Finite Group Databasebeta

← #95123

Subgroups — #95123

Order 512.

914 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 914 conjugacy classes of subgroups — too many to draw. Only its 166 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#20715C212yesmaximal
2C4×#381C211yesmaximal, characteristic
3#7032C214yesmaximal
4C23×C42C2211yescharacteristic
5C2×#432—22no—
6C2×#381C412yes—
7C2×C4×C22⋊C4C412yes—
8#1817—22no—
9#1797—22no—
10#1797C414yescomplemented
11#1779C2214yes—
12#1749—22no—
13#1749C414yescomplemented
14#1434—22no—
15C22⋊#95C2212yes—
16C4×#95—22no—
17C24×C4C2×C411yescharacteristic
18C24×C4C2×C411yescharacteristic
19C24×C4C2311yescharacteristic
20C22×C22⋊C4C2×C414yes—
21C22×C42—22no—
22C22×C42—44no—
23C22×C42C2×C412yes—
24C22×C42C2×C412yes—
25C22×C42D412yes—
26C22×C42D411yescharacteristic
27C22×C42D411yescharacteristic
28C22×C42D412yes—
29C22×C42Q811yescharacteristic
30C22×C42Q811yescharacteristic
31#432—48no—
32C2×C4⋊C8—24no—
33C2×C22⋊C8—24no—
34C2×C22⋊C8—24no—
35C2×C22⋊C8C2×C418yes—
36C2×#97—24no—
37#381—28no—
38C4×C22⋊C4—44no—
39C2×#95—24no—
40C2×#95—22no—
41C2×#95—24no—
42#336—24no—
43#336—24no—
44C25C22×C411yescharacteristic
45C23×C4—24no—
46C23×C4—24no—
47C23×C4—24no—
48C23×C4—44no—
49C23×C4—44no—
50C23×C4—42no—
51C23×C4—42no—
52C23×C4—42no—
53C23×C4C4211yescharacteristic
54C23×C4C4212yes—
55C23×C4C4211yescharacteristic
56C23×C4C22⋊C412yes—
57C23×C4C22⋊C411yescharacteristic
58C23×C4C22⋊C411yescharacteristic
59C23×C4C4⋊C411yescharacteristic
60C23×C4C4⋊C411yescharacteristic
61C23×C4C4⋊C411yescharacteristic
62C23×C4C4⋊C411yescharacteristic
63C23×C4C22×C412yes—
64C23×C4C2×D411yescharacteristic
65C23×C4C2×D412yes—
66C23×C4C2×Q811yescharacteristic
67C22×C8—28no—
68C2×C22⋊C4—24no—
69C2×C22⋊C4—44no—
70C2×C22⋊C4—44no—
71C2×C22⋊C4M1618yes—
72C2×C42—28no—
73C2×C42—44no—
74C2×C42—44no—
75C2×C42—42no—
76C2×C42—42no—
77C2×C42—42no—
78C2×C42—42no—
79C2×C42—44no—
80C2×C42—48no—
81C2×C42—44no—
82C2×C42C22⋊C412yes—
83C2×C42C22⋊C412yes—
84C2×C42C22⋊C412yes—
85C2×C42C22⋊C412yes—
86C2×C42C4⋊C412yes—
87C2×C42C4⋊C412yes—
88C2×C42C4⋊C412yes—
89C2×C42C4⋊C412yes—
90C4⋊C8—48no—
91C22⋊C8—48no—
92C22⋊C8—48no—
93C22⋊C8—48no—
94#97—48no—
95#95—216no—
96#95—48no—
97#95—44no—
98C24—22no—
99C24—22no—
100C24—42no—
101C24—42no—
102C24C2×C4211yescharacteristic
103C24C2×C22⋊C411yescharacteristic
104C24C2×C4⋊C411yescharacteristic
105C24C2×M1614yes—
106C22×C4—24no—
107C22×C4—24no—
108C22×C4—24no—
109C22×C4—22no—
110C22×C4—24no—
111C22×C4—24no—
112C22×C4—24no—
113C22×C4—22no—
114C22×C4—22no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—24no—
120C22×C4—24no—
121C22×C4—24no—
