Finite Group Databasebeta

← #241167

Subgroups — #241167

Order 512.

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

normal subgroups only

This group has 878 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#13854C211yesmaximal, characteristic
4#10571C218yesmaximal
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⋊C4C2211yescharacteristic
16C2×C4×C4⋊C4C2211yescharacteristic
17C2×#376C2211yescharacteristic
18#1948—22no—
19#1947—22no—
20#1853C2214yes—
21#1852C2214yes—
22#1776—24no—
23#1442—22no—
24#1442C2214yes—
25C4×#95—22no—
26#1294C2218yes—
27#1280C2214yes—
28C2×#125—22no—
29C2×#117—22no—
30C22×C4⋊C4D412yes—
31C22×C4⋊C4C2312yes—
32C22×C42—21no—
33C22×C42—21no—
34C22×C42D412yes—
35C22×C42C2311yescharacteristic
36#432—24no—
37#432Q818yes—
38C2×#107—24no—
39C2×#106—24no—
40C2×C4⋊C8—22no—
41C2×C22⋊C8C2314yes—
42C2×#97—22no—
43#400—28no—
44#398—28no—
45#389—24no—
46#389—24no—
47#389—24no—
48#387—28no—
49#385—28no—
50#384—24no—
51#384D412yes—
52#384D412yes—
53#382—24no—
54#382—22no—
55#382—22no—
56#380—22no—
57#380C2314yes—
58C4×C4⋊C4—22no—
59C4×C4⋊C4—22no—
60C4×C4⋊C4—22no—
61C4×C4⋊C4—22no—
62C4×C4⋊C4—22no—
63C4×C4⋊C4D414yes—
64#376—22no—
65#376Q814yes—
66C2×#95—21no—
67C2×#95—24no—
68C2×#95—24no—
69C2×#95—21no—
70C2×#95C2312yes—
71C2×#95C2312yes—
72#340—24no—
73#340—24no—
74C23×C4—22no—
75C23×C4—22no—
76C23×C4C2×D411yescharacteristic
77C23×C4C2×D411yescharacteristic
78C23×C4C2×D414yes—
79C23×C4C2411yescharacteristic
80C22×C8—24no—
81#125—84no—
82#117—48no—
83C2×C4⋊C4—22no—
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—28no—
92C2×C4⋊C4—28no—
93C2×C4⋊C4—44no—
94C2×C4⋊C4—44no—
95C2×C4⋊C4C2×D414yes—
96C2×C4⋊C4C2×D414yes—
97C2×C22⋊C4—24no—
98C2×C42—24no—
99C2×C42—22no—
100C2×C42—24no—
101C2×C42—22no—
102C2×C42—42no—
103C2×C42—42no—
104C2×C42—44no—
105C2×C42—44no—
106C2×C42—41no—
107C2×C42—41no—
108C2×C42C2×D412yes—
109C2×C42C2×D412yes—
110C2×C42C2×Q814yes—
111C2×C42C2×Q812yes—
112C2×C42#4112yes—
113#107—48no—
114#106—48no—
115C4⋊C8—28no—
116C22⋊C8—24no—
117C22⋊C8C2×Q818yes—
118#97—28no—
119#95—28no—
120#95—28no—
121#95—24no—
122#95—24no—
123#95—24no—
124#95—24no—
125#95—22no—
126#95—48no—
127#95—48no—
128#95—42no—
129#95C2×Q814yes—
130C24C22×D411yescharacteristic
131C22×C4—24no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—24no—
135C22×C4—24no—
136C22×C4—24no—
137C22×C4—24no—
138C22×C4—24no—
139C22×C4—24no—
140C22×C4—24no—
141C22×C4—42no—
142C22×C4—44no—
143C22×C4—44no—
144C22×C4—44no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4—44no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4—44no—
151C22×C4—44no—
152C22×C4#12011yescharacteristic
153C22×C4#12012yes—
154C22×C4#12011yescharacteristic
155C22×C4C22⋊Q812yes—
156C22×C4C22⋊Q812yes—
157C22×C4C4⋊Q812yes—
158C22×C4C4⋊Q812yes—
159C22×C4C22×D412yes—
160C22×C4C22×Q812yes—
161C22×C4C22×Q811yescharacteristic
162C22×C4C2×#4111yescharacteristic
163C2×C8—28no—
164C2×C8—44no—
165C2×C8—44no—
166C4⋊C4—24no—
167C4⋊C4—24no—
168C4⋊C4—44no—
169C4⋊C4—44no—
170C4⋊C4—44no—
171C4⋊C4—44no—
172C4⋊C4—42no—
173C4⋊C4—42no—
174C4⋊C4—44no—
175C4⋊C4—44no—
176C4⋊C4—44no—
177C4⋊C4—44no—
178C4⋊C4—88no—
179C4⋊C4—88no—
180C22⋊C4—48no—
181C42—48no—
182C42—48no—
183C42—44no—
184C42—82no—
185C42—82no—
186C42—82no—
187C42C22⋊Q814yes—
188C42C22⋊Q814yes—
189C42C4⋊Q814yes—
190C42C4⋊Q814yes—
191C23—21no—
192C23—21no—
193C23—22no—
194C23C2×#12011yescharacteristic
195C23C2×C22⋊Q811yescharacteristic
196C23C2×C4⋊Q811yescharacteristic
197C23#57514yes—
198C2×C4—24no—
199C2×C4—24no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—24no—
207C2×C4—24no—
208C2×C4—42no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—44no—
212C2×C4—44no—
213C2×C4—48no—
214C2×C4—48no—
215C2×C4—44no—
216C2×C4—44no—
217C2×C4—44no—
218C2×C4—44no—
219C2×C4—44no—
220C2×C4—44no—
221C2×C4—44no—
222C2×C4—44no—
223C2×C4—44no—
224C2×C4—44no—
225C2×C4—44no—
226C2×C4—84no—
227C2×C4—84no—
228C2×C4—84no—
229C2×C4—82no—
230C2×C4—82no—
231C2×C4—84no—
232C2×C4#39514yes—
233C2×C4#39512yes—
234C2×C4#39512yes—
235C2×C4#45412yes—
236C2×C4#45512yes—
237C2×C4C2×C22⋊Q812yes—
238C2×C4C2×C4⋊Q812yes—
239C8—48no—
240C22—24no—
241C22—21no—
242C22—21no—
243C22—21no—
244C22—21no—
245C22—24no—
246C22C2×#39511yescharacteristic
247C22#300512yes—
248C22C2×#45511yescharacteristic
249C22C2×#45411yescharacteristic
250C22#302512yes—
251C22#308012yes—
252C22#316612yes—
253C4—24no—
254C4—22no—
255C4—22no—
256C4—22no—
257C4—84no—
258C4—84no—
259C4—84no—
260C4—84no—
261C4—84no—
262C4#206114yes—
263C2—22no—
264C2—22no—
265C2#1818212yes—
266C2#1819112yes—
267C2#2355311yescharacteristic
268C2C2×#206111yescharacteristic
269C2#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
#138542
#105712
#80222
#79122