Finite Group Databasebeta

← #95085

Subgroups — #95085

Order 512.

866 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 866 conjugacy classes of subgroups — too many to draw. Only its 182 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
2C2×C4×C22⋊C8C211yesmaximal, characteristic
3#7020C214yesmaximal, complemented
4C23×C42C2211yescharacteristic
5C2×#432—22no—
6C22×C22⋊C8C412yes—
7C22×C4×C8C412yes—
8#1817C414yes—
9#1797—22no—
10#1797C2214yes—
11#1779—22no—
12#1749—22no—
13#1749C414yes—
14C4×C22⋊C8—22no—
15C4×C22⋊C8—22no—
16C2×#336C2212yes—
17C24×C4C2×C411yescharacteristic
18C24×C4D411yescharacteristic
19C24×C4Q811yescharacteristic
20C23×C8C2×C414yes—
21C22×C42—44no—
22C22×C42C2×C411yescharacteristic
23C22×C42C2×C412yes—
24C22×C42C2×C412yes—
25C22×C42D412yes—
26C22×C42D411yescharacteristic
27C22×C42D412yes—
28C22×C42Q811yescharacteristic
29C22×C42C2311yescharacteristic
30#432—48no—
31C2×C4⋊C8—24no—
32C2×C22⋊C8—24no—
33C2×C22⋊C8—28no—
34C2×C22⋊C8—24no—
35C2×C22⋊C8—24no—
36C2×C22⋊C8—24no—
37C2×#97—24no—
38C2×C4×C8—22no—
39C2×C4×C8—22no—
40C2×C4×C8—44no—
41#336—24no—
42#336—24no—
43#336C2×C418yes—
44C25C4⋊C411yescharacteristic
45C23×C4—24no—
46C23×C4—24no—
47C23×C4—24no—
48C23×C4—42no—
49C23×C4—42no—
50C23×C4—42no—
51C23×C4—42no—
52C23×C4—42no—
53C23×C4—42no—
54C23×C4C4212yes—
55C23×C4C22⋊C412yes—
56C23×C4C22⋊C412yes—
57C23×C4C22⋊C412yes—
58C23×C4C4⋊C411yescharacteristic
59C23×C4C4⋊C411yescharacteristic
60C23×C4C4⋊C412yes—
61C23×C4C4⋊C412yes—
62C23×C4C4⋊C411yescharacteristic
63C23×C4C22×C411yescharacteristic
64C23×C4C2×D411yescharacteristic
65C23×C4C2×Q811yescharacteristic
66C22×C8—24no—
67C22×C8—24no—
68C22×C8—28no—
69C22×C8—44no—
70C22×C8—44no—
71C22×C8M1618yes—
72C2×C42—24no—
73C2×C42—28no—
74C2×C42—44no—
75C2×C42—44no—
76C2×C42—42no—
77C2×C42—44no—
78C2×C42—42no—
79C2×C42—42no—
80C2×C42C4211yescharacteristic
81C2×C42C4211yescharacteristic
82C2×C42C22⋊C411yescharacteristic
83C2×C42C22⋊C412yes—
84C2×C42C22⋊C412yes—
85C2×C42C22⋊C411yescharacteristic
86C2×C42C4⋊C412yes—
87C2×C42C4⋊C412yes—
88C2×C42C22×C412yes—
89C2×C42C2×D412yes—
90C4⋊C8—48no—
91C22⋊C8—216no—
92C22⋊C8—48no—
93C22⋊C8—48no—
94C22⋊C8—48no—
95#97—48no—
96C4×C8—44no—
97C4×C8—44no—
98C24—22no—
99C24—22no—
100C24—41no—
101C24—41no—
102C24—41no—
103C24—41no—
104C24#9512yes—
105C24#10814yes—
106C24C2×C4⋊C411yescharacteristic
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—22no—
115C22×C4—24no—
116C22×C4—24no—
117C22×C4—22no—
118C22×C4—22no—
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—24no—
129C22×C4—42no—
130C22×C4—42no—
131C22×C4—42no—
132C22×C4—41no—
133C22×C4—41no—
134C22×C4—41no—
