Finite Group Databasebeta

← C23×C3×D3×C9 #13049345

Subgroups — C23×C3×D3×C9 #13049345

Order 1296.

1580 conjugacy classes of subgroups; only the 830 normal subgroups are shown.

normal subgroups only

This group has 1580 conjugacy classes of subgroups — too many to draw. Only its 830 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×C32×C9C211yesmaximal, characteristic, complemented
2C22×C3×D3×C9C2114yesmaximal, direct product factor
3C23×C32×D3C311yesmaximal, characteristic
4C24×C3×C9—31nomaximal
5C23×D3×C9C313yesmaximal, direct product factor
6C22×C32×C9C2217yescomplemented
7C2×C3×D3×C9C22128yesdirect product factor
8C23×C33C2×C311yescharacteristic
9C22×C32×D3C2×C3114yes—
10C23×C3×C9—21no—
11C23×C3×C9—23no—
12C23×C3×C9—314no—
13C23×C3×C9D311yescharacteristic, direct product factor
14C23×C3×C9C2×C313yescomplemented
15C22×D3×C9C2×C3142yesdirect product factor
16C2×C32×C9C2317yescomplemented
17C3×D3×C9C2318yesdirect product factor
18C24×C32—31no—
19C23×C3×D3C913yesdirect product factor
20C23×C3×D3C3211yescharacteristic
21C24×C9—33no—
22C22×C33C22×C317yes—
23C2×C32×D3C22×C3128yes—
24C22×C3×C9—27no—
25C22×C3×C9—221no—
26C22×C3×C9—328no—
27C22×C3×C9C2×D317yesdirect product factor
28C22×C3×C9C22×C3121yescomplemented
29C2×D3×C9C22×C3184yesdirect product factor
30C32×C9C2411yescharacteristic, complemented, Sylow
31C23×C32—23no—
32C23×C32—21no—
33C23×C32—314no—
34C23×C32C2×C913yescomplemented
35C23×C32C3×D311yescharacteristic
36C23×C32C2×C3211yescharacteristic
37C22×C3×D3C2×C9142yesdirect product factor
38C22×C3×D3C2×C32114yes—
39C23×C9—23no—
40C23×C9—342no—
41C23×C9C3×D313yesdirect product factor
42C2×C33C23×C317yes—
43C32×D3C23×C318yes—
44C2×C3×C9—27no—
45C2×C3×C9—221no—
46C2×C3×C9—38no—
47C2×C3×C9C22×D317yesdirect product factor
48C2×C3×C9C23×C3121yescomplemented
49D3×C9C23×C3124yesdirect product factor
50C24×C3—33no—
51C24×C3—31no—
52C23×D3C3×C911yescharacteristic, direct product factor
53C22×C32—221no—
54C22×C32—27no—
55C22×C32—328no—
56C22×C32C22×C9121yescomplemented
57C22×C32C2×C3×D317yes—
58C22×C32C22×C3217yes—
59C2×C3×D3C22×C9184yesdirect product factor
60C2×C3×D3C22×C32128yes—
61C22×C9—221no—
62C22×C9—384no—
63C22×C9C2×C3×D3121yesdirect product factor
64C33C24×C311yescharacteristic
65C3×C9—21no—
66C3×C9—23no—
67C3×C9C23×D311yescharacteristic, direct product factor
68C3×C9C24×C313yescomplemented
69C23×C3—23no—
70C23×C3—21no—
71C23×C3—342no—
72C23×C3—314no—
73C23×C3D3×C913yesdirect product factor
74C23×C3C2×C3×C911yescharacteristic, complemented
75C23×C3C32×D311yescharacteristic
76C22×D3C2×C3×C9114yesdirect product factor
77C2×C32—221no—
78C2×C32—27no—
79C2×C32—38no—
80C2×C32C23×C9121yescomplemented
81C2×C32C22×C3×D317yes—
82C2×C32C23×C3217yes—
83C3×D3C23×C9124yesdirect product factor
84C3×D3C23×C3218yes—
85C2×C9—221no—
86C2×C9—324no—
87C2×C9C22×C3×D3121yesdirect product factor
88C24—31noSylow
89C22×C3—221no—
90C22×C3—27no—
91C22×C3—384no—
92C22×C3—328no—
93C22×C3C2×D3×C9121yesdirect product factor
94C22×C3C22×C3×C917yescomplemented
95C22×C3C2×C32×D317yes—
96C2×D3C22×C3×C9128yesdirect product factor
97C32—21no—
98C32—23no—
99C32C24×C913yescomplemented
100C32C23×C3×D311yescharacteristic
101C32C24×C3211yescharacteristic
102C9—23no—
103C9C23×C3×D313yesdirect product factor
104C23—314no—
105C23C3×D3×C911yescharacteristic, direct product factor
106C2×C3—221no—
107C2×C3—27no—
108C2×C3—38no—
109C2×C3—324no—
110C2×C3C22×D3×C9121yesdirect product factor
111C2×C3C23×C3×C917yescomplemented
112C2×C3C22×C32×D317yes—
113D3C23×C3×C918yesdirect product factor
114C22—328no—
115C22C2×C3×D3×C917yesdirect product factor
116C3—21no—
117C3—23no—
118C3C23×D3×C913yesdirect product factor
119C3C24×C3×C911yescharacteristic, complemented
120C3C23×C32×D311yescharacteristic
121C2—38no—
122C2C22×C3×D3×C917yesdirect product factor
subgroupG/H
center, hypercenterC23×C3×C9D3
commutator, hypocenterC3C24×C3×C9
FrattiniC3C23×C32×D3
FittingC23×C32×C9C2
socleC23×C33C2×C3
prime radicalC23×C32×D3C3
cosocleC32C24×C32
LCSG ▷ C3
UCSC1 ◁ C23×C3×C9

