Finite Group Databasebeta

← #37554820

Subgroups — #37554820

Order 23488.

428 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 115 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
1C22×D1468C211yesmaximal, characteristic, complemented
2C2×D4×D367C212yesmaximal, complemented
3C2×D4×D367C211yesmaximal, characteristic, complemented
4#26044257C211yesmaximal, characteristic, complemented
5#26044236C211yesmaximal, characteristic, complemented
6#26044234C212yesmaximal, complemented
7#26044231C212yesmaximal, complemented
8#26044229C212yesmaximal, complemented
9#26044223C211yesmaximal, characteristic, complemented
10C4×C22⋊D367C211yesmaximal, characteristic, complemented
11#121×C367C211yesmaximal, characteristic, complemented
12C23×D367C2212yescomplemented
13C23×D367C2212yescomplemented
14C2×D1468—24no—
15C2×D1468C2211yescharacteristic
16C2×D1468C2212yescomplemented
17C2×D1468C2212yescomplemented
18C2×D1468C2211yescharacteristic, complemented
19C2×C22⋊D367C2211yescharacteristic
20C2×C22⋊D367C2212yescomplemented
21C2×C22⋊D367C2212yescomplemented
22C2×C22⋊D367C2212yescomplemented
23D4×D367—24no—
24D4×D367—28no—
25C2×C4×D367C2211yescharacteristic, complemented
26C2×C4×D367C2212yescomplemented
27Dic367⋊C4C2211yescharacteristic, complemented
28C22⋊Dic367C2211yescharacteristic, complemented
29(C2×D367)⋊C4C2211yescharacteristic, complemented
30(C2×D367)⋊C4C2212yescomplemented
31(C2×D367)⋊C4C2212yescomplemented
32C4×Dic367C2211yescharacteristic, complemented
33C2×D4×C367C2212yescomplemented
34C2×D4×C367C2211yescharacteristic, complemented
35C22×C4×C367C2211yescharacteristic, complemented
36C4⋊C4×C367C2211yescharacteristic, complemented
37C22⋊C4×C367C2212yescomplemented
38C22×D367—24no—
39C22×D367—24no—
40C22×D367—24no—
41C22×D367—24no—
42C22×D367—42no—
43C22×D367—42no—
44C22×D367C2312yes—
45C22×D367C2311yescharacteristic
46C22×D367C2312yes—
47D1468—24no—
48D1468—24no—
49D1468—22no—
50D1468D414yescomplemented
51C22⋊D367—24no—
52C22⋊D367—24no—
53C22⋊D367—42no—
54C22⋊D367D414yescomplemented
55C2×Dic367C2311yescharacteristic
56C2×Dic367C2311yescharacteristic
57C2×Dic367C2311yescharacteristic
58C4×D367—22no—
59C4×D367—24no—
60C23×C367C2312yes—
61C23×C367C2311yescharacteristic
62D4×C367—24no—
63D4×C367—22no—
64C2×C4×C367—22no—
65C2×C4×C367C2311yescharacteristic
66C2×C4×C367C2312yes—
67C2×C4×C367C2311yescharacteristic
68C22×C367—21no—
69C22×C367—21no—
70C22×C367—22no—
71C22×C367—22no—
72C22×C367—22no—
73C22×C367C2×D412yescomplemented
74C22×C367C2411yescharacteristic
75C2×D367—21no—
76C2×D367—22no—
77C2×D367—21no—
78C2×D367—22no—
79C2×D367—22no—
80C2×D367—22no—
81C2×D367—22no—
82C2×D367—48no—
83C2×D367—48no—
84C2×D367C2×D414yescomplemented
85C2×D367C2×D412yescomplemented
86C4×C367—21no—
87C4×C367—22no—
88C4×C367C2×D412yescomplemented
89Dic367—21no—
90Dic367—21no—
91Dic367C2×D412yescomplemented
92C2×C367—22no—
93C2×C367—42no—
94C2×C367C22×D411yescharacteristic
95C2×C367C22×D411yescharacteristic
96C2×C367H(5,𝔽2)11yescharacteristic
97D367—22no—
98D367—24no—
99D367—42no—
100C367D4211yescharacteristic, complemented, Sylow
101D42—3671nomaximal, Sylow
102C22×D4—3672no—
103C22×D4—3671no—
104C22×D4—3671no—
105#127—3671no—
106#121—3671no—
107#121—3672no—
108#121D36711yescharacteristic, complemented
109#120—3672no—
110#120—3672no—
111C4×D4—3671no—
112C4×D4—3671no—
113C24—3672no—
114C24—3672no—
