Finite Group Databasebeta

← #193074

Subgroups — #193074

Order 512.

918 conjugacy classes of subgroups.

The subgroup lattice has 287 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34372C211yesmaximal, characteristic
2#21325C211yesmaximal, characteristic
3#20823C212yesmaximal
4C22×#330C211yesmaximal, characteristic
5#8915C212yesmaximal, complemented
6#8913C212yesmaximal, complemented
7#8912C214yesmaximal, complemented
8#8029C212yesmaximal
9C22×#125C2211yescharacteristic
10C2×C4×M16C2211yescharacteristic
11C22×#97C2212yes—
12#2382C2212yes—
13#2382C2212yes—
14#2371—22no—
15#2345—22no—
16#2345—22no—
17#2336—24no—
18#2313—22no—
19#2285C2211yescharacteristic
20#2283C2212yes—
21#1992C2212yes—
22#1992C2212yes—
23#1991—22no—
24#1847C2214yes—
25#1836C2214yes—
26#1808—22no—
27#1807—22no—
28#1776—24no—
29#1736—22no—
30C2×#330C418yescomplemented
31C2×#330C2212yes—
32C2×#330C2211yescharacteristic
33C2×#330C2211yescharacteristic
34#1295C2214yes—
35#1295C2214yes—
36C22×M16—21no—
37C23×C8—22no—
38C2×#125—22no—
39C2×#125—24no—
40C2×#125C2×C414yes—
41C2×#125D412yes—
42C2×#125Q814yes—
43C2×#125C2311yescharacteristic
44C2×#125C2311yescharacteristic
45C2×#117—21no—
46C22×C4⋊C4—22no—
47C22×C4⋊C4D412yes—
48C22×C42C2311yescharacteristic
49C2×#107—24no—
50C2×#106—24no—
51C4×M16—24no—
52C2×#97—22no—
53C2×#97—24no—
54C2×#97C2×C418yes—
55C2×#97C2312yes—
56C2×#97C2312yes—
57C2×#97C2312yes—
58C2×C4×C8—21no—
59#389—24no—
60#389—24no—
61#387—24no—
62#387—24no—
63#387—24no—
64#387—24no—
65#387—24no—
66#387—24no—
67#387—24no—
68#387—24no—
69#384C2312yes—
70#383—22no—
71#382—21no—
72#382C2314yes—
73#381—22no—
74#380—22no—
75#380—22no—
76#380—22no—
77#380D412yes—
78#380D412yes—
79C4×C4⋊C4—24no—
80C4×C4⋊C4—24no—
81C4×C4⋊C4—22no—
82C4×C4⋊C4D414yes—
83C4×C22⋊C4—22no—
84C4×C22⋊C4—22no—
85C4×C22⋊C4—22no—
86#376—22no—
87#340—24no—
88#340—24no—
89#340—24no—
90#340—24no—
91#330C2×C418yes—
92#330C2×C414yes—
93#330C2×C414yes—
94C23×C4—22no—
95C23×C4C2×D411yescharacteristic
96C23×C4C2×D412yes—
97C2×M16—24no—
98C2×M16—42no—
99C2×M16—42no—
100C22×C8—22no—
101C22×C8—28no—
102C22×C8—22no—
103C22×C8—22no—
104C22×C8—48no—
105#125—28no—
106#125—28no—
107#125—24no—
108#125—22no—
109#125—22no—
110#125C4⋊C418yes—
111#125C22×C412yes—
112#125C22×C412yes—
113#125C2×Q812yes—
114#125C2×Q812yes—
115#117—44no—
116C2×C4⋊C4—28no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—28no—
121C2×C4⋊C4—24no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4C4⋊C418yes—
129C2×C4⋊C4C2×D412yes—
130C2×C4⋊C4C2×D412yes—
131C2×C4⋊C4C2×D414yes—
132C2×C4⋊C4C2×D414yes—
133C2×C22⋊C4—22no—
134C2×C22⋊C4—22no—
135C2×C22⋊C4—41no—
136C2×C42—24no—
137C2×C42—24no—
138C2×C42—24no—
139C2×C42—41no—
140C2×C42C22×C412yes—
141C2×C42C2×D411yescharacteristic
