Finite Group Databasebeta

← #27164665

Subgroups — #27164665

Order 12736.

334 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 105 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×#15691886C211yesmaximal, characteristic, complemented
2C2×D4×D199C211yesmaximal, characteristic, complemented
3#20263942C211yesmaximal, characteristic, complemented
4#20263937C212yesmaximal, complemented
5#20263935C211yesmaximal, characteristic, complemented
6#20263926C212yesmaximal, complemented
7#20263923C212yesmaximal, complemented
8#20263917C211yesmaximal, characteristic, complemented
9C4×C22⋊D199C211yesmaximal, characteristic, complemented
10C22⋊C4×D199C212yesmaximal, complemented
11#121×C199C211yesmaximal, characteristic, complemented
12C23×D199C2212yes—
13#15691886—24no—
14C2×Dic398C2211yescharacteristic, complemented
15C2×D796C2211yescharacteristic
16C2×C22⋊D199C2212yes—
17C2×C22⋊D199C2211yescharacteristic
18C2×C22⋊D199C2212yescomplemented
19D4×D199—24no—
20C2×C4×D199C2211yescharacteristic
21C2×C4×D199C2212yescomplemented
22C2×C4×D199C2212yescomplemented
23C4⋊Dic199C2212yescomplemented
24Dic199⋊C4C2211yescharacteristic, complemented
25C22⋊Dic199C2212yescomplemented
26C22⋊Dic199C2212yescomplemented
27C22⋊Dic199C2211yescharacteristic, complemented
28(C2×D199)⋊C4C2211yescharacteristic
29(C2×D199)⋊C4C2212yes—
30(C2×D199)⋊C4C2212yescomplemented
31C4×Dic199C2211yescharacteristic, complemented
32C2×D4×C199C2211yescharacteristic
33C2×D4×C199C2212yescomplemented
34C22×C4×C199C2211yescharacteristic, complemented
35C4⋊C4×C199C2211yescharacteristic, complemented
36C22⋊C4×C199C2212yescomplemented
37C22×D199—24no—
38C22×D199—24no—
39C22×D199—42no—
40C22×D199C2312yes—
41C22×D199C2311yescharacteristic
42Dic398—22no—
43D796D414yescomplemented
44C22⋊D199—24no—
45C22⋊D199—24no—
46C22⋊D199—22no—
47C2×Dic199C2311yescharacteristic
48C2×Dic199C2311yescharacteristic
49C2×Dic199C2312yes—
50C2×Dic199C2311yescharacteristic
51C4×D199—24no—
52C4×D199—22no—
53C4×D199—42no—
54C23×C199C2312yes—
55C23×C199C2311yescharacteristic
56D4×C199—22no—
57D4×C199—42no—
58C2×C4×C199—22no—
59C2×C4×C199C2311yescharacteristic
60C2×C4×C199C2312yes—
61C2×C4×C199C2311yescharacteristic
62C22×C199—21no—
63C22×C199—22no—
64C22×C199—21no—
65C22×C199—22no—
66C22×C199—21no—
67C22×C199—22no—
68C22×C199C2411yescharacteristic
69C2×D199—21no—
70C2×D199—22no—
71C2×D199—22no—
72C2×D199—21no—
73C2×D199—48no—
74C2×D199C2×D414yes—
75C2×D199#4112yescomplemented
76C4×C199—21no—
77C4×C199—22no—
78C4×C199C2×D412yes—
79Dic199—21no—
80Dic199—22no—
81Dic199—21no—
82Dic199—21no—
83C2×C199—42no—
84C2×C199—41no—
85C2×C199C22×D411yescharacteristic
86C2×C199C2×#4111yescharacteristic
87C2×C199H(5,𝔽2)11yescharacteristic
88D199—22no—
89D199—24no—
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#121D19911yescharacteristic, complemented
