Finite Group Databasebeta

← C23×C32×#41 #11770639

Subgroups — C23×C32×#41 #11770639

Order 1152.

19560 conjugacy classes of subgroups.

The subgroup lattice has 120 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C32×#41C2156yesmaximal, direct product factor
2C23×C32×Q8C211yesmaximal, characteristic, complemented
3C23×C32×D4C213yesmaximal, complemented
4C24×C32×C4C213yesmaximal, complemented
5C23×C3×#41C314yesmaximal, direct product factor
6C25×C32C2213yes—
7C2×C32×#41C221448yesdirect product factor
8C22×C32×Q8C22128yescomplemented
9C22×C32×D4C22184yescomplemented
10C23×C32×C4C2213yes—
11C23×C32×C4C2211yescharacteristic
12C23×C32×C4C22184yescomplemented
13C22×C3×#41C2×C31224yesdirect product factor
14C23×C3×Q8C2×C314yescomplemented
15C23×C3×D4C2×C3112yescomplemented
16C24×C3×C4C2×C3112yescomplemented
17C24×C32—224no—
18C24×C32C23142yes—
19C24×C32C2311yescharacteristic
20C32×#41C231512yesdirect product factor
21C2×C32×Q8C231112yescomplemented
22C2×C32×D4C231336yescomplemented
23C22×C32×C4C23142yes—
24C22×C32×C4C23114yes—
25C22×C32×C4C231336yescomplemented
26C23×#41C3211yescharacteristic, direct product factor, Sylow
27C25×C3C22×C3112yes—
28C2×C3×#41C22×C311792yesdirect product factor
29C22×C3×Q8C22×C31112yescomplemented
30C22×C3×D4C22×C31336yescomplemented
31C23×C3×C4C22×C314yes—
32C23×C3×C4C22×C3112yes—
33C23×C3×C4C22×C31336yescomplemented
34C23×C32—2168no—
35C23×C32#4118yesdirect product factor
36C23×C32C24184yes—
37C23×C32C2417yes—
38C32×Q8C24164yescomplemented
39C32×D4C241192yescomplemented
40C2×C32×C4C24128yes—
41C2×C32×C4C24184yes—
42C2×C32×C4C241192yescomplemented
43C22×#41C2×C32156yesdirect product factor
44C23×Q8C2×C3211yescharacteristic, complemented
45C23×D4C2×C3213yescomplemented
46C24×C4C2×C3213yescomplemented
47C24×C3—296no—
48C24×C3C23×C31168yes—
49C24×C3C23×C314yes—
50C3×#41C23×C312048yesdirect product factor
51C2×C3×Q8C23×C31448yescomplemented
52C2×C3×D4C23×C311344yescomplemented
53C22×C3×C4C23×C3156yes—
54C22×C3×C4C23×C31168yes—
55C22×C3×C4C23×C311344yescomplemented
56C22×C32—2168no—
57C22×C32C2×#41128yesdirect product factor
58C22×C32C25124yes—
59C22×C32C2517yes—
60C32×C4C2518yes—
61C32×C4C25124yes—
62C25C22×C3213yes—
63C2×#41C22×C321448yesdirect product factor
64C22×Q8C22×C32128yescomplemented
65C22×D4C22×C32184yescomplemented
66C23×C4C22×C3211yescharacteristic
67C23×C4C22×C3213yes—
68C23×C4C22×C32184yescomplemented
69C23×C3—2672no—
70C23×C3C3×#41132yesdirect product factor
71C23×C3C24×C31336yes—
72C23×C3C24×C3128yes—
73C3×Q8C24×C31256yescomplemented
74C3×D4C24×C31768yescomplemented
75C2×C3×C4C24×C31112yes—
76C2×C3×C4C24×C31336yes—
77C2×C3×C4C24×C31768yescomplemented
78C2×C32—224no—
79C2×C32C22×#41114yesdirect product factor
80C2×C32C2611yescharacteristic
81C24—224no—
82C24C23×C32142yes—
83C24C23×C3211yescharacteristic
84#41C23×C321512yesdirect product factor
85C2×Q8C23×C321112yescomplemented
