Finite Group Databasebeta

← #319339

Subgroups — #319339

Order 512.

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

normal subgroups only

This group has 866 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#33207C211yesmaximal, characteristic, complemented
2#27889C212yesmaximal, complemented
3#25671C212yesmaximal
4#21195C211yesmaximal, characteristic
5#10685C211yesmaximal, characteristic, complemented
6#10677C211yesmaximal, characteristic, complemented
7#10316C211yesmaximal, characteristic, complemented
8#10312C211yesmaximal, characteristic, complemented
9#9828C212yesmaximal, complemented
10#9827C212yesmaximal, complemented
11#8144C211yesmaximal, characteristic, complemented
12C2×#532C2211yescharacteristic
13C2×#498C2212yes—
14C2×#498C2211yescharacteristic
15C2×#486C2212yes—
16C2×#416—22no—
17C22×C4×C8C2211yescharacteristic
18#2857C2212yes—
19#2285C2211yescharacteristic
20#2035—24no—
21#2034—24no—
22#1990C2212yes—
23#1990C2212yes—
24#1958—22no—
25#1957C2212yes—
26#1957C2211yescharacteristic
27#1957C2211yescharacteristic
28#1956C2212yes—
29#1956C2212yes—
30#1956C2211yescharacteristic
31#1956C2211yescharacteristic
32#1941—22no—
33#1927—24no—
34#1864—24no—
35#1844C2212yes—
36#1844C2211yescharacteristic
37#1834C2212yes—
38#1807—22no—
39C4×C22⋊C8C2212yes—
40C2×#322C2212yes—
41#1273C2211yescharacteristic
42#1273C2211yescharacteristic
43C22×#41—21no—
44C23×C8D412yes—
45#532—22no—
46#532—22no—
47#532—42no—
48C2×C4⋊Q8C2312yes—
49C2×C4⋊Q8C2311yescharacteristic
50C2×#127C2311yescharacteristic
51C2×#124—21no—
52C2×C22⋊Q8—22no—
53C2×#121—22no—
54C2×C4×D4—22no—
55C2×#117—21no—
56C22×C42C2311yescharacteristic
57#498—24no—
58#498—42no—
59#498—42no—
60#486—24no—
61#486—42no—
62C2×#106—22no—
63C2×#106—22no—
64C2×#106—24no—
65C2×#106—24no—
66#416—48no—
67C2×#103—22no—
68C2×#103—22no—
69C2×#103—24no—
70C2×#103—24no—
71C2×#102—24no—
72C2×#102—24no—
73C2×#102—22no—
74C2×#102—22no—
75C2×C22⋊C8D414yes—
76C2×C4×C8—42no—
77C2×C4×C8D412yes—
78C2×C4×C8C2312yes—
79C2×C4×C8C2311yescharacteristic
80C2×C4×C8C2312yes—
81C2×C4×C8C2311yescharacteristic
82#391—22no—
83#389—22no—
84#389—22no—
85#384C2312yes—
86#384C2311yescharacteristic
87#384C2311yescharacteristic
88#382—21no—
89#380—22no—
90#380—22no—
91#380—22no—
92C4×C4⋊C4—22no—
93#340—24no—
94#340—24no—
95#340—24no—
96#336D414yes—
97#322—42no—
98C2×#41—24no—
99C2×#41—42no—
100C2×#41—44no—
101C2×#41—44no—
102C22×Q8—21no—
103C22×Q8—22no—
104C22×D4—22no—
105C22×D4—21no—
106C22×D4—41no—
107C23×C4—41no—
108C23×C4C2×D412yes—
109C23×C4C2×D411yescharacteristic
110C22×C8—42no—
111C22×C8—42no—
112C22×C8SD1618yes—
113C22×C8C2×D414yes—
114C22×C8C2×D414yes—
115C22×C8C2×D412yes—
116C22×C8C2×D412yes—
117C4⋊Q8—21no—
118C4⋊Q8—22no—
119C4⋊Q8—22no—
120C4⋊Q8—22no—
121C4⋊Q8—22no—
122C4⋊Q8—22no—
123C4⋊Q8#4112yes—
124#127—21no—
125#127—22no—
126#127#4112yes—
127#124—24no—
128#124—42no—
129C22⋊Q8—84no—
130#121—44no—
131#121—44no—
132C4×D4—44no—
133C4×D4—44no—
134#117—44no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—42no—
146C2×C22⋊C4—42no—
147C2×C22⋊C4—41no—
148C2×C22⋊C4—42no—
149C2×C42—21no—
150C2×C42—21no—
151C2×C42—22no—
152C2×C42—41no—
153C2×C42C2×D412yes—
154C2×C42C2×D411yescharacteristic
155C2×C42C2411yescharacteristic
156#106—48no—
157#106—44no—
158#106—44no—
159#106—84no—
160#103—48no—