122C22×C4—24no—
123C22×C4—24no—
124C22×C4—28no—
125C22×C4—42no—
126C22×C4—41no—
127C22×C4—41no—
128C22×C4—42no—
129C22×C4—41no—
130C22×C4—41no—
131C22×C4—42no—
132C22×C4—42no—
133C22×C4—42no—
134C22×C4—42no—
135C22×C4—41no—
136C22×C4—41no—
137C22×C4—42no—
138C22×C4—42no—
139C22×C4—42no—
140C22×C4—416no—
141C22×C4—48no—
142C22×C4—48no—
143C22×C4—416no—
144C22×C4—48no—
145C22×C4—48no—
146C22×C4—44no—
147C22×C4—44no—
148C22×C4#9512yes—
149C22×C4#9512yes—
150C22×C4#9511yescharacteristic
151C22×C4#9511yescharacteristic
152C22×C4#9511yescharacteristic
153C22×C4#9511yescharacteristic
154C22×C4#9914yes—
155C22×C4C2×C22⋊C412yes—
156C22×C4C2×C4⋊C412yes—
157C2×C8—48no—
158C2×C8—48no—
159C2×C8—48no—
160C22⋊C4—48no—
161C22⋊C4—48no—
162C42—24no—
163C42—24no—
164C42—24no—
165C42—24no—
166C42—42no—
167C42—42no—
168C42—42no—
169C42—42no—
170C42—88no—
171C23—22no—
172C23—22no—
173C23—24no—
174C23—24no—
175C23—21no—
176C23—21no—
177C23—22no—
178C23—22no—
179C23—21no—
180C23—21no—
181C23—22no—
182C23—22no—
183C23—48no—
184C23—44no—
185C23—44no—
186C23—48no—
187C23C2×#9511yescharacteristic
188C23C4×M1612yes—
189C23#40712yes—
190C23C2×#9912yes—
191C23#41312yes—
192C23#42312yes—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—28no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—44no—
211C2×C4—48no—
212C2×C4—48no—
213C2×C4—44no—
214C2×C4—44no—
215C2×C4—44no—
216C2×C4—48no—
217C2×C4—44no—
218C2×C4—44no—
219C2×C4—44no—
220C2×C4—48no—
221C2×C4—44no—
222C2×C4—44no—
223C2×C4—416no—
224C2×C4—48no—
225C2×C4—84no—
226C2×C4—84no—
227C2×C4—88no—
228C2×C4#34212yes—
229C2×C4#34212yes—
230C2×C4#34414yes—
231C2×C4#41014yes—
232C8—88no—
233C22—22no—
234C22—24no—
235C22—22no—
236C22—24no—
237C22—22no—
238C22—22no—
239C22—24no—
240C22—24no—
241C22—48no—
242C22—48no—
243C22—41no—
244C22—41no—
245C22—42no—
246C22—42no—
247C22—41no—
248C22—41no—
249C22#146514yes—
250C22#172811yescharacteristic
251C22C2×#34211yescharacteristic
252C22#174012yes—
253C22#174211yescharacteristic
254C22C2×#34412yes—
255C4—24no—
256C4—22no—
257C4—42no—
258C4—41no—
259C4—41no—
260C4—42no—
261C4—42no—
262C4—88no—
263C4#130614yes—
264C2—24no—
265C2—42no—
266C2—42no—
267C2#791111yescharacteristic
268C2#793012yes—
269C2#810011yescharacteristic
270C2#811012yes—
271C2C2×#130611yescharacteristic
subgroupG/H
center, socleC23C2×#95
commutatorC24C2×C42
Frattini, cosocleC24×C4C23
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C24×C4 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
10. #1797C4semidirect
13. #1749C4semidirect

Maximal subgroups

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

subgroupindex
#207152
C4×#3812
#70322