Finite Group Databasebeta

← C9×#1009 #11722635

Subgroups — C9×#1009 #11722635

Order 864.

538 conjugacy classes of subgroups; only the 255 normal subgroups are shown.

normal subgroups only

This group has 538 conjugacy classes of subgroups — too many to draw. Only its 255 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
1C3×C9×#41C211yesmaximal, characteristic, complemented
2C9×#239C212yesmaximal, complemented
3D3×D4×C9C216yesmaximal, complemented
4C9×#235C213yesmaximal, complemented
5C2×C9×D12C213yesmaximal, complemented
6C3×#1009C311yesmaximal, characteristic
7C9×H(5,𝔽2)—31nomaximal
8C22×D3×C9C2216yescomplemented
9C3×Q8×C9C2211yescharacteristic, complemented
10C3×D4×C9C2213yescomplemented
11C2×C3×C4×C9C2213yescomplemented
12C9×#67C2216yescomplemented
13C9×D12C2213yescomplemented
14C9×D12C2216yescomplemented
15C4×D3×C9C2216yescomplemented
16C9×Dic6C2211yescharacteristic, complemented
17C32×#41C2×C311yescharacteristic
18C3×#239C2×C312yes—
19C3×D3×D4C2×C316yes—
20C3×#235C2×C313yes—
21C2×C3×D12C2×C313yes—
22C9×#41—21no—
23C9×#41—32no—
24C9×#41—33no—
25C9×#41D311yescharacteristic, complemented
26C2×D4×C9—36no—
27C2×D4×C9—33no—
28C22×C3×C9C2313yes—
29C2×D3×C9—46no—
30C2×D3×C9C2316yes—
31C3×C4×C9C2311yescharacteristic
32C3×C4×C9C2313yes—
33C9×Dic3C2312yes—
34C3×H(5,𝔽2)—31no—
35#1009C911yescharacteristic, direct product factor
36C22×C3×D3C22×C316yes—
37C32×Q8C22×C311yescharacteristic
38C32×D4C22×C313yes—
39C2×C32×C4C22×C313yes—
40C3×#67C22×C316yes—
41C3×D12C22×C313yes—
42C3×D12C22×C316yes—
43C3×C4×D3C22×C316yes—
44C3×Dic6C22×C311yescharacteristic
45C23×C9—36no—
46Q8×C9—21no—
47Q8×C9—31no—
48Q8×C9C2×D311yescharacteristic, complemented
49D4×C9—23no—
50D4×C9—36no—
51D4×C9—36no—
52D4×C9—33no—
53D4×C9C2×D313yescomplemented
54C2×C4×C9—23no—
55C2×C4×C9—36no—
56C2×C4×C9C2×D313yescomplemented
57C2×C3×C9—23no—
58C2×C3×C9C2411yescharacteristic
59D3×C9—26no—
60C3×#41—21no—
61C3×#41—32no—
62C3×#41—33no—
63C3×#41C2×C911yescharacteristic, complemented
64C3×#41C3×D311yescharacteristic
65C2×C3×D4—36no—
66C2×C3×D4—33no—
67#239C2×C912yescomplemented
68D3×D4C2×C916yescomplemented
69#235C2×C913yescomplemented
70C2×D12C2×C913yescomplemented
71C22×C32C23×C313yes—
72C2×C3×D3—46no—
73C2×C3×D3C23×C316yes—
74C32×C4C23×C311yescharacteristic
75C32×C4C23×C313yes—
76C3×Dic3C23×C312yes—
77C22×C9—23no—
78C22×C9—36no—
79C22×C9—126no—
80C22×C9C22×D313yes—
81C4×C9—21no—
82C4×C9—23no—
83C4×C9—32no—
84C4×C9C22×D313yes—
85C4×C9C22×D311yescharacteristic
86H(5,𝔽2)—31noSylow
87C3×C9H(5,𝔽2)11yescharacteristic, complemented, Sylow
88C23×C3—36no—
89C22×D3C22×C916yescomplemented
90C3×Q8—21no—
91C3×Q8—31no—
92C3×Q8C22×C911yescharacteristic, complemented
93C3×Q8C2×C3×D311yescharacteristic
94C3×D4—23no—
95C3×D4—36no—
96C3×D4—36no—
97C3×D4—33no—
98C3×D4C22×C913yescomplemented
99C3×D4C2×C3×D313yes—
100C2×C3×C4—23no—
101C2×C3×C4—36no—
102C2×C3×C4C22×C913yescomplemented
103C2×C3×C4C2×C3×D313yes—
104#67C22×C916yescomplemented
105D12C22×C913yescomplemented
106D12C22×C916yescomplemented
107C4×D3C22×C916yescomplemented
108Dic6C22×C911yescharacteristic, complemented
109C2×C32—23no—
110C2×C32C24×C311yescharacteristic
111C3×D3—26no—
112C2×C9—23no—
113C2×C9—21no—
114C2×C9—43no—
115C2×C9—66no—
116C2×C9C23×D311yescharacteristic
117#41—32no—
118#41—33no—
119#41D3×C911yescharacteristic, complemented
120C2×D4—36no—
121C2×D4—33no—
122C22×C3—23no—
123C22×C3—36no—
124C22×C3—126no—
125C22×C3C23×C913yes—
126C22×C3C22×C3×D313yes—
127C2×D3—46no—
128C2×D3C23×C916yes—
129C3×C4—21no—
130C3×C4—23no—
131C3×C4—32no—
132C3×C4C23×C911yescharacteristic
133C3×C4C23×C913yes—
134C3×C4C22×C3×D313yes—
135C3×C4C22×C3×D311yescharacteristic
136Dic3C23×C912yes—
137C32C3×H(5,𝔽2)11yescharacteristic
138C9—21no—
139C9#100911yescharacteristic, direct product factor
140C23—36no—
141Q8—31no—
142Q8C2×D3×C911yescharacteristic, complemented
143D4—36no—
144D4—36no—
145D4—33no—
146D4C2×D3×C913yescomplemented
147C2×C4—36no—
148C2×C4C2×D3×C913yescomplemented
149C2×C3—23no—
150C2×C3—23no—
151C2×C3—21no—
152C2×C3—43no—
153C2×C3—66no—
154C2×C3C24×C911yescharacteristic
155C2×C3C23×C3×D311yescharacteristic
156D3—26no—
157C22—36no—
158C22—126no—
159C22C22×D3×C913yes—
160C4—32no—
161C4C22×D3×C913yes—
162C4C22×D3×C911yescharacteristic
163C3—21no—
164C3C9×H(5,𝔽2)11yescharacteristic, complemented
165C3C3×#100911yescharacteristic
166C2—23no—
167C2—66no—
168C2C23×D3×C911yescharacteristic
subgroupG/H
centerC2×C9C23×D3
commutatorC2×C3C24×C9
FrattiniC2×C3C23×C3×D3
FittingC3×C9×#41C2
socle, cosocleC2×C32C24×C3
prime radicalC3×#1009C3
hypercenterC9×#41D3
hypocenterC3C9×H(5,𝔽2)
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C9 ◁ C9×#41

