Finite Group Databasebeta

← #45662763

Subgroups — #45662763

Order 32896.

316 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 93 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#30656794C211yesmaximal, characteristic, complemented
2C4×#16861602C211yesmaximal, characteristic, complemented
3D4⋊#14523339C214yesmaximal, complemented
4C2×C4×D4×C257C211yesmaximal, characteristic
5C2×#16861602—21no—
6C4⋊#14523339C2212yescomplemented
7C22⋊#14523339—21no—
8#22551503—21no—
9#22551503—21no—
10C4×#14523339C2212yescomplemented
11C22×D4×C257C411yescharacteristic
12C23×C4×C257—21no—
13C4×D4×C257—21no—
14C4×D4×C257—21no—
15C4×D4×C257C412yes—
16C4×D4×C257C2212yes—
17C2×C4⋊C4×C257C411yescharacteristic
18C2×C22⋊C4×C257—21no—
19C2×C42×C257C2211yescharacteristic
20#16861602—24no—
21C2×#14523339—22no—
22C2×#14523339—22no—
23C24×C257—21no—
24C2×D4×C257—24no—
25C2×D4×C257—21no—
26C2×D4×C257C2×C412yes—
27C22×C4×C257—21no—
28C22×C4×C257—22no—
29C22×C4×C257—22no—
30C22×C4×C257—44no—
31C22×C4×C257C2×C411yescharacteristic
32C22×C4×C257D411yescharacteristic
33C22×C4×C257D411yescharacteristic
34C4⋊C4×C257—21no—
35C4⋊C4×C257C2×C412yes—
36C22⋊C4×C257—22no—
37C22⋊C4×C257—22no—
38C42×C257C2×C411yescharacteristic
39C42×C257D411yescharacteristic
40C42×C257D411yescharacteristic
41C42×C257C2311yescharacteristic
42#14523339—24no—
43#14523339—42no—
44C23×C257—24no—
45C23×C257—22no—
46C23×C257—42no—
47C23×C257—42no—
48C23×C257C22⋊C411yescharacteristic
49D4×C257—24no—
50D4×C257—22no—
51D4×C257M1614yescomplemented
52C2×C4×C257—22no—
53C2×C4×C257—22no—
54C2×C4×C257—22no—
55C2×C4×C257—22no—
56C2×C4×C257—44no—
57C2×C4×C257—42no—
58C2×C4×C257—44no—
59C2×C4×C257C22⋊C411yescharacteristic
60C2×C4×C257C22⋊C411yescharacteristic
61C2×C4×C257C22⋊C411yescharacteristic
62C2×C4×C257C22×C411yescharacteristic
63C2×C4×C257C2×D411yescharacteristic
64C2×C4×C257C2×D411yescharacteristic
65C22×C257—21no—
66C22×C257—21no—
67C22×C257—24no—
68C22×C257—48no—
69C22×C257—42no—
70C22×C257—42no—
71C22×C257#10412yes—
72C22×C257C2×C22⋊C411yescharacteristic
73C4×C257—21no—
74C4×C257—21no—
75C4×C257—21no—
76C4×C257—21no—
77C4×C257—42no—
78C4×C257C2×M1612yes—
79C4×C257AGL(1,ℤ/8)12yes—
80C2×C257—22no—
81C2×C257—44no—
82C2×C257#40711yescharacteristic
83C2×C257#41811yescharacteristic
84C2×C257C2×#10411yescharacteristic
85C257#149011yescharacteristic, complemented, Sylow
86#1490—2571nomaximal, Sylow
87C2×C4×D4D25711yescharacteristic
88#432—2571no—
89C4×M16—2571no—
90D4⋊C8—2574no—
91C22×D4Dic25711yescharacteristic
92C23×C4—21no—
93C2×M16—5141no—
94C4×D4—21no—
95C4×D4—21no—
96C4×D4Dic25712yes—
97C4×D4C2×D25712yes—
98C2×C4⋊C4Dic25711yescharacteristic
99C2×C22⋊C4—21no—
100C2×C42C2×D25711yescharacteristic
101C4⋊C8—2572no—
102C22⋊C8—5141no—
103#97—5141no—
104#97—5141no—
105C4×C8—2572no—
106C24—21no—
107C2×D4—24no—
108C2×D4—21no—
109C2×D4C2×Dic25712yes—
110C22×C4—21no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—44no—
114C22×C4C2×Dic25711yescharacteristic
115C22×C4C22⋊D25711yescharacteristic
116C22×C4C22⋊D25711yescharacteristic
117M16—5144no—
118C2×C8—5142no—
119C2×C8—5142no—
120C4⋊C4—21no—
121C4⋊C4C2×Dic25712yes—
122C22⋊C4—22no—
123C22⋊C4—22no—
124C42C2×Dic25711yescharacteristic
125C42C22⋊D25711yescharacteristic
126C42C22⋊D25711yescharacteristic
127C42C22×D25711yescharacteristic
128C23—24no—
129C23—22no—
130C23—42no—
131C23—42no—
132C23C22⋊Dic25711yescharacteristic
133D4—24no—
134D4—22no—
135D4#1686160214yescomplemented
136C2×C4—22no—
137C2×C4—22no—
138C2×C4—22no—
139C2×C4—22no—
140C2×C4—44no—
141C2×C4—42no—
142C2×C4—44no—
143C2×C4C22⋊Dic25711yescharacteristic
144C2×C4C22⋊Dic25711yescharacteristic
145C2×C4C22⋊Dic25711yescharacteristic
146C2×C4C22×Dic25711yescharacteristic
147C2×C4C2×C22⋊D25711yescharacteristic
148C2×C4C2×C22⋊D25711yescharacteristic
149C8—5144no—
150C8—10282no—
151C22—21no—
152C22—21no—
153C22—24no—
154C22—48no—
155C22—42no—
156C22—42no—
157C22#2255151912yes—
158C22C2×C22⋊Dic25711yescharacteristic
159C4—21no—
160C4—21no—
161C4—21no—
162C4—21no—
163C4—42no—
164C4C2×#1686160212yes—
165C4#2255161812yes—
166C2—22no—
167C2—44no—
168C2#3065670111yescharacteristic
169C2#3065674211yescharacteristic
170C2C2×#2255151911yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊D257
commutatorC2×C4×C257C22×C4
FrattiniC42C22×D257
FittingC2×C4×D4×C257C2
socleC22×C257C2×C22⋊C4
prime radicalC22×D4×C257C4
cosocleC42×C257C23
hypercenterC2×C4×D4D257
hypocenterC257#1490
LCSG ▷ C2×C4×C257 ▷ C2×C257 ▷ C257
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. #30656794C2semidirect
2. C4×#16861602C2semidirect
3. D4⋊#14523339C2semidirect
6. C4⋊#14523339C22semidirect
10. C4×#14523339C22semidirect
51. D4×C257M16semidirect
85. C257#1490semidirect
135. D4#16861602semidirect

Maximal subgroups

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

subgroupindex
#306567942
C4×#168616022
D4⋊#145233392
C2×C4×D4×C2572
#1490257