Finite Group Databasebeta

← #237960

Subgroups — #237960

Order 512.

789 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 789 conjugacy classes of subgroups — too many to draw. Only its 182 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#35344C211yesmaximal, characteristic, complemented
2#21418C211yesmaximal, characteristic
3#17571C212yesmaximal
4#10369C212yesmaximal, complemented
5#10368C212yesmaximal, complemented
6#10363C212yesmaximal, complemented
7#10362C212yesmaximal, complemented
8#7956C212yesmaximal, complemented
9#7866C211yesmaximal, characteristic, complemented
10#3444C2211yescharacteristic
11#2920C2211yescharacteristic
12C2×#431C2212yes—
13#2864C2212yes—
14#2862C2212yes—
15#2628—22no—
16#2607—24no—
17#2605—24no—
18#2331—22no—
19#2283C2212yes—
20C4×#117C2211yescharacteristic
21#1994C2211yescharacteristic
22#1994C2211yescharacteristic
23#1962C2211yescharacteristic
24#1962C2211yescharacteristic
25#1897—22no—
26#1759—22no—
27#1518C2214yes—
28#1517C2214yes—
29#1516—24no—
30#1500—24no—
31#1286C2214yes—
32#1286C2214yes—
33#1276C2212yes—
34#1276C2212yes—
35#533Q812yescomplemented
36#533C2311yescharacteristic
37#532Q812yescomplemented
38#532C2311yescharacteristic
39C2×C4⋊Q8—22no—
40#529—22no—
41#529—22no—
42#529—22no—
43#525—21no—
44#525—22no—
45C4×#41—21no—
46C4×#41—22no—
47C2×#117D412yes—
48C22×C42C2311yescharacteristic
49#432—22no—
50#431—22no—
51#431—22no—
52#431—22no—
53#431—22no—
54#431C2314yes—
55#423C2311yescharacteristic
56#423C2311yescharacteristic
57C2×C4⋊C8—22no—
58C2×#104—22no—
59C2×#104—22no—
60#407—21no—
61C2×C22⋊C8—22no—
62C4×M16—21no—
63C2×C4×C8—22no—
64#400—24no—
65#399—24no—
66#395—24no—
67#393—24no—
68#391—28no—
69#391—28no—
70#389—24no—
71#387—28no—
72#387—28no—
73#383—22no—
74#382—22no—
75#382C2314yes—
76#381D414yes—
77#380—22no—
78C4×C4⋊C4—22no—
79C4×C4⋊C4—22no—
80C4×C22⋊C4—22no—
81C4×C22⋊C4—22no—
82#376D412yes—
83#376D414yes—
84C43C2311yescharacteristic
85C43C2311yescharacteristic
86#339—22no—
87#339—22no—
88#328—28no—
89#327—28no—
90C2×#41—22no—
91C2×#41—21no—
92C22×Q8—24no—
93C23×C4C2×D411yescharacteristic
94C23×C4C2×D412yes—
95C2×M16—21no—
96C2×M16—21no—
97C22×C8—42no—
98C4⋊Q8—44no—
99C4⋊Q8—44no—
100C4⋊Q8C2×Q812yes—
101C4⋊Q8C2×Q811yescharacteristic
102#127C2×Q811yescharacteristic
103#126—42no—
104#125—22no—
105#125—21no—
106#124—22no—
107#124—21no—
108#123—41no—
109#123—42no—
110C22⋊Q8—22no—
111C22⋊Q8—22no—
112C22⋊Q8—22no—
113#121—22no—
114#121—22no—
115#121—22no—
116#120—41no—
117C4×Q8—22no—
118C4×Q8—22no—
119C4×Q8—22no—
120C4×D4—22no—
121C4×D4—22no—
122C4×D4—22no—
123C4×D4—42no—
124C4×D4—42no—
125C4×D4—42no—
126#117—24no—
127#117—22no—
128#117—21no—
129#117—42no—
130#117—41no—
131#117—42no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—28no—
136C2×C22⋊C4—22no—
137C2×C22⋊C4—24no—
138C2×C42—24no—
139C2×C42—22no—
140C2×C42—22no—
141C2×C42—41no—
142C2×C42C2×D412yes—
143C2×C42C2×D412yes—
144C2×C42C2×D411yescharacteristic
145C2×C42C2×D412yes—
146C2×C42C2×Q812yes—
147C2×C42#4112yes—
148C2×C42#4112yes—
149C2×C42#4112yes—
150C2×C42C2411yescharacteristic
151C4⋊C8—24no—
152C4⋊C8—22no—
153C4⋊C8—22no—
154C4⋊C8—42no—
155#104—44no—
156#104—44no—
