Finite Group Databasebeta

← #30682503

Subgroups — #30682503

Order 16832.

334 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 103 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×#16862375C211yesmaximal, characteristic, complemented
2C22×C22⋊D263C211yesmaximal, characteristic, complemented
3#22554466C212yesmaximal, complemented
4#22554456C212yesmaximal, complemented
5#22554443C212yesmaximal, complemented
6#22554442C212yesmaximal, complemented
7#22554440C211yesmaximal, characteristic, complemented
8#22554439C211yesmaximal, characteristic, complemented
9#22554438C211yesmaximal, characteristic, complemented
10#22554426C211yesmaximal, characteristic, complemented
11#120×C263C211yesmaximal, characteristic, complemented
12C23×D263C2211yescharacteristic
13#16862375—24no—
14C2×Dic526C2211yescharacteristic, complemented
15C2×D1052C2212yescomplemented
16C2×C22⋊D263—24no—
17C2×C22⋊D263C2211yescharacteristic
18C2×C22⋊D263C2212yes—
19C2×C22⋊D263C2212yes—
20C2×C22⋊D263C2211yescharacteristic, complemented
21C2×C22⋊D263C2212yescomplemented
22C2×C22⋊D263C2212yescomplemented
23C22×Dic263C2211yescharacteristic
24C22×Dic263C2211yescharacteristic, complemented
25C2×C4×D263C2211yescharacteristic, complemented
26Dic263⋊C4C2212yescomplemented
27C22⋊Dic263C2211yescharacteristic
28C22⋊Dic263C2212yescomplemented
29(C2×D263)⋊C4C2212yescomplemented
30(C2×D263)⋊C4C2212yescomplemented
31C4×Dic263C2212yescomplemented
32C24×C263C2211yescharacteristic
33C2×D4×C263C2211yescharacteristic
34C2×D4×C263C2212yescomplemented
35C22⋊C4×C263C2211yescharacteristic
36C22⋊C4×C263C2212yescomplemented
37C22×D263—22no—
38C22×D263—42no—
39C22×D263C2312yes—
40C22×D263C2311yescharacteristic
41Dic526—22no—
42D1052—42no—
43C22⋊D263—22no—
44C22⋊D263—24no—
45C22⋊D263—24no—
46C22⋊D263—22no—
47C22⋊D263—42no—
48C22⋊D263—42no—
49C2×Dic263—22no—
50C2×Dic263D414yescomplemented
51C2×Dic263C2312yes—
52C2×Dic263C2311yescharacteristic
53C2×Dic263C2311yescharacteristic
54C2×Dic263C2311yescharacteristic
55C4×D263—22no—
56C23×C263—22no—
57C23×C263—42no—
58C23×C263C2312yes—
59C23×C263C2311yescharacteristic
60C23×C263C2311yescharacteristic
61D4×C263—22no—
62D4×C263—42no—
63C2×C4×C263C2312yes—
64C2×C4×C263C2311yescharacteristic
65C22×C263—22no—
66C22×C263—21no—
67C22×C263—22no—
68C22×C263—22no—
69C22×C263—22no—
70C22×C263—22no—
71C22×C263—44no—
72C22×C263C2×D412yes—
73C22×C263C2411yescharacteristic
74C2×D263—21no—
75C2×D263—22no—
76C2×D263—22no—
77C2×D263—22no—
78C2×D263—22no—
79C2×D263—44no—
80C4×C263—21no—
81C4×C263—22no—
82Dic263—21no—
83Dic263—22no—
84Dic263C2×D412yes—
85Dic263C2×D412yes—
86C2×C263—22no—
87C2×C263—41no—
88C2×C263—42no—
89C2×C263C22×D411yescharacteristic
90C2×C263H(5,𝔽2)12yes—
91D263—41no—
92D263—42no—
93C263#53511yescharacteristic, complemented, Sylow
94#535—2631nomaximal, Sylow
95C2×#41—2631no—
96C22×D4—2631no—
97#127—2632no—
