Finite Group Databasebeta

← #237830

Subgroups — #237830

Order 512.

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

normal subgroups only

This group has 852 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#35181C211yesmaximal, characteristic
2#21838C212yesmaximal
3#14289C211yesmaximal, characteristic
4#10347C214yesmaximal
5#10346C214yesmaximal
6#8018C212yesmaximal
7#7912C211yesmaximal, characteristic
8C22×#125C2211yescharacteristic
9C2×#432C2212yes—
10#2851—22no—
11#2646C2214yes—
12#2571—22no—
13#2570—24no—
14#2552—22no—
15#2317—22no—
16#2285C2212yes—
17C2×#383C2211yescharacteristic
18C2×C4×C4⋊C4C2211yescharacteristic
19#1992—22no—
20#1992—22no—
21#1989—24no—
22#1824C2214yes—
23#1823C2214yes—
24#1445C2212yes—
25#1445C2212yes—
26#1441—22no—
27#1436—22no—
28#1294C2218yes—
29#1280C2214yes—
30C2×#125—22no—
31C2×#125—22no—
32C2×#125—22no—
33C2×#125—22no—
34C2×#125—24no—
35C2×#125Q814yes—
36C2×#117—22no—
37C22×C4⋊C4—21no—
38C22×C4⋊C4—21no—
39C22×C4⋊C4D412yes—
40C22×C4⋊C4C2311yescharacteristic
41C22×C4⋊C4C2311yescharacteristic
42C22×C42D412yes—
43C22×C42C2311yescharacteristic
44#432—24no—
45#432—24no—
46C2×C4⋊C8—22no—
47C2×C22⋊C8C2314yes—
48C2×#97—22no—
49#400—28no—
50#398—28no—
51#389—24no—
52#389—24no—
53#389—24no—
54#387—28no—
55#385—28no—
56#384—24no—
57#384D412yes—
58#384D412yes—
59#383—24no—
60#382—24no—
61#382—22no—
62#382—22no—
63#380—22no—
64#380C2314yes—
65C4×C4⋊C4—22no—
66C4×C4⋊C4—22no—
67C4×C4⋊C4—22no—
68C4×C4⋊C4D414yes—
69C2×#95—21no—
70C2×#95—24no—
71C2×#95—21no—
72C2×#95—24no—
73C2×#95C2312yes—
74C2×#95C2312yes—
75#340—24no—
76#340—24no—
77#340—24no—
78#340—24no—
79C23×C4—21no—
80C23×C4—21no—
81C23×C4—22no—
82C23×C4C2×D411yescharacteristic
83C23×C4C2×D414yes—
84C23×C4C2×D411yescharacteristic
85C23×C4C2411yescharacteristic
86C22×C8—24no—
87#125—28no—
88#125—28no—
89#125—44no—
90#125—44no—
91#125—84no—
92#117—48no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—24no—
104C2×C4⋊C4—24no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—28no—
108C2×C4⋊C4—28no—
109C2×C4⋊C4—44no—
110C2×C4⋊C4—41no—
111C2×C4⋊C4—41no—
112C2×C4⋊C4—42no—
113C2×C4⋊C4—42no—
114C2×C4⋊C4—42no—
115C2×C4⋊C4C2×D414yes—
116C2×C4⋊C4C2×D414yes—
117C2×C4⋊C4C2×Q812yes—
118C2×C4⋊C4C2×Q812yes—
119C2×C22⋊C4—24no—
120C2×C42—24no—
121C2×C42—22no—
122C2×C42—22no—
123C2×C42—82no—
124C2×C42C2×D412yes—
125C2×C42C2×D412yes—
126C2×C42C2×Q812yes—
127C2×C42#4114yes—
128C2×C42#4112yes—
129C4⋊C8—44no—
130C22⋊C8—24no—
131C22⋊C8—24no—
132#97—44no—
133#95—28no—
134#95—28no—
135#95—24no—
136#95—24no—
137#95—24no—
138#95—24no—
139#95—48no—
140#95—42no—
141#95—42no—
142#95—84no—
143C24C22×D411yescharacteristic
144C22×C4—24no—
145C22×C4—24no—
146C22×C4—24no—
147C22×C4—24no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—24no—
151C22×C4—24no—
152C22×C4—41no—
153C22×C4—41no—
154C22×C4—42no—
155C22×C4—44no—
156C22×C4—44no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—44no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—44no—
167C22×C4—42no—
168C22×C4—44no—
169C22×C4#12011yescharacteristic
170C22×C4#12012yes—
171C22×C4#12011yescharacteristic
172C22×C4C22⋊Q812yes—
173C22×C4C22⋊Q812yes—
174C22×C4#12412yes—
175C22×C4#12412yes—
176C22×C4C22×D412yes—
177C22×C4C22×Q811yescharacteristic
178C22×C4C2×#4112yes—
179C22×C4C2×#4111yescharacteristic
180C2×C8—44no—
181C2×C8—44no—
182C2×C8—44no—
183C4⋊C4—24no—
184C4⋊C4—24no—
185C4⋊C4—44no—
186C4⋊C4—44no—
187C4⋊C4—44no—
188C4⋊C4—44no—
189C4⋊C4—44no—
190C4⋊C4—44no—
191C4⋊C4—44no—
192C4⋊C4—44no—
193C4⋊C4—42no—
194C4⋊C4—42no—
195C4⋊C4—48no—
196C4⋊C4—44no—
197C4⋊C4—44no—
198C4⋊C4—88no—
199C4⋊C4—88no—
200C22⋊C4—48no—
201C42—44no—
202C42—48no—
203C42—82no—
204C42C22⋊Q814yes—
205C42C22⋊Q814yes—
206C42#12414yes—
207C42#12414yes—
208C23—21no—
209C23—21no—
210C23—22no—
211C23C2×#12011yescharacteristic
212C23C2×C22⋊Q811yescharacteristic
213C23C2×#12411yescharacteristic
214C23#57514yes—
215C2×C4—24no—
216C2×C4—24no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—42no—
230C2×C4—44no—
231C2×C4—48no—
232C2×C4—44no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—44no—
236C2×C4—44no—
237C2×C4—44no—
238C2×C4—44no—
239C2×C4—44no—
240C2×C4—44no—
241C2×C4—44no—
242C2×C4—44no—
243C2×C4—44no—
244C2×C4—82no—
245C2×C4—82no—
246C2×C4—82no—
247C2×C4—81no—
248C2×C4—81no—
249C2×C4—84no—
250C2×C4—82no—
251C2×C4—82no—
252C2×C4—84no—
253C2×C4—84no—
254C2×C4—82no—
255C2×C4#39314yes—
256C2×C4#39312yes—
257C2×C4#39312yes—
258C2×C4C2×C22⋊Q812yes—
259C2×C4C2×#12412yes—
260C8—84no—
261C22—21no—
262C22—21no—
263C22—21no—
264C22—21no—
265C22—24no—
266C22—24no—
267C22C2×#39311yescharacteristic
268C22#300512yes—
269C22#301911yescharacteristic
270C22#302011yescharacteristic
271C22#302812yes—
272C22#308012yes—
273C22#314212yes—
274C4—24no—
275C4—22no—
276C4—22no—
277C4—22no—
278C4—22no—
279C4—82no—
280C4—82no—
281C4—84no—
282C4—84no—
283C4—84no—
284C4—162no—
285C2—22no—
286C2—22no—
287C2#1836512yes—
288C2#1837512yes—
289C2#2345311yescharacteristic
290C2#2346211yescharacteristic
291C2#2347011yescharacteristic
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
#218382
#142892
#103472
#103462
#80182
#79122