157C22⋊C8—24no—
158C22⋊C8—42no—
159C22⋊C8—42no—
160#97—41no—
161C4×C8—24no—
162C4×C8—42no—
163C4×C8—41no—
164#95—24no—
165#95—24no—
166#95—28no—
167#95—24no—
168#95—24no—
169#95—24no—
170#95—24no—
171#95C2×D418yes—
172C24#12011yescharacteristic
173#41—42no—
174#41—42no—
175#41—84no—
176C2×Q8—21no—
177C2×Q8—22no—
178C2×Q8—48no—
179C2×Q8—88no—
180C2×D4—21no—
181C2×D4—22no—
182C2×D4—41no—
183C2×D4—42no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—28no—
188C22×C4—22no—
189C22×C4—41no—
190C22×C4—42no—
191C22×C4—82no—
192C22×C4#12011yescharacteristic
193C22×C4#12012yes—
194C22×C4C22⋊Q812yes—
195C22×C4C22⋊Q812yes—
196C22×C4C22⋊Q812yes—
197C22×C4#12412yes—
198C22×C4#12412yes—
199C22×C4#12412yes—
200C22×C4C22×D412yes—
201C22×C4C22×D411yescharacteristic
202M16—42no—
203M16—42no—
204C2×C8—44no—
205C2×C8—41no—
206C2×C8—41no—
207C2×C8—84no—
208C4⋊C4—22no—
209C4⋊C4—22no—
210C4⋊C4—22no—
211C4⋊C4—44no—
212C4⋊C4—48no—
213C4⋊C4—42no—
214C4⋊C4—44no—
215C4⋊C4—44no—
216C4⋊C4—42no—
217C4⋊C4—42no—
218C4⋊C4—88no—
219C22⋊C4—44no—
220C22⋊C4—41no—
221C22⋊C4—42no—
222C22⋊C4—42no—
223C22⋊C4—42no—
224C22⋊C4—42no—
225C22⋊C4—84no—
226C42—22no—
227C42—22no—
228C42—22no—
229C42—41no—
230C42—42no—
231C42—84no—
232C42—82no—
233C42—82no—
234C42C22⋊Q812yes—
235C42C22⋊Q812yes—
236C42C22⋊Q812yes—
237C42#12412yes—
238C42#12411yescharacteristic
239C42#12411yescharacteristic
240C42#12412yes—
241C42#13514yes—
242C42C22×Q811yescharacteristic
243C42C2×#4111yescharacteristic
244C42C2×#4112yes—
245C23—22no—
246C23—42no—
247C23—41no—
248C23#39312yes—
249C23C2×#12011yescharacteristic
250Q8—42no—
251Q8—82no—
252Q8—168no—
253D4—42no—
254D4—82no—
255D4—84no—
256D4—82no—
257C2×C4—22no—
258C2×C4—24no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—24no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—44no—
269C2×C4—44no—
270C2×C4—48no—
271C2×C4—48no—
272C2×C4—41no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—48no—
276C2×C4—84no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4#39312yes—
280C2×C4#39312yes—
281C2×C4#39312yes—
282C2×C4C2×C22⋊Q812yes—
283C2×C4C2×C22⋊Q811yescharacteristic
284C2×C4C2×#12412yes—
285C2×C4C2×#12411yescharacteristic
286C2×C4C2×#13512yes—
287C2×C4#57512yes—
288C8—84no—
289C8—81no—
290C8—81no—
291C22—22no—
292C22—22no—
293C22—22no—
294C22—22no—
295C22—82no—
296C22—81no—
297C22—82no—
298C22C2×#39311yescharacteristic
299C22#300212yes—
300C4—22no—
301C4—22no—
302C4—22no—
303C4—22no—
304C4—22no—
305C4—44no—
306C4—82no—
307C4—88no—
308C4—81no—
309C4—82no—
310C4#307612yes—
311C4#313111yescharacteristic
312C4#313511yescharacteristic
313C2—22no—
314C2—42no—
315C2—161no—
316C2#2345111yescharacteristic
317C2#2347212yes—
subgroupG/H
center, socleC22C2×#393
commutator, Frattini, cosocleC2×C42C24
prime radical#3444C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35344C2semidirect
4. #10369C2semidirect
5. #10368C2semidirect
6. #10363C2semidirect
7. #10362C2semidirect
8. #7956C2semidirect
9. #7866C2semidirect
35. #533Q8semidirect
37. #532Q8semidirect

Maximal subgroups

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

subgroupindex
#353442
#214182
#175712
#103692
#103682
#103632
#103622
#79562
#78662