Finite Group Databasebeta

← #146498

Subgroups — #146498

Order 512.

817 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 817 conjugacy classes of subgroups — too many to draw. Only its 184 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#60126C211yesmaximal, characteristic
2#20710C212yesmaximal
3#12577C212yesmaximal
4C2×#1391C212yesmaximal
5#3428C2211yescharacteristic
6#2925—24no—
7#2924—24no—
8#2923—22no—
9#2922—22no—
10#2905C411yescharacteristic, complemented
11C2×#424C411yescharacteristic, complemented
12C2×#410C2214yes—
13#2879C2211yescharacteristic
14C2×#408C2211yescharacteristic
15#2873C411yescharacteristic, complemented
16C2×#405C411yescharacteristic, complemented
17#1828—24no—
18#1828C418yescomplemented
19#1740—24no—
20#1391—22no—
21#1391—22no—
22#1391—24no—
23C22×#41C2311yescharacteristic
24C22×M16C2×C411yescharacteristic
25C22×M16C2×C411yescharacteristic
26C22×M16C2×C411yescharacteristic
27C23×C8C2×C411yescharacteristic
28#520—22no—
29#520—22no—
30#519Q812yes—
31#518C2×C414yes—
32#518D412yes—
33C4×#41—22no—
34C4×#41—22no—
35C2×#117—24no—
36C2×#117C2×C411yescharacteristic
37C2×#117C2×C411yescharacteristic
38C2×#117D414yes—
39#433—28no—
40#433—28no—
41#432—28no—
42#431—28no—
43#424—22no—
44#424—21no—
45#424—21no—
46#424—22no—
47#424—22no—
48#423—22no—
49C2×C4⋊C8—22no—
50#410—24no—
51#410—24no—
52#410—24no—
53#410—24no—
54C2×#99—24no—
55C2×#99C2×C418yes—
56#408—22no—
57#408—22no—
58#408D412yes—
59#408D412yes—
60#408D414yes—
61#407—22no—
62C2×C22⋊C8—22no—
63#405—22no—
64#405—22no—
65#405—22no—
66#405—22no—
67C4×M16—22no—
68C2×#97—21no—
69C2×C4×C8—21no—
70#342—216no—
71C2×#41—42no—
72C2×#41—42no—
73C2×#41C2×D412yes—
74C2×#41C2×D412yes—
75C2×#41#4112yes—
76C2×#41#4112yes—
77C22×Q8C2×Q811yescharacteristic
78C22×D4C22×C412yes—
79C22×D4C2×D411yescharacteristic
80C23×C4C22×C411yescharacteristic
81C23×C4C2×D412yes—
82C2×M16—21no—
83C2×M16—21no—
84C2×M16—21no—
85C2×M16—21no—
86C2×M16—42no—
87C2×M16—42no—
88C2×M16—42no—
89C2×M16C22⋊C412yes—
90C2×M16C22⋊C412yes—
91C22×C8—21no—
92C22×C8—21no—
93C22×C8—21no—
94C22×C8—21no—
95C22×C8—42no—
96C22×C8C22⋊C411yescharacteristic
97C22×C8C22⋊C411yescharacteristic
98C22×C8C22⋊C412yes—
99C4×Q8—24no—
100C4×Q8—24no—
101C4×D4—28no—
102C4×D4—28no—
103C4×D4—24no—
104C4×D4—24no—
105#117—21no—
106#117—21no—
107#117—21no—
108#117—21no—
109#117—24no—
110#117—24no—
111#117—44no—
112#117—44no—
113#117—44no—
114#117—44no—
115#117C22⋊C412yes—
116#117C22⋊C412yes—
117#117C4⋊C412yes—
118#117C4⋊C412yes—
119#117C4⋊C418yes—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—44no—
124C2×C22⋊C4—28no—
125C2×C22⋊C4C4212yes—
126C2×C22⋊C4C4212yes—
127C2×C22⋊C4C22⋊C418yes—
128C2×C42—21no—
129C2×C42—21no—
130C2×C42—24no—
131C2×C42—44no—
132C4⋊C8—24no—
133C4⋊C8—24no—
134C4⋊C8—28no—
135#99—28no—
136#99—28no—
137#99—44no—
138#99—44no—
139C22⋊C8—24no—
140C22⋊C8—24no—
141C22⋊C8—28no—
142#97—24no—
143#97—24no—
144C4×C8—24no—
145C4×C8—24no—
146C24C2×C4211yescharacteristic
