Finite Group Databasebeta

← C24×#11758226 #30593897

Subgroups — C24×#11758226 #30593897

Order 15616.

65470 conjugacy classes of subgroups; only the 32651 normal subgroups are shown.

normal subgroups only

This group has 65470 conjugacy classes of subgroups — too many to draw. Only its 32651 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
1C23×#11758226C21120yesmaximal, direct product factor
2C24×D244C213yesmaximal, complemented
3C24×C4×D61C213yesmaximal, complemented
4C24×Q8×C61C211yesmaximal, characteristic, complemented
5C25×D61C2213yes—
6C22×#11758226C2212240yesdirect product factor
7C23×D244C221180yescomplemented
8C24×Dic61C2211yescharacteristic
9C23×C4×D61C221180yescomplemented
10C23×Q8×C61C22160yescomplemented
11C24×C4×C61C2213yes—
12C24×D61—248no—
13C24×D61C23190yes—
14C2×#11758226C2317680yesdirect product factor
15C22×D244C2311680yescomplemented
16C23×Dic61C23130yes—
17C22×C4×D61C2311680yescomplemented
18C25×C61C2311yescharacteristic
19C22×Q8×C61C231560yescomplemented
20C23×C4×C61C23190yes—
21C24×C61#41116yescomplemented
22C24×C61C24115yes—
23C23×D61—2720no—
24C23×D61C241420yes—
25C2×Q8×C61C241960yescomplemented
26C22×C4×C61C241420yes—
27C22×Dic61C241140yes—
28#11758226C2414096yesdirect product factor
29C2×D244C2412880yescomplemented
30C2×C4×D61C2412880yescomplemented
31C23×C61C2×#411120yescomplemented
32C23×C61C25135yes—
33C22×D61—21680no—
34C22×D61C251360yes—
35Q8×C61C251256yescomplemented
36C2×C4×C61C251360yes—
37C2×Dic61C251120yes—
38D244C251768yescomplemented
39C4×D61C251768yescomplemented
40C24×#41—611nomaximal, Sylow
41C22×C61C22×#411140yescomplemented
42C22×C61C26115yes—
43C2×D61—2720no—
44C2×D61C26148yes—
45C4×C61C26148yes—
46Dic61C26116yes—
47C23×#41—61120no—
48C24×Q8D6111yescharacteristic, complemented
49C24×D4—613no—
50C25×C4—613no—
51C2×C61C23×#41130yescomplemented
52C2×C61C2711yescharacteristic
53D61—248no—
54C26—613no—
55C22×#41—612240no—
56C23×Q8C2×D61160yescomplemented
57C23×D4—61180no—
58C24×C4—611no—
59C24×C4—61180no—
60C24×C4C2×D6113yes—
61C61C24×#4111yescharacteristic, complemented, Sylow
62C25—6190no—
63C25—12248no—
64C25C22×D6111yescharacteristic
65C2×#41—617680no—
66C22×Q8C22×D611560yescomplemented
67C22×D4—611680no—
68C23×C4—6130no—
69C23×C4—611680no—
70C23×C4C22×D61190yes—
71C24—61420no—
72C24—122720no—
73C24#11758226116yesdirect product factor
74C24C23×D61115yes—
75#41—614096no—
76C2×Q8C23×D611960yescomplemented
77C2×D4—612880no—
78C22×C4—61140no—
79C22×C4—612880no—
80C22×C4C23×D611420yes—
81C23—61360no—
82C23—1221680no—
83C23C2×#117582261120yesdirect product factor
84C23C24×D61135yes—
85Q8C24×D611256yescomplemented
86D4—61768no—
87C2×C4—61120no—
88C2×C4—61768no—
89C2×C4C24×D611360yes—
90C22—6148no—
91C22—122720no—
92C22C22×#117582261140yesdirect product factor
93C22C25×D61115yes—
94C4—6116no—
95C4C25×D61148yes—
96C2—12248no—
97C2C23×#11758226130yesdirect product factor
98C2C26×D6111yescharacteristic
subgroupG/H
centerC25C22×D61
commutator, cosocleC2×C61C27
FrattiniC2C26×D61
FittingC24×Q8×C61C2
socleC25×C61C23
hypercenterC24×Q8D61
hypocenterC61C24×#41
LCSG ▷ C2×C61 ▷ C61
UCSC1 ◁ C25 ◁ C24×Q8

Complements

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

NH ≅ G/Ntype
1. C23×#11758226C2direct (N × H)
2. C24×D244C2semidirect
3. C24×C4×D61C2semidirect
4. C24×Q8×C61C2semidirect
6. C22×#11758226C22direct (N × H)
7. C23×D244C22semidirect
9. C23×C4×D61C22semidirect
10. C23×Q8×C61C22semidirect
14. C2×#11758226C23direct (N × H)
15. C22×D244C23semidirect
17. C22×C4×D61C23semidirect
19. C22×Q8×C61C23semidirect
21. C24×C61#41semidirect
25. C2×Q8×C61C24semidirect
28. #11758226C24direct (N × H)
29. C2×D244C24semidirect
30. C2×C4×D61C24semidirect
31. C23×C61C2×#41semidirect
35. Q8×C61C25semidirect
38. D244C25semidirect
39. C4×D61C25semidirect
41. C22×C61C22×#41semidirect
48. C24×Q8D61semidirect
51. C2×C61C23×#41semidirect
56. C23×Q8C2×D61semidirect
61. C61C24×#41semidirect
66. C22×Q8C22×D61semidirect
73. C24#11758226direct (N × H)
76. C2×Q8C23×D61semidirect
83. C23C2×#11758226direct (N × H)
85. Q8C24×D61semidirect
92. C22C22×#11758226direct (N × H)
97. C2C23×#11758226direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#117582262
C24×D2442
C24×C4×D612
C24×Q8×C612
C24×#4161