Finite Group Databasebeta

← #30657296

Subgroups — #30657296

Order 16448.

346 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 111 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
1C22×C22⋊D257C211yesmaximal, characteristic, complemented
2#22551575C212yesmaximal, complemented
3#22551573C212yesmaximal, complemented
4#22551564C214yesmaximal, complemented
5#22551561C212yesmaximal, complemented
6#22551560C212yesmaximal, complemented
7C2×C22⋊Dic257C211yesmaximal, characteristic, complemented
8C22×D4×C257C211yesmaximal, characteristic, complemented
9C23×D257C2211yescharacteristic, complemented
10C2×C22⋊D257—24no—
11C2×C22⋊D257C2212yes—
12C2×C22⋊D257C2212yescomplemented
13C2×C22⋊D257C2214yescomplemented
14C22×Dic257C2211yescharacteristic, complemented
15C22×Dic257C2212yescomplemented
16#16861597C2214yescomplemented
17C22⋊Dic257C2212yescomplemented
18C22⋊Dic257C2214yescomplemented
19C22⋊Dic257C2212yescomplemented
20#16861595C2214yescomplemented
21C24×C257C2211yescharacteristic
22C24×C257C2211yescharacteristic, complemented
23C2×D4×C257—24no—
24C2×D4×C257C2212yes—
25C2×D4×C257C2212yescomplemented
26C22×C4×C257C2211yescharacteristic, complemented
27C22×D257—22no—
28C22×D257—42no—
29C22×D257C2312yes—
30C22⋊D257—24no—
31C22⋊D257—24no—
32C22⋊D257—44no—
33C2×Dic257—22no—
34C2×Dic257—42no—
35C2×Dic257C2314yes—
36C2×Dic257C2312yes—
37C23×C257—22no—
38C23×C257—42no—
39C23×C257—42no—
40C23×C257D414yescomplemented
41C23×C257C2312yes—
42C23×C257C2311yescharacteristic
43C23×C257C2311yescharacteristic
44C23×C257C2311yescharacteristic
45D4×C257—24no—
46D4×C257—24no—
47C2×C4×C257—22no—
48C2×C4×C257C2312yes—
49C22×C257—22no—
50C22×C257—22no—
51C22×C257—22no—
52C22×C257—22no—
53C22×C257—24no—
54C22×C257—44no—
55C22×C257—44no—
56C22×C257C2×D412yes—
57C22×C257C2×D412yes—
58C22×C257C2×D412yes—
59C22×C257C2411yescharacteristic
60C2×D257—22no—
61C2×D257—24no—
62C2×D257—44no—
63C4×C257—22no—
64Dic257—22no—
65Dic257—24no—
66C2×C257—22no—
67C2×C257—22no—
68C2×C257—22no—
69C2×C257—42no—
70C2×C257C22×D411yescharacteristic
71C2×C257H(5,𝔽2)12yes—
72D257—42no—
73C257#53411yescharacteristic, complemented, Sylow
74#534—2571nomaximal, Sylow
75C22×D4—2571no—
76C22×D4D25711yescharacteristic, complemented
77#123—2572no—
78#123—2572no—
79#121—2574no—
80#120—2572no—
81#120—2572no—
82C2×C22⋊C4—2571no—
83C24—2571no—
84C24C2×D25711yescharacteristic
85C24C2×D25711yescharacteristic, complemented
86C2×D4—2572no—
87C2×D4—2572no—
88C2×D4—2574no—
89C2×D4—24no—
90C2×D4—5144no—
91C2×D4C2×D25712yes—
92C2×D4C2×D25712yescomplemented
93C22×C4—2571no—
94C22×C4—2572no—
95C22×C4C2×D25711yescharacteristic, complemented
96C4⋊C4—2574no—
97C22⋊C4—2572no—
98C22⋊C4—2572no—
99C22⋊C4—2574no—
100C22⋊C4—2574no—
101C23—2572no—
102C23—22no—
103C23—5142no—
104C23—42no—
105C23—42no—
106C23—10282no—
107C23C22⋊D25714yescomplemented
108C23C22×D25712yes—
109C23C22×D25711yescharacteristic
110C23C22×D25711yescharacteristic
111C23C22×D25711yescharacteristic
112D4—24no—
113D4—24no—
114D4—5144no—
115D4—5144no—
116D4—10284no—
117C2×C4—2574no—
118C2×C4—2572no—
119C2×C4—22no—
120C2×C4—5142no—
121C2×C4—10282no—
122C2×C4C22×D25712yes—
123C22—22no—
124C22—22no—
125C22—22no—
126C22—22no—
127C22—24no—
128C22—5142no—
129C22—5144no—
130C22—44no—
131C22—44no—
132C22—10284no—
133C22C2×C22⋊D25712yes—
134C22C2×C22⋊D25712yes—
135C22C2×C22⋊D25712yes—
136C22C23×D25711yescharacteristic
137C4—22no—
138C4—5142no—
139C4—5144no—
140C2—22no—
141C2—22no—
142C2—22no—
143C2—42no—
144C2—10282no—
145C2C22×C22⋊D25711yescharacteristic
146C2#2255163712yes—
subgroupG/H
center, FrattiniC22C23×D257
commutator, socle, cosocleC22×C257C24
FittingC22×D4×C257C2
hypercenterC22×D4D257
hypocenterC257#534
LCSG ▷ C22×C257 ▷ C257
UCSC1 ◁ C22 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. C22×C22⋊D257C2semidirect
2. #22551575C2semidirect
3. #22551573C2semidirect
4. #22551564C2semidirect
5. #22551561C2semidirect
6. #22551560C2semidirect
7. C2×C22⋊Dic257C2semidirect
8. C22×D4×C257C2semidirect
9. C23×D257C22semidirect
12. C2×C22⋊D257C22semidirect
13. C2×C22⋊D257C22semidirect
14. C22×Dic257C22semidirect
15. C22×Dic257C22semidirect
16. #16861597C22semidirect
17. C22⋊Dic257C22semidirect
18. C22⋊Dic257C22semidirect
19. C22⋊Dic257C22semidirect
20. #16861595C22semidirect
22. C24×C257C22semidirect
25. C2×D4×C257C22semidirect
26. C22×C4×C257C22semidirect
40. C23×C257D4semidirect
73. C257#534semidirect
76. C22×D4D257semidirect
85. C24C2×D257semidirect
92. C2×D4C2×D257semidirect
95. C22×C4C2×D257semidirect
107. C23C22⋊D257semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊D2572
#225515752
#225515732
#225515642
#225515612
#225515602
C2×C22⋊Dic2572
C22×D4×C2572
#534257