Complements

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

NH ≅ G/Ntype
1. C23×C32×C9C2semidirect
2. C22×C3×D3×C9C2direct (N × H)
5. C23×D3×C9C3direct (N × H)
6. C22×C32×C9C22semidirect
7. C2×C3×D3×C9C22direct (N × H)
13. C23×C3×C9D3direct (N × H)
14. C23×C3×C9C2×C3semidirect
15. C22×D3×C9C2×C3direct (N × H)
16. C2×C32×C9C23semidirect
17. C3×D3×C9C23direct (N × H)
19. C23×C3×D3C9direct (N × H)
27. C22×C3×C9C2×D3direct (N × H)
28. C22×C3×C9C22×C3semidirect
29. C2×D3×C9C22×C3direct (N × H)
30. C32×C9C24semidirect
34. C23×C32C2×C9semidirect
37. C22×C3×D3C2×C9direct (N × H)
41. C23×C9C3×D3direct (N × H)
47. C2×C3×C9C22×D3direct (N × H)
48. C2×C3×C9C23×C3semidirect
49. D3×C9C23×C3direct (N × H)
52. C23×D3C3×C9direct (N × H)
56. C22×C32C22×C9semidirect
59. C2×C3×D3C22×C9direct (N × H)
63. C22×C9C2×C3×D3direct (N × H)
67. C3×C9C23×D3direct (N × H)
68. C3×C9C24×C3semidirect
73. C23×C3D3×C9direct (N × H)
74. C23×C3C2×C3×C9semidirect
76. C22×D3C2×C3×C9direct (N × H)
80. C2×C32C23×C9semidirect
83. C3×D3C23×C9direct (N × H)
87. C2×C9C22×C3×D3direct (N × H)
93. C22×C3C2×D3×C9direct (N × H)
94. C22×C3C22×C3×C9semidirect
96. C2×D3C22×C3×C9direct (N × H)
99. C32C24×C9semidirect
103. C9C23×C3×D3direct (N × H)
105. C23C3×D3×C9direct (N × H)
110. C2×C3C22×D3×C9direct (N × H)
111. C2×C3C23×C3×C9semidirect
113. D3C23×C3×C9direct (N × H)
115. C22C2×C3×D3×C9direct (N × H)
118. C3C23×D3×C9direct (N × H)
119. C3C24×C3×C9semidirect
122. C2C22×C3×D3×C9direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C32×C92
C22×C3×D3×C92
C23×C32×D33
C24×C3×C93
C23×D3×C93