Finite Group Databasebeta

← #28412046

Subgroups — #28412046

Order 14656.

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

normal subgroups only

This group has 334 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
1C2×#16814665C211yesmaximal, characteristic, complemented
2C22×C22⋊D229C211yesmaximal, characteristic, complemented
3#20300984C212yesmaximal, complemented
4#20300974C212yesmaximal, complemented
5#20300961C212yesmaximal, complemented
6#20300960C212yesmaximal, complemented
7#20300958C211yesmaximal, characteristic, complemented
8#20300957C211yesmaximal, characteristic, complemented
9#20300956C211yesmaximal, characteristic, complemented
10#20300944C211yesmaximal, characteristic, complemented
11#120×C229C211yesmaximal, characteristic, complemented
12C23×D229C2211yescharacteristic
13#16814665—24no—
14C2×Dic458C2211yescharacteristic, complemented
15C2×D916C2212yescomplemented
16C2×C22⋊D229—24no—
17C2×C22⋊D229C2211yescharacteristic
18C2×C22⋊D229C2212yes—
19C2×C22⋊D229C2212yes—
20C2×C22⋊D229C2211yescharacteristic, complemented
21C2×C22⋊D229C2212yescomplemented
22C2×C22⋊D229C2212yescomplemented
23C22×Dic229C2211yescharacteristic
24C22×Dic229C2211yescharacteristic, complemented
25C2×C4×D229C2211yescharacteristic, complemented
26#16814645C2212yescomplemented
27C22⋊Dic229C2211yescharacteristic
28C22⋊Dic229C2212yescomplemented
29#16814643C2212yescomplemented
30#16814643C2212yescomplemented
31C4×Dic229C2212yescomplemented
32C24×C229C2211yescharacteristic
33C2×D4×C229C2211yescharacteristic
34C2×D4×C229C2212yescomplemented
35C22⋊C4×C229C2211yescharacteristic
36C22⋊C4×C229C2212yescomplemented
37C22×D229—22no—
38C22×D229—42no—
39C22×D229C2312yes—
40C22×D229C2311yescharacteristic
41Dic458—22no—
42D916—42no—
43C22⋊D229—22no—
44C22⋊D229—24no—
45C22⋊D229—24no—
46C22⋊D229—22no—
47C22⋊D229—42no—
48C22⋊D229—42no—
49C2×Dic229—22no—
50C2×Dic229D414yescomplemented
51C2×Dic229C2312yes—
52C2×Dic229C2311yescharacteristic
53C2×Dic229C2311yescharacteristic
54C2×Dic229C2311yescharacteristic
55C4×D229—22no—
56C23×C229—22no—
57C23×C229—42no—
58C23×C229C2312yes—
59C23×C229C2311yescharacteristic
60C23×C229C2311yescharacteristic
61D4×C229—22no—
62D4×C229—42no—
63C2×C4×C229C2312yes—
64C2×C4×C229C2311yescharacteristic
65C22×C229—22no—
66C22×C229—21no—
67C22×C229—22no—
68C22×C229—22no—
69C22×C229—22no—
70C22×C229—22no—
71C22×C229—44no—
72C22×C229C2×D412yes—
73C22×C229C2411yescharacteristic
74C2×D229—21no—
75C2×D229—22no—
76C2×D229—22no—
77C2×D229—22no—
78C2×D229—22no—
79C2×D229—44no—
80C4×C229—21no—
81C4×C229—22no—
82Dic229—21no—
83Dic229—22no—
84Dic229C2×D412yes—
85Dic229C2×D412yes—
86C2×C229—22no—
87C2×C229—41no—
88C2×C229—42no—
89C2×C229C22×D411yescharacteristic
90C2×C229H(5,𝔽2)12yes—
91D229—41no—
92D229—42no—
93C229#53511yescharacteristic, complemented, Sylow
94#535—2291nomaximal, Sylow
95C2×#41—2291no—
