Finite Group Databasebeta

← #95126

Subgroups — #95126

Order 512.

1030 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 1030 conjugacy classes of subgroups — too many to draw. Only its 170 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
2#13756C211yesmaximal, characteristic
3#7032C214yesmaximal
4C22×#117C412yes—
5C23×C42C2211yescharacteristic
6C2×#432—22no—
7#1817—22no—
8#1797—22no—
9#1797C414yescomplemented
10#1779C2214yes—
11#1749—22no—
12#1749C414yescomplemented
13C4⋊#95—22no—
14C22⋊#95C412yes—
15C22⋊#95C2212yes—
16C4×#95—22no—
17C24×C4C2×C411yescharacteristic
18C24×C4C2×C411yescharacteristic
19C24×C4C2311yescharacteristic
20C2×#117—28no—
21C2×#117—44no—
22C2×#117—44no—
23C22×C4⋊C4—22no—
24C22×C22⋊C4C2×C414yes—
25C22×C42—22no—
26C22×C42—44no—
27C22×C42C2×C412yes—
28C22×C42C2×C412yes—
29C22×C42D412yes—
30C22×C42D412yes—
31C22×C42D412yes—
32C22×C42Q812yes—
33#432—48no—
34C2×C4⋊C8—24no—
35C2×C22⋊C8—24no—
36C2×C22⋊C8—24no—
37C2×C22⋊C8C2×C418yes—
38C2×#97—24no—
39C2×#95—24no—
40C2×#95—24no—
41#336—24no—
42#336—24no—
43C25C22×C411yescharacteristic
44C23×C4—24no—
45C23×C4—24no—
46C23×C4—22no—
47C23×C4—22no—
48C23×C4—24no—
49C23×C4—44no—
50C23×C4—44no—
51C23×C4—42no—
52C23×C4—42no—
53C23×C4C4212yes—
54C23×C4C4211yescharacteristic
55C23×C4C4211yescharacteristic
56C23×C4C22⋊C412yes—
57C23×C4C22⋊C411yescharacteristic
58C23×C4C22⋊C411yescharacteristic
59C23×C4C4⋊C412yes—
60C23×C4C4⋊C412yes—
61C23×C4C22×C412yes—
62C23×C4C2×D411yescharacteristic
63C23×C4C2×D412yes—
64C23×C4C2×Q811yescharacteristic
65C22×C8—28no—
66#117—432no—
67C2×C4⋊C4—28no—
68C2×C4⋊C4—44no—
69C2×C4⋊C4—44no—
70C2×C22⋊C4—24no—
71C2×C22⋊C4—44no—
72C2×C22⋊C4—44no—
73C2×C22⋊C4M1618yes—
74C2×C42—28no—
75C2×C42—28no—
76C2×C42—44no—
77C2×C42—44no—
78C2×C42—44no—
79C2×C42—44no—
80C2×C42—44no—
81C2×C42—44no—
82C2×C42—44no—
83C2×C42C22⋊C412yes—
84C2×C42C22⋊C412yes—
85C2×C42C22⋊C412yes—
86C2×C42C22⋊C412yes—
87C2×C42C4⋊C412yes—
88C2×C42C4⋊C412yes—
89C2×C42C4⋊C412yes—
90C2×C42C4⋊C412yes—
91C4⋊C8—48no—
92C22⋊C8—48no—
93C22⋊C8—48no—
94C22⋊C8—48no—
95#97—48no—
96#95—48no—
97#95—48no—
98C24—22no—
99C24—21no—
100C24—21no—
101C24—42no—
102C24—42no—
103C24C2×C4211yescharacteristic
104C24C2×C22⋊C411yescharacteristic
105C24C2×C4⋊C411yescharacteristic
106C24C2×M1614yes—
107C22×C4—216no—
108C22×C4—216no—
109C22×C4—24no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—24no—
116C22×C4—24no—
117C22×C4—24no—
118C22×C4—24no—
119C22×C4—24no—
120C22×C4—24no—
121C22×C4—24no—
122C22×C4—24no—
123C22×C4—24no—
124C22×C4—24no—
125C22×C4—24no—
126C22×C4—24no—
127C22×C4—24no—
128C22×C4—28no—
129C22×C4—41no—
130C22×C4—41no—
131C22×C4—42no—
132C22×C4—42no—
133C22×C4—41no—
134C22×C4—41no—
135C22×C4—42no—
136C22×C4—42no—
137C22×C4—42no—
138C22×C4—42no—
139C22×C4—42no—
140C22×C4—42no—
141C22×C4—416no—
142C22×C4—416no—
143C22×C4—48no—
144C22×C4—48no—
145C22×C4—44no—
146C22×C4—44no—
147C22×C4#9512yes—
148C22×C4#9512yes—
149C22×C4#9512yes—
150C22×C4#9512yes—
151C22×C4#9914yes—
152C22×C4#10012yes—
153C22×C4#10112yes—
154C22×C4C2×C22⋊C412yes—
155C22×C4C2×C4⋊C412yes—
156C2×C8—48no—
157C2×C8—48no—
158C2×C8—48no—
159C4⋊C4—416no—
160C22⋊C4—48no—
161C22⋊C4—48no—
162C42—24no—
163C42—24no—
164C42—24no—
165C42—24no—
166C42—42no—
167C42—42no—
168C42—42no—
169C42—42no—
170C42—416no—
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—44no—
187C23—44no—
188C23C2×#9511yescharacteristic
189C23C4×M1612yes—
190C23#40712yes—
191C23C2×#9912yes—
192C23C2×#10011yescharacteristic
193C23C2×#10111yescharacteristic
194C23#42312yes—
195C2×C4—216no—
196C2×C4—216no—
197C2×C4—24no—
198C2×C4—24no—
199C2×C4—24no—
200C2×C4—24no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—44no—
216C2×C4—44no—
217C2×C4—48no—
218C2×C4—48no—
219C2×C4—48no—
220C2×C4—48no—
221C2×C4—44no—
222C2×C4—44no—
223C2×C4—416no—
224C2×C4—48no—
225C2×C4—84no—
226C2×C4—88no—
227C2×C4—84no—
228C2×C4#34214yes—
229C2×C4#34314yes—
230C2×C4#34414yes—
231C2×C4#41014yes—
232C8—88no—
233C22—22no—
234C22—22no—
235C22—22no—
236C22—22no—
237C22—24no—
238C22—22no—
239C22—22no—
240C22—24no—
241C22—24no—
242C22—48no—
243C22—48no—
244C22—41no—
245C22—41no—
246C22—42no—
247C22—42no—
248C22—41no—
249C22—41no—
250C22#146212yes—
251C22#146312yes—
252C22#172811yescharacteristic
253C22C2×#34211yescharacteristic
254C22#174012yes—
255C22C2×#34311yescharacteristic
256C22C2×#34412yes—
257C4—24no—
258C4—24no—
259C4—24no—
260C4—24no—
261C4—42no—
262C4—42no—
263C4—88no—
264C2—22no—
265C2—22no—
266C2—42no—
267C2—42no—
268C2#790911yescharacteristic
269C2#793012yes—
270C2#810211yescharacteristic
271C2#810512yes—
272C2#811511yescharacteristic
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
9. #1797C4semidirect
12. #1749C4semidirect

Maximal subgroups

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

subgroupindex
#207152
#137562
#70322