Finite Group Databasebeta

← #27161724

Subgroups — #27161724

Order 12608.

334 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 113 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×#15691357C211yesmaximal, characteristic, complemented
2C2×D4×D197C211yesmaximal, characteristic, complemented
3#20263314C211yesmaximal, characteristic, complemented
4#20263310C212yesmaximal, complemented
5#20263306C211yesmaximal, characteristic, complemented
6#20263301C211yesmaximal, characteristic, complemented
7#20263300C212yesmaximal, complemented
8#20263296C212yesmaximal, complemented
9C4×D788C211yesmaximal, characteristic, complemented
10C2×#15691335C212yesmaximal, complemented
11C4×D4×C197C211yesmaximal, characteristic, complemented
12C23×D197C2212yes—
13#15691357—24no—
14C2×Dic394C2211yescharacteristic, complemented
15C2×D788C2211yescharacteristic
16C2×D788C2211yescharacteristic
17C2×D788C2212yescomplemented
18C2×C22⋊D197C2212yes—
19C2×C22⋊D197C2212yescomplemented
20D4×D197—24no—
21C22×Dic197C2212yescomplemented
22C2×C4×D197C2211yescharacteristic
23C2×C4×D197C2211yescharacteristic, complemented
24C4⋊Dic197C2211yescharacteristic, complemented
25C4⋊Dic197C2212yescomplemented
26#15691335C2211yescharacteristic
27#15691335C2212yes—
28#15691335C2212yescomplemented
29#15691335C2212yescomplemented
30#15691335C2211yescharacteristic, complemented
31#15691335C2212yescomplemented
32C2×D4×C197C2211yescharacteristic
33C22×C4×C197C2212yescomplemented
34C4⋊C4×C197C2211yescharacteristic, complemented
35C22⋊C4×C197C2212yescomplemented
36C42×C197C2211yescharacteristic, complemented
37C22×D197—24no—
38C22×D197—24no—
39C22×D197—42no—
40C22×D197C2312yes—
41C22×D197C2311yescharacteristic
42C22×D197C2311yescharacteristic
43Dic394—22no—
44D788—22no—
45D788—22no—
46D788—42no—
47C22⋊D197—24no—
48C22⋊D197—24no—
49C2×Dic197—24no—
50C2×Dic197C2311yescharacteristic
51C2×Dic197C2312yes—
52C2×Dic197C2311yescharacteristic
53C4×D197—22no—
54C4×D197—22no—
55C23×C197C2312yes—
56D4×C197D414yescomplemented
57C2×C4×C197—42no—
58C2×C4×C197C2311yescharacteristic
59C2×C4×C197C2311yescharacteristic
60C2×C4×C197C2312yes—
61C2×C4×C197C2311yescharacteristic
62C22×C197—22no—
63C22×C197—22no—
64C22×C197C2×D414yes—
65C22×C197C2411yescharacteristic
66C2×D197—21no—
67C2×D197—21no—
68C2×D197—21no—
69C2×D197—22no—
70C2×D197—21no—
71C2×D197—22no—
72C2×D197—21no—
73C2×D197—22no—
74C2×D197—48no—
75C2×D197#4112yescomplemented
76C4×C197—21no—
77C4×C197—21no—
78C4×C197—22no—
79C4×C197C2×D412yes—
80Dic197—21no—
81Dic197—21no—
82Dic197—22no—
83C2×C197—24no—
84C2×C197C22×D411yescharacteristic
85C2×C197C2×#4111yescharacteristic
86C2×C197H(5,𝔽2)11yescharacteristic
87D197—22no—
88D197—41no—
89D197—42no—
90C197#54611yescharacteristic, complemented, Sylow
91#546—1971nomaximal, Sylow
92C2×#41—1971no—
93C22×D4—1971no—
94#124—1971no—
95#123—1972no—
96C22⋊Q8—1971no—
97#121—1972no—
98#121—1971no—
99#120—1972no—
100C4×D4—1971no—
