Finite Group Databasebeta

← #27161589

Subgroups — #27161589

Order 12608.

330 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 109 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
2C22×C4×D197C211yesmaximal, characteristic, complemented
3#20263308C212yesmaximal, complemented
4#20263306C211yesmaximal, characteristic, complemented
5#20263305C211yesmaximal, characteristic, complemented
6#20263297C211yesmaximal, characteristic, complemented
7#20263294C212yesmaximal, complemented
8D4×Dic197C212yesmaximal, complemented
9#20263286C212yesmaximal, complemented
10#20263284C211yesmaximal, characteristic, complemented
11#121×C197C211yesmaximal, characteristic, complemented
12C23×D197C2211yescharacteristic
13#15691357—24no—
14C2×Dic394C2211yescharacteristic, complemented
15C2×C22⋊D197C2212yes—
16C2×C22⋊D197C2212yes—
17C22×Dic197C2211yescharacteristic
18C22×Dic197C2212yescomplemented
19C2×C4×D197—24no—
20C2×C4×D197C2211yescharacteristic
21C2×C4×D197C2211yescharacteristic
22C2×C4×D197C2211yescharacteristic
23C2×C4×D197C2211yescharacteristic
24C4⋊Dic197C2211yescharacteristic, complemented
25C4⋊Dic197C2212yescomplemented
26#15691337C2212yescomplemented
27C22⋊Dic197C2212yes—
28C22⋊Dic197C2212yescomplemented
29#15691335C2212yes—
30#15691335C2212yes—
31C4×Dic197C2212yescomplemented
32C2×D4×C197C2212yes—
33C2×D4×C197C2211yescharacteristic, complemented
34C22×C4×C197C2211yescharacteristic
35C4⋊C4×C197C2211yescharacteristic, complemented
36C22⋊C4×C197C2212yescomplemented
37C22×D197—22no—
38C22×D197—22no—
39C22×D197—22no—
40C22×D197C2311yescharacteristic
41C22×D197C2311yescharacteristic
42Dic394—22no—
43C22⋊D197—24no—
44C22⋊D197—24no—
45C2×Dic197—22no—
46C2×Dic197—24no—
47C2×Dic197C2312yes—
48C2×Dic197C2312yes—
49C2×Dic197C2311yescharacteristic
50C2×Dic197C2311yescharacteristic
51C4×D197—22no—
52C4×D197—22no—
53C4×D197—22no—
54C4×D197D414yescomplemented
55C23×C197C2312yes—
56C23×C197C2311yescharacteristic
57D4×C197—24no—
58D4×C197—22no—
59C2×C4×C197—22no—
60C2×C4×C197C2311yescharacteristic
61C2×C4×C197C2312yes—
62C2×C4×C197C2311yescharacteristic
63C22×C197—21no—
64C22×C197—21no—
65C22×C197—22no—
66C22×C197—22no—
67C22×C197—22no—
68C22×C197#4112yescomplemented
69C22×C197C2411yescharacteristic
70C2×D197—28no—
71C2×D197—21no—
72C2×D197—21no—
73C2×D197—21no—
74C2×D197—21no—
75C2×D197C2×D412yes—
76C2×D197#4112yescomplemented
77C4×C197—21no—
78C4×C197—22no—
79C4×C197C2×D412yes—
80Dic197—21no—
81Dic197—22no—
82Dic197—22no—
83Dic197C2×D412yes—
84C2×C197—22no—
85C2×C197—42no—
86C2×C197C22×D411yescharacteristic
87C2×C197C2×#4111yescharacteristic
88C2×C197C2×#4111yescharacteristic
89D197—22no—
90D197—22no—
91C197#52511yescharacteristic, complemented, Sylow
92#525—1971nomaximal, Sylow
93C2×#41—1971no—
94C23×C4—1971no—
95#123—1972no—
96C22⋊Q8—1971no—
97C22⋊Q8—1971no—
98#121—1971no—
