Finite Group Databasebeta

← #27161640

Subgroups — #27161640

Order 12608.

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×D788C211yesmaximal, characteristic, complemented
2C2×D4×D197C211yesmaximal, characteristic, complemented
3#20263311C212yesmaximal, complemented
4#20263309C211yesmaximal, characteristic, complemented
5#20263301C212yesmaximal, complemented
6#20263299C212yesmaximal, complemented
7#20263296C211yesmaximal, characteristic, complemented
8#20263296C212yesmaximal, complemented
9#20263294C211yesmaximal, characteristic, complemented
10C22⋊C4×D197C211yesmaximal, characteristic, complemented
11#123×C197C211yesmaximal, characteristic, complemented
12C23×D197C2211yescharacteristic
13C23×D197C2211yescharacteristic, complemented
14C23×D197C2211yescharacteristic, complemented
15C2×D788—24no—
16C2×D788C2212yes—
17C2×D788C2212yescomplemented
18C2×D788C2211yescharacteristic, complemented
19C2×D788C2212yescomplemented
20C2×C22⋊D197C2211yescharacteristic
21C2×C22⋊D197C2211yescharacteristic, complemented
22C2×C22⋊D197C2212yescomplemented
23D4×D197—24no—
24C2×C4×D197C2211yescharacteristic, complemented
25C2×C4×D197C2212yescomplemented
26#15691337C2212yescomplemented
27C22⋊Dic197C2211yescharacteristic, complemented
28#15691335C2212yescomplemented
29#15691335C2212yescomplemented
30#15691335C2212yescomplemented
31#15691335C2212yescomplemented
32C2×D4×C197C2211yescharacteristic
33C22×C4×C197C2211yescharacteristic, complemented
34C4⋊C4×C197C2212yescomplemented
35C22⋊C4×C197C2211yescharacteristic, complemented
36C22⋊C4×C197C2212yescomplemented
37C22×D197—22no—
38C22×D197—22no—
39C22×D197—42no—
40C22×D197—42no—
41C22×D197—42no—
42C22×D197D414yescomplemented
43C22×D197C2311yescharacteristic
44C22×D197C2312yes—
45C22×D197C2311yescharacteristic
46C22×D197C2311yescharacteristic
47D788—22no—
48D788—24no—
49D788—24no—
50D788—42no—
51C22⋊D197—22no—
52C22⋊D197—22no—
53C22⋊D197—42no—
54C2×Dic197C2312yes—
55C2×Dic197C2311yescharacteristic
56C4×D197—22no—
57C4×D197—42no—
58C23×C197C2311yescharacteristic
59C23×C197C2311yescharacteristic
60D4×C197—22no—
61C2×C4×C197—22no—
62C2×C4×C197C2312yes—
63C2×C4×C197C2312yes—
64C2×C4×C197C2311yescharacteristic
65C22×C197—22no—
66C22×C197—21no—
67C22×C197—22no—
68C22×C197C2×D412yes—
69C22×C197C2411yescharacteristic
70C2×D197—22no—
71C2×D197—22no—
72C2×D197—21no—
73C2×D197—22no—
74C2×D197—22no—
75C2×D197—22no—
76C2×D197—22no—
77C2×D197—44no—
78C2×D197—44no—
79C2×D197—44no—
80C2×D197C2×D412yes—
81C2×D197C2×D412yes—
82C4×C197—22no—
83C4×C197—21no—
84C4×C197—22no—
85Dic197—21no—
86Dic197—22no—
87C2×C197—22no—
88C2×C197—41no—
89C2×C197C22×D411yescharacteristic
90C2×C197H(5,𝔽2)12yes—
91D197—22no—
92D197—22no—
93D197—41no—
94D197—42no—
95C197#53411yescharacteristic, complemented, Sylow
96#534—1971nomaximal, Sylow
97C22×D4—1971no—
98C22×D4—1971no—
99#123—1971no—
100#123—1972no—
101#123D19711yescharacteristic, complemented
102#121—1972no—
