Finite Group Databasebeta

← #95214

Subgroups — #95214

Order 512.

924 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 924 conjugacy classes of subgroups — too many to draw. Only its 158 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#20714C211yesmaximal, characteristic
2#14683C212yesmaximal
3#7035C214yesmaximal
4C23×C42C2211yescharacteristic
5C2×#431C412yescomplemented
6C2×#383—22no—
7#1817C412yescomplemented
8#1797—22no—
9#1779C2212yes—
10C4×C22⋊C8—22no—
11#1434—22no—
12#1434C414yescomplemented
13#1434C414yescomplemented
14C22⋊#95—22no—
15C22⋊#95—22no—
16C22⋊#95C2214yes—
17C24×C4D411yescharacteristic
18C24×C4D411yescharacteristic
19C24×C4C2311yescharacteristic
20C22×C22⋊C4—24no—
21C22×C42—44no—
22C22×C42C2×C411yescharacteristic
23C22×C42C2×C411yescharacteristic
24C22×C42C2×C412yes—
25C22×C42C2×C412yes—
26C22×C42D412yes—
27C22×C42D412yes—
28C22×C42Q811yescharacteristic
29C22×C42Q811yescharacteristic
30#431—44no—
31C2×C4⋊C8—22no—
32C2×C22⋊C8—24no—
33C2×C22⋊C8C2×C414yes—
34C2×C4×C8—22no—
35#383—48no—
36C2×#95—24no—
37C2×#95—24no—
38C2×#95—24no—
39C2×#95—24no—
40C2×#95—24no—
41C2×#95C2×C418yes—
42#336—24no—
43C25C2×D411yescharacteristic
44C23×C4—24no—
45C23×C4—28no—
46C23×C4—42no—
47C23×C4—42no—
48C23×C4—42no—
49C23×C4—42no—
50C23×C4—44no—
51C23×C4—44no—
52C23×C4C22⋊C412yes—
53C23×C4C22⋊C412yes—
54C23×C4C22⋊C412yes—
55C23×C4C22⋊C412yes—
56C23×C4C4⋊C411yescharacteristic
57C23×C4C4⋊C411yescharacteristic
58C23×C4C4⋊C411yescharacteristic
59C23×C4C4⋊C411yescharacteristic
60C23×C4C22×C411yescharacteristic
61C23×C4C22×C412yes—
62C23×C4C2×D412yes—
63C23×C4C2×Q811yescharacteristic
64C22×C8—24no—
65C2×C22⋊C4—48no—
66C2×C22⋊C4—48no—
67C2×C22⋊C4—48no—
68C2×C42—28no—
69C2×C42—24no—
70C2×C42—44no—
71C2×C42—44no—
72C2×C42—42no—
73C2×C42—44no—
74C2×C42—42no—
75C2×C42—42no—
76C2×C42C4211yescharacteristic
77C2×C42C4211yescharacteristic
78C2×C42C4212yes—
79C2×C42C22⋊C412yes—
80C2×C42C22⋊C411yescharacteristic
81C2×C42C22⋊C411yescharacteristic
82C2×C42C4⋊C412yes—
83C2×C42C4⋊C412yes—
84C2×C42C4⋊C412yes—
85C2×C42C4⋊C412yes—
86C4⋊C8—44no—
87C22⋊C8—48no—
88C22⋊C8—44no—
89C4×C8—44no—
90#95—48no—
91#95—48no—
92#95—48no—
93#95—48no—
94#95—48no—
95#95—48no—
96C24—24no—
97C24—24no—
98C24—41no—
99C24—41no—
100C24C2×C22⋊C412yes—
101C24C2×C4⋊C411yescharacteristic
102C24#13514yes—
103C22×C4—22no—
104C22×C4—22no—
105C22×C4—22no—
106C22×C4—24no—
107C22×C4—24no—
108C22×C4—24no—
109C22×C4—24no—
110C22×C4—24no—
111C22×C4—24no—
112C22×C4—22no—
113C22×C4—24no—
114C22×C4—24no—
115C22×C4—24no—
116C22×C4—22no—
