Finite Group Databasebeta

← #124351

Subgroups — #124351

Order 512.

391 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 391 conjugacy classes of subgroups — too many to draw. Only its 88 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#21413C211yesmaximal, characteristic
2#12850C211yesmaximal, characteristic
3#12549C211yesmaximal, characteristic
4#7135C212yesmaximal, complemented
5#7108C212yesmaximal, complemented
6C2×#541C411yescharacteristic
7C22×M32—21no—
8C2×#405C2211yescharacteristic
9#2314C411yescharacteristic
10C2×#349—21no—
11#1992C411yescharacteristic
12#1991C2211yescharacteristic
13#1960C411yescharacteristic
14#1956C2211yescharacteristic
15#1894—21no—
16#1892—21no—
17#1763—21no—
18C4×#103—21no—
19#1376—21no—
20#1376C2212yes—
21#1376C2212yes—
22#1375—21no—
23C22×M16D411yescharacteristic
24C23×C8D411yescharacteristic
25#541—22no—
26#541—42no—
27#541C814yes—
28C2×C4⋊Q8C2×C411yescharacteristic
29C2×#126—41no—
30C2×#125—41no—
31C2×#125C2×C411yescharacteristic
32C2×C22⋊Q8—21no—
33C2×C4×Q8—21no—
34C2×#117C2×C411yescharacteristic
35C2×M32—24no—
36C2×M32—22no—
37C2×M32—22no—
38C2×M32—42no—
39C2×M32—41no—
40C22×C16—41no—
41C2×#103—21no—
42C2×#103—21no—
43C2×#103—21no—
44C2×#103—21no—
45#405—42no—
46C2×#97C2×C411yescharacteristic
47C2×#97D411yescharacteristic
48C2×C4×C8D411yescharacteristic
49C2×C4×C8C2311yescharacteristic
50#388—41no—
51#387—41no—
52#384C2×C411yescharacteristic
53#382C2×C411yescharacteristic
54#380—21no—
55C4×C4⋊C4—21no—
56#349—42no—
57#340—21no—
58#340—21no—
59#340—21no—
60#340—21no—
61C22×Q8—21no—
62C23×C4C22⋊C411yescharacteristic
63C2×M16—22no—
64C2×M16—22no—
65C2×M16—41no—
66C22×C8—22no—
67C22×C8—22no—
68C22×C8—21no—
69C22×C8—21no—
70C22×C8C22⋊C411yescharacteristic
71C22×C8C22⋊C411yescharacteristic
72C22×C8C2×D411yescharacteristic
73C22×C8C2×D411yescharacteristic
74C4⋊Q8—21no—
75C4⋊Q8—41no—
76C4⋊Q8C2×C812yes—
77#126—44no—
78#126—44no—
79#125—21no—
80#125—41no—
81#125—44no—
82#125—44no—
83#125C2×C812yes—
84C22⋊Q8—24no—
85C22⋊Q8—42no—
86C4×Q8—24no—
87C4×Q8—42no—
88#117—22no—
89#117C2×C812yes—
90#117M1612yes—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—41no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—41no—
102C2×C22⋊C4—41no—
103C2×C42—41no—
104C2×C42—41no—
105C2×C42C22⋊C411yescharacteristic
106C2×C42C22×C411yescharacteristic
107M32—24no—
108M32—44no—
109M32—42no—
110M32—42no—
111M32—42no—
112C2×C16—44no—
113C2×C16—42no—
114C2×C16—42no—
115#103—42no—
116#103—42no—
117#103—42no—
118#103—41no—
119#103—41no—
120#97—41no—
121#97—41no—
122C4×C8—41no—
123C4×C8—41no—
124#95—41no—
125#95—41no—
126#95—41no—
127C24#10011yescharacteristic
128C2×Q8—22no—
129C2×Q8—41no—
130C2×Q8—41no—
131C2×Q8—41no—
132C2×Q8—41no—
133C2×Q8—41no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—41no—
141C22×C4—41no—
142C22×C4—41no—
143C22×C4—41no—
144C22×C4C22⋊C812yes—
145C22×C4#9912yes—
146C22×C4#10011yescharacteristic
147C22×C4C2×C22⋊C411yescharacteristic
148M16—44no—
149C2×C8—22no—
150C2×C8—22no—
151C2×C8—24no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8#10412yes—
162C2×C8#13512yes—
163C2×C8AGL(1,ℤ/8)11yescharacteristic
164C2×C8#13711yescharacteristic
165C4⋊C4—22no—
166C4⋊C4—22no—
167C4⋊C4—22no—
168C4⋊C4—42no—
169C4⋊C4—42no—
170C4⋊C4—44no—
171C4⋊C4—42no—
172C4⋊C4—42no—
173C4⋊C4—44no—
174C4⋊C4—41no—
175C4⋊C4—41no—
176C4⋊C4—41no—
177C4⋊C4—41no—
178C4⋊C4—41no—
179C4⋊C4—161no—
180C22⋊C4—22no—
181C22⋊C4—22no—
182C22⋊C4—44no—
183C22⋊C4—161no—
184C42—21no—
185C42—21no—
186C42—44no—
187C42—161no—
188C42C22⋊C812yes—
189C42C22×C811yescharacteristic
190C42C2×M1611yescharacteristic
191C16—44no—
192C16—82no—
193C23—21no—
194C23—21no—
195C23—41no—
196C23—41no—
197C23#32312yes—
198C23C2×#10011yescharacteristic
199Q8—42no—
200Q8—42no—
201Q8—81no—
202Q8—81no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—81no—
219C2×C4—81no—
220C2×C4—81no—
221C2×C4—81no—
222C2×C4—81no—
223C2×C4—81no—
224C2×C4—81no—
225C2×C4—81no—
226C2×C4—81no—
227C2×C4—81no—
228C2×C4#32312yes—
229C2×C4C2×C22⋊C811yescharacteristic
230C2×C4C2×#9911yescharacteristic
231C2×C4#41611yescharacteristic
232C2×C4#41711yescharacteristic
233C2×C4C2×#10411yescharacteristic
234C2×C4#42111yescharacteristic
235C8—21no—
236C8—22no—
237C8—21no—
238C8—22no—
239C8—41no—
240C22—21no—
241C22—21no—
242C22—22no—
243C22—42no—
244C22—41no—
245C22—41no—
246C22—81no—
247C22C2×#32311yescharacteristic
248C22#153011yescharacteristic
249C22#155111yescharacteristic
250C4—22no—
251C4—42no—
252C4—41no—
253C4—41no—
254C4—82no—
255C4—82no—
256C4—161no—
257C4—161no—
258C4#147611yescharacteristic
259C4#147711yescharacteristic
260C4#150411yescharacteristic
261C4#151611yescharacteristic
262C2—21no—
263C2—21no—
264C2—42no—
265C2#757811yescharacteristic
266C2#1199011yescharacteristic
267C2#1199111yescharacteristic
subgroupG/H
center, socleC22#1551
commutatorC42C22×C8
Frattini, cosocleC2×C4×C8C23
prime radicalC24#100
LCSG ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#405 ◁ G
LECSG ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7135C2semidirect
5. #7108C2semidirect

Maximal subgroups

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

subgroupindex
#214132
#128502
#125492
#71352
#71082