Finite Group Databasebeta

← #193072

Subgroups — #193072

Order 512.

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

normal subgroups only

This group has 972 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#20764C212yesmaximal
4C2×#1526C211yesmaximal, characteristic
5#8915C218yesmaximal, complemented
6#8028C212yesmaximal
7C2×#533C2211yescharacteristic
8C2×C4×M16C2211yescharacteristic
9C2×C4×M16C2212yes—
10#2382C2214yes—
11#2365—22no—
12#2355—22no—
13Q8×C4⋊C4—22no—
14#2323—24no—
15#2314—22no—
16#2285C2212yes—
17C2×#384C2211yescharacteristic
18#1992C2214yes—
19#1990—22no—
20#1836C2218yes—
21#1834—22no—
22#1834—22no—
23#1736—24no—
24#1736—22no—
25#1526C418yescomplemented
26C2×#330C2214yes—
27#1295C2218yes—
28C22×M16—21no—
29C22×M16—22no—
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—21no—
41C22×C4⋊C4—22no—
42C22×C42C2311yescharacteristic
43C4×M16—24no—
44C4×M16—24no—
45C4×M16C2×C418yes—
46C2×#97C2312yes—
47C2×#97C2314yes—
48C2×C4×C8—22no—
49C2×C4×C8—21no—
50#391—24no—
51#389—24no—
52#389—28no—
53#388—28no—
54#387—28no—
55#387—28no—
56#384—22no—
57#384—22no—
58#384—21no—
59#384D412yes—
60#384D412yes—
61#384C2312yes—
62#382—22no—
63#382C2314yes—
64#380—22no—
65#380—22no—
66C4×C4⋊C4—24no—
67C4×C4⋊C4—24no—
68C4×C4⋊C4—22no—
69C4×C4⋊C4—22no—
70#340—28no—
71#340—28no—
72#340—28no—
73#330C2×C4116yescomplemented
74C22×Q8—22no—
75C23×C4—41no—
76C23×C4C2×D411yescharacteristic
77C23×C4C2×D412yes—
78C2×M16—24no—
79C2×M16—28no—
80C2×M16—42no—
81C2×M16—44no—
82C2×M16—44no—
83C2×M16—42no—
84C22×C8—22no—
85C22×C8—24no—
86C4⋊Q8—24no—
87C4⋊Q8C4⋊C418yes—
88#125—24no—
89#125C22×C414yes—
90#125C2×Q814yes—
91C22⋊Q8—28no—
92C22⋊Q8—28no—
93C4×Q8—28no—
94C4×Q8—28no—
95C4×Q8—88no—
96#117—24no—
97#117—48no—
98#117C4⋊C418yes—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—24no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—24no—
107C2×C4⋊C4—28no—
108C2×C4⋊C4—44no—
109C2×C4⋊C4—48no—
110C2×C4⋊C4—42no—
111C2×C4⋊C4—41no—
112C2×C4⋊C4C2×D414yes—
113C2×C4⋊C4C2×D414yes—
114C2×C4⋊C4C2×D414yes—
115C2×C22⋊C4—22no—
116C2×C22⋊C4—24no—
117C2×C42—22no—
118C2×C42—24no—
119C2×C42—28no—
120C2×C42C22×C412yes—
121C2×C42C2×D411yescharacteristic
122C2×C42C2×D412yes—
123C2×C42C2×Q812yes—
124C2×C42#4114yes—
125C2×C42C2411yescharacteristic
126#97—24no—
127#97—24no—
128#97C22×C418yes—
129C4×C8—28no—
130C4×C8—42no—
131#95—28no—
132#95—28no—
133#95—24no—
134#95—24no—
135C24#12011yescharacteristic
136C2×Q8—28no—
137C2×Q8—44no—
138C2×Q8—44no—
139C22×C4—22no—
140C22×C4—24no—
141C22×C4—24no—
142C22×C4—22no—
143C22×C4—24no—
144C22×C4—24no—
145C22×C4—24no—
146C22×C4—42no—
147C22×C4—82no—
148C22×C4—84no—
149C22×C4C2×C4⋊C412yes—
150C22×C4C4×D414yes—
151C22×C4#12012yes—
152C22×C4#12011yescharacteristic
153C22×C4C22⋊Q814yes—
154C22×C4#12312yes—
155C22×C4C22×D411yescharacteristic
156C22×C4C22×D412yes—
157M16—48no—
158M16—416no—
159C2×C8—24no—
160C2×C8—28no—
161C2×C8—44no—
162C2×C8—42no—
163C2×C8—44no—
164C2×C8—42no—
165C4⋊C4—28no—
166C4⋊C4—28no—
167C4⋊C4—24no—
168C4⋊C4—28no—
169C4⋊C4—44no—
170C4⋊C4—44no—
171C4⋊C4—44no—
172C4⋊C4—48no—
173C4⋊C4—44no—
174C4⋊C4—44no—
175C4⋊C4—44no—
176C4⋊C4—44no—
177C4⋊C4—88no—
178C4⋊C4—82no—
179C4⋊C4—84no—
180C4⋊C4C2×C4⋊C418yes—
181C22⋊C4—28no—
182C22⋊C4—84no—
183C42—28no—
184C42—48no—
185C42—88no—
186C42C2×C4⋊C412yes—
187C42C4×D414yes—
188C42C22⋊Q814yes—
189C42#12312yes—
190C42C23×C411yescharacteristic
191C42C22×Q811yescharacteristic
192C42C2×#4112yes—
193C23—22no—
194C23—21no—
195C23—22no—
196C23—21no—
197C23#38512yes—
198C23C2×#12011yescharacteristic
199Q8—416no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—21no—
204C2×C4—24no—
205C2×C4—24no—
206C2×C4—28no—
207C2×C4—28no—
208C2×C4—28no—
209C2×C4—44no—
210C2×C4—44no—
211C2×C4—44no—
212C2×C4—44no—
213C2×C4—44no—
214C2×C4—44no—
215C2×C4—44no—
216C2×C4—44no—
217C2×C4—82no—
218C2×C4—84no—
219C2×C4—164no—
220C2×C4#38512yes—
221C2×C4#38514yes—
222C2×C4#45412yes—
223C2×C4#45414yes—
224C2×C4#45514yes—
225C2×C4#45612yes—
226C2×C4C22×C4⋊C411yescharacteristic
227C2×C4C2×C4×D412yes—
228C2×C4C2×C22⋊Q812yes—
229C2×C4C2×#12311yescharacteristic
230C8—44no—
231C8—48no—
232C22—22no—
233C22—24no—
234C22—41no—
235C22—42no—
236C22—41no—
237C22C2×#38511yescharacteristic
238C22C2×#45512yes—
239C22C2×#45411yescharacteristic
240C22C2×#45412yes—
241C22C2×#45611yescharacteristic
242C4—21no—
243C4—21no—
244C4—22no—
245C4—22no—
246C4—48no—
247C4—84no—
248C4—84no—
249C4—162no—
250C4#185714yes—
251C4#186018yes—
252C2—42no—
253C2#1908011yescharacteristic
254C2#1908112yes—
255C2#1908311yescharacteristic
256C2C2×#185711yescharacteristic
257C2C2×#186012yes—
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. #8915C2semidirect
25. #1526C4semidirect
73. #330C2×C4semidirect

Maximal subgroups

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

subgroupindex
#343742
#213222
#207642
C2×#15262
#89152
#80282