Finite Group Databasebeta

← #45663437

Subgroups — #45663437

Order 32896.

314 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 57 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#30657372C211yesmaximal, characteristic
2#30657109C211yesmaximal, characteristic, complemented
3#30657028C212yesmaximal, complemented
4#30656464C212yesmaximal, complemented
5#331×C257C211yesmaximal, characteristic, complemented
6C2×D4×D257—22no—
7C2×#16861606—22no—
8#22551592C2211yescharacteristic, complemented
9#22551563—22no—
10#22551560—22no—
11C4×D1028C2212yes—
12#22551524—22no—
13#22551514—22no—
14C4×#14523339C2211yescharacteristic, complemented
15#127×C257C2211yescharacteristic
16C4⋊C8×C257C2212yescomplemented
17C23×D257—22no—
18C2×Dic514—21no—
19C2×D1028D412yes—
20C2×C22⋊D257—24no—
21D4×D257—44no—
22D4×D257—44no—
23C2×C4×D257—22no—
24#16861606—44no—
25C2×#14523339—21no—
26C4⋊Dic257—22no—
27C4⋊Dic257D412yes—
28C22⋊Dic257—22no—
29#16861595—22no—
30C2×D4×C257—22no—
31C2×D4×C257D412yes—
32C2×C8×C257—22no—
33C42×C257C2311yescharacteristic
34C22×D257—22no—
35C22×D257—48no—
36C22×D257—82no—
37Dic514—42no—
38D1028—22no—
39D1028SD1614yescomplemented
40C22⋊D257—48no—
41C2×Dic257—22no—
42C2×Dic257—41no—
43C4×D257—42no—
44C4×D257—42no—
45#14523339—22no—
46C23×C257—22no—
47D4×C257—42no—
48D4×C257—42no—
49D4×C257—42no—
50C2×C4×C257C2×D412yes—
51C2×C4×C257C2×D411yescharacteristic
52C8×C257—42no—
53C22×C257—44no—
54C22×C257—42no—
55C22×C257#12011yescharacteristic
56C2×D257—24no—
57C2×D257—42no—
58C2×D257—42no—
59C2×D257—88no—
60C4×C257—21no—
61C4×C257C2×SD1612yes—
62C4×C257AGL(1,ℤ/8)12yes—
63Dic257—42no—
64Dic257—81no—
65C2×C257—82no—
66C2×C257#45012yes—
67C2×C257#45311yescharacteristic
68D257—44no—
69C257#162911yescharacteristic, complemented, Sylow
70#1629—2571nomaximal, Sylow
71D42—2571no—
72#492—2571no—
73#476—2572no—
74#331D25711yescharacteristic, complemented
75D4⋊C8—2572no—
76C22×D4—5142no—
77C2×SD16—5142no—
78C4⋊Q8—2571no—
79#127C2×D25711yescharacteristic
80#121—5142no—
81#120—5142no—
82C4×D4—2572no—
83#106—5142no—
84C4⋊C8C2×D25712yescomplemented
85#102—5142no—
86C4×C8—2571no—
87C24—5142no—
88C2×Q8—5141no—
89C2×D4—2572no—
90C2×D4—22no—
91C2×D4—5144no—
92C2×D4—10284no—
93C2×D4—10284no—
94C2×D4D102812yes—
95C22×C4—5142no—
96SD16—10284no—
97C2×C8—22no—
98C2×C8—5141no—
99C4⋊C4—2572no—
100C4⋊C4—5142no—
101C22⋊C4—5142no—
102C22⋊C4—5142no—
103C42C22×D25711yescharacteristic
104C23—22no—
105C23—5142no—
106C23—10288no—
107C23—20562no—
108Q8—10282no—
109D4—2574no—
110D4—5142no—
111D4—42no—
112D4—42no—
113D4—42no—
114D4—10288no—
115C2×C4—5142no—
116C2×C4—10282no—
117C2×C4—10282no—
118C2×C4—10281no—
119C2×C4D4×D25712yes—
120C2×C4C2×D102811yescharacteristic
121C8—5142no—
122C8—42no—
123C22—5144no—
124C22—44no—
125C22—42no—
126C22—10282no—
127C22—10282no—
128C22—20568no—
129C22#2255156211yescharacteristic
130C4—21no—
131C4—10282no—
132C4—20561no—
133C4SD16×D25712yes—
134C4#2255161612yes—
135C2—10284no—
136C2—82no—
137C2#3065688912yes—
138C2#3065690711yescharacteristic
subgroupG/H
centerC22#22551562
commutator, cosocleC42×C257C23
FrattiniC42C22×D257
Fitting#331×C257C2
socleC22×C257#120
prime radical#30657372C2
hypercenter#331D257
hypocenterC257#1629
LCSG ▷ C42×C257 ▷ C22×C257 ▷ C257
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
2. #30657109C2semidirect
3. #30657028C2semidirect
4. #30656464C2semidirect
5. #331×C257C2semidirect
8. #22551592C22semidirect
14. C4×#14523339C22semidirect
16. C4⋊C8×C257C22semidirect
39. D1028SD16semidirect
69. C257#1629semidirect
74. #331D257semidirect
84. C4⋊C8C2×D257semidirect

Maximal subgroups

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

subgroupindex
#306573722
#306571092
#306570282
#306564642
#331×C2572
#1629257