Finite Group Databasebeta

← #30682556

Subgroups — #30682556

Order 16832.

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

normal subgroups only

This group has 216 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#22554469C211yesmaximal, characteristic, complemented
2#22554465C211yesmaximal, characteristic, complemented
3#22554463C211yesmaximal, characteristic, complemented
4#22554457C211yesmaximal, characteristic, complemented
5#22554456C211yesmaximal, characteristic, complemented
6#22554456C211yesmaximal, characteristic, complemented
7#22554452C211yesmaximal, characteristic, complemented
8#22554449C211yesmaximal, characteristic, complemented
9#22554448C211yesmaximal, characteristic, complemented
10#22554447C211yesmaximal, characteristic, complemented
11#22554442C211yesmaximal, characteristic
12#22554435C211yesmaximal, characteristic, complemented
13C4×C22⋊D263C211yesmaximal, characteristic, complemented
14#22554426C211yesmaximal, characteristic, complemented
15#123×C263C211yesmaximal, characteristic, complemented
16C2×Dic526C2211yescharacteristic
17C2×Dic526C2211yescharacteristic
18C2×Dic526C2211yescharacteristic
19C2×C22⋊D263C2211yescharacteristic
20C2×C22⋊D263C2211yescharacteristic
21C22×Dic263C2211yescharacteristic
22C2×C4×D263C2211yescharacteristic
23C4⋊Dic263C2211yescharacteristic
24C4⋊Dic263C2211yescharacteristic
25Dic263⋊C4C2211yescharacteristic
26Dic263⋊C4C2211yescharacteristic
27Dic263⋊C4C2211yescharacteristic
28Dic263⋊C4C2211yescharacteristic
29Dic263⋊C4C2211yescharacteristic
30Dic263⋊C4C2211yescharacteristic
31C22⋊Dic263C2211yescharacteristic
32C22⋊Dic263C2211yescharacteristic
33C22⋊Dic263C2211yescharacteristic
34C22⋊Dic263C2211yescharacteristic
35C22⋊Dic263C2211yescharacteristic
36(C2×D263)⋊C4C2211yescharacteristic
37(C2×D263)⋊C4C2211yescharacteristic
38(C2×D263)⋊C4C2211yescharacteristic
39(C2×D263)⋊C4C2211yescharacteristic
40C4×Dic263C2211yescharacteristic
41C4×Dic263C2211yescharacteristic
42C4×Dic263C2211yescharacteristic
43C4×Dic263C2211yescharacteristic
44C2×D4×C263C2211yescharacteristic
45C22×C4×C263C2211yescharacteristic
46C4⋊C4×C263C2211yescharacteristic
47C4⋊C4×C263C2211yescharacteristic
48C22⋊C4×C263C2211yescharacteristic
49C22⋊C4×C263C2211yescharacteristic
50C22⋊C4×C263C2211yescharacteristic
51C22×D263C2311yescharacteristic
52Dic526—22no—
53Dic526—41no—
54Dic526—41no—
55C22⋊D263—22no—
56C22⋊D263—41no—
57C2×Dic263—22no—
58C2×Dic263C2311yescharacteristic
59C2×Dic263C2311yescharacteristic
60C2×Dic263C2311yescharacteristic
61C2×Dic263C2311yescharacteristic
62C2×Dic263C2311yescharacteristic
63C2×Dic263C2311yescharacteristic
64C2×Dic263C2311yescharacteristic
65C4×D263—41no—
66C23×C263C2311yescharacteristic
67C23×C263C2311yescharacteristic
68D4×C263—41no—
69C2×C4×C263—41no—
70C2×C4×C263C2311yescharacteristic
71C2×C4×C263C2311yescharacteristic
72C2×C4×C263C2311yescharacteristic
73C2×C4×C263C2311yescharacteristic
74C2×C4×C263C2311yescharacteristic
75C22×C263—21no—
76C22×C263—21no—
77C22×C263—21no—
78C22×C263—21no—
79C22×C263—21no—
80C22×C263—21no—
81C22×C263C2411yescharacteristic
82C2×D263—21no—
83C2×D263—21no—
84C2×D263—21no—
85C4×C263—21no—
86C4×C263—21no—
87C4×C263—21no—
88C4×C263—21no—
89C4×C263—21no—
90Dic263—21no—
