Finite Group Databasebeta

← #193069

Subgroups — #193069

Order 512.

1060 conjugacy classes of subgroups; only the 274 normal subgroups are shown.

normal subgroups only

This group has 1060 conjugacy classes of subgroups — too many to draw. Only its 274 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#34374C211yesmaximal, characteristic
2#21322C211yesmaximal, characteristic
3#20800C212yesmaximal
4C2×#1526C211yesmaximal, characteristic
5#8913C218yesmaximal, complemented
6#8028C212yesmaximal
7C2×#533C2211yescharacteristic
8C2×#428—24no—
9C2×C4×M16C2211yescharacteristic
10C2×C4×M16C2212yes—
11#2382C2214yes—
12#2365—22no—
13#2355—22no—
14Q8×C4⋊C4—22no—
15#2323—24no—
16#2314—22no—
17#2285C2212yes—
18C2×#384C2211yescharacteristic
19#1992C2214yes—
20#1990—22no—
21#1847C2218yes—
22#1845—22no—
23#1844—22no—
24#1736—22no—
25#1526C418yescomplemented
26C2×#330C2214yes—
27#1295C2218yes—
28C22×M16—22no—
29C22×M16—21no—
30#533—24no—
31#533C2×C414yes—
32#533Q814yes—
33C2×C4⋊Q8D412yes—
34C2×#125C2312yes—
35C2×C22⋊Q8—22no—
36C2×C4×Q8—24no—
37C2×C4×Q8—22no—
38C2×#117D412yes—
39C2×#117D414yes—
40C22×C4⋊C4—22no—
41C22×C4⋊C4—21no—
42C22×C42C2311yescharacteristic
43#428—416no—
44C2×#107—28no—
45C2×#106—28no—
46C4×M16—24no—
47C4×M16—24no—
48C4×M16C2×C418yes—
49C2×#97C2312yes—
50C2×#97C2314yes—
51C2×C4×C8—22no—
52C2×C4×C8—21no—
53#391—24no—
54#389—24no—
55#389—28no—
56#388—28no—
57#387—28no—
58#387—28no—
59#384—22no—
60#384—22no—
61#384—21no—
62#384D412yes—
63#384D412yes—
64#384C2312yes—
65#382—22no—
66#382C2314yes—
67#380—22no—
68#380—22no—
69C4×C4⋊C4—24no—
70C4×C4⋊C4—24no—
71C4×C4⋊C4—22no—
72C4×C4⋊C4—22no—
73#340—28no—
74#330C2×C4116yescomplemented
75C22×Q8—22no—
76C23×C4—41no—
77C23×C4C2×D411yescharacteristic
78C23×C4C2×D412yes—
79C2×M16—28no—
80C2×M16—28no—
81C2×M16—28no—
82C2×M16—24no—
83C2×M16—42no—
84C2×M16—42no—
85C22×C8—24no—
86C22×C8—22no—
87C4⋊Q8—24no—
88C4⋊Q8C4⋊C418yes—
89#125—24no—
90#125C22×C414yes—
91#125C2×Q814yes—
92C22⋊Q8—28no—
93C22⋊Q8—28no—
94C4×Q8—28no—
95C4×Q8—28no—
96C4×Q8—88no—
97#117—24no—
98#117—48no—
99#117C4⋊C418yes—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—24no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—24no—
108C2×C4⋊C4—28no—
109C2×C4⋊C4—44no—
110C2×C4⋊C4—48no—
111C2×C4⋊C4—42no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4C2×D414yes—
114C2×C4⋊C4C2×D414yes—
115C2×C4⋊C4C2×D414yes—
116C2×C22⋊C4—22no—
117C2×C22⋊C4—24no—
118C2×C42—22no—
119C2×C42—24no—
120C2×C42—28no—
121C2×C42C22×C412yes—
122C2×C42C2×D411yescharacteristic
123C2×C42C2×D412yes—
124C2×C42C2×Q812yes—
125C2×C42#4114yes—
126C2×C42C2411yescharacteristic
