Finite Group Databasebeta

← C22×C3×#585 #10681707

Subgroups — C22×C3×#585 #10681707

Order 768.

60366 conjugacy classes of subgroups.

The subgroup lattice has 126 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#585C2196yesmaximal, direct product factor
2C22×C3×2−1+4C216yesmaximal, complemented
3C22×C3×H(5,𝔽2)C2110yesmaximal, complemented
4C23×C3×#41C2115yesmaximal, complemented
5C22×#585C311yesmaximal, characteristic, direct product factor, Sylow
6C3×#585C2211024yesdirect product factor
7C2×C3×2−1+4C221288yescomplemented
8C2×C3×H(5,𝔽2)C221480yescomplemented
9C22×C3×#41C22120yescomplemented
10C22×C3×#41C22160yescomplemented
11C22×C3×#41C221720yescomplemented
12C23×C3×Q8C22115yescomplemented
13C23×C3×D4C22145yescomplemented
14C24×C3×C4C22115yescomplemented
15C2×#585C2×C3196yesdirect product factor
16C22×2−1+4C2×C316yescomplemented
17C22×H(5,𝔽2)C2×C3110yescomplemented
18C23×#41C2×C3115yescomplemented
19C25×C3C23115yes—
20C3×2−1+4C2311536yescomplemented
21C3×H(5,𝔽2)C2312560yescomplemented
22C2×C3×#41C231480yescomplemented
23C2×C3×#41C2311440yescomplemented
24C2×C3×#41C2313840yescomplemented
25C22×C3×Q8C23120yes—
26C22×C3×Q8C231360yescomplemented
27C22×C3×D4C23160yes—
28C22×C3×D4C2311080yescomplemented
29C23×C3×C4C23145yes—
30C23×C3×C4C23115yes—
31C23×C3×C4C231360yescomplemented
32#585C22×C311024yesdirect product factor
33C2×2−1+4C22×C31288yescomplemented
34C2×H(5,𝔽2)C22×C31480yescomplemented
35C22×#41C22×C3120yescomplemented
36C22×#41C22×C3160yescomplemented
37C22×#41C22×C31720yescomplemented
38C23×Q8C22×C3115yescomplemented
39C23×D4C22×C3145yescomplemented
40C24×C4C22×C3115yescomplemented
41C24×C3—460no—
42C24×C3C24115yes—
43C24×C3C241180yes—
44C3×#41C2411280yescomplemented
45C3×#41C2413840yescomplemented
46C2×C3×Q8C241240yes—
47C2×C3×Q8C241960yescomplemented
48C2×C3×D4C241720yes—
49C2×C3×D4C2412880yescomplemented
50C22×C3×C4C24115yes—
51C22×C3×C4C241180yes—
52C22×C3×C4C241540yes—
53C22×C3×C4C2411yescharacteristic
54C22×C3×C4C241960yescomplemented
55C25C23×C3115yes—
562−1+4C23×C311536yescomplemented
57H(5,𝔽2)C23×C312560yescomplemented
58C2×#41C23×C31480yescomplemented
59C2×#41C23×C311440yescomplemented
60C2×#41C23×C313840yescomplemented
61C22×Q8C23×C3120yes—
62C22×Q8C23×C31360yescomplemented
63C22×D4C23×C3160yes—
64C22×D4C23×C311080yescomplemented
65C23×C4C23×C3115yes—
66C23×C4C23×C3145yes—
67C23×C4C23×C31360yescomplemented
68C23×C3—260no—
69C23×C3—4360no—
70C23×C3C25190yes—
71C23×C3C251240yes—
72C23×C3C2511yescharacteristic
73C3×Q8C251320yes—
74C3×D4C251960yes—
75C2×C3×C4C2516yes—
76C2×C3×C4C25190yes—
77C2×C3×C4C251240yes—
78C2×C3×C4C251720yes—
79C24—460no—
80C24C24×C3115yes—
81C24C24×C31180yes—
82#41C24×C311280yescomplemented
83#41C24×C313840yescomplemented
84C2×Q8C24×C31240yes—
85C2×Q8C24×C31960yescomplemented
86C2×D4C24×C31720yes—
