Finite Group Databasebeta

← #375666

Subgroups — #375666

Order 512.

1774 conjugacy classes of subgroups; only the 250 normal subgroups are shown.

normal subgroups only

This group has 1774 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
1#35350C211yesmaximal, characteristic
2#20392C212yesmaximal, complemented
3#11166C212yesmaximal, complemented
4#10965C212yesmaximal
5#10935C214yesmaximal, complemented
6#10915C214yesmaximal, complemented
7C22×#120C2211yescharacteristic
8C2×#518C2211yescharacteristic
9C22×#99C2214yes—
10#2482C414yescomplemented
11#2418—28no—
12#2405C414yescomplemented
13#2315C2212yes—
14#2308C2211yescharacteristic
15C2×#379C2212yes—
16C2×#379C2214yes—
17#2273C2214yes—
18C4×#99—28no—
19C2×#342C2218yes—
20C2×#342C2214yes—
21C2×#342C2214yes—
22C26C2311yescharacteristic
23C23×D4C2×C412yes—
24C23×D4C2311yescharacteristic
25C2×#123—24no—
26C2×#120—24no—
27C2×#120—44no—
28#518—44no—
29C2×C4×D4D418yes—
30C2×#117C2×C412yes—
31C22×C22⋊C4—24no—
32C22×C22⋊C4—44no—
33C22×C22⋊C4C2×C412yes—
34C22×C22⋊C4C2×C412yes—
35C22×C22⋊C4C2311yescharacteristic
36C22×C22⋊C4C2312yes—
37C22×C22⋊C4C2318yes—
38C22×C22⋊C4C2312yes—
39C2×#99—28no—
40C2×#99—216no—
41#399—24no—
42#397—24no—
43#394—28no—
44#388—28no—
45#386—24no—
46#386—24no—
47#385—24no—
48#385C2×C418yes—
49#381C2×C412yes—
50#380—24no—
51#379—24no—
52#379—216no—
53#379C2×C412yes—
54#379C2×C418yes—
55C4×C22⋊C4—28no—
56C4×C22⋊C4—48no—
57#342—28no—
58#342—216no—
59#342—44no—
60C25—22no—
61C25—24no—
62C25—28no—
63C25—84no—
64C25C22×C412yes—
65C25C22×C411yescharacteristic
66C25C2411yescharacteristic
67C22×D4—28no—
68C22×D4—24no—
69C22×D4—24no—
70C22×D4—44no—
71C22×D4—84no—
72C22×D4C2×D414yes—
73C23×C4—22no—
74C23×C4—28no—
75C23×C4C22⋊C414yescomplemented
76C23×C4C22×C411yescharacteristic
77C23×C4C22×C412yes—
78C23×C4C2×D414yes—
79#123—88no—
80#120—416no—
81C4×D4—416no—
82#117—44no—
83C2×C4⋊C4—28no—
84C2×C4⋊C4C22⋊C414yescomplemented
85C2×C22⋊C4—28no—
86C2×C22⋊C4—28no—
87C2×C22⋊C4—28no—
88C2×C22⋊C4—28no—
89C2×C22⋊C4—28no—
90C2×C22⋊C4—28no—
91C2×C22⋊C4—22no—
92C2×C22⋊C4—22no—
93C2×C22⋊C4—28no—
94C2×C22⋊C4—24no—
95C2×C22⋊C4—28no—
96C2×C22⋊C4—24no—
97C2×C22⋊C4—24no—
98C2×C22⋊C4—28no—
99C2×C22⋊C4—44no—
100C2×C22⋊C4—44no—
101C2×C22⋊C4—416no—
102C2×C22⋊C4—88no—
103C2×C22⋊C4—816no—
104C2×C22⋊C4C22⋊C414yescomplemented
105C2×C22⋊C4C22×C414yes—
106C2×C22⋊C4C22×C414yes—
107C2×C22⋊C4C2×D414yes—
108C2×C22⋊C4#41116yes—
109C2×C42—48no—
110C2×C42C22⋊C414yescomplemented
111#99—232no—
112#95—28no—
113#95—24no—
114#95—48no—
115C24—22no—
116C24—24no—
117C24—22no—
118C24—24no—
119C24—28no—
120C24—44no—
121C24—48no—
122C24—416no—
123C24—44no—
124C24—44no—
125C24—44no—
126C24—44no—
127C24—44no—
128C24—48no—
129C24—82no—
130C24—82no—
131C24—84no—
132C24—88no—
133C24—832no—
134C24—82no—
135C24C2×C22⋊C412yes—
