Finite Group Databasebeta

← C22×C3×#611 #11722645

Subgroups — C22×C3×#611 #11722645

Order 864.

842 conjugacy classes of subgroups; only the 264 normal subgroups are shown.

normal subgroups only

This group has 842 conjugacy classes of subgroups — too many to draw. Only its 264 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×C3×D9C211yesmaximal, characteristic, complemented
3C2×C3×#611C2112yesmaximal, direct product factor
4C22×C3×Dic9C211yesmaximal, characteristic, complemented
5C22×C3×#67—31nomaximal
6C22×#611C311yesmaximal, characteristic, direct product factor
7C23×C3×C9—24no—
8C23×C3×C9C2211yescharacteristic
9C23×C3×C9C2216yescomplemented
10C22×C3×D9—24no—
11C22×C3×D9C2216yescomplemented
12C3×#611C22116yesdirect product factor
13C2×C3×Dic9C2216yescomplemented
14C24×C32D311yescharacteristic
15C23×C3×D3—31no—
16C2×C3×#67—312no—
17C22×C3×Dic3—31no—
18C24×C9—21no—
19C24×C9C2×C311yescharacteristic, complemented
20C23×D9C2×C311yescharacteristic, complemented
21C2×#611C2×C3112yesdirect product factor
22C22×Dic9C2×C311yescharacteristic, complemented
23C22×C3×C9—212no—
24C22×C3×C9D414yescomplemented
25C22×C3×C9C2313yes—
26C22×C3×C9C2314yescomplemented
27C2×C3×D9—212no—
28C2×C3×D9C2314yescomplemented
29C3×Dic9C2314yescomplemented
30C22×C3×D4—91no—
31C22×#67—31no—
32C23×C32—24no—
33C23×C32C2×D311yescharacteristic
34C23×C32C2×D316yes—
35C22×C3×D3—36no—
36C22×C3×D3—64no—
37C3×#67—316no—
38C2×C3×Dic3—36no—
39C23×C9—28no—
40C23×C9—24no—
41C23×C9—21no—
42C23×C9—26no—
43C23×C9C22×C311yescharacteristic
44C23×C9C22×C316yescomplemented
45C22×D9—24no—
46C22×D9C22×C316yescomplemented
47#611C22×C3116yesdirect product factor
48C2×Dic9C22×C316yescomplemented
49C2×C3×C9—24no—
50C2×C3×C9C2×D416yescomplemented
51C2×C3×C9C2411yescharacteristic
52C3×D9—24no—
53C24×C3—21no—
54C24×C3—91no—
55C24×C3D911yescharacteristic, complemented
56C24×C3C3×D311yescharacteristic
57C23×D3—31no—
58C2×C3×D4—912no—
59C22×C3×C4—91no—
60C2×#67—312no—
61C22×Dic3—31no—
62C22×C32—212no—
63C22×C32#6714yes—
64C22×C32C22×D313yes—
65C22×C32C22×D314yes—
66C2×C3×D3—34no—
67C2×C3×D3—612no—
68C3×Dic3—34no—
69C22×C9—224no—
70C22×C9—212no—
71C22×C9—24no—
72C22×C9—23no—
73C22×C9—24no—
74C22×C9C3×D414yescomplemented
75C22×C9C23×C313yes—
76C22×C9C23×C314yescomplemented
77C2×D9—212no—
78C2×D9C23×C314yescomplemented
79Dic9C23×C314yescomplemented
80C22×D4—91noSylow
81C3×C9C22×D411yescharacteristic, complemented, Sylow
82C23×C3—28no—
83C23×C3—24no—
84C23×C3—24no—
85C23×C3—21no—
86C23×C3—26no—
87C23×C3—96no—
88C23×C3—184no—
89C23×C3C2×D911yescharacteristic
90C23×C3C2×D916yescomplemented
91C23×C3C2×C3×D311yescharacteristic
92C23×C3C2×C3×D316yes—
93C22×D3—36no—
94C22×D3—64no—
95C3×D4—916no—
96C2×C3×C4—96no—
97#67—316no—
98C2×Dic3—36no—
99C2×C32—24no—
100C2×C32C2×#6716yes—
101C2×C32C23×D311yescharacteristic
102C3×D3—64no—
103C2×C9—28no—
104C2×C9—24no—
105C2×C9—21no—
106C2×C9—26no—
107C2×C9C2×C3×D416yescomplemented
108C2×C9C24×C311yescharacteristic
109D9—24no—
110C24—91no—
111C24C3×D911yescharacteristic, complemented
112C2×D4—912no—
113C22×C4—91no—
114C22×C3—224no—
115C22×C3—212no—
116C22×C3—212no—
117C22×C3—24no—
118C22×C3—23no—
119C22×C3—24no—
120C22×C3—94no—
121C22×C3—1812no—
122C22×C3#61114yesdirect product factor
123C22×C3C22×D913yes—
124C22×C3C22×D914yescomplemented
125C22×C3C3×#6714yes—
126C22×C3C22×C3×D313yes—
127C22×C3C22×C3×D314yes—
128C2×D3—34no—
129C2×D3—612no—
130C3×C4—94no—
131Dic3—34no—
132C32C22×#6711yescharacteristic
133C9—21no—
134C9C22×C3×D411yescharacteristic, complemented
135C23—24no—
136C23—96no—
137C23—184no—
138C23C2×C3×D911yescharacteristic
139C23C2×C3×D916yescomplemented
140D4—916no—
141C2×C4—96no—
142C2×C3—28no—
143C2×C3—24no—
144C2×C3—24no—
145C2×C3—21no—
146C2×C3—26no—
147C2×C3—184no—
148C2×C3C2×#61116yesdirect product factor
149C2×C3C23×D911yescharacteristic
150C2×C3C2×C3×#6716yes—
151C2×C3C23×C3×D311yescharacteristic
152D3—64no—
153C22—212no—
154C22—94no—
155C22—1812no—
156C22C3×#61114yesdirect product factor
157C22C22×C3×D913yes—
158C22C22×C3×D914yescomplemented
159C4—94no—
160C3—21no—
161C3C22×#61111yescharacteristic, direct product factor
162C3C22×C3×#6711yescharacteristic
163C2—24no—
164C2—184no—
165C2C2×C3×#61116yesdirect product factor
166C2C23×C3×D911yescharacteristic
subgroupG/H
centerC23×C3C2×D9
commutator, cosocleC2×C9C24×C3
FrattiniC2×C3C23×C3×D3
FittingC24×C3×C9C2
socleC23×C32C2×D3
hypercenterC24×C3D9
hypocenterC9C22×C3×D4
LCSG ▷ C2×C9 ▷ C9
UCSC1 ◁ C23×C3 ◁ C24×C3