99#120—1992no—
100C4×D4—1991no—
101C4×D4—1991no—
102C2×C22⋊C4—1992no—
103C24—1992no—
104#41—3984no—
105C2×Q8—1991no—
106C2×D4—3984no—
107C2×D4—1991no—
108C2×D4—1992no—
109C2×D4—1992no—
110C2×D4—1991no—
111C2×D4C2×D19911yescharacteristic
112C2×D4C2×D19912yescomplemented
113C22×C4—1992no—
114C22×C4—1991no—
115C22×C4—1992no—
116C22×C4C2×D19911yescharacteristic, complemented
117C4⋊C4—1991no—
118C4⋊C4—1992no—
119C4⋊C4C2×D19911yescharacteristic, complemented
120C22⋊C4—1992no—
121C22⋊C4—1991no—
122C22⋊C4—1992no—
123C22⋊C4—1991no—
124C22⋊C4—1992no—
125C22⋊C4—1992no—
126C22⋊C4C2×D19912yescomplemented
127C42—1991no—
128C23—7962no—
129C23—3984no—
130C23—3984no—
131C23—1991no—
132C23—1992no—
133C23C22×D19912yes—
134C23C22×D19911yescharacteristic
135Q8—3982no—
136D4—22no—
137D4—42no—
138D4—3984no—
139D4—3984no—
140D4—3982no—
141D4—1994no—
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×C4C22×D19912yes—
151C2×C4C22×D19911yescharacteristic
152C2×C4C22×D19911yescharacteristic
153C22—21no—
154C22—22no—
155C22—21no—
156C22—22no—
157C22—21no—
158C22—22no—
159C22—7968no—
160C22—3981no—
161C22—3982no—
162C22—3982no—
163C22—3981no—
164C22—1994no—
165C22—1992no—
166C22C23×D19911yescharacteristic
167C4—21no—
168C4—22no—
169C4—3981no—
170C4—3982no—
171C4—3981no—
172C4—3981no—
173C4D4×D19912yes—
174C2—42no—
175C2—41no—
176C2—3982no—
177C2—3984no—
178C2C2×D4×D19911yescharacteristic
179C2#41×D19911yescharacteristic
180C2#2026400011yescharacteristic
subgroupG/H
center, FrattiniC22C23×D199
commutator, socle, cosocleC22×C199C24
Fitting#121×C199C2
hypercenter#121D199
hypocenterC199#546
LCSG ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×#15691886C2semidirect
2. C2×D4×D199C2semidirect
3. #20263942C2semidirect
4. #20263937C2semidirect
5. #20263935C2semidirect
6. #20263926C2semidirect
7. #20263923C2semidirect
8. #20263917C2semidirect
9. C4×C22⋊D199C2semidirect
10. C22⋊C4×D199C2semidirect
11. #121×C199C2semidirect
14. C2×Dic398C22semidirect
18. C2×C22⋊D199C22semidirect
21. C2×C4×D199C22semidirect
22. C2×C4×D199C22semidirect
23. C4⋊Dic199C22semidirect
24. Dic199⋊C4C22semidirect
25. C22⋊Dic199C22semidirect
26. C22⋊Dic199C22semidirect
27. C22⋊Dic199C22semidirect
30. (C2×D199)⋊C4C22semidirect
31. C4×Dic199C22semidirect
33. C2×D4×C199C22semidirect
34. C22×C4×C199C22semidirect
35. C4⋊C4×C199C22semidirect
36. C22⋊C4×C199C22semidirect
43. D796D4semidirect
75. C2×D199#41semidirect
90. C199#546semidirect
98. #121D199semidirect
112. C2×D4C2×D199semidirect
116. C22×C4C2×D199semidirect
119. C4⋊C4C2×D199semidirect
126. C22⋊C4C2×D199semidirect

Maximal subgroups

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

subgroupindex
C2×#156918862
C2×D4×D1992
#202639422
#202639372
#202639352
#202639262
#202639232
#202639172
C4×C22⋊D1992
C22⋊C4×D1992
#121×C1992
#546199