Finite Group Databasebeta

← #27164828

Subgroups — #27164828

Order 12736.

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

normal subgroups only

This group has 258 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
1#20264002C211yesmaximal, characteristic, complemented
2#20264000C211yesmaximal, characteristic, complemented
3#20263988C211yesmaximal, characteristic, complemented
4#20263984C212yesmaximal, complemented
5#20263979C212yesmaximal, complemented
6#20263975C211yesmaximal, characteristic, complemented
7#20263974C212yesmaximal, complemented
8C2×#15691874C211yesmaximal, characteristic, complemented
9SD16×D199C212yesmaximal, complemented
10#20263963C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×C199C211yesmaximal, characteristic, complemented
12#15691886—21no—
13#15691886C2211yescharacteristic
14#15691886C2211yescharacteristic
15#15691885—21no—
16#15691885—22no—
17#15691885C2212yes—
18#15691885C2212yescomplemented
19C2×Dic398—21no—
20C2×Dic398C2211yescharacteristic, complemented
21Q8×D199C2212yescomplemented
22C2×C22⋊D199—22no—
23D4×D199—21no—
24D4×D199C2212yes—
25Dic796C2212yescomplemented
26#15691876C2211yescharacteristic, complemented
27#15691875C2212yescomplemented
28#15691874C2212yescomplemented
29#15691874C2212yescomplemented
30#15691874C2211yescharacteristic, complemented
31#15691873C2211yescharacteristic, complemented
32#15691871C2211yescharacteristic, complemented
33#15691869C2212yescomplemented
34C2×C199⋊C8C2211yescharacteristic, complemented
35C8×D199C2212yescomplemented
36#41×C199C2211yescharacteristic, complemented
37C2×D4×C199C2211yescharacteristic
38SD16×C199C2212yescomplemented
39D8×C199C2212yescomplemented
40M16×C199C2211yescharacteristic, complemented
41C22×D199—22no—
42Dic398—21no—
43Dic398—22no—
44Dic398D411yescharacteristic
45Dic398C2312yes—
46Dic398C2311yescharacteristic
47D796D411yescharacteristic
48D796C2311yescharacteristic
49C22⋊D199—21no—
50C22⋊D199—22no—
51C22⋊D199—22no—
52C22⋊D199—22no—
53C22⋊D199D412yes—
54C2×Dic199—21no—
55C2×Dic199—22no—
56C2×Dic199—22no—
57C4×D199—22no—
58C4×D199—21no—
59C4×D199C2312yes—
60C199⋊C8C2311yescharacteristic
61C199⋊C8C2311yescharacteristic
62C23×C199—21no—
63Q8×C199C2311yescharacteristic
64D4×C199—21no—
65D4×C199—21no—
66D4×C199C2311yescharacteristic
67D4×C199C2312yes—
68C2×C4×C199—21no—
69C2×C4×C199C2311yescharacteristic
70C8×C199C2312yes—
71C22×C199—21no—
72C22×C199—22no—
73C22×C199—81no—
74C22×C199C2×D411yescharacteristic
75C2×D199—21no—
76C2×D199—82no—
77C2×D199C2×D412yes—
78C4×C199—21no—
79C4×C199C2×D411yescharacteristic
80C4×C199C2411yescharacteristic
81Dic199—21no—
82Dic199—22no—
83Dic199C2×D412yes—
84C2×C199—21no—
85C2×C199—42no—
86C2×C199—41no—
87C2×C199C22×D411yescharacteristic
88D199—22no—
89D199—41no—
90C199#57711yescharacteristic, complemented, Sylow
91#577—1991nomaximal, Sylow
922−1+4—1991no—
93H(5,𝔽2)—1991no—
94#137—1992no—
95#137—1991no—
96AGL(1,ℤ/8)—1992no—
