Finite Group Databasebeta

← #194598

Subgroups — #194598

Order 512.

1096 conjugacy classes of subgroups; only the 258 normal subgroups are shown.

normal subgroups only

This group has 1096 conjugacy classes of subgroups — too many to draw. Only its 258 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×#3162C211yesmaximal, characteristic
2#34374C211yesmaximal, characteristic
3#21870C212yesmaximal
4C2×#1537C211yesmaximal, characteristic
5#9000C218yesmaximal, complemented
6#8034C212yesmaximal
7C2×#533C2211yescharacteristic
8C2×#533C2211yescharacteristic
9#3162—24no—
10#3162C418yescomplemented
11C2×#501—22no—
12C2×#501C2214yes—
13C2×#432C2212yes—
14C2×#427—22no—
15C22×#97C2211yescharacteristic
16#2382C2214yes—
17#2365—22no—
18#2355—22no—
19Q8×C4⋊C4—22no—
20#2323—24no—
21#2314—22no—
22#2285C2212yes—
23#1993C2214yes—
24#1960C2214yes—
25#1873—24no—
26#1803—22no—
27#1803—22no—
28#1537—24no—
29C2×#330C2214yes—
30#1295C2218yes—
31C23×C8—21no—
32#533—22no—
33#533—24no—
34#533—24no—
35#533C2×C414yes—
36#533C2×C414yes—
37#533D414yes—
38C2×C4⋊Q8D412yes—
39C2×C4⋊Q8C2312yes—
40C2×#125—21no—
41C2×#125C2312yes—
42C2×C22⋊Q8—22no—
43C2×C22⋊Q8—22no—
44C2×C4×Q8—24no—
45C2×C4×Q8—22no—
46C2×C4×Q8—22no—
47C2×#117—21no—
48C2×#117D412yes—
49C2×#117D414yes—
50C22×C42C2311yescharacteristic
51#501—216no—
52#501—28no—
53#501C2×C4116yes—
54#432—24no—
55#432—24no—
56#427—216no—
57C2×#107—24no—
58C2×#106—24no—
59C2×C4⋊C8—22no—
60C2×#103—24no—
61C2×#103—24no—
62C2×C22⋊C8—22no—
63C2×#97—21no—
64C2×#97—22no—
65C2×#97C2×C414yes—
66C2×#97C2314yes—
67C2×#97C2312yes—
68#391—24no—
69#389—24no—
70#389—28no—
71#388—28no—
72#387—28no—
73#387—28no—
74#384—22no—
75#384D412yes—
76#384D412yes—
77#382—22no—
78#382C2314yes—
79#380—22no—
80#380—22no—
81C4×C4⋊C4—24no—
82C4×C4⋊C4—24no—
83C4×C4⋊C4—22no—
84C4×C4⋊C4—22no—
85#340—24no—
86#340—24no—
87#330—28no—
88C22×Q8—22no—
89C22×Q8—22no—
90C23×C4C2×D411yescharacteristic
91C23×C4C2×D412yes—
92C22×C8—24no—
93C22×C8—24no—
94C22×C8—24no—
95C22×C8—22no—
96C22×C8—24no—
97C4⋊Q8—24no—
98C4⋊Q8—22no—
99C4⋊Q8—22no—
100C4⋊Q8—22no—
101C4⋊Q8C22⋊C418yes—
102C4⋊Q8C22×C414yes—
103#125—24no—
104#125—24no—
105#125—42no—
106#125C22×C414yes—
107#125C2×D414yes—
108C22⋊Q8—28no—
109C22⋊Q8—28no—
110C22⋊Q8—28no—
111C22⋊Q8—44no—
112C4×Q8—28no—
113C4×Q8—28no—
114C4×Q8—28no—
115C4×Q8—44no—
116C4×Q8—88no—
117#117—24no—
118#117—24no—
119#117—42no—
120#117—84no—
121#117C22⋊C418yes—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—28no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4C2×D414yes—
133C2×C4⋊C4C2×D414yes—
134C2×C4⋊C4C2×D414yes—
135C2×C22⋊C4—22no—
136C2×C22⋊C4—24no—
137C2×C22⋊C4—41no—
138C2×C42—22no—
139C2×C42—24no—
140C2×C42—28no—
141C2×C42—41no—
142C2×C42C22×C412yes—
143C2×C42C2×D411yescharacteristic
