Finite Group Databasebeta

← #27164817

Subgroups — #27164817

Order 12736.

262 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 99 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
2#41×D199C211yesmaximal, characteristic, complemented
3#20263987C212yesmaximal, complemented
4#20263986C211yesmaximal, characteristic, complemented
5#20263983C211yesmaximal, characteristic, complemented
6#20263982C212yesmaximal, complemented
7#20263980C211yesmaximal, characteristic, complemented
8#20263978C212yesmaximal, complemented
9#20263976C212yesmaximal, complemented
10M16×D199C211yesmaximal, characteristic, complemented
11#137×C199C211yesmaximal, characteristic, complemented
12#15691887—21no—
13#15691887—21no—
14#15691887C2212yes—
15#15691887C2212yescomplemented
16#15691887C2211yescharacteristic, complemented
17#15691886—21no—
18#15691886C2211yescharacteristic
19#15691885—21no—
20#15691885C2211yescharacteristic, complemented
21Q8×D199C2211yescharacteristic, complemented
22C2×D796—21no—
23C2×D796C2211yescharacteristic, complemented
24D4×D199—21no—
25D4×D199C2211yescharacteristic, complemented
26C2×C4×D199—21no—
27C2×C4×D199—21no—
28C2×C4×D199C2211yescharacteristic
29#15691876C2212yescomplemented
30#15691875C2212yescomplemented
31#15691873C2212yescomplemented
32#15691873C2212yescomplemented
33D1592C2212yescomplemented
34#15691871C2212yescomplemented
35#15691870C2211yescharacteristic, complemented
36#15691869C2212yescomplemented
37C8×D199C2212yescomplemented
38#41×C199C2211yescharacteristic
39C2×Q8×C199C2211yescharacteristic, complemented
40Q16×C199C2212yescomplemented
41SD16×C199C2212yescomplemented
42M16×C199C2211yescharacteristic, complemented
43C22×D199—21no—
44C22×D199—21no—
45C22×D199D411yescharacteristic
46Dic398—21no—
47Dic398C2311yescharacteristic
48D796—21no—
49D796—22no—
50D796—22no—
51D796—21no—
52D796C2312yes—
53D796C2311yescharacteristic
54C22⋊D199—21no—
55C22⋊D199—21no—
56C22⋊D199—21no—
57C2×Dic199—21no—
58C2×Dic199D411yescharacteristic
59C4×D199—21no—
60C4×D199—21no—
61C4×D199—21no—
62C4×D199—22no—
63C4×D199—22no—
64C4×D199D411yescharacteristic
65C4×D199D411yescharacteristic
66C4×D199C2311yescharacteristic
67C4×D199C2311yescharacteristic
68C199⋊C8C2312yes—
69Q8×C199—21no—
70Q8×C199C2312yes—
71Q8×C199C2311yescharacteristic
72D4×C199—21no—
73D4×C199C2311yescharacteristic
74C2×C4×C199—21no—
75C2×C4×C199—21no—
76C2×C4×C199C2311yescharacteristic
77C8×C199C2312yes—
78C22×C199—21no—
79C22×C199C2×D411yescharacteristic
80C2×D199—21no—
81C2×D199—22no—
82C2×D199—41no—
83C2×D199—42no—
84C2×D199—42no—
85C2×D199C2×D411yescharacteristic
86C2×D199C2×D411yescharacteristic
87C4×C199—21no—
88C4×C199—22no—
89C4×C199C2×D411yescharacteristic
90C4×C199C2411yescharacteristic
91Dic199—21no—
92Dic199C2×D411yescharacteristic
93Dic199C2×D411yescharacteristic
94C2×C199—21no—
95C2×C199—41no—
96C2×C199C22×D411yescharacteristic
97D199—21no—
98D199—21no—
99D199—41no—
100D199—42no—
101C199#57511yescharacteristic, complemented, Sylow
102#575—1991nomaximal, Sylow
103C2×#41—1991no—
104C2×#41—1991no—
105#137—1991no—
106#137—1992no—
107#137D19911yescharacteristic, complemented
108AGL(1,ℤ/8)—1991no—
109AGL(1,ℤ/8)—1992no—
110AGL(1,ℤ/8)—1991no—
111#135—1992no—
112#135—1992no—
113C2×M16—1991no—
114#41—3981no—
115#41—3981no—
116#41—3981no—
117#41—3981no—
118#41—1992no—
119#41—1991no—
120#41—1992no—
121#41—1991no—
122#41—1991no—
123#41C2×D19911yescharacteristic
124C2×Q8—1991no—
125C2×Q8C2×D19911yescharacteristic, complemented
126C2×D4—3981no—
127C2×D4—3981no—
128C2×D4—1991no—
129C2×D4—1991no—
130C22×C4—3981no—
131C22×C4—3981no—
132C22×C4—1991no—
133Q16—1992no—
134Q16C2×D19912yescomplemented
135SD16—1992no—
136SD16—1992no—
137SD16—1992no—
138SD16C2×D19912yescomplemented
139D8—1992no—
140D8—1992no—
141M16—1991no—
142M16—1992no—
143M16C2×D19911yescharacteristic, complemented
144C2×C8—1992no—
145C23—3981no—
146C23—3981no—
147C23—1991no—
148Q8—21no—
149Q8—3981no—
150Q8—1991no—
151Q8C22×D19911yescharacteristic
152Q8C22×D19912yes—
153D4—21no—
154D4—3981no—
155D4—3982no—
156D4—3982no—
157D4—3981no—
158D4—3981no—
159D4—3981no—
160D4—3981no—
161D4—1992no—
162D4—1991no—
163D4C22×D19911yescharacteristic
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—3981no—
167C2×C4—3981no—
168C2×C4—3982no—
169C2×C4—3982no—
170C2×C4—3981no—
171C2×C4—3981no—
172C2×C4—1991no—
173C2×C4—1991no—
174C2×C4—1991no—
175C2×C4—1991no—
176C2×C4—1991no—
177C2×C4C22×D19911yescharacteristic
178C8—1992no—
179C8C22×D19912yes—
180C22—21no—
181C22—7962no—
182C22—7961no—
183C22—7962no—
184C22—3982no—
185C22—3981no—
186C22—1991no—
187C22—1991no—
188C22D4×D19911yescharacteristic
189C4—21no—
190C4—22no—
191C4—3981no—
192C4—1991no—
193C4—1991no—
194C4D4×D19911yescharacteristic
195C4C23×D19911yescharacteristic
196C2—21no—
197C2—41no—
198C2—7962no—
199C2—7961no—
200C2—3981no—
201C2—3981no—
202C2C2×D4×D19911yescharacteristic
subgroupG/H
centerC2C2×D4×D199
commutator, cosocleC4×C199C24
FrattiniC4C23×D199
Fitting#137×C199C2
socleC2×C199C22×D4
hypercenter#137D199
hypocenterC199#575
LCSG ▷ C4×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C2 ◁ C2×C4 ◁ #137

