Finite Group Databasebeta

← #27164663

Subgroups — #27164663

Order 12736.

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×#15691885C211yesmaximal, characteristic, complemented
2C2×D4×D199C211yesmaximal, characteristic, complemented
3#20263943C211yesmaximal, characteristic, complemented
4#20263939C212yesmaximal, complemented
5#20263935C211yesmaximal, characteristic, complemented
6#20263930C211yesmaximal, characteristic, complemented
7#20263929C212yesmaximal, complemented
8#20263925C212yesmaximal, complemented
9C4×D796C211yesmaximal, characteristic, complemented
10C2×(C2×D199)⋊C4C212yesmaximal, complemented
11C4×D4×C199C211yesmaximal, characteristic, complemented
12C23×D199C2212yes—
13#15691885—24no—
14C2×Dic398C2211yescharacteristic, complemented
15C2×D796C2211yescharacteristic
16C2×D796C2211yescharacteristic
17C2×D796C2212yescomplemented
18C2×C22⋊D199C2212yes—
19C2×C22⋊D199C2212yescomplemented
20D4×D199—24no—
21C22×Dic199C2212yescomplemented
22C2×C4×D199C2211yescharacteristic
23C2×C4×D199C2211yescharacteristic, complemented
24C4⋊Dic199C2211yescharacteristic, complemented
25C4⋊Dic199C2212yescomplemented
26(C2×D199)⋊C4C2211yescharacteristic
27(C2×D199)⋊C4C2212yes—
28(C2×D199)⋊C4C2212yescomplemented
29(C2×D199)⋊C4C2212yescomplemented
30(C2×D199)⋊C4C2212yescomplemented
31(C2×D199)⋊C4C2211yescharacteristic, complemented
32C2×D4×C199C2211yescharacteristic
33C22×C4×C199C2212yescomplemented
34C4⋊C4×C199C2211yescharacteristic, complemented
35C22⋊C4×C199C2212yescomplemented
36C42×C199C2211yescharacteristic, complemented
37C22×D199—24no—
38C22×D199—24no—
39C22×D199—42no—
40C22×D199C2312yes—
41C22×D199C2311yescharacteristic
42C22×D199C2311yescharacteristic
43Dic398—22no—
44D796—22no—
45D796—22no—
46D796—42no—
47C22⋊D199—24no—
48C22⋊D199—24no—
49C2×Dic199—24no—
50C2×Dic199C2311yescharacteristic
51C2×Dic199C2312yes—
52C2×Dic199C2311yescharacteristic
53C4×D199—22no—
54C4×D199—22no—
55C23×C199C2312yes—
56D4×C199D414yescomplemented
57C2×C4×C199—42no—
58C2×C4×C199C2311yescharacteristic
59C2×C4×C199C2311yescharacteristic
60C2×C4×C199C2312yes—
61C2×C4×C199C2311yescharacteristic
62C22×C199—22no—
63C22×C199—22no—
64C22×C199C2×D414yes—
65C22×C199C2411yescharacteristic
66C2×D199—21no—
67C2×D199—21no—
68C2×D199—21no—
69C2×D199—22no—
70C2×D199—21no—
71C2×D199—22no—
72C2×D199—21no—
73C2×D199—22no—
74C2×D199—48no—
75C2×D199#4112yescomplemented
76C4×C199—21no—
77C4×C199—21no—
78C4×C199—22no—
79C4×C199C2×D412yes—
80Dic199—21no—
81Dic199—21no—
82Dic199—22no—
83C2×C199—24no—
84C2×C199C22×D411yescharacteristic
85C2×C199C2×#4111yescharacteristic
86C2×C199H(5,𝔽2)11yescharacteristic
87D199—22no—
88D199—41no—
89D199—42no—
90C199#54611yescharacteristic, complemented, Sylow
91#546—1991nomaximal, Sylow
92C2×#41—1991no—
93C22×D4—1991no—
94#124—1991no—
95#123—1992no—
96C22⋊Q8—1991no—
97#121—1992no—
98#121—1991no—
99#120—1992no—
100C4×D4—1991no—
101C4×D4D19911yescharacteristic, complemented
