Finite Group Databasebeta

← #30682585

Subgroups — #30682585

Order 16832.

334 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 105 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×#16862376C211yesmaximal, characteristic, complemented
2C2×D4×D263C211yesmaximal, characteristic, complemented
3#22554457C211yesmaximal, characteristic, complemented
4#22554452C212yesmaximal, complemented
5#22554450C211yesmaximal, characteristic, complemented
6#22554441C212yesmaximal, complemented
7#22554438C212yesmaximal, complemented
8#22554432C211yesmaximal, characteristic, complemented
9C4×C22⋊D263C211yesmaximal, characteristic, complemented
10C22⋊C4×D263C212yesmaximal, complemented
11#121×C263C211yesmaximal, characteristic, complemented
12C23×D263C2212yes—
13#16862376—24no—
14C2×Dic526C2211yescharacteristic, complemented
15C2×D1052C2211yescharacteristic
16C2×C22⋊D263C2211yescharacteristic
17C2×C22⋊D263C2212yes—
18C2×C22⋊D263C2212yescomplemented
19D4×D263—24no—
20C2×C4×D263C2211yescharacteristic
21C2×C4×D263C2212yescomplemented
22C2×C4×D263C2212yescomplemented
23C4⋊Dic263C2212yescomplemented
24Dic263⋊C4C2211yescharacteristic, complemented
25C22⋊Dic263C2212yescomplemented
26C22⋊Dic263C2211yescharacteristic, complemented
27C22⋊Dic263C2212yescomplemented
28(C2×D263)⋊C4C2211yescharacteristic
29(C2×D263)⋊C4C2212yes—
30(C2×D263)⋊C4C2212yescomplemented
31C4×Dic263C2211yescharacteristic, complemented
32C2×D4×C263C2211yescharacteristic
33C2×D4×C263C2212yescomplemented
34C22×C4×C263C2211yescharacteristic, complemented
35C4⋊C4×C263C2211yescharacteristic, complemented
36C22⋊C4×C263C2212yescomplemented
37C22×D263—24no—
38C22×D263—24no—
39C22×D263—42no—
40C22×D263C2312yes—
41C22×D263C2311yescharacteristic
42Dic526—22no—
43D1052D414yescomplemented
44C22⋊D263—24no—
45C22⋊D263—24no—
46C22⋊D263—22no—
47C2×Dic263C2311yescharacteristic
48C2×Dic263C2311yescharacteristic
49C2×Dic263C2312yes—
50C2×Dic263C2311yescharacteristic
51C4×D263—24no—
52C4×D263—22no—
53C4×D263—42no—
54C23×C263C2312yes—
55C23×C263C2311yescharacteristic
56D4×C263—22no—
57D4×C263—42no—
58C2×C4×C263—22no—
59C2×C4×C263C2311yescharacteristic
60C2×C4×C263C2312yes—
61C2×C4×C263C2311yescharacteristic
62C22×C263—21no—
63C22×C263—22no—
64C22×C263—21no—
65C22×C263—22no—
66C22×C263—21no—
67C22×C263—22no—
68C22×C263C2411yescharacteristic
69C2×D263—21no—
70C2×D263—22no—
71C2×D263—22no—
72C2×D263—21no—
73C2×D263—48no—
74C2×D263C2×D414yes—
75C2×D263#4112yescomplemented
76C4×C263—21no—
77C4×C263—22no—
78C4×C263C2×D412yes—
79Dic263—21no—
80Dic263—22no—
81Dic263—21no—
82Dic263—21no—
83C2×C263—42no—
84C2×C263—41no—
85C2×C263C22×D411yescharacteristic
86C2×C263C2×#4111yescharacteristic
87C2×C263H(5,𝔽2)11yescharacteristic
88D263—22no—
89D263—24no—
90C263#54611yescharacteristic, complemented, Sylow
91#546—2631nomaximal, Sylow
92C2×#41—2631no—
93C22×D4—2631no—
94#124—2631no—
95#123—2632no—
96C22⋊Q8—2631no—
97#121—2632no—
