Finite Group Databasebeta

← #27164825

Subgroups — #27164825

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#20264001C211yesmaximal, characteristic, complemented
3#20263987C212yesmaximal, complemented
4#20263985C211yesmaximal, characteristic, complemented
5#20263980C211yesmaximal, characteristic, complemented
6#20263979C212yesmaximal, complemented
7#20263976C212yesmaximal, complemented
8C2×#15691875C211yesmaximal, characteristic, complemented
9SD16×D199C212yesmaximal, complemented
10#20263961C211yesmaximal, characteristic, complemented
11#135×C199C211yesmaximal, characteristic, complemented
12#15691887C2212yes—
13#15691886—21no—
14#15691886—21no—
15#15691886C2211yescharacteristic
16#15691885—22no—
17#15691885C2212yescomplemented
18C2×Dic398—21no—
19C2×Dic398C2211yescharacteristic, complemented
20Q8×D199C2212yescomplemented
21C2×D796—21no—
22C2×D796C2211yescharacteristic
23D4×D199—22no—
24D4×D199C2212yes—
25#15691876C2212yescomplemented
26#15691875C2212yescomplemented
27#15691875C2212yescomplemented
28#15691874C2212yescomplemented
29#15691873C2212yescomplemented
30#15691871C2212yescomplemented
31#15691870C2211yescharacteristic, complemented
32#15691869C2212yescomplemented
33C8×D199C2212yescomplemented
34#41×C199C2211yescharacteristic
35#41×C199C2211yescharacteristic, complemented
36Q16×C199C2211yescharacteristic, complemented
37SD16×C199C2211yescharacteristic, complemented
38SD16×C199C2211yescharacteristic, complemented
39D8×C199C2211yescharacteristic, complemented
40C2×C8×C199C2211yescharacteristic, complemented
41C22×D199—21no—
42C22×D199—22no—
43Dic398—21no—
44Dic398—22no—
45Dic398D411yescharacteristic
46Dic398C2312yes—
47D796—22no—
48D796—21no—
49D796D411yescharacteristic
50D796C2312yes—
51C22⋊D199—22no—
52C22⋊D199—22no—
53C22⋊D199D412yes—
54C2×Dic199—21no—
55C2×Dic199—22no—
56C4×D199—22no—
57C4×D199—22no—
58C4×D199C2312yes—
59C199⋊C8C2312yes—
60Q8×C199C2311yescharacteristic
61Q8×C199C2311yescharacteristic
62D4×C199—21no—
63D4×C199—21no—
64D4×C199C2311yescharacteristic
65D4×C199C2311yescharacteristic
66C2×C4×C199—21no—
67C2×C4×C199—21no—
68C2×C4×C199C2311yescharacteristic
69C8×C199C2311yescharacteristic
70C8×C199C2311yescharacteristic
71C22×C199—21no—
72C22×C199—21no—
73C22×C199C2×D411yescharacteristic
74C2×D199—22no—
75C2×D199—82no—
76C2×D199—81no—
77C2×D199C2×D412yes—
78C4×C199—21no—
79C4×C199—21no—
80C4×C199C2×D411yescharacteristic
81C4×C199C2411yescharacteristic
82Dic199—22no—
83Dic199C2×D412yes—
84C2×C199—21no—
85C2×C199—41no—
86C2×C199—41no—
87C2×C199C22×D411yescharacteristic
88D199—22no—
89D199—42no—
90C199#57711yescharacteristic, complemented, Sylow
91#577—1991nomaximal, Sylow
922−1+4—1991no—
93H(5,𝔽2)—1991no—
94#137—1991no—
95#137—1992no—
96AGL(1,ℤ/8)—1991no—
97AGL(1,ℤ/8)—1992no—
98#135—1992no—
99#135D19911yescharacteristic, complemented
100C2×SD16—1992no—
101C2×SD16—1991no—
102#131—1991no—
103#41—3981no—
104#41—3981no—
105#41—3982no—
106#41—1992no—
107#41—1992no—
108#41—1991no—
109#41C2×D19911yescharacteristic
110#41C2×D19911yescharacteristic, complemented
111C2×Q8—3981no—
112C2×Q8—1992no—
113C2×Q8—1991no—
114C2×D4—3981no—
115C2×D4—3982no—
116C2×D4—1992no—
117C2×D4—1991no—
118Q16—1992no—
119Q16C2×D19911yescharacteristic, complemented
120SD16—1992no—
121SD16—1992no—
122SD16—1992no—
123SD16—1992no—
124SD16C2×D19911yescharacteristic, complemented
125SD16C2×D19911yescharacteristic, complemented
126D8—1992no—
127D8C2×D19911yescharacteristic, complemented
128M16—1991no—
129M16—1992no—
130C2×C8—1992no—
131C2×C8C2×D19911yescharacteristic, complemented
132C23—3982no—
133C23—3981no—
134Q8—3982no—
135Q8—3981no—
136Q8—1992no—
137Q8—1991no—
138Q8C22×D19911yescharacteristic
139Q8C22×D19911yescharacteristic
140D4—21no—
141D4—21no—
142D4—3982no—
143D4—3982no—
144D4—3982no—
145D4—3981no—
146D4—1992no—
147D4—1991no—
148D4—1992no—
149D4C22×D19911yescharacteristic
150D4C22×D19911yescharacteristic
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—3982no—
154C2×C4—3982no—
155C2×C4—3981no—
156C2×C4—3982no—
157C2×C4—1992no—
158C2×C4C22×D19911yescharacteristic
159C8—1992no—
160C8C22×D19911yescharacteristic
161C8C22×D19911yescharacteristic
162C22—21no—
163C22—21no—
164C22—15921no—
165C22—15922no—
166C22—3982no—
167C22—1992no—
168C22D4×D19911yescharacteristic
169C4—21no—
170C4—21no—
171C4—3982no—
172C4—1992no—
173C4D4×D19911yescharacteristic
174C4C23×D19911yescharacteristic
175C2—21no—
176C2—41no—
177C2—41no—
178C2—7962no—
179C2—3982no—
180C2C2×D4×D19911yescharacteristic
subgroupG/H
centerC2C2×D4×D199
commutator, cosocleC4×C199C24
FrattiniC4C23×D199
Fitting#135×C199C2
socleC2×C199C22×D4
hypercenter#135D199
hypocenterC199#577
LCSG ▷ C4×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C2 ◁ C2×C4 ◁ #135

