Finite Group Databasebeta

← #27164531

Subgroups — #27164531

Order 12736.

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×#15691887C211yesmaximal, characteristic, complemented
2C22×C4×D199C211yesmaximal, characteristic, complemented
3#20263940C212yesmaximal, complemented
4#20263931C211yesmaximal, characteristic, complemented
5#20263930C211yesmaximal, characteristic, complemented
6#20263929C211yesmaximal, characteristic, complemented
7#20263925C212yesmaximal, complemented
8#20263917C212yesmaximal, complemented
9(C22⋊D199)⋊C4C212yesmaximal, complemented
10#20263913C211yesmaximal, characteristic, complemented
11C22⋊Q8×C199C211yesmaximal, characteristic, complemented
12C23×D199C2211yescharacteristic
13#15691887—24no—
14C2×D796C2212yes—
15C2×D796C2212yes—
16C2×D796C2211yescharacteristic, complemented
17C2×C22⋊D199C2212yes—
18C22×Dic199C2211yescharacteristic
19C2×C4×D199—24no—
20C2×C4×D199C2211yescharacteristic
21C2×C4×D199C2211yescharacteristic
22C2×C4×D199C2211yescharacteristic
23C2×C4×D199C2211yescharacteristic
24C2×C4×D199C2212yescomplemented
25C4⋊Dic199C2211yescharacteristic, complemented
26Dic199⋊C4C2212yescomplemented
27(C2×D199)⋊C4C2212yes—
28(C2×D199)⋊C4C2212yes—
29(C2×D199)⋊C4C2212yescomplemented
30(C2×D199)⋊C4C2212yescomplemented
31C4×Dic199C2212yescomplemented
32C2×Q8×C199C2211yescharacteristic, complemented
33C22×C4×C199C2211yescharacteristic
34C4⋊C4×C199C2212yescomplemented
35C4⋊C4×C199C2211yescharacteristic, complemented
36C22⋊C4×C199C2212yes—
37C22×D199—22no—
38C22×D199—22no—
39C22×D199—22no—
40C22×D199C2312yes—
41C22×D199C2311yescharacteristic
42C22×D199C2311yescharacteristic
43D796—24no—
44D796—24no—
45D796—22no—
46C22⋊D199—24no—
47C2×Dic199—22no—
48C2×Dic199C2312yes—
49C2×Dic199C2311yescharacteristic
50C2×Dic199C2311yescharacteristic
51C4×D199—22no—
52C4×D199—22no—
53C4×D199—22no—
54C4×D199—24no—
55C4×D199D414yescomplemented
56C23×C199C2311yescharacteristic
57Q8×C199—22no—
58C2×C4×C199—22no—
59C2×C4×C199C2312yes—
60C2×C4×C199C2312yes—
61C2×C4×C199C2311yescharacteristic
62C2×C4×C199C2311yescharacteristic
63C22×C199—21no—
64C22×C199—21no—
65C22×C199#4112yescomplemented
66C22×C199C2411yescharacteristic
67C2×D199—28no—
68C2×D199—21no—
69C2×D199—21no—
70C2×D199—21no—
71C2×D199—21no—
72C2×D199—22no—
73C2×D199—22no—
74C2×D199—22no—
75C2×D199C2×D412yes—
76C2×D199#4112yescomplemented
77C4×C199—21no—
78C4×C199—22no—
79C4×C199—22no—
80C4×C199C2×D412yes—
81Dic199—21no—
82Dic199—22no—
83Dic199C2×D412yes—
84C2×C199—22no—
85C2×C199C22×D411yescharacteristic
86C2×C199C2×#4111yescharacteristic
87C2×C199C2×#4111yescharacteristic
88D199—22no—
89D199—22no—
90D199—42no—
91C199#52511yescharacteristic, complemented, Sylow
92#525—1991nomaximal, Sylow
93C2×#41—1991no—
94C23×C4—1991no—
95#123—1992no—
96C22⋊Q8—1991no—
97C22⋊Q8D19911yescharacteristic, complemented
98#121—1991no—
