Finite Group Databasebeta

← #14580896

Subgroups — #14580896

Order 2624.

346 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 103 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×C22⋊D41C211yesmaximal, characteristic, complemented
2C2×D4×D41C211yesmaximal, characteristic, complemented
3#13050148C211yesmaximal, characteristic, complemented
4#13050146C212yesmaximal, complemented
5#13050145C211yesmaximal, characteristic, complemented
6#13050137C212yesmaximal, complemented
7#13050135C212yesmaximal, complemented
8#13050133C211yesmaximal, characteristic, complemented
9#13050132C212yesmaximal, complemented
10C22⋊C4×D41C211yesmaximal, characteristic, complemented
11#120×C41C211yesmaximal, characteristic, complemented
12C24×C41C2211yescharacteristic, complemented
13C23×D41C2211yescharacteristic
14C23×D41C2211yescharacteristic, complemented
15C2×D4×C41C2211yescharacteristic
16C2×D4×C41C2212yescomplemented
17C2×C22⋊D41—24no—
18C2×C22⋊D41C2212yes—
19C2×C22⋊D41C2211yescharacteristic
20C2×C22⋊D41C2211yescharacteristic, complemented
21C2×C22⋊D41C2212yescomplemented
22C2×C22⋊D41C2212yescomplemented
23C22×Dic41C2211yescharacteristic, complemented
24D4×D41—24no—
25C2×D164C2211yescharacteristic, complemented
26C2×C4×D41C2211yescharacteristic, complemented
27C2×C4×D41C2212yescomplemented
28C22⋊C4×C41C2211yescharacteristic, complemented
29C22⋊C4×C41C2212yescomplemented
30C22⋊Dic41C2211yescharacteristic, complemented
31C22⋊Dic41C2212yescomplemented
32C22⋊Dic41C2212yescomplemented
33#10620281C2212yescomplemented
34#10620281C2212yescomplemented
35C4⋊Dic41C2212yescomplemented
36#10620279C2212yescomplemented
37C23×C41—22no—
38C23×C41—42no—
39C23×C41C2311yescharacteristic
40C23×C41C2312yes—
41C23×C41C2311yescharacteristic
42C22×D41—22no—
43C22×D41—42no—
44C22×D41—42no—
45C22×D41D414yescomplemented
46C22×D41C2311yescharacteristic
47C22×D41C2311yescharacteristic
48C22×D41C2311yescharacteristic
49D4×C41—22no—
50D4×C41—42no—
51C2×C4×C41C2311yescharacteristic
52C2×C4×C41C2312yes—
53C22⋊D41—22no—
54C22⋊D41—22no—
55C22⋊D41—24no—
56C22⋊D41—24no—
57C22⋊D41—42no—
58C2×Dic41—22no—
59C2×Dic41C2312yes—
60C2×Dic41C2312yes—
61C2×Dic41C2311yescharacteristic
62D164—22no—
63C4×D41—22no—
64C4×D41—42no—
65C22×C41—22no—
66C22×C41—22no—
67C22×C41—21no—
68C22×C41—22no—
69C22×C41—22no—
70C22×C41—22no—
71C22×C41—44no—
72C22×C41C2×D412yes—
73C22×C41C2411yescharacteristic
74C2×D41—22no—
75C2×D41—22no—
76C2×D41—21no—
77C2×D41—22no—
78C2×D41—44no—
79C2×D41—44no—
80C2×D41C2×D412yes—
81C2×D41C2×D412yes—
82C4×C41—21no—
83C4×C41—22no—
84Dic41—21no—
85Dic41—22no—
86Dic41—22no—
87C2×C41—22no—
88C2×C41—41no—
89C2×C41—42no—
90C2×C41C22×D411yescharacteristic
91C2×C41H(5,𝔽2)12yes—
92D41—22no—
93D41—22no—
94D41—41no—
95#534—411nomaximal, Sylow
96C41#53411yescharacteristic, complemented, Sylow
97C22×D4—411no—
98C22×D4—411no—
99#123—411no—
100#123—412no—
101#123—411no—
102#121—412no—
