Finite Group Databasebeta

← C22×C9×#67 #11722646

Subgroups — C22×C9×#67 #11722646

Order 864.

842 conjugacy classes of subgroups; only the 315 normal subgroups are shown.

normal subgroups only

This group has 842 conjugacy classes of subgroups — too many to draw. Only its 315 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
1C24×C3×C9C211yesmaximal, characteristic, complemented
2C23×D3×C9C211yesmaximal, characteristic, complemented
3C2×C9×#67C2112yesmaximal, direct product factor
4C22×C9×Dic3C211yesmaximal, characteristic, complemented
5C22×C3×#67C311yesmaximal, characteristic
6C22×D4×C9—31nomaximal
7C23×C3×C9—24no—
8C23×C3×C9C2211yescharacteristic
9C23×C3×C9C2216yescomplemented
10C22×D3×C9—24no—
11C22×D3×C9C2216yescomplemented
12C9×#67C22116yesdirect product factor
13C2×C9×Dic3C2216yescomplemented
14C24×C32C2×C311yescharacteristic
15C23×C3×D3C2×C311yescharacteristic
16C2×C3×#67C2×C3112yes—
17C22×C3×Dic3C2×C311yescharacteristic
18C24×C9—21no—
19C24×C9—31no—
20C24×C9D311yescharacteristic, complemented
21C2×D4×C9—312no—
22C22×C4×C9—31no—
23C22×C3×C9—212no—
24C22×C3×C9D414yescomplemented
25C22×C3×C9C2313yes—
26C22×C3×C9C2314yescomplemented
27C2×D3×C9—212no—
28C2×D3×C9C2314yescomplemented
29C9×Dic3C2314yescomplemented
30C22×C3×D4—31no—
31C22×#67C911yescharacteristic, direct product factor
32C23×C32—24no—
33C23×C32C22×C311yescharacteristic
34C23×C32C22×C316yes—
35C22×C3×D3—24no—
36C22×C3×D3C22×C316yes—
37C3×#67C22×C3116yes—
38C2×C3×Dic3C22×C316yes—
39C23×C9—28no—
40C23×C9—24no—
41C23×C9—21no—
42C23×C9—26no—
43C23×C9—36no—
44C23×C9—64no—
45C23×C9C2×D311yescharacteristic
46C23×C9C2×D316yescomplemented
47D4×C9—316no—
48C2×C4×C9—36no—
49C2×C3×C9—24no—
50C2×C3×C9C2×D416yescomplemented
51C2×C3×C9C2411yescharacteristic
52D3×C9—24no—
53C24×C3—21no—
54C24×C3—31no—
55C24×C3C2×C911yescharacteristic, complemented
56C24×C3C3×D311yescharacteristic
57C23×D3C2×C911yescharacteristic, complemented
58C2×C3×D4—312no—
59C22×C3×C4—31no—
60C2×#67C2×C9112yesdirect product factor
61C22×Dic3C2×C911yescharacteristic, complemented
62C22×C32—212no—
63C22×C32C3×D414yes—
64C22×C32C23×C313yes—
65C22×C32C23×C314yes—
66C2×C3×D3—212no—
67C2×C3×D3C23×C314yes—
68C3×Dic3C23×C314yes—
69C22×C9—224no—
70C22×C9—212no—
71C22×C9—24no—
72C22×C9—23no—
73C22×C9—24no—
74C22×C9—34no—
75C22×C9—612no—
76C22×C9#6714yesdirect product factor
77C22×C9C22×D313yes—
78C22×C9C22×D314yescomplemented
79C4×C9—34no—
80C22×D4—31noSylow
81C3×C9C22×D411yescharacteristic, complemented, Sylow
82C23×C3—28no—
83C23×C3—24no—
84C23×C3—24no—
85C23×C3—21no—
86C23×C3—26no—
87C23×C3—36no—
88C23×C3—64no—
89C23×C3C22×C911yescharacteristic
90C23×C3C22×C916yescomplemented
91C23×C3C2×C3×D311yescharacteristic
92C23×C3C2×C3×D316yes—
93C22×D3—24no—
94C22×D3C22×C916yescomplemented
95C3×D4—316no—
96C2×C3×C4—36no—
97#67C22×C9116yesdirect product factor
98C2×Dic3C22×C916yescomplemented
99C2×C32—24no—
100C2×C32C2×C3×D416yes—
101C2×C32C24×C311yescharacteristic
102C3×D3—24no—
103C2×C9—28no—
104C2×C9—24no—
105C2×C9—21no—
106C2×C9—26no—
107C2×C9—64no—
108C2×C9C2×#6716yesdirect product factor
109C2×C9C23×D311yescharacteristic
110C24—31no—
111C24D3×C911yescharacteristic, complemented
112C2×D4—312no—
113C22×C4—31no—
114C22×C3—224no—
115C22×C3—212no—
116C22×C3—212no—
117C22×C3—24no—
118C22×C3—23no—
119C22×C3—24no—
120C22×C3—34no—
121C22×C3—612no—
122C22×C3D4×C914yescomplemented
123C22×C3C23×C913yes—
124C22×C3C23×C914yescomplemented
125C22×C3C3×#6714yes—
126C22×C3C22×C3×D313yes—
127C22×C3C22×C3×D314yes—
128C2×D3—212no—
129C2×D3C23×C914yescomplemented
130C3×C4—34no—
131Dic3C23×C914yescomplemented
132C32C22×C3×D411yescharacteristic
133C9—21no—
134C9C22×#6711yescharacteristic, direct product factor
135C23—24no—
136C23—36no—
137C23—64no—
138C23C2×D3×C911yescharacteristic
139C23C2×D3×C916yescomplemented
140D4—316no—
141C2×C4—36no—
142C2×C3—28no—
143C2×C3—24no—
144C2×C3—24no—
145C2×C3—21no—
146C2×C3—26no—
147C2×C3—64no—
148C2×C3C2×D4×C916yescomplemented
149C2×C3C24×C911yescharacteristic
150C2×C3C2×C3×#6716yes—
151C2×C3C23×C3×D311yescharacteristic
152D3—24no—
153C22—212no—
154C22—34no—
155C22—612no—
156C22C9×#6714yesdirect product factor
157C22C22×D3×C913yes—
158C22C22×D3×C914yescomplemented
159C4—34no—
160C3—21no—
161C3C22×D4×C911yescharacteristic, complemented
162C3C22×C3×#6711yescharacteristic
163C2—24no—
164C2—64no—
165C2C2×C9×#6716yesdirect product factor
166C2C23×D3×C911yescharacteristic
subgroupG/H
centerC23×C9C2×D3
commutatorC2×C3C24×C9
FrattiniC2×C3C23×C3×D3
FittingC24×C3×C9C2
socleC23×C32C22×C3
prime radicalC22×C3×#67C3
cosocleC2×C32C24×C3
hypercenterC24×C9D3
hypocenterC3C22×D4×C9
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C23×C9 ◁ C24×C9

