Finite Group Databasebeta

← #87568

Subgroups — #87568

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
1C2×C3×H(5,𝔽2)C211yesmaximal, characteristic, complemented
2#5966C2116yesmaximal, complemented
3D3×H(5,𝔽2)C2116yesmaximal, complemented
4#5964C2112yesmaximal, complemented
5C2×#1004C2118yesmaximal, complemented
6H(7,𝔽2)—31nomaximal, Sylow
7C3×H(5,𝔽2)C22116yescomplemented
8C2×C3×#41C2216yescomplemented
9C22×C3×D4C2219yescomplemented
10C22×#67C22124yescomplemented
11#1010C22148yescomplemented
12#1009C22148yescomplemented
13D3×#41C22196yescomplemented
14#1007C2218yescomplemented
15#1004C22172yescomplemented
16#1004C221144yescomplemented
17C2×#237C22172yescomplemented
18C2×D3×D4C22172yescomplemented
19C2×#235C22136yescomplemented
20#585—316no—
21#585—312no—
22C2×H(5,𝔽2)—316no—
23C2×H(5,𝔽2)—318no—
24C2×H(5,𝔽2)D311yescharacteristic, complemented
25C24×C3C2316yescomplemented
26C23×D3C23124yescomplemented
27C3×#41C23148yescomplemented
28C2×C3×Q8C2312yescomplemented
29C2×C3×D4C23172yescomplemented
30C2×C3×D4C23118yescomplemented
31C22×C3×C4C2319yescomplemented
32C2×#67C23172yescomplemented
33C2×#67C231144yescomplemented
34C22×Dic3C23124yescomplemented
35#239C23132yescomplemented
36D3×Q8C23132yescomplemented
37#237C231288yescomplemented
38D3×D4C231288yescomplemented
39#235C23148yescomplemented
40#235C231144yescomplemented
41C2×D12C23136yescomplemented
42C2×C4×D3C23172yescomplemented
43C2×Dic6C23136yescomplemented
442−1+4—348no—
452−1+4—38no—
46H(5,𝔽2)—348no—
47H(5,𝔽2)—3144no—
48H(5,𝔽2)—372no—
49H(5,𝔽2)C2×D3116yescomplemented
50C2×#41—396no—
51C2×#41—372no—
52C2×#41—336no—
53C2×#41C2×D316yescomplemented
54C22×D4—324no—
55C22×D4—372no—
56C22×D4C2×D319yescomplemented
57C23×C3—86no—
58C23×C3C24124yes—
59C23×C3C2419yes—
60C22×D3—824no—
61C22×D3C24172yes—
62C3×Q8C2418yes—
63C3×D4C24172yes—
64C2×C3×C4C24136yes—
65C2×C3×C4C2416yes—
66#67C24116yes—
67#67C241144yes—
68C2×Dic3C24172yes—
69D12C24148yes—
70C4×D3C24196yes—
71Dic6C24148yes—
72C24—324no—
73C24C22×D316yescomplemented
74#41—332no—
75#41—3288no—
76#41—3144no—
77#41—348no—
78#41C22×D3148yescomplemented
79C2×Q8—332no—
80C2×Q8—336no—
81C2×Q8C22×D312yescomplemented
82C2×D4—3144no—
83C2×D4—372no—
84C2×D4—3288no—
85C2×D4—336no—
86C2×D4C22×D3172yescomplemented
87C2×D4C22×D3118yescomplemented
88C22×C4—324no—
89C22×C4—372no—
90C22×C4C22×D319yescomplemented
91C22×C3—424no—
92C22×C3—49no—
93C22×C3C25118yes—
94C22×C3C2511yescharacteristic
95C2×D3—472no—
96C2×D3C25116yes—
97C3×C4C25112yes—
98Dic3C25116yes—
99C23—372no—
100C23—86no—
101C23—2424no—
102C23C23×D3124yes—
103C23C23×D319yes—
104Q8—348no—
105Q8C23×D318yes—
106D4—3144no—
107D4—316no—
108D4—348no—
109D4C23×D3172yes—
110C2×C4—372no—
111C2×C4—396no—
112C2×C4C23×D3136yes—
113C2×C4C23×D316yes—
114C2×C3—218no—
115C2×C3—21no—
116C2×C3C2611yescharacteristic
117D3—216no—
118C22—316no—
119C22—49no—
120C22—424no—
121C22—1272no—
122C22C24×D3118yes—
123C22C24×D311yescharacteristic
124C4—316no—
125C4C24×D3112yes—
126C3H(7,𝔽2)11yescharacteristic, complemented, Sylow
127C2—21no—
128C2—218no—
129C2—616no—
130C2C25×D311yescharacteristic
subgroupG/H
center, FrattiniC2C25×D3
commutator, socle, cosocleC2×C3C26
FittingC2×C3×H(5,𝔽2)C2
hypercenterC2×H(5,𝔽2)D3
hypocenterC3H(7,𝔽2)
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C2×C3×H(5,𝔽2)C2semidirect
2. #5966C2semidirect
3. D3×H(5,𝔽2)C2semidirect
4. #5964C2semidirect
5. C2×#1004C2semidirect
7. C3×H(5,𝔽2)C22semidirect
8. C2×C3×#41C22semidirect
9. C22×C3×D4C22semidirect
10. C22×#67C22semidirect
11. #1010C22semidirect
12. #1009C22semidirect
13. D3×#41C22semidirect
14. #1007C22semidirect
15. #1004C22semidirect
16. #1004C22semidirect
17. C2×#237C22semidirect
18. C2×D3×D4C22semidirect
19. C2×#235C22semidirect
24. C2×H(5,𝔽2)D3semidirect
25. C24×C3C23semidirect
26. C23×D3C23semidirect
27. C3×#41C23semidirect
28. C2×C3×Q8C23semidirect
29. C2×C3×D4C23semidirect
30. C2×C3×D4C23semidirect
31. C22×C3×C4C23semidirect
32. C2×#67C23semidirect
33. C2×#67C23semidirect
34. C22×Dic3C23semidirect
35. #239C23semidirect
36. D3×Q8C23semidirect
37. #237C23semidirect
38. D3×D4C23semidirect
39. #235C23semidirect
40. #235C23semidirect
41. C2×D12C23semidirect
42. C2×C4×D3C23semidirect
43. C2×Dic6C23semidirect
49. H(5,𝔽2)C2×D3semidirect
53. C2×#41C2×D3semidirect
56. C22×D4C2×D3semidirect
73. C24C22×D3semidirect
78. #41C22×D3semidirect
81. C2×Q8C22×D3semidirect
86. C2×D4C22×D3semidirect
87. C2×D4C22×D3semidirect
90. C22×C4C22×D3semidirect
126. C3H(7,𝔽2)semidirect

Maximal subgroups

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

subgroupindex
C2×C3×H(5,𝔽2)2
#59662
D3×H(5,𝔽2)2
#59642
C2×#10042
H(7,𝔽2)3