Finite Group Databasebeta

← C9×#1004 #11722612

Subgroups — C9×#1004 #11722612

Order 864.

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

normal subgroups only

This group has 552 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
1C2×C3×D4×C9C211yesmaximal, characteristic, complemented
2C2×C9×#67C214yesmaximal, complemented
3C9×#237C214yesmaximal, complemented
4D3×D4×C9C214yesmaximal, complemented
5C9×#235C212yesmaximal, complemented
6C3×#1004C311yesmaximal, characteristic
7C9×H(5,𝔽2)—31nomaximal
8C23×C3×C9C2212yescomplemented
9C22×D3×C9C2214yescomplemented
10C3×D4×C9C2214yescomplemented
11C2×C3×C4×C9C2211yescharacteristic, complemented
12C9×#67C2214yescomplemented
13C9×#67C2218yescomplemented
14C2×C9×Dic3C2214yescomplemented
15C9×D12C2212yescomplemented
16C4×D3×C9C2214yescomplemented
17C9×Dic6C2212yescomplemented
18C2×C32×D4C2×C311yescharacteristic
19C2×C3×#67C2×C314yes—
20C3×#237C2×C314yes—
21C3×D3×D4C2×C314yes—
22C3×#235C2×C312yes—
23C9×#41—34no—
24C9×#41—32no—
25C2×D4×C9—21no—
26C2×D4×C9—34no—
27C2×D4×C9—34no—
28C2×D4×C9D311yescharacteristic, complemented
29C22×C3×C9—42no—
30C22×C3×C9C2314yes—
31C22×C3×C9C2311yescharacteristic
32C2×D3×C9—44no—
33C2×D3×C9C2314yes—
34C3×C4×C9C2312yes—
35C9×Dic3C2314yes—
36C3×H(5,𝔽2)—31no—
37#1004C911yescharacteristic, direct product factor
38C23×C32C22×C312yes—
39C22×C3×D3C22×C314yes—
40C32×D4C22×C314yes—
41C2×C32×C4C22×C311yescharacteristic
42C3×#67C22×C314yes—
43C3×#67C22×C318yes—
44C2×C3×Dic3C22×C314yes—
45C3×D12C22×C312yes—
46C3×C4×D3C22×C314yes—
47C3×Dic6C22×C312yes—
48C23×C9—22no—
49C23×C9—34no—
50C23×C9C2×D312yescomplemented
51Q8×C9—32no—
52D4×C9—24no—
53D4×C9—38no—
54D4×C9—34no—
55D4×C9—32no—
56D4×C9C2×D314yescomplemented
57C2×C4×C9—21no—
58C2×C4×C9—34no—
59C2×C4×C9—34no—
60C2×C4×C9C2×D311yescharacteristic, complemented
61C2×C3×C9—21no—
62C2×C3×C9—24no—
63C2×C3×C9C2411yescharacteristic
64D3×C9—24no—
65C3×#41—34no—
66C3×#41—32no—
67C2×C3×D4—21no—
68C2×C3×D4—34no—
69C2×C3×D4—34no—
70C2×C3×D4C2×C911yescharacteristic, complemented
71C2×C3×D4C3×D311yescharacteristic
72C2×#67C2×C914yescomplemented
73#237C2×C914yescomplemented
74D3×D4C2×C914yescomplemented
75#235C2×C912yescomplemented
76C22×C32—42no—
77C22×C32C23×C314yes—
78C22×C32C23×C311yescharacteristic
79C2×C3×D3—44no—
80C2×C3×D3C23×C314yes—
81C32×C4C23×C312yes—
82C3×Dic3C23×C314yes—
83C22×C9—21no—
84C22×C9—24no—
85C22×C9—34no—
86C22×C9—42no—
87C22×C9—44no—
88C22×C9—124no—
89C22×C9C22×D314yes—
90C22×C9C22×D311yescharacteristic
91C4×C9—22no—
92C4×C9—34no—
93C4×C9C22×D312yes—
94H(5,𝔽2)—31noSylow
95C3×C9H(5,𝔽2)11yescharacteristic, complemented, Sylow
96C23×C3—22no—
97C23×C3—34no—
98C23×C3C22×C912yescomplemented
99C23×C3C2×C3×D312yes—
100C22×D3C22×C914yescomplemented
101C3×Q8—32no—
102C3×D4—24no—
103C3×D4—38no—
104C3×D4—34no—
105C3×D4—32no—
106C3×D4C22×C914yescomplemented
107C3×D4C2×C3×D314yes—
108C2×C3×C4—21no—
109C2×C3×C4—34no—
110C2×C3×C4—34no—
111C2×C3×C4C22×C911yescharacteristic, complemented
112C2×C3×C4C2×C3×D311yescharacteristic
113#67C22×C914yescomplemented
114#67C22×C918yescomplemented
115C2×Dic3C22×C914yescomplemented
116D12C22×C912yescomplemented
117C4×D3C22×C914yescomplemented
118Dic6C22×C912yescomplemented
119C2×C32—21no—
120C2×C32—24no—
121C2×C32C24×C311yescharacteristic
122C3×D3—24no—
123C2×C9—22no—
124C2×C9—21no—
125C2×C9—24no—
126C2×C9—21no—
127C2×C9—44no—
128C2×C9—64no—
129C2×C9C23×D311yescharacteristic
130#41—34no—
131#41—32no—
132C2×D4—34no—
133C2×D4—34no—
134C2×D4D3×C911yescharacteristic, complemented
135C22×C3—21no—
136C22×C3—24no—
137C22×C3—34no—
138C22×C3—42no—
139C22×C3—42no—
140C22×C3—44no—
141C22×C3—124no—
142C22×C3C23×C914yes—
143C22×C3C23×C911yescharacteristic
144C22×C3C22×C3×D314yes—
145C22×C3C22×C3×D311yescharacteristic
146C2×D3—44no—
147C2×D3C23×C914yes—
148C3×C4—22no—
149C3×C4—34no—
150C3×C4C23×C912yes—
151C3×C4C22×C3×D312yes—
152Dic3C23×C914yes—
153C32C3×H(5,𝔽2)11yescharacteristic
154C9—21no—
155C9#100411yescharacteristic, direct product factor
156C23—34no—
157C23C2×D3×C912yescomplemented
158Q8—32no—
159D4—38no—
160D4—34no—
161D4—32no—
162D4C2×D3×C914yescomplemented
163C2×C4—34no—
164C2×C4—34no—
165C2×C4C2×D3×C911yescharacteristic, complemented
166C2×C3—22no—
167C2×C3—21no—
168C2×C3—24no—
169C2×C3—24no—
170C2×C3—21no—
171C2×C3—21no—
172C2×C3—44no—
173C2×C3—64no—
174C2×C3C24×C911yescharacteristic
175C2×C3C23×C3×D311yescharacteristic
176D3—24no—
177C22—34no—
178C22—42no—
179C22—124no—
180C22C22×D3×C914yes—
181C22C22×D3×C911yescharacteristic
182C4—34no—
183C4C22×D3×C912yes—
184C3—21no—
185C3C9×H(5,𝔽2)11yescharacteristic, complemented
186C3C3×#100411yescharacteristic
187C2—21no—
188C2—24no—
189C2—64no—
190C2C23×D3×C911yescharacteristic
subgroupG/H
centerC2×C9C23×D3
commutatorC2×C3C24×C9
FrattiniC2×C3C23×C3×D3
FittingC2×C3×D4×C9C2
socle, cosocleC2×C32C24×C3
prime radicalC3×#1004C3
hypercenterC2×D4×C9D3
hypocenterC3C9×H(5,𝔽2)
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C9 ◁ C2×D4×C9

