Finite Group Databasebeta

← #237794

Subgroups — #237794

Order 512.

1050 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 1050 conjugacy classes of subgroups — too many to draw. Only its 194 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#35020C211yesmaximal, characteristic
2#20702C212yesmaximal
3#13853C211yesmaximal, characteristic
4#10347C214yesmaximal, complemented
5#10346C214yesmaximal, complemented
6#8019C212yesmaximal
7#7911C211yesmaximal, characteristic
8C22×#117C2211yescharacteristic
9C2×#407C2212yes—
10#2851—22no—
11#2646C2214yes—
12#2569—24no—
13#2442—24no—
14#2315—22no—
15C2×#380C2212yes—
16C2×C4×C22⋊C4C2211yescharacteristic
17C2×C4×C22⋊C4C2211yescharacteristic
18#1825—24no—
19#1824C2214yes—
20#1823C2214yes—
21#1736—24no—
22#1445C2212yes—
23#1445C2212yes—
24#1440—22no—
25C4×#95—22no—
26#1294C2218yes—
27#1280C2214yes—
28C24×C4C2311yescharacteristic
29C22×M16—22no—
30C2×#125—22no—
31C2×#117—22no—
32C2×#117—22no—
33C2×#117—22no—
34C2×#117—24no—
35C2×#117Q814yes—
36C22×C4⋊C4D414yes—
37C22×C4⋊C4C2311yescharacteristic
38C22×C4⋊C4C2311yescharacteristic
39C22×C22⋊C4—22no—
40C22×C22⋊C4—21no—
41C22×C22⋊C4—21no—
42C22×C22⋊C4D412yes—
43C22×C42—21no—
44C22×C42—21no—
45C22×C42D412yes—
46#407—24no—
47#407—24no—
48C2×C22⋊C8—22no—
49C2×C22⋊C8C2314yes—
50#400—28no—
51#398—28no—
52#389—24no—
53#387—28no—
54#385—28no—
55#385—28no—
56#382—22no—
57#380—28no—
58#380—22no—
59#380—22no—
60#380D414yes—
61#380C2314yes—
62C4×C22⋊C4—22no—
63C4×C22⋊C4—22no—
64C4×C22⋊C4—24no—
65C2×#95—22no—
66C2×#95—24no—
67C2×#95—24no—
68C2×#95C2312yes—
69C2×#95C2312yes—
70#340—28no—
71#340—28no—
72C25C2×D411yescharacteristic
73C23×C4—21no—
74C23×C4—21no—
75C23×C4—22no—
76C23×C4—24no—
77C23×C4—44no—
78C23×C4C2×D411yescharacteristic
79C23×C4C2×D412yes—
80C23×C4C2×D414yes—
81C23×C4C2×D412yes—
82C23×C4C2×Q812yes—
83C23×C4#4114yes—
84C23×C4#4112yes—
85C23×C4C2411yescharacteristic
86C2×M16—48no—
87C2×M16—44no—
88C22×C8—24no—
89#125—84no—
90#117—28no—
91#117—28no—
92#117—44no—
93#117—44no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—24no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—28no—
102C2×C4⋊C4—28no—
103C2×C4⋊C4—48no—
104C2×C4⋊C4C2×D418yes—
105C2×C4⋊C4C2×Q812yes—
106C2×C4⋊C4C2×Q812yes—
107C2×C22⋊C4—24no—
108C2×C22⋊C4—22no—
109C2×C22⋊C4—22no—
110C2×C22⋊C4—24no—
111C2×C22⋊C4—24no—
112C2×C22⋊C4—44no—
113C2×C22⋊C4—44no—
114C2×C22⋊C4—48no—
115C2×C22⋊C4—42no—
116C2×C22⋊C4—42no—
117C2×C22⋊C4—42no—
118C2×C42—22no—
119C2×C42—24no—
120C2×C42—22no—
121C2×C42—41no—
122C2×C42—41no—
123C2×C42—42no—
124C2×C42—44no—
125C2×C42—44no—
126C2×C42—42no—
127C22⋊C8—24no—
128C22⋊C8—24no—
129C22⋊C8—44no—
130#95—28no—
131#95—28no—
132#95—24no—
133#95—24no—
134#95—24no—
135#95—24no—
136#95—48no—
137#95—48no—
138C24—24no—
139C24—42no—
140C24—42no—
141C24#12012yes—
142C24C22⋊Q812yes—
143C24#12412yes—
144C24C22×D411yescharacteristic
