Finite Group Databasebeta

← #27164584

Subgroups — #27164584

Order 12736.

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×#15691886C211yesmaximal, characteristic, complemented
2C2×D4×D199C211yesmaximal, characteristic, complemented
3#20263951C212yesmaximal, complemented
4#20263941C212yesmaximal, complemented
5#20263930C211yesmaximal, characteristic, complemented
6#20263929C211yesmaximal, characteristic, complemented
7#20263926C211yesmaximal, characteristic, complemented
8#20263925C212yesmaximal, complemented
9#20263923C212yesmaximal, complemented
10#20263913C211yesmaximal, characteristic, complemented
11#121×C199C211yesmaximal, characteristic, complemented
12C23×D199C2212yes—
13#15691886—24no—
14C2×Dic398C2211yescharacteristic, complemented
15C2×D796C2211yescharacteristic
16C2×D796C2211yescharacteristic
17C2×D796C2212yescomplemented
18C2×C22⋊D199C2212yes—
19C2×C22⋊D199C2211yescharacteristic
20C2×C22⋊D199C2211yescharacteristic, complemented
21C2×C22⋊D199C2212yescomplemented
22C2×C22⋊D199C2212yescomplemented
23D4×D199—24no—
24C2×C4×D199C2211yescharacteristic
25C2×C4×D199C2211yescharacteristic, complemented
26C4⋊Dic199C2211yescharacteristic, complemented
27C22⋊Dic199C2212yescomplemented
28(C2×D199)⋊C4C2212yes—
29(C2×D199)⋊C4C2212yescomplemented
30(C2×D199)⋊C4C2212yescomplemented
31C4×Dic199C2212yescomplemented
32C2×D4×C199C2211yescharacteristic
33C2×D4×C199C2212yescomplemented
34C22×C4×C199C2211yescharacteristic, complemented
35C4⋊C4×C199C2211yescharacteristic, complemented
36C22⋊C4×C199C2212yescomplemented
37C22×D199—24no—
38C22×D199—42no—
39C22×D199—42no—
40C22×D199C2312yes—
41C22×D199C2311yescharacteristic
42C22×D199C2311yescharacteristic
43Dic398—22no—
44D796—22no—
45D796—22no—
46D796—42no—
47C22⋊D199—22no—
48C22⋊D199—24no—
49C22⋊D199—22no—
50C22⋊D199—42no—
51C22⋊D199—42no—
52C2×Dic199C2312yes—
53C2×Dic199C2311yescharacteristic
54C2×Dic199C2311yescharacteristic
55C4×D199—22no—
56C4×D199D414yescomplemented
57C23×C199C2312yes—
58C23×C199C2311yescharacteristic
59D4×C199—22no—
60D4×C199—42no—
61C2×C4×C199—22no—
62C2×C4×C199C2311yescharacteristic
63C2×C4×C199C2312yes—
64C2×C4×C199C2311yescharacteristic
65C22×C199—21no—
66C22×C199—22no—
67C22×C199—21no—
68C22×C199—22no—
69C22×C199—21no—
70C22×C199—22no—
71C22×C199C2411yescharacteristic
72C2×D199—21no—
73C2×D199—21no—
74C2×D199—21no—
75C2×D199—22no—
76C2×D199—21no—
77C2×D199—22no—
78C2×D199—21no—
79C2×D199—22no—
80C2×D199—48no—
81C2×D199C2×D412yes—
82C4×C199—21no—
83C4×C199—22no—
84C4×C199C2×D412yes—
85Dic199—21no—
86Dic199—22no—
87Dic199C2×D412yes—
88C2×C199—42no—
89C2×C199—41no—
90C2×C199C22×D411yescharacteristic
91C2×C199H(5,𝔽2)11yescharacteristic
92C2×C199H(5,𝔽2)11yescharacteristic
93D199—22no—
94D199—41no—
95D199—42no—
96C199#53511yescharacteristic, complemented, Sylow
97#535—1991nomaximal, Sylow
98C2×#41—1991no—
99C22×D4—1991no—
100#127—1992no—
101#124—1992no—
102#121—1991no—