99#121—1991no—
100#120—1992no—
101C4×D4—1992no—
102C4×D4—1992no—
103#117—1991no—
104C24—1991no—
105#41—3984no—
106C2×Q8C2×D19911yescharacteristic, complemented
107C2×D4—1992no—
108C2×D4—1992no—
109C2×D4—1992no—
110C2×D4—1991no—
111C22×C4—3984no—
112C22×C4—1991no—
113C22×C4—1991no—
114C22×C4—1991no—
115C22×C4—1991no—
116C22×C4—1992no—
117C22×C4—1991no—
118C22×C4C2×D19911yescharacteristic
119C4⋊C4—1992no—
120C4⋊C4—1991no—
121C4⋊C4C2×D19911yescharacteristic, complemented
122C4⋊C4C2×D19912yescomplemented
123C22⋊C4—1992no—
124C22⋊C4—1992no—
125C22⋊C4—1992no—
126C22⋊C4—1992no—
127C22⋊C4C2×D19912yes—
128C42—1992no—
129C23—3982no—
130C23—3982no—
131C23—3982no—
132C23—1992no—
133C23—1991no—
134C23—1991no—
135C23C22×D19911yescharacteristic
136Q8—22no—
137D4—3984no—
138D4—3984no—
139D4—3984no—
140D4—3982no—
141C2×C4—22no—
142C2×C4—3982no—
143C2×C4—3982no—
144C2×C4—3984no—
145C2×C4—3982no—
146C2×C4—3982no—
147C2×C4—1992no—
148C2×C4—1991no—
149C2×C4—1991no—
150C2×C4—1994no—
151C2×C4C22×D19912yes—
152C2×C4C22×D19911yescharacteristic
153C2×C4C22×D19912yes—
154C2×C4C22×D19911yescharacteristic
155C22—21no—
156C22—21no—
157C22—3988no—
158C22—3982no—
159C22—3982no—
160C22—3982no—
161C22—3981no—
162C22—3981no—
163C22—3981no—
164C22—3981no—
165C22—1992no—
166C22—1992no—
167C22#1569188712yescomplemented
168C22C23×D19911yescharacteristic
169C4—21no—
170C4—22no—
171C4—22no—
172C4—3981no—
173C4—3982no—
174C4—1992no—
175C4D4×D19912yes—
176C2—22no—
177C2—7962no—
178C2—3982no—
179C2—3982no—
180C2C2×D4×D19911yescharacteristic
181C2#41×D19911yescharacteristic
182C2C2×#1569188711yescharacteristic
subgroupG/H
center, FrattiniC22C23×D199
commutator, socle, cosocleC22×C199C24
FittingC22⋊Q8×C199C2
hypercenterC22⋊Q8D199
hypocenterC199#525
LCSG ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#15691887C2semidirect
2. C22×C4×D199C2semidirect
3. #20263940C2semidirect
4. #20263931C2semidirect
5. #20263930C2semidirect
6. #20263929C2semidirect
7. #20263925C2semidirect
8. #20263917C2semidirect
9. (C22⋊D199)⋊C4C2semidirect
10. #20263913C2semidirect
11. C22⋊Q8×C199C2semidirect
16. C2×D796C22semidirect
24. C2×C4×D199C22semidirect
25. C4⋊Dic199C22semidirect
26. Dic199⋊C4C22semidirect
29. (C2×D199)⋊C4C22semidirect
30. (C2×D199)⋊C4C22semidirect
31. C4×Dic199C22semidirect
32. C2×Q8×C199C22semidirect
34. C4⋊C4×C199C22semidirect
35. C4⋊C4×C199C22semidirect
55. C4×D199D4semidirect
65. C22×C199#41semidirect
76. C2×D199#41semidirect
91. C199#525semidirect
97. C22⋊Q8D199semidirect
106. C2×Q8C2×D199semidirect
121. C4⋊C4C2×D199semidirect
122. C4⋊C4C2×D199semidirect
167. C22#15691887semidirect

Maximal subgroups

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

subgroupindex
C2×#156918872
C22×C4×D1992
#202639402
#202639312
#202639302
#202639292
#202639252
#202639172
(C22⋊D199)⋊C42
#202639132
C22⋊Q8×C1992
#525199