Finite Group Databasebeta

← #27164650

Subgroups — #27164650

Order 12736.

428 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 115 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⋊D199C212yesmaximal, complemented
2C2×D4×D199C212yesmaximal, complemented
3#20263951C211yesmaximal, characteristic, complemented
4#20263928C212yesmaximal, complemented
5#20263927C212yesmaximal, complemented
6#20263925C212yesmaximal, complemented
7#20263923C211yesmaximal, characteristic, complemented
8(C22⋊D199)⋊C4C212yesmaximal, complemented
9#120×C199C211yesmaximal, characteristic, complemented
10C23×D199C2211yescharacteristic, complemented
11C23×D199C2212yescomplemented
12C2×D796C2212yescomplemented
13C2×D796C2211yescharacteristic, complemented
14C2×C22⋊D199—28no—
15C2×C22⋊D199C2212yes—
16C2×C22⋊D199C2212yescomplemented
17C2×C22⋊D199C2212yescomplemented
18C2×C22⋊D199C2212yescomplemented
19C2×C22⋊D199C2212yescomplemented
20D4×D199—28no—
21C22×Dic199C2212yescomplemented
22C2×C4×D199C2212yescomplemented
23Dic199⋊C4C2212yescomplemented
24C22⋊Dic199C2211yescharacteristic, complemented
25(C2×D199)⋊C4C2212yescomplemented
26(C2×D199)⋊C4C2212yescomplemented
27C4×Dic199C2211yescharacteristic, complemented
28C24×C199C2211yescharacteristic, complemented
29C2×D4×C199C2212yescomplemented
30C2×D4×C199C2211yescharacteristic, complemented
31C22⋊C4×C199C2212yescomplemented
32C22⋊C4×C199C2211yescharacteristic, complemented
33C22×D199—24no—
34C22×D199—24no—
35C22×D199—24no—
36C22×D199—41no—
37C22×D199—42no—
38C22×D199C2312yes—
39C22×D199C2312yes—
40D796—24no—
41D796—41no—
42C22⋊D199—24no—
43C22⋊D199—24no—
44C22⋊D199—24no—
45C22⋊D199—24no—
46C22⋊D199D418yescomplemented
47C2×Dic199—24no—
48C2×Dic199C2312yes—
49C2×Dic199C2312yes—
50C4×D199—24no—
51C23×C199—24no—
52C23×C199—41no—
53C23×C199C2311yescharacteristic
54C23×C199C2311yescharacteristic
55C23×C199C2312yes—
56D4×C199—24no—
57D4×C199—41no—
58C2×C4×C199C2311yescharacteristic
59C2×C4×C199C2312yes—
60C22×C199—22no—
61C22×C199—22no—
62C22×C199—22no—
63C22×C199—22no—
64C22×C199—21no—
65C22×C199—21no—
66C22×C199—44no—
67C22×C199C2×D414yescomplemented
68C22×C199C2411yescharacteristic
69C2×D199—22no—
70C2×D199—22no—
71C2×D199—22no—
72C2×D199—22no—
73C2×D199—22no—
74C2×D199—44no—
75C2×D199—48no—
76C2×D199C2×D414yescomplemented
77C4×C199—22no—
78C4×C199—21no—
79Dic199—22no—
80Dic199C2×D414yescomplemented
81C2×C199—24no—
82C2×C199—41no—
83C2×C199—41no—
84C2×C199C22×D412yes—
85C2×C199H(5,𝔽2)11yescharacteristic
86D199—24no—
87D199—42no—
88C199D4211yescharacteristic, complemented, Sylow
89D42—1991nomaximal, Sylow
90C22×D4—1992no—
91C22×D4—1992no—
92#127—1991no—
93#121—1992no—
94#121—1992no—
95#120—1991no—
96#120—1992no—
97#120D19911yescharacteristic, complemented
98C4×D4—1992no—
99C24—1991no—
100C24—1992no—
101C24C2×D19911yescharacteristic, complemented
