4—34no—
72C22⋊C4C2×D312yescomplemented
73C42C2×D311yescharacteristic, complemented
74C22×C3—22no—
75C22×C3—22no—
76C22×C3C2×D414yescomplemented
77C22×C3C2411yescharacteristic
78C2×D3—22no—
79C2×D3—22no—
80C2×D3—24no—
81C2×D3—24no—
82C2×D3—24no—
83C2×D3—416no—
84C2×D3C2×D414yescomplemented
85C3×C4—22no—
86C3×C4—21no—
87C3×C4C2×D412yescomplemented
88C3×C4C2×D412yescomplemented
89Dic3—22no—
90C23—32no—
91C23—34no—
92C23—68no—
93C23—68no—
94C23—124no—
95C23C22×D312yes—
96D4—34no—
97D4—68no—
98D4—68no—
99D4—64no—
100D4—64no—
101D4—122no—
102D4D1214yesdirect product factor
103C2×C4—24no—
104C2×C4—32no—
105C2×C4—64no—
106C2×C4C22×D311yescharacteristic
107C2×C4C22×D312yes—
108C2×C4C22×D311yescharacteristic
109C2×C4C22×D311yescharacteristic
110C2×C3—24no—
111C2×C3C22×D411yescharacteristic
112C2×C3C22×D411yescharacteristic
113C2×C3H(5,𝔽2)11yescharacteristic
114D3—24no—
115D3—44no—
116C22—22no—
117C22—22no—
118C22—34no—
119C22—62no—
120C22—62no—
121C22—64no—
122C22—64no—
123C22—64no—
124C22—1216no—
125C22C2×D1214yescomplemented
126C22C23×D311yescharacteristic
127C4—22no—
128C4—21no—
129C4—62no—
130C4C2×D1212yescomplemented
131C4D3×D412yescomplemented
132C3D4211yescharacteristic, complemented, Sylow
133C2—24no—
134C2—64no—
135C2—124no—
136C2C22×D1211yescharacteristic
137C2C2×D3×D411yescharacteristic
138C2#100911yescharacteristic
subgroupG/H
center, FrattiniC22C23×D3
commutator, socle, cosocleC22×C3C24
FittingC3×C4×D4C2
hypercenterC4×D4D3
hypocenterC3D42
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×D3×D4C2semidirect
2. C22×D12C2semidirect
3. C3×C4×D4C2semidirect
4. #930C2semidirect
5. #895C2semidirect
6. #882C2semidirect
7. #874C2semidirect
8. C4×D12C2semidirect
10. C23×D3C22semidirect
12. C22×C3×C4C22semidirect
13. C2×#67C22semidirect
17. C2×D12C22semidirect
18. C2×D12C22semidirect
19. C2×D12C22semidirect
20. C2×D12C22semidirect
21. C2×C4×D3C22semidirect
22. C3×C4⋊C4C22semidirect
23. C3×C22⋊C4C22semidirect
24. C3×C42C22semidirect
25. (C2×D3)⋊C4C22semidirect
26. (C2×D3)⋊C4C22semidirect
27. C4⋊Dic3C22semidirect
35. C4×D4D3semidirect
42. C3×D4D4semidirect
54. D12D4direct (N × H)
67. C22×C4C2×D3semidirect
69. C4⋊C4C2×D3semidirect
72. C22⋊C4C2×D3semidirect
73. C42C2×D3semidirect
76. C22×C3C2×D4semidirect
84. C2×D3C2×D4semidirect
87. C3×C4C2×D4semidirect
88. C3×C4C2×D4semidirect
102. D4D12direct (N × H)
125. C22C2×D12semidirect
130. C4C2×D12semidirect
131. C4D3×D4semidirect
132. C3D42semidirect

Maximal subgroups

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

subgroupindex
C2×D3×D42
C22×D122
C3×C4×D42
#9302
#8952
#8822
#8742
C4×D122
D423