Finite Group Databasebeta

← C2×D4×#11745400 #28417499

Subgroups — C2×D4×#11745400 #28417499

Order 14912.

426 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 156 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×#11745400C212yesmaximal, complemented
2D4×#11745400C218yesmaximal, direct product factor
3C2×C4⋊#11745400C211yesmaximal, characteristic, complemented
4C2×C22⋊#11745400C212yesmaximal, complemented
5C2×C4×#11745400C211yesmaximal, characteristic, complemented
6C2×D4×D233C211yesmaximal, characteristic
7C22×#11745400—28no—
8C22×#11745400C2211yescharacteristic
9C22×#11745400C2212yes—
10C22×#11745400C2211yescharacteristic
11C22×#11745400C2218yescomplemented
12C4⋊#11745400C2214yescomplemented
13C22⋊#11745400C2218yescomplemented
14C4×#11745400C2214yescomplemented
15C23×D233C2212yes—
16C2×D932C411yescharacteristic, complemented
17C2×C22⋊D233C412yescomplemented
18D4×D233C414yescomplemented
19D4×D233C2214yes—
20C2×C4×D233C2211yescharacteristic
21C2×D4×C233C411yescharacteristic, complemented
22C2×#11745400—24no—
23C2×#11745400—22no—
24C2×#11745400—216no—
25C2×#11745400D414yesdirect product factor
26C2×#11745400C2312yes—
27C2×#11745400C2314yes—
28C2×#11745400C2312yes—
29C22×D233—24no—
30C22×D233—24no—
31C22×D233C2×C414yescomplemented
32C22×D233C2×C412yescomplemented
33C22×D233C2314yes—
34C22×D233C2311yescharacteristic
35D932C2×C414yescomplemented
36C22⋊D233C2×C418yescomplemented
37C2×Dic233C2×C411yescharacteristic, complemented
38C4×D233C2×C412yescomplemented
39C4×D233C2312yes—
40C23×C233C2×C412yescomplemented
41D4×C233C2×C414yescomplemented
42C2×C4×C233C2×C411yescharacteristic, complemented
43C22×C233—24no—
44C22×C233C22×C411yescharacteristic
45C22×C233C22×C414yescomplemented
46C2×D233—24no—
47C2×D233—28no—
48C2×D233—24no—
49C2×D233—24no—
50C2×D233C22×C411yescharacteristic
51C2×D233C22×C414yescomplemented
52C2×D233C2×D412yes—
53C2×D233#4112yes—
54C2×D233C2411yescharacteristic
55#11745400—24no—
56#11745400—22no—
57#11745400C2×D414yesdirect product factor
58C4×C233C22×C412yescomplemented
59Dic233C22×C412yescomplemented
60C2×C233—24no—
61C2×C233C4×D412yescomplemented
62C2×C233C23×C411yescharacteristic
63D233—24no—
64D233C4×D412yescomplemented
65D233C22×D411yescharacteristic
66D233C2×#4111yescharacteristic
67C233C2×C4×D411yescharacteristic, complemented, Sylow
68C2×C4×D4—2331nomaximal, Sylow
69C22×D4—2331no—
70C23×C4—2332no—
71C4×D4—2338no—
72C2×C4⋊C4—2331no—
73C2×C22⋊C4—2332no—
74C2×C42—2331no—
75C24—2332no—
76C2×D4—2332no—
77C2×D4—2334no—
78C2×D4—2334no—
79C2×D4—2331no—
80C2×D4#1174540011yescharacteristic, direct product factor
81C22×C4—4668no—
82C22×C4—2331no—
83C22×C4—2332no—
84C22×C4—2338no—
85C22×C4—2331no—
86C22×C4—2331no—
87C4⋊C4—2334no—
88C22⋊C4—2338no—
89C42—2334no—
90C23—4664no—
91C23—4664no—
92C23—2332no—
93C23—2334no—
94C23—2334no—
95C23—2331no—
96C23C2×#1174540012yescomplemented
97D4—2338no—
98D4—2334no—
99D4C2×#1174540014yesdirect product factor
100C2×C4—4664no—
101C2×C4—4662no—
102C2×C4—46616no—
103C2×C4—2332no—
104C2×C4—2334no—
105C2×C4—2332no—
106C2×C4—2334no—
107C2×C4—2331no—
108C2×C4—2332no—
109C2×C4—2332no—
110C2×C4C2×#1174540011yescharacteristic, complemented
111C22—24no—
112C22—4664no—
113C22—4664no—
114C22—4668no—
115C22—4664no—
116C22—2334no—
117C22—2332no—
118C22—2332no—
119C22—2331no—
120C22—2331no—
121C22C22×#1174540011yescharacteristic
122C22C22×#1174540014yescomplemented
123C4—4664no—
124C4—4662no—
125C4—2334no—
126C4—2332no—
127C4C22×#1174540012yescomplemented
128C2—24no—
129C2—4664no—
130C2—2331no—
131C2—2331no—
132C2—2332no—
133C2D4×#1174540012yesdirect product factor
134C2C23×#1174540011yescharacteristic
subgroupG/H
centerC22C22×#11745400
commutatorC2×C233C23×C4
FrattiniC2C23×#11745400
FittingC2×D4×C233C4
socleC22×C233C22×C4
prime radicalC2×D4×D233C2
cosocleC2×D233C24
hypercenterC2×D4#11745400
hypocenterC233C2×C4×D4
LCSG ▷ C2×C233 ▷ C233
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C23×#11745400C2semidirect
2. D4×#11745400C2direct (N × H)
3. C2×C4⋊#11745400C2semidirect
4. C2×C22⋊#11745400C2semidirect
5. C2×C4×#11745400C2semidirect
11. C22×#11745400C22semidirect
12. C4⋊#11745400C22semidirect
13. C22⋊#11745400C22semidirect
14. C4×#11745400C22semidirect
16. C2×D932C4semidirect
17. C2×C22⋊D233C4semidirect
18. D4×D233C4semidirect
21. C2×D4×C233C4semidirect
25. C2×#11745400D4direct (N × H)
31. C22×D233C2×C4semidirect
32. C22×D233C2×C4semidirect
35. D932C2×C4semidirect
36. C22⋊D233C2×C4semidirect
37. C2×Dic233C2×C4semidirect
38. C4×D233C2×C4semidirect
40. C23×C233C2×C4semidirect
41. D4×C233C2×C4semidirect
42. C2×C4×C233C2×C4semidirect
45. C22×C233C22×C4semidirect
51. C2×D233C22×C4semidirect
57. #11745400C2×D4direct (N × H)
58. C4×C233C22×C4semidirect
59. Dic233C22×C4semidirect
61. C2×C233C4×D4semidirect
64. D233C4×D4semidirect
67. C233C2×C4×D4semidirect
80. C2×D4#11745400direct (N × H)
96. C23C2×#11745400semidirect
99. D4C2×#11745400direct (N × H)
110. C2×C4C2×#11745400semidirect
122. C22C22×#11745400semidirect
127. C4C22×#11745400semidirect
133. C2D4×#11745400direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#117454002
D4×#117454002
C2×C4⋊#117454002
C2×C22⋊#117454002
C2×C4×#117454002
C2×D4×D2332
C2×C4×D4233