Finite Group Databasebeta

← #30682677

Subgroups — #30682677

Order 16832.

252 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 91 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#22554466C211yesmaximal, characteristic, complemented
2#22554463C212yesmaximal, complemented
3#22554452C212yesmaximal, complemented
4#22554451C211yesmaximal, characteristic, complemented
5#22554443C212yesmaximal, complemented
6#22554442C212yesmaximal, complemented
7#22554441C212yesmaximal, complemented
8#22554439C212yesmaximal, complemented
9#120×C263C211yesmaximal, characteristic, complemented
10C2×D1052C2211yescharacteristic, complemented
11C2×C22⋊D263C2212yes—
12C2×C22⋊D263C2212yes—
13C2×C22⋊D263C2212yes—
14C2×C22⋊D263C2212yescomplemented
15C22×Dic263C2211yescharacteristic, complemented
16C2×C4×D263C2212yescomplemented
17C4⋊Dic263C2212yescomplemented
18Dic263⋊C4C2212yescomplemented
19Dic263⋊C4C2212yescomplemented
20C22⋊Dic263C2212yes—
21C22⋊Dic263C2212yes—
22C22⋊Dic263C2212yes—
23C22⋊Dic263C2211yescharacteristic, complemented
24(C2×D263)⋊C4C2212yescomplemented
25C4×Dic263C2211yescharacteristic, complemented
26C24×C263C2211yescharacteristic
27C2×D4×C263C2212yes—
28C2×D4×C263C2211yescharacteristic
29C22⋊C4×C263C2212yes—
30C22⋊C4×C263C2211yescharacteristic
31C22×D263C2312yes—
32D1052—41no—
33C22⋊D263—42no—
34C22⋊D263—42no—
35C22⋊D263—42no—
36C22⋊D263—42no—
37C2×Dic263—41no—
38C2×Dic263C2312yes—
39C2×Dic263C2312yes—
40C2×Dic263C2312yes—
41C4×D263—42no—
42C23×C263—42no—
43C23×C263—41no—
44C23×C263C2312yes—
45C23×C263C2311yescharacteristic
46C23×C263C2311yescharacteristic
47D4×C263—42no—
48D4×C263—41no—
49C2×C4×C263C2311yescharacteristic
50C2×C4×C263C2312yes—
51C22×C263—22no—
52C22×C263—21no—
53C22×C263—21no—
54C22×C263—22no—
55C22×C263—22no—
56C22×C263—22no—
57C22×C263—22no—
58C22×C263—44no—
59C22×C263C2411yescharacteristic
60C2×D263—22no—
61C2×D263—22no—
62C2×D263—22no—
63C4×C263—22no—
64C4×C263—21no—
65Dic263—22no—
66Dic263—22no—
67Dic263—22no—
68C2×C263—41no—
69C2×C263—42no—
70C2×C263—41no—
71C2×C263H(5,𝔽2)12yes—
72C2×C263H(5,𝔽2)11yescharacteristic
73D263—42no—
74C263#56011yescharacteristic, complemented, Sylow
75#560—2631nomaximal, Sylow
76#127—2631no—
77#126—2632no—
78#123—2631no—
79#123—2632no—
80#121—2632no—
81#121—2632no—
82#121—2632no—
83#120—2632no—
84#120D26311yescharacteristic, complemented
85C24C2×D26311yescharacteristic
86C2×D4—2632no—
87C2×D4—2632no—
88C2×D4—2632no—
89C2×D4—2632no—
90C2×D4—2631no—
91C2×D4C2×D26312yes—
92C2×D4C2×D26311yescharacteristic
93C22×C4—2631no—
94C22×C4—2632no—
95C4⋊C4—2632no—
96C4⋊C4—2632no—
97C4⋊C4—2632no—
98C22⋊C4—2632no—
99C22⋊C4—2632no—
100C22⋊C4—2632no—
101C22⋊C4—2631no—
102C22⋊C4—2632no—
103C22⋊C4C2×D26312yes—
104C22⋊C4C2×D26311yescharacteristic
105C42—2631no—
106C23—42no—
107C23—41no—
108C23—2632no—
109C23C22×D26312yes—
110C23C22×D26311yescharacteristic
111C23C22×D26311yescharacteristic
112D4—42no—
113D4—41no—
114D4—10521no—
115D4—10522no—
116D4—10522no—
117D4—10522no—
118D4—10522no—
119C2×C4—2632no—
120C2×C4—2632no—
121C2×C4—2632no—
122C2×C4—10522no—
123C2×C4—10521no—
124C2×C4C22×D26311yescharacteristic
125C2×C4C22×D26312yes—
126C22—21no—
127C22—22no—
128C22—22no—
129C22—21no—
130C22—22no—
131C22—22no—
132C22—22no—
133C22—44no—
134C22—5262no—
135C22—5262no—
136C22—5262no—
137C22C23×D26311yescharacteristic
138C4—22no—
139C4—21no—
140C4—5262no—
141C4—5262no—
142C4—5262no—
143C2—42no—
144C2—41no—
145C2—41no—
146C2—10522no—
147C2#2255451512yes—
148C2#2255451511yescharacteristic
subgroupG/H
center, FrattiniC22C23×D263
commutator, socle, cosocleC22×C263C24
Fitting#120×C263C2
hypercenter#120D263
hypocenterC263#560
LCSG ▷ C22×C263 ▷ C263
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. #22554466C2semidirect
2. #22554463C2semidirect
3. #22554452C2semidirect
4. #22554451C2semidirect
5. #22554443C2semidirect
6. #22554442C2semidirect
7. #22554441C2semidirect
8. #22554439C2semidirect
9. #120×C263C2semidirect
10. C2×D1052C22semidirect
14. C2×C22⋊D263C22semidirect
15. C22×Dic263C22semidirect
16. C2×C4×D263C22semidirect
17. C4⋊Dic263C22semidirect
18. Dic263⋊C4C22semidirect
19. Dic263⋊C4C22semidirect
23. C22⋊Dic263C22semidirect
24. (C2×D263)⋊C4C22semidirect
25. C4×Dic263C22semidirect
74. C263#560semidirect
84. #120D263semidirect

Maximal subgroups

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

subgroupindex
#225544662
#225544632
#225544522
#225544512
#225544432
#225544422
#225544412
#225544392
#120×C2632
#560263