103#121—412no—
104#120—411no—
105#120—412no—
106#120D4111yescharacteristic, complemented
107C2×C22⋊C4—411no—
108C24—411no—
109C24—411no—
110C24C2×D4111yescharacteristic, complemented
111C2×D4—412no—
112C2×D4—411no—
113C2×D4—411no—
114C2×D4—412no—
115C2×D4—412no—
116C2×D4—411no—
117C2×D4—824no—
118C2×D4—824no—
119C2×D4C2×D4111yescharacteristic
120C2×D4C2×D4112yescomplemented
121C22×C4—411no—
122C22×C4—411no—
123C22×C4—412no—
124C4⋊C4—412no—
125C4⋊C4—412no—
126C22⋊C4—411no—
127C22⋊C4—412no—
128C22⋊C4—412no—
129C22⋊C4—412no—
130C22⋊C4—412no—
131C22⋊C4C2×D4111yescharacteristic, complemented
132C22⋊C4C2×D4112yescomplemented
133C23—22no—
134C23—42no—
135C23—411no—
136C23—411no—
137C23—414no—
138C23—411no—
139C23—822no—
140C23—1642no—
141C23—1642no—
142C23C22×D4111yescharacteristic
143C23C22×D4112yes—
144C23C22×D4111yescharacteristic
145D4—22no—
146D4—42no—
147D4—822no—
148D4—822no—
149D4—824no—
150D4—824no—
151D4—822no—
152D4—1642no—
153C2×C4—412no—
154C2×C4—411no—
155C2×C4—412no—
156C2×C4—822no—
157C2×C4—822no—
158C2×C4—1642no—
159C2×C4C22×D4112yes—
160C2×C4C22×D4111yescharacteristic
161C22—22no—
162C22—21no—
163C22—22no—
164C22—22no—
165C22—22no—
166C22—22no—
167C22—44no—
168C22—412no—
169C22—412no—
170C22—822no—
171C22—822no—
172C22—821no—
173C22—822no—
174C22—1644no—
175C22—1644no—
176C22D4×D4112yes—
177C22C23×D4111yescharacteristic
178C4—21no—
179C4—22no—
180C4—821no—
181C4—822no—
182C4—822no—
183C2—22no—
184C2—42no—
185C2—41no—
186C2—822no—
187C2—822no—
188C2—1641no—
189C2C2×D4×D4111yescharacteristic
190C2#1305020912yes—
subgroupG/H
center, FrattiniC22C23×D41
commutator, socle, cosocleC22×C41C24
Fitting#120×C41C2
hypercenter#120D41
hypocenterC41#534
LCSG ▷ C22×C41 ▷ C41
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C22×C22⋊D41C2semidirect
2. C2×D4×D41C2semidirect
3. #13050148C2semidirect
4. #13050146C2semidirect
5. #13050145C2semidirect
6. #13050137C2semidirect
7. #13050135C2semidirect
8. #13050133C2semidirect
9. #13050132C2semidirect
10. C22⋊C4×D41C2semidirect
11. #120×C41C2semidirect
12. C24×C41C22semidirect
14. C23×D41C22semidirect
16. C2×D4×C41C22semidirect
20. C2×C22⋊D41C22semidirect
21. C2×C22⋊D41C22semidirect
22. C2×C22⋊D41C22semidirect
23. C22×Dic41C22semidirect
25. C2×D164C22semidirect
26. C2×C4×D41C22semidirect
27. C2×C4×D41C22semidirect
28. C22⋊C4×C41C22semidirect
29. C22⋊C4×C41C22semidirect
30. C22⋊Dic41C22semidirect
31. C22⋊Dic41C22semidirect
32. C22⋊Dic41C22semidirect
33. #10620281C22semidirect
34. #10620281C22semidirect
35. C4⋊Dic41C22semidirect
36. #10620279C22semidirect
45. C22×D41D4semidirect
96. C41#534semidirect
106. #120D41semidirect
110. C24C2×D41semidirect
120. C2×D4C2×D41semidirect
131. C22⋊C4C2×D41semidirect
132. C22⋊C4C2×D41semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊D412
C2×D4×D412
#130501482
#130501462
#130501452
#130501372
#130501352
#130501332
#130501322
C22⋊C4×D412
#120×C412
#53441