Finite Group Databasebeta

← #391728

Subgroups — #391728

Order 512.

1485 conjugacy classes of subgroups; only the 250 normal subgroups are shown.

normal subgroups only

This group has 1485 conjugacy classes of subgroups — too many to draw. Only its 250 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
1C2×#2853C211yesmaximal, characteristic
2#14685C212yesmaximal, complemented
3#11203C214yesmaximal, complemented
4#11197C212yesmaximal, complemented
5#11043C214yesmaximal
6#10966C212yesmaximal
7C22×#123C2211yescharacteristic
8C23×C42C2211yescharacteristic
9C22×#99C2214yes—
10#2853—24no—
11#2853—24no—
12C2×#388C414yescomplemented
13C2×#387C414yescomplemented
14C2×#386C2212yes—
15C2×#385C2212yes—
16C2×#381C2214yes—
17C2×#381C2211yescharacteristic
18C2×#380C2214yes—
19C2×#342C2218yes—
20C2×#342C2214yes—
21C4⋊#95—24no—
22#1434—24no—
23C22⋊#95C2214yes—
24C23×D4C2311yescharacteristic
25C24×C4C2311yescharacteristic
26C24×C4C2312yes—
27C2×#123—24no—
28C2×#123—44no—
29C22×C4⋊C4—24no—
30C22×C4⋊C4C2×C412yes—
31C22×C4⋊C4C2×C414yes—
32C22×C22⋊C4C2×C412yes—
33C22×C22⋊C4C2×C414yes—
34C22×C22⋊C4C2318yes—
35C22×C22⋊C4C2311yescharacteristic
36C22×C22⋊C4C2312yes—
37C22×C42—44no—
38C22×C42C2×C414yes—
39C22×C42D414yes—
40C22×C42D414yes—
41C2×#99—48no—
42C2×#99—44no—
43#388—24no—
44#388—24no—
45#388—24no—
46#388—24no—
47#387—24no—
48#387—24no—
49#387—24no—
50#387—24no—
51#386—22no—
52#386—24no—
53#386—22no—
54#385—22no—
55#385—24no—
56#385—22no—
57#381—24no—
58#381—48no—
59#380—48no—
60C2×#95—28no—
61C2×#95—24no—
62C2×#95—28no—
63C2×#95C2×C414yes—
64C2×#95C2×C414yes—
65C2×#95C2×C414yes—
66#342—44no—
67#342—48no—
68C25C22×C411yescharacteristic
69C25C2411yescharacteristic
70C22×D4—22no—
71C22×D4—24no—
72C22×D4—24no—
73C22×D4—24no—
74C23×C4—24no—
75C23×C4—28no—
76C23×C4—24no—
77C23×C4—24no—
78C23×C4—28no—
79C23×C4—48no—
80C23×C4C22×C411yescharacteristic
81C23×C4C22×C412yes—
82C23×C4C22×C414yes—
83C23×C4C22×C412yes—
84C23×C4C22×C414yes—
85C23×C4C2×D412yes—
86C23×C4C2×D414yes—
87C23×C4C2×D414yes—
88C23×C4C2×D412yes—
89#123—416no—
90C2×C4⋊C4—24no—
91C2×C4⋊C4—24no—
92C2×C4⋊C4—24no—
93C2×C4⋊C4—44no—
94C2×C4⋊C4—48no—
95C2×C4⋊C4—416no—
96C2×C4⋊C4—48no—
97C2×C22⋊C4—22no—
98C2×C22⋊C4—22no—
99C2×C22⋊C4—22no—
100C2×C22⋊C4—24no—
101C2×C22⋊C4—24no—
102C2×C22⋊C4—24no—
103C2×C22⋊C4—24no—
104C2×C22⋊C4—24no—
105C2×C22⋊C4—24no—
106C2×C22⋊C4—28no—
107C2×C22⋊C4—28no—
108C2×C22⋊C4—28no—
109C2×C22⋊C4—28no—
110C2×C22⋊C4—28no—
111C2×C22⋊C4—28no—
112C2×C22⋊C4—44no—
113C2×C22⋊C4—48no—
114C2×C42—48no—
115C2×C42—48no—
116C2×C42—48no—
117C2×C42C22⋊C414yes—
118C2×C42C22⋊C414yes—
119C2×C42C22⋊C414yes—
120C2×C42C22⋊C414yes—
121#99—416no—
122#95—24no—
123#95—24no—
124#95—24no—
125#95—24no—
126#95—24no—
127#95—24no—
128#95—416no—
129#95—48no—
130#95—416no—
131C24—21no—
132C24—22no—
133C24—24no—
134C24—44no—
