Finite Group Databasebeta

← #109667

Subgroups — #109667

Order 512.

1100 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 1100 conjugacy classes of subgroups — too many to draw. Only its 178 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#13965C211yesmaximal, characteristic
2#13749C211yesmaximal, characteristic
3C2×#1297C212yesmaximal
4#8019C212yesmaximal
5#7909C211yesmaximal, characteristic
6C2×#407C2212yes—
7C2×#385—22no—
8C2×#380C2211yescharacteristic
9C2×C4×C22⋊C4—22no—
10C2×C4×C22⋊C4C2211yescharacteristic
11C2×C4×C22⋊C4C2212yes—
12C22×#95C2211yescharacteristic
13#1444C414yescomplemented
14C4⋊#95—22no—
15C4×#95—24no—
16#1297C418yescomplemented
17#1294—24no—
18#1280—22no—
19C24×C4D412yes—
20C24×C4C2311yescharacteristic
21C22×M16—22no—
22C22×C4⋊C4—24no—
23C22×C4⋊C4C2×C412yes—
24C22×C22⋊C4—22no—
25C22×C22⋊C4—21no—
26C22×C22⋊C4—21no—
27C22×C42—24no—
28C22×C42—22no—
29C22×C42—21no—
30#407—22no—
31#407—22no—
32#407—22no—
33#407C2×C414yes—
34C2×C22⋊C8—22no—
35C2×C22⋊C8—22no—
36#385—48no—
37#380—22no—
38#380—22no—
39C4×C22⋊C4—216no—
40C4×C22⋊C4—22no—
41C4×C22⋊C4—22no—
42C4×C22⋊C4—22no—
43C4×C22⋊C4C2×C414yes—
44C4×C22⋊C4C2×C414yes—
45C4×C22⋊C4Q814yes—
46C2×#95—24no—
47C2×#95—22no—
48C2×#95—22no—
49C2×#95—24no—
50C2×#95—21no—
51C2×#95C2×C412yes—
52C2×#95C2×C412yes—
53C2×#95D412yes—
54C2×#95D414yes—
55C25C2×D411yescharacteristic
56C23×C4—22no—
57C23×C4—24no—
58C23×C4—22no—
59C23×C4—22no—
60C23×C4—28no—
61C23×C4—22no—
62C23×C4—44no—
63C23×C4—82no—
64C23×C4—82no—
65C23×C4C22⋊C414yes—
66C23×C4C22×C411yescharacteristic
67C23×C4C22×C412yes—
68C23×C4C2×D412yes—
69C23×C4C2×D411yescharacteristic
70C23×C4C2×Q812yes—
71C23×C4#4112yes—
72C23×C4#4112yes—
73C23×C4#4112yes—
74C2×M16—24no—
75C2×M16—42no—
76C2×M16—84no—
77C22×C8—42no—
78C2×C4⋊C4—24no—
79C2×C4⋊C4—48no—
80C2×C4⋊C4—48no—
81C2×C4⋊C4—84no—
82C2×C4⋊C4—82no—
83C2×C22⋊C4—24no—
84C2×C22⋊C4—216no—
85C2×C22⋊C4—24no—
86C2×C22⋊C4—24no—
87C2×C22⋊C4—24no—
88C2×C22⋊C4—24no—
89C2×C22⋊C4—44no—
90C2×C22⋊C4—41no—
91C2×C22⋊C4—41no—
92C2×C42—216no—
93C2×C42—24no—
94C2×C42—24no—
95C2×C42—24no—
96C2×C42—84no—
97C2×C42—82no—
98C2×C42—84no—
99C2×C42—84no—
100C22⋊C8—24no—
101C22⋊C8—24no—
102C22⋊C8—42no—
103C22⋊C8—42no—
104#95—216no—
105#95—22no—
106#95—24no—
107#95—24no—
108#95—24no—
109#95—48no—
110#95—42no—
111#95C4214yescomplemented
112#95C4⋊C418yescomplemented
113C24—24no—
114C24—22no—
115C24—22no—
116C24—42no—
117C24C2×C22⋊C411yescharacteristic
118C24C4×D412yes—
119C24#12011yescharacteristic
120C24C22⋊Q812yes—
121C24#12311yescharacteristic
122C22×C4—24no—
123C22×C4—216no—
124C22×C4—24no—
125C22×C4—24no—
126C22×C4—24no—
127C22×C4—24no—
128C22×C4—28no—
129C22×C4—24no—
130C22×C4—28no—
131C22×C4—24no—
132C22×C4—44no—
133C22×C4—42no—
134C22×C4—42no—
135C22×C4—44no—
136C22×C4—48no—
137C22×C4—44no—
138C22×C4—44no—
139C22×C4—44no—
140C22×C4—44no—
141C22×C4—44no—
142C22×C4—44no—
