Finite Group Databasebeta

← #28412042

Subgroups — #28412042

Order 14656.

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×D916C211yesmaximal, characteristic, complemented
2C2×D4×D229C211yesmaximal, characteristic, complemented
3#20300973C212yesmaximal, complemented
4#20300971C211yesmaximal, characteristic, complemented
5#20300963C212yesmaximal, complemented
6#20300961C212yesmaximal, complemented
7#20300958C211yesmaximal, characteristic, complemented
8#20300958C212yesmaximal, complemented
9#20300956C211yesmaximal, characteristic, complemented
10C22⋊C4×D229C211yesmaximal, characteristic, complemented
11#123×C229C211yesmaximal, characteristic, complemented
12C23×D229C2211yescharacteristic
13C23×D229C2211yescharacteristic, complemented
14C23×D229C2211yescharacteristic, complemented
15C2×D916—24no—
16C2×D916C2212yes—
17C2×D916C2212yescomplemented
18C2×D916C2211yescharacteristic, complemented
19C2×D916C2212yescomplemented
20C2×C22⋊D229C2211yescharacteristic
21C2×C22⋊D229C2211yescharacteristic, complemented
22C2×C22⋊D229C2212yescomplemented
23D4×D229—24no—
24C2×C4×D229C2211yescharacteristic, complemented
25C2×C4×D229C2212yescomplemented
26#16814645C2212yescomplemented
27C22⋊Dic229C2211yescharacteristic, complemented
28#16814643C2212yescomplemented
29#16814643C2212yescomplemented
30#16814643C2212yescomplemented
31#16814643C2212yescomplemented
32C2×D4×C229C2211yescharacteristic
33C22×C4×C229C2211yescharacteristic, complemented
34C4⋊C4×C229C2212yescomplemented
35C22⋊C4×C229C2211yescharacteristic, complemented
36C22⋊C4×C229C2212yescomplemented
37C22×D229—22no—
38C22×D229—22no—
39C22×D229—42no—
40C22×D229—42no—
41C22×D229—42no—
42C22×D229D414yescomplemented
43C22×D229C2311yescharacteristic
44C22×D229C2312yes—
45C22×D229C2311yescharacteristic
46C22×D229C2311yescharacteristic
47D916—22no—
48D916—24no—
49D916—24no—
50D916—42no—
51C22⋊D229—22no—
52C22⋊D229—22no—
53C22⋊D229—42no—
54C2×Dic229C2312yes—
55C2×Dic229C2311yescharacteristic
56C4×D229—22no—
57C4×D229—42no—
58C23×C229C2311yescharacteristic
59C23×C229C2311yescharacteristic
60D4×C229—22no—
61C2×C4×C229—22no—
62C2×C4×C229C2312yes—
63C2×C4×C229C2312yes—
64C2×C4×C229C2311yescharacteristic
65C22×C229—22no—
66C22×C229—21no—
67C22×C229—22no—
68C22×C229C2×D412yes—
69C22×C229C2411yescharacteristic
70C2×D229—22no—
71C2×D229—22no—
72C2×D229—21no—
73C2×D229—22no—
74C2×D229—22no—
75C2×D229—22no—
76C2×D229—22no—
77C2×D229—44no—
78C2×D229—44no—
79C2×D229—44no—
80C2×D229C2×D412yes—
81C2×D229C2×D412yes—
82C4×C229—22no—
83C4×C229—21no—
84C4×C229—22no—
85Dic229—21no—
86Dic229—22no—
87C2×C229—22no—
88C2×C229—41no—
89C2×C229C22×D411yescharacteristic
90C2×C229H(5,𝔽2)12yes—
91D229—22no—
92D229—22no—
93D229—41no—
94D229—42no—
95C229#53411yescharacteristic, complemented, Sylow
96#534—2291nomaximal, Sylow
97C22×D4—2291no—
98C22×D4—2291no—
99#123—2291no—
100#123—2292no—
101#123D22911yescharacteristic, complemented
102#121—2292no—
103#121—2292no—
104#120—2291no—
105#120—2292no—
