Finite Group Databasebeta

← #30682579

Subgroups — #30682579

Order 16832.

334 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 107 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×#16862377C211yesmaximal, characteristic, complemented
2C22×D1052C211yesmaximal, characteristic, complemented
3#22554459C211yesmaximal, characteristic, complemented
4#22554455C212yesmaximal, complemented
5#22554445C211yesmaximal, characteristic, complemented
6#22554445C212yesmaximal, complemented
7#22554440C212yesmaximal, complemented
8#22554432C211yesmaximal, characteristic, complemented
9C4×C22⋊D263C211yesmaximal, characteristic, complemented
10C22⋊C4×D263C212yesmaximal, complemented
11C22⋊Q8×C263C211yesmaximal, characteristic, complemented
12C23×D263C2212yes—
13#16862377—24no—
14C2×D1052—24no—
15C2×D1052C2212yes—
16C2×D1052C2211yescharacteristic
17C2×D1052C2211yescharacteristic
18C2×D1052C2212yescomplemented
19C2×D1052C2212yescomplemented
20C2×C22⋊D263C2211yescharacteristic
21C2×C4×D263C2212yescomplemented
22C2×C4×D263C2211yescharacteristic, complemented
23C2×C4×D263C2212yescomplemented
24Dic263⋊C4C2211yescharacteristic, complemented
25C22⋊Dic263C2211yescharacteristic
26(C2×D263)⋊C4C2212yescomplemented
27(C2×D263)⋊C4C2212yescomplemented
28(C2×D263)⋊C4C2211yescharacteristic, complemented
29(C2×D263)⋊C4C2212yescomplemented
30(C2×D263)⋊C4C2212yescomplemented
31C4×Dic263C2211yescharacteristic, complemented
32C2×Q8×C263C2211yescharacteristic, complemented
33C22×C4×C263C2211yescharacteristic
34C4⋊C4×C263C2211yescharacteristic, complemented
35C4⋊C4×C263C2212yescomplemented
36C22⋊C4×C263C2212yes—
37C22×D263—24no—
38C22×D263—24no—
39C22×D263—42no—
40C22×D263C2312yes—
41C22×D263C2311yescharacteristic
42C22×D263C2312yes—
43D1052—24no—
44D1052—24no—
45D1052—22no—
46D1052—42no—
47D1052D414yescomplemented
48C22⋊D263—22no—
49C2×Dic263C2311yescharacteristic
50C2×Dic263C2311yescharacteristic
51C2×Dic263C2311yescharacteristic
52C4×D263—24no—
53C4×D263—22no—
54C4×D263—42no—
55C23×C263C2311yescharacteristic
56Q8×C263—22no—
57C2×C4×C263—22no—
58C2×C4×C263C2312yes—
59C2×C4×C263C2312yes—
60C2×C4×C263C2311yescharacteristic
61C2×C4×C263C2311yescharacteristic
62C22×C263—21no—
63C22×C263—21no—
64C22×C263#4112yescomplemented
65C22×C263C2411yescharacteristic
66C2×D263—22no—
67C2×D263—22no—
68C2×D263—21no—
69C2×D263—22no—
70C2×D263—21no—
71C2×D263—22no—
72C2×D263—21no—
73C2×D263—22no—
74C2×D263—48no—
75C2×D263C2×D414yes—
76C4×C263—21no—
77C4×C263—22no—
78C4×C263—22no—
79C4×C263C2×D412yes—
80Dic263—21no—
81Dic263—21no—
82Dic263—21no—
83C2×C263—22no—
84C2×C263C22×D411yescharacteristic
85C2×C263C2×#4111yescharacteristic
86C2×C263H(5,𝔽2)11yescharacteristic
87D263—24no—
88D263—41no—
89D263—42no—
90C263#54611yescharacteristic, complemented, Sylow
91#546—2631nomaximal, Sylow
92C2×#41—2631no—
93C22×D4—2631no—
94#124—2631no—
95#123—2632no—
96C22⋊Q8D26311yescharacteristic, complemented
97#121—2631no—
98#121—2632no—
