Finite Group Databasebeta

← #28411991

Subgroups — #28411991

Order 14656.

330 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 109 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×C4×D229C211yesmaximal, characteristic, complemented
3#20300970C212yesmaximal, complemented
4#20300968C211yesmaximal, characteristic, complemented
5#20300967C211yesmaximal, characteristic, complemented
6#20300959C211yesmaximal, characteristic, complemented
7#20300956C212yesmaximal, complemented
8D4×Dic229C212yesmaximal, complemented
9#20300948C212yesmaximal, complemented
10#20300946C211yesmaximal, characteristic, complemented
11#121×C229C211yesmaximal, characteristic, complemented
12C23×D229C2211yescharacteristic
13#16814665—24no—
14C2×Dic458C2211yescharacteristic, complemented
15C2×C22⋊D229C2212yes—
16C2×C22⋊D229C2212yes—
17C22×Dic229C2211yescharacteristic
18C22×Dic229C2212yescomplemented
19C2×C4×D229—24no—
20C2×C4×D229C2211yescharacteristic
21C2×C4×D229C2211yescharacteristic
22C2×C4×D229C2211yescharacteristic
23C2×C4×D229C2211yescharacteristic
24C4⋊Dic229C2211yescharacteristic, complemented
25C4⋊Dic229C2212yescomplemented
26#16814645C2212yescomplemented
27C22⋊Dic229C2212yes—
28C22⋊Dic229C2212yescomplemented
29#16814643C2212yes—
30#16814643C2212yes—
31C4×Dic229C2212yescomplemented
32C2×D4×C229C2212yes—
33C2×D4×C229C2211yescharacteristic, complemented
34C22×C4×C229C2211yescharacteristic
35C4⋊C4×C229C2211yescharacteristic, complemented
36C22⋊C4×C229C2212yescomplemented
37C22×D229—22no—
38C22×D229—22no—
39C22×D229—22no—
40C22×D229C2311yescharacteristic
41C22×D229C2311yescharacteristic
42Dic458—22no—
43C22⋊D229—24no—
44C22⋊D229—24no—
45C2×Dic229—22no—
46C2×Dic229—24no—
47C2×Dic229C2312yes—
48C2×Dic229C2312yes—
49C2×Dic229C2311yescharacteristic
50C2×Dic229C2311yescharacteristic
51C4×D229—22no—
52C4×D229—22no—
53C4×D229—22no—
54C4×D229D414yescomplemented
55C23×C229C2312yes—
56C23×C229C2311yescharacteristic
57D4×C229—24no—
58D4×C229—22no—
59C2×C4×C229—22no—
60C2×C4×C229C2311yescharacteristic
61C2×C4×C229C2312yes—
62C2×C4×C229C2311yescharacteristic
63C22×C229—21no—
64C22×C229—21no—
65C22×C229—22no—
66C22×C229—22no—
67C22×C229—22no—
68C22×C229#4112yescomplemented
69C22×C229C2411yescharacteristic
70C2×D229—28no—
71C2×D229—21no—
72C2×D229—21no—
73C2×D229—21no—
74C2×D229—21no—
75C2×D229C2×D412yes—
76C2×D229#4112yescomplemented
77C4×C229—21no—
78C4×C229—22no—
79C4×C229C2×D412yes—
80Dic229—21no—
81Dic229—22no—
82Dic229—22no—
83Dic229C2×D412yes—
84C2×C229—22no—
85C2×C229—42no—
86C2×C229C22×D411yescharacteristic
87C2×C229C2×#4111yescharacteristic
88C2×C229C2×#4111yescharacteristic
89D229—22no—
90D229—22no—
91C229#52511yescharacteristic, complemented, Sylow
92#525—2291nomaximal, Sylow
93C2×#41—2291no—
94C23×C4—2291no—
95#123—2292no—
96C22⋊Q8—2291no—
97C22⋊Q8—2291no—
98#121—2291no—