135C22×C4—42no—
136C22×C4—42no—
137C22×C4—42no—
138C22×C4—41no—
139C22×C4—41no—
140C22×C4—41no—
141C22×C4—416no—
142C22×C4—416no—
143C22×C4—48no—
144C22×C4—48no—
145C22×C4#9512yes—
146C22×C4#9511yescharacteristic
147C22×C4#9511yescharacteristic
148C22×C4#9511yescharacteristic
149C22×C4#9511yescharacteristic
150C22×C4C2×C4211yescharacteristic
151C22×C4C2×C22⋊C412yes—
152C22×C4C2×C22⋊C411yescharacteristic
153C22×C4C2×C4⋊C412yes—
154C22×C4C2×M1614yes—
155C22×C4#13514yes—
156C2×C8—28no—
157C2×C8—48no—
158C2×C8—44no—
159C2×C8—44no—
160C2×C8—48no—
161C2×C8—48no—
162C2×C8—48no—
163C2×C8—48no—
164C42—24no—
165C42—22no—
166C42—22no—
167C42—24no—
168C42—22no—
169C42—24no—
170C42—42no—
171C42—42no—
172C42—42no—
173C42#9914yes—
174C23—21no—
175C23—21no—
176C23—21no—
177C23—21no—
178C23—28no—
179C23—22no—
180C23—22no—
181C23—22no—
182C23—22no—
183C23—22no—
184C23—22no—
185C23—44no—
186C23—44no—
187C23—44no—
188C23—44no—
189C23—44no—
190C23—44no—
191C23#34114yes—
192C23C2×#9511yescharacteristic
193C23#42312yes—
194C23#42712yes—
195C23C2×#10812yes—
196C2×C4—28no—
197C2×C4—28no—
198C2×C4—22no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—24no—
211C2×C4—24no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—44no—
219C2×C4—48no—
220C2×C4—48no—
221C2×C4—48no—
222C2×C4—48no—
223C2×C4—48no—
224C2×C4—44no—
225C2×C4—44no—
226C2×C4—48no—
227C2×C4H(3,ℤ/4)12yes—
228C2×C4#33812yes—
229C2×C4#34212yes—
230C2×C4#34212yes—
231C2×C4C4×M1612yes—
232C2×C4#40712yes—
233C2×C4C2×#9912yes—
234C2×C4#41312yes—
235C2×C4#41614yes—
236C8—48no—
237C8—88no—
238C22—22no—
239C22—22no—
240C22—22no—
241C22—22no—
242C22—22no—
243C22—22no—
244C22—24no—
245C22—24no—
246C22—24no—
247C22—416no—
248C22—42no—
249C22—42no—
250C22—42no—
251C22—42no—
252C22#156814yes—
253C22#172811yescharacteristic
254C22C2×H(3,ℤ/4)11yescharacteristic
255C22C2×#33811yescharacteristic
256C22#173511yescharacteristic
257C22C2×#34111yescharacteristic
258C22C2×#34211yescharacteristic
259C22#174211yescharacteristic
260C4—24no—
261C4—22no—
262C4—22no—
263C4—22no—
264C4—22no—
265C4—22no—
266C4—42no—
267C4—42no—
268C4#146514yes—
269C2—24no—
270C2—44no—
271C2#786211yescharacteristic
272C2#787511yescharacteristic
273C2#791111yescharacteristic
274C2#798211yescharacteristic
275C2#798611yescharacteristic
276C2#800011yescharacteristic
277C2#800811yescharacteristic
subgroupG/H
center, socleC23C2×#95
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
prime radicalC25C4⋊C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7020C2semidirect

Maximal subgroups

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

subgroupindex
#207152
C2×C4×C22⋊C82
#70202