144C2×C42C2×D412yes—
145C2×C42C2×D412yes—
146C2×C42#4114yes—
147C2×C42C2411yescharacteristic
148#107—216no—
149#106—216no—
150C4⋊C8—44no—
151#103—84no—
152#103—84no—
153C22⋊C8—44no—
154#97—24no—
155#97—22no—
156#97—24no—
157#97—24no—
158#97C22×C414yes—
159#95—28no—
160#95—28no—
161#95—24no—
162#95—24no—
163C24#12011yescharacteristic
164C2×Q8—28no—
165C2×Q8—24no—
166C2×Q8—44no—
167C2×Q8—44no—
168C2×Q8—42no—
169C2×Q8—42no—
170C2×Q8—42no—
171C2×Q8—42no—
172C2×Q8—42no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—24no—
176C22×C4—24no—
177C22×C4—24no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4C2×C22⋊C412yes—
182C22×C4C4×D414yes—
183C22×C4#12012yes—
184C22×C4#12011yescharacteristic
185C22×C4#12114yes—
186C22×C4#12312yes—
187C22×C4C22×D411yescharacteristic
188C22×C4C22×D412yes—
189C2×C8—24no—
190C2×C8—24no—
191C2×C8—216no—
192C2×C8—24no—
193C2×C8—44no—
194C2×C8—44no—
195C2×C8—44no—
196C4⋊C4—28no—
197C4⋊C4—28no—
198C4⋊C4—24no—
199C4⋊C4—28no—
200C4⋊C4—24no—
201C4⋊C4—24no—
202C4⋊C4—44no—
203C4⋊C4—44no—
204C4⋊C4—44no—
205C4⋊C4—44no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C4⋊C4—48no—
209C4⋊C4—44no—
210C4⋊C4—44no—
211C4⋊C4—88no—
212C4⋊C4C2×C22⋊C418yes—
213C22⋊C4—28no—
214C22⋊C4—44no—
215C22⋊C4—84no—
216C42—28no—
217C42—48no—
218C42—44no—
219C42—88no—
220C42C2×C22⋊C412yes—
221C42C4×D414yes—
222C42#12114yes—
223C42#12312yes—
224C42C23×C411yescharacteristic
225C42C22×D411yescharacteristic
226C42C2×#4112yes—
227C23—21no—
228C23—21no—
229C23—22no—
230C23—22no—
231C23#38612yes—
232C23C2×#12011yescharacteristic
233Q8—416no—
234Q8—44no—
235Q8—44no—
236Q8—82no—
237Q8—82no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—24no—
245C2×C4—24no—
246C2×C4—28no—
247C2×C4—44no—
248C2×C4—44no—
249C2×C4—44no—
250C2×C4—44no—
251C2×C4—44no—
252C2×C4—44no—
253C2×C4—44no—
254C2×C4—44no—
255C2×C4—44no—
256C2×C4—44no—
257C2×C4—44no—
258C2×C4—82no—
259C2×C4—82no—
260C2×C4#38614yes—
261C2×C4#38612yes—
262C2×C4#45612yes—
263C2×C4#45612yes—
264C2×C4C22×C22⋊C411yescharacteristic
265C2×C4C2×C4×D412yes—
266C2×C4C2×#12112yes—
267C2×C4C2×#12311yescharacteristic
268C8—28no—
269C8—84no—
270C22—24no—
271C22—22no—
272C22—41no—
273C22—42no—
274C22—41no—
275C22C2×#38611yescharacteristic
276C22C2×#45611yescharacteristic
277C22C2×#45611yescharacteristic
278C22#301912yes—
279C22#302012yes—
280C4—22no—
281C4—22no—
282C4—24no—
283C4—21no—
284C4—21no—
285C4—48no—
286C4—84no—
287C4—84no—
288C4—84no—
289C4#188614yes—
290C2—42no—
291C2#1906511yescharacteristic
292C2#1906612yes—
293C2#1906711yescharacteristic
294C2#2094312yes—
295C2C2×#188611yescharacteristic
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. #9000C2semidirect
10. #3162C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31622
#343742
#218702
C2×#15372
#90002
#80342