99#120—2632no—
100C4×D4—2631no—
101C4×D4—2631no—
102C2×C22⋊C4—2632no—
103C24—2632no—
104#41—5264no—
105C2×Q8C2×D26311yescharacteristic, complemented
106C2×D4—2631no—
107C2×D4—2631no—
108C2×D4—2632no—
109C2×D4—2632no—
110C2×D4—2631no—
111C2×D4—2632no—
112C2×D4—5264no—
113C22×C4—2632no—
114C22×C4—2631no—
115C22×C4—2632no—
116C22×C4C2×D26311yescharacteristic
117C4⋊C4—2631no—
118C4⋊C4C2×D26311yescharacteristic, complemented
119C4⋊C4C2×D26312yescomplemented
120C22⋊C4—2631no—
121C22⋊C4—2632no—
122C22⋊C4—2632no—
123C22⋊C4—2632no—
124C22⋊C4—2631no—
125C22⋊C4—2632no—
126C22⋊C4C2×D26312yes—
127C42—2631no—
128C23—2632no—
129C23—2632no—
130C23—2631no—
131C23—5264no—
132C23—5264no—
133C23—10522no—
134C23C22×D26311yescharacteristic
135Q8—22no—
136D4—2634no—
137D4—5264no—
138D4—5264no—
139D4—5262no—
140D4—5262no—
141D4—10522no—
142C2×C4—22no—
143C2×C4—2631no—
144C2×C4—2631no—
145C2×C4—2631no—
146C2×C4—5264no—
147C2×C4—5262no—
148C2×C4—10522no—
149C2×C4C22×D26312yes—
150C2×C4C22×D26312yes—
151C2×C4C22×D26311yescharacteristic
152C2×C4C22×D26311yescharacteristic
153C22—21no—
154C22—21no—
155C22—2634no—
156C22—5262no—
157C22—5262no—
158C22—5261no—
159C22—5262no—
160C22—5261no—
161C22—5262no—
162C22—5261no—
163C22—5262no—
164C22—10528no—
165C22#1686237712yescomplemented
166C22C23×D26311yescharacteristic
167C4—21no—
168C4—22no—
169C4—22no—
170C4—5261no—
171C4—5261no—
172C4—5261no—
173C4D4×D26312yes—
174C2—22no—
175C2—5264no—
176C2—10522no—
177C2—10521no—
178C2C2×D4×D26311yescharacteristic
179C2C2×#1686237711yescharacteristic
180C2#2255451611yescharacteristic
subgroupG/H
center, FrattiniC22C23×D263
commutator, socle, cosocleC22×C263C24
FittingC22⋊Q8×C263C2
hypercenterC22⋊Q8D263
hypocenterC263#546
LCSG ▷ C22×C263 ▷ C263
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#16862377C2semidirect
2. C22×D1052C2semidirect
3. #22554459C2semidirect
4. #22554455C2semidirect
5. #22554445C2semidirect
6. #22554445C2semidirect
7. #22554440C2semidirect
8. #22554432C2semidirect
9. C4×C22⋊D263C2semidirect
10. C22⋊C4×D263C2semidirect
11. C22⋊Q8×C263C2semidirect
18. C2×D1052C22semidirect
19. C2×D1052C22semidirect
21. C2×C4×D263C22semidirect
22. C2×C4×D263C22semidirect
23. C2×C4×D263C22semidirect
24. Dic263⋊C4C22semidirect
26. (C2×D263)⋊C4C22semidirect
27. (C2×D263)⋊C4C22semidirect
28. (C2×D263)⋊C4C22semidirect
29. (C2×D263)⋊C4C22semidirect
30. (C2×D263)⋊C4C22semidirect
31. C4×Dic263C22semidirect
32. C2×Q8×C263C22semidirect
34. C4⋊C4×C263C22semidirect
35. C4⋊C4×C263C22semidirect
47. D1052D4semidirect
64. C22×C263#41semidirect
90. C263#546semidirect
96. C22⋊Q8D263semidirect
105. C2×Q8C2×D263semidirect
118. C4⋊C4C2×D263semidirect
119. C4⋊C4C2×D263semidirect
165. C22#16862377semidirect

Maximal subgroups

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

subgroupindex
C2×#168623772
C22×D10522
#225544592
#225544552
#225544452
#225544402
#225544322
C4×C22⋊D2632
C22⋊C4×D2632
C22⋊Q8×C2632
#546263