127#107—416no—
128#106—416no—
129#97—24no—
130#97—24no—
131#97C22×C418yes—
132C4×C8—28no—
133C4×C8—42no—
134#95—28no—
135#95—28no—
136#95—24no—
137#95—24no—
138C24#12011yescharacteristic
139C2×Q8—28no—
140C2×Q8—44no—
141C2×Q8—44no—
142C22×C4—24no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—24no—
146C22×C4—24no—
147C22×C4—24no—
148C22×C4—24no—
149C22×C4—42no—
150C22×C4—84no—
151C22×C4—82no—
152C22×C4C2×C4⋊C412yes—
153C22×C4C4×D414yes—
154C22×C4#12012yes—
155C22×C4#12011yescharacteristic
156C22×C4C22⋊Q814yes—
157C22×C4#12312yes—
158C22×C4C22×D411yescharacteristic
159C22×C4C22×D412yes—
160M16—232no—
161M16—48no—
162C2×C8—28no—
163C2×C8—28no—
164C2×C8—28no—
165C2×C8—24no—
166C2×C8—42no—
167C2×C8—42no—
168C4⋊C4—28no—
169C4⋊C4—28no—
170C4⋊C4—24no—
171C4⋊C4—28no—
172C4⋊C4—44no—
173C4⋊C4—44no—
174C4⋊C4—48no—
175C4⋊C4—44no—
176C4⋊C4—44no—
177C4⋊C4—44no—
178C4⋊C4—44no—
179C4⋊C4—44no—
180C4⋊C4—88no—
181C4⋊C4—84no—
182C4⋊C4—82no—
183C4⋊C4C2×C4⋊C418yes—
184C22⋊C4—28no—
185C22⋊C4—84no—
186C42—28no—
187C42—48no—
188C42—88no—
189C42C2×C4⋊C412yes—
190C42C4×D414yes—
191C42C22⋊Q814yes—
192C42#12312yes—
193C42C23×C411yescharacteristic
194C42C22×Q811yescharacteristic
195C42C2×#4112yes—
196C23—22no—
197C23—21no—
198C23—21no—
199C23—22no—
200C23#38512yes—
201C23C2×#12011yescharacteristic
202Q8—416no—
203C2×C4—22no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—24no—
208C2×C4—24no—
209C2×C4—28no—
210C2×C4—28no—
211C2×C4—28no—
212C2×C4—44no—
213C2×C4—44no—
214C2×C4—44no—
215C2×C4—44no—
216C2×C4—44no—
217C2×C4—44no—
218C2×C4—44no—
219C2×C4—44no—
220C2×C4—84no—
221C2×C4—82no—
222C2×C4—164no—
223C2×C4#38512yes—
224C2×C4#38514yes—
225C2×C4#45314yes—
226C2×C4#45412yes—
227C2×C4#45612yes—
228C2×C4#45614yes—
229C2×C4C22×C4⋊C411yescharacteristic
230C2×C4C2×C4×D412yes—
231C2×C4C2×C22⋊Q812yes—
232C2×C4C2×#12311yescharacteristic
233C8—216no—
234C8—44no—
235C22—22no—
236C22—24no—
237C22—41no—
238C22—42no—
239C22—41no—
240C22C2×#38511yescharacteristic
241C22C2×#45312yes—
242C22C2×#45411yescharacteristic
243C22C2×#45611yescharacteristic
244C22C2×#45612yes—
245C4—21no—
246C4—21no—
247C4—22no—
248C4—22no—
249C4—48no—
250C4—84no—
251C4—84no—
252C4—162no—
253C4#185714yes—
254C4#185918yes—
255C2—42no—
256C2#1907511yescharacteristic
257C2#1907812yes—
258C2#1908411yescharacteristic
259C2C2×#185711yescharacteristic
260C2C2×#185912yes—
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. #8913C2semidirect
25. #1526C4semidirect
74. #330C2×C4semidirect

Maximal subgroups

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

subgroupindex
#343742
#213222
#208002
C2×#15262
#89132
#80282