Finite Group Databasebeta

← #124709

Subgroups — #124709

Order 512.

364 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 72 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#12790C211yesmaximal, characteristic
3#12606C211yesmaximal, characteristic
4#7130C212yesmaximal, complemented
5#7117C212yesmaximal, complemented
6C2×#541C411yescharacteristic
7C2×#405C2211yescharacteristic
8#2314C411yescharacteristic
9C2×#364—21no—
10C2×#349—21no—
11#2112—21no—
12C2×#347—21no—
13#1992C2211yescharacteristic
14#1991C411yescharacteristic
15#1960C2211yescharacteristic
16#1956C411yescharacteristic
17#1894—21no—
18#1892—21no—
19#1763—21no—
20C4×#103—21no—
21#1375C2212yes—
22#1375C2212yes—
23C22×M16D411yescharacteristic
24C23×C8D411yescharacteristic
25#541—22no—
26#541—22no—
27#541—42no—
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—22no—
36C2×M32—22no—
37C2×#103—21no—
38C2×#103—21no—
39C2×#103—21no—
40C2×#103—21no—
41#405—42no—
42C2×#97D411yescharacteristic
43C2×#97C2311yescharacteristic
44C2×C4×C8C2×C411yescharacteristic
45C2×C4×C8D411yescharacteristic
46#388—41no—
47#387—41no—
48#384C2×C411yescharacteristic
49#382C2×C411yescharacteristic
50#380—21no—
51C4×C4⋊C4—21no—
52#364—42no—
53#349—42no—
54C22⋊C16—24no—
55#347—42no—
56#340—21no—
57#340—21no—
58#340—21no—
59#340—21no—
60C22×Q8—21no—
61C23×C4C22⋊C411yescharacteristic
62C2×M16—21no—
63C2×M16—21no—
64C2×M16—41no—
65C2×M16—41no—
66C22×C8—22no—
67C22×C8—22no—
68C22×C8—41no—
69C22×C8C22⋊C411yescharacteristic
70C22×C8C22⋊C411yescharacteristic
71C22×C8C2×D411yescharacteristic
72C22×C8C2×D411yescharacteristic
73C4⋊Q8—21no—
74C4⋊Q8—41no—
75C4⋊Q8M1612yes—
76#126—44no—
77#126—44no—
78#125—21no—
79#125—41no—
80#125—44no—
81#125—44no—
82#125M1612yes—
83C22⋊Q8—24no—
84C22⋊Q8—42no—
85C4×Q8—24no—
86C4×Q8—42no—
87#117—22no—
88#117—21no—
89#117—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—41no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—41no—
101C2×C22⋊C4—41no—
102C2×C42—41no—
103C2×C42—41no—
104C2×C42C22⋊C411yescharacteristic
105C2×C42C22×C411yescharacteristic
106M32—42no—
107M32—42no—
108M32—42no—
109M32—42no—
110C2×C16—24no—
111C2×C16—42no—
112#103—42no—
113#103—42no—
114#103—42no—
115#103—41no—
116#103—41no—
117#97—22no—
118#97—41no—
119C4×C8—22no—
120C4×C8—41no—
121#95—41no—
122#95—41no—
123#95—41no—
124C24#10011yescharacteristic
125C2×Q8—22no—
126C2×Q8—41no—
127C2×Q8—41no—
128C2×Q8—41no—
129C2×Q8—41no—
130C2×Q8—41no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—41no—
140C22×C4—41no—
141C22×C4—41no—
142C22×C4—41no—
143C22×C4#10011yescharacteristic
144C22×C4C2×C22⋊C411yescharacteristic
145M16—42no—
146M16—42no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—44no—
157C2×C8#10412yes—
158C2×C8#13512yes—
159C2×C8AGL(1,ℤ/8)11yescharacteristic
160C2×C8#13711yescharacteristic
161C4⋊C4—22no—
162C4⋊C4—22no—
163C4⋊C4—22no—
164C4⋊C4—42no—
165C4⋊C4—42no—
166C4⋊C4—44no—
167C4⋊C4—44no—
168C4⋊C4—42no—
169C4⋊C4—42no—
170C4⋊C4—41no—
171C4⋊C4—41no—
172C4⋊C4—41no—
173C4⋊C4—41no—
174C4⋊C4—41no—
175C4⋊C4—161no—
176C22⋊C4—22no—
177C22⋊C4—22no—
178C22⋊C4—44no—
179C22⋊C4—161no—
180C42—21no—
181C42—21no—
182C42—44no—
183C42—161no—
184C42#9912yes—
185C42C2×M1611yescharacteristic
186C42C2×M1611yescharacteristic
187C16—44no—
188C16—82no—
189C23—21no—
190C23—21no—
191C23—21no—
192C23—41no—
193C23—41no—
194C23C2×#10011yescharacteristic
195Q8—42no—
196Q8—42no—
197Q8—81no—
198Q8—81no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—81no—
216C2×C4—81no—
217C2×C4—81no—
218C2×C4—81no—
219C2×C4—81no—
220C2×C4—81no—
221C2×C4—81no—
222C2×C4—81no—
223C2×C4—81no—
224C2×C4—81no—
225C2×C4#40711yescharacteristic
226C2×C4C2×#9911yescharacteristic
227C2×C4#41611yescharacteristic
228C2×C4#41711yescharacteristic
229C2×C4C2×#10411yescharacteristic
230C2×C4#42111yescharacteristic
231C8—22no—
232C8—22no—
233C8—21no—
234C8—21no—
235C8—41no—
236C22—21no—
237C22—21no—
238C22—42no—
239C22—41no—
240C22—41no—
241C22—41no—
242C22—81no—
243C22#146211yescharacteristic
244C22#153011yescharacteristic
245C22#155111yescharacteristic
246C4—42no—
247C4—41no—
248C4—41no—
249C4—41no—
250C4—82no—
251C4—82no—
252C4—161no—
253C4—161no—
254C4#148811yescharacteristic
255C4#148911yescharacteristic
256C4#150111yescharacteristic
257C4#151311yescharacteristic
258C2—21no—
259C2—21no—
260C2—81no—
261C2#761211yescharacteristic
262C2#1191211yescharacteristic
263C2#1191411yescharacteristic
subgroupG/H
center, socleC22#1530
commutatorC2×C42C22×C4
Frattini, cosocleC2×#97C23
prime radicalC24#100
LCSG ▷ C2×C42 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#405 ◁ G
LECSG ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7130C2semidirect
5. #7117C2semidirect

Maximal subgroups

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

subgroupindex
#214132
#127902
#126062
#71302
#71172