Finite Group Databasebeta

← #6481972

Subgroups — #6481972

Order 512.

1474 conjugacy classes of subgroups.

The subgroup lattice has 377 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37643C211yesmaximal, characteristic, complemented
2#34188C211yesmaximal, characteristic
3#33156C214yesmaximal, complemented
4#29964C214yesmaximal
5#26092C214yesmaximal, complemented
6#26086C214yesmaximal, complemented
7#26025C214yesmaximal, complemented
8#26023C214yesmaximal, complemented
9#21982C212yesmaximal, complemented
10#21952C211yesmaximal, characteristic, complemented
11#20737C212yesmaximal
12C2×C4×#41C2211yescharacteristic
13C2×#431C2212yes—
14#2845—22no—
15#2629—24no—
16#2629C2214yes—
17#2552—22no—
18#2552C2214yes—
19#2552C2214yes—
20#2355C2214yes—
21#2354C2214yes—
22#2352—24no—
23#2314—22no—
24#2311C2212yes—
25#2285—22no—
26#2285C2212yes—
27#2285C2211yescharacteristic
28C2×#384C2211yescharacteristic
29#2274C2212yes—
30#2095C2214yes—
31#2093C2214yes—
32#2068C2218yes—
33#2067C2218yes—
34#2026C2214yes—
35#2025C2214yes—
36#1989C2218yes—
37#1947C2218yes—
38#1947C2214yes—
39#1947C2214yes—
40#1939C2214yes—
41#1939C2214yes—
42#1921C2218yes—
43#1918C2218yes—
44#1852C2218yes—
45#1845C2214yes—
46#1834C2214yes—
47#1825C2214yes—
48#1803C2212yes—
49#1803C2212yes—
50#1802C2212yes—
51#1802C2212yes—
52#1798C2214yes—
53#1788C2214yes—
54C4×#103C2214yes—
55C4×#102C2214yes—
56C22×#41C2311yescharacteristic
57C2×#125—24no—
58C2×#125C2314yes—
59C4×#41—24no—
60C4×#41Q818yes—
61C2×C4×Q8C2312yes—
62C2×C4×D4—22no—
63C2×C4×D4C2312yes—
64C2×#117—22no—
65C2×#117—24no—
66C2×#117—21no—
67C2×#117C2311yescharacteristic
68C2×#117C2312yes—
69C22×C4⋊C4—22no—
70C22×C4⋊C4C2312yes—
71C22×C42—21no—
72C22×C42C2311yescharacteristic
73#431—28no—
74C2×#107C2318yes—
75C2×#107C2318yes—
76C2×C4⋊C8C2312yes—
77C2×C4⋊C8C2312yes—
78C2×C4⋊C8C2318yes—
79C2×#103C2314yes—
80C2×#103C2314yes—
81C2×#102C2314yes—
82C2×#102C2314yes—
83C2×C22⋊C8C2314yes—
84C2×C4×C8C2314yes—
85#398—28no—
86#398—28no—
87#398—28no—
88#396—28no—
89#391C2314yes—
90#390C2314yes—
91#389—24no—
92#389—24no—
93#389C2318yes—
94#388—28no—
95#387—28no—
96#386—24no—
97#384—24no—
98#384—21no—
99#384—21no—
100#384D412yes—
101#384C2312yes—
102#384C2312yes—
103#384C2314yes—
104#384C2314yes—
105#382—24no—
106#382—22no—
107#382D412yes—
108#380—22no—
109#380—24no—
110#380—24no—
111#380D414yes—
112#380C2314yes—
113C4×C4⋊C4—24no—
114C4×C4⋊C4—24no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4C2318yes—
117C4×C4⋊C4C2314yes—
118C4×C4⋊C4C2314yes—
119C4×C22⋊C4—24no—
120#340C2314yes—
121#340C2314yes—
122#340C2314yes—
123#340C2314yes—
124#340C2318yes—
125#340C2318yes—
126#336C2314yes—
127#336C2314yes—
128C2×#41—22no—
129C2×#41—42no—
130C2×#41—42no—
131C2×#41C2×Q814yes—
132C22×Q8C2411yescharacteristic
133C22×D4—21no—
134C22×D4C2411yescharacteristic
135C23×C4—22no—
136C23×C4—21no—
137C23×C4—22no—
138C23×C4C2×D412yes—
139C23×C4C2411yescharacteristic
140C22×C8C2414yes—
141C22×C8C2414yes—
142#125—48no—
143#125—88no—
144C4×Q8—24no—
145C4×Q8C2×Q818yes—
146C4×D4—216no—
147C4×D4—24no—
148C4×D4C2×Q818yes—
149#117—22no—
150#117—28no—
151#117—44no—
152#117—84no—
153#117—88no—
154#117C2×Q814yes—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—28no—
164C2×C4⋊C4—28no—
165C2×C4⋊C4—28no—
166C2×C4⋊C4—44no—
167C2×C4⋊C4—84no—
168C2×C4⋊C4C2×D414yes—
169C2×C4⋊C4C2×D414yes—
170C2×C4⋊C4C2×D414yes—
171C2×C4⋊C4C2414yes—
172C2×C4⋊C4C2414yes—
173C2×C4⋊C4C2414yes—
174C2×C4⋊C4C2412yes—
175C2×C22⋊C4—21no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—28no—
179C2×C22⋊C4C2×D414yes—
180C2×C42—21no—
181C2×C42—22no—
182C2×C42—28no—
