Finite Group Databasebeta

← #27164662

Subgroups — #27164662

Order 12736.

334 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 115 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
2#20263942C211yesmaximal, characteristic, complemented
3#20263936C212yesmaximal, complemented
4#20263934C211yesmaximal, characteristic, complemented
5#20263927C212yesmaximal, complemented
6#20263926C211yesmaximal, characteristic, complemented
7#20263924C212yesmaximal, complemented
8D4×Dic199C211yesmaximal, characteristic, complemented
9C4×C22⋊D199C211yesmaximal, characteristic, complemented
10C2×C22⋊Dic199C212yesmaximal, complemented
11C22×D4×C199C211yesmaximal, characteristic, complemented
12#15691885—24no—
13C2×Dic398C2211yescharacteristic, complemented
14C2×C22⋊D199C2211yescharacteristic
15C2×C22⋊D199C2212yescomplemented
16C2×C22⋊D199C2212yescomplemented
17C22×Dic199C2212yescomplemented
18C22×Dic199C2212yescomplemented
19C2×C4×D199C2211yescharacteristic, complemented
20C4⋊Dic199C2211yescharacteristic, complemented
21Dic199⋊C4C2211yescharacteristic, complemented
22Dic199⋊C4C2212yescomplemented
23C22⋊Dic199C2211yescharacteristic
24C22⋊Dic199C2212yes—
25C22⋊Dic199C2212yescomplemented
26C22⋊Dic199C2212yescomplemented
27C22⋊Dic199C2212yescomplemented
28C22⋊Dic199C2212yescomplemented
29(C2×D199)⋊C4C2211yescharacteristic, complemented
30C4×Dic199C2211yescharacteristic, complemented
31C24×C199C2212yes—
32C2×D4×C199—24no—
33C2×D4×C199C2212yes—
34C2×D4×C199C2211yescharacteristic
35C2×D4×C199C2211yescharacteristic
36C22×C4×C199C2211yescharacteristic
37C22×D199C2311yescharacteristic
38Dic398—22no—
39C22⋊D199—24no—
40C22⋊D199—22no—
41C22⋊D199—42no—
42C2×Dic199—24no—
43C2×Dic199—42no—
44C2×Dic199C2311yescharacteristic
45C2×Dic199C2311yescharacteristic
46C2×Dic199C2312yes—
47C2×Dic199C2311yescharacteristic
48C2×Dic199C2312yes—
49C4×D199—22no—
50C23×C199—24no—
51C23×C199—24no—
52C23×C199—42no—
53C23×C199C2312yes—
54C23×C199C2312yes—
55C23×C199C2311yescharacteristic
56D4×C199—24no—
57D4×C199—22no—
58D4×C199D414yescomplemented
59C2×C4×C199—22no—
60C2×C4×C199C2311yescharacteristic
61C2×C4×C199C2311yescharacteristic
62C22×C199—21no—
63C22×C199—22no—
64C22×C199—21no—
65C22×C199—22no—
66C22×C199—22no—
67C22×C199—22no—
68C22×C199—22no—
69C22×C199—48no—
70C22×C199C2×D414yes—
71C22×C199#4112yescomplemented
72C22×C199C2411yescharacteristic
73C2×D199—21no—
74C2×D199—21no—
75C2×D199—21no—
76C4×C199—21no—
77C4×C199C2×D412yes—
78Dic199—21no—
79Dic199—22no—
80Dic199—21no—
81Dic199—21no—
82Dic199—22no—
83C2×C199—22no—
84C2×C199—24no—
85C2×C199—42no—
86C2×C199C22×D411yescharacteristic
87C2×C199C2×#4111yescharacteristic
88C2×C199H(5,𝔽2)11yescharacteristic
89D199—41no—
90C199#54611yescharacteristic, complemented, Sylow
91#546—1991nomaximal, Sylow
92C2×#41—1991no—
93C22×D4D19911yescharacteristic, complemented
94#124—1991no—
95#123—1992no—