96C22×D4—2291no—
97#127—2292no—
98#124—2292no—
99#121—2292no—
100#121—2292no—
101#120—2291no—
102#120—2291no—
103#120—2291no—
104#120D22911yescharacteristic, complemented
105#117—2291no—
106C24—2291no—
107C24C2×D22911yescharacteristic
108#41—4584no—
109C2×Q8—2291no—
110C2×D4—4584no—
111C2×D4—2292no—
112C2×D4—2292no—
113C2×D4—2291no—
114C2×D4—2292no—
115C2×D4—2292no—
116C2×D4—2291no—
117C2×D4—2292no—
118C2×D4C2×D22911yescharacteristic
119C2×D4C2×D22912yescomplemented
120C22×C4—2291no—
121C22×C4—2291no—
122C22×C4—2291no—
123C4⋊C4—2292no—
124C22⋊C4—2291no—
125C22⋊C4—2292no—
126C22⋊C4—2292no—
127C22⋊C4—2292no—
128C22⋊C4C2×D22911yescharacteristic
129C22⋊C4C2×D22912yescomplemented
130C42—2292no—
131C23—22no—
132C23—42no—
133C23—9162no—
134C23—4582no—
135C23—2292no—
136C23—2291no—
137C23C22×D22912yes—
138C23C22×D22911yescharacteristic
139C23C22×D22911yescharacteristic
140Q8—4582no—
141D4—22no—
142D4—42no—
143D4—9162no—
144D4—9162no—
145D4—9162no—
146D4—4582no—
147D4—4584no—
148D4—4584no—
149D4—4582no—
150C2×C4—4582no—
151C2×C4—4582no—
152C2×C4—2292no—
153C2×C4—2291no—
154C2×C4—2294no—
155C2×C4—2291no—
156C2×C4—2291no—
157C2×C4C22×D22912yes—
158C2×C4C22×D22911yescharacteristic
159C22—22no—
160C22—21no—
161C22—22no—
162C22—22no—
163C22—22no—
164C22—22no—
165C22—44no—
166C22—9164no—
167C22—4581no—
168C22—4582no—
169C22—4582no—
170C22—4582no—
171C22—4582no—
172C22D4×D22912yes—
173C22C23×D22911yescharacteristic
174C4—21no—
175C4—22no—
176C4—4581no—
177C4—4582no—
178C4—2292no—
179C4—2292no—
180C2—22no—
181C2—42no—
182C2—41no—
183C2—9162no—
184C2—9161no—
185C2C2×D4×D22911yescharacteristic
186C2#2030103312yes—
subgroupG/H
center, FrattiniC22C23×D229
commutator, socle, cosocleC22×C229C24
Fitting#120×C229C2
hypercenter#120D229
hypocenterC229#535
LCSG ▷ C22×C229 ▷ C229
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C2×#16814665C2semidirect
2. C22×C22⋊D229C2semidirect
3. #20300984C2semidirect
4. #20300974C2semidirect
5. #20300961C2semidirect
6. #20300960C2semidirect
7. #20300958C2semidirect
8. #20300957C2semidirect
9. #20300956C2semidirect
10. #20300944C2semidirect
11. #120×C229C2semidirect
14. C2×Dic458C22semidirect
15. C2×D916C22semidirect
20. C2×C22⋊D229C22semidirect
21. C2×C22⋊D229C22semidirect
22. C2×C22⋊D229C22semidirect
24. C22×Dic229C22semidirect
25. C2×C4×D229C22semidirect
26. #16814645C22semidirect
28. C22⋊Dic229C22semidirect
29. #16814643C22semidirect
30. #16814643C22semidirect
31. C4×Dic229C22semidirect
34. C2×D4×C229C22semidirect
36. C22⋊C4×C229C22semidirect
50. C2×Dic229D4semidirect
93. C229#535semidirect
104. #120D229semidirect
119. C2×D4C2×D229semidirect
129. C22⋊C4C2×D229semidirect

Maximal subgroups

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

subgroupindex
C2×#168146652
C22×C22⋊D2292
#203009842
#203009742
#203009612
#203009602
#203009582
#203009572
#203009562
#203009442
#120×C2292
#535229