Finite Group Databasebeta

← #6510734

Subgroups — #6510734

Order 512.

1866 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35002C211yesmaximal, characteristic, complemented
2C2×#2504C211yesmaximal, characteristic
3#29109C214yesmaximal, complemented
4#27969C214yesmaximal, complemented
5#27711C2116yesmaximal, complemented
6#21758C212yesmaximal
7C2×#1824C212yesmaximal, complemented
8#20526C211yesmaximal, characteristic, complemented
9C2×#543—21no—
10C22×#121C2211yescharacteristic
11C23×C4⋊C4C2211yescharacteristic
12C22×C22⋊C8C2212yes—
13#2857C2214yes—
14#2659—22no—
15#2549—24no—
16#2504—24no—
17#2504C2218yes—
18#2440C2218yescomplemented
19C2×#397—22no—
20#2315—21no—
21C2×#390C2212yes—
22C2×#386—21no—
23C2×#386—21no—
24C2×#382C2212yes—
25C2×#380C2212yes—
26C2×C4×C22⋊C4C2211yescharacteristic
27#2101C22116yescomplemented
28#2071C2218yescomplemented
29#2023C2218yescomplemented
30#1993C2218yescomplemented
31#1936C22116yescomplemented
32#1893C2218yes—
33#1874C2218yes—
34#1836C2218yescomplemented
35#1824C2218yes—
36#1824C2218yescomplemented
37#1823C2214yes—
38#1786C2214yes—
39#1752C2218yescomplemented
40#1749C2214yes—
41C2×#340C2218yescomplemented
42C23×D4—21no—
43C23×D4C2311yescharacteristic
44C24×C4—21no—
45C24×C4C2311yescharacteristic
46C23×C8C2314yes—
47#543—48no—
48C2×C4⋊Q8C2314yes—
49C2×C22⋊Q8—24no—
50C2×C22⋊Q8—28no—
51C2×#121—22no—
52C2×#121—28no—
53C2×#121C2314yes—
54C22×C4⋊C4—28no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2314yes—
60C22×C4⋊C4C2314yes—
61C22×C22⋊C4—22no—
62C22×C22⋊C4—21no—
63C22×C22⋊C4—22no—
64C22×C22⋊C4—22no—
65C22×C22⋊C4D411yescharacteristic
66C22×C22⋊C4D411yescharacteristic
67C22×C42C2312yes—
68C2×#102C23116yes—
69C2×C22⋊C8—28no—
70C2×C22⋊C8C2314yes—
71C2×C22⋊C8C2314yes—
72C2×#97C2318yes—
73#398—28no—
74#397—216no—
75#395—28no—
76#391—24no—
77#390—24no—
78#390C2318yes—
79#389—24no—
80#389C2318yes—
81#386—28no—
82#386—28no—
83#385—28no—
84#384C2314yes—
85#382—24no—
86#382C2318yes—
87#380—24no—
88#380—22no—
89#380—22no—
90#380C2314yes—
91#380C2314yes—
92C4×C22⋊C4D414yes—
93C4×C22⋊C4C2314yes—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95D412yes—
98#340C23116yes—
99#340C23116yes—
100#340C23116yes—
101#336C2318yes—
102C25—21no—
103C25C2×D411yescharacteristic
104C22×Q8—24no—
105C22×D4—24no—
106C22×D4—21no—
107C22×D4—24no—
108C22×D4—44no—
109C22×D4—44no—
110C22×D4—44no—
111C22×D4—44no—
112C22×D4C2412yes—
113C23×C4—22no—
114C23×C4—21no—
115C23×C4—24no—
116C23×C4—24no—
117C23×C4—44no—
118C23×C4—44no—
119C23×C4—84no—
120C23×C4C2×D412yes—
121C23×C4C2×D412yes—
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D412yes—
124C23×C4#4118yes—
125C23×C4C2412yes—
126C23×C4C2411yescharacteristic
127C22×C8—28no—
128C22×C8—48no—
129C22×C8C2418yes—
130C4⋊Q8—48no—
131C22⋊Q8—432no—
132C22⋊Q8—416no—
133#121—28no—
134#121—28no—
135#121—48no—
136C2×C4⋊C4—28no—
137C2×C4⋊C4—28no—
138C2×C4⋊C4—28no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—28no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—48no—
149C2×C4⋊C4—44no—
150C2×C4⋊C4—48no—
151C2×C4⋊C4C2414yes—
152C2×C4⋊C4C2412yes—
153C2×C4⋊C4C2418yes—
154C2×C22⋊C4—24no—
155C2×C22⋊C4—24no—
156C2×C22⋊C4—24no—
157C2×C22⋊C4—24no—
158C2×C22⋊C4—24no—
159C2×C22⋊C4—28no—
160C2×C22⋊C4—28no—
161C2×C22⋊C4—48no—
162C2×C22⋊C4—44no—
163C2×C22⋊C4—84no—
164C2×C22⋊C4—84no—
165C2×C22⋊C4C2×D414yes—
166C2×C22⋊C4C2×D414yes—
167C2×C42—44no—
168C2×C42C2×D414yes—
169C2×C42C2414yes—
170#102—416no—
171C22⋊C8—28no—
172C22⋊C8—28no—
173#97—48no—
174#95—24no—
175#95—24no—
176#95—28no—
177#95—28no—
178#95—28no—
179#95C2×D418yes—
180C24—22no—
181C24—42no—
