Finite Group Databasebeta

← #241169

Subgroups — #241169

Order 512.

966 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 966 conjugacy classes of subgroups — too many to draw. Only its 210 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#35181C211yesmaximal, characteristic
2#21365C212yesmaximal
3#13856C211yesmaximal, characteristic
4#10572C218yesmaximal
5#8022C212yesmaximal
6#7912C211yesmaximal, characteristic
7C2×#432C2212yes—
8#2851—22no—
9#2646C2214yes—
10#2571—22no—
11#2570—24no—
12#2552—22no—
13#2317—22no—
14#2285C2212yes—
15C2×C4×C4⋊C4—22no—
16C2×C4×C4⋊C4C2211yescharacteristic
17C2×C4×C4⋊C4C2211yescharacteristic
18C2×#376C2211yescharacteristic
19#1948—22no—
20#1947—22no—
21#1853C2214yes—
22#1852C2214yes—
23#1776—24no—
24#1444—22no—
25#1444C2214yes—
26#1294C2218yes—
27#1280C2214yes—
28C2×#125—22no—
29C2×#117—22no—
30C22×C4⋊C4—24no—
31C22×C4⋊C4D412yes—
32C22×C4⋊C4C2312yes—
33C22×C42—21no—
34C22×C42—21no—
35C22×C42D412yes—
36C22×C42C2311yescharacteristic
37#432—24no—
38#432Q818yes—
39C2×#107—24no—
40C2×#106—24no—
41C2×C4⋊C8—22no—
42C2×C22⋊C8C2314yes—
43C2×#97—22no—
44#400—28no—
45#398—28no—
46#389—24no—
47#389—24no—
48#389—24no—
49#387—28no—
50#385—28no—
51#384—24no—
52#384D412yes—
53#384D412yes—
54#382—24no—
55#382—22no—
56#382—22no—
57#380—22no—
58#380C2314yes—
59C4×C4⋊C4—22no—
60C4×C4⋊C4—22no—
61C4×C4⋊C4—22no—
62C4×C4⋊C4—22no—
63C4×C4⋊C4—22no—
64C4×C4⋊C4—48no—
65C4×C4⋊C4D414yes—
66#376—22no—
67#376Q814yes—
68C2×#95—21no—
69C2×#95—21no—
70C2×#95—24no—
71C2×#95C2312yes—
72C2×#95C2312yes—
73#340—24no—
74#340—24no—
75C23×C4—22no—
76C23×C4—22no—
77C23×C4C2×D411yescharacteristic
78C23×C4C2×D411yescharacteristic
79C23×C4C2×D414yes—
80C23×C4C2411yescharacteristic
81C22×C8—24no—
82#125—84no—
83#117—48no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—28no—
93C2×C4⋊C4—28no—
94C2×C4⋊C4—44no—
95C2×C4⋊C4—44no—
96C2×C4⋊C4—48no—
97C2×C4⋊C4—48no—
98C2×C4⋊C4—48no—
99C2×C4⋊C4C2×D414yes—
100C2×C4⋊C4C2×D414yes—
101C2×C22⋊C4—24no—
102C2×C42—24no—
103C2×C42—24no—
104C2×C42—24no—
105C2×C42—24no—
106C2×C42—22no—
107C2×C42—22no—
108C2×C42—44no—
109C2×C42—44no—
110C2×C42—41no—
111C2×C42—41no—
112C2×C42C2×D412yes—
113C2×C42C2×D412yes—
114C2×C42C2×Q814yes—
115C2×C42C2×Q812yes—
116C2×C42#4112yes—
117#107—48no—
118#106—48no—
119C4⋊C8—28no—
120C22⋊C8—24no—
121C22⋊C8C2×Q818yes—
122#97—28no—
123#95—28no—
124#95—28no—
125#95—24no—
126#95—24no—
127#95—24no—
128#95—24no—
129#95—22no—
130#95—48no—
131#95—42no—
132#95C2×Q814yes—
133C24C22×D411yescharacteristic