103#121—1972no—
104#120—1971no—
105#120—1972no—
106#120—1971no—
107C2×C22⋊C4—1971no—
108C24—1971no—
109C24—1971no—
110C24—1971no—
111C2×D4—3944no—
112C2×D4—3944no—
113C2×D4—1972no—
114C2×D4—1971no—
115C2×D4—1971no—
116C2×D4—1972no—
117C2×D4—1972no—
118C2×D4—1971no—
119C2×D4—1972no—
120C2×D4C2×D19711yescharacteristic
121C22×C4—1972no—
122C22×C4—1971no—
123C22×C4C2×D19711yescharacteristic, complemented
124C4⋊C4—1972no—
125C4⋊C4C2×D19712yescomplemented
126C22⋊C4—1971no—
127C22⋊C4—1972no—
128C22⋊C4—1972no—
129C22⋊C4—1972no—
130C22⋊C4—1972no—
131C22⋊C4C2×D19711yescharacteristic, complemented
132C22⋊C4C2×D19712yescomplemented
133C23—7882no—
134C23—7882no—
135C23—7882no—
136C23—3942no—
137C23—3942no—
138C23—1974no—
139C23—1972no—
140C23—1971no—
141C23—1971no—
142C23—1971no—
143C23C22×D19711yescharacteristic
144C23C22×D19711yescharacteristic
145D4—22no—
146D4—7882no—
147D4—7882no—
148D4—3944no—
149D4—3944no—
150D4—3942no—
151D4—3942no—
152D4—3942no—
153C2×C4—22no—
154C2×C4—7882no—
155C2×C4—3942no—
156C2×C4—1972no—
157C2×C4—1971no—
158C2×C4C22×D19712yes—
159C2×C4C22×D19711yescharacteristic
160C2×C4C22×D19712yes—
161C22—22no—
162C22—21no—
163C22—22no—
164C22—7884no—
165C22—7884no—
166C22—7884no—
167C22—3942no—
168C22—3941no—
169C22—3942no—
170C22—3942no—
171C22—3942no—
172C22—3942no—
173C22—3942no—
174C22—1972no—
175C22—1972no—
176C22D4×D19712yes—
177C22C23×D19711yescharacteristic
178C4—22no—
179C4—21no—
180C4—22no—
181C4—3941no—
182C4—3942no—
183C2—22no—
184C2—41no—
185C2—7882no—
186C2—7881no—
187C2—3942no—
188C2—3942no—
189C2C2×D4×D19711yescharacteristic
190C2#2026337212yes—
subgroupG/H
center, FrattiniC22C23×D197
commutator, socle, cosocleC22×C197C24
Fitting#123×C197C2
hypercenter#123D197
hypocenterC197#534
LCSG ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
1. C22×D788C2semidirect
2. C2×D4×D197C2semidirect
3. #20263311C2semidirect
4. #20263309C2semidirect
5. #20263301C2semidirect
6. #20263299C2semidirect
7. #20263296C2semidirect
8. #20263296C2semidirect
9. #20263294C2semidirect
10. C22⋊C4×D197C2semidirect
11. #123×C197C2semidirect
13. C23×D197C22semidirect
14. C23×D197C22semidirect
17. C2×D788C22semidirect
18. C2×D788C22semidirect
19. C2×D788C22semidirect
21. C2×C22⋊D197C22semidirect
22. C2×C22⋊D197C22semidirect
24. C2×C4×D197C22semidirect
25. C2×C4×D197C22semidirect
26. #15691337C22semidirect
27. C22⋊Dic197C22semidirect
28. #15691335C22semidirect
29. #15691335C22semidirect
30. #15691335C22semidirect
31. #15691335C22semidirect
33. C22×C4×C197C22semidirect
34. C4⋊C4×C197C22semidirect
35. C22⋊C4×C197C22semidirect
36. C22⋊C4×C197C22semidirect
42. C22×D197D4semidirect
95. C197#534semidirect
101. #123D197semidirect
123. C22×C4C2×D197semidirect
125. C4⋊C4C2×D197semidirect
131. C22⋊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
C22×D7882
C2×D4×D1972
#202633112
#202633092
#202633012
#202632992
#202632962
#202632942
C22⋊C4×D1972
#123×C1972
#534197