Finite Group Databasebeta

← #27161645

Subgroups — #27161645

Order 12608.

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×#15691358C211yesmaximal, characteristic, complemented
2C2×D4×D197C211yesmaximal, characteristic, complemented
3#20263322C212yesmaximal, complemented
4#20263312C212yesmaximal, complemented
5#20263301C211yesmaximal, characteristic, complemented
6#20263300C211yesmaximal, characteristic, complemented
7#20263297C211yesmaximal, characteristic, complemented
8#20263296C212yesmaximal, complemented
9#20263294C212yesmaximal, complemented
10#20263284C211yesmaximal, characteristic, complemented
11#121×C197C211yesmaximal, characteristic, complemented
12C23×D197C2212yes—
13#15691358—24no—
14C2×Dic394C2211yescharacteristic, complemented
15C2×D788C2211yescharacteristic
16C2×D788C2211yescharacteristic
17C2×D788C2212yescomplemented
18C2×C22⋊D197C2211yescharacteristic
19C2×C22⋊D197C2212yes—
20C2×C22⋊D197C2211yescharacteristic, complemented
21C2×C22⋊D197C2212yescomplemented
22C2×C22⋊D197C2212yescomplemented
23D4×D197—24no—
24C2×C4×D197C2211yescharacteristic
25C2×C4×D197C2211yescharacteristic, complemented
26C4⋊Dic197C2211yescharacteristic, complemented
27C22⋊Dic197C2212yescomplemented
28#15691335C2212yes—
29#15691335C2212yescomplemented
30#15691335C2212yescomplemented
31C4×Dic197C2212yescomplemented
32C2×D4×C197C2211yescharacteristic
33C2×D4×C197C2212yescomplemented
34C22×C4×C197C2211yescharacteristic, complemented
35C4⋊C4×C197C2211yescharacteristic, complemented
36C22⋊C4×C197C2212yescomplemented
37C22×D197—24no—
38C22×D197—42no—
39C22×D197—42no—
40C22×D197C2312yes—
41C22×D197C2311yescharacteristic
42C22×D197C2311yescharacteristic
43Dic394—22no—
44D788—22no—
45D788—22no—
46D788—42no—
47C22⋊D197—22no—
48C22⋊D197—24no—
49C22⋊D197—22no—
50C22⋊D197—42no—
51C22⋊D197—42no—
52C2×Dic197C2312yes—
53C2×Dic197C2311yescharacteristic
54C2×Dic197C2311yescharacteristic
55C4×D197—22no—
56C4×D197D414yescomplemented
57C23×C197C2312yes—
58C23×C197C2311yescharacteristic
59D4×C197—22no—
60D4×C197—42no—
61C2×C4×C197—22no—
62C2×C4×C197C2311yescharacteristic
63C2×C4×C197C2312yes—
64C2×C4×C197C2311yescharacteristic
65C22×C197—21no—
66C22×C197—22no—
67C22×C197—21no—
68C22×C197—22no—
69C22×C197—21no—
70C22×C197—22no—
71C22×C197C2411yescharacteristic
72C2×D197—21no—
73C2×D197—21no—
74C2×D197—21no—
75C2×D197—22no—
76C2×D197—21no—
77C2×D197—22no—
78C2×D197—21no—
79C2×D197—22no—
80C2×D197—48no—
81C2×D197C2×D412yes—
82C4×C197—21no—
83C4×C197—22no—
84C4×C197C2×D412yes—
85Dic197—21no—
86Dic197—22no—
87Dic197C2×D412yes—
88C2×C197—42no—
89C2×C197—41no—
90C2×C197C22×D411yescharacteristic
91C2×C197H(5,𝔽2)11yescharacteristic
92C2×C197H(5,𝔽2)11yescharacteristic
93D197—22no—
94D197—41no—
95D197—42no—
96C197#53511yescharacteristic, complemented, Sylow
97#535—1971nomaximal, Sylow
98C2×#41—1971no—
99C22×D4—1971no—
100#127—1972no—
101#124—1972no—
102#121—1971no—
103#121—1971no—
104#121—1971no—
