Finite Group Databasebeta

← #27161637

Subgroups — #27161637

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
1C2×D4×D197C212yesmaximal, complemented
2#20263311C212yesmaximal, complemented
3#20263308C212yesmaximal, complemented
4#20263300C211yesmaximal, characteristic, complemented
5#20263298C212yesmaximal, complemented
6#20263296C212yesmaximal, complemented
7#20263294C212yesmaximal, complemented
8C2×#15691335C211yesmaximal, characteristic, complemented
9#121×C197C211yesmaximal, characteristic, complemented
10C23×D197C2211yescharacteristic
11C23×D197C2212yescomplemented
12C2×D788C2212yes—
13C2×D788C2211yescharacteristic, complemented
14C2×C22⋊D197C2212yes—
15C2×C22⋊D197C2212yescomplemented
16C2×C22⋊D197C2212yescomplemented
17D4×D197—28no—
18C22×Dic197C2211yescharacteristic, complemented
19C2×C4×D197C2212yescomplemented
20C4⋊Dic197C2211yescharacteristic, complemented
21#15691337C2212yescomplemented
22C22⋊Dic197C2212yescomplemented
23#15691335C2212yescomplemented
24#15691335C2212yescomplemented
25#15691335C2212yescomplemented
26#15691335C2212yescomplemented
27C2×D4×C197C2212yescomplemented
28C2×D4×C197C2211yescharacteristic, complemented
29C22×C4×C197C2211yescharacteristic, complemented
30C4⋊C4×C197C2211yescharacteristic, complemented
31C22⋊C4×C197C2212yescomplemented
32C22×D197—24no—
33C22×D197—41no—
34C22×D197—41no—
35C22×D197—42no—
36C22×D197—42no—
37C22×D197D414yescomplemented
38C22×D197C2312yes—
39C22×D197C2312yes—
40D788—24no—
41D788—41no—
42C22⋊D197—24no—
43C22⋊D197—24no—
44C22⋊D197—42no—
45C2×Dic197—41no—
46C2×Dic197C2312yes—
47C2×Dic197C2312yes—
48C4×D197—24no—
49C23×C197C2312yes—
50C23×C197C2311yescharacteristic
51D4×C197—24no—
52D4×C197—41no—
53C2×C4×C197—41no—
54C2×C4×C197C2311yescharacteristic
55C2×C4×C197C2312yes—
56C2×C4×C197C2311yescharacteristic
57C22×C197—21no—
58C22×C197—21no—
59C22×C197—22no—
60C22×C197—22no—
61C22×C197—22no—
62C22×C197C2×D412yes—
63C22×C197C2411yescharacteristic
64C2×D197—22no—
65C2×D197—22no—
66C2×D197—22no—
67C2×D197—22no—
68C2×D197—22no—
69C2×D197—44no—
70C2×D197—48no—
71C2×D197C2×D414yes—
72C4×C197—22no—
73C4×C197—21no—
74C4×C197—21no—
75Dic197—22no—
76Dic197—22no—
77C2×C197—22no—
78C2×C197—42no—
79C2×C197C22×D411yescharacteristic
80C2×C197H(5,𝔽2)11yescharacteristic
81C2×C197H(5,𝔽2)11yescharacteristic
82D197—24no—
83D197—42no—
84C197#53411yescharacteristic, complemented, Sylow
85#534—1971nomaximal, Sylow
86C22×D4—1972no—
87#123—1972no—
88#123—1972no—
89#121—1972no—
90#121—1971no—
91#121D19711yescharacteristic, complemented
92#120—1972no—
93#120—1972no—
94C2×C22⋊C4—1971no—
95C24—1972no—
96C24—1971no—
97C2×D4—3948no—
98C2×D4—1972no—
99C2×D4—1972no—
100C2×D4—1972no—
101C2×D4—1971no—
102C2×D4—1972no—
103C2×D4C2×D19712yescomplemented
104C2×D4C2×D19711yescharacteristic, complemented
105C22×C4—1971no—
106C22×C4—1972no—
107C22×C4C2×D19711yescharacteristic, complemented
108C4⋊C4—1972no—
109C4⋊C4—1971no—
110C4⋊C4C2×D19711yescharacteristic, complemented
111C22⋊C4—1972no—
112C22⋊C4—1972no—
113C22⋊C4—1972no—
114C22⋊C4—1972no—
115C22⋊C4—1972no—
116C22⋊C4C2×D19712yescomplemented
117C23—7881no—
118C23—7881no—
119C23—7882no—
120C23—7882no—
121C23—3944no—
122C23—1974no—
123C23—1972no—
124C23—1972no—
125C23C22×D19712yes—
126C23C22×D19711yescharacteristic
127D4—24no—
128D4—41no—
129D4—7881no—
130D4—7882no—
131D4—3944no—
132D4—3944no—
133D4—3944no—
134C2×C4—41no—
135C2×C4—7881no—
136C2×C4—3944no—
137C2×C4—1972no—
138C2×C4—1972no—
139C2×C4C22×D19711yescharacteristic
140C2×C4C22×D19711yescharacteristic
141C2×C4C22×D19712yes—
142C22—21no—
143C22—21no—
144C22—22no—
145C22—22no—
146C22—22no—
147C22—7884no—
148C22—7888no—
149C22—3942no—
150C22—3942no—
151C22—3942no—
152C22—3942no—
153C22—3942no—
154C22—1974no—
155C22D4×D19712yes—
156C22C23×D19711yescharacteristic
157C4—22no—
158C4—21no—
159C4—21no—
160C4—3942no—
161C4—3942no—
162C2—22no—
163C2—42no—
164C2—7882no—
165C2—3944no—
166C2C2×D4×D19711yescharacteristic
167C2#2026337111yescharacteristic
168C2#2026337211yescharacteristic
subgroupG/H
center, FrattiniC22C23×D197
commutator, socle, cosocleC22×C197C24
Fitting#121×C197C2
hypercenter#121D197
hypocenterC197#534
LCSG ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×D4×D197C2semidirect
2. #20263311C2semidirect
3. #20263308C2semidirect
4. #20263300C2semidirect
5. #20263298C2semidirect
6. #20263296C2semidirect
7. #20263294C2semidirect
8. C2×#15691335C2semidirect
9. #121×C197C2semidirect
11. C23×D197C22semidirect
13. C2×D788C22semidirect
15. C2×C22⋊D197C22semidirect
16. C2×C22⋊D197C22semidirect
18. C22×Dic197C22semidirect
19. C2×C4×D197C22semidirect
20. C4⋊Dic197C22semidirect
21. #15691337C22semidirect
22. C22⋊Dic197C22semidirect
23. #15691335C22semidirect
24. #15691335C22semidirect
25. #15691335C22semidirect
26. #15691335C22semidirect
27. C2×D4×C197C22semidirect
28. C2×D4×C197C22semidirect
29. C22×C4×C197C22semidirect
30. C4⋊C4×C197C22semidirect
31. C22⋊C4×C197C22semidirect
37. C22×D197D4semidirect
84. C197#534semidirect
91. #121D197semidirect
103. C2×D4C2×D197semidirect
104. C2×D4C2×D197semidirect
107. C22×C4C2×D197semidirect
110. C4⋊C4C2×D197semidirect
116. C22⋊C4C2×D197semidirect

Maximal subgroups

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

subgroupindex
C2×D4×D1972
#202633112
#202633082
#202633002
#202632982
#202632962
#202632942
C2×#156913352
#121×C1972
#534197