136C24C23×C411yescharacteristic
137C24C22×D412yes—
138C24C2×#4118yes—
139C24H(5,𝔽2)14yes—
140C2×D4—28no—
141C2×D4—28no—
142C2×D4—28no—
143C2×D4—44no—
144C2×D4—48no—
145C2×D4—44no—
146C2×D4—48no—
147C2×D4—48no—
148C2×D4—84no—
149C2×D4—88no—
150C2×D4—88no—
151C22×C4—24no—
152C22×C4—22no—
153C22×C4—28no—
154C22×C4—24no—
155C22×C4—22no—
156C22×C4—24no—
157C22×C4—216no—
158C22×C4—44no—
159C22×C4—48no—
160C22×C4—44no—
161C22×C4—48no—
162C22×C4—84no—
163C22×C4—82no—
164C22×C4—84no—
165C22×C4—84no—
166C22×C4—816no—
167C22×C4C2×C22⋊C412yes—
168C22×C4C2×C22⋊C418yes—
169C4⋊C4—44no—
170C4⋊C4—88no—
171C22⋊C4—28no—
172C22⋊C4—28no—
173C22⋊C4—28no—
174C22⋊C4—216no—
175C22⋊C4—216no—
176C22⋊C4—44no—
177C22⋊C4—48no—
178C22⋊C4—416no—
179C22⋊C4—48no—
180C22⋊C4—44no—
181C22⋊C4—44no—
182C22⋊C4—48no—
183C22⋊C4—88no—
184C22⋊C4—84no—
185C22⋊C4—832no—
186C42—44no—
187C42—168no—
188C23—24no—
189C23—28no—
190C23—22no—
191C23—24no—
192C23—44no—
193C23—48no—
194C23—42no—
195C23—44no—
196C23—42no—
197C23—44no—
198C23—48no—
199C23—44no—
200C23—416no—
201C23—416no—
202C23—416no—
203C23—42no—
204C23—48no—
205C23—44no—
206C23—44no—
207C23—84no—
208C23—816no—
209C23—88no—
210C23—816no—
211C23—816no—
212C23—832no—
213C23—88no—
214C23—88no—
215C23—88no—
216C23—88no—
217C23—164no—
218C23C22×C22⋊C411yescharacteristic
219C23C4×#4114yes—
220C23#51812yes—
221C23#52514yes—
222C23#53214yes—
223C23#53414yes—
224C23#55114yes—
225C23#55318yes—
226C23#55914yes—
227D4—416no—
228D4—816no—
229C2×C4—22no—
230C2×C4—28no—
231C2×C4—24no—
232C2×C4—44no—
233C2×C4—48no—
234C2×C4—48no—
235C2×C4—44no—
236C2×C4—42no—
237C2×C4—416no—
238C2×C4—44no—
239C2×C4—88no—
240C2×C4—816no—
241C2×C4—84no—
242C2×C4—816no—
243C2×C4—164no—
244C2×C4—164no—
245C2×C4—168no—
246C22—22no—
247C22—28no—
248C22—24no—
249C22—44no—
250C22—44no—
251C22—48no—
252C22—44no—
253C22—44no—
254C22—44no—
255C22—48no—
256C22—416no—
257C22—44no—
258C22—44no—
259C22—42no—
260C22—84no—
261C22—84no—
262C22—832no—
263C22—88no—
264C22—82no—
265C22—82no—
266C22—164no—
267C22#229712yes—
268C22#230811yescharacteristic
269C22#237414yes—
270C22#260218yes—
271C4—48no—
272C4—42no—
273C4—84no—
274C4—816no—
275C4—82no—
276C2—22no—
277C2—28no—
278C2—24no—
279C2—84no—
280C2—82no—
281C2#1416711yescharacteristic
282C2#2177612yes—
subgroupG/H
center, socleC22#2308
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC22×#120C22
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C25 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20392C2semidirect
3. #11166C2semidirect
5. #10935C2semidirect
6. #10915C2semidirect
10. #2482C4semidirect
12. #2405C4semidirect
75. C23×C4C22⋊C4semidirect
84. C2×C4⋊C4C22⋊C4semidirect
104. C2×C22⋊C4C22⋊C4semidirect
110. C2×C42C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
#353502
#203922
#111662
#109652
#109352
#109152