Finite Group Databasebeta

← C2×C3×#585 #87583

Subgroups — C2×C3×#585 #87583

Order 384.

5956 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#585C2132yesmaximal, direct product factor
2C2×C3×2−1+4C216yesmaximal, complemented
3C2×C3×H(5,𝔽2)C2110yesmaximal, complemented
4C22×C3×#41C2115yesmaximal, complemented
5C2×#585C311yesmaximal, characteristic, direct product factor, Sylow
6C3×2−1+4C22196yescomplemented
7C3×H(5,𝔽2)C221160yescomplemented
8C2×C3×#41C22120yescomplemented
9C2×C3×#41C22160yescomplemented
10C2×C3×#41C221240yescomplemented
11C22×C3×Q8C22115yescomplemented
12C22×C3×D4C22145yescomplemented
13C23×C3×C4C22115yescomplemented
14#585C2×C3132yesdirect product factor
15C2×2−1+4C2×C316yescomplemented
16C2×H(5,𝔽2)C2×C3110yescomplemented
17C22×#41C2×C3115yescomplemented
18C24×C3C23115yes—
19C3×#41C231160yescomplemented
20C3×#41C231480yescomplemented
21C2×C3×Q8C23120yes—
22C2×C3×Q8C231120yescomplemented
23C2×C3×D4C23160yes—
24C2×C3×D4C231360yescomplemented
25C22×C3×C4C23145yes—
26C22×C3×C4C23115yes—
27C22×C3×C4C231120yescomplemented
282−1+4C22×C3196yescomplemented
29H(5,𝔽2)C22×C31160yescomplemented
30C2×#41C22×C3120yescomplemented
31C2×#41C22×C3160yescomplemented
32C2×#41C22×C31240yescomplemented
33C22×Q8C22×C3115yescomplemented
34C22×D4C22×C3145yescomplemented
35C23×C4C22×C3115yescomplemented
36C23×C3—430no—
37C23×C3C24115yes—
38C23×C3C24160yes—
39C3×Q8C24180yes—
40C3×D4C241240yes—
41C2×C3×C4C24115yes—
42C2×C3×C4C24160yes—
43C2×C3×C4C241180yes—
44C2×C3×C4C2411yescharacteristic
45C24C23×C3115yes—
46#41C23×C31160yescomplemented
47#41C23×C31480yescomplemented
48C2×Q8C23×C3120yes—
49C2×Q8C23×C31120yescomplemented
50C2×D4C23×C3160yes—
51C2×D4C23×C31360yescomplemented
52C22×C4C23×C3115yes—
53C22×C4C23×C3145yes—
54C22×C4C23×C31120yescomplemented
55C22×C3—230no—
56C22×C3—460no—
57C22×C3C25130yes—
58C22×C3C2511yescharacteristic
59C3×C4C2512yes—
60C3×C4C25130yes—
61C23—430no—
62C23C24×C3115yes—
63C23C24×C3160yes—
64Q8C24×C3180yes—
65D4C24×C31240yes—
66C2×C4C24×C311yescharacteristic
67C2×C4C24×C3115yes—
68C2×C4C24×C3160yes—
69C2×C4C24×C31180yes—
70C2×C3—230no—
71C2×C3#58512yesdirect product factor
72C2×C3C2611yescharacteristic
73C22—230no—
74C22—460no—
75C22C25×C3130yes—
76C22C25×C311yescharacteristic
77C4C25×C312yes—
78C4C25×C3130yes—
79C3C2×#58511yescharacteristic, direct product factor, Sylow
80C2—230no—
81C2C3×#58512yesdirect product factor
82C2C26×C311yescharacteristic
subgroupG/H
centerC2×C3×C4C24
commutator, Frattini, cosocleC2C26×C3
socleC22×C3C25
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C2×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#585C2direct (N × H)
2. C2×C3×2−1+4C2semidirect
3. C2×C3×H(5,𝔽2)C2semidirect
4. C22×C3×#41C2semidirect
5. C2×#585C3direct (N × H)
6. C3×2−1+4C22semidirect
7. C3×H(5,𝔽2)C22semidirect
8. C2×C3×#41C22semidirect
9. C2×C3×#41C22semidirect
10. C2×C3×#41C22semidirect
11. C22×C3×Q8C22semidirect
12. C22×C3×D4C22semidirect
13. C23×C3×C4C22semidirect
14. #585C2×C3direct (N × H)
15. C2×2−1+4C2×C3semidirect
16. C2×H(5,𝔽2)C2×C3semidirect
17. C22×#41C2×C3semidirect
19. C3×#41C23semidirect
20. C3×#41C23semidirect
22. C2×C3×Q8C23semidirect
24. C2×C3×D4C23semidirect
27. C22×C3×C4C23semidirect
28. 2−1+4C22×C3semidirect
29. H(5,𝔽2)C22×C3semidirect
30. C2×#41C22×C3semidirect
31. C2×#41C22×C3semidirect
32. C2×#41C22×C3semidirect
33. C22×Q8C22×C3semidirect
34. C22×D4C22×C3semidirect
35. C23×C4C22×C3semidirect
46. #41C23×C3semidirect
47. #41C23×C3semidirect
49. C2×Q8C23×C3semidirect
51. C2×D4C23×C3semidirect
54. C22×C4C23×C3semidirect
71. C2×C3#585direct (N × H)
79. C3C2×#585direct (N × H)
81. C2C3×#585direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#5852
C2×C3×2−1+42
C2×C3×H(5,𝔽2)2
C22×C3×#412
C2×#5853