Finite Group Databasebeta

← #27164184

Subgroups — #27164184

Order 12736.

168 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 39 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#20264001C211yesmaximal, characteristic, complemented
2#20263981C211yesmaximal, characteristic, complemented
3#20263951C211yesmaximal, characteristic, complemented
4#20263882C211yesmaximal, characteristic, complemented
5#20263881C211yesmaximal, characteristic, complemented
6#20263871C211yesmaximal, characteristic, complemented
7AGL(1,ℤ/8)×C199C211yesmaximal, characteristic, complemented
8#15691887—21no—
9#15691886—21no—
10#15691886C2211yescharacteristic
11C2×D796—21no—
12C2×D796C2211yescharacteristic
13C2×C22⋊D199—22no—
14D4×D199—22no—
15D4×D199—21no—
16#15691874—21no—
17#15691871—21no—
18#15691870C2211yescharacteristic, complemented
19C4×Dic199C2211yescharacteristic, complemented
20#41×C199C2211yescharacteristic
21C2×D4×C199C2211yescharacteristic, complemented
22SD16×C199—21no—
23D8×C199—21no—
24M16×C199C2211yescharacteristic, complemented
25C22×D199—22no—
26C22×D199D412yes—
27Dic398D411yescharacteristic, complemented
28D796—22no—
29D796—21no—
30D796—21no—
31D796D411yescharacteristic, complemented
32C22⋊D199—22no—
33C22⋊D199—21no—
34C22⋊D199—44no—
35C2×Dic199—21no—
36C4×D199—22no—
37C4×D199—21no—
38C199⋊C8—21no—
39C23×C199—21no—
40Q8×C199D411yescharacteristic, complemented
41D4×C199—21no—
42D4×C199—21no—
43D4×C199—21no—
44D4×C199D411yescharacteristic, complemented
45C2×C4×C199—21no—
46C2×C4×C199C2311yescharacteristic
47C8×C199—21no—
48C22×C199—21no—
49C22×C199—42no—
50C22×C199—41no—
51C22×C199C2×D411yescharacteristic
52C2×D199—22no—
53C2×D199—21no—
54C2×D199—42no—
55C2×D199—82no—
56C4×C199—21no—
57C4×C199C2×D411yescharacteristic
58C4×C199C2×D411yescharacteristic
59Dic199—22no—
60Dic199—21no—
61C2×C199—21no—
62C2×C199—41no—
63C2×C199—81no—
64C2×C199#12011yescharacteristic
65D199—42no—
66D199—41no—
67C199#45311yescharacteristic, complemented, Sylow
68#453—1991nomaximal, Sylow
69H(5,𝔽2)—1991no—
70AGL(1,ℤ/8)—1991no—
71AGL(1,ℤ/8)D19911yescharacteristic, complemented
72#127—1991no—
73#104—1991no—
74#104—1991no—
75#100—1991no—
76#41—3981no—
77#41—3981no—
78#41—1991no—
79#41C2×D19911yescharacteristic
80C2×D4—3981no—
81C2×D4—3982no—
82C2×D4—3982no—
83C2×D4—3981no—
84C2×D4—1991no—
85C2×D4C2×D19911yescharacteristic, complemented
86SD16—21no—
87SD16—3981no—
88D8—21no—
89D8—3981no—
90M16—1991no—
91M16C2×D19911yescharacteristic, complemented
92C42—1991no—
93C23—21no—
94C23—3982no—
95C23—1992no—
96Q8—1991no—
97Q8C22⋊D19911yescharacteristic, complemented
98D4—21no—
99D4—21no—
100D4—21no—
101D4—7964no—
102D4—3982no—
103D4—3982no—
104D4—3981no—
105D4—3981no—
106D4—3981no—
107D4—1991no—
108D4C22⋊D19911yescharacteristic, complemented
109C2×C4—21no—
110C2×C4—3982no—
111C2×C4—3981no—
112C2×C4—3981no—
113C2×C4C22×D19911yescharacteristic
114C8—21no—
115C8—3981no—
116C22—21no—
117C22—42no—
118C22—41no—
119C22—7962no—
120C22—15922no—
121C22—3982no—
122C22—3981no—
123C22D4×D19911yescharacteristic
124C4—21no—
125C4—3982no—
126C4—3981no—
127C4D4×D19911yescharacteristic
128C4C2×C22⋊D19911yescharacteristic
129C2—21no—
130C2—41no—
131C2—81no—
132C2—7962no—
133C2—7961no—
134C2#2026392311yescharacteristic
subgroupG/H
centerC2#20263923
commutator, cosocleC2×C4×C199C23
FrattiniC2×C4C22×D199
FittingAGL(1,ℤ/8)×C199C2
socleC2×C199#120
hypercenterAGL(1,ℤ/8)D199
hypocenterC199#453
LCSG ▷ C2×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. #20264001C2semidirect
2. #20263981C2semidirect
3. #20263951C2semidirect
4. #20263882C2semidirect
5. #20263881C2semidirect
6. #20263871C2semidirect
7. AGL(1,ℤ/8)×C199C2semidirect
18. #15691870C22semidirect
19. C4×Dic199C22semidirect
21. C2×D4×C199C22semidirect
24. M16×C199C22semidirect
27. Dic398D4semidirect
31. D796D4semidirect
40. Q8×C199D4semidirect
44. D4×C199D4semidirect
67. C199#453semidirect
71. AGL(1,ℤ/8)D199semidirect
85. C2×D4C2×D199semidirect
91. M16C2×D199semidirect
97. Q8C22⋊D199semidirect
108. D4C22⋊D199semidirect

Maximal subgroups

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

subgroupindex
#202640012
#202639812
#202639512
#202638822
#202638812
#202638712
AGL(1,ℤ/8)×C1992
#453199