Finite Group Databasebeta

← #27164579

Subgroups — #27164579

Order 12736.

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×D796C211yesmaximal, characteristic, complemented
2C2×D4×D199C211yesmaximal, characteristic, complemented
3#20263940C212yesmaximal, complemented
4#20263938C211yesmaximal, characteristic, complemented
5#20263930C212yesmaximal, complemented
6#20263928C212yesmaximal, complemented
7#20263925C211yesmaximal, characteristic, complemented
8#20263925C212yesmaximal, complemented
9#20263923C211yesmaximal, characteristic, complemented
10C22⋊C4×D199C211yesmaximal, characteristic, complemented
11#123×C199C211yesmaximal, characteristic, complemented
12C23×D199C2211yescharacteristic
13C23×D199C2211yescharacteristic, complemented
14C23×D199C2211yescharacteristic, complemented
15C2×D796—24no—
16C2×D796C2212yes—
17C2×D796C2211yescharacteristic, complemented
18C2×D796C2212yescomplemented
19C2×D796C2212yescomplemented
20C2×C22⋊D199C2211yescharacteristic
21C2×C22⋊D199C2211yescharacteristic, complemented
22C2×C22⋊D199C2212yescomplemented
23D4×D199—24no—
24C2×C4×D199C2211yescharacteristic, complemented
25C2×C4×D199C2212yescomplemented
26Dic199⋊C4C2212yescomplemented
27C22⋊Dic199C2211yescharacteristic, complemented
28(C2×D199)⋊C4C2212yescomplemented
29(C2×D199)⋊C4C2212yescomplemented
30(C2×D199)⋊C4C2212yescomplemented
31(C2×D199)⋊C4C2212yescomplemented
32C2×D4×C199C2211yescharacteristic
33C22×C4×C199C2211yescharacteristic, complemented
34C4⋊C4×C199C2212yescomplemented
35C22⋊C4×C199C2211yescharacteristic, complemented
36C22⋊C4×C199C2212yescomplemented
37C22×D199—22no—
38C22×D199—22no—
39C22×D199—42no—
40C22×D199—42no—
41C22×D199—42no—
42C22×D199D414yescomplemented
43C22×D199C2311yescharacteristic
44C22×D199C2312yes—
45C22×D199C2311yescharacteristic
46C22×D199C2311yescharacteristic
47D796—22no—
48D796—24no—
49D796—24no—
50D796—42no—
51C22⋊D199—22no—
52C22⋊D199—22no—
53C22⋊D199—42no—
54C2×Dic199C2312yes—
55C2×Dic199C2311yescharacteristic
56C4×D199—22no—
57C4×D199—42no—
58C23×C199C2311yescharacteristic
59C23×C199C2311yescharacteristic
60D4×C199—22no—
61C2×C4×C199—22no—
62C2×C4×C199C2312yes—
63C2×C4×C199C2312yes—
64C2×C4×C199C2311yescharacteristic
65C22×C199—22no—
66C22×C199—21no—
67C22×C199—22no—
68C22×C199C2×D412yes—
69C22×C199C2411yescharacteristic
70C2×D199—22no—
71C2×D199—22no—
72C2×D199—21no—
73C2×D199—22no—
74C2×D199—22no—
75C2×D199—22no—
76C2×D199—22no—
77C2×D199—44no—
78C2×D199—44no—
79C2×D199—44no—
80C2×D199C2×D412yes—
81C2×D199C2×D412yes—
82C4×C199—22no—
83C4×C199—21no—
84C4×C199—22no—
85Dic199—21no—
86Dic199—22no—
87C2×C199—22no—
88C2×C199—41no—
89C2×C199C22×D411yescharacteristic
90C2×C199H(5,𝔽2)12yes—
91D199—22no—
92D199—22no—
93D199—41no—
94D199—42no—
95C199#53411yescharacteristic, complemented, Sylow
96#534—1991nomaximal, Sylow
97C22×D4—1991no—
98C22×D4—1991no—
99#123—1991no—
100#123—1992no—
101#123D19911yescharacteristic, complemented
102#121—1992no—
103#121—1992no—
104#120—1991no—
