Finite Group Databasebeta

← #95117

Subgroups — #95117

Order 512.

950 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 950 conjugacy classes of subgroups — too many to draw. Only its 210 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#20714C212yesmaximal
2C4×#381C211yesmaximal, characteristic
3#7032C214yesmaximal
4C23×C42C2211yescharacteristic
5C2×#431—22no—
6C2×#381C412yes—
7C2×C4×C22⋊C4C412yes—
8#1817—22no—
9#1797—22no—
10#1797C414yescomplemented
11#1779C2214yes—
12C4×C22⋊C8—22no—
13C4×C22⋊C8C414yescomplemented
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×C42D412yes—
27C22×C42D412yes—
28C22×C42Q812yes—
29#431—48no—
30C2×C4⋊C8—24no—
31C2×C22⋊C8—24no—
32C2×C22⋊C8—24no—
33C2×C22⋊C8C2×C418yes—
34C2×C4×C8—24no—
35#381—24no—
36#381C818yescomplemented
37C4×C22⋊C4—44no—
38C2×#95—24no—
39C2×#95—22no—
40C2×#95—24no—
41#336—24no—
42#336—24no—
43C25C22×C411yescharacteristic
44C23×C4—24no—
45C23×C4—24no—
46C23×C4—44no—
47C23×C4—44no—
48C23×C4—42no—
49C23×C4—42no—
50C23×C4—42no—
51C23×C4C4212yes—
52C23×C4C4211yescharacteristic
53C23×C4C4211yescharacteristic
54C23×C4C22⋊C411yescharacteristic
55C23×C4C22⋊C412yes—
56C23×C4C22⋊C411yescharacteristic
57C23×C4C4⋊C412yes—
58C23×C4C4⋊C412yes—
59C23×C4C2×C814yes—
60C23×C4M1614yes—
61C23×C4C22×C412yes—
62C23×C4C2×D411yescharacteristic
63C23×C4C2×D412yes—
64C23×C4C2×Q811yescharacteristic
65C22×C8—28no—
66C2×C22⋊C4—24no—
67C2×C22⋊C4—44no—
68C2×C22⋊C4—44no—
69C2×C22⋊C4C2×C818yes—
70C2×C42—28no—
71C2×C42—44no—
72C2×C42—44no—
73C2×C42—44no—
74C2×C42—44no—
75C2×C42—44no—
76C2×C42—44no—
77C2×C42—48no—
78C2×C42C22⋊C412yes—
79C2×C42C22⋊C412yes—
80C2×C42C22⋊C412yes—
81C2×C42C22⋊C412yes—
82C2×C42C4⋊C412yes—
83C2×C42C4⋊C412yes—
84C2×C42C4⋊C412yes—
85C2×C42C4⋊C412yes—
86C4⋊C8—48no—
87C22⋊C8—48no—
88C22⋊C8—48no—
89C22⋊C8—48no—
90C4×C8—48no—
91#95—216no—
92#95—48no—
93#95—44no—
94C24—22no—
95C24—22no—
96C24—42no—
97C24—42no—
98C24C2×C4211yescharacteristic
99C24C2×C22⋊C411yescharacteristic
100C24C2×C4⋊C411yescharacteristic
101C24C22×C812yes—
102C24C2×M1612yes—
103C22×C4—24no—
104C22×C4—22no—
105C22×C4—24no—
106C22×C4—24no—
107C22×C4—24no—
108C22×C4—24no—
109C22×C4—24no—
110C22×C4—24no—
111C22×C4—24no—
112C22×C4—24no—
113C22×C4—24no—
114C22×C4—24no—
115C22×C4—28no—
116C22×C4—41no—
117C22×C4—41no—
118C22×C4—42no—
119C22×C4—42no—
120C22×C4—41no—
121C22×C4—41no—
122C22×C4—41no—
123C22×C4—41no—
124C22×C4—42no—
125C22×C4—42no—
126C22×C4—42no—
127C22×C4—42no—
128C22×C4—48no—
129C22×C4—48no—
130C22×C4—42no—
131C22×C4—42no—
132C22×C4—42no—
133C22×C4—416no—
134C22×C4—416no—
135C22×C4—48no—
136C22×C4—48no—
137C22×C4—44no—
138C22×C4—44no—
139C22×C4#9512yes—
140C22×C4#9512yes—
141C22×C4#9512yes—
142C22×C4#9512yes—
143C22×C4C4×C812yes—
144C22×C4C4×C812yes—
145C22×C4#9712yes—
146C22×C4#9712yes—
147C22×C4C22⋊C818yes—
148C22×C4#9914yes—
149C22×C4C4⋊C818yes—
150C22×C4C2×C22⋊C412yes—
151C22×C4C2×C4⋊C412yes—
152C2×C8—48no—
153C2×C8—48no—
154C2×C8—48no—
155C22⋊C4—48no—
156C22⋊C4—48no—
157C42—24no—
158C42—24no—
159C42—24no—
160C42—24no—
161C42—42no—
162C42—42no—
163C42—42no—
164C42—42no—
165C42—88no—
166C23—22no—
167C23—22no—
168C23—24no—
169C23—24no—
170C23—21no—
171C23—21no—
172C23—22no—
173C23—22no—
174C23—21no—
175C23—21no—
176C23—22no—
177C23—22no—
178C23—44no—
179C23—44no—
180C23—44no—
181C23—44no—
182C23—48no—
183C23C2×#9511yescharacteristic
184C23C2×C4×C811yescharacteristic
185C23C2×#9711yescharacteristic
186C23C2×C22⋊C812yes—
187C23C2×#9912yes—
188C23#41312yes—
189C23C2×C4⋊C812yes—
190C2×C4—28no—
191C2×C4—216no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—44no—
210C2×C4—48no—
211C2×C4—44no—
212C2×C4—48no—
213C2×C4—48no—
214C2×C4—48no—
215C2×C4—48no—
216C2×C4—44no—
217C2×C4—416no—
218C2×C4—48no—
219C2×C4—84no—
220C2×C4—84no—
221C2×C4—88no—
222C2×C4#33614yes—
223C2×C4#33614yes—
224C2×C4#34214yes—
225C2×C4#34414yes—
226C2×C4#41014yes—
227C8—88no—
228C22—22no—
229C22—24no—
230C22—22no—
231C22—24no—
232C22—21no—
233C22—21no—
234C22—22no—
235C22—22no—
236C22—22no—
237C22—24no—
238C22—48no—
239C22—48no—
240C22—41no—
241C22—41no—
242C22—42no—
243C22—42no—
244C22—41no—
245C22—41no—
246C22#145914yes—
247C22C2×#33611yescharacteristic
248C22C2×#34211yescharacteristic
249C22#174012yes—
250C22#174211yescharacteristic
251C22C2×#34412yes—
252C4—24no—
253C4—24no—
254C4—24no—
255C4—24no—
256C4—24no—
257C4—42no—
258C4—88no—
259C2—24no—
260C2—41no—
261C2—41no—
262C2—42no—
263C2#790711yescharacteristic
264C2#792412yes—
265C2#810211yescharacteristic
266C2#811012yes—
267C2#811411yescharacteristic
subgroupG/H
center, socleC23C2×#95
commutatorC23C2×C4×C8
Frattini, cosocleC24×C4C23
prime radicalC25C22×C4
LCSG ▷ C23 ▷ 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. C4×C22⋊C8C4semidirect
36. #381C8semidirect

Maximal subgroups

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

subgroupindex
#207142
C4×#3812
#70322