Finite Group Databasebeta

← #237752

Subgroups — #237752

Order 512.

1070 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 1070 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#20738C212yesmaximal
3#13959C211yesmaximal, characteristic
4#10347C214yesmaximal, complemented
5#10344C214yesmaximal
6#8018C212yesmaximal, complemented
7#7910C211yesmaximal, characteristic, complemented
8C22×C4×D4C2211yescharacteristic
9C2×#432C2212yes—
10#2647—24no—
11#2646—22no—
12#2646C2214yes—
13#2402—24no—
14#2296—22no—
15#2285C2212yes—
16C2×#383C2211yescharacteristic
17C2×#376C2211yescharacteristic
18#1824C2214yes—
19#1802—24no—
20#1786C2214yes—
21#1762—24no—
22#1445C2212yes—
23#1438C2212yes—
24#1437—22no—
25C22⋊#95—22no—
26#1294C2218yes—
27#1280C2214yes—
28C23×D4C2311yescharacteristic
29C24×C4—21no—
30C2×C4×D4—22no—
31C2×C4×D4—22no—
32C2×C4×D4—22no—
33C2×C4×D4—24no—
34C2×C4×D4Q814yes—
35C2×#117—22no—
36C22×C4⋊C4C2311yescharacteristic
37C22×C22⋊C4—21no—
38C22×C22⋊C4—24no—
39C22×C42D412yes—
40C22×C42C2311yescharacteristic
41#432—24no—
42#432—24no—
43C2×C4⋊C8—22no—
44C2×#102—28no—
45C2×C22⋊C8C2314yes—
46C2×#97—22no—
47#400—28no—
48#400—28no—
49#398—28no—
50#398—28no—
51#389—24no—
52#389—24no—
53#385—28no—
54#384D414yes—
55#383—24no—
56#382—22no—
57#380—22no—
58#380C2314yes—
59C4×C4⋊C4—28no—
60C4×C4⋊C4—22no—
61C4×C4⋊C4D414yes—
62#376—22no—
63#376—22no—
64C2×#95—21no—
65C2×#95—21no—
66C2×#95—24no—
67C2×#95D412yes—
68C2×#95C2312yes—
69C2×#95C2312yes—
70#340—28no—
71C25—21no—
72C22×D4—22no—
73C22×D4—22no—
74C22×D4—21no—
75C22×D4—44no—
76C22×D4C2×Q812yes—
77C23×C4—21no—
78C23×C4—24no—
79C23×C4—22no—
80C23×C4—42no—
81C23×C4—42no—
82C23×C4—84no—
83C23×C4C2×D411yescharacteristic
84C23×C4C2×D414yes—
85C23×C4C2×D411yescharacteristic
86C23×C4C2411yescharacteristic
87C22×C8—24no—
88C4×D4—28no—
89C4×D4—28no—
90C4×D4—44no—
91C4×D4—44no—
92#117—48no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—24no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—28no—
98C2×C4⋊C4—28no—
99C2×C4⋊C4—24no—
100C2×C4⋊C4C2×D418yes—
101C2×C4⋊C4C2×Q812yes—
102C2×C22⋊C4—24no—
103C2×C22⋊C4—24no—
104C2×C22⋊C4—41no—
105C2×C22⋊C4—41no—
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×C42C2×D412yes—
115C2×C42C2×D412yes—
116C2×C42C2×Q812yes—
117C2×C42#4114yes—
118C2×C42#4112yes—
119C4⋊C8—44no—
120#102—416no—
121C22⋊C8—24no—
122C22⋊C8—24no—
123#97—44no—
124#95—28no—
125#95—28no—
126#95—24no—
127#95—24no—
128#95—24no—
129#95—24no—
130#95—24no—
131#95—42no—
132#95—42no—
133#95—84no—
134C24—24no—
135C24—22no—
136C24—44no—
137C24—42no—
138C24—41no—
139C24—41no—
140C24—44no—
141C24C22×D411yescharacteristic
142C2×D4—28no—
143C2×D4—24no—
144C2×D4—44no—
145C2×D4—44no—
146C2×D4—48no—
147C2×D4—44no—
148C2×D4—42no—
