Finite Group Databasebeta

← #30682524

Subgroups — #30682524

Order 16832.

250 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 93 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
1#22554465C211yesmaximal, characteristic, complemented
2#22554463C211yesmaximal, characteristic, complemented
3#22554456C211yesmaximal, characteristic, complemented
4#22554456C211yesmaximal, characteristic, complemented
5#22554455C211yesmaximal, characteristic, complemented
6#22554450C211yesmaximal, characteristic, complemented
7#22554444C211yesmaximal, characteristic, complemented
8#22554443C211yesmaximal, characteristic, complemented
9#22554441C211yesmaximal, characteristic, complemented
10#22554440C211yesmaximal, characteristic, complemented
11#22554438C211yesmaximal, characteristic, complemented
12#22554432C211yesmaximal, characteristic, complemented
13C4×C22⋊D263C211yesmaximal, characteristic, complemented
14C22⋊C4×D263C211yesmaximal, characteristic, complemented
15#123×C263C211yesmaximal, characteristic, complemented
16C23×D263C2211yescharacteristic
17C2×Dic526C2211yescharacteristic, complemented
18C2×D1052C2211yescharacteristic
19C2×D1052C2211yescharacteristic
20C2×C22⋊D263C2211yescharacteristic
21C2×C22⋊D263C2211yescharacteristic
22C2×C22⋊D263C2211yescharacteristic
23C2×C22⋊D263C2211yescharacteristic, complemented
24C2×C4×D263C2211yescharacteristic
25C2×C4×D263C2211yescharacteristic
26C2×C4×D263C2211yescharacteristic, complemented
27C4⋊Dic263C2211yescharacteristic, complemented
28C4⋊Dic263C2211yescharacteristic, complemented
29Dic263⋊C4C2211yescharacteristic, complemented
30Dic263⋊C4C2211yescharacteristic, complemented
31C22⋊Dic263C2211yescharacteristic
32C22⋊Dic263C2211yescharacteristic
33C22⋊Dic263C2211yescharacteristic, complemented
34(C2×D263)⋊C4C2211yescharacteristic
35(C2×D263)⋊C4C2211yescharacteristic
36(C2×D263)⋊C4C2211yescharacteristic
37(C2×D263)⋊C4C2211yescharacteristic
38(C2×D263)⋊C4C2211yescharacteristic
39(C2×D263)⋊C4C2211yescharacteristic
40(C2×D263)⋊C4C2211yescharacteristic, complemented
41(C2×D263)⋊C4C2211yescharacteristic, complemented
42C4×Dic263C2211yescharacteristic, complemented
43C4×Dic263C2211yescharacteristic, complemented
44C2×D4×C263C2211yescharacteristic
45C22×C4×C263C2211yescharacteristic, complemented
46C4⋊C4×C263C2211yescharacteristic, complemented
47C4⋊C4×C263C2211yescharacteristic, complemented
48C22⋊C4×C263C2211yescharacteristic
49C22⋊C4×C263C2211yescharacteristic
50C22⋊C4×C263C2211yescharacteristic, complemented
51C22×D263—22no—
52C22×D263—41no—
53C22×D263—41no—
54C22×D263C2311yescharacteristic
55C22×D263C2311yescharacteristic
56C22×D263C2311yescharacteristic
57Dic526—41no—
58D1052—22no—
59D1052—41no—
60C22⋊D263—22no—
61C22⋊D263—41no—
62C22⋊D263—41no—
63C22⋊D263—41no—
64C2×Dic263C2311yescharacteristic
65C2×Dic263C2311yescharacteristic
66C2×Dic263C2311yescharacteristic
67C2×Dic263C2311yescharacteristic
68C2×Dic263C2311yescharacteristic
69C4×D263—41no—
70C4×D263—41no—
71C4×D263—41no—
72C23×C263C2311yescharacteristic
73C23×C263C2311yescharacteristic
74D4×C263—41no—
75C2×C4×C263—41no—
76C2×C4×C263C2311yescharacteristic
77C2×C4×C263C2311yescharacteristic
78C2×C4×C263C2311yescharacteristic
79C2×C4×C263C2311yescharacteristic
80C2×C4×C263C2311yescharacteristic
81C22×C263—21no—
82C22×C263—21no—
83C22×C263—21no—
84C22×C263—21no—
85C22×C263—21no—
86C22×C263—21no—
87C22×C263C2411yescharacteristic
88C2×D263—21no—
89C2×D263—21no—
90C2×D263—21no—
91C2×D263—21no—
92C2×D263—21no—
93C2×D263—21no—
94C2×D263—21no—
95C2×D263—21no—
96C2×D263—44no—
97C2×D263#4112yes—
98C4×C263—21no—
99C4×C263—21no—
100C4×C263—21no—
101C4×C263—21no—
102C4×C263—21no—
103Dic263—21no—
104Dic263—21no—
105Dic263—21no—
106Dic263—21no—
107Dic263—21no—
108C2×C263—41no—
109C2×C263—41no—
110C2×C263C2×#4111yescharacteristic
111C2×C263H(5,𝔽2)11yescharacteristic
112C2×C263H(5,𝔽2)11yescharacteristic