103#121—1991no—
104#121—1991no—
105#121D19911yescharacteristic, complemented
106#120—1992no—
107#120—1992no—
108#117—1991no—
109C24—1992no—
110#41—3984no—
111C2×Q8—1991no—
112C2×D4—3984no—
113C2×D4—1992no—
114C2×D4—1992no—
115C2×D4—1991no—
116C2×D4—1992no—
117C2×D4—1991no—
118C2×D4—1992no—
119C2×D4—1991no—
120C2×D4—1991no—
121C2×D4C2×D19911yescharacteristic
122C2×D4C2×D19912yescomplemented
123C22×C4—1991no—
124C22×C4—1991no—
125C22×C4C2×D19911yescharacteristic, complemented
126C4⋊C4—1991no—
127C4⋊C4C2×D19911yescharacteristic, complemented
128C22⋊C4—1992no—
129C22⋊C4—1992no—
130C22⋊C4—1992no—
131C22⋊C4—1992no—
132C22⋊C4C2×D19912yescomplemented
133C42—1992no—
134C23—7962no—
135C23—7962no—
136C23—3984no—
137C23—1992no—
138C23—1991no—
139C23—1991no—
140C23C22×D19912yes—
141C23C22×D19911yescharacteristic
142Q8—3982no—
143D4—22no—
144D4—42no—
145D4—7962no—
146D4—7962no—
147D4—7962no—
148D4—3982no—
149D4—3982no—
150D4—3984no—
151D4—3982no—
152D4—3982no—
153C2×C4—22no—
154C2×C4—3982no—
155C2×C4—1992no—
156C2×C4—1991no—
157C2×C4—1991no—
158C2×C4—1994no—
159C2×C4C22×D19912yes—
160C2×C4C22×D19911yescharacteristic
161C2×C4C22×D19911yescharacteristic
162C22—21no—
163C22—22no—
164C22—21no—
165C22—22no—
166C22—21no—
167C22—22no—
168C22—7968no—
169C22—3981no—
170C22—3982no—
171C22—3981no—
172C22—3982no—
173C22—3981no—
174C22—3982no—
175C22—3981no—
176C22—3981no—
177C22—1992no—
178C22C23×D19911yescharacteristic
179C4—21no—
180C4—22no—
181C4—3981no—
182C4—3982no—
183C4—1992no—
184C4D4×D19912yes—
185C2—42no—
186C2—41no—
187C2—7962no—
188C2—7961no—
189C2—3982no—
190C2C2×D4×D19911yescharacteristic
191C2#2026400011yescharacteristic
192C2#2026400111yescharacteristic
subgroupG/H
center, FrattiniC22C23×D199
commutator, socle, cosocleC22×C199C24
Fitting#121×C199C2
hypercenter#121D199
hypocenterC199#535
LCSG ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×#15691886C2semidirect
2. C2×D4×D199C2semidirect
3. #20263951C2semidirect
4. #20263941C2semidirect
5. #20263930C2semidirect
6. #20263929C2semidirect
7. #20263926C2semidirect
8. #20263925C2semidirect
9. #20263923C2semidirect
10. #20263913C2semidirect
11. #121×C199C2semidirect
14. C2×Dic398C22semidirect
17. C2×D796C22semidirect
20. C2×C22⋊D199C22semidirect
21. C2×C22⋊D199C22semidirect
22. C2×C22⋊D199C22semidirect
25. C2×C4×D199C22semidirect
26. C4⋊Dic199C22semidirect
27. C22⋊Dic199C22semidirect
29. (C2×D199)⋊C4C22semidirect
30. (C2×D199)⋊C4C22semidirect
31. C4×Dic199C22semidirect
33. C2×D4×C199C22semidirect
34. C22×C4×C199C22semidirect
35. C4⋊C4×C199C22semidirect
36. C22⋊C4×C199C22semidirect
56. C4×D199D4semidirect
96. C199#535semidirect
105. #121D199semidirect
122. C2×D4C2×D199semidirect
125. C22×C4C2×D199semidirect
127. C4⋊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
C2×#156918862
C2×D4×D1992
#202639512
#202639412
#202639302
#202639292
#202639262
#202639252
#202639232
#202639132
#121×C1992
#535199