134C22×C4—24no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—24no—
138C22×C4—24no—
139C22×C4—24no—
140C22×C4—24no—
141C22×C4—24no—
142C22×C4—24no—
143C22×C4—24no—
144C22×C4—24no—
145C22×C4—24no—
146C22×C4—42no—
147C22×C4—44no—
148C22×C4—44no—
149C22×C4—44no—
150C22×C4—44no—
151C22×C4—44no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4—44no—
155C22×C4#12011yescharacteristic
156C22×C4#12012yes—
157C22×C4#12011yescharacteristic
158C22×C4C22⋊Q812yes—
159C22×C4C22⋊Q812yes—
160C22×C4C4⋊Q812yes—
161C22×C4C4⋊Q812yes—
162C22×C4C22×D412yes—
163C22×C4C22×Q812yes—
164C22×C4C22×Q811yescharacteristic
165C22×C4C2×#4111yescharacteristic
166C2×C8—28no—
167C2×C8—44no—
168C2×C8—44no—
169C4⋊C4—24no—
170C4⋊C4—24no—
171C4⋊C4—44no—
172C4⋊C4—44no—
173C4⋊C4—44no—
174C4⋊C4—44no—
175C4⋊C4—42no—
176C4⋊C4—42no—
177C4⋊C4—44no—
178C4⋊C4—44no—
179C4⋊C4—432no—
180C4⋊C4—44no—
181C4⋊C4—44no—
182C4⋊C4—88no—
183C4⋊C4—88no—
184C22⋊C4—48no—
185C42—216no—
186C42—48no—
187C42—44no—
188C42—82no—
189C42—82no—
190C42—82no—
191C42C22⋊Q814yes—
192C42C22⋊Q814yes—
193C42C4⋊Q814yes—
194C42C4⋊Q814yes—
195C23—21no—
196C23—21no—
197C23—22no—
198C23C2×#12011yescharacteristic
199C23C2×C22⋊Q811yescharacteristic
200C23C2×C4⋊Q811yescharacteristic
201C23#57514yes—
202C2×C4—24no—
203C2×C4—24no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—24no—
211C2×C4—24no—
212C2×C4—28no—
213C2×C4—42no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—48no—
217C2×C4—48no—
218C2×C4—44no—
219C2×C4—44no—
220C2×C4—48no—
221C2×C4—48no—
222C2×C4—44no—
223C2×C4—44no—
224C2×C4—44no—
225C2×C4—44no—
226C2×C4—44no—
227C2×C4—44no—
228C2×C4—44no—
229C2×C4—44no—
230C2×C4—44no—
231C2×C4—44no—
232C2×C4—84no—
233C2×C4—82no—
234C2×C4—82no—
235C2×C4—84no—
236C2×C4#39514yes—
237C2×C4#39512yes—
238C2×C4#39512yes—
239C2×C4#45312yes—
240C2×C4#45612yes—
241C2×C4C2×C22⋊Q812yes—
242C2×C4C2×C4⋊Q812yes—
243C8—48no—
244C22—21no—
245C22—21no—
246C22—24no—
247C22—21no—
248C22—21no—
249C22—24no—
250C22C2×#39511yescharacteristic
251C22#300512yes—
252C22C2×#45311yescharacteristic
253C22C2×#45611yescharacteristic
254C22#302512yes—
255C22#308012yes—
256C22#316612yes—
257C4—24no—
258C4—22no—
259C4—22no—
260C4—22no—
261C4—48no—
262C4—84no—
263C4—84no—
264C4—84no—
265C4—84no—
266C4#206014yes—
267C2—22no—
268C2—22no—
269C2#1818312yes—
270C2#1819012yes—
271C2#2355311yescharacteristic
272C2C2×#206011yescharacteristic
273C2#2356411yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#351812
#213652
#138562
#105722
#80222
#79122