91Dic263—21no—
92Dic263—21no—
93Dic263—21no—
94Dic263—21no—
95Dic263—21no—
96Dic263#4112yes—
97C2×C263—41no—
98C2×C263—41no—
99C2×C263C2×#4111yescharacteristic
100C2×C263H(5,𝔽2)11yescharacteristic
101C2×C2632−1+411yescharacteristic
102D263—41no—
103C263#54211yescharacteristic, complemented, Sylow
104#542—2631nomaximal, Sylow
105C4⋊Q8—2631no—
106#126—2631no—
107#126—2631no—
108#124—2631no—
109#124—2631no—
110#124—2631no—
111#123—2631no—
112#123D26311yescharacteristic, complemented
113C22⋊Q8—2631no—
114C22⋊Q8—2631no—
115C22⋊Q8—2631no—
116#121—2631no—
117C4×Q8—2631no—
118C4×D4—2631no—
119#117—2631no—
120C2×Q8—2631no—
121C2×Q8—2631no—
122C2×Q8—2631no—
123C2×D4—2631no—
124C2×D4—2631no—
125C2×D4C2×D26311yescharacteristic
126C22×C4—2631no—
127C22×C4—2631no—
128C22×C4C2×D26311yescharacteristic
129C4⋊C4—2631no—
130C4⋊C4—2631no—
131C4⋊C4—2631no—
132C4⋊C4—2631no—
133C4⋊C4—2631no—
134C4⋊C4—2631no—
135C4⋊C4—2631no—
136C4⋊C4—2631no—
137C4⋊C4C2×D26311yescharacteristic
138C4⋊C4C2×D26311yescharacteristic
139C22⋊C4—2631no—
140C22⋊C4—2631no—
141C22⋊C4—2631no—
142C22⋊C4—2631no—
143C22⋊C4—2631no—
144C22⋊C4—2631no—
145C22⋊C4—2631no—
146C22⋊C4—2631no—
147C22⋊C4—2631no—
148C22⋊C4C2×D26311yescharacteristic
149C22⋊C4C2×D26311yescharacteristic
150C22⋊C4C2×D26311yescharacteristic
151C42—2631no—
152C42—2631no—
153C42—2631no—
154C42—2631no—
155C23—2631no—
156C23C22×D26311yescharacteristic
157C23C22×D26311yescharacteristic
158Q8—5262no—
159Q8—10521no—
160Q8—10521no—
161D4—41no—
162D4—5262no—
163D4—10521no—
164C2×C4—41no—
165C2×C4—2631no—
166C2×C4—2631no—
167C2×C4—2631no—
168C2×C4—2631no—
169C2×C4—2631no—
170C2×C4—2631no—
171C2×C4—2631no—
172C2×C4—5262no—
173C2×C4—10521no—
174C2×C4C22×D26311yescharacteristic
175C2×C4C22×D26311yescharacteristic
176C2×C4C22×D26311yescharacteristic
177C2×C4C22×D26311yescharacteristic
178C2×C4C22×D26311yescharacteristic
179C22—21no—
180C22—21no—
181C22—21no—
182C22—21no—
183C22—21no—
184C22—21no—
185C22—5261no—
186C22—5261no—
187C22—5261no—
188C22C23×D26311yescharacteristic
189C4—21no—
190C4—21no—
191C4—21no—
192C4—21no—
193C4—21no—
194C4—2632no—
195C4—5261no—
196C4—5261no—
197C4—5261no—
198C4—5261no—
199C4—5261no—
200C4—5261no—
201C2—41no—
202C2—41no—
203C2—10521no—
204C2#41×D26311yescharacteristic
205C2#2255451511yescharacteristic
206C2#2255451711yescharacteristic
subgroupG/H
center, FrattiniC22C23×D263
commutator, socle, cosocleC22×C263C24
Fitting#123×C263C2
prime radical#22554442C2
hypercenter#123D263
hypocenterC263#542
LCSG ▷ C22×C263 ▷ C263
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
1. #22554469C2semidirect
2. #22554465C2semidirect
3. #22554463C2semidirect
4. #22554457C2semidirect
5. #22554456C2semidirect
6. #22554456C2semidirect
7. #22554452C2semidirect
8. #22554449C2semidirect
9. #22554448C2semidirect
10. #22554447C2semidirect
12. #22554435C2semidirect
13. C4×C22⋊D263C2semidirect
14. #22554426C2semidirect
15. #123×C263C2semidirect
103. C263#542semidirect
112. #123D263semidirect

Maximal subgroups

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

subgroupindex
#225544692
#225544652
#225544632
#225544572
#225544562
#225544522
#225544492
#225544482
#225544472
#225544422
#225544352
C4×C22⋊D2632
#225544262
#123×C2632
#542263