Finite Group Databasebeta

← #6532032

Subgroups — #6532032

Order 512.

1510 conjugacy classes of subgroups.

The subgroup lattice has 381 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35280C211yesmaximal, characteristic
2#34374C211yesmaximal, characteristic, complemented
3#27620C214yesmaximal, complemented
4#27595C214yesmaximal
5#25062C214yesmaximal, complemented
6#25056C214yesmaximal, complemented
7#24995C214yesmaximal, complemented
8#24993C214yesmaximal, complemented
9#21982C212yesmaximal, complemented
10#21819C211yesmaximal, characteristic, complemented
11#20737C212yesmaximal
12C2×#533C2211yescharacteristic
13C2×C4×#41C2211yescharacteristic
14C2×#431C2212yes—
15#2857—22no—
16#2629—22no—
17#2582C2214yes—
18#2538C2214yes—
19#2417—24no—
20#2382—22no—
21#2365—22no—
22#2355C2214yes—
23Q8×C4⋊C4C2214yes—
24#2323—24no—
25#2316C2212yes—
26#2314C2214yes—
27#2311C2212yes—
28#2285C2212yes—
29C2×#384C2211yescharacteristic
30#2101C2218yes—
31#2095C2214yes—
32#2093C2214yes—
33#2083C2218yes—
34#2074C2214yes—
35#2073C2214yes—
36#1984C2214yes—
37#1984C2214yes—
38#1947C2214yes—
39#1947C2214yes—
40#1928C2218yes—
41#1925C2218yes—
42#1921C2218yes—
43#1918C2218yes—
44#1852C2218yes—
45#1845C2214yes—
46#1834C2214yes—
47#1825C2214yes—
48#1818C2214yes—
49#1803C2212yes—
50#1803C2212yes—
51#1802C2212yes—
52#1802C2212yes—
53#1788C2214yes—
54C4×#103C2214yes—
55C4×#102C2214yes—
56C22×#41C2311yescharacteristic
57#533—28no—
58C2×C4⋊Q8D412yes—
59C2×#125D412yes—
60C2×C22⋊Q8—24no—
61C2×C22⋊Q8D414yes—
62C4×#41—28no—
63C2×C4×Q8—24no—
64C2×C4×Q8C2312yes—
65C2×C4×Q8C2314yes—
66C2×C4×D4—22no—
67C2×C4×D4C2312yes—
68C2×#117—22no—
69C2×#117—21no—
70C2×#117C2311yescharacteristic
71C2×#117C2312yes—
72C22×C4⋊C4—22no—
73C22×C4⋊C4C2312yes—
74C22×C42—21no—
75C22×C42C2311yescharacteristic
76#431—28no—
77C2×#107C2318yes—
78C2×#107C2318yes—
79C2×C4⋊C8C2312yes—
80C2×C4⋊C8C2312yes—
81C2×#103C2314yes—
82C2×#103C2314yes—
83C2×#102C2314yes—
84C2×#102C2314yes—
85C2×C22⋊C8C2314yes—
86C2×C4×C8C2314yes—
87#398—28no—
88#396—28no—
89#391—24no—
90#391—24no—
91#391C2314yes—
92#390C2314yes—
93#389—24no—
94#389—24no—
95#389—28no—
96#389—24no—
97#389C2318yes—
98#388—28no—
99#387—28no—
100#387—28no—
101#386—24no—
102#384—21no—
103#384—21no—
104#384C2312yes—
105#384C2312yes—
106#384C2314yes—
107#384C2314yes—
108#382—22no—
109#382C2314yes—
110#380—22no—
111#380—24no—
112#380—24no—
113#380—24no—
114#380C2314yes—
115C4×C4⋊C4—24no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4C2318yes—
118C4×C4⋊C4C2314yes—
119#340C2314yes—
120#340C2314yes—
121#340C2314yes—
122#340C2314yes—
123#340C2318yes—
124#340C2318yes—
125#336C2314yes—
126#336C2314yes—
127#336C2318yes—
128C2×#41—22no—
129C2×#41—22no—
130C2×#41—24no—
131C2×#41—42no—
132C22×Q8C2×D414yes—
133C22×Q8C2411yescharacteristic
134C22×D4—21no—
135C22×D4C2411yescharacteristic
136C23×C4—21no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4C2×D412yes—
140C23×C4C2411yescharacteristic
141C22×C8C2414yes—
142C22×C8C2414yes—
143C4⋊Q8—28no—
144#125—28no—
145C22⋊Q8—216no—
146C22⋊Q8—88no—
147C4×Q8—28no—
148C4×Q8—28no—
149C4×Q8—24no—
150C4×Q8—24no—
151C4×Q8—88no—
152C4×D4—24no—
153C4×D4—24no—
154C4×D4—48no—
155#117—28no—
156#117—24no—
157#117—44no—
158#117—84no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—28no—
168C2×C4⋊C4—28no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—84no—
171C2×C4⋊C4C2×D414yes—
172C2×C4⋊C4C2×D418yes—
173C2×C4⋊C4C2×D414yes—
174C2×C4⋊C4C2414yes—
175C2×C4⋊C4C2412yes—
176C2×C4⋊C4C2414yes—
177C2×C4⋊C4C2414yes—
178C2×C22⋊C4—21no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—24no—
181C2×C22⋊C4—28no—
182C2×C22⋊C4C2×D414yes—
183C2×C42—21no—
184C2×C42—24no—
185C2×C42—22no—
186C2×C42—28no—
