Finite Group Databasebeta

← #237780

Subgroups — #237780

Order 512.

1024 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 1024 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, complemented
2#21808C212yesmaximal
3#14169C211yesmaximal, characteristic
4#10345C214yesmaximal, complemented
5#10344C214yesmaximal
6#8022C212yesmaximal, complemented
7#7910C211yesmaximal, characteristic, complemented
8C22×#124C2211yescharacteristic
9C2×#432C2212yes—
10#2647—24no—
11#2646—22no—
12#2646C2214yes—
13#2402—24no—
14#2296—22no—
15#2285C2212yes—
16C2×#376C2211yescharacteristic
17C2×#376C2211yescharacteristic
18#1983—24no—
19#1959—24no—
20#1787C2214yes—
21#1786C2214yes—
22#1446C2212yes—
23#1438C2212yes—
24#1437—22no—
25#1437—22no—
26#1294C2218yes—
27#1280C2214yes—
28C23×Q8C2311yescharacteristic
29C23×D4C2311yescharacteristic
30C2×#124—22no—
31C2×#124—22no—
32C2×#124—22no—
33C2×#124—24no—
34C2×#124Q814yes—
35C2×#117—22no—
36C22×C22⋊C4—21no—
37C22×C22⋊C4—21no—
38C22×C22⋊C4—24no—
39C22×C42—21no—
40C22×C42D412yes—
41C22×C42C2311yescharacteristic
42#432—24no—
43#432—24no—
44C2×C4⋊C8—22no—
45C2×#103—28no—
46C2×#102—28no—
47C2×C22⋊C8C2314yes—
48C2×#97—22no—
49#400—28no—
50#400—28no—
51#398—28no—
52#398—28no—
53#389—24no—
54#389—24no—
55#385—28no—
56#384D414yes—
57#382—22no—
58#380—22no—
59#380C2314yes—
60C4×C4⋊C4—28no—
61C4×C4⋊C4—22no—
62C4×C4⋊C4D414yes—
63#376—22no—
64#376—22no—
65#376—24no—
66C2×#95—21no—
67C2×#95—24no—
68C2×#95D412yes—
69C2×#95C2312yes—
70C2×#95C2312yes—
71C25—21no—
72C22×Q8—21no—
73C22×Q8—22no—
74C22×Q8—22no—
75C22×Q8—82no—
76C22×Q8C2×Q812yes—
77C22×D4—22no—
78C22×D4—22no—
79C22×D4—21no—
80C22×D4—82no—
81C22×D4C2×Q812yes—
82C23×C4—21no—
83C23×C4—22no—
84C23×C4C2×D411yescharacteristic
85C23×C4C2×D414yes—
86C23×C4C2×D411yescharacteristic
87C23×C4C2411yescharacteristic
88C22×C8—24no—
89#124—28no—
90#124—28no—
91#124—44no—
92#124—44no—
93#117—48no—
94C2×C4⋊C4—24no—
95C2×C4⋊C4—28no—
96C2×C4⋊C4—28no—
97C2×C4⋊C4—24no—
98C2×C4⋊C4C2×D418yes—
99C2×C22⋊C4—24no—
100C2×C22⋊C4—24no—
101C2×C22⋊C4—24no—
102C2×C22⋊C4—42no—
103C2×C22⋊C4—41no—
104C2×C22⋊C4—41no—
105C2×C22⋊C4—42no—
106C2×C22⋊C4—42no—
107C2×C22⋊C4—48no—
108C2×C22⋊C4—48no—
109C2×C22⋊C4—84no—
110C2×C42—24no—
111C2×C42—22no—
112C2×C42—22no—
113C2×C42—44no—
114C2×C42—44no—
115C2×C42—42no—
116C2×C42C2×D412yes—
117C2×C42C2×D412yes—
118C2×C42C2×Q812yes—
119C2×C42#4114yes—
120C2×C42#4112yes—
121C4⋊C8—44no—
122#103—88no—
123#102—88no—
124C22⋊C8—24no—
125C22⋊C8—24no—
126#97—44no—
127#95—28no—
128#95—28no—
129#95—24no—
130#95—24no—
131#95—24no—
132#95—24no—
133#95—24no—
134#95—42no—
135#95—84no—
136C24—22no—
137C24—44no—
138C24—44no—
139C24—41no—
140C24—41no—
141C24—44no—
142C24C22×D411yescharacteristic
143C2×Q8—24no—
144C2×Q8—48no—