113D263—22no—
114D263—41no—
115D263—41no—
116C263#53811yescharacteristic, complemented, Sylow
117#538—2631nomaximal, Sylow
118#126—2631no—
119#126—2631no—
120#124—2631no—
121#124—2631no—
122#123—2631no—
123#123D26311yescharacteristic, complemented
124C22⋊Q8—2631no—
125#121—2631no—
126#121—2631no—
127#121—2631no—
128#120—2631no—
129#120—2631no—
130C4×D4—2631no—
131C4×D4—2631no—
132C2×C22⋊C4—2631no—
133C24—2631no—
134C2×Q8—2631no—
135C2×D4—2631no—
136C2×D4—2631no—
137C2×D4—2631no—
138C2×D4—2631no—
139C2×D4—2631no—
140C2×D4—2631no—
141C2×D4C2×D26311yescharacteristic
142C22×C4—2631no—
143C22×C4—2631no—
144C22×C4—2631no—
145C22×C4C2×D26311yescharacteristic, complemented
146C4⋊C4—2631no—
147C4⋊C4—2631no—
148C4⋊C4—2631no—
149C4⋊C4—2631no—
150C4⋊C4C2×D26311yescharacteristic, complemented
151C4⋊C4C2×D26311yescharacteristic, complemented
152C22⋊C4—2631no—
153C22⋊C4—2631no—
154C22⋊C4—2631no—
155C22⋊C4—2631no—
156C22⋊C4—2631no—
157C22⋊C4—2631no—
158C22⋊C4—2631no—
159C22⋊C4—2631no—
160C22⋊C4—2631no—
161C22⋊C4—2631no—
162C22⋊C4—2631no—
163C22⋊C4C2×D26311yescharacteristic
164C22⋊C4C2×D26311yescharacteristic
165C22⋊C4C2×D26311yescharacteristic, complemented
166C42—2631no—
167C42—2631no—
168C23—2631no—
169C23—2631no—
170C23—2631no—
171C23—5262no—
172C23—10521no—
173C23—10521no—
174C23C22×D26311yescharacteristic
175C23C22×D26311yescharacteristic
176Q8—10521no—
177D4—41no—
178D4—5262no—
179D4—5262no—
180D4—10521no—
181D4—10521no—
182D4—10521no—
183D4—10521no—
184C2×C4—41no—
185C2×C4—2631no—
186C2×C4—2631no—
187C2×C4—2631no—
188C2×C4—2631no—
189C2×C4—2631no—
190C2×C4—10521no—
191C2×C4—10521no—
192C2×C4—10521no—
193C2×C4C22×D26311yescharacteristic
194C2×C4C22×D26311yescharacteristic
195C2×C4C22×D26311yescharacteristic
196C2×C4C22×D26311yescharacteristic
197C2×C4C22×D26311yescharacteristic
198C22—21no—
199C22—21no—
200C22—21no—
201C22—21no—
202C22—21no—
203C22—21no—
204C22—2632no—
205C22—5261no—
206C22—5261no—
207C22—5261no—
208C22—5261no—
209C22—5261no—
210C22—5261no—
211C22—5261no—
212C22—5261no—
213C22—10524no—
214C22C23×D26311yescharacteristic
215C4—21no—
216C4—21no—
217C4—21no—
218C4—21no—
219C4—21no—
220C4—5261no—
221C4—5261no—
222C4—5261no—
223C4—5261no—
224C4—5261no—
225C2—41no—
226C2—41no—
227C2—5262no—
228C2—10521no—
229C2—10521no—
230C2#41×D26311yescharacteristic
231C2#2255451511yescharacteristic
232C2#2255451611yescharacteristic
subgroupG/H
center, FrattiniC22C23×D263
commutator, socle, cosocleC22×C263C24
Fitting#123×C263C2
hypercenter#123D263
hypocenterC263#538
LCSG ▷ C22×C263 ▷ C263
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
1. #22554465C2semidirect
2. #22554463C2semidirect
3. #22554456C2semidirect
4. #22554456C2semidirect
5. #22554455C2semidirect
6. #22554450C2semidirect
7. #22554444C2semidirect
8. #22554443C2semidirect
9. #22554441C2semidirect
10. #22554440C2semidirect
11. #22554438C2semidirect
12. #22554432C2semidirect
13. C4×C22⋊D263C2semidirect
14. C22⋊C4×D263C2semidirect
15. #123×C263C2semidirect
17. C2×Dic526C22semidirect
23. C2×C22⋊D263C22semidirect
26. C2×C4×D263C22semidirect
27. C4⋊Dic263C22semidirect
28. C4⋊Dic263C22semidirect
29. Dic263⋊C4C22semidirect
30. Dic263⋊C4C22semidirect
33. C22⋊Dic263C22semidirect
40. (C2×D263)⋊C4C22semidirect
41. (C2×D263)⋊C4C22semidirect
42. C4×Dic263C22semidirect
43. C4×Dic263C22semidirect
45. C22×C4×C263C22semidirect
46. C4⋊C4×C263C22semidirect
47. C4⋊C4×C263C22semidirect
50. C22⋊C4×C263C22semidirect
116. C263#538semidirect
123. #123D263semidirect
145. C22×C4C2×D263semidirect
150. C4⋊C4C2×D263semidirect
151. C4⋊C4C2×D263semidirect
165. C22⋊C4C2×D263semidirect

Maximal subgroups

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

subgroupindex
#225544652
#225544632
#225544562
#225544552
#225544502
#225544442
#225544432
#225544412
#225544402
#225544382
#225544322
C4×C22⋊D2632
C22⋊C4×D2632
#123×C2632
#538263