Finite Group Databasebeta

← #237754

Subgroups — #237754

Order 512.

918 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 918 conjugacy classes of subgroups — too many to draw. Only its 186 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#34638C211yesmaximal, characteristic
2#20740C212yesmaximal
3#13961C211yesmaximal, characteristic
4#10346C214yesmaximal
5#10345C214yesmaximal
6#8018C212yesmaximal
7#7910C211yesmaximal, characteristic
8C22×C4×Q8C2211yescharacteristic
9C2×#432C2212yes—
10#2647—24no—
11#2646—22no—
12#2646C2214yes—
13#2402—24no—
14#2296—22no—
15#2285C2212yes—
16C2×#383C2211yescharacteristic
17C2×#376C2211yescharacteristic
18#1823C2214yes—
19#1803—24no—
20#1787C2214yes—
21#1763—24no—
22#1446C2212yes—
23#1445C2212yes—
24#1436—22no—
25#1434—22no—
26#1294C2218yes—
27#1280C2214yes—
28C23×Q8C2311yescharacteristic
29C2×C4×Q8—22no—
30C2×C4×Q8—22no—
31C2×C4×Q8—22no—
32C2×C4×Q8—24no—
33C2×C4×Q8Q814yes—
34C2×#117—22no—
35C22×C4⋊C4—21no—
36C22×C4⋊C4C2311yescharacteristic
37C22×C42—21no—
38C22×C42D412yes—
39C22×C42C2311yescharacteristic
40#432—24no—
41#432—24no—
42C2×C4⋊C8—22no—
43C2×#103—28no—
44C2×C22⋊C8C2314yes—
45C2×#97—22no—
46#400—28no—
47#400—28no—
48#398—28no—
49#398—28no—
50#389—24no—
51#389—24no—
52#385—28no—
53#384D414yes—
54#383—24no—
55#382—22no—
56#380—22no—
57#380C2314yes—
58C4×C4⋊C4—28no—
59C4×C4⋊C4—22no—
60C4×C4⋊C4D414yes—
61#376—22no—
62#376—22no—
63C2×#95—21no—
64C2×#95—24no—
65C2×#95—24no—
66C2×#95—21no—
67C2×#95D412yes—
68C2×#95C2312yes—
69C2×#95C2312yes—
70#340—28no—
71C22×Q8—21no—
72C22×Q8—22no—
73C22×Q8—22no—
74C22×Q8—44no—
75C22×Q8C2×Q812yes—
76C23×C4—21no—
77C23×C4—21no—
78C23×C4—22no—
79C23×C4C2×D411yescharacteristic
80C23×C4C2×D414yes—
81C23×C4C2×D411yescharacteristic
82C23×C4C2411yescharacteristic
83C22×C8—24no—
84C4×Q8—28no—
85C4×Q8—28no—
86C4×Q8—44no—
87C4×Q8—44no—
88#117—48no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—24no—
92C2×C4⋊C4—24no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—28no—
95C2×C4⋊C4—28no—
96C2×C4⋊C4—24no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—42no—
100C2×C4⋊C4C2×D418yes—
101C2×C4⋊C4C2×Q812yes—
102C2×C22⋊C4—24no—
103C2×C42—24no—
104C2×C42—22no—
105C2×C42—22no—
106C2×C42—24no—
107C2×C42—44no—
108C2×C42—42no—
109C2×C42—42no—
110C2×C42C2×D412yes—
111C2×C42C2×D412yes—
112C2×C42C2×Q812yes—
113C2×C42#4114yes—
114C2×C42#4112yes—
115C4⋊C8—44no—
116#103—416no—
117C22⋊C8—24no—
118C22⋊C8—24no—
119#97—44no—
120#95—28no—
121#95—28no—
122#95—24no—
123#95—24no—
124#95—24no—
125#95—24no—
126#95—24no—
127#95—48no—
128#95—48no—
129#95—42no—
130#95—42no—
131C24C22×D411yescharacteristic
132C2×Q8—28no—
133C2×Q8—24no—
134C2×Q8—44no—
135C2×Q8—44no—
136C2×Q8—44no—
137C2×Q8—48no—
138C2×Q8—42no—
139C22×C4—24no—
140C22×C4—24no—
141C22×C4—22no—
142C22×C4—24no—
143C22×C4—28no—
144C22×C4—22no—
145C22×C4—24no—
146C22×C4—24no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4—41no—
151C22×C4—41no—
152C22×C4—44no—
153C22×C4—44no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—44no—
157C22×C4—44no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—44no—
162C22×C4#12012yes—
163C22×C4#12011yescharacteristic
164C22×C4#12011yescharacteristic
165C22×C4C22⋊Q812yes—
166C22×C4C22⋊Q812yes—
167C22×C4#12412yes—
168C22×C4#12412yes—
169C22×C4C22×D412yes—
170C22×C4C22×Q811yescharacteristic
171C22×C4C2×#4112yes—
172C22×C4C2×#4111yescharacteristic
173C2×C8—48no—
174C2×C8—44no—
175C4⋊C4—24no—
176C4⋊C4—48no—
177C4⋊C4—48no—
178C4⋊C4—42no—
179C4⋊C4—44no—
180C4⋊C4—44no—
181C4⋊C4—84no—
182C4⋊C4—88no—
183C4⋊C4—88no—
184C22⋊C4—48no—
185C42—44no—
186C42—44no—
187C42—48no—
188C42—48no—
189C42C22⋊Q814yes—
190C42C22⋊Q814yes—
191C42#12414yes—
192C42#12414yes—
193C23—22no—
194C23—22no—
195C23C2×#12011yescharacteristic
196C23C2×C22⋊Q811yescharacteristic
197C23C2×#12411yescharacteristic
198C23#57514yes—
199Q8—28no—
200Q8—48no—
201Q8—44no—
202C2×C4—24no—
203C2×C4—24no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—24no—
207C2×C4—24no—
208C2×C4—24no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—44no—
216C2×C4—42no—
217C2×C4—44no—
218C2×C4—48no—
219C2×C4—44no—
220C2×C4—42no—
221C2×C4—44no—
222C2×C4—48no—
223C2×C4—48no—
224C2×C4—44no—
225C2×C4—48no—
226C2×C4—44no—
227C2×C4—82no—
228C2×C4—82no—
229C2×C4—84no—
230C2×C4—82no—
231C2×C4—82no—
232C2×C4—84no—
233C2×C4—84no—
234C2×C4—82no—
235C2×C4#39314yes—
236C2×C4#39312yes—
237C2×C4#39312yes—
238C2×C4C2×C22⋊Q812yes—
239C2×C4C2×#12412yes—
240C8—84no—
241C22—21no—
242C22—21no—
243C22—21no—
244C22—21no—
245C22—24no—
246C22—24no—
247C22C2×#39311yescharacteristic
248C22#300512yes—
249C22#302212yes—
250C22#302512yes—
251C22#308012yes—
252C22#314212yes—
253C4—22no—
254C4—22no—
255C4—24no—
256C4—22no—
257C4—22no—
258C4—44no—
259C4—82no—
260C4—88no—
261C4—84no—
262C4—162no—
263C2—22no—
264C2—22no—
265C2#1815714yes—
266C2#2345311yescharacteristic
267C2#2346411yescharacteristic
268C2#2346711yescharacteristic
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
#346382
#207402
#139612
#103462
#103452
#80182
#79102