99#121D19711yescharacteristic, complemented
100#120—1972no—
101C4×D4—1972no—
102C4×D4—1972no—
103#117—1971no—
104C24—1971no—
105#41—3944no—
106C2×Q8—1971no—
107C2×D4—1972no—
108C2×D4—1972no—
109C2×D4C2×D19712yes—
110C2×D4C2×D19711yescharacteristic, complemented
111C22×C4—3944no—
112C22×C4—1971no—
113C22×C4—1972no—
114C22×C4—1971no—
115C22×C4—1971no—
116C22×C4—1971no—
117C22×C4—1971no—
118C22×C4C2×D19711yescharacteristic
119C4⋊C4—1972no—
120C4⋊C4—1971no—
121C4⋊C4—1972no—
122C4⋊C4C2×D19711yescharacteristic, complemented
123C22⋊C4—1972no—
124C22⋊C4—1972no—
125C22⋊C4—1972no—
126C22⋊C4—1972no—
127C22⋊C4C2×D19712yescomplemented
128C42—1972no—
129C23—3942no—
130C23—3942no—
131C23—3942no—
132C23—1971no—
133C23—1971no—
134C23C22×D19712yes—
135C23C22×D19711yescharacteristic
136Q8—3942no—
137D4—24no—
138D4—22no—
139D4—3944no—
140D4—3944no—
141C2×C4—22no—
142C2×C4—3942no—
143C2×C4—3942no—
144C2×C4—3942no—
145C2×C4—3944no—
146C2×C4—3942no—
147C2×C4—1972no—
148C2×C4—1971no—
149C2×C4—1972no—
150C2×C4—1971no—
151C2×C4—1974no—
152C2×C4C22×D19712yes—
153C2×C4C22×D19711yescharacteristic
154C2×C4C22×D19711yescharacteristic
155C22—21no—
156C22—21no—
157C22—22no—
158C22—22no—
159C22—22no—
160C22—3948no—
161C22—3941no—
162C22—3941no—
163C22—3941no—
164C22—3941no—
165C22—1972no—
166C22—1972no—
167C22#1569135712yescomplemented
168C22C23×D19711yescharacteristic
169C4—21no—
170C4—22no—
171C4—3941no—
172C4—3942no—
173C4—3942no—
174C4—1972no—
175C4D4×D19712yes—
176C2—22no—
177C2—42no—
178C2—3942no—
179C2—3942no—
180C2C2×D4×D19711yescharacteristic
181C2#41×D19711yescharacteristic
182C2C2×#1569135711yescharacteristic
subgroupG/H
center, FrattiniC22C23×D197
commutator, socle, cosocleC22×C197C24
Fitting#121×C197C2
hypercenter#121D197
hypocenterC197#525
LCSG ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×#15691357C2semidirect
2. C22×C4×D197C2semidirect
3. #20263308C2semidirect
4. #20263306C2semidirect
5. #20263305C2semidirect
6. #20263297C2semidirect
7. #20263294C2semidirect
8. D4×Dic197C2semidirect
9. #20263286C2semidirect
10. #20263284C2semidirect
11. #121×C197C2semidirect
14. C2×Dic394C22semidirect
18. C22×Dic197C22semidirect
24. C4⋊Dic197C22semidirect
25. C4⋊Dic197C22semidirect
26. #15691337C22semidirect
28. C22⋊Dic197C22semidirect
31. C4×Dic197C22semidirect
33. C2×D4×C197C22semidirect
35. C4⋊C4×C197C22semidirect
36. C22⋊C4×C197C22semidirect
54. C4×D197D4semidirect
68. C22×C197#41semidirect
76. C2×D197#41semidirect
91. C197#525semidirect
99. #121D197semidirect
110. C2×D4C2×D197semidirect
122. C4⋊C4C2×D197semidirect
127. C22⋊C4C2×D197semidirect
167. C22#15691357semidirect

Maximal subgroups

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

subgroupindex
C2×#156913572
C22×C4×D1972
#202633082
#202633062
#202633052
#202632972
#202632942
D4×Dic1972
#202632862
#202632842
#121×C1972
#525197