143C22×C4—44no—
144C22×C4—44no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4—44no—
148C22×C4—42no—
149C22×C4—44no—
150C22×C4—82no—
151C22×C4—82no—
152C22×C4—82no—
153C22×C4—82no—
154C22×C4—816no—
155C22×C4—82no—
156C22×C4—82no—
157C22×C4#10418yes—
158C22×C4C2×C4211yescharacteristic
159C22×C4C2×C4⋊C412yes—
160C22×C4#11712yes—
161C22×C4#11711yescharacteristic
162C22×C4C4×D412yes—
163C22×C4C4×D412yes—
164C22×C4C4×D412yes—
165C22×C4C4×Q812yes—
166C22×C4C4×Q812yes—
167C22×C4#12112yes—
168C22×C4C22⋊Q812yes—
169C22×C4#12312yes—
170C22×C4#12411yescharacteristic
171C22×C4#12512yes—
172C22×C4#12612yes—
173C22×C4C4⋊Q811yescharacteristic
174M16—84no—
175M16—84no—
176C2×C8—44no—
177C2×C8—84no—
178C4⋊C4—84no—
179C4⋊C4—816no—
180C22⋊C4—216no—
181C22⋊C4—216no—
182C22⋊C4—216no—
183C22⋊C4—82no—
184C22⋊C4—82no—
185C42—84no—
186C42—84no—
187C42—84no—
188C42—816no—
189C23—21no—
190C23—21no—
191C23—22no—
192C23—22no—
193C23—22no—
194C23—24no—
195C23—24no—
196C23—21no—
197C23—21no—
198C23—22no—
199C23—22no—
200C23—24no—
201C23—24no—
202C23—42no—
203C23—42no—
204C23—48no—
205C23—48no—
206C23C4×C22⋊C411yescharacteristic
207C23#38512yes—
208C23#38611yescharacteristic
209C23#38712yes—
210C23#39111yescharacteristic
211C23C2×#10414yes—
212C23#45312yes—
213C23#45612yes—
214C2×C4—24no—
215C2×C4—24no—
216C2×C4—216no—
217C2×C4—28no—
218C2×C4—28no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—41no—
225C2×C4—42no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—41no—
231C2×C4—42no—
232C2×C4—41no—
233C2×C4—44no—
234C2×C4—48no—
235C2×C4—416no—
236C2×C4—48no—
237C2×C4—48no—
238C2×C4—416no—
239C2×C4—48no—
240C2×C4—48no—
241C2×C4—48no—
242C2×C4—44no—
243C2×C4—44no—
244C2×C4—44no—
245C2×C4—48no—
246C2×C4—84no—
247C2×C4—88no—
248C2×C4—84no—
249C2×C4—84no—
250C2×C4—82no—
251C2×C4—82no—
252C2×C4—82no—
253C2×C4C4×C4⋊C412yes—
254C2×C4#38712yes—
255C2×C4#38812yes—
256C2×C4#38912yes—
257C8—84no—
258C22—22no—
259C22—21no—
260C22—21no—
261C22—21no—
262C22—22no—
263C22—21no—
264C22—24no—
265C22—24no—
266C22—24no—
267C22—24no—
268C22—22no—
269C22—22no—
270C22—22no—
271C22—48no—
272C22—48no—
273C22—42no—
274C22—42no—
275C22#143611yescharacteristic
276C22C4×#10412yes—
277C22#185914yes—
278C22#188611yescharacteristic
279C22#188711yescharacteristic
280C22#189612yes—
281C22#191312yes—
282C22#199412yes—
283C4—22no—
284C4—22no—
285C4—22no—
286C4—22no—
287C4—416no—
288C4—48no—
289C4—48no—
290C4—162no—
291C2—24no—
292C2—22no—
293C2—22no—
294C2—42no—
295C2#940812yes—
296C2#940912yes—
297C2#1087811yescharacteristic
298C2#1089111yescharacteristic
299C2#1091511yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC22×C4C2×C42
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
13. #1444C4semidirect
16. #1297C4semidirect
111. #95C42semidirect
112. #95C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#139652
#137492
C2×#12972
#80192
#79092