183C2×C42—24no—
184C2×C42—28no—
185C2×C42C2×D411yescharacteristic
186C2×C42C2×D411yescharacteristic
187C2×C42C2×Q814yes—
188C2×C42#4114yes—
189C2×C42C2414yes—
190C2×C42C2412yes—
191#107—48no—
192#107—48no—
193C4⋊C8—24no—
194C4⋊C8—24no—
195C4⋊C8—48no—
196#103—44no—
197#103—44no—
198#102—44no—
199#102—44no—
200C22⋊C8—28no—
201C4×C8—28no—
202#95—28no—
203#95—28no—
204#95—24no—
205#95—24no—
206#95—28no—
207#95C2×D418yes—
208C24—21no—
209C24C22×D411yescharacteristic
210#41—48no—
211#41—48no—
212C2×Q8—21no—
213C2×Q8—22no—
214C2×Q8—22no—
215C2×Q8C22×Q812yes—
216C2×D4—24no—
217C2×D4—21no—
218C2×D4—22no—
219C2×D4—22no—
220C2×D4—42no—
221C2×D4—42no—
222C2×D4C22×Q812yes—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—21no—
228C22×C4—22no—
229C22×C4—24no—
230C22×C4—28no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—44no—
237C22×C4—44no—
238C22×C4—44no—
239C22×C4—84no—
240C22×C4C22⋊Q818yes—
241C22×C4C22×D418yes—
242C22×C4C22×D411yescharacteristic
243C22×C4C22×D412yes—
244C22×C4C22×D412yes—
245C22×C4C22×Q812yes—
246C22×C4C2×#4112yes—
247C22×C4C2511yescharacteristic
248C2×C8—24no—
249C2×C8—24no—
250C2×C8—24no—
251C2×C8—24no—
252C2×C8—24no—
253C2×C8—24no—
254C4⋊C4—28no—
255C4⋊C4—28no—
256C4⋊C4—28no—
257C4⋊C4—22no—
258C4⋊C4—24no—
259C4⋊C4—24no—
260C4⋊C4—24no—
261C4⋊C4—24no—
262C4⋊C4—28no—
263C4⋊C4—44no—
264C4⋊C4—44no—
265C4⋊C4—48no—
266C4⋊C4—44no—
267C4⋊C4—44no—
268C4⋊C4—84no—
269C4⋊C4—82no—
270C4⋊C4—82no—
271C4⋊C4—88no—
272C4⋊C4—88no—
273C4⋊C4—88no—
274C4⋊C4C22×Q814yes—
275C22⋊C4—24no—
276C22⋊C4—28no—
277C22⋊C4—44no—
278C22⋊C4—88no—
279C22⋊C4—84no—
280C42—24no—
281C42—28no—
282C42—28no—
283C42—48no—
284C42—88no—
285C42C22⋊Q814yes—
286C42C22⋊Q814yes—
287C42C22×Q814yes—
288C42C2×#4114yes—
289C23—21no—
290C23—21no—
291C23—22no—
292C23—22no—
293C23—22no—
294C23—41no—
295C23—41no—
296C23—42no—
297C23—42no—
298C23C2×C22⋊Q812yes—
299C23C23×D411yescharacteristic
300Q8—24no—
301Q8—42no—
302D4—24no—
303D4—48no—
304D4—42no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—216no—
310C2×C4—24no—
311C2×C4—24no—
312C2×C4—24no—
313C2×C4—22no—
314C2×C4—28no—
315C2×C4—28no—
316C2×C4—28no—
317C2×C4—48no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—416no—
323C2×C4—44no—
324C2×C4—44no—
325C2×C4—44no—
326C2×C4—44no—
327C2×C4—44no—
328C2×C4—48no—
329C2×C4—44no—
330C2×C4—44no—
331C2×C4—164no—
332C2×C4C2×C22⋊Q812yes—
333C2×C4C2×C22⋊Q814yes—
334C2×C4C2×C22⋊Q814yes—
335C2×C4#57614yes—
336C2×C4#57814yes—
337C2×C4C23×Q811yescharacteristic
338C2×C4C22×#4111yescharacteristic
339C2×C4#58512yes—
340C2×C4#58512yes—
341C8—44no—
342C8—44no—
343C22—24no—
344C22—22no—
345C22—42no—
346C22—41no—
347C22—41no—
348C22—44no—
349C22—42no—
350C22—82no—
351C22—81no—
352C22—81no—
353C22C22×C22⋊Q811yescharacteristic
354C22#348511yescharacteristic
355C22#348711yescharacteristic
356C22C2×#57612yes—
357C22C2×#57812yes—
358C4—24no—
359C4—22no—
360C4—44no—
361C4—42no—
362C4—48no—
363C4—84no—
364C4—84no—
365C4—84no—
366C4#308318yes—
367C4#345812yes—
368C4#349312yes—
369C2—42no—
370C2—82no—
371C2#3515111yescharacteristic
372C2C2×#308312yes—
373C2#6179112yes—
374C2#6193211yescharacteristic
375C2#6193611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2285 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #37643C2semidirect
3. #33156C2semidirect
5. #26092C2semidirect
6. #26086C2semidirect
7. #26025C2semidirect
8. #26023C2semidirect
9. #21982C2semidirect
10. #21952C2semidirect

Maximal subgroups

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

subgroupindex
#376432
#341882
#331562
#299642
#260922
#260862
#260252
#260232
#219822
#219522
#207372