Finite Group Databasebeta

← C2×#5966 #87566

Subgroups — C2×#5966 #87566

Order 384.

5956 conjugacy classes of subgroups; only the 3205 normal subgroups are shown.

normal subgroups only

This group has 5956 conjugacy classes of subgroups — too many to draw. Only its 3205 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#5966C2132yesmaximal, direct product factor
3C2×#1010C216yesmaximal, complemented
4C2×D3×#41C216yesmaximal, complemented
5C2×#1004C219yesmaximal, complemented
6C22×#237C219yesmaximal, complemented
7C2×#585—31nomaximal, Sylow
8C3×H(5,𝔽2)C22116yescomplemented
9C2×C3×#41C2216yescomplemented
10C22×C3×D4C2219yescomplemented
11C22×#67C22118yescomplemented
12C23×Dic3C2216yescomplemented
13#1010C22196yescomplemented
14D3×#41C22196yescomplemented
15C2×#239C2212yescomplemented
16C2×D3×Q8C2216yescomplemented
17#1004C221144yescomplemented
18C2×#237C22118yescomplemented
19C2×#237C22136yescomplemented
20C2×#237C221144yescomplemented
21C2×D3×D4C22118yescomplemented
22C2×#235C22118yescomplemented
23C22×C4×D3C2219yescomplemented
24C22×Dic6C2219yescomplemented
25#585—332no—
26C2×2−1+4—36no—
27C2×H(5,𝔽2)—39no—
28C2×H(5,𝔽2)D311yescharacteristic, complemented
29C22×#41—36no—
30C22×#41—39no—
31C24×C3C2316yes—
32C23×D3C2319yes—
33C3×#41C23148yescomplemented
34C2×C3×Q8C2312yes—
35C2×C3×D4C23118yes—
36C2×C3×D4C23172yescomplemented
37C22×C3×C4C2319yes—
38C2×#67C23136yes—
39C2×#67C231144yescomplemented
40C22×Dic3C23118yes—
41C22×Dic3C2319yes—
42C22×Dic3C23148yescomplemented
43#239C23116yescomplemented
44D3×Q8C23148yescomplemented
45#237C231144yescomplemented
46#237C231288yescomplemented
47D3×D4C231144yescomplemented
48#235C231144yescomplemented
49C2×D12C2316yes—
50C2×C4×D3C2316yes—
51C2×C4×D3C23118yes—
52C2×C4×D3C23172yescomplemented
53C2×Dic6C23118yes—
54C2×Dic6C23172yescomplemented
552−1+4—396no—
56H(5,𝔽2)—3144no—
57H(5,𝔽2)C2×D3116yescomplemented
58C2×#41—396no—
59C2×#41—32no—
60C2×#41—318no—
61C2×#41—336no—
62C2×#41—3144no—
63C2×#41—318no—
64C2×#41C2×D316yescomplemented
65C22×Q8—36no—
66C22×Q8—39no—
67C22×D4—318no—
68C22×D4—318no—
69C22×D4C2×D319yescomplemented
70C23×C4—36no—
71C23×C4—39no—
72C23×C3—412no—
73C23×C3C2419yes—
74C23×C3C24124yes—
75C22×D3—418no—
76C22×D3C2416yes—
77C22×D3C24136yes—
78C3×Q8C2418yes—
79C3×D4C24172yes—
80C2×C3×C4C2416yes—
81C2×C3×C4C24136yes—
82#67C241144yes—
83C2×Dic3C2419yes—
84C2×Dic3C24172yes—
85C2×Dic3C2411yescharacteristic
86C2×Dic3C24136yes—
87D12C24124yes—
88C4×D3C24124yes—
89C4×D3C24172yes—
90Dic6C24172yes—
91C24—39no—
92C24C22×D316yes—
93#41—316no—
94#41—3144no—
95#41—3288no—
96#41—3144no—
97#41C22×D3148yescomplemented
98C2×Q8—348no—
99C2×Q8—318no—
