Finite Group Databasebeta

← #149552

Subgroups — #149552

Order 512.

462 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 80 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#20688C211yesmaximal, characteristic
2#11425C212yesmaximal
3#7349C214yesmaximal
4C22×#117C411yescharacteristic
5#2280C411yescharacteristic
6#2280C2211yescharacteristic
7#1820—22no—
8#1819C412yes—
9#1819C2212yes—
10#1740—22no—
11#1458—22no—
12#1317—22no—
13#1316C2214yes—
14C24×C4C2×C411yescharacteristic
15C2×#117—21no—
16C2×#117—21no—
17C2×#117—21no—
18C2×#117—41no—
19C2×#117—41no—
20C2×#117—41no—
21C2×#117—41no—
22C2×#117C812yescomplemented
23C2×#117C812yescomplemented
24C2×#117D412yes—
25C22×C4⋊C4—21no—
26C22×C22⋊C4C2×C412yes—
27C22×C42—21no—
28#431—42no—
29#381—21no—
30#381D412yes—
31#380—41no—
32#380—41no—
33#379C2×C412yes—
34#379C2×C411yescharacteristic
35#379C2311yescharacteristic
36C4×C22⋊C4—41no—
37C4×C22⋊C4—41no—
38#342—24no—
39#342—24no—
40#324—42no—
41#323—24no—
42C25C22×C411yescharacteristic
43C23×C4—21no—
44C23×C4—21no—
45C23×C4—21no—
46C23×C4—21no—
47C23×C4—22no—
48C23×C4—42no—
49C23×C4—42no—
50C23×C4C22⋊C411yescharacteristic
51C23×C4C22⋊C411yescharacteristic
52C23×C4C2×C811yescharacteristic
53C23×C4C2×C811yescharacteristic
54C23×C4M1611yescharacteristic
55C23×C4M1611yescharacteristic
56#117—42no—
57#117—42no—
58#117—41no—
59#117—41no—
60#117—41no—
61#117—41no—
62#117—42no—
63#117—42no—
64#117—42no—
65#117—42no—
66#117—42no—
67#117—42no—
68#117—42no—
69#117—42no—
70C2×C4⋊C4—22no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—41no—
74C2×C4⋊C4—41no—
75C2×C4⋊C4—41no—
76C2×C4⋊C4—41no—
77C2×C4⋊C4—41no—
78C2×C22⋊C4—22no—
79C2×C22⋊C4—22no—
80C2×C22⋊C4—42no—
81C2×C22⋊C4—42no—
82C2×C22⋊C4—41no—
83C2×C22⋊C4—41no—
84C2×C22⋊C4—42no—
85C2×C22⋊C4C22⋊C412yes—
86C2×C22⋊C4C2×C814yes—
87C2×C22⋊C4C2×D412yes—
88C2×C42—22no—
89C2×C42—22no—
90C2×C42—22no—
91C2×C42—42no—
92C2×C42—42no—
93C2×C42—41no—
94C4⋊C8—82no—
95C22⋊C8—44no—
96C4×C8—82no—
97#95—42no—
98#95—42no—
99C24—21no—
100C24—21no—
101C24—42no—
102C24—82no—
103C24C2×C22⋊C411yescharacteristic
104C24C22×C811yescharacteristic
105C24C2×M1611yescharacteristic
106C22×C4—22no—
107C22×C4—22no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—24no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—44no—
122C22×C4—42no—
123C22×C4—44no—
124C22×C4—42no—
125C22×C4—41no—
126C22×C4—42no—
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—42no—
136C22×C4—42no—
137C22×C4—84no—
138C22×C4C22⋊C811yescharacteristic
139C22×C4C22⋊C811yescharacteristic
140C22×C4C22⋊C811yescharacteristic
141C22×C4C22⋊C811yescharacteristic
142C22×C4#9911yescharacteristic
143C22×C4#9911yescharacteristic
144C22×C4#10011yescharacteristic
145C22×C4#10011yescharacteristic
146C2×C8—84no—
147C4⋊C4—42no—
148C4⋊C4—42no—
149C4⋊C4—42no—
150C4⋊C4—42no—
151C4⋊C4—82no—
152C4⋊C4—82no—
153C22⋊C4—42no—
154C22⋊C4—42no—
155C22⋊C4—44no—
156C22⋊C4—42no—
157C22⋊C4—42no—
158C22⋊C4—44no—
159C22⋊C4—82no—
160C22⋊C4—82no—
161C22⋊C4—84no—
162C42—44no—
163C42—42no—
164C42—42no—
165C42—42no—
166C42—42no—
167C42—81no—
168C42—81no—
169C23—21no—
170C23—22no—
171C23—21no—
172C23—41no—
173C23—42no—
174C23—42no—
175C23—42no—
176C23—42no—
177C23—41no—
178C23—42no—
179C23—81no—
180C23—81no—
181C23—81no—
182C23—81no—
183C23—84no—
184C23—84no—
185C23C2×C22⋊C811yescharacteristic
186C23C2×#9911yescharacteristic
187C23C2×#10011yescharacteristic
188C2×C4—22no—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—41no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—44no—
214C2×C4—84no—
215C2×C4—84no—
216C2×C4—82no—
217C2×C4—82no—
218C2×C4—84no—
219C2×C4—82no—
220C2×C4—84no—
221C2×C4#32311yescharacteristic
222C2×C4#32311yescharacteristic
223C2×C4#32311yescharacteristic
224C2×C4#32311yescharacteristic
225C2×C4#35212yes—
226C2×C4#35212yes—
227C8—164no—
228C22—22no—
229C22—21no—
230C22—21no—
231C22—41no—
232C22—41no—
233C22—41no—
234C22—41no—
235C22—41no—
236C22—41no—
237C22—42no—
238C22—42no—
239C22—41no—
240C22—41no—
241C22—82no—
242C22—84no—
243C22—84no—
244C22—82no—
245C22C2×#32311yescharacteristic
246C22C2×#35212yes—
247C4—21no—
248C4—21no—
249C4—41no—
250C4—41no—
251C4—82no—
252C4—82no—
253C4—84no—
254C4—162no—
255C4#131711yescharacteristic
256C4#131712yes—
257C4#131711yescharacteristic
258C2—21no—
259C2—22no—
260C2—41no—
261C2—42no—
262C2—41no—
263C2—81no—
264C2C2×#131711yescharacteristic
subgroupG/H
centerC4#1317
commutatorC24C22×C8
Frattini, cosocle#379C23
socleC2C2×#1317
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C22×C4 ◁ C23×C4 ◁ #2280 ◁ G
LECSG ▷ #379 ▷ C24 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
22. C2×#117C8semidirect
23. C2×#117C8semidirect

Maximal subgroups

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

subgroupindex
#206882
#114252
#73492