Finite Group Databasebeta

← #41003399

Subgroups — #41003399

Order 26944.

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⋊D421C211yesmaximal, characteristic, complemented
2#27212349C212yesmaximal, complemented
3#27212347C212yesmaximal, complemented
4#27212338C214yesmaximal, complemented
5#27212335C212yesmaximal, complemented
6#27212334C212yesmaximal, complemented
7C2×C22⋊Dic421C211yesmaximal, characteristic, complemented
8C22×D4×C421C211yesmaximal, characteristic, complemented
9C23×D421C2211yescharacteristic, complemented
10C2×C22⋊D421—24no—
11C2×C22⋊D421C2212yes—
12C2×C22⋊D421C2212yescomplemented
13C2×C22⋊D421C2214yescomplemented
14C22×Dic421C2211yescharacteristic, complemented
15C22×Dic421C2212yescomplemented
16#20269750C2214yescomplemented
17C22⋊Dic421C2212yescomplemented
18C22⋊Dic421C2212yescomplemented
19C22⋊Dic421C2214yescomplemented
20#20269748C2214yescomplemented
21C24×C421C2211yescharacteristic
22C24×C421C2211yescharacteristic, complemented
23C2×D4×C421—24no—
24C2×D4×C421C2212yes—
25C2×D4×C421C2212yescomplemented
26C22×C4×C421C2211yescharacteristic, complemented
27C22×D421—22no—
28C22×D421—42no—
29C22×D421C2312yes—
30C22⋊D421—24no—
31C22⋊D421—24no—
32C22⋊D421—44no—
33C2×Dic421—22no—
34C2×Dic421—42no—
35C2×Dic421C2314yes—
36C2×Dic421C2312yes—
37C23×C421—22no—
38C23×C421—42no—
39C23×C421—42no—
40C23×C421D414yescomplemented
41C23×C421C2312yes—
42C23×C421C2311yescharacteristic
43C23×C421C2311yescharacteristic
44C23×C421C2311yescharacteristic
45D4×C421—24no—
46D4×C421—24no—
47C2×C4×C421—22no—
48C2×C4×C421C2312yes—
49C22×C421—22no—
50C22×C421—22no—
51C22×C421—22no—
52C22×C421—22no—
53C22×C421—24no—
54C22×C421—44no—
55C22×C421—44no—
56C22×C421C2×D412yes—
57C22×C421C2×D412yes—
58C22×C421C2×D412yes—
59C22×C421C2411yescharacteristic
60C2×D421—22no—
61C2×D421—24no—
62C2×D421—44no—
63C4×C421—22no—
64Dic421—22no—
65Dic421—24no—
66C2×C421—22no—
67C2×C421—22no—
68C2×C421—22no—
69C2×C421—42no—
70C2×C421C22×D411yescharacteristic
71C2×C421H(5,𝔽2)12yes—
72D421—42no—
73C421#53411yescharacteristic, complemented, Sylow
74#534—4211nomaximal, Sylow
75C22×D4—4211no—
76C22×D4D42111yescharacteristic, complemented
77#123—4212no—
78#123—4212no—
79#121—4214no—
80#120—4212no—
81#120—4212no—
82C2×C22⋊C4—4211no—
83C24—4211no—
84C24C2×D42111yescharacteristic
85C24C2×D42111yescharacteristic, complemented
86C2×D4—24no—
87C2×D4—8424no—
88C2×D4—4212no—
89C2×D4—4212no—
90C2×D4—4214no—
91C2×D4C2×D42112yes—
92C2×D4C2×D42112yescomplemented
93C22×C4—4211no—
94C22×C4—4212no—
95C22×C4C2×D42111yescharacteristic, complemented
96C4⋊C4—4214no—
97C22⋊C4—4212no—
98C22⋊C4—4212no—
99C22⋊C4—4214no—
100C22⋊C4—4214no—
101C23—22no—
102C23—42no—
103C23—42no—
104C23—8422no—
105C23—16842no—
106C23—4212no—
107C23C22⋊D42114yescomplemented
108C23C22×D42112yes—
109C23C22×D42111yescharacteristic
110C23C22×D42111yescharacteristic
111C23C22×D42111yescharacteristic
112D4—24no—
113D4—24no—
114D4—8424no—
115D4—8424no—
116D4—16844no—
117C2×C4—22no—
118C2×C4—8422no—
119C2×C4—16842no—
120C2×C4—4214no—
121C2×C4—4212no—
122C2×C4C22×D42112yes—
123C22—22no—
124C22—22no—
125C22—22no—
126C22—22no—
127C22—24no—
128C22—44no—
129C22—44no—
130C22—8422no—
131C22—8424no—
132C22—16844no—
133C22C2×C22⋊D42112yes—
134C22C2×C22⋊D42112yes—
135C22C2×C22⋊D42112yes—
136C22C23×D42111yescharacteristic
137C4—22no—
138C4—8422no—
139C4—8424no—
140C2—22no—
141C2—22no—
142C2—22no—
143C2—42no—
144C2—16842no—
145C2C22×C22⋊D42111yescharacteristic
146C2#2721241112yes—
subgroupG/H
center, FrattiniC22C23×D421
commutator, socle, cosocleC22×C421C24
FittingC22×D4×C421C2
hypercenterC22×D4D421
hypocenterC421#534
LCSG ▷ C22×C421 ▷ C421
UCSC1 ◁ C22 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. C22×C22⋊D421C2semidirect
2. #27212349C2semidirect
3. #27212347C2semidirect
4. #27212338C2semidirect
5. #27212335C2semidirect
6. #27212334C2semidirect
7. C2×C22⋊Dic421C2semidirect
8. C22×D4×C421C2semidirect
9. C23×D421C22semidirect
12. C2×C22⋊D421C22semidirect
13. C2×C22⋊D421C22semidirect
14. C22×Dic421C22semidirect
15. C22×Dic421C22semidirect
16. #20269750C22semidirect
17. C22⋊Dic421C22semidirect
18. C22⋊Dic421C22semidirect
19. C22⋊Dic421C22semidirect
20. #20269748C22semidirect
22. C24×C421C22semidirect
25. C2×D4×C421C22semidirect
26. C22×C4×C421C22semidirect
40. C23×C421D4semidirect
73. C421#534semidirect
76. C22×D4D421semidirect
85. C24C2×D421semidirect
92. C2×D4C2×D421semidirect
95. C22×C4C2×D421semidirect
107. C23C22⋊D421semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊D4212
#272123492
#272123472
#272123382
#272123352
#272123342
C2×C22⋊Dic4212
C22×D4×C4212
#534421