117C22×C4—24no—
118C22×C4—22no—
119C22×C4—42no—
120C22×C4—42no—
121C22×C4—41no—
122C22×C4—41no—
123C22×C4—42no—
124C22×C4—41no—
125C22×C4—42no—
126C22×C4—42no—
127C22×C4—42no—
128C22×C4—42no—
129C22×C4—41no—
130C22×C4—42no—
131C22×C4—42no—
132C22×C4—42no—
133C22×C4—42no—
134C22×C4—42no—
135C22×C4—416no—
136C22×C4—416no—
137C22×C4—416no—
138C22×C4—48no—
139C22×C4—48no—
140C22×C4—48no—
141C22×C4—48no—
142C22×C4—48no—
143C22×C4—48no—
144C22×C4—48no—
145C22×C4#9512yes—
146C22×C4#9511yescharacteristic
147C22×C4#9512yes—
148C22×C4#9511yescharacteristic
149C22×C4#9511yescharacteristic
150C22×C4#9511yescharacteristic
151C22×C4C2×C4211yescharacteristic
152C22×C4C2×C22⋊C411yescharacteristic
153C22×C4C2×C4⋊C412yes—
154C2×C8—48no—
155C2×C8—44no—
156C22⋊C4—816no—
157C42—22no—
158C42—22no—
159C42—24no—
160C42—24no—
161C42—22no—
162C42—24no—
163C42—42no—
164C42—42no—
165C42—42no—
166C42#9914yes—
167C23—22no—
168C23—22no—
169C23—22no—
170C23—22no—
171C23—21no—
172C23—22no—
173C23—21no—
174C23—22no—
175C23—216no—
176C23—44no—
177C23—44no—
178C23—44no—
179C23—48no—
180C23C2×#9511yescharacteristic
181C23#41014yes—
182C23#41614yes—
183C23#42114yes—
184C23#42712yes—
185C2×C4—28no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—48no—
206C2×C4—48no—
207C2×C4—48no—
208C2×C4—48no—
209C2×C4—44no—
210C2×C4—44no—
211C2×C4—44no—
212C2×C4—48no—
213C2×C4—48no—
214C2×C4—44no—
215C2×C4—44no—
216C2×C4—88no—
217C2×C4—88no—
218C2×C4—88no—
219C2×C4—88no—
220C2×C4—88no—
221C2×C4—88no—
222C2×C4—88no—
223C2×C4H(3,ℤ/4)12yes—
224C2×C4#34212yes—
225C2×C4#34212yes—
226C2×C4C2×#9912yes—
227C8—84no—
228C22—24no—
229C22—24no—
230C22—24no—
231C22—24no—
232C22—24no—
233C22—24no—
234C22—22no—
235C22—24no—
236C22—416no—
237C22—42no—
238C22—41no—
239C22—41no—
240C22#152118yes—
241C22C2×H(3,ℤ/4)11yescharacteristic
242C22#173312yes—
243C22#173511yescharacteristic
244C22C2×#34211yescharacteristic
245C22#174012yes—
246C4—22no—
247C4—22no—
248C4—22no—
249C4—22no—
250C4—42no—
251C4—42no—
252C4—41no—
253C4—41no—
254C4—42no—
255C4—168no—
256C2—28no—
257C2—42no—
258C2#794412yes—
259C2#798211yescharacteristic
260C2#802512yes—
261C2#804911yescharacteristic
262C2#810011yescharacteristic
subgroupG/H
center, socleC23C2×#95
commutatorC22×C4C2×C42
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ 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
5. C2×#431C4semidirect
7. #1817C4semidirect
12. #1434C4semidirect
13. #1434C4semidirect

Maximal subgroups

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

subgroupindex
#207142
#146832
#70352