135C24—44no—
136C24C23×C411yescharacteristic
137C24C22×D412yes—
138C24H(5,𝔽2)14yes—
139C24H(5,𝔽2)14yes—
140C242−1+414yes—
141C2×D4—44no—
142C2×D4—42no—
143C2×D4—42no—
144C2×D4—44no—
145C2×D4—44no—
146C2×D4—48no—
147C2×D4—44no—
148C22×C4—24no—
149C22×C4—28no—
150C22×C4—24no—
151C22×C4—28no—
152C22×C4—28no—
153C22×C4—24no—
154C22×C4—28no—
155C22×C4—28no—
156C22×C4—24no—
157C22×C4—24no—
158C22×C4—28no—
159C22×C4—28no—
160C22×C4—28no—
161C22×C4—24no—
162C22×C4—21no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—24no—
166C22×C4—24no—
167C22×C4—28no—
168C22×C4—44no—
169C22×C4—44no—
170C22×C4—44no—
171C22×C4—432no—
172C22×C4—44no—
173C22×C4—48no—
174C22×C4—44no—
175C22×C4—44no—
176C22×C4—44no—
177C22×C4—44no—
178C22×C4—416no—
179C22×C4—416no—
180C22×C4—48no—
181C22×C4—48no—
182C22×C4—48no—
183C22×C4C2×C22⋊C414yes—
184C22×C4C2×C22⋊C414yes—
185C22×C4C2×C22⋊C414yes—
186C4⋊C4—48no—
187C4⋊C4—48no—
188C4⋊C4—48no—
189C4⋊C4—816no—
190C22⋊C4—48no—
191C22⋊C4—48no—
192C22⋊C4—44no—
193C22⋊C4—42no—
194C22⋊C4—42no—
195C22⋊C4—44no—
196C22⋊C4—48no—
197C22⋊C4—48no—
198C22⋊C4—48no—
199C22⋊C4—48no—
200C22⋊C4—48no—
201C42—44no—
202C42—44no—
203C42—44no—
204C42—44no—
205C23—24no—
206C23—24no—
207C23—24no—
208C23—48no—
209C23—48no—
210C23—48no—
211C23—41no—
212C23—42no—
213C23—44no—
214C23—44no—
215C23—88no—
216C23—84no—
217C23C22×C22⋊C411yescharacteristic
218C23#51812yes—
219C23#52014yes—
220C23#53414yes—
221C23#53512yes—
222C23#53612yes—
223C23#53714yes—
224C23#56014yes—
225C23#56218yes—
226C23#56314yes—
227D4—84no—
228D4—84no—
229C2×C4—216no—
230C2×C4—24no—
231C2×C4—24no—
232C2×C4—24no—
233C2×C4—24no—
234C2×C4—28no—
235C2×C4—24no—
236C2×C4—24no—
237C2×C4—416no—
238C2×C4—416no—
239C2×C4—48no—
240C2×C4—48no—
241C2×C4—48no—
242C2×C4—48no—
243C2×C4—42no—
244C2×C4—41no—
245C2×C4—44no—
246C2×C4—42no—
247C2×C4—44no—
248C2×C4—44no—
249C2×C4—44no—
250C2×C4—44no—
251C2×C4—88no—
252C2×C4—84no—
253C2×C4—88no—
254C2×C4—816no—
255C2×C4—816no—
256C2×C4—88no—
257C2×C4—88no—
258C2×C4#41018yes—
259C22—24no—
260C22—28no—
261C22—28no—
262C22—24no—
263C22—24no—
264C22—24no—
265C22—24no—
266C22—44no—
267C22—48no—
268C22—44no—
269C22—81no—
270C22—84no—
271C22—82no—
272C22—164no—
273C22#230811yescharacteristic
274C22#231112yes—
275C22#237614yes—
276C22#281314yes—
277C22#281414yes—
278C22C2×#41014yes—
279C4—28no—
280C4—44no—
281C4—44no—
282C4—81no—
283C4—84no—
284C4—82no—
285C4—168no—
286C2—24no—
287C2—24no—
288C2—24no—
289C2—161no—
290C2#1465811yescharacteristic
291C2#2174812yes—
292C2#2187514yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #14685C2semidirect
3. #11203C2semidirect
4. #11197C2semidirect
12. C2×#388C4semidirect
13. C2×#387C4semidirect

Maximal subgroups

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

subgroupindex
C2×#28532
#146852
#112032
#111972
#110432
#109662