Finite Group Databasebeta

← C22×#585 #63100

Subgroups — C22×#585 #63100

Order 256.

30183 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#585C2196yesmaximal, direct product factor
2C22×2−1+4C216yesmaximal, complemented
3C22×H(5,𝔽2)C2110yesmaximal, complemented
4C23×#41C2115yesmaximal, complemented
5#585C2211024yesdirect product factor
6C2×2−1+4C221288yescomplemented
7C2×H(5,𝔽2)C221480yescomplemented
8C22×#41C22120yescomplemented
9C22×#41C22160yescomplemented
10C22×#41C221720yescomplemented
11C23×Q8C22115yescomplemented
12C23×D4C22145yescomplemented
13C24×C4C22115yescomplemented
14C25C23115yes—
152−1+4C2311536yescomplemented
16H(5,𝔽2)C2312560yescomplemented
17C2×#41C231480yescomplemented
18C2×#41C2311440yescomplemented
19C2×#41C2313840yescomplemented
20C22×Q8C23120yes—
21C22×Q8C231360yescomplemented
22C22×D4C23160yes—
23C22×D4C2311080yescomplemented
24C23×C4C23145yes—
25C23×C4C23115yes—
26C23×C4C231360yescomplemented
27C24—460no—
28C24C24115yes—
29C24C241180yes—
30#41C2411280yescomplemented
31#41C2413840yescomplemented
32C2×Q8C241240yes—
33C2×Q8C241960yescomplemented
34C2×D4C241720yes—
35C2×D4C2412880yescomplemented
36C22×C4C24115yes—
37C22×C4C241180yes—
38C22×C4C241540yes—
39C22×C4C2411yescharacteristic
40C22×C4C241960yescomplemented
41C23—260no—
42C23—4360no—
43C23C25190yes—
44C23C251240yes—
45C23C2511yescharacteristic
46Q8C251320yes—
47D4C251960yes—
48C2×C4C2516yes—
49C2×C4C25190yes—
50C2×C4C251240yes—
51C2×C4C251720yes—
52C22—2180no—
53C22—4240no—
54C22#58514yesdirect product factor
55C22C26160yes—
56C22C2613yes—
57C4C2614yes—
58C4C26160yes—
59C2—260no—
60C2C2×#58516yesdirect product factor
61C2C2711yescharacteristic
subgroupG/H
centerC22×C4C24
commutator, Frattini, cosocleC2C27
socleC23C25
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ G
LECSG ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#585C2direct (N × H)
2. C22×2−1+4C2semidirect
3. C22×H(5,𝔽2)C2semidirect
4. C23×#41C2semidirect
5. #585C22direct (N × H)
6. C2×2−1+4C22semidirect
7. C2×H(5,𝔽2)C22semidirect
8. C22×#41C22semidirect
9. C22×#41C22semidirect
10. C22×#41C22semidirect
11. C23×Q8C22semidirect
12. C23×D4C22semidirect
13. C24×C4C22semidirect
15. 2−1+4C23semidirect
16. H(5,𝔽2)C23semidirect
17. C2×#41C23semidirect
18. C2×#41C23semidirect
19. C2×#41C23semidirect
21. C22×Q8C23semidirect
23. C22×D4C23semidirect
26. C23×C4C23semidirect
30. #41C24semidirect
31. #41C24semidirect
33. C2×Q8C24semidirect
35. C2×D4C24semidirect
40. C22×C4C24semidirect
54. C22#585direct (N × H)
60. C2C2×#585direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#5852
C22×2−1+42
C22×H(5,𝔽2)2
C23×#412