Finite Group Databasebeta

← #30682451

Subgroups — #30682451

Order 16832.

330 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 109 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×C4×D263C211yesmaximal, characteristic, complemented
3#22554455C212yesmaximal, complemented
4#22554446C211yesmaximal, characteristic, complemented
5#22554445C211yesmaximal, characteristic, complemented
6#22554444C211yesmaximal, characteristic, complemented
7#22554440C212yesmaximal, complemented
8#22554432C212yesmaximal, complemented
9(C22⋊D263)⋊C4C212yesmaximal, complemented
10#22554428C211yesmaximal, characteristic, complemented
11C22⋊Q8×C263C211yesmaximal, characteristic, complemented
12C23×D263C2211yescharacteristic
13#16862377—24no—
14C2×D1052C2212yes—
15C2×D1052C2212yes—
16C2×D1052C2211yescharacteristic, complemented
17C2×C22⋊D263C2212yes—
18C22×Dic263C2211yescharacteristic
19C2×C4×D263—24no—
20C2×C4×D263C2211yescharacteristic
21C2×C4×D263C2211yescharacteristic
22C2×C4×D263C2211yescharacteristic
23C2×C4×D263C2211yescharacteristic
24C2×C4×D263C2212yescomplemented
25C4⋊Dic263C2211yescharacteristic, complemented
26Dic263⋊C4C2212yescomplemented
27(C2×D263)⋊C4C2212yes—
28(C2×D263)⋊C4C2212yes—
29(C2×D263)⋊C4C2212yescomplemented
30(C2×D263)⋊C4C2212yescomplemented
31C4×Dic263C2212yescomplemented
32C2×Q8×C263C2211yescharacteristic, complemented
33C22×C4×C263C2211yescharacteristic
34C4⋊C4×C263C2211yescharacteristic, complemented
35C4⋊C4×C263C2212yescomplemented
36C22⋊C4×C263C2212yes—
37C22×D263—22no—
38C22×D263—22no—
39C22×D263—22no—
40C22×D263C2312yes—
41C22×D263C2311yescharacteristic
42C22×D263C2311yescharacteristic
43D1052—24no—
44D1052—24no—
45D1052—22no—
46C22⋊D263—24no—
47C2×Dic263—22no—
48C2×Dic263C2312yes—
49C2×Dic263C2311yescharacteristic
50C2×Dic263C2311yescharacteristic
51C4×D263—22no—
52C4×D263—22no—
53C4×D263—22no—
54C4×D263—24no—
55C4×D263D414yescomplemented
56C23×C263C2311yescharacteristic
57Q8×C263—22no—
58C2×C4×C263—22no—
59C2×C4×C263C2312yes—
60C2×C4×C263C2312yes—
61C2×C4×C263C2311yescharacteristic
62C2×C4×C263C2311yescharacteristic
63C22×C263—21no—
64C22×C263—21no—
65C22×C263#4112yescomplemented
66C22×C263C2411yescharacteristic
67C2×D263—28no—
68C2×D263—21no—
69C2×D263—21no—
70C2×D263—21no—
71C2×D263—21no—
72C2×D263—22no—
73C2×D263—22no—
74C2×D263—22no—
75C2×D263C2×D412yes—
76C2×D263#4112yescomplemented
77C4×C263—21no—
78C4×C263—22no—
79C4×C263—22no—
80C4×C263C2×D412yes—
81Dic263—21no—
82Dic263—22no—
83Dic263C2×D412yes—
84C2×C263—22no—
85C2×C263C22×D411yescharacteristic
86C2×C263C2×#4111yescharacteristic
87C2×C263C2×#4111yescharacteristic
88D263—22no—
89D263—22no—
90D263—42no—
91C263#52511yescharacteristic, complemented, Sylow
92#525—2631nomaximal, Sylow
93C2×#41—2631no—
94C23×C4—2631no—
95#123—2632no—
96C22⋊Q8—2631no—
