Finite Group Databasebeta

← #6434675

Subgroups — #6434675

Order 512.

3265 conjugacy classes of subgroups.

The subgroup lattice has 351 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#125C211yesmaximal, characteristic, complemented
2C2×#3092C212yesmaximal
3#36626C211yesmaximal, characteristic
4#29643C218yesmaximal, complemented
5#29640C218yesmaximal, complemented
6#23553C214yesmaximal, complemented
7#23453C214yesmaximal, complemented
8#20700C212yesmaximal, complemented
9#20682C211yesmaximal, characteristic, complemented
10C22×#125—22no—
11C22×#125—24no—
12C22×#125—216no—
13C22×#125C2214yes—
14C22×#125C2218yes—
15C2×#525C2211yescharacteristic
16C23×C4⋊C4—22no—
17C23×C4⋊C4C2212yes—
18C23×C42C2211yescharacteristic
19#3092—216no—
20C2×#484C2214yes—
21C2×#481C2214yes—
22C2×#483C2214yes—
23C2×#480C2214yes—
24C2×#428C2214yes—
25C2×#417C2214yes—
26C2×#407C2212yes—
27#2855C2214yes—
28#2854C2214yes—
29#2853—22no—
30#2849—22no—
31C4×#123—24no—
32#2297C2212yes—
33#2296C2212yes—
34#2280C2211yescharacteristic
35#2059C22132yescomplemented
36#2026C22116yescomplemented
37#2025C22116yescomplemented
38#1824C2218yes—
39#1823C2218yes—
40#1817C2218yes—
41#1736C2218yes—
42#1728C2214yes—
43C22×#41C2311yescharacteristic
44C24×C4—22no—
45C24×C4D412yes—
46C24×C4C2311yescharacteristic
47C22×M16C2314yes—
48C2×#125—264no—
49C2×#125—232no—
50C2×#125—232no—
51C2×#125—232no—
52C2×#125—232no—
53C2×#125—28no—
54C2×#125—216no—
55C2×#125C23116yes—
56#525—28no—
57C2×#123—24no—
58C2×#123—21no—
59C2×C22⋊Q8C2312yes—
60C2×#121C2312yes—
61C2×#120—21no—
62C2×C4×D4—22no—
63C2×#117C2311yescharacteristic
64C2×#117C2312yes—
65C22×C4⋊C4—232no—
66C22×C4⋊C4—28no—
67C22×C4⋊C4—28no—
68C22×C4⋊C4—28no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—28no—
73C22×C4⋊C4D4116yes—
74C22×C4⋊C4C2314yes—
75C22×C4⋊C4C2314yes—
76C22×C4⋊C4C2314yes—
77C22×C42—28no—
78C22×C42D412yes—
79C22×C42D414yes—
80C22×C42C2314yes—
81C22×C42C2312yes—
82#484—216no—
83#483—216no—
84#481—216no—
85#480—216no—
86#428—216no—
87C2×#107C2318yes—
88C2×#106C2318yes—
89#417—216no—
90C2×#103C2318yes—
91C2×#102C2318yes—
92#407—28no—
93C2×C22⋊C8C2314yes—
94C2×C22⋊C8C2314yes—
95#391C2314yes—
96#390C2314yes—
97#389C2318yes—
98#388—28no—
99#387—28no—
100#386—24no—
101#386—24no—
102#385—24no—
103#385—24no—
104#381—24no—
105#381—21no—
106#380C2314yes—
107#379—21no—
108C4×C4⋊C4—24no—
109C4×C22⋊C4—22no—
110C4×C22⋊C4—24no—
111#340C23132yes—
112#336C23116yes—
113C25C2×D411yescharacteristic
114C2×#41—24no—
115C2×#41—28no—
116C22×Q8C2411yescharacteristic
117C22×D4—22no—
118C22×D4—21no—
119C22×D4C2411yescharacteristic
120C23×C4—28no—
121C23×C4—216no—
122C23×C4—28no—
123C23×C4—24no—
124C23×C4—24no—
125C23×C4—24no—
126C23×C4—21no—
127C23×C4—48no—
128C23×C4—416no—
129C23×C4C2×D414yes—
130C23×C4C2×D412yes—
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D418yes—
133C23×C4C2×D414yes—
134C23×C4C2×D414yes—
135C23×C4C2412yes—
136C23×C4C2411yescharacteristic
137C2×M16—28no—
138C2×M16—216no—
139C22×C8C2418yes—
140#125—2128no—
141#125—264no—
142#125—264no—
143#123—416no—
144#123—44no—
145C22⋊Q8—24no—
146C22⋊Q8—24no—
147#121—24no—
148#121—24no—
149#120—44no—
150C4×D4—48no—
151#117—24no—
152#117—28no—
153C2×C4⋊C4—264no—
154C2×C4⋊C4—264no—
155C2×C4⋊C4—264no—
156C2×C4⋊C4—216no—
157C2×C4⋊C4—216no—
158C2×C4⋊C4—216no—
159C2×C4⋊C4—232no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—28no—
164C2×C4⋊C4—28no—
165C2×C4⋊C4—28no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—28no—
