Finite Group Databasebeta

← #27161720

Subgroups — #27161720

Order 12608.

334 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 107 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×#15691359C211yesmaximal, characteristic, complemented
2C22×D788C211yesmaximal, characteristic, complemented
3#20263315C211yesmaximal, characteristic, complemented
4#20263311C212yesmaximal, complemented
5#20263301C211yesmaximal, characteristic, complemented
6#20263301C212yesmaximal, complemented
7#20263296C212yesmaximal, complemented
8#20263288C211yesmaximal, characteristic, complemented
9C4×C22⋊D197C211yesmaximal, characteristic, complemented
10C22⋊C4×D197C212yesmaximal, complemented
11C22⋊Q8×C197C211yesmaximal, characteristic, complemented
12C23×D197C2212yes—
13#15691359—24no—
14C2×D788—24no—
15C2×D788C2212yes—
16C2×D788C2211yescharacteristic
17C2×D788C2211yescharacteristic
18C2×D788C2212yescomplemented
19C2×D788C2212yescomplemented
20C2×C22⋊D197C2211yescharacteristic
21C2×C4×D197C2212yescomplemented
22C2×C4×D197C2211yescharacteristic, complemented
23C2×C4×D197C2212yescomplemented
24#15691337C2211yescharacteristic, complemented
25C22⋊Dic197C2211yescharacteristic
26#15691335C2212yescomplemented
27#15691335C2212yescomplemented
28#15691335C2211yescharacteristic, complemented
29#15691335C2212yescomplemented
30#15691335C2212yescomplemented
31C4×Dic197C2211yescharacteristic, complemented
32C2×Q8×C197C2211yescharacteristic, complemented
33C22×C4×C197C2211yescharacteristic
34C4⋊C4×C197C2211yescharacteristic, complemented
35C4⋊C4×C197C2212yescomplemented
36C22⋊C4×C197C2212yes—
37C22×D197—24no—
38C22×D197—24no—
39C22×D197—42no—
40C22×D197C2312yes—
41C22×D197C2311yescharacteristic
42C22×D197C2312yes—
43D788—24no—
44D788—24no—
45D788—22no—
46D788—42no—
47D788D414yescomplemented
48C22⋊D197—22no—
49C2×Dic197C2311yescharacteristic
50C2×Dic197C2311yescharacteristic
51C2×Dic197C2311yescharacteristic
52C4×D197—24no—
53C4×D197—22no—
54C4×D197—42no—
55C23×C197C2311yescharacteristic
56Q8×C197—22no—
57C2×C4×C197—22no—
58C2×C4×C197C2312yes—
59C2×C4×C197C2312yes—
60C2×C4×C197C2311yescharacteristic
61C2×C4×C197C2311yescharacteristic
62C22×C197—21no—
63C22×C197—21no—
64C22×C197#4112yescomplemented
65C22×C197C2411yescharacteristic
66C2×D197—22no—
67C2×D197—22no—
68C2×D197—21no—
69C2×D197—22no—
70C2×D197—21no—
71C2×D197—22no—
72C2×D197—21no—
73C2×D197—22no—
74C2×D197—48no—
75C2×D197C2×D414yes—
76C4×C197—21no—
77C4×C197—22no—
78C4×C197—22no—
79C4×C197C2×D412yes—
80Dic197—21no—
81Dic197—21no—
82Dic197—21no—
83C2×C197—22no—
84C2×C197C22×D411yescharacteristic
85C2×C197C2×#4111yescharacteristic
86C2×C197H(5,𝔽2)11yescharacteristic
87D197—24no—
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⋊Q8D19711yescharacteristic, complemented
97#121—1971no—
98#121—1972no—
99#120—1972no—
100C4×D4—1971no—
101C4×D4—1971no—