142C2×C42C2×D412yes—
143C2×C42C2×Q812yes—
144C2×C42#4114yes—
145C2×C42C2411yescharacteristic
146#107—48no—
147#106—48no—
148#97—28no—
149#97—22no—
150#97—22no—
151#97—24no—
152#97C22×C414yes—
153#97C22×C414yes—
154C4×C8—42no—
155#95—24no—
156#95—24no—
157#95—24no—
158#95—24no—
159#95—22no—
160#95—22no—
161#95—24no—
162#95—24no—
163#95—24no—
164#95—42no—
165C24#12011yescharacteristic
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—28no—
171C22×C4—24no—
172C22×C4—24no—
173C22×C4—44no—
174C22×C4—42no—
175C22×C4—84no—
176C22×C4C2×C4⋊C412yes—
177C22×C4C4×D414yes—
178C22×C4#12011yescharacteristic
179C22×C4#12012yes—
180C22×C4C22⋊Q814yes—
181C22×C4#12312yes—
182C22×C4C22×D411yescharacteristic
183C22×C4C22×D412yes—
184M16—48no—
185C2×C8—28no—
186C2×C8—24no—
187C2×C8—24no—
188C2×C8—24no—
189C2×C8—416no—
190C2×C8—44no—
191C2×C8—42no—
192C2×C8—42no—
193C4⋊C4—24no—
194C4⋊C4—24no—
195C4⋊C4—24no—
196C4⋊C4—24no—
197C4⋊C4—22no—
198C4⋊C4—22no—
199C4⋊C4—28no—
200C4⋊C4—28no—
201C4⋊C4—28no—
202C4⋊C4—44no—
203C4⋊C4—44no—
204C4⋊C4—44no—
205C4⋊C4—44no—
206C4⋊C4—44no—
207C4⋊C4—84no—
208C4⋊C4—84no—
209C4⋊C4C2×C4⋊C414yes—
210C4⋊C4C2×C4⋊C414yes—
211C22⋊C4—44no—
212C22⋊C4—44no—
213C22⋊C4—82no—
214C42—84no—
215C42—82no—
216C42—84no—
217C42—84no—
218C42C2×C4⋊C412yes—
219C42C4×D414yes—
220C42C22⋊Q814yes—
221C42#12312yes—
222C42C23×C411yescharacteristic
223C42C22×Q811yescharacteristic
224C42C2×#4112yes—
225C23—24no—
226C23#38512yes—
227C23#45412yes—
228C23#45612yes—
229C23C2×#12011yescharacteristic
230C2×C4—24no—
231C2×C4—24no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—24no—
237C2×C4—24no—
238C2×C4—24no—
239C2×C4—24no—
240C2×C4—44no—
241C2×C4—44no—
242C2×C4—44no—
243C2×C4—44no—
244C2×C4—44no—
245C2×C4—44no—
246C2×C4—44no—
247C2×C4—44no—
248C2×C4—44no—
249C2×C4—44no—
250C2×C4—88no—
251C2×C4—82no—
252C2×C4—84no—
253C2×C4#38514yes—
254C2×C4#38512yes—
255C2×C4C22×C4⋊C411yescharacteristic
256C2×C4C2×C4×D412yes—
257C2×C4C2×C22⋊Q812yes—
258C2×C4C2×#12311yescharacteristic
259C8—28no—
260C8—44no—
261C8—44no—
262C22—24no—
263C22—24no—
264C22—24no—
265C22#185714yes—
266C22C2×#38511yescharacteristic
267C22C2×#45411yescharacteristic
268C22C2×#45611yescharacteristic
269C22#301814yes—
270C4—21no—
271C4—21no—
272C4—22no—
273C4—22no—
274C4—22no—
275C4—24no—
276C4—44no—
277C4—44no—
278C4—84no—
279C4—84no—
280C4—162no—
281C2—24no—
282C2#1908212yes—
283C2#1908512yes—
284C2C2×#185711yescharacteristic
285C2#2086312yes—
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
6. #8913C2semidirect
7. #8912C2semidirect
30. C2×#330C4semidirect

Maximal subgroups

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

subgroupindex
#343722
#213252
#208232
C22×#3302
#89152
#89132
#89122
#80292