Complements

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

NH ≅ G/Ntype
1. C24×C3×C9C2semidirect
2. C23×C3×D9C2semidirect
3. C2×C3×#611C2direct (N × H)
4. C22×C3×Dic9C2semidirect
6. C22×#611C3direct (N × H)
9. C23×C3×C9C22semidirect
11. C22×C3×D9C22semidirect
12. C3×#611C22direct (N × H)
13. C2×C3×Dic9C22semidirect
19. C24×C9C2×C3semidirect
20. C23×D9C2×C3semidirect
21. C2×#611C2×C3direct (N × H)
22. C22×Dic9C2×C3semidirect
24. C22×C3×C9D4semidirect
26. C22×C3×C9C23semidirect
28. C2×C3×D9C23semidirect
29. C3×Dic9C23semidirect
44. C23×C9C22×C3semidirect
46. C22×D9C22×C3semidirect
47. #611C22×C3direct (N × H)
48. C2×Dic9C22×C3semidirect
50. C2×C3×C9C2×D4semidirect
55. C24×C3D9semidirect
74. C22×C9C3×D4semidirect
76. C22×C9C23×C3semidirect
78. C2×D9C23×C3semidirect
79. Dic9C23×C3semidirect
81. C3×C9C22×D4semidirect
90. C23×C3C2×D9semidirect
107. C2×C9C2×C3×D4semidirect
111. C24C3×D9semidirect
122. C22×C3#611direct (N × H)
124. C22×C3C22×D9semidirect
134. C9C22×C3×D4semidirect
139. C23C2×C3×D9semidirect
148. C2×C3C2×#611direct (N × H)
156. C22C3×#611direct (N × H)
158. C22C22×C3×D9semidirect
161. C3C22×#611direct (N × H)
165. C2C2×C3×#611direct (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×C3×D92
C2×C3×#6112
C22×C3×Dic92
C22×C3×#673
C22×#6113