Finite Group Databasebeta

← #193254

Subgroups — #193254

Order 512.

1122 conjugacy classes of subgroups; only the 282 normal subgroups are shown.

normal subgroups only

This group has 1122 conjugacy classes of subgroups — too many to draw. Only its 282 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×#2369C211yesmaximal, characteristic
2#21365C212yesmaximal
3#21202C211yesmaximal, characteristic
4C2×#1558C211yesmaximal, characteristic
5#8927C218yesmaximal, complemented
6#7904C212yesmaximal
7C22×C4⋊Q8C2211yescharacteristic
8C2×#432C2212yes—
9C2×#419—22no—
10C2×C4×M16C2211yescharacteristic
11#2369—24no—
12#2369—22no—
13#2369C2214yes—
14C2×#391—22no—
15C2×#389—22no—
16#2285C2211yescharacteristic
17C2×C4×C4⋊C4—22no—
18C2×C4×C4⋊C4C2212yes—
19#1960—22no—
20#1956C2214yes—
21#1948C2214yes—
22#1947C2214yes—
23#1853—22no—
24#1852—22no—
25#1776—24no—
26#1558C418yescomplemented
27C2×#333C2214yes—
28#1279C2218yes—
29C23×Q8D412yes—
30C22×M16—21no—
31C2×C4⋊Q8—22no—
32C2×C4⋊Q8—24no—
33C2×C4⋊Q8C2×C414yes—
34C2×C4⋊Q8D412yes—
35C2×C4⋊Q8Q814yes—
36C2×C4⋊Q8C2312yes—
37C2×#117—21no—
38C22×C4⋊C4—22no—
39C22×C4⋊C4—22no—
40C22×C4⋊C4D412yes—
41C22×C4⋊C4D412yes—
42C22×C42—22no—
43C22×C42—22no—
44C22×C42C2311yescharacteristic
45#432—24no—
46#432C2×C418yes—
47C2×#107—24no—
48C2×#106—24no—
49C2×C4⋊C8C2314yes—
50#419—48no—
51C2×#103—28no—
52C2×C22⋊C8—22no—
53C4×M16—24no—
54C2×#97—22no—
55C2×#97—21no—
56C2×C4×C8C2312yes—
57#391—28no—
58#391—28no—
59#389—28no—
60#389—28no—
61#384C2312yes—
62#382—21no—
63#380—22no—
64C4×C4⋊C4—28no—
65C4×C4⋊C4—28no—
66C4×C4⋊C4—22no—
67C4×C4⋊C4—22no—
68C4×C4⋊C4—22no—
69C4×C4⋊C4D414yes—
70C4×C4⋊C4C2314yes—
71C2×#95—24no—
72C2×#95—22no—
73#340—24no—
74#340—24no—
75#333C2×C4116yescomplemented
76C22×Q8—22no—
77C22×Q8—24no—
78C22×Q8—44no—
79C22×Q8—44no—
80C22×Q8C4⋊C418yes—
81C22×Q8C2×D414yes—
82C23×C4—22no—
83C23×C4—22no—
84C23×C4—22no—
85C23×C4C2×D411yescharacteristic
86C23×C4C2×D412yes—
87C2×M16—24no—
88C2×M16—42no—
89C2×M16—42no—
90C22×C8—22no—
91C22×C8—24no—
92C4⋊Q8—28no—
93C4⋊Q8—28no—
94C4⋊Q8—24no—
95C4⋊Q8—24no—
96C4⋊Q8C4⋊C418yes—
97C4⋊Q8C22×C414yes—
98C4⋊Q8C2×Q814yes—
99#117—44no—
100C2×C4⋊C4—28no—
101C2×C4⋊C4—24no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—28no—
104C2×C4⋊C4—24no—
105C2×C4⋊C4—28no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—28no—
111C2×C4⋊C4—44no—
112C2×C4⋊C4—84no—
113C2×C4⋊C4C2×D414yes—
114C2×C4⋊C4C2×D414yes—
115C2×C22⋊C4—41no—
116C2×C42—24no—
117C2×C42—24no—
118C2×C42—28no—
119C2×C42—48no—
120C2×C42—41no—
121C2×C42—84no—
122C2×C42C22×C412yes—
123C2×C42C2×D411yescharacteristic
124C2×C42C2×D412yes—
125C2×C42C2×Q812yes—
126C2×C42#4114yes—
127C2×C42C2411yescharacteristic
128#107—48no—
129#106—48no—
130C4⋊C8—24no—