147C24C2×C22⋊C412yes—
148#41—42no—
149#41—42no—
150#41—42no—
151#41—42no—
152#41—42no—
153#41—44no—
154#41—42no—
155C2×Q8—42no—
156C2×Q8C22⋊Q812yes—
157C2×Q8C22⋊Q812yes—
158C2×D4—44no—
159C2×D4—42no—
160C2×D4C4×D414yes—
161C2×D4C4×D414yes—
162C2×D4#12112yes—
163C2×D4#12112yes—
164C22×C4—24no—
165C22×C4—24no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—28no—
171C22×C4—48no—
172C22×C4C2×C22⋊C411yescharacteristic
173C22×C4C2×C22⋊C411yescharacteristic
174C22×C4C2×C22⋊C411yescharacteristic
175C22×C4C2×C4⋊C411yescharacteristic
176C22×C4C2×C4⋊C412yes—
177C22×C4#11711yescharacteristic
178C22×C4#11711yescharacteristic
179C22×C4#12014yes—
180C22×C4#12314yes—
181M16—42no—
182M16—42no—
183M16—42no—
184M16—42no—
185M16—42no—
186M16—42no—
187C2×C8—21no—
188C2×C8—22no—
189C2×C8—22no—
190C2×C8—22no—
191C2×C8—21no—
192C2×C8—42no—
193C2×C8—42no—
194C2×C8—42no—
195C2×C8—42no—
196C2×C8—41no—
197C2×C8—41no—
198C2×C8—42no—
199C4⋊C4—22no—
200C4⋊C4—22no—
201C4⋊C4—22no—
202C4⋊C4—22no—
203C4⋊C4—28no—
204C4⋊C4—84no—
205C4⋊C4—84no—
206C22⋊C4—22no—
207C22⋊C4—22no—
208C22⋊C4—28no—
209C22⋊C4—48no—
210C22⋊C4—48no—
211C22⋊C4#9514yes—
212C22⋊C4#9514yes—
213C42—22no—
214C42—22no—
215C42—22no—
216C42—22no—
217C42—28no—
218C42—84no—
219C42—84no—
220C23—24no—
221C23—22no—
222C23—81no—
223C23—82no—
224C23C2×#9511yescharacteristic
225C23C4×C22⋊C411yescharacteristic
226C23C4×C22⋊C411yescharacteristic
227C23#38614yes—
228Q8—42no—
229Q8—42no—
230D4—44no—
231D4—44no—
232D4—42no—
233D4—42no—
234C2×C4—22no—
235C2×C4—24no—
236C2×C4—24no—
237C2×C4—24no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—24no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—24no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—24no—
247C2×C4—24no—
248C2×C4—82no—
249C2×C4—82no—
250C2×C4—88no—
251C2×C4—88no—
252C2×C4—168no—
253C2×C4#37911yescharacteristic
254C2×C4#38011yescharacteristic
255C2×C4#38011yescharacteristic
256C2×C4#38111yescharacteristic
257C2×C4#38514yes—
258C2×C4#41012yes—
259C2×C4#41012yes—
260C8—42no—
261C8—42no—
262C8—41no—
263C8—41no—
264C8—42no—
265C22—24no—
266C22—21no—
267C22—21no—
268C22—41no—
269C22—41no—
270C22—41no—
271C22—44no—
272C22—84no—
273C22—82no—
274C22C22⋊#9511yescharacteristic
275C22#174012yes—
276C4—21no—
277C4—21no—
278C4—24no—
279C4—44no—
280C4—44no—
281C4—168no—
282C4#178211yescharacteristic
283C4#178211yescharacteristic
284C4#182012yes—
285C2—21no—
286C2—22no—
287C2—44no—
288C2—41no—
289C2—41no—
290C2#1072611yescharacteristic
subgroupG/H
centerC4#1782
commutatorC24C2×C42
Frattini, prime radical, cosocleC22×#41C23
socleC2#10726
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C22×C4 ◁ #3428 ◁ G
LECSG ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
10. #2905C4semidirect
11. C2×#424C4semidirect
15. #2873C4semidirect
16. C2×#405C4semidirect
18. #1828C4semidirect

Maximal subgroups

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

subgroupindex
#601262
#207102
#125772
C2×#13912