149C22×C4—24no—
150C22×C4—24no—
151C22×C4—28no—
152C22×C4—22no—
153C22×C4—24no—
154C22×C4—24no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4—44no—
158C22×C4—44no—
159C22×C4—44no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—48no—
163C22×C4—42no—
164C22×C4—44no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—44no—
168C22×C4—88no—
169C22×C4—88no—
170C22×C4—88no—
171C22×C4#12012yes—
172C22×C4#12011yescharacteristic
173C22×C4#12011yescharacteristic
174C22×C4C22⋊Q812yes—
175C22×C4C22⋊Q812yes—
176C22×C4#12412yes—
177C22×C4#12412yes—
178C22×C4C22×D412yes—
179C22×C4C22×Q811yescharacteristic
180C22×C4C2×#4112yes—
181C22×C4C2×#4111yescharacteristic
182C2×C8—48no—
183C2×C8—44no—
184C4⋊C4—24no—
185C4⋊C4—48no—
186C4⋊C4—48no—
187C4⋊C4—42no—
188C4⋊C4—84no—
189C4⋊C4—88no—
190C4⋊C4—88no—
191C22⋊C4—48no—
192C22⋊C4—44no—
193C22⋊C4—44no—
194C22⋊C4—816no—
195C42—44no—
196C42—44no—
197C42—48no—
198C42C22⋊Q814yes—
199C42C22⋊Q814yes—
200C42#12414yes—
201C42#12414yes—
202C23—22no—
203C23—22no—
204C23—24no—
205C23—48no—
206C23—44no—
207C23—44no—
208C23—42no—
209C23—48no—
210C23—44no—
211C23—42no—
212C23—44no—
213C23—82no—
214C23—84no—
215C23—84no—
216C23—82no—
217C23—82no—
218C23—82no—
219C23C2×#12011yescharacteristic
220C23C2×C22⋊Q811yescharacteristic
221C23C2×#12411yescharacteristic
222C23#57514yes—
223D4—28no—
224D4—48no—
225D4—44no—
226C2×C4—24no—
227C2×C4—24no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—24no—
231C2×C4—24no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—42no—
237C2×C4—48no—
238C2×C4—44no—
239C2×C4—42no—
240C2×C4—44no—
241C2×C4—48no—
242C2×C4—48no—
243C2×C4—44no—
244C2×C4—48no—
245C2×C4—44no—
246C2×C4—816no—
247C2×C4—82no—
248C2×C4—82no—
249C2×C4—84no—
250C2×C4—84no—
251C2×C4—82no—
252C2×C4—82no—
253C2×C4—84no—
254C2×C4—82no—
255C2×C4#39312yes—
256C2×C4#39312yes—
257C2×C4#39314yes—
258C2×C4C2×C22⋊Q812yes—
259C2×C4C2×#12412yes—
260C8—84no—
261C22—21no—
262C22—21no—
263C22—21no—
264C22—21no—
265C22—24no—
266C22—24no—
267C22—48no—
268C22—44no—
269C22—88no—
270C22—84no—
271C22—82no—
272C22—82no—
273C22—88no—
274C22C2×#39311yescharacteristic
275C22#300512yes—
276C22#302112yes—
277C22#302512yes—
278C22#308012yes—
279C22#314212yes—
280C4—22no—
281C4—22no—
282C4—24no—
283C4—22no—
284C4—22no—
285C4—82no—
286C4—88no—
287C4—84no—
288C4—162no—
289C2—22no—
290C2—22no—
291C2—84no—
292C2#1815614yes—
293C2#2345311yescharacteristic
294C2#2346311yescharacteristic
295C2#2346711yescharacteristic
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. #10347C2semidirect
6. #8018C2semidirect
7. #7910C2semidirect

Maximal subgroups

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

subgroupindex
#346382
#207382
#139592
#103472
#103442
#80182
#79102