131C4⋊C8C22×C418yes—
132#103—416no—
133C22⋊C8—28no—
134#97—28no—
135#97—42no—
136C4×C8—24no—
137#95—216no—
138#95—24no—
139#95—24no—
140#95—42no—
141C24#12011yescharacteristic
142C2×Q8—24no—
143C2×Q8—28no—
144C2×Q8—44no—
145C2×Q8—416no—
146C2×Q8—44no—
147C2×Q8—88no—
148C2×Q8—82no—
149C2×Q8—82no—
150C2×Q8C2×C4⋊C418yes—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—24no—
154C22×C4—24no—
155C22×C4—28no—
156C22×C4—24no—
157C22×C4—24no—
158C22×C4—28no—
159C22×C4—24no—
160C22×C4—44no—
161C22×C4—44no—
162C22×C4—44no—
163C22×C4—44no—
164C22×C4—44no—
165C22×C4—44no—
166C22×C4—44no—
167C22×C4—42no—
168C22×C4C2×C4⋊C412yes—
169C22×C4C4×D414yes—
170C22×C4#12011yescharacteristic
171C22×C4#12012yes—
172C22×C4C22⋊Q814yes—
173C22×C4#12312yes—
174C22×C4C22×D411yescharacteristic
175C22×C4C22×D412yes—
176M16—48no—
177C2×C8—24no—
178C2×C8—28no—
179C2×C8—42no—
180C2×C8—42no—
181C2×C8—44no—
182C2×C8—44no—
183C4⋊C4—28no—
184C4⋊C4—28no—
185C4⋊C4—28no—
186C4⋊C4—24no—
187C4⋊C4—216no—
188C4⋊C4—24no—
189C4⋊C4—24no—
190C4⋊C4—44no—
191C4⋊C4—44no—
192C4⋊C4—44no—
193C4⋊C4—44no—
194C4⋊C4—88no—
195C4⋊C4#13718yes—
196C22⋊C4—82no—
197C42—48no—
198C42—48no—
199C42—82no—
200C42—88no—
201C42C2×C4⋊C412yes—
202C42C4×D414yes—
203C42C22⋊Q814yes—
204C42#12312yes—
205C42C23×C411yescharacteristic
206C42C22×Q811yescharacteristic
207C42C2×#4112yes—
208C23—22no—
209C23—22no—
210C23#38512yes—
211C23#45312yes—
212C23#45512yes—
213C23C2×#12011yescharacteristic
214Q8—48no—
215Q8—84no—
216Q8—88no—
217C2×C4—24no—
218C2×C4—24no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—28no—
226C2×C4—28no—
227C2×C4—28no—
228C2×C4—28no—
229C2×C4—416no—
230C2×C4—416no—
231C2×C4—44no—
232C2×C4—44no—
233C2×C4—44no—
234C2×C4—44no—
235C2×C4—44no—
236C2×C4—44no—
237C2×C4—44no—
238C2×C4—88no—
239C2×C4—82no—
240C2×C4—82no—
241C2×C4—82no—
242C2×C4#38514yes—
243C2×C4#38512yes—
244C2×C4C22×C4⋊C411yescharacteristic
245C2×C4C2×C4×D412yes—
246C2×C4C2×C22⋊Q812yes—
247C2×C4C2×#12311yescharacteristic
248C2×C4C2×#13714yes—
249C2×C4#57514yes—
250C8—44no—
251C8—48no—
252C22—24no—
253C22—22no—
254C22—22no—
255C22—24no—
256C22#185614yes—
257C22C2×#38511yescharacteristic
258C22#300412yes—
259C22#300512yes—
260C22C2×#45311yescharacteristic
261C22C2×#45511yescharacteristic
262C4—21no—
263C4—21no—
264C4—22no—
265C4—24no—
266C4—48no—
267C4—48no—
268C4—84no—
269C4—162no—
270C4C4×#13714yes—
271C4#307814yes—
272C2—24no—
273C2#1852512yes—
274C2#1855111yescharacteristic
275C2#1855211yescharacteristic
276C2C2×#185611yescharacteristic
277C2#2087412yes—
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #8927C2semidirect
26. #1558C4semidirect
75. #333C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23692
#213652
#212022
C2×#15582
#89272
#79042