Complements

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

NH ≅ G/Ntype
1. C24×C3×C9C2semidirect
2. C23×D3×C9C2semidirect
3. C2×C9×#67C2direct (N × H)
4. C22×C9×Dic3C2semidirect
9. C23×C3×C9C22semidirect
11. C22×D3×C9C22semidirect
12. C9×#67C22direct (N × H)
13. C2×C9×Dic3C22semidirect
20. C24×C9D3semidirect
24. C22×C3×C9D4semidirect
26. C22×C3×C9C23semidirect
28. C2×D3×C9C23semidirect
29. C9×Dic3C23semidirect
31. C22×#67C9direct (N × H)
46. C23×C9C2×D3semidirect
50. C2×C3×C9C2×D4semidirect
55. C24×C3C2×C9semidirect
57. C23×D3C2×C9semidirect
60. C2×#67C2×C9direct (N × H)
61. C22×Dic3C2×C9semidirect
76. C22×C9#67direct (N × H)
78. C22×C9C22×D3semidirect
81. C3×C9C22×D4semidirect
90. C23×C3C22×C9semidirect
94. C22×D3C22×C9semidirect
97. #67C22×C9direct (N × H)
98. C2×Dic3C22×C9semidirect
108. C2×C9C2×#67direct (N × H)
111. C24D3×C9semidirect
122. C22×C3D4×C9semidirect
124. C22×C3C23×C9semidirect
129. C2×D3C23×C9semidirect
131. Dic3C23×C9semidirect
134. C9C22×#67direct (N × H)
139. C23C2×D3×C9semidirect
148. C2×C3C2×D4×C9semidirect
156. C22C9×#67direct (N × H)
158. C22C22×D3×C9semidirect
161. C3C22×D4×C9semidirect
165. C2C2×C9×#67direct (N × H)

Maximal subgroups

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

subgroupindex
C24×C3×C92
C23×D3×C92
C2×C9×#672
C22×C9×Dic32
C22×C3×#673
C22×D4×C93