Complements

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

NH ≅ G/Ntype
1. C2×#15691887C2semidirect
2. #41×D199C2semidirect
3. #20263987C2semidirect
4. #20263986C2semidirect
5. #20263983C2semidirect
6. #20263982C2semidirect
7. #20263980C2semidirect
8. #20263978C2semidirect
9. #20263976C2semidirect
10. M16×D199C2semidirect
11. #137×C199C2semidirect
15. #15691887C22semidirect
16. #15691887C22semidirect
20. #15691885C22semidirect
21. Q8×D199C22semidirect
23. C2×D796C22semidirect
25. D4×D199C22semidirect
29. #15691876C22semidirect
30. #15691875C22semidirect
31. #15691873C22semidirect
32. #15691873C22semidirect
33. D1592C22semidirect
34. #15691871C22semidirect
35. #15691870C22semidirect
36. #15691869C22semidirect
37. C8×D199C22semidirect
39. C2×Q8×C199C22semidirect
40. Q16×C199C22semidirect
41. SD16×C199C22semidirect
42. M16×C199C22semidirect
101. C199#575semidirect
107. #137D199semidirect
125. C2×Q8C2×D199semidirect
134. Q16C2×D199semidirect
138. SD16C2×D199semidirect
143. M16C2×D199semidirect

Maximal subgroups

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

subgroupindex
C2×#156918872
#41×D1992
#202639872
#202639862
#202639832
#202639822
#202639802
#202639782
#202639762
M16×D1992
#137×C1992
#575199