182C24—42no—
183C24—44no—
184C24—44no—
185C24—42no—
186C24—42no—
187C24—42no—
188C24#12414yes—
189C24AGL(1,ℤ/8)14yes—
190C24#13714yes—
191C24C22×D411yescharacteristic
192C24C22×D412yes—
193C2×Q8—48no—
194C2×Q8—84no—
195C2×Q8—84no—
196C2×D4—24no—
197C2×D4—28no—
198C2×D4—44no—
199C2×D4—48no—
200C2×D4—48no—
201C2×D4—48no—
202C2×D4—42no—
203C2×D4—88no—
204C2×D4—816no—
205C22×C4—24no—
206C22×C4—28no—
207C22×C4—24no—
208C22×C4—24no—
209C22×C4—22no—
210C22×C4—24no—
211C22×C4—28no—
212C22×C4—416no—
213C22×C4—44no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—44no—
220C22×C4—44no—
221C22×C4—42no—
222C22×C4—44no—
223C22×C4—48no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—48no—
227C22×C4—88no—
228C22×C4—84no—
229C22×C4—84no—
230C22×C4—816no—
231C22×C4#12414yes—
232C22×C4#12418yes—
233C22×C4C22×D414yes—
234C22×C4C22×D414yes—
235C22×C4C22×D411yescharacteristic
236C22×C4C22×D412yes—
237C22×C4C2×#4114yes—
238C22×C4C2×#4114yes—
239C22×C4C2×#4114yes—
240C22×C4C2511yescharacteristic
241C2×C8—28no—
242C2×C8—216no—
243C2×C8—88no—
244C4⋊C4—28no—
245C4⋊C4—24no—
246C4⋊C4—216no—
247C4⋊C4—48no—
248C4⋊C4—416no—
249C4⋊C4—48no—
250C4⋊C4—44no—
251C4⋊C4—42no—
252C4⋊C4—84no—
253C4⋊C4—84no—
254C4⋊C4—84no—
255C22⋊C4—28no—
256C22⋊C4—28no—
257C22⋊C4—48no—
258C22⋊C4—48no—
259C22⋊C4—48no—
260C22⋊C4—88no—
261C22⋊C4—88no—
262C42—28no—
263C42—44no—
264C23—22no—
265C23—24no—
266C23—22no—
267C23—24no—
268C23—44no—
269C23—44no—
270C23—48no—
271C23—48no—
272C23—44no—
273C23—42no—
274C23—44no—
275C23—42no—
276C23—42no—
277C23—84no—
278C23—88no—
279C23—88no—
280C23—84no—
281C23—84no—
282C23#489116yes—
283C23C2×#12412yes—
284C23C2×#12414yes—
285C23C2×AGL(1,ℤ/8)14yes—
286C23C2×AGL(1,ℤ/8)12yes—
287C23C2×#13714yes—
288C23C2×#13712yes—
289C23C23×D411yescharacteristic
290Q8—168no—
291D4—48no—
292D4—84no—
293D4—816no—
294C2×C4—22no—
295C2×C4—24no—
296C2×C4—28no—
297C2×C4—28no—
298C2×C4—28no—
299C2×C4—24no—
300C2×C4—24no—
301C2×C4—24no—
302C2×C4—24no—
303C2×C4—24no—
304C2×C4—24no—
305C2×C4—44no—
306C2×C4—48no—
307C2×C4—44no—
308C2×C4—44no—
309C2×C4—48no—
310C2×C4—42no—
311C2×C4—44no—
312C2×C4—48no—
313C2×C4—48no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—84no—
317C2×C4—84no—
318C2×C4—88no—
319C2×C4—816no—
320C2×C4—84no—
321C2×C4—168no—
322C2×C4C2×#12414yes—
323C2×C4C2×#12412yes—
324C2×C4C22×#4112yes—
325C2×C4#58514yes—
326C8—48no—
327C22—24no—
328C22—28no—
329C22—24no—
330C22—28no—
331C22—22no—
332C22—24no—
333C22—22no—
334C22—24no—
335C22—416no—
336C22—84no—
337C22—84no—
338C22—82no—
339C22—82no—
340C22—82no—
341C22—168no—
342C22C2×#48914yes—
343C22C2×#48918yes—
344C22C22×#12411yescharacteristic
345C22#348512yes—
346C22C22×AGL(1,ℤ/8)11yescharacteristic
347C22C22×#13711yescharacteristic
348C22#358512yes—
349C4—24no—
350C4—22no—
351C4—44no—
352C4—44no—
353C4—88no—
354C4—82no—
355C4—84no—
356C4#349114yes—
357C2—24no—
358C2—28no—
359C2—162no—
360C2#3545511yescharacteristic
361C2C22×#48911yescharacteristic
362C2#6061911yescharacteristic
363C2#6170712yes—
364C2#6170812yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35002C2semidirect
3. #29109C2semidirect
4. #27969C2semidirect
5. #27711C2semidirect
7. C2×#1824C2semidirect
8. #20526C2semidirect
18. #2440C22semidirect
27. #2101C22semidirect
28. #2071C22semidirect
29. #2023C22semidirect
30. #1993C22semidirect
31. #1936C22semidirect
34. #1836C22semidirect
36. #1824C22semidirect
39. #1752C22semidirect
41. C2×#340C22semidirect

Maximal subgroups

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

subgroupindex
#350022
C2×#25042
#291092
#279692
#277112
#217582
C2×#18242
#205262