Complements

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

NH ≅ G/Ntype
1. #20264002C2semidirect
2. #20264001C2semidirect
3. #20263987C2semidirect
4. #20263985C2semidirect
5. #20263980C2semidirect
6. #20263979C2semidirect
7. #20263976C2semidirect
8. C2×#15691875C2semidirect
9. SD16×D199C2semidirect
10. #20263961C2semidirect
11. #135×C199C2semidirect
17. #15691885C22semidirect
19. C2×Dic398C22semidirect
20. Q8×D199C22semidirect
25. #15691876C22semidirect
26. #15691875C22semidirect
27. #15691875C22semidirect
28. #15691874C22semidirect
29. #15691873C22semidirect
30. #15691871C22semidirect
31. #15691870C22semidirect
32. #15691869C22semidirect
33. C8×D199C22semidirect
35. #41×C199C22semidirect
36. Q16×C199C22semidirect
37. SD16×C199C22semidirect
38. SD16×C199C22semidirect
39. D8×C199C22semidirect
40. C2×C8×C199C22semidirect
90. C199#577semidirect
99. #135D199semidirect
110. #41C2×D199semidirect
119. Q16C2×D199semidirect
124. SD16C2×D199semidirect
125. SD16C2×D199semidirect
127. D8C2×D199semidirect
131. C2×C8C2×D199semidirect

Maximal subgroups

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

subgroupindex
#202640022
#202640012
#202639872
#202639852
#202639802
#202639792
#202639762
C2×#156918752
SD16×D1992
#202639612
#135×C1992
#577199