102C2×C22⋊C4—1992no—
103C24—1992no—
104#41—3984no—
105C2×Q8—1991no—
106C2×D4—3984no—
107C2×D4—1992no—
108C2×D4—1992no—
109C2×D4—1992no—
110C2×D4—1991no—
111C2×D4—1991no—
112C2×D4C2×D19911yescharacteristic
113C22×C4—1992no—
114C22×C4—1991no—
115C22×C4—1991no—
116C22×C4C2×D19912yescomplemented
117C4⋊C4—1991no—
118C4⋊C4—1992no—
119C4⋊C4C2×D19911yescharacteristic, complemented
120C22⋊C4—1991no—
121C22⋊C4—1992no—
122C22⋊C4—1992no—
123C22⋊C4—1992no—
124C22⋊C4—1991no—
125C22⋊C4—1992no—
126C22⋊C4C2×D19912yescomplemented
127C42C2×D19911yescharacteristic, complemented
128C23—7962no—
129C23—3984no—
130C23—3984no—
131C23—1992no—
132C23—1991no—
133C23—1991no—
134C23C22×D19912yes—
135Q8—3982no—
136D4—7962no—
137D4—3982no—
138D4—3984no—
139D4—3984no—
140D4—3982no—
141D4D79614yescomplemented
142C2×C4—42no—
143C2×C4—3984no—
144C2×C4—3982no—
145C2×C4—3982no—
146C2×C4—1991no—
147C2×C4—1992no—
148C2×C4—1991no—
149C2×C4C22×D19911yescharacteristic
150C2×C4C22×D19911yescharacteristic
151C2×C4C22×D19912yes—
152C2×C4C22×D19911yescharacteristic
153C22—22no—
154C22—22no—
155C22—7968no—
156C22—3981no—
157C22—3982no—
158C22—3981no—
159C22—3982no—
160C22—3981no—
161C22—3982no—
162C22—3981no—
163C22—3981no—
164C22—1992no—
165C22C2×D79614yes—
166C22C23×D19911yescharacteristic
167C4—21no—
168C4—21no—
169C4—22no—
170C4—3981no—
171C4—3981no—
172C4—3982no—
173C4C2×D79612yes—
174C2—24no—
175C2—7962no—
176C2—7961no—
177C2—3982no—
178C2C22×D79611yescharacteristic
179C2#41×D19911yescharacteristic
180C2#2026400011yescharacteristic
subgroupG/H
center, FrattiniC22C23×D199
commutator, socle, cosocleC22×C199C24
FittingC4×D4×C199C2
hypercenterC4×D4D199
hypocenterC199#546
LCSG ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×#15691885C2semidirect
2. C2×D4×D199C2semidirect
3. #20263943C2semidirect
4. #20263939C2semidirect
5. #20263935C2semidirect
6. #20263930C2semidirect
7. #20263929C2semidirect
8. #20263925C2semidirect
9. C4×D796C2semidirect
10. C2×(C2×D199)⋊C4C2semidirect
11. C4×D4×C199C2semidirect
14. C2×Dic398C22semidirect
17. C2×D796C22semidirect
19. C2×C22⋊D199C22semidirect
21. C22×Dic199C22semidirect
23. C2×C4×D199C22semidirect
24. C4⋊Dic199C22semidirect
25. C4⋊Dic199C22semidirect
28. (C2×D199)⋊C4C22semidirect
29. (C2×D199)⋊C4C22semidirect
30. (C2×D199)⋊C4C22semidirect
31. (C2×D199)⋊C4C22semidirect
33. C22×C4×C199C22semidirect
34. C4⋊C4×C199C22semidirect
35. C22⋊C4×C199C22semidirect
36. C42×C199C22semidirect
56. D4×C199D4semidirect
75. C2×D199#41semidirect
90. C199#546semidirect
101. C4×D4D199semidirect
116. C22×C4C2×D199semidirect
119. C4⋊C4C2×D199semidirect
126. C22⋊C4C2×D199semidirect
127. C42C2×D199semidirect
141. D4D796semidirect

Maximal subgroups

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

subgroupindex
C2×#156918852
C2×D4×D1992
#202639432
#202639392
#202639352
#202639302
#202639292
#202639252
C4×D7962
C2×(C2×D199)⋊C42
C4×D4×C1992
#546199