97C22⋊Q8D26311yescharacteristic, complemented
98#121—2631no—
99#121—2631no—
100#120—2632no—
101C4×D4—2632no—
102C4×D4—2632no—
103#117—2631no—
104C24—2631no—
105#41—5264no—
106C2×Q8C2×D26311yescharacteristic, complemented
107C2×D4—2632no—
108C2×D4—2632no—
109C2×D4—2632no—
110C2×D4—2631no—
111C22×C4—2631no—
112C22×C4—2631no—
113C22×C4—2631no—
114C22×C4—2632no—
115C22×C4—2631no—
116C22×C4—2631no—
117C22×C4—5264no—
118C22×C4C2×D26311yescharacteristic
119C4⋊C4—2631no—
120C4⋊C4—2632no—
121C4⋊C4C2×D26311yescharacteristic, complemented
122C4⋊C4C2×D26312yescomplemented
123C22⋊C4—2632no—
124C22⋊C4—2632no—
125C22⋊C4—2632no—
126C22⋊C4—2632no—
127C22⋊C4C2×D26312yes—
128C42—2632no—
129C23—2631no—
130C23—2631no—
131C23—2632no—
132C23—5262no—
133C23—5262no—
134C23—5262no—
135C23C22×D26311yescharacteristic
136Q8—22no—
137D4—5264no—
138D4—5264no—
139D4—5264no—
140D4—5262no—
141C2×C4—22no—
142C2×C4—2631no—
143C2×C4—2634no—
144C2×C4—2632no—
145C2×C4—2631no—
146C2×C4—5262no—
147C2×C4—5264no—
148C2×C4—5262no—
149C2×C4—5262no—
150C2×C4—5262no—
151C2×C4C22×D26312yes—
152C2×C4C22×D26311yescharacteristic
153C2×C4C22×D26312yes—
154C2×C4C22×D26311yescharacteristic
155C22—21no—
156C22—21no—
157C22—2632no—
158C22—2632no—
159C22—5268no—
160C22—5261no—
161C22—5261no—
162C22—5261no—
163C22—5261no—
164C22—5262no—
165C22—5262no—
166C22—5262no—
167C22#1686237712yescomplemented
168C22C23×D26311yescharacteristic
169C4—21no—
170C4—22no—
171C4—22no—
172C4—2632no—
173C4—5261no—
174C4—5262no—
175C4D4×D26312yes—
176C2—22no—
177C2—5262no—
178C2—5262no—
179C2—10522no—
180C2C2×D4×D26311yescharacteristic
181C2#41×D26311yescharacteristic
182C2C2×#1686237711yescharacteristic
subgroupG/H
center, FrattiniC22C23×D263
commutator, socle, cosocleC22×C263C24
FittingC22⋊Q8×C263C2
hypercenterC22⋊Q8D263
hypocenterC263#525
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×C4×D263C2semidirect
3. #22554455C2semidirect
4. #22554446C2semidirect
5. #22554445C2semidirect
6. #22554444C2semidirect
7. #22554440C2semidirect
8. #22554432C2semidirect
9. (C22⋊D263)⋊C4C2semidirect
10. #22554428C2semidirect
11. C22⋊Q8×C263C2semidirect
16. C2×D1052C22semidirect
24. C2×C4×D263C22semidirect
25. C4⋊Dic263C22semidirect
26. Dic263⋊C4C22semidirect
29. (C2×D263)⋊C4C22semidirect
30. (C2×D263)⋊C4C22semidirect
31. C4×Dic263C22semidirect
32. C2×Q8×C263C22semidirect
34. C4⋊C4×C263C22semidirect
35. C4⋊C4×C263C22semidirect
55. C4×D263D4semidirect
65. C22×C263#41semidirect
76. C2×D263#41semidirect
91. C263#525semidirect
97. C22⋊Q8D263semidirect
106. C2×Q8C2×D263semidirect
121. C4⋊C4C2×D263semidirect
122. C4⋊C4C2×D263semidirect
167. 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×C4×D2632
#225544552
#225544462
#225544452
#225544442
#225544402
#225544322
(C22⋊D263)⋊C42
#225544282
C22⋊Q8×C2632
#525263