Finite Group Databasebeta

← #87574

Subgroups — #87574

Order 384.

5992 conjugacy classes of subgroups; only the 3201 normal subgroups are shown.

normal subgroups only

This group has 5992 conjugacy classes of subgroups — too many to draw. Only its 3201 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
1C3×#585C211yesmaximal, characteristic, complemented
2#5968C2112yesmaximal, complemented
3D3×H(5,𝔽2)C2120yesmaximal, complemented
4#5964C2115yesmaximal, complemented
5C2×#1009C2115yesmaximal, complemented
6H(7,𝔽2)—31nomaximal, Sylow
7C3×2−1+4C2216yescomplemented
8C3×H(5,𝔽2)C22110yescomplemented
9C2×C3×#41C22115yescomplemented
10#1010C22120yescomplemented
11#1009C22160yescomplemented
12#1009C221120yescomplemented
13D3×#41C221120yescomplemented
14#1007C22130yescomplemented
15C2×#239C22130yescomplemented
16#1004C22190yescomplemented
17C2×D3×D4C22190yescomplemented
18C2×#235C22145yescomplemented
19C22×D12C22115yescomplemented
20#585—312no—
21#585—315no—
22#585D311yescharacteristic, complemented
23C2×H(5,𝔽2)—320no—
24C2×H(5,𝔽2)—315no—
25C23×D3C23130yescomplemented
26C3×#41C23160yescomplemented
27C3×#41C23120yescomplemented
28C2×C3×Q8C23115yescomplemented
29C2×C3×D4C23145yescomplemented
30C22×C3×C4C23115yescomplemented
31C2×#67C23190yescomplemented
32#239C231120yescomplemented
33D3×Q8C23140yescomplemented
34#237C231120yescomplemented
35D3×D4C231360yescomplemented
36#235C23160yescomplemented
37#235C231180yescomplemented
38C2×D12C23145yescomplemented
39C2×D12C23190yescomplemented
40C2×C4×D3C23190yescomplemented
41C2×Dic6C23115yescomplemented
422−1+4—320no—
432−1+4—330no—
442−1+4C2×D316yescomplemented
45H(5,𝔽2)—3120no—
46H(5,𝔽2)—360no—
47H(5,𝔽2)—390no—
48H(5,𝔽2)C2×D3110yescomplemented
49C2×#41—3120no—
50C2×#41—330no—
51C2×#41—345no—
52C2×#41C2×D3115yescomplemented
53C22×D4—390no—
54C22×D4—315no—
55C23×C3C24115yes—
56C22×D3—830no—
57C22×D3C24190yes—
58C3×Q8C24120yes—
59C3×D4C24160yes—
60C2×C3×C4C24115yes—
61C2×C3×C4C24145yes—
62#67C241120yes—
63C2×Dic3C24130yes—
64D12C24110yes—
65D12C24190yes—
66C4×D3C241120yes—
67Dic6C2416yes—
68Dic6C24130yes—
69C24—330no—
70#41—3120no—
71#41—3120no—
72#41—3180no—
73#41—360no—
74#41C22×D3160yescomplemented
75#41C22×D3120yescomplemented
76C2×Q8—340no—
77C2×Q8—315no—
78C2×Q8C22×D3115yescomplemented
79C2×D4—390no—
80C2×D4—3360no—
81C2×D4—390no—
82C2×D4—345no—
83C2×D4C22×D3145yescomplemented
84C22×C4—390no—
85C22×C4C22×D3115yescomplemented
86C22×C3—415no—
87C22×C3C25115yes—
88C2×D3—490no—
89C2×D3C25120yes—
90C3×C4C2511yescharacteristic
91C3×C4C25115yes—
92Dic3C25112yes—
93C23—390no—
94C23—2430no—
95C23C23×D3115yes—
96Q8—330no—
97Q8—36no—
98Q8C23×D3120yes—
99D4—3120no—
100D4—390no—
101D4—310no—
102D4C23×D3160yes—
103C2×C4—330no—
104C2×C4—3120no—
105C2×C4C23×D3145yes—
106C2×C4C23×D3115yes—
107C2×C3—215no—
108C2×C3C2611yescharacteristic
109D3—220no—
110C22—320no—
111C22—415no—
112C22—1290no—
113C22C24×D3115yes—
114C4—312no—
115C4C24×D3115yes—
116C4C24×D311yescharacteristic
117C3H(7,𝔽2)11yescharacteristic, complemented, Sylow
118C2—215no—
119C2—620no—
120C2C25×D311yescharacteristic
subgroupG/H
center, FrattiniC2C25×D3
commutator, socle, cosocleC2×C3C26
FittingC3×#585C2
hypercenter#585D3
hypocenterC3H(7,𝔽2)
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ #585

Complements

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

NH ≅ G/Ntype
1. C3×#585C2semidirect
2. #5968C2semidirect
3. D3×H(5,𝔽2)C2semidirect
4. #5964C2semidirect
5. C2×#1009C2semidirect
7. C3×2−1+4C22semidirect
8. C3×H(5,𝔽2)C22semidirect
9. C2×C3×#41C22semidirect
10. #1010C22semidirect
11. #1009C22semidirect
12. #1009C22semidirect
13. D3×#41C22semidirect
14. #1007C22semidirect
15. C2×#239C22semidirect
16. #1004C22semidirect
17. C2×D3×D4C22semidirect
18. C2×#235C22semidirect
19. C22×D12C22semidirect
22. #585D3semidirect
25. C23×D3C23semidirect
26. C3×#41C23semidirect
27. C3×#41C23semidirect
28. C2×C3×Q8C23semidirect
29. C2×C3×D4C23semidirect
30. C22×C3×C4C23semidirect
31. C2×#67C23semidirect
32. #239C23semidirect
33. D3×Q8C23semidirect
34. #237C23semidirect
35. D3×D4C23semidirect
36. #235C23semidirect
37. #235C23semidirect
38. C2×D12C23semidirect
39. C2×D12C23semidirect
40. C2×C4×D3C23semidirect
41. C2×Dic6C23semidirect
44. 2−1+4C2×D3semidirect
48. H(5,𝔽2)C2×D3semidirect
52. C2×#41C2×D3semidirect
74. #41C22×D3semidirect
75. #41C22×D3semidirect
78. C2×Q8C22×D3semidirect
83. C2×D4C22×D3semidirect
85. C22×C4C22×D3semidirect
117. C3H(7,𝔽2)semidirect

Maximal subgroups

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

subgroupindex
C3×#5852
#59682
D3×H(5,𝔽2)2
#59642
C2×#10092
H(7,𝔽2)3