145C22×C4—24no—
146C22×C4—28no—
147C22×C4—24no—
148C22×C4—24no—
149C22×C4—28no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—24no—
153C22×C4—24no—
154C22×C4—44no—
155C22×C4—48no—
156C22×C4—44no—
157C22×C4—44no—
158C22×C4—42no—
159C22×C4—44no—
160C22×C4—44no—
161C22×C4—44no—
162C22×C4—44no—
163C22×C4—42no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—42no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—44no—
170C22×C4—44no—
171C22×C4—48no—
172C22×C4—84no—
173C22×C4—84no—
174C22×C4#12011yescharacteristic
175C22×C4#12011yescharacteristic
176C22×C4C22⋊Q814yes—
177C22×C4C22⋊Q812yes—
178C22×C4C22⋊Q814yes—
179C22×C4#12414yes—
180C22×C4#12412yes—
181C22×C4#12414yes—
182C22×C4C22×D412yes—
183C22×C4C22×Q811yescharacteristic
184C22×C4C2×#4112yes—
185C22×C4C2×#4111yescharacteristic
186M16—88no—
187C2×C8—48no—
188C2×C8—44no—
189C4⋊C4—24no—
190C4⋊C4—24no—
191C4⋊C4—48no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C4⋊C4—48no—
195C4⋊C4—88no—
196C4⋊C4—88no—
197C22⋊C4—44no—
198C22⋊C4—44no—
199C22⋊C4—48no—
200C22⋊C4—48no—
201C22⋊C4—48no—
202C22⋊C4—48no—
203C42—44no—
204C42—44no—
205C42—44no—
206C42—82no—
207C42—84no—
208C23—22no—
209C23—22no—
210C23—22no—
211C23—22no—
212C23—24no—
213C23—24no—
214C23—24no—
215C23—24no—
216C23—42no—
217C23—42no—
218C23—42no—
219C23—42no—
220C23—84no—
221C23—84no—
222C23#39314yes—
223C23C2×#12011yescharacteristic
224C23C2×C22⋊Q811yescharacteristic
225C23C2×#12411yescharacteristic
226C23#57514yes—
227C2×C4—22no—
228C2×C4—24no—
229C2×C4—24no—
230C2×C4—28no—
231C2×C4—28no—
232C2×C4—24no—
233C2×C4—44no—
234C2×C4—44no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—42no—
238C2×C4—44no—
239C2×C4—48no—
240C2×C4—44no—
241C2×C4—48no—
242C2×C4—44no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—44no—
246C2×C4—48no—
247C2×C4—44no—
248C2×C4—48no—
249C2×C4—44no—
250C2×C4—48no—
251C2×C4—82no—
252C2×C4—84no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4—84no—
259C2×C4—168no—
260C2×C4#39312yes—
261C2×C4#39312yes—
262C2×C4#45814yes—
263C2×C4C2×C22⋊Q812yes—
264C2×C4C2×#12412yes—
265C8—84no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—21no—
270C22—24no—
271C22—24no—
272C22—24no—
273C22—24no—
274C22—24no—
275C22—44no—
276C22—44no—
277C22—84no—
278C22—84no—
279C22C2×#39311yescharacteristic
280C22#300512yes—
281C22#301812yes—
282C22C2×#45812yes—
283C22#308012yes—
284C22#314212yes—
285C4—22no—
286C4—22no—
287C4—22no—
288C4—84no—
289C4—82no—
290C4—82no—
291C4—84no—
292C4—88no—
293C4#203014yes—
294C2—22no—
295C2—22no—
296C2—44no—
297C2#1817514yes—
298C2#2345311yescharacteristic
299C2#2346111yescharacteristic
300C2C2×#203011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #10347C2semidirect
5. #10346C2semidirect

Maximal subgroups

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

subgroupindex
#350202
#207022
#138532
#103472
#103462
#80192
#79112