Finite Group Databasebeta

← #28416994

Subgroups — #28416994

Order 14912.

346 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 111 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
1C22×C22⋊D233C211yesmaximal, characteristic, complemented
2#21418518C212yesmaximal, complemented
3#21418516C212yesmaximal, complemented
4#21418507C214yesmaximal, complemented
5#21418504C212yesmaximal, complemented
6#21418503C212yesmaximal, complemented
7C2×C22⋊Dic233C211yesmaximal, characteristic, complemented
8C22×D4×C233C211yesmaximal, characteristic, complemented
9C23×D233C2211yescharacteristic, complemented
10C2×C22⋊D233—24no—
11C2×C22⋊D233C2212yes—
12C2×C22⋊D233C2212yescomplemented
13C2×C22⋊D233C2214yescomplemented
14C22×Dic233C2211yescharacteristic, complemented
15C22×Dic233C2212yescomplemented
16#16836573C2214yescomplemented
17C22⋊Dic233C2212yescomplemented
18C22⋊Dic233C2214yescomplemented
19C22⋊Dic233C2212yescomplemented
20#16836571C2214yescomplemented
21C24×C233C2211yescharacteristic
22C24×C233C2211yescharacteristic, complemented
23C2×D4×C233—24no—
24C2×D4×C233C2212yes—
25C2×D4×C233C2212yescomplemented
26C22×C4×C233C2211yescharacteristic, complemented
27C22×D233—22no—
28C22×D233—42no—
29C22×D233C2312yes—
30C22⋊D233—24no—
31C22⋊D233—24no—
32C22⋊D233—44no—
33C2×Dic233—22no—
34C2×Dic233—42no—
35C2×Dic233C2314yes—
36C2×Dic233C2312yes—
37C23×C233—22no—
38C23×C233—42no—
39C23×C233—42no—
40C23×C233D414yescomplemented
41C23×C233C2312yes—
42C23×C233C2311yescharacteristic
43C23×C233C2311yescharacteristic
44C23×C233C2311yescharacteristic
45D4×C233—24no—
46D4×C233—24no—
47C2×C4×C233—22no—
48C2×C4×C233C2312yes—
49C22×C233—22no—
50C22×C233—22no—
51C22×C233—22no—
52C22×C233—22no—
53C22×C233—24no—
54C22×C233—44no—
55C22×C233—44no—
56C22×C233C2×D412yes—
57C22×C233C2×D412yes—
58C22×C233C2×D412yes—
59C22×C233C2411yescharacteristic
60C2×D233—22no—
61C2×D233—24no—
62C2×D233—44no—
63C4×C233—22no—
64Dic233—22no—
65Dic233—24no—
66C2×C233—22no—
67C2×C233—22no—
68C2×C233—22no—
69C2×C233—42no—
70C2×C233C22×D411yescharacteristic
71C2×C233H(5,𝔽2)12yes—
72D233—42no—
73C233#53411yescharacteristic, complemented, Sylow
74#534—2331nomaximal, Sylow
75C22×D4—2331no—
76C22×D4D23311yescharacteristic, complemented
77#123—2332no—
78#123—2332no—
79#121—2334no—
80#120—2332no—
81#120—2332no—
82C2×C22⋊C4—2331no—
83C24—2331no—
84C24C2×D23311yescharacteristic
85C24C2×D23311yescharacteristic, complemented
86C2×D4—24no—
87C2×D4—4664no—
88C2×D4—2334no—
89C2×D4—2332no—
90C2×D4—2332no—
91C2×D4C2×D23312yes—
92C2×D4C2×D23312yescomplemented
93C22×C4—2332no—
94C22×C4—2331no—
95C22×C4C2×D23311yescharacteristic, complemented
96C4⋊C4—2334no—
97C22⋊C4—2332no—
98C22⋊C4—2332no—
99C22⋊C4—2334no—
100C22⋊C4—2334no—
101C23—22no—
102C23—42no—
103C23—42no—
104C23—9322no—
105C23—4662no—
106C23—2332no—
107C23C22⋊D23314yescomplemented
108C23C22×D23312yes—
109C23C22×D23311yescharacteristic
110C23C22×D23311yescharacteristic
111C23C22×D23311yescharacteristic
112D4—24no—
113D4—24no—
114D4—9324no—
115D4—4664no—
116D4—4664no—
117C2×C4—22no—
118C2×C4—9322no—
119C2×C4—4662no—
120C2×C4—2334no—
121C2×C4—2332no—
122C2×C4C22×D23312yes—
123C22—22no—
124C22—22no—
125C22—22no—
126C22—22no—
127C22—24no—
128C22—44no—
129C22—44no—
130C22—9324no—
131C22—4662no—
132C22—4664no—
133C22C2×C22⋊D23312yes—
134C22C2×C22⋊D23312yes—
135C22C2×C22⋊D23312yes—
136C22C23×D23311yescharacteristic
137C4—22no—
138C4—4662no—
139C4—4664no—
140C2—22no—
141C2—22no—
142C2—22no—
143C2—42no—
144C2—9322no—
145C2C22×C22⋊D23311yescharacteristic
146C2#2141858012yes—
subgroupG/H
center, FrattiniC22C23×D233
commutator, socle, cosocleC22×C233C24
FittingC22×D4×C233C2
hypercenterC22×D4D233
hypocenterC233#534
LCSG ▷ C22×C233 ▷ C233
UCSC1 ◁ C22 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. C22×C22⋊D233C2semidirect
2. #21418518C2semidirect
3. #21418516C2semidirect
4. #21418507C2semidirect
5. #21418504C2semidirect
6. #21418503C2semidirect
7. C2×C22⋊Dic233C2semidirect
8. C22×D4×C233C2semidirect
9. C23×D233C22semidirect
12. C2×C22⋊D233C22semidirect
13. C2×C22⋊D233C22semidirect
14. C22×Dic233C22semidirect
15. C22×Dic233C22semidirect
16. #16836573C22semidirect
17. C22⋊Dic233C22semidirect
18. C22⋊Dic233C22semidirect
19. C22⋊Dic233C22semidirect
20. #16836571C22semidirect
22. C24×C233C22semidirect
25. C2×D4×C233C22semidirect
26. C22×C4×C233C22semidirect
40. C23×C233D4semidirect
73. C233#534semidirect
76. C22×D4D233semidirect
85. C24C2×D233semidirect
92. C2×D4C2×D233semidirect
95. C22×C4C2×D233semidirect
107. C23C22⋊D233semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊D2332
#214185182
#214185162
#214185072
#214185042
#214185032
C2×C22⋊Dic2332
C22×D4×C2332
#534233