105#121D19711yescharacteristic, complemented
106#120—1972no—
107#120—1972no—
108#117—1971no—
109C24—1972no—
110#41—3944no—
111C2×Q8—1971no—
112C2×D4—3944no—
113C2×D4—1972no—
114C2×D4—1972no—
115C2×D4—1971no—
116C2×D4—1972no—
117C2×D4—1971no—
118C2×D4—1972no—
119C2×D4—1971no—
120C2×D4—1971no—
121C2×D4C2×D19711yescharacteristic
122C2×D4C2×D19712yescomplemented
123C22×C4—1971no—
124C22×C4—1971no—
125C22×C4C2×D19711yescharacteristic, complemented
126C4⋊C4—1971no—
127C4⋊C4C2×D19711yescharacteristic, complemented
128C22⋊C4—1972no—
129C22⋊C4—1972no—
130C22⋊C4—1972no—
131C22⋊C4—1972no—
132C22⋊C4C2×D19712yescomplemented
133C42—1972no—
134C23—7882no—
135C23—7882no—
136C23—3944no—
137C23—1972no—
138C23—1971no—
139C23—1971no—
140C23C22×D19712yes—
141C23C22×D19711yescharacteristic
142Q8—3942no—
143D4—22no—
144D4—42no—
145D4—7882no—
146D4—7882no—
147D4—7882no—
148D4—3942no—
149D4—3942no—
150D4—3944no—
151D4—3942no—
152D4—3942no—
153C2×C4—22no—
154C2×C4—3942no—
155C2×C4—1972no—
156C2×C4—1971no—
157C2×C4—1971no—
158C2×C4—1974no—
159C2×C4C22×D19712yes—
160C2×C4C22×D19711yescharacteristic
161C2×C4C22×D19711yescharacteristic
162C22—21no—
163C22—22no—
164C22—21no—
165C22—22no—
166C22—21no—
167C22—22no—
168C22—7888no—
169C22—3941no—
170C22—3942no—
171C22—3941no—
172C22—3942no—
173C22—3941no—
174C22—3942no—
175C22—3941no—
176C22—3941no—
177C22—1972no—
178C22C23×D19711yescharacteristic
179C4—21no—
180C4—22no—
181C4—3941no—
182C4—3942no—
183C4—1972no—
184C4D4×D19712yes—
185C2—42no—
186C2—41no—
187C2—7882no—
188C2—7881no—
189C2—3942no—
190C2C2×D4×D19711yescharacteristic
191C2#2026337111yescharacteristic
192C2#2026337211yescharacteristic
subgroupG/H
center, FrattiniC22C23×D197
commutator, socle, cosocleC22×C197C24
Fitting#121×C197C2
hypercenter#121D197
hypocenterC197#535
LCSG ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×#15691358C2semidirect
2. C2×D4×D197C2semidirect
3. #20263322C2semidirect
4. #20263312C2semidirect
5. #20263301C2semidirect
6. #20263300C2semidirect
7. #20263297C2semidirect
8. #20263296C2semidirect
9. #20263294C2semidirect
10. #20263284C2semidirect
11. #121×C197C2semidirect
14. C2×Dic394C22semidirect
17. C2×D788C22semidirect
20. C2×C22⋊D197C22semidirect
21. C2×C22⋊D197C22semidirect
22. C2×C22⋊D197C22semidirect
25. C2×C4×D197C22semidirect
26. C4⋊Dic197C22semidirect
27. C22⋊Dic197C22semidirect
29. #15691335C22semidirect
30. #15691335C22semidirect
31. C4×Dic197C22semidirect
33. C2×D4×C197C22semidirect
34. C22×C4×C197C22semidirect
35. C4⋊C4×C197C22semidirect
36. C22⋊C4×C197C22semidirect
56. C4×D197D4semidirect
96. C197#535semidirect
105. #121D197semidirect
122. C2×D4C2×D197semidirect
125. C22×C4C2×D197semidirect
127. C4⋊C4C2×D197semidirect
132. C22⋊C4C2×D197semidirect

Maximal subgroups

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

subgroupindex
C2×#156913582
C2×D4×D1972
#202633222
#202633122
#202633012
#202633002
#202632972
#202632962
#202632942
#202632842
#121×C1972
#535197