161#103—44no—
162#103—44no—
163#103—84no—
164#102—48no—
165#102—48no—
166#102—44no—
167#102—44no—
168C22⋊C8—44no—
169C4×C8—22no—
170C4×C8—22no—
171C4×C8—22no—
172C4×C8—22no—
173C4×C8—22no—
174C4×C8—22no—
175C4×C8—42no—
176#95—42no—
177#95—42no—
178#95—42no—
179C24—41no—
180C24#12711yescharacteristic
181#41—416no—
182#41—88no—
183C2×Q8—22no—
184C2×Q8—41no—
185C2×Q8—42no—
186C2×Q8—42no—
187C2×Q8—42no—
188C2×Q8—42no—
189C2×Q8—44no—
190C2×Q8—44no—
191C2×D4—22no—
192C2×D4—24no—
193C2×D4—44no—
194C2×D4—41no—
195C2×D4—42no—
196C2×D4—42no—
197C2×D4—44no—
198C2×D4—42no—
199C2×D4—44no—
200C2×D4—82no—
201C2×D4—82no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—24no—
208C22×C4—24no—
209C22×C4—44no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—82no—
216C22×C4—82no—
217C22×C4#12712yes—
218C22×C4#12711yescharacteristic
219C22×C4C2×SD1614yes—
220C22×C4C22×D411yescharacteristic
221C22×C4C22×D412yes—
222C2×C8—22no—
223C2×C8—22no—
224C2×C8—22no—
225C2×C8—22no—
226C2×C8—24no—
227C2×C8—24no—
228C2×C8—24no—
229C2×C8—24no—
230C2×C8—24no—
231C2×C8—24no—
232C2×C8—48no—
233C2×C8C2×SD1614yes—
234C2×C8C2×SD1614yes—
235C4⋊C4—24no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—44no—
244C4⋊C4—44no—
245C4⋊C4—44no—
246C4⋊C4—44no—
247C4⋊C4—44no—
248C4⋊C4—84no—
249C22⋊C4—48no—
250C22⋊C4—84no—
251C22⋊C4—82no—
252C42—21no—
253C42—21no—
254C42—22no—
255C42—22no—
256C42—82no—
257C42C2×#4111yescharacteristic
258C42C2×#4111yescharacteristic
259C42H(5,𝔽2)12yes—
260C23—22no—
261C23—21no—
262C23—22no—
263C23—21no—
264C23—42no—
265C23—81no—
266C23—82no—
267C23—81no—
268C23—82no—
269C23#49212yes—
270C23C2×#12711yescharacteristic
271Q8—44no—
272Q8—82no—
273Q8—84no—
274Q8—84no—
275D4—44no—
276D4—48no—
277D4—88no—
278D4—82no—
279D4—84no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—24no—
283C2×C4—24no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—42no—
290C2×C4—88no—
291C2×C4—81no—
292C2×C4—81no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4#49212yes—
304C2×C4#52511yescharacteristic
305C2×C4#53411yescharacteristic
306C2×C4#54612yes—
307C2×C4#54612yes—
308C2×C4C22×SD1612yes—
309C2×C4#57512yes—
310C2×C4#57514yes—
311C2×C4#57714yes—
312C8—24no—
313C8—24no—
314C8—44no—
315C8—44no—
316C22—22no—
317C22—22no—
318C22—22no—
319C22—42no—
320C22—42no—
321C22—82no—
322C22—84no—
323C22—161no—
324C22—161no—
325C22—162no—
326C22#253911yescharacteristic
327C22C2×#49211yescharacteristic
328C22#315012yes—
329C22#315311yescharacteristic
330C22#315412yes—
331C4—22no—
332C4—42no—
333C4—82no—
334C4—161no—
335C4—162no—
336C4—161no—
337C4—162no—
338C4#310112yes—
339C4#318712yes—
340C4#330714yes—
341C4#332114yes—
342C2—42no—
343C2—162no—
344C2#1960811yescharacteristic
345C2#1965411yescharacteristic
346C2#1969212yes—
347C2#2830211yescharacteristic
348C2#2835112yes—
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
LCSG ▷ C2×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
1. #33207C2semidirect
2. #27889C2semidirect
5. #10685C2semidirect
6. #10677C2semidirect
7. #10316C2semidirect
8. #10312C2semidirect
9. #9828C2semidirect
10. #9827C2semidirect
11. #8144C2semidirect

Maximal subgroups

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

subgroupindex
#332072
#278892
#256712
#211952
#106852
#106772
#103162
#103122
#98282
#98272
#81442