86C2×D4C23×C321336yescomplemented
87C22×C4C23×C32114yes—
88C22×C4C23×C32142yes—
89C22×C4C23×C321336yescomplemented
90C22×C3—2672no—
91C22×C3C2×C3×#411112yesdirect product factor
92C22×C3C25×C3196yes—
93C22×C3C25×C3128yes—
94C3×C4C25×C3132yes—
95C3×C4C25×C3196yes—
96C32C23×#4111yescharacteristic, direct product factor, Sylow
97C23—2168no—
98C23C32×#4118yesdirect product factor
99C23C24×C32184yes—
100C23C24×C3217yes—
101Q8C24×C32164yescomplemented
102D4C24×C321192yescomplemented
103C2×C4C24×C32128yes—
104C2×C4C24×C32184yes—
105C2×C4C24×C321192yescomplemented
106C2×C3—296no—
107C2×C3C22×C3×#41156yesdirect product factor
108C2×C3C26×C314yes—
109C22—2168no—
110C22C2×C32×#41128yesdirect product factor
111C22C25×C32124yes—
112C22C25×C3217yes—
113C4C25×C3218yes—
114C4C25×C32124yes—
115C3C23×C3×#4114yesdirect product factor
116C2—224no—
117C2C22×C32×#41114yesdirect product factor
118C2C26×C3211yescharacteristic
subgroupG/H
centerC23×C32×C4C22
commutator, Frattini, cosocleC2C26×C32
socleC24×C32C23
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C23×C32×C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×C32×#41C2direct (N × H)
2. C23×C32×Q8C2semidirect
3. C23×C32×D4C2semidirect
4. C24×C32×C4C2semidirect
5. C23×C3×#41C3direct (N × H)
7. C2×C32×#41C22direct (N × H)
8. C22×C32×Q8C22semidirect
9. C22×C32×D4C22semidirect
12. C23×C32×C4C22semidirect
13. C22×C3×#41C2×C3direct (N × H)
14. C23×C3×Q8C2×C3semidirect
15. C23×C3×D4C2×C3semidirect
16. C24×C3×C4C2×C3semidirect
20. C32×#41C23direct (N × H)
21. C2×C32×Q8C23semidirect
22. C2×C32×D4C23semidirect
25. C22×C32×C4C23semidirect
26. C23×#41C32direct (N × H)
28. C2×C3×#41C22×C3direct (N × H)
29. C22×C3×Q8C22×C3semidirect
30. C22×C3×D4C22×C3semidirect
33. C23×C3×C4C22×C3semidirect
35. C23×C32#41direct (N × H)
38. C32×Q8C24semidirect
39. C32×D4C24semidirect
42. C2×C32×C4C24semidirect
43. C22×#41C2×C32direct (N × H)
44. C23×Q8C2×C32semidirect
45. C23×D4C2×C32semidirect
46. C24×C4C2×C32semidirect
50. C3×#41C23×C3direct (N × H)
51. C2×C3×Q8C23×C3semidirect
52. C2×C3×D4C23×C3semidirect
55. C22×C3×C4C23×C3semidirect
57. C22×C32C2×#41direct (N × H)
63. C2×#41C22×C32direct (N × H)
64. C22×Q8C22×C32semidirect
65. C22×D4C22×C32semidirect
68. C23×C4C22×C32semidirect
70. C23×C3C3×#41direct (N × H)
73. C3×Q8C24×C3semidirect
74. C3×D4C24×C3semidirect
77. C2×C3×C4C24×C3semidirect
79. C2×C32C22×#41direct (N × H)
84. #41C23×C32direct (N × H)
85. C2×Q8C23×C32semidirect
86. C2×D4C23×C32semidirect
89. C22×C4C23×C32semidirect
91. C22×C3C2×C3×#41direct (N × H)
96. C32C23×#41direct (N × H)
98. C23C32×#41direct (N × H)
101. Q8C24×C32semidirect
102. D4C24×C32semidirect
105. C2×C4C24×C32semidirect
107. C2×C3C22×C3×#41direct (N × H)
110. C22C2×C32×#41direct (N × H)
115. C3C23×C3×#41direct (N × H)
117. C2C22×C32×#41direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C32×#412
C23×C32×Q82
C23×C32×D42
C24×C32×C42
C23×C3×#413