Finite Group Databasebeta

← C2×#27165782 #38786944

Subgroups — C2×#27165782 #38786944

Order 25664.

898 conjugacy classes of subgroups; only the 447 normal subgroups are shown.

normal subgroups only

This group has 898 conjugacy classes of subgroups — too many to draw. Only its 447 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#27165782C2116yesmaximal, direct product factor
2C2×#20264366C212yesmaximal, complemented
3C2×#20264365C214yesmaximal, complemented
4C22×C22⋊D401C214yesmaximal, complemented
5C2×D4×D401C214yesmaximal, complemented
6C22×D4×C401C211yesmaximal, characteristic, complemented
7C23×D401C2214yescomplemented
8#20264366C22116yescomplemented
9#20264365C22132yescomplemented
10C2×Dic802C2212yescomplemented
11C2×D1604C2212yescomplemented
12C2×C22⋊D401C2218yescomplemented
13C2×C22⋊D401C2214yescomplemented
14C2×C22⋊D401C22132yescomplemented
15D4×D401C22132yescomplemented
16C22×Dic401C2214yescomplemented
17C2×C4×D401C2214yescomplemented
18C24×C401C2212yescomplemented
19C2×D4×C401C2214yescomplemented
20C2×D4×C401C2218yescomplemented
21C22×C4×C401C2211yescharacteristic, complemented
22C22×D401—48no—
23C22×D401C2314yes—
24C22×D401C23116yescomplemented
25Dic802C2318yescomplemented
26D1604C2318yescomplemented
27C22⋊D401C23132yescomplemented
28C22⋊D401C23116yescomplemented
29C2×Dic401C2314yes—
30C2×Dic401C23116yescomplemented
31C4×D401C23116yescomplemented
32C23×C401—44no—
33C23×C401C2311yescharacteristic
34C23×C401C2314yes—
35C23×C401C2318yescomplemented
36D4×C401C23116yescomplemented
37C2×C4×C401C2312yes—
38C2×C4×C401C2314yescomplemented
39C22×C401—22no—
40C22×C401—28no—
41C22×C401—48no—
42C22×C401C2412yes—
43C22×C401C2418yes—
44C22×C401C2411yescharacteristic
45C2×D401—28no—
46C2×D401—416no—
47C2×D401C2418yes—
48C4×C401C2414yes—
49Dic401C2418yes—
50C2×C401—28no—
51C2×C401—22no—
52C2×C401H(5,𝔽2)12yescomplemented
53C2×C401C2511yescharacteristic
54D401—28no—
55C401C2×H(5,𝔽2)11yescharacteristic, complemented, Sylow
56C2×H(5,𝔽2)—4011nomaximal, Sylow
57H(5,𝔽2)—40116no—
58C2×#41—4014no—
59C2×#41—4012no—
60C22×D4—4014no—
61C22×D4—4014no—
62C22×D4D40111yescharacteristic, complemented
63C24—4014no—
64C24C2×D40112yescomplemented
65#41—40132no—
66#41—40116no—
67C2×Q8—4012no—
68C2×D4—4018no—
69C2×D4—4014no—
70C2×D4—40132no—
71C2×D4—40132no—
72C2×D4—4012no—
73C2×D4C2×D40114yescomplemented
74C2×D4C2×D40118yescomplemented
75C22×C4—4014no—
76C22×C4—4014no—
77C22×C4C2×D40111yescharacteristic, complemented
78C23—44no—
79C23—16048no—
80C23—4014no—
81C23—40116no—
82C23C22×D40114yes—
83C23C22×D40111yescharacteristic
84C23C22×D40118yescomplemented
85Q8—4018no—
86D4—40132no—
87D4—40116no—
88D4—4018no—
89D4C22×D401116yescomplemented
90C2×C4—40116no—
91C2×C4—4014no—
92C2×C4—40116no—
93C2×C4C22×D40112yes—
94C2×C4C22×D40114yescomplemented
95C22—22no—
96C22—28no—
97C22—48no—
98C22—8028no—
99C22—160416no—
100C22—4018no—
101C22C23×D40118yes—
102C22C23×D40112yes—
103C22C23×D40111yescharacteristic
104C4—4018no—
105C4C23×D40114yes—
106C2—22no—
107C2—28no—
108C2—8028no—
109C2#2716578212yesdirect product factor
110C2C24×D40111yescharacteristic
subgroupG/H
centerC22C23×D401
commutator, cosocleC2×C401C25
FrattiniC2C24×D401
FittingC22×D4×C401C2
socleC22×C401C24
hypercenterC22×D4D401
hypocenterC401C2×H(5,𝔽2)
LCSG ▷ C2×C401 ▷ C401
UCSC1 ◁ C22 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. #27165782C2direct (N × H)
2. C2×#20264366C2semidirect
3. C2×#20264365C2semidirect
4. C22×C22⋊D401C2semidirect
5. C2×D4×D401C2semidirect
6. C22×D4×C401C2semidirect
7. C23×D401C22semidirect
8. #20264366C22semidirect
9. #20264365C22semidirect
10. C2×Dic802C22semidirect
11. C2×D1604C22semidirect
12. C2×C22⋊D401C22semidirect
13. C2×C22⋊D401C22semidirect
14. C2×C22⋊D401C22semidirect
15. D4×D401C22semidirect
16. C22×Dic401C22semidirect
17. C2×C4×D401C22semidirect
18. C24×C401C22semidirect
19. C2×D4×C401C22semidirect
20. C2×D4×C401C22semidirect
21. C22×C4×C401C22semidirect
24. C22×D401C23semidirect
25. Dic802C23semidirect
26. D1604C23semidirect
27. C22⋊D401C23semidirect
28. C22⋊D401C23semidirect
30. C2×Dic401C23semidirect
31. C4×D401C23semidirect
35. C23×C401C23semidirect
36. D4×C401C23semidirect
38. C2×C4×C401C23semidirect
52. C2×C401H(5,𝔽2)semidirect
55. C401C2×H(5,𝔽2)semidirect
62. C22×D4D401semidirect
64. C24C2×D401semidirect
73. C2×D4C2×D401semidirect
74. C2×D4C2×D401semidirect
77. C22×C4C2×D401semidirect
84. C23C22×D401semidirect
89. D4C22×D401semidirect
94. C2×C4C22×D401semidirect
109. C2#27165782direct (N × H)

Maximal subgroups

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

subgroupindex
#271657822
C2×#202643662
C2×#202643652
C22×C22⋊D4012
C2×D4×D4012
C22×D4×C4012
C2×H(5,𝔽2)401