97AGL(1,ℤ/8)D19911yescharacteristic, complemented
98#135—1991no—
99#135—1992no—
100C2×SD16—1991no—
101C2×SD16—1992no—
102#131—1991no—
103#41—3981no—
104#41—3981no—
105#41—3982no—
106#41—1992no—
107#41—1992no—
108#41—1991no—
109#41—1991no—
110#41C2×D19911yescharacteristic, complemented
111C2×Q8—3981no—
112C2×Q8—1992no—
113C2×Q8—1991no—
114C2×D4—3981no—
115C2×D4—3982no—
116C2×D4—1992no—
117C2×D4C2×D19911yescharacteristic
118Q16—1991no—
119Q16—1992no—
120SD16—1991no—
121SD16—1992no—
122SD16—1992no—
123SD16—1991no—
124SD16—1992no—
125SD16C2×D19912yescomplemented
126D8—1991no—
127D8C2×D19912yescomplemented
128M16—1992no—
129M16C2×D19911yescharacteristic, complemented
130C2×C8—1991no—
131C2×C8—1992no—
132C23—21no—
133C23—3982no—
134Q8—3982no—
135Q8—3981no—
136Q8—1992no—
137Q8—1991no—
138Q8—1991no—
139Q8C22×D19911yescharacteristic
140D4—21no—
141D4—21no—
142D4—3982no—
143D4—3982no—
144D4—3981no—
145D4—3982no—
146D4—1992no—
147D4—1991no—
148D4—1991no—
149D4C22×D19911yescharacteristic
150D4C22×D19912yes—
151C2×C4—21no—
152C2×C4—3982no—
153C2×C4—3981no—
154C2×C4—3982no—
155C2×C4—3982no—
156C2×C4—3981no—
157C2×C4—1992no—
158C2×C4C22×D19911yescharacteristic
159C8—1991no—
160C8—1991no—
161C8C22×D19912yes—
162C22—21no—
163C22—22no—
164C22—81no—
165C22—15922no—
166C22—3981no—
167C22—1992no—
168C22D4×D19911yescharacteristic
169C4—21no—
170C4—3981no—
171C4—3982no—
172C4—1992no—
173C4D4×D19911yescharacteristic
174C4C23×D19911yescharacteristic
175C2—21no—
176C2—42no—
177C2—41no—
178C2—7961no—
179C2—3982no—
180C2C2×D4×D19911yescharacteristic
subgroupG/H
centerC2C2×D4×D199
commutator, cosocleC4×C199C24
FrattiniC4C23×D199
FittingAGL(1,ℤ/8)×C199C2
socleC2×C199C22×D4
hypercenterAGL(1,ℤ/8)D199
hypocenterC199#577
LCSG ▷ C4×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #20264002C2semidirect
2. #20264000C2semidirect
3. #20263988C2semidirect
4. #20263984C2semidirect
5. #20263979C2semidirect
6. #20263975C2semidirect
7. #20263974C2semidirect
8. C2×#15691874C2semidirect
9. SD16×D199C2semidirect
10. #20263963C2semidirect
11. AGL(1,ℤ/8)×C199C2semidirect
18. #15691885C22semidirect
20. C2×Dic398C22semidirect
21. Q8×D199C22semidirect
25. Dic796C22semidirect
26. #15691876C22semidirect
27. #15691875C22semidirect
28. #15691874C22semidirect
29. #15691874C22semidirect
30. #15691874C22semidirect
31. #15691873C22semidirect
32. #15691871C22semidirect
33. #15691869C22semidirect
34. C2×C199⋊C8C22semidirect
35. C8×D199C22semidirect
36. #41×C199C22semidirect
38. SD16×C199C22semidirect
39. D8×C199C22semidirect
40. M16×C199C22semidirect
90. C199#577semidirect
97. AGL(1,ℤ/8)D199semidirect
110. #41C2×D199semidirect
125. SD16C2×D199semidirect
127. D8C2×D199semidirect
129. M16C2×D199semidirect

Maximal subgroups

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

subgroupindex
#202640022
#202640002
#202639882
#202639842
#202639792
#202639752
#202639742
C2×#156918742
SD16×D1992
#202639632
AGL(1,ℤ/8)×C1992
#577199