115C2×D4—3671no—
116C2×D4—3672no—
117C2×D4—3672no—
118C2×D4—3672no—
119C2×D4—3671no—
120C2×D4—3672no—
121C2×D4—3672no—
122C2×D4—3671no—
123C2×D4—7344no—
124C2×D4—7348no—
125C2×D4—7344no—
126C2×D4C2×D36712yescomplemented
127C2×D4C2×D36711yescharacteristic, complemented
128C22×C4—3672no—
129C22×C4—3671no—
130C22×C4C2×D36711yescharacteristic, complemented
131C4⋊C4—3671no—
132C4⋊C4C2×D36711yescharacteristic, complemented
133C22⋊C4—3671no—
134C22⋊C4—3672no—
135C22⋊C4—3671no—
136C22⋊C4—3672no—
137C22⋊C4C2×D36712yescomplemented
138C42—3671no—
139C23—3671no—
140C23—3672no—
141C23—3672no—
142C23—14682no—
143C23—14682no—
144C23—7344no—
145C23—7344no—
146C23—7344no—
147C23—7344no—
148C23C22×D36712yes—
149C23C22×D36711yescharacteristic
150D4—24no—
151D4—22no—
152D4—3674no—
153D4—3674no—
154D4—14682no—
155D4—7344no—
156D4—7344no—
157D4—7344no—
158D4—7344no—
159D4—7342no—
160C2×C4—22no—
161C2×C4—3671no—
162C2×C4—3671no—
163C2×C4—3671no—
164C2×C4—7344no—
165C2×C4—7342no—
166C2×C4C22×D36712yes—
167C2×C4C22×D36711yescharacteristic
168C2×C4C22×D36711yescharacteristic
169C22—21no—
170C22—21no—
171C22—22no—
172C22—22no—
173C22—22no—
174C22—3672no—
175C22—3674no—
176C22—14688no—
177C22—14688no—
178C22—7342no—
179C22—7342no—
180C22—7342no—
181C22—7342no—
182C22—7341no—
183C22—7342no—
184C22—7341no—
185C22D4×D36712yescomplemented
186C22C23×D36711yescharacteristic
187C4—21no—
188C4—22no—
189C4—3672no—
190C4—7341no—
191C4—7341no—
192C4D4×D36712yescomplemented
193C2—22no—
194C2—42no—
195C2—14682no—
196C2—7344no—
197C2—7342no—
198C2C2×D4×D36711yescharacteristic
199C2C2×D4×D36711yescharacteristic
200C2#2604430711yescharacteristic
subgroupG/H
center, FrattiniC22C23×D367
commutator, socle, cosocleC22×C367C24
Fitting#121×C367C2
hypercenter#121D367
hypocenterC367D42
LCSG ▷ C22×C367 ▷ C367
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C22×D1468C2semidirect
2. C2×D4×D367C2semidirect
3. C2×D4×D367C2semidirect
4. #26044257C2semidirect
5. #26044236C2semidirect
6. #26044234C2semidirect
7. #26044231C2semidirect
8. #26044229C2semidirect
9. #26044223C2semidirect
10. C4×C22⋊D367C2semidirect
11. #121×C367C2semidirect
12. C23×D367C22semidirect
13. C23×D367C22semidirect
16. C2×D1468C22semidirect
17. C2×D1468C22semidirect
18. C2×D1468C22semidirect
20. C2×C22⋊D367C22semidirect
21. C2×C22⋊D367C22semidirect
22. C2×C22⋊D367C22semidirect
25. C2×C4×D367C22semidirect
26. C2×C4×D367C22semidirect
27. Dic367⋊C4C22semidirect
28. C22⋊Dic367C22semidirect
29. (C2×D367)⋊C4C22semidirect
30. (C2×D367)⋊C4C22semidirect
31. (C2×D367)⋊C4C22semidirect
32. C4×Dic367C22semidirect
33. C2×D4×C367C22semidirect
34. C2×D4×C367C22semidirect
35. C22×C4×C367C22semidirect
36. C4⋊C4×C367C22semidirect
37. C22⋊C4×C367C22semidirect
50. D1468D4semidirect
54. C22⋊D367D4semidirect
73. C22×C367C2×D4semidirect
84. C2×D367C2×D4semidirect
85. C2×D367C2×D4semidirect
88. C4×C367C2×D4semidirect
91. Dic367C2×D4semidirect
100. C367D42semidirect
108. #121D367semidirect
126. C2×D4C2×D367semidirect
127. C2×D4C2×D367semidirect
130. C22×C4C2×D367semidirect
132. C4⋊C4C2×D367semidirect
137. C22⋊C4C2×D367semidirect
185. C22D4×D367semidirect
192. C4D4×D367semidirect

Maximal subgroups

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

subgroupindex
C22×D14682
C2×D4×D3672
#260442572
#260442362
#260442342
#260442312
#260442292
#260442232
C4×C22⋊D3672
#121×C3672
D42367