98#124—2632no—
99#121—2632no—
100#121—2632no—
101#120—2631no—
102#120—2631no—
103#120—2631no—
104#120D26311yescharacteristic, complemented
105#117—2631no—
106C24—2631no—
107C24C2×D26311yescharacteristic
108#41—5264no—
109C2×Q8—2631no—
110C2×D4—2631no—
111C2×D4—2632no—
112C2×D4—2632no—
113C2×D4—2631no—
114C2×D4—2632no—
115C2×D4—2632no—
116C2×D4—2632no—
117C2×D4—5264no—
118C2×D4C2×D26311yescharacteristic
119C2×D4C2×D26312yescomplemented
120C22×C4—2631no—
121C22×C4—2631no—
122C22×C4—2631no—
123C4⋊C4—2632no—
124C22⋊C4—2631no—
125C22⋊C4—2632no—
126C22⋊C4—2632no—
127C22⋊C4—2632no—
128C22⋊C4C2×D26311yescharacteristic
129C22⋊C4C2×D26312yescomplemented
130C42—2632no—
131C23—22no—
132C23—42no—
133C23—2632no—
134C23—2631no—
135C23—5262no—
136C23—10522no—
137C23C22×D26312yes—
138C23C22×D26311yescharacteristic
139C23C22×D26311yescharacteristic
140Q8—5262no—
141D4—22no—
142D4—42no—
143D4—5264no—
144D4—5264no—
145D4—5262no—
146D4—5262no—
147D4—10522no—
148D4—10522no—
149D4—10522no—
150C2×C4—2634no—
151C2×C4—2631no—
152C2×C4—2631no—
153C2×C4—2632no—
154C2×C4—2631no—
155C2×C4—5262no—
156C2×C4—5262no—
157C2×C4C22×D26312yes—
158C2×C4C22×D26311yescharacteristic
159C22—22no—
160C22—21no—
161C22—22no—
162C22—22no—
163C22—22no—
164C22—22no—
165C22—44no—
166C22—5261no—
167C22—5262no—
168C22—5262no—
169C22—5262no—
170C22—5262no—
171C22—10524no—
172C22D4×D26312yes—
173C22C23×D26311yescharacteristic
174C4—21no—
175C4—22no—
176C4—2632no—
177C4—2632no—
178C4—5261no—
179C4—5262no—
180C2—22no—
181C2—42no—
182C2—41no—
183C2—10522no—
184C2—10521no—
185C2C2×D4×D26311yescharacteristic
186C2#2255451512yes—
subgroupG/H
center, FrattiniC22C23×D263
commutator, socle, cosocleC22×C263C24
Fitting#120×C263C2
hypercenter#120D263
hypocenterC263#535
LCSG ▷ C22×C263 ▷ C263
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C2×#16862375C2semidirect
2. C22×C22⋊D263C2semidirect
3. #22554466C2semidirect
4. #22554456C2semidirect
5. #22554443C2semidirect
6. #22554442C2semidirect
7. #22554440C2semidirect
8. #22554439C2semidirect
9. #22554438C2semidirect
10. #22554426C2semidirect
11. #120×C263C2semidirect
14. C2×Dic526C22semidirect
15. C2×D1052C22semidirect
20. C2×C22⋊D263C22semidirect
21. C2×C22⋊D263C22semidirect
22. C2×C22⋊D263C22semidirect
24. C22×Dic263C22semidirect
25. C2×C4×D263C22semidirect
26. Dic263⋊C4C22semidirect
28. C22⋊Dic263C22semidirect
29. (C2×D263)⋊C4C22semidirect
30. (C2×D263)⋊C4C22semidirect
31. C4×Dic263C22semidirect
34. C2×D4×C263C22semidirect
36. C22⋊C4×C263C22semidirect
50. C2×Dic263D4semidirect
93. C263#535semidirect
104. #120D263semidirect
119. C2×D4C2×D263semidirect
129. C22⋊C4C2×D263semidirect

Maximal subgroups

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

subgroupindex
C2×#168623752
C22×C22⋊D2632
#225544662
#225544562
#225544432
#225544422
#225544402
#225544392
#225544382
#225544262
#120×C2632
#535263