187C2×C42—44no—
188C2×C42C2×D411yescharacteristic
189C2×C42C2×D411yescharacteristic
190C2×C42#4118yes—
191C2×C42C2414yes—
192C2×C42C2412yes—
193#107—48no—
194#107—48no—
195C4⋊C8—24no—
196C4⋊C8—24no—
197#103—28no—
198#103—44no—
199#102—28no—
200#102—44no—
201C22⋊C8—28no—
202C4×C8—28no—
203#95—24no—
204#95—28no—
205#95—24no—
206#95—28no—
207#95—28no—
208#95—28no—
209C24—21no—
210C24C22×D411yescharacteristic
211#41—48no—
212#41—48no—
213C2×Q8—28no—
214C2×Q8—28no—
215C2×Q8—28no—
216C2×Q8—21no—
217C2×Q8—22no—
218C2×Q8—22no—
219C2×Q82−1+412yes—
220C2×D4—24no—
221C2×D4—22no—
222C2×D4—21no—
223C2×D4—22no—
224C2×D4—42no—
225C2×D4—42no—
226C2×D42−1+412yes—
227C22×C4—24no—
228C22×C4—24no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—22no—
232C22×C4—24no—
233C22×C4—21no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—22no—
237C22×C4—24no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—44no—
241C22×C4—44no—
242C22×C4—44no—
243C22×C4—44no—
244C22×C4—84no—
245C22×C4#12318yes—
246C22×C4C22×D418yes—
247C22×C4C22×D411yescharacteristic
248C22×C4C22×D412yes—
249C22×C4C22×D412yes—
250C22×C4C2×#4114yes—
251C22×C4C2511yescharacteristic
252C2×C8—24no—
253C2×C8—24no—
254C2×C8—24no—
255C2×C8—24no—
256C2×C8—24no—
257C2×C8—24no—
258C4⋊C4—28no—
259C4⋊C4—216no—
260C4⋊C4—28no—
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—24no—
264C4⋊C4—24no—
265C4⋊C4—24no—
266C4⋊C4—24no—
267C4⋊C4—28no—
268C4⋊C4—44no—
269C4⋊C4—44no—
270C4⋊C4—48no—
271C4⋊C4—44no—
272C4⋊C4—44no—
273C4⋊C4—84no—
274C4⋊C4—82no—
275C4⋊C4—82no—
276C4⋊C4—88no—
277C4⋊C4—88no—
278C22⋊C4—28no—
279C22⋊C4—42no—
280C22⋊C4—44no—
281C22⋊C4—88no—
282C22⋊C4—84no—
283C42—28no—
284C42—42no—
285C42—44no—
286C42—48no—
287C42—88no—
288C42#12314yes—
289C42#12314yes—
290C42C2×#4114yes—
291C42C2×#4114yes—
292C23—21no—
293C23—21no—
294C23—22no—
295C23—22no—
296C23—22no—
297C23—41no—
298C23—41no—
299C23—42no—
300C23—42no—
301C23C2×#12312yes—
302C23C23×D411yescharacteristic
303Q8—24no—
304Q8—416no—
305Q8—42no—
306D4—24no—
307D4—48no—
308D4—42no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—216no—
314C2×C4—24no—
315C2×C4—24no—
316C2×C4—28no—
317C2×C4—22no—
318C2×C4—28no—
319C2×C4—28no—
320C2×C4—48no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—42no—
324C2×C4—44no—
325C2×C4—44no—
326C2×C4—44no—
327C2×C4—44no—
328C2×C4—44no—
329C2×C4—44no—
330C2×C4—48no—
331C2×C4—42no—
332C2×C4—44no—
333C2×C4—42no—
334C2×C4—88no—
335C2×C4—164no—
336C2×C4C2×#12312yes—
337C2×C4C2×#12314yes—
338C2×C4C2×#12314yes—
339C2×C4#57614yes—
340C2×C4#57814yes—
341C2×C4C22×#4112yes—
342C2×C4C2×2−1+411yescharacteristic
343C2×C4C2×2−1+411yescharacteristic
344C2×C4#58512yes—
345C8—44no—
346C8—44no—
347C22—22no—
348C22—24no—
349C22—42no—
350C22—41no—
351C22—41no—
352C22—42no—
353C22—44no—
354C22—82no—
355C22—81no—
356C22—81no—
357C22C22×#12311yescharacteristic
358C22#348611yescharacteristic
359C22#348711yescharacteristic
360C22C2×#57612yes—
361C22C2×#57812yes—
362C4—22no—
363C4—24no—
364C4—42no—
365C4—48no—
366C4—82no—
367C4—84no—
368C4—84no—
369C4—84no—
370C4#309818yes—
371C4#349512yes—
372C4#349512yes—
373C2—42no—
374C2—82no—
375C2#3567211yescharacteristic
376C2C2×#309812yes—
377C2#6187012yes—
378C2#6193211yescharacteristic
379C2#6193611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#533 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34374C2semidirect
3. #27620C2semidirect
5. #25062C2semidirect
6. #25056C2semidirect
7. #24995C2semidirect
8. #24993C2semidirect
9. #21982C2semidirect
10. #21819C2semidirect

Maximal subgroups

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

subgroupindex
#352802
#343742
#276202
#275952
#250622
#250562
#249952
#249932
#219822
#218192
#207372