Complements

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

NH ≅ G/Ntype
1. C2×C3×D4×C9C2semidirect
2. C2×C9×#67C2semidirect
3. C9×#237C2semidirect
4. D3×D4×C9C2semidirect
5. C9×#235C2semidirect
8. C23×C3×C9C22semidirect
9. C22×D3×C9C22semidirect
10. C3×D4×C9C22semidirect
11. C2×C3×C4×C9C22semidirect
12. C9×#67C22semidirect
13. C9×#67C22semidirect
14. C2×C9×Dic3C22semidirect
15. C9×D12C22semidirect
16. C4×D3×C9C22semidirect
17. C9×Dic6C22semidirect
28. C2×D4×C9D3semidirect
37. #1004C9direct (N × H)
50. C23×C9C2×D3semidirect
56. D4×C9C2×D3semidirect
60. C2×C4×C9C2×D3semidirect
70. C2×C3×D4C2×C9semidirect
72. C2×#67C2×C9semidirect
73. #237C2×C9semidirect
74. D3×D4C2×C9semidirect
75. #235C2×C9semidirect
95. C3×C9H(5,𝔽2)semidirect
98. C23×C3C22×C9semidirect
100. C22×D3C22×C9semidirect
106. C3×D4C22×C9semidirect
111. C2×C3×C4C22×C9semidirect
113. #67C22×C9semidirect
114. #67C22×C9semidirect
115. C2×Dic3C22×C9semidirect
116. D12C22×C9semidirect
117. C4×D3C22×C9semidirect
118. Dic6C22×C9semidirect
134. C2×D4D3×C9semidirect
155. C9#1004direct (N × H)
157. C23C2×D3×C9semidirect
162. D4C2×D3×C9semidirect
165. C2×C4C2×D3×C9semidirect
185. 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
C2×C3×D4×C92
C2×C9×#672
C9×#2372
D3×D4×C92
C9×#2352
C3×#10043
C9×H(5,𝔽2)3