Finite Group Databasebeta

← C2×D4×#11757210 #29508781

Subgroups — C2×D4×#11757210 #29508781

Order 15424.

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×#11757210C212yesmaximal, complemented
2D4×#11757210C218yesmaximal, direct product factor
3C2×C4⋊#11757210C211yesmaximal, characteristic, complemented
4C2×C22⋊#11757210C212yesmaximal, complemented
5C2×C4×#11757210C211yesmaximal, characteristic, complemented
6C2×D4×D241C211yesmaximal, characteristic
7C22×#11757210—28no—
8C22×#11757210C2211yescharacteristic
9C22×#11757210C2212yes—
10C22×#11757210C2211yescharacteristic
11C22×#11757210C2218yescomplemented
12C4⋊#11757210C2214yescomplemented
13C22⋊#11757210C2218yescomplemented
14C4×#11757210C2214yescomplemented
15C23×D241C2212yes—
16C2×D964C411yescharacteristic, complemented
17C2×C22⋊D241C412yescomplemented
18D4×D241C414yescomplemented
19D4×D241C2214yes—
20C2×C4×D241C2211yescharacteristic
21C2×D4×C241C411yescharacteristic, complemented
22C2×#11757210—24no—
23C2×#11757210—22no—
24C2×#11757210—216no—
25C2×#11757210D414yesdirect product factor
26C2×#11757210C2312yes—
27C2×#11757210C2314yes—
28C2×#11757210C2312yes—
29C22×D241—24no—
30C22×D241—24no—
31C22×D241C2×C414yescomplemented
32C22×D241C2×C412yescomplemented
33C22×D241C2314yes—
34C22×D241C2311yescharacteristic
35D964C2×C414yescomplemented
36C22⋊D241C2×C418yescomplemented
37C2×Dic241C2×C411yescharacteristic, complemented
38C4×D241C2×C412yescomplemented
39C4×D241C2312yes—
40C23×C241C2×C412yescomplemented
41D4×C241C2×C414yescomplemented
42C2×C4×C241C2×C411yescharacteristic, complemented
43C22×C241—24no—
44C22×C241C22×C411yescharacteristic
45C22×C241C22×C414yescomplemented
46C2×D241—24no—
47C2×D241—28no—
48C2×D241—24no—
49C2×D241—24no—
50C2×D241C22×C411yescharacteristic
51C2×D241C22×C414yescomplemented
52C2×D241C2×D412yes—
53C2×D241#4112yes—
54C2×D241C2411yescharacteristic
55#11757210—24no—
56#11757210—22no—
57#11757210C2×D414yesdirect product factor
58C4×C241C22×C412yescomplemented
59Dic241C22×C412yescomplemented
60C2×C241—24no—
61C2×C241C4×D412yescomplemented
62C2×C241C23×C411yescharacteristic
63D241—24no—
64D241C4×D412yescomplemented
65D241C22×D411yescharacteristic
66D241C2×#4111yescharacteristic
67C241C2×C4×D411yescharacteristic, complemented, Sylow
68C2×C4×D4—2411nomaximal, Sylow
69C22×D4—2411no—
70C23×C4—2412no—
71C4×D4—2418no—
72C2×C4⋊C4—2411no—
73C2×C22⋊C4—2412no—
74C2×C42—2411no—
75C24—2412no—
76C2×D4—2412no—
77C2×D4—2414no—
78C2×D4—2414no—
79C2×D4—2411no—
80C2×D4#1175721011yescharacteristic, direct product factor
81C22×C4—4828no—
82C22×C4—2411no—
83C22×C4—2412no—
84C22×C4—2418no—
85C22×C4—2411no—
86C22×C4—2411no—
87C4⋊C4—2414no—
88C22⋊C4—2418no—
89C42—2414no—
90C23—4824no—
91C23—4824no—
92C23—2412no—
93C23—2414no—
94C23—2414no—
95C23—2411no—
96C23C2×#1175721012yescomplemented
97D4—2418no—
98D4—2414no—
99D4C2×#1175721014yesdirect product factor
100C2×C4—4824no—
101C2×C4—4822no—
102C2×C4—48216no—
103C2×C4—2412no—
104C2×C4—2414no—
105C2×C4—2412no—
106C2×C4—2414no—
107C2×C4—2411no—
108C2×C4—2412no—
109C2×C4—2412no—
110C2×C4C2×#1175721011yescharacteristic, complemented
111C22—24no—
112C22—4824no—
113C22—4824no—
114C22—4828no—
115C22—4824no—
116C22—2414no—
117C22—2412no—
118C22—2412no—
119C22—2411no—
120C22—2411no—
121C22C22×#1175721011yescharacteristic
122C22C22×#1175721014yescomplemented
123C4—4824no—
124C4—4822no—
125C4—2414no—
126C4—2412no—
127C4C22×#1175721012yescomplemented
128C2—24no—
129C2—4824no—
130C2—2411no—
131C2—2411no—
132C2—2412no—
133C2D4×#1175721012yesdirect product factor
134C2C23×#1175721011yescharacteristic
subgroupG/H
centerC22C22×#11757210
commutatorC2×C241C23×C4
FrattiniC2C23×#11757210
FittingC2×D4×C241C4
socleC22×C241C22×C4
prime radicalC2×D4×D241C2
cosocleC2×D241C24
hypercenterC2×D4#11757210
hypocenterC241C2×C4×D4
LCSG ▷ C2×C241 ▷ C241
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C23×#11757210C2semidirect
2. D4×#11757210C2direct (N × H)
3. C2×C4⋊#11757210C2semidirect
4. C2×C22⋊#11757210C2semidirect
5. C2×C4×#11757210C2semidirect
11. C22×#11757210C22semidirect
12. C4⋊#11757210C22semidirect
13. C22⋊#11757210C22semidirect
14. C4×#11757210C22semidirect
16. C2×D964C4semidirect
17. C2×C22⋊D241C4semidirect
18. D4×D241C4semidirect
21. C2×D4×C241C4semidirect
25. C2×#11757210D4direct (N × H)
31. C22×D241C2×C4semidirect
32. C22×D241C2×C4semidirect
35. D964C2×C4semidirect
36. C22⋊D241C2×C4semidirect
37. C2×Dic241C2×C4semidirect
38. C4×D241C2×C4semidirect
40. C23×C241C2×C4semidirect
41. D4×C241C2×C4semidirect
42. C2×C4×C241C2×C4semidirect
45. C22×C241C22×C4semidirect
51. C2×D241C22×C4semidirect
57. #11757210C2×D4direct (N × H)
58. C4×C241C22×C4semidirect
59. Dic241C22×C4semidirect
61. C2×C241C4×D4semidirect
64. D241C4×D4semidirect
67. C241C2×C4×D4semidirect
80. C2×D4#11757210direct (N × H)
96. C23C2×#11757210semidirect
99. D4C2×#11757210direct (N × H)
110. C2×C4C2×#11757210semidirect
122. C22C22×#11757210semidirect
127. C4C22×#11757210semidirect
133. C2D4×#11757210direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#117572102
D4×#117572102
C2×C4⋊#117572102
C2×C22⋊#117572102
C2×C4×#117572102
C2×D4×D2412
C2×C4×D4241