Finite Group Databasebeta

← #27164815

Subgroups — #27164815

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
1#41×D199C212yesmaximal, complemented
2#20263987C212yesmaximal, complemented
3#20263984C212yesmaximal, complemented
4#20263982C212yesmaximal, complemented
5#20263979C212yesmaximal, complemented
6#20263977C211yesmaximal, characteristic, complemented
7#20263975C212yesmaximal, complemented
8C2×#15691869C211yesmaximal, characteristic, complemented
9#135×C199C211yesmaximal, characteristic, complemented
10#15691887C2212yes—
11#15691886—22no—
12#15691886C2212yescomplemented
13#15691885—22no—
14#15691885C2212yes—
15Q8×D199C2212yescomplemented
16D4×D199—22no—
17D4×D199C2212yescomplemented
18C2×C4×D199—22no—
19C2×C4×D199C2211yescharacteristic
20Dic796C2211yescharacteristic, complemented
21#15691876C2212yescomplemented
22#15691875C2212yescomplemented
23#15691874C2212yescomplemented
24#15691873C2212yescomplemented
25D1592C2211yescharacteristic, complemented
26#15691871C2212yescomplemented
27#15691869C2212yescomplemented
28#15691869C2212yescomplemented
29C2×C199⋊C8C2211yescharacteristic, complemented
30#41×C199C2212yescomplemented
31Q16×C199C2211yescharacteristic, complemented
32SD16×C199C2212yescomplemented
33D8×C199C2211yescharacteristic, complemented
34C2×C8×C199C2211yescharacteristic, complemented
35C22×D199—22no—
36C22×D199D411yescharacteristic
37Dic398—22no—
38Dic398C2312yes—
39D796—22no—
40D796C2312yes—
41C22⋊D199—22no—
42C22⋊D199—22no—
43C22⋊D199—22no—
44C2×Dic199—22no—
45C2×Dic199D411yescharacteristic
46C4×D199—22no—
47C4×D199—22no—
48C4×D199—22no—
49C4×D199D412yes—
50C4×D199C2312yes—
51C199⋊C8C2312yes—
52Q8×C199C2312yes—
53D4×C199—22no—
54D4×C199C2312yes—
55C2×C4×C199—22no—
56C2×C4×C199C2311yescharacteristic
57C8×C199C2311yescharacteristic
58C8×C199C2311yescharacteristic
59C22×C199—22no—
60C22×C199C2×D411yescharacteristic
61C2×D199—22no—
62C2×D199—41no—
63C2×D199—44no—
64C2×D199C2×D412yes—
65C4×C199—22no—
66C4×C199C2×D411yescharacteristic
67C4×C199C2411yescharacteristic
68Dic199—22no—
69Dic199C2×D412yes—
70C2×C199—21no—
71C2×C199—42no—
72C2×C199C22×D411yescharacteristic
73D199—22no—
74D199—42no—
75C199#57511yescharacteristic, complemented, Sylow
76#575—1991nomaximal, Sylow
77C2×#41—1992no—
78#137—1992no—
79#137—1992no—
80AGL(1,ℤ/8)—1992no—
81AGL(1,ℤ/8)—1992no—
82#135—1992no—
83#135—1991no—
84#135D19911yescharacteristic, complemented
85C2×M16—1991no—
86#41—3982no—
87#41—3982no—
88#41—1992no—
89#41—1992no—
90#41—1992no—
91#41C2×D19912yescomplemented
92C2×Q8—1992no—
93C2×D4—3982no—
94C2×D4—1992no—
95C22×C4—3982no—
96C22×C4—1991no—
97Q16—1992no—
98Q16—1991no—
99Q16C2×D19911yescharacteristic, complemented
100SD16—1992no—
101SD16—1992no—
102SD16—1992no—
103SD16C2×D19912yescomplemented
104D8—1992no—
105D8—1991no—
106D8C2×D19911yescharacteristic, complemented
107M16—1992no—
108M16—1992no—
109C2×C8—1991no—
110C2×C8C2×D19911yescharacteristic, complemented
111C23—3982no—
112C23—1991no—
113Q8—3982no—
114Q8—1992no—
115Q8C22×D19912yes—
116D4—22no—
117D4—3982no—
118D4—3982no—
119D4—3982no—
120D4—3982no—
121D4—1992no—
122D4C22×D19912yes—
123C2×C4—22no—
124C2×C4—3982no—
125C2×C4—3982no—
126C2×C4—3982no—
127C2×C4—3982no—
128C2×C4—1991no—
129C2×C4—1992no—
130C2×C4—1992no—
131C2×C4C22×D19911yescharacteristic
132C8—1992no—
133C8C22×D19911yescharacteristic
134C8C22×D19911yescharacteristic
135C22—22no—
136C22—7961no—
137C22—7964no—
138C22—3982no—
139C22—1992no—
140C22D4×D19911yescharacteristic
141C4—22no—
142C4—3982no—
143C4—1992no—
144C4D4×D19911yescharacteristic
145C4C23×D19911yescharacteristic
146C2—21no—
147C2—42no—
148C2—7962no—
149C2—3982no—
150C2C2×D4×D19911yescharacteristic
subgroupG/H
centerC4D4×D199
commutator, cosocleC4×C199C24
FrattiniC4C23×D199
Fitting#135×C199C2
socleC2×C199C22×D4
hypercenter#135D199
hypocenterC199#575
LCSG ▷ C4×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C4 ◁ C2×C4 ◁ #135

Complements

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

NH ≅ G/Ntype
1. #41×D199C2semidirect
2. #20263987C2semidirect
3. #20263984C2semidirect
4. #20263982C2semidirect
5. #20263979C2semidirect
6. #20263977C2semidirect
7. #20263975C2semidirect
8. C2×#15691869C2semidirect
9. #135×C199C2semidirect
12. #15691886C22semidirect
15. Q8×D199C22semidirect
17. D4×D199C22semidirect
20. Dic796C22semidirect
21. #15691876C22semidirect
22. #15691875C22semidirect
23. #15691874C22semidirect
24. #15691873C22semidirect
25. D1592C22semidirect
26. #15691871C22semidirect
27. #15691869C22semidirect
28. #15691869C22semidirect
29. C2×C199⋊C8C22semidirect
30. #41×C199C22semidirect
31. Q16×C199C22semidirect
32. SD16×C199C22semidirect
33. D8×C199C22semidirect
34. C2×C8×C199C22semidirect
75. C199#575semidirect
84. #135D199semidirect
91. #41C2×D199semidirect
99. Q16C2×D199semidirect
103. SD16C2×D199semidirect
106. D8C2×D199semidirect
110. C2×C8C2×D199semidirect

Maximal subgroups

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

subgroupindex
#41×D1992
#202639872
#202639842
#202639822
#202639792
#202639772
#202639752
C2×#156918692
#135×C1992
#575199