106#120—2291no—
107C2×C22⋊C4—2291no—
108C24—2291no—
109C24—2291no—
110C24—2291no—
111C2×D4—4584no—
112C2×D4—4584no—
113C2×D4—2292no—
114C2×D4—2291no—
115C2×D4—2291no—
116C2×D4—2292no—
117C2×D4—2292no—
118C2×D4—2291no—
119C2×D4—2292no—
120C2×D4C2×D22911yescharacteristic
121C22×C4—2292no—
122C22×C4—2291no—
123C22×C4C2×D22911yescharacteristic, complemented
124C4⋊C4—2292no—
125C4⋊C4C2×D22912yescomplemented
126C22⋊C4—2291no—
127C22⋊C4—2292no—
128C22⋊C4—2292no—
129C22⋊C4—2292no—
130C22⋊C4—2292no—
131C22⋊C4C2×D22911yescharacteristic, complemented
132C22⋊C4C2×D22912yescomplemented
133C23—9162no—
134C23—9162no—
135C23—9162no—
136C23—4582no—
137C23—4582no—
138C23—2294no—
139C23—2292no—
140C23—2291no—
141C23—2291no—
142C23—2291no—
143C23C22×D22911yescharacteristic
144C23C22×D22911yescharacteristic
145D4—22no—
146D4—9162no—
147D4—9162no—
148D4—4584no—
149D4—4584no—
150D4—4582no—
151D4—4582no—
152D4—4582no—
153C2×C4—22no—
154C2×C4—9162no—
155C2×C4—4582no—
156C2×C4—2292no—
157C2×C4—2291no—
158C2×C4C22×D22912yes—
159C2×C4C22×D22911yescharacteristic
160C2×C4C22×D22912yes—
161C22—22no—
162C22—21no—
163C22—22no—
164C22—9164no—
165C22—9164no—
166C22—9164no—
167C22—4582no—
168C22—4581no—
169C22—4582no—
170C22—4582no—
171C22—4582no—
172C22—4582no—
173C22—4582no—
174C22—2292no—
175C22—2292no—
176C22D4×D22912yes—
177C22C23×D22911yescharacteristic
178C4—22no—
179C4—21no—
180C4—22no—
181C4—4581no—
182C4—4582no—
183C2—22no—
184C2—41no—
185C2—9162no—
186C2—9161no—
187C2—4582no—
188C2—4582no—
189C2C2×D4×D22911yescharacteristic
190C2#2030103412yes—
subgroupG/H
center, FrattiniC22C23×D229
commutator, socle, cosocleC22×C229C24
Fitting#123×C229C2
hypercenter#123D229
hypocenterC229#534
LCSG ▷ C22×C229 ▷ C229
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
1. C22×D916C2semidirect
2. C2×D4×D229C2semidirect
3. #20300973C2semidirect
4. #20300971C2semidirect
5. #20300963C2semidirect
6. #20300961C2semidirect
7. #20300958C2semidirect
8. #20300958C2semidirect
9. #20300956C2semidirect
10. C22⋊C4×D229C2semidirect
11. #123×C229C2semidirect
13. C23×D229C22semidirect
14. C23×D229C22semidirect
17. C2×D916C22semidirect
18. C2×D916C22semidirect
19. C2×D916C22semidirect
21. C2×C22⋊D229C22semidirect
22. C2×C22⋊D229C22semidirect
24. C2×C4×D229C22semidirect
25. C2×C4×D229C22semidirect
26. #16814645C22semidirect
27. C22⋊Dic229C22semidirect
28. #16814643C22semidirect
29. #16814643C22semidirect
30. #16814643C22semidirect
31. #16814643C22semidirect
33. C22×C4×C229C22semidirect
34. C4⋊C4×C229C22semidirect
35. C22⋊C4×C229C22semidirect
36. C22⋊C4×C229C22semidirect
42. C22×D229D4semidirect
95. C229#534semidirect
101. #123D229semidirect
123. C22×C4C2×D229semidirect
125. C4⋊C4C2×D229semidirect
131. C22⋊C4C2×D229semidirect
132. C22⋊C4C2×D229semidirect

Maximal subgroups

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

subgroupindex
C22×D9162
C2×D4×D2292
#203009732
#203009712
#203009632
#203009612
#203009582
#203009562
C22⋊C4×D2292
#123×C2292
#534229