87C2×D4C24×C312880yescomplemented
88C22×C4C24×C311yescharacteristic
89C22×C4C24×C3115yes—
90C22×C4C24×C31180yes—
91C22×C4C24×C31540yes—
92C22×C4C24×C31960yescomplemented
93C22×C3—2180no—
94C22×C3—4240no—
95C22×C3#58514yesdirect product factor
96C22×C3C26160yes—
97C22×C3C2613yes—
98C3×C4C2614yes—
99C3×C4C26160yes—
100C23—260no—
101C23—4360no—
102C23C25×C3190yes—
103C23C25×C31240yes—
104C23C25×C311yescharacteristic
105Q8C25×C31320yes—
106D4C25×C31960yes—
107C2×C4C25×C316yes—
108C2×C4C25×C3190yes—
109C2×C4C25×C31240yes—
110C2×C4C25×C31720yes—
111C2×C3—260no—
112C2×C3C2×#58516yesdirect product factor
113C2×C3C2711yescharacteristic
114C22—2180no—
115C22—4240no—
116C22C3×#58514yesdirect product factor
117C22C26×C3160yes—
118C22C26×C313yes—
119C4C26×C314yes—
120C4C26×C3160yes—
121C3C22×#58511yescharacteristic, direct product factor, Sylow
122C2—260no—
123C2C2×C3×#58516yesdirect product factor
124C2C27×C311yescharacteristic
subgroupG/H
centerC22×C3×C4C24
commutator, Frattini, cosocleC2C27×C3
socleC23×C3C25
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C22×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#585C2direct (N × H)
2. C22×C3×2−1+4C2semidirect
3. C22×C3×H(5,𝔽2)C2semidirect
4. C23×C3×#41C2semidirect
5. C22×#585C3direct (N × H)
6. C3×#585C22direct (N × H)
7. C2×C3×2−1+4C22semidirect
8. C2×C3×H(5,𝔽2)C22semidirect
9. C22×C3×#41C22semidirect
10. C22×C3×#41C22semidirect
11. C22×C3×#41C22semidirect
12. C23×C3×Q8C22semidirect
13. C23×C3×D4C22semidirect
14. C24×C3×C4C22semidirect
15. C2×#585C2×C3direct (N × H)
16. C22×2−1+4C2×C3semidirect
17. C22×H(5,𝔽2)C2×C3semidirect
18. C23×#41C2×C3semidirect
20. C3×2−1+4C23semidirect
21. C3×H(5,𝔽2)C23semidirect
22. C2×C3×#41C23semidirect
23. C2×C3×#41C23semidirect
24. C2×C3×#41C23semidirect
26. C22×C3×Q8C23semidirect
28. C22×C3×D4C23semidirect
31. C23×C3×C4C23semidirect
32. #585C22×C3direct (N × H)
33. C2×2−1+4C22×C3semidirect
34. C2×H(5,𝔽2)C22×C3semidirect
35. C22×#41C22×C3semidirect
36. C22×#41C22×C3semidirect
37. C22×#41C22×C3semidirect
38. C23×Q8C22×C3semidirect
39. C23×D4C22×C3semidirect
40. C24×C4C22×C3semidirect
44. C3×#41C24semidirect
45. C3×#41C24semidirect
47. C2×C3×Q8C24semidirect
49. C2×C3×D4C24semidirect
54. C22×C3×C4C24semidirect
56. 2−1+4C23×C3semidirect
57. H(5,𝔽2)C23×C3semidirect
58. C2×#41C23×C3semidirect
59. C2×#41C23×C3semidirect
60. C2×#41C23×C3semidirect
62. C22×Q8C23×C3semidirect
64. C22×D4C23×C3semidirect
67. C23×C4C23×C3semidirect
82. #41C24×C3semidirect
83. #41C24×C3semidirect
85. C2×Q8C24×C3semidirect
87. C2×D4C24×C3semidirect
92. C22×C4C24×C3semidirect
95. C22×C3#585direct (N × H)
112. C2×C3C2×#585direct (N × H)
116. C22C3×#585direct (N × H)
121. C3C22×#585direct (N × H)
123. C2C2×C3×#585direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5852
C22×C3×2−1+42
C22×C3×H(5,𝔽2)2
C23×C3×#412
C22×#5853