98#121D26311yescharacteristic, complemented
99#120—2632no—
100C4×D4—2631no—
101C4×D4—2631no—
102C2×C22⋊C4—2632no—
103C24—2632no—
104#41—5264no—
105C2×Q8—2631no—
106C2×D4—2631no—
107C2×D4—2632no—
108C2×D4—2632no—
109C2×D4—2631no—
110C2×D4—5264no—
111C2×D4C2×D26311yescharacteristic
112C2×D4C2×D26312yescomplemented
113C22×C4—2632no—
114C22×C4—2631no—
115C22×C4—2632no—
116C22×C4C2×D26311yescharacteristic, complemented
117C4⋊C4—2631no—
118C4⋊C4—2632no—
119C4⋊C4C2×D26311yescharacteristic, complemented
120C22⋊C4—2632no—
121C22⋊C4—2631no—
122C22⋊C4—2632no—
123C22⋊C4—2631no—
124C22⋊C4—2632no—
125C22⋊C4—2632no—
126C22⋊C4C2×D26312yescomplemented
127C42—2631no—
128C23—2631no—
129C23—2632no—
130C23—5264no—
131C23—5264no—
132C23—10522no—
133C23C22×D26312yes—
134C23C22×D26311yescharacteristic
135Q8—5262no—
136D4—22no—
137D4—42no—
138D4—2634no—
139D4—5264no—
140D4—5264no—
141D4—5262no—
142C2×C4—22no—
143C2×C4—2631no—
144C2×C4—2631no—
145C2×C4—2632no—
146C2×C4—2631no—
147C2×C4—5264no—
148C2×C4—5262no—
149C2×C4—10522no—
150C2×C4C22×D26312yes—
151C2×C4C22×D26311yescharacteristic
152C2×C4C22×D26311yescharacteristic
153C22—21no—
154C22—22no—
155C22—21no—
156C22—22no—
157C22—21no—
158C22—22no—
159C22—2634no—
160C22—2632no—
161C22—5261no—
162C22—5262no—
163C22—5262no—
164C22—5261no—
165C22—10528no—
166C22C23×D26311yescharacteristic
167C4—21no—
168C4—22no—
169C4—5261no—
170C4—5262no—
171C4—5261no—
172C4—5261no—
173C4D4×D26312yes—
174C2—42no—
175C2—41no—
176C2—5262no—
177C2—5264no—
178C2C2×D4×D26311yescharacteristic
179C2#41×D26311yescharacteristic
180C2#2255451511yescharacteristic
subgroupG/H
center, FrattiniC22C23×D263
commutator, socle, cosocleC22×C263C24
Fitting#121×C263C2
hypercenter#121D263
hypocenterC263#546
LCSG ▷ C22×C263 ▷ C263
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×#16862376C2semidirect
2. C2×D4×D263C2semidirect
3. #22554457C2semidirect
4. #22554452C2semidirect
5. #22554450C2semidirect
6. #22554441C2semidirect
7. #22554438C2semidirect
8. #22554432C2semidirect
9. C4×C22⋊D263C2semidirect
10. C22⋊C4×D263C2semidirect
11. #121×C263C2semidirect
14. C2×Dic526C22semidirect
18. C2×C22⋊D263C22semidirect
21. C2×C4×D263C22semidirect
22. C2×C4×D263C22semidirect
23. C4⋊Dic263C22semidirect
24. Dic263⋊C4C22semidirect
25. C22⋊Dic263C22semidirect
26. C22⋊Dic263C22semidirect
27. C22⋊Dic263C22semidirect
30. (C2×D263)⋊C4C22semidirect
31. C4×Dic263C22semidirect
33. C2×D4×C263C22semidirect
34. C22×C4×C263C22semidirect
35. C4⋊C4×C263C22semidirect
36. C22⋊C4×C263C22semidirect
43. D1052D4semidirect
75. C2×D263#41semidirect
90. C263#546semidirect
98. #121D263semidirect
112. C2×D4C2×D263semidirect
116. C22×C4C2×D263semidirect
119. C4⋊C4C2×D263semidirect
126. C22⋊C4C2×D263semidirect

Maximal subgroups

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

subgroupindex
C2×#168623762
C2×D4×D2632
#225544572
#225544522
#225544502
#225544412
#225544382
#225544322
C4×C22⋊D2632
C22⋊C4×D2632
#121×C2632
#546263