101C4×D4D19711yescharacteristic, complemented
102C2×C22⋊C4—1972no—
103C24—1972no—
104#41—3944no—
105C2×Q8—1971no—
106C2×D4—3944no—
107C2×D4—1972no—
108C2×D4—1972no—
109C2×D4—1972no—
110C2×D4—1971no—
111C2×D4—1971no—
112C2×D4C2×D19711yescharacteristic
113C22×C4—1972no—
114C22×C4—1971no—
115C22×C4—1971no—
116C22×C4C2×D19712yescomplemented
117C4⋊C4—1971no—
118C4⋊C4—1972no—
119C4⋊C4C2×D19711yescharacteristic, complemented
120C22⋊C4—1971no—
121C22⋊C4—1972no—
122C22⋊C4—1972no—
123C22⋊C4—1972no—
124C22⋊C4—1971no—
125C22⋊C4—1972no—
126C22⋊C4C2×D19712yescomplemented
127C42C2×D19711yescharacteristic, complemented
128C23—7882no—
129C23—3944no—
130C23—3944no—
131C23—1972no—
132C23—1971no—
133C23—1971no—
134C23C22×D19712yes—
135Q8—3942no—
136D4—7882no—
137D4—3942no—
138D4—3944no—
139D4—3944no—
140D4—3942no—
141D4D78814yescomplemented
142C2×C4—42no—
143C2×C4—3944no—
144C2×C4—3942no—
145C2×C4—3942no—
146C2×C4—1971no—
147C2×C4—1972no—
148C2×C4—1971no—
149C2×C4C22×D19711yescharacteristic
150C2×C4C22×D19711yescharacteristic
151C2×C4C22×D19712yes—
152C2×C4C22×D19711yescharacteristic
153C22—22no—
154C22—22no—
155C22—7888no—
156C22—3941no—
157C22—3942no—
158C22—3941no—
159C22—3942no—
160C22—3941no—
161C22—3942no—
162C22—3941no—
163C22—3941no—
164C22—1972no—
165C22C2×D78814yes—
166C22C23×D19711yescharacteristic
167C4—21no—
168C4—21no—
169C4—22no—
170C4—3941no—
171C4—3941no—
172C4—3942no—
173C4C2×D78812yes—
174C2—24no—
175C2—7882no—
176C2—7881no—
177C2—3942no—
178C2C22×D78811yescharacteristic
179C2#41×D19711yescharacteristic
180C2#2026337111yescharacteristic
subgroupG/H
center, FrattiniC22C23×D197
commutator, socle, cosocleC22×C197C24
FittingC4×D4×C197C2
hypercenterC4×D4D197
hypocenterC197#546
LCSG ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×#15691357C2semidirect
2. C2×D4×D197C2semidirect
3. #20263314C2semidirect
4. #20263310C2semidirect
5. #20263306C2semidirect
6. #20263301C2semidirect
7. #20263300C2semidirect
8. #20263296C2semidirect
9. C4×D788C2semidirect
10. C2×#15691335C2semidirect
11. C4×D4×C197C2semidirect
14. C2×Dic394C22semidirect
17. C2×D788C22semidirect
19. C2×C22⋊D197C22semidirect
21. C22×Dic197C22semidirect
23. C2×C4×D197C22semidirect
24. C4⋊Dic197C22semidirect
25. C4⋊Dic197C22semidirect
28. #15691335C22semidirect
29. #15691335C22semidirect
30. #15691335C22semidirect
31. #15691335C22semidirect
33. C22×C4×C197C22semidirect
34. C4⋊C4×C197C22semidirect
35. C22⋊C4×C197C22semidirect
36. C42×C197C22semidirect
56. D4×C197D4semidirect
75. C2×D197#41semidirect
90. C197#546semidirect
101. C4×D4D197semidirect
116. C22×C4C2×D197semidirect
119. C4⋊C4C2×D197semidirect
126. C22⋊C4C2×D197semidirect
127. C42C2×D197semidirect
141. D4D788semidirect

Maximal subgroups

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

subgroupindex
C2×#156913572
C2×D4×D1972
#202633142
#202633102
#202633062
#202633012
#202633002
#202632962
C4×D7882
C2×#156913352
C4×D4×C1972
#546197