Finite Group Databasebeta

← #27164189

Subgroups — #27164189

Order 12736.

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

normal subgroups only

This group has 146 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#20264002C211yesmaximal, characteristic, complemented
2#20263981C211yesmaximal, characteristic, complemented
3#20263942C211yesmaximal, characteristic, complemented
4#20263882C211yesmaximal, characteristic, complemented
5#20263881C211yesmaximal, characteristic, complemented
6#20263873C211yesmaximal, characteristic, complemented
7AGL(1,ℤ/8)×C199C211yesmaximal, characteristic, complemented
8#15691886—21no—
9#15691886C2211yescharacteristic
10#15691885—22no—
11#15691885—21no—
12C2×Dic398—21no—
13C2×Dic398C2211yescharacteristic
14Q8×D199—21no—
15#15691874—21no—
16#15691871—21no—
17#15691870C2211yescharacteristic
18C22⋊Dic199—22no—
19C4×Dic199C2211yescharacteristic
20#41×C199C2211yescharacteristic
21C2×D4×C199C2211yescharacteristic
22SD16×C199—21no—
23D8×C199—21no—
24M16×C199C2211yescharacteristic
25Dic398—22no—
26Dic398—21no—
27Dic398—21no—
28Dic398D411yescharacteristic
29D796D411yescharacteristic
30C22⋊D199—22no—
31C22⋊D199—21no—
32C2×Dic199—22no—
33C2×Dic199—21no—
34C2×Dic199D412yes—
35C4×D199—22no—
36C4×D199—21no—
37C199⋊C8—21no—
38C23×C199—21no—
39Q8×C199D411yescharacteristic
40D4×C199—21no—
41D4×C199—21no—
42D4×C199—21no—
43D4×C199D411yescharacteristic
44C2×C4×C199—21no—
45C2×C4×C199C2311yescharacteristic
46C8×C199—21no—
47C22×C199—21no—
48C22×C199—42no—
49C22×C199—41no—
50C22×C199C2×D411yescharacteristic
51C2×D199—21no—
52C4×C199—21no—
53C4×C199C2×D411yescharacteristic
54C4×C199C2×D411yescharacteristic
55Dic199—21no—
56Dic199—22no—
57Dic199—41no—
58C2×C199—21no—
59C2×C199—41no—
60C2×C199—81no—
61C2×C199#12011yescharacteristic
62D199—41no—
63C199#45411yescharacteristic, complemented, Sylow
64#454—1991nomaximal, Sylow
652−1+4—1991no—
66AGL(1,ℤ/8)—1991no—
67AGL(1,ℤ/8)D19911yescharacteristic, complemented
68#124—1991no—
69#104—1991no—
70#104—1991no—
71#101—1991no—
72#41—3981no—
73#41—3981no—
74#41—3982no—
75#41—1991no—
76#41C2×D19911yescharacteristic
77C2×Q8—3981no—
78C2×Q8—3981no—
79C2×Q8—1991no—
80C2×D4C2×D19911yescharacteristic
81SD16—21no—
82SD16—3981no—
83D8—21no—
84D8—3981no—
85M16—1991no—
86M16C2×D19911yescharacteristic
87C22⋊C4—3982no—
88C42—1991no—
89C23—21no—
90Q8—3982no—
91Q8—3981no—
92Q8—3981no—
93Q8—1991no—
94Q8C22⋊D19911yescharacteristic
95D4—21no—
96D4—21no—
97D4—21no—
98D4—3982no—
99D4—3981no—
100D4—1991no—
101D4C22⋊D19911yescharacteristic
102C2×C4—21no—
103C2×C4—3982no—
104C2×C4—3981no—
105C2×C4—3982no—
106C2×C4—3981no—
107C2×C4—1992no—
108C2×C4C22×D19911yescharacteristic
109C8—21no—
110C8—3981no—
111C22—21no—
112C22—42no—
113C22—41no—
114C22—3981no—
115C22D4×D19911yescharacteristic
116C4—21no—
117C4—7961no—
118C4—3981no—
119C4—3982no—
120C4D4×D19911yescharacteristic
121C4C2×C22⋊D19911yescharacteristic
122C2—21no—
123C2—41no—
124C2—81no—
125C2—7961no—
126C2#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#454
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. #20264002C2semidirect
2. #20263981C2semidirect
3. #20263942C2semidirect
4. #20263882C2semidirect
5. #20263881C2semidirect
6. #20263873C2semidirect
7. AGL(1,ℤ/8)×C199C2semidirect
63. C199#454semidirect
67. AGL(1,ℤ/8)D199semidirect

Maximal subgroups

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

subgroupindex
#202640022
#202639812
#202639422
#202638822
#202638812
#202638732
AGL(1,ℤ/8)×C1992
#454199