145C2×Q8—42no—
146C2×Q8—82no—
147C2×Q8—82no—
148C2×Q8—84no—
149C2×D4—24no—
150C2×D4—48no—
151C2×D4—42no—
152C2×D4—82no—
153C2×D4—84no—
154C2×D4—82no—
155C22×C4—24no—
156C22×C4—24no—
157C22×C4—24no—
158C22×C4—28no—
159C22×C4—22no—
160C22×C4—24no—
161C22×C4—24no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—44no—
165C22×C4—44no—
166C22×C4—44no—
167C22×C4—44no—
168C22×C4—42no—
169C22×C4—41no—
170C22×C4—42no—
171C22×C4—41no—
172C22×C4—44no—
173C22×C4#12012yes—
174C22×C4#12011yescharacteristic
175C22×C4#12011yescharacteristic
176C22×C4C22⋊Q812yes—
177C22×C4C22⋊Q812yes—
178C22×C4#12412yes—
179C22×C4#12412yes—
180C22×C4C22×D412yes—
181C22×C4C22×Q811yescharacteristic
182C22×C4C2×#4112yes—
183C22×C4C2×#4111yescharacteristic
184C2×C8—48no—
185C2×C8—44no—
186C4⋊C4—48no—
187C4⋊C4—48no—
188C4⋊C4—84no—
189C4⋊C4—88no—
190C4⋊C4—88no—
191C22⋊C4—48no—
192C22⋊C4—44no—
193C22⋊C4—44no—
194C22⋊C4—48no—
195C22⋊C4—816no—
196C42—44no—
197C42—44no—
198C42—48no—
199C42—48no—
200C42C22⋊Q814yes—
201C42C22⋊Q814yes—
202C42#12414yes—
203C42#12414yes—
204C23—22no—
205C23—22no—
206C23—44no—
207C23—44no—
208C23—44no—
209C23—42no—
210C23—84no—
211C23—82no—
212C23—84no—
213C23—88no—
214C23—82no—
215C23—82no—
216C23—82no—
217C23—82no—
218C23—82no—
219C23C2×#12011yescharacteristic
220C23C2×C22⋊Q811yescharacteristic
221C23C2×#12411yescharacteristic
222C23#57514yes—
223Q8—48no—
224Q8—84no—
225D4—48no—
226D4—84no—
227C2×C4—24no—
228C2×C4—24no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—24no—
232C2×C4—24no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—42no—
238C2×C4—44no—
239C2×C4—48no—
240C2×C4—48no—
241C2×C4—44no—
242C2×C4—42no—
243C2×C4—44no—
244C2×C4—48no—
245C2×C4—48no—
246C2×C4—44no—
247C2×C4—48no—
248C2×C4—44no—
249C2×C4—84no—
250C2×C4—82no—
251C2×C4—82no—
252C2×C4—82no—
253C2×C4—82no—
254C2×C4—84no—
255C2×C4#39312yes—
256C2×C4#39312yes—
257C2×C4#39314yes—
258C2×C4C2×C22⋊Q812yes—
259C2×C4C2×#12412yes—
260C8—84no—
261C22—24no—
262C22—24no—
263C22—21no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—88no—
268C22—82no—
269C22—82no—
270C22—82no—
271C22—164no—
272C22—164no—
273C22C2×#39311yescharacteristic
274C22#300512yes—
275C22#301812yes—
276C22#302712yes—
277C22#308012yes—
278C22#314212yes—
279C4—22no—
280C4—22no—
281C4—24no—
282C4—22no—
283C4—22no—
284C4—84no—
285C4—88no—
286C4—84no—
287C4—82no—
288C2—22no—
289C2—22no—
290C2—162no—
291C2#1833414yes—
292C2#2345311yescharacteristic
293C2#2346111yescharacteristic
294C2#2346911yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #34638C2semidirect
4. #10345C2semidirect
6. #8022C2semidirect
7. #7910C2semidirect

Maximal subgroups

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

subgroupindex
#346382
#218082
#141692
#103452
#103442
#80222
#79102