102C2×C22⋊C4—1972no—
103C24—1972no—
104#41—3944no—
105C2×Q8C2×D19711yescharacteristic, complemented
106C2×D4—3944no—
107C2×D4—1971no—
108C2×D4—1972no—
109C2×D4—1972no—
110C2×D4—1971no—
111C2×D4—1972no—
112C2×D4—1971no—
113C22×C4—1972no—
114C22×C4—1972no—
115C22×C4—1971no—
116C22×C4C2×D19711yescharacteristic
117C4⋊C4—1971no—
118C4⋊C4C2×D19711yescharacteristic, complemented
119C4⋊C4C2×D19712yescomplemented
120C22⋊C4—1971no—
121C22⋊C4—1972no—
122C22⋊C4—1972no—
123C22⋊C4—1972no—
124C22⋊C4—1971no—
125C22⋊C4—1972no—
126C22⋊C4C2×D19712yes—
127C42—1971no—
128C23—7882no—
129C23—3944no—
130C23—3944no—
131C23—1972no—
132C23—1971no—
133C23—1972no—
134C23C22×D19711yescharacteristic
135Q8—22no—
136D4—7882no—
137D4—3944no—
138D4—3944no—
139D4—3942no—
140D4—3942no—
141D4—1974no—
142C2×C4—22no—
143C2×C4—7882no—
144C2×C4—3944no—
145C2×C4—3942no—
146C2×C4—1971no—
147C2×C4—1971no—
148C2×C4—1971no—
149C2×C4C22×D19712yes—
150C2×C4C22×D19712yes—
151C2×C4C22×D19711yescharacteristic
152C2×C4C22×D19711yescharacteristic
153C22—21no—
154C22—21no—
155C22—7888no—
156C22—3941no—
157C22—3942no—
158C22—3941no—
159C22—3942no—
160C22—3941no—
161C22—3942no—
162C22—3942no—
163C22—3942no—
164C22—1974no—
165C22#1569135912yescomplemented
166C22C23×D19711yescharacteristic
167C4—21no—
168C4—22no—
169C4—22no—
170C4—3941no—
171C4—3941no—
172C4—3941no—
173C4D4×D19712yes—
174C2—22no—
175C2—7882no—
176C2—7881no—
177C2—3944no—
178C2C2×D4×D19711yescharacteristic
179C2C2×#1569135911yescharacteristic
180C2#2026337211yescharacteristic
subgroupG/H
center, FrattiniC22C23×D197
commutator, socle, cosocleC22×C197C24
FittingC22⋊Q8×C197C2
hypercenterC22⋊Q8D197
hypocenterC197#546
LCSG ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#15691359C2semidirect
2. C22×D788C2semidirect
3. #20263315C2semidirect
4. #20263311C2semidirect
5. #20263301C2semidirect
6. #20263301C2semidirect
7. #20263296C2semidirect
8. #20263288C2semidirect
9. C4×C22⋊D197C2semidirect
10. C22⋊C4×D197C2semidirect
11. C22⋊Q8×C197C2semidirect
18. C2×D788C22semidirect
19. C2×D788C22semidirect
21. C2×C4×D197C22semidirect
22. C2×C4×D197C22semidirect
23. C2×C4×D197C22semidirect
24. #15691337C22semidirect
26. #15691335C22semidirect
27. #15691335C22semidirect
28. #15691335C22semidirect
29. #15691335C22semidirect
30. #15691335C22semidirect
31. C4×Dic197C22semidirect
32. C2×Q8×C197C22semidirect
34. C4⋊C4×C197C22semidirect
35. C4⋊C4×C197C22semidirect
47. D788D4semidirect
64. C22×C197#41semidirect
90. C197#546semidirect
96. C22⋊Q8D197semidirect
105. C2×Q8C2×D197semidirect
118. C4⋊C4C2×D197semidirect
119. C4⋊C4C2×D197semidirect
165. C22#15691359semidirect

Maximal subgroups

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

subgroupindex
C2×#156913592
C22×D7882
#202633152
#202633112
#202633012
#202632962
#202632882
C4×C22⋊D1972
C22⋊C4×D1972
C22⋊Q8×C1972
#546197