168C2×C4⋊C4C2×D4116yes—
169C2×C4⋊C4C2×D4116yes—
170C2×C4⋊C4C2×D4116yes—
171C2×C4⋊C4C2412yes—
172C2×C4⋊C4C2414yes—
173C2×C4⋊C4C2418yes—
174C2×C22⋊C4—21no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—28no—
180C2×C42—216no—
181C2×C42—28no—
182C2×C42—28no—
183C2×C42—28no—
184C2×C42—28no—
185C2×C42—21no—
186C2×C42—22no—
187C2×C42C2×D414yes—
188C2×C42C2×D418yes—
189C2×C42C2414yes—
190#107—216no—
191#106—216no—
192#103—216no—
193#102—216no—
194C22⋊C8—28no—
195C22⋊C8—28no—
196#95—24no—
197#95—28no—
198#95—28no—
199C24—24no—
200C24—28no—
201C24—21no—
202C24#12014yes—
203C24C22×D411yescharacteristic
204C24C22×D412yes—
205#41—416no—
206C2×Q8—22no—
207C2×Q8—24no—
208C2×D4—22no—
209C2×D4—24no—
210C2×D4—44no—
211C2×D4—42no—
212C2×D4—44no—
213C2×D4—48no—
214C22×C4—24no—
215C22×C4—232no—
216C22×C4—28no—
217C22×C4—28no—
218C22×C4—28no—
219C22×C4—28no—
220C22×C4—28no—
221C22×C4—216no—
222C22×C4—216no—
223C22×C4—216no—
224C22×C4—28no—
225C22×C4—28no—
226C22×C4—28no—
227C22×C4—21no—
228C22×C4—24no—
229C22×C4—22no—
230C22×C4—48no—
231C22×C4—48no—
232C22×C4—432no—
233C22×C4—464no—
234C22×C4—416no—
235C22×C4—416no—
236C22×C4—416no—
237C22×C4—42no—
238C22×C4—44no—
239C22×C4#12018yes—
240C22×C4#12014yes—
241C22×C4C22×D418yes—
242C22×C4C22×D412yes—
243C22×C4C22×D414yes—
244C22×C4C22×D414yes—
245C22×C4C22×D411yescharacteristic
246C22×C4C22×D414yes—
247C22×C42−1+418yes—
248C22×C4C2511yescharacteristic
249M16—416no—
250C2×C8—28no—
251C2×C8—216no—
252C4⋊C4—2128no—
253C4⋊C4—216no—
254C4⋊C4—216no—
255C4⋊C4—216no—
256C4⋊C4—28no—
257C4⋊C4—22no—
258C4⋊C4—22no—
259C4⋊C4—28no—
260C4⋊C4—416no—
261C4⋊C4—44no—
262C4⋊C4C22×D4116yes—
263C22⋊C4—416no—
264C22⋊C4—42no—
265C22⋊C4—44no—
266C22⋊C4—44no—
267C22⋊C4—48no—
268C22⋊C4—44no—
269C42—216no—
270C42—28no—
271C42—28no—
272C42—42no—
273C42—44no—
274C23—232no—
275C23—28no—
276C23—24no—
277C23—216no—
278C23—24no—
279C23—24no—
280C23—41no—
281C23—42no—
282C23—44no—
283C23C2×#12012yes—
284C23C2×#12014yes—
285C23#53614yes—
286C23#57518yes—
287C23C23×D411yescharacteristic
288Q8—44no—
289D4—44no—
290D4—84no—
291D4—88no—
292C2×C4—28no—
293C2×C4—24no—
294C2×C4—216no—
295C2×C4—28no—
296C2×C4—28no—
297C2×C4—28no—
298C2×C4—216no—
299C2×C4—28no—
300C2×C4—28no—
301C2×C4—416no—
302C2×C4—416no—
303C2×C4—432no—
304C2×C4—432no—
305C2×C4—416no—
306C2×C4—432no—
307C2×C4—416no—
308C2×C4—42no—
309C2×C4—41no—
310C2×C4—44no—
311C2×C4—42no—
312C2×C4—48no—
313C2×C4—44no—
314C2×C4—84no—
315C2×C4C2×#12014yes—
316C2×C4C2×#12012yes—
317C2×C4#53614yes—
318C2×C4#536116yes—
319C2×C4C23×D412yes—
320C2×C4C2×2−1+414yes—
321C8—48no—
322C22—232no—
323C22—24no—
324C22—28no—
325C22—216no—
326C22—24no—
327C22—24no—
328C22—81no—
329C22—84no—
330C22—82no—
331C22#261414yes—
332C22#300518yes—
333C22C22×#12011yescharacteristic
334C22C2×#53612yes—
335C22C2×#57514yes—
336C4—24no—
337C4—48no—
338C4—416no—
339C4—81no—
340C4—84no—
341C4—82no—
342C4#261414yes—
343C4C2×#53614yes—
344C2—24no—
345C2—28no—
346C2—161no—
347C2C2×#261411yescharacteristic
348C2C2×#300512yes—
349C2#6086114yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C23×#125C2semidirect
4. #29643C2semidirect
5. #29640C2semidirect
6. #23553C2semidirect
7. #23453C2semidirect
8. #20700C2semidirect
9. #20682C2semidirect
35. #2059C22semidirect
36. #2026C22semidirect
37. #2025C22semidirect

Maximal subgroups

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

subgroupindex
C23×#1252
C2×#30922
#366262
#296432
#296402
#235532
#234532
#207002
#206822