102C2×D4—3988no—
103C2×D4—3988no—
104C2×D4—1992no—
105C2×D4—1992no—
106C2×D4—1992no—
107C2×D4—1992no—
108C2×D4—1992no—
109C2×D4—1991no—
110C2×D4—1992no—
111C2×D4C2×D19912yescomplemented
112C2×D4C2×D19911yescharacteristic, complemented
113C22×C4—1992no—
114C22×C4—1992no—
115C4⋊C4—1992no—
116C22⋊C4—1991no—
117C22⋊C4—1992no—
118C22⋊C4—1992no—
119C22⋊C4C2×D19912yescomplemented
120C22⋊C4C2×D19911yescharacteristic, complemented
121C42—1991no—
122C23—24no—
123C23—41no—
124C23—7962no—
125C23—7961no—
126C23—3984no—
127C23—3984no—
128C23—3984no—
129C23—1992no—
130C23—1992no—
131C23C22×D19911yescharacteristic
132C23C22×D19911yescharacteristic
133C23C22×D19912yes—
134D4—24no—
135D4—41no—
136D4—7961no—
137D4—3984no—
138D4—3984no—
139D4—3984no—
140D4—3984no—
141D4—3984no—
142D4—1998no—
143C2×C4—3984no—
144C2×C4—3984no—
145C2×C4—1992no—
146C2×C4—1992no—
147C2×C4C22×D19911yescharacteristic
148C2×C4C22×D19912yes—
149C22—22no—
150C22—22no—
151C22—22no—
152C22—22no—
153C22—21no—
154C22—21no—
155C22—44no—
156C22—7968no—
157C22—7964no—
158C22—3982no—
159C22—3982no—
160C22—3982no—
161C22—3982no—
162C22—3982no—
163C22—1994no—
164C22D4×D19914yescomplemented
165C22C23×D19911yescharacteristic
166C4—22no—
167C4—21no—
168C4—3982no—
169C4—1994no—
170C2—24no—
171C2—41no—
172C2—41no—
173C2—7962no—
174C2—3984no—
175C2C2×D4×D19912yes—
176C2#2026400011yescharacteristic
subgroupG/H
center, FrattiniC22C23×D199
commutator, socle, cosocleC22×C199C24
Fitting#120×C199C2
hypercenter#120D199
hypocenterC199D42
LCSG ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C22×C22⋊D199C2semidirect
2. C2×D4×D199C2semidirect
3. #20263951C2semidirect
4. #20263928C2semidirect
5. #20263927C2semidirect
6. #20263925C2semidirect
7. #20263923C2semidirect
8. (C22⋊D199)⋊C4C2semidirect
9. #120×C199C2semidirect
10. C23×D199C22semidirect
11. C23×D199C22semidirect
12. C2×D796C22semidirect
13. C2×D796C22semidirect
16. C2×C22⋊D199C22semidirect
17. C2×C22⋊D199C22semidirect
18. C2×C22⋊D199C22semidirect
19. C2×C22⋊D199C22semidirect
21. C22×Dic199C22semidirect
22. C2×C4×D199C22semidirect
23. Dic199⋊C4C22semidirect
24. C22⋊Dic199C22semidirect
25. (C2×D199)⋊C4C22semidirect
26. (C2×D199)⋊C4C22semidirect
27. C4×Dic199C22semidirect
28. C24×C199C22semidirect
29. C2×D4×C199C22semidirect
30. C2×D4×C199C22semidirect
31. C22⋊C4×C199C22semidirect
32. C22⋊C4×C199C22semidirect
46. C22⋊D199D4semidirect
67. C22×C199C2×D4semidirect
76. C2×D199C2×D4semidirect
80. Dic199C2×D4semidirect
88. C199D42semidirect
97. #120D199semidirect
101. C24C2×D199semidirect
111. C2×D4C2×D199semidirect
112. C2×D4C2×D199semidirect
119. C22⋊C4C2×D199semidirect
120. C22⋊C4C2×D199semidirect
164. C22D4×D199semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊D1992
C2×D4×D1992
#202639512
#202639282
#202639272
#202639252
#202639232
(C22⋊D199)⋊C42
#120×C1992
D42199