99#121D22911yescharacteristic, complemented
100#120—2292no—
101C4×D4—2292no—
102C4×D4—2292no—
103#117—2291no—
104C24—2291no—
105#41—4584no—
106C2×Q8—2291no—
107C2×D4—2292no—
108C2×D4—2292no—
109C2×D4C2×D22912yes—
110C2×D4C2×D22911yescharacteristic, complemented
111C22×C4—4584no—
112C22×C4—2291no—
113C22×C4—2292no—
114C22×C4—2291no—
115C22×C4—2291no—
116C22×C4—2291no—
117C22×C4—2291no—
118C22×C4C2×D22911yescharacteristic
119C4⋊C4—2292no—
120C4⋊C4—2291no—
121C4⋊C4—2292no—
122C4⋊C4C2×D22911yescharacteristic, complemented
123C22⋊C4—2292no—
124C22⋊C4—2292no—
125C22⋊C4—2292no—
126C22⋊C4—2292no—
127C22⋊C4C2×D22912yescomplemented
128C42—2292no—
129C23—4582no—
130C23—4582no—
131C23—4582no—
132C23—2291no—
133C23—2291no—
134C23C22×D22912yes—
135C23C22×D22911yescharacteristic
136Q8—4582no—
137D4—24no—
138D4—22no—
139D4—4584no—
140D4—4584no—
141C2×C4—22no—
142C2×C4—4582no—
143C2×C4—4582no—
144C2×C4—4582no—
145C2×C4—4584no—
146C2×C4—4582no—
147C2×C4—2292no—
148C2×C4—2291no—
149C2×C4—2292no—
150C2×C4—2291no—
151C2×C4—2294no—
152C2×C4C22×D22912yes—
153C2×C4C22×D22911yescharacteristic
154C2×C4C22×D22911yescharacteristic
155C22—21no—
156C22—21no—
157C22—22no—
158C22—22no—
159C22—22no—
160C22—4588no—
161C22—4581no—
162C22—4581no—
163C22—4581no—
164C22—4581no—
165C22—2292no—
166C22—2292no—
167C22#1681466512yescomplemented
168C22C23×D22911yescharacteristic
169C4—21no—
170C4—22no—
171C4—4581no—
172C4—4582no—
173C4—4582no—
174C4—2292no—
175C4D4×D22912yes—
176C2—22no—
177C2—42no—
178C2—4582no—
179C2—4582no—
180C2C2×D4×D22911yescharacteristic
181C2#41×D22911yescharacteristic
182C2C2×#1681466511yescharacteristic
subgroupG/H
center, FrattiniC22C23×D229
commutator, socle, cosocleC22×C229C24
Fitting#121×C229C2
hypercenter#121D229
hypocenterC229#525
LCSG ▷ C22×C229 ▷ C229
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×#16814665C2semidirect
2. C22×C4×D229C2semidirect
3. #20300970C2semidirect
4. #20300968C2semidirect
5. #20300967C2semidirect
6. #20300959C2semidirect
7. #20300956C2semidirect
8. D4×Dic229C2semidirect
9. #20300948C2semidirect
10. #20300946C2semidirect
11. #121×C229C2semidirect
14. C2×Dic458C22semidirect
18. C22×Dic229C22semidirect
24. C4⋊Dic229C22semidirect
25. C4⋊Dic229C22semidirect
26. #16814645C22semidirect
28. C22⋊Dic229C22semidirect
31. C4×Dic229C22semidirect
33. C2×D4×C229C22semidirect
35. C4⋊C4×C229C22semidirect
36. C22⋊C4×C229C22semidirect
54. C4×D229D4semidirect
68. C22×C229#41semidirect
76. C2×D229#41semidirect
91. C229#525semidirect
99. #121D229semidirect
110. C2×D4C2×D229semidirect
122. C4⋊C4C2×D229semidirect
127. C22⋊C4C2×D229semidirect
167. C22#16814665semidirect

Maximal subgroups

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

subgroupindex
C2×#168146652
C22×C4×D2292
#203009702
#203009682
#203009672
#203009592
#203009562
D4×Dic2292
#203009482
#203009462
#121×C2292
#525229