96C22⋊Q8—1991no—
97#121—1991no—
98#121—1992no—
99#120—1992no—
100C4×D4—1991no—
101C4×D4—1991no—
102C2×C22⋊C4—1992no—
103C24C2×D19912yes—
104#41—3984no—
105C2×Q8—1991no—
106C2×D4—24no—
107C2×D4—1992no—
108C2×D4—1991no—
109C2×D4—1992no—
110C2×D4C2×D19912yes—
111C2×D4C2×D19911yescharacteristic
112C2×D4C2×D19911yescharacteristic
113C22×C4—1992no—
114C22×C4—1992no—
115C22×C4—1991no—
116C22×C4C2×D19911yescharacteristic
117C4⋊C4—1991no—
118C4⋊C4—1992no—
119C4⋊C4—1991no—
120C22⋊C4—1991no—
121C22⋊C4—1992no—
122C22⋊C4—1992no—
123C22⋊C4—1992no—
124C22⋊C4—1992no—
125C22⋊C4—1992no—
126C22⋊C4—1991no—
127C42—1991no—
128C23—24no—
129C23—24no—
130C23—42no—
131C23—1991no—
132C23C22×D19912yes—
133C23C22×D19912yes—
134C23C22×D19911yescharacteristic
135Q8—3982no—
136D4—24no—
137D4—22no—
138D4—7962no—
139D4—3984no—
140D4—3982no—
141D4C22⋊D19914yescomplemented
142C2×C4—22no—
143C2×C4—7962no—
144C2×C4—3984no—
145C2×C4—3982no—
146C2×C4—1991no—
147C2×C4—1991no—
148C2×C4—1992no—
149C2×C4—1991no—
150C2×C4—1992no—
151C2×C4C22×D19911yescharacteristic
152C2×C4C22×D19911yescharacteristic
153C22—21no—
154C22—22no—
155C22—22no—
156C22—21no—
157C22—22no—
158C22—22no—
159C22—22no—
160C22—48no—
161C22—3981no—
162C22—3981no—
163C22—3981no—
164C22C2×C22⋊D19914yes—
165C22#1569188512yescomplemented
166C22C23×D19911yescharacteristic
167C4—21no—
168C4—3981no—
169C4—3982no—
170C4—3981no—
171C4—3981no—
172C4—3982no—
173C4C2×C22⋊D19912yes—
174C2—22no—
175C2—24no—
176C2—42no—
177C2—7961no—
178C2C22×C22⋊D19911yescharacteristic
179C2C2×#1569188511yescharacteristic
180C2#2026400011yescharacteristic
subgroupG/H
center, FrattiniC22C23×D199
commutator, socle, cosocleC22×C199C24
FittingC22×D4×C199C2
hypercenterC22×D4D199
hypocenterC199#546
LCSG ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. C2×#15691885C2semidirect
2. #20263942C2semidirect
3. #20263936C2semidirect
4. #20263934C2semidirect
5. #20263927C2semidirect
6. #20263926C2semidirect
7. #20263924C2semidirect
8. D4×Dic199C2semidirect
9. C4×C22⋊D199C2semidirect
10. C2×C22⋊Dic199C2semidirect
11. C22×D4×C199C2semidirect
13. C2×Dic398C22semidirect
15. C2×C22⋊D199C22semidirect
16. C2×C22⋊D199C22semidirect
17. C22×Dic199C22semidirect
18. C22×Dic199C22semidirect
19. C2×C4×D199C22semidirect
20. C4⋊Dic199C22semidirect
21. Dic199⋊C4C22semidirect
22. Dic199⋊C4C22semidirect
25. C22⋊Dic199C22semidirect
26. C22⋊Dic199C22semidirect
27. C22⋊Dic199C22semidirect
28. C22⋊Dic199C22semidirect
29. (C2×D199)⋊C4C22semidirect
30. C4×Dic199C22semidirect
58. D4×C199D4semidirect
71. C22×C199#41semidirect
90. C199#546semidirect
93. C22×D4D199semidirect
141. D4C22⋊D199semidirect
165. C22#15691885semidirect

Maximal subgroups

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

subgroupindex
C2×#156918852
#202639422
#202639362
#202639342
#202639272
#202639262
#202639242
D4×Dic1992
C4×C22⋊D1992
C2×C22⋊Dic1992
C22×D4×C1992
#546199