100C2×Q8—372no—
101C2×Q8C22×D312yes—
102C2×D4—336no—
103C2×D4—3144no—
104C2×D4—3144no—
105C2×D4—36no—
106C2×D4C22×D3118yes—
107C2×D4C22×D3172yescomplemented
108C22×C4—39no—
109C22×C4—318no—
110C22×C4—348no—
111C22×C4—36no—
112C22×C4—318no—
113C22×C4—372no—
114C22×C4C22×D319yes—
115C22×C3—218no—
116C22×C3—424no—
117C22×C3C25118yes—
118C22×C3C2511yescharacteristic
119C2×D3—212no—
120C2×D3—436no—
121C2×D3C25112yes—
122C3×C4C25112yes—
123Dic3C25118yes—
124Dic3C2512yes—
125C23—36no—
126C23—336no—
127C23—412no—
128C23—1218no—
129C23C23×D319yes—
130C23C23×D3124yes—
131Q8—372no—
132Q8C23×D318yes—
133D4—3144no—
134D4—324no—
135D4C23×D3172yes—
136C2×C4—336no—
137C2×C4—372no—
138C2×C4—31no—
139C2×C4—39no—
140C2×C4—324no—
141C2×C4—372no—
142C2×C4C23×D3136yes—
143C2×C4C23×D316yes—
144C2×C3—218no—
145C2×C3#58512yescomplemented
146C2×C3C2611yescharacteristic
147D3—212no—
148C22—218no—
149C22—312no—
150C22—424no—
151C22—612no—
152C22—1236no—
153C22C24×D3118yes—
154C22C24×D311yescharacteristic
155C4—32no—
156C4—318no—
157C4C24×D3112yes—
158C3C2×#58511yescharacteristic, complemented, Sylow
159C2—218no—
160C2—612no—
161C2#596612yesdirect product factor
162C2C25×D311yescharacteristic
subgroupG/H
centerC22C24×D3
commutator, cosocleC2×C3C26
FrattiniC2C25×D3
FittingC2×C3×H(5,𝔽2)C2
socleC22×C3C25
hypercenterC2×H(5,𝔽2)D3
hypocenterC3C2×#585
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ 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. #5966C2direct (N × H)
3. C2×#1010C2semidirect
4. C2×D3×#41C2semidirect
5. C2×#1004C2semidirect
6. C22×#237C2semidirect
8. C3×H(5,𝔽2)C22semidirect
9. C2×C3×#41C22semidirect
10. C22×C3×D4C22semidirect
11. C22×#67C22semidirect
12. C23×Dic3C22semidirect
13. #1010C22semidirect
14. D3×#41C22semidirect
15. C2×#239C22semidirect
16. C2×D3×Q8C22semidirect
17. #1004C22semidirect
18. C2×#237C22semidirect
19. C2×#237C22semidirect
20. C2×#237C22semidirect
21. C2×D3×D4C22semidirect
22. C2×#235C22semidirect
23. C22×C4×D3C22semidirect
24. C22×Dic6C22semidirect
28. C2×H(5,𝔽2)D3semidirect
33. C3×#41C23semidirect
36. C2×C3×D4C23semidirect
39. C2×#67C23semidirect
42. C22×Dic3C23semidirect
43. #239C23semidirect
44. D3×Q8C23semidirect
45. #237C23semidirect
46. #237C23semidirect
47. D3×D4C23semidirect
48. #235C23semidirect
52. C2×C4×D3C23semidirect
54. C2×Dic6C23semidirect
57. H(5,𝔽2)C2×D3semidirect
64. C2×#41C2×D3semidirect
69. C22×D4C2×D3semidirect
97. #41C22×D3semidirect
107. C2×D4C22×D3semidirect
145. C2×C3#585semidirect
158. C3C2×#585semidirect
161. C2#5966direct (N × H)

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
C2×#10102
C2×D3×#412
C2×#10042
C22×#2372
C2×#5853