105#120—1992no—
106#120—1991no—
107C2×C22⋊C4—1991no—
108C24—1991no—
109C24—1991no—
110C24—1991no—
111C2×D4—3984no—
112C2×D4—3984no—
113C2×D4—1992no—
114C2×D4—1991no—
115C2×D4—1991no—
116C2×D4—1992no—
117C2×D4—1992no—
118C2×D4—1991no—
119C2×D4—1992no—
120C2×D4C2×D19911yescharacteristic
121C22×C4—1992no—
122C22×C4—1991no—
123C22×C4C2×D19911yescharacteristic, complemented
124C4⋊C4—1992no—
125C4⋊C4C2×D19912yescomplemented
126C22⋊C4—1991no—
127C22⋊C4—1992no—
128C22⋊C4—1992no—
129C22⋊C4—1992no—
130C22⋊C4—1992no—
131C22⋊C4C2×D19911yescharacteristic, complemented
132C22⋊C4C2×D19912yescomplemented
133C23—7962no—
134C23—7962no—
135C23—7962no—
136C23—3982no—
137C23—3982no—
138C23—1994no—
139C23—1992no—
140C23—1991no—
141C23—1991no—
142C23—1991no—
143C23C22×D19911yescharacteristic
144C23C22×D19911yescharacteristic
145D4—22no—
146D4—7962no—
147D4—7962no—
148D4—3984no—
149D4—3984no—
150D4—3982no—
151D4—3982no—
152D4—3982no—
153C2×C4—22no—
154C2×C4—7962no—
155C2×C4—3982no—
156C2×C4—1992no—
157C2×C4—1991no—
158C2×C4C22×D19912yes—
159C2×C4C22×D19911yescharacteristic
160C2×C4C22×D19912yes—
161C22—22no—
162C22—21no—
163C22—22no—
164C22—7964no—
165C22—7964no—
166C22—7964no—
167C22—3982no—
168C22—3981no—
169C22—3982no—
170C22—3982no—
171C22—3982no—
172C22—3982no—
173C22—3982no—
174C22—1992no—
175C22—1992no—
176C22D4×D19912yes—
177C22C23×D19911yescharacteristic
178C4—22no—
179C4—21no—
180C4—22no—
181C4—3981no—
182C4—3982no—
183C2—22no—
184C2—41no—
185C2—7962no—
186C2—7961no—
187C2—3982no—
188C2—3982no—
189C2C2×D4×D19911yescharacteristic
190C2#2026400112yes—
subgroupG/H
center, FrattiniC22C23×D199
commutator, socle, cosocleC22×C199C24
Fitting#123×C199C2
hypercenter#123D199
hypocenterC199#534
LCSG ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
1. C22×D796C2semidirect
2. C2×D4×D199C2semidirect
3. #20263940C2semidirect
4. #20263938C2semidirect
5. #20263930C2semidirect
6. #20263928C2semidirect
7. #20263925C2semidirect
8. #20263925C2semidirect
9. #20263923C2semidirect
10. C22⋊C4×D199C2semidirect
11. #123×C199C2semidirect
13. C23×D199C22semidirect
14. C23×D199C22semidirect
17. C2×D796C22semidirect
18. C2×D796C22semidirect
19. C2×D796C22semidirect
21. C2×C22⋊D199C22semidirect
22. C2×C22⋊D199C22semidirect
24. C2×C4×D199C22semidirect
25. C2×C4×D199C22semidirect
26. Dic199⋊C4C22semidirect
27. C22⋊Dic199C22semidirect
28. (C2×D199)⋊C4C22semidirect
29. (C2×D199)⋊C4C22semidirect
30. (C2×D199)⋊C4C22semidirect
31. (C2×D199)⋊C4C22semidirect
33. C22×C4×C199C22semidirect
34. C4⋊C4×C199C22semidirect
35. C22⋊C4×C199C22semidirect
36. C22⋊C4×C199C22semidirect
42. C22×D199D4semidirect
95. C199#534semidirect
101. #123D199semidirect
123. C22×C4C2×D199semidirect
125. C4⋊C4C2×D199semidirect
131. C22⋊C4C2×D199semidirect
132. C22⋊C4C2×D199semidirect

Maximal subgroups

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

subgroupindex
C22×D7962
C2×D4×D1992
#202639402
#202639382
#202639302
#202639282
#202639252
#202639232
C22⋊C4×D1992
#123×C1992
#534199