Complements

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

NH ≅ G/Ntype
1. C3×C9×#41C2semidirect
2. C9×#239C2semidirect
3. D3×D4×C9C2semidirect
4. C9×#235C2semidirect
5. C2×C9×D12C2semidirect
8. C22×D3×C9C22semidirect
9. C3×Q8×C9C22semidirect
10. C3×D4×C9C22semidirect
11. C2×C3×C4×C9C22semidirect
12. C9×#67C22semidirect
13. C9×D12C22semidirect
14. C9×D12C22semidirect
15. C4×D3×C9C22semidirect
16. C9×Dic6C22semidirect
25. C9×#41D3semidirect
35. #1009C9direct (N × H)
48. Q8×C9C2×D3semidirect
53. D4×C9C2×D3semidirect
56. C2×C4×C9C2×D3semidirect
63. C3×#41C2×C9semidirect
67. #239C2×C9semidirect
68. D3×D4C2×C9semidirect
69. #235C2×C9semidirect
70. C2×D12C2×C9semidirect
87. C3×C9H(5,𝔽2)semidirect
89. C22×D3C22×C9semidirect
92. C3×Q8C22×C9semidirect
98. C3×D4C22×C9semidirect
102. C2×C3×C4C22×C9semidirect
104. #67C22×C9semidirect
105. D12C22×C9semidirect
106. D12C22×C9semidirect
107. C4×D3C22×C9semidirect
108. Dic6C22×C9semidirect
119. #41D3×C9semidirect
139. C9#1009direct (N × H)
142. Q8C2×D3×C9semidirect
146. D4C2×D3×C9semidirect
148. C2×C4C2×D3×C9semidirect
164. C3C9×H(5,𝔽2)semidirect

Maximal subgroups

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

subgroupindex
C3×C9×#412
C9×#2392
D3×D4×C92
C9×#2352
C2×C9×D122
C3×#10093
C9×H(5,𝔽2)3