Finite Group Databasebeta

← #6528504

Subgroups — #6528504

Order 512.

1592 conjugacy classes of subgroups.

The subgroup lattice has 403 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35181C211yesmaximal, characteristic, complemented
2C2×#2566C211yesmaximal, characteristic
3#29110C212yesmaximal
4#29104C212yesmaximal
5#27965C212yesmaximal, complemented
6#27964C212yesmaximal, complemented
7#27829C218yesmaximal, complemented
8#27743C218yesmaximal, complemented
9#21838C212yesmaximal, complemented
10#20738C212yesmaximal
11#20534C211yesmaximal, characteristic, complemented
12C22×#125C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C2×#432C2212yes—
15#2851C2212yes—
16#2851C2212yes—
17#2646—22no—
18#2571C2214yescomplemented
19#2570—24no—
20#2566—24no—
21#2566C2218yes—
22#2552C2214yescomplemented
23C2×#396—22no—
24#2317—22no—
25C2×#390C2211yescharacteristic
26C2×#390C2211yescharacteristic
27C2×#385—22no—
28#2285C2212yes—
29C2×#384C2211yescharacteristic
30C2×#382C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32#2101C2218yescomplemented
33#2099C2218yescomplemented
34#2073C2214yes—
35#2073C2214yes—
36#2025C2214yes—
37#2025C2214yes—
38#1992C2214yescomplemented
39#1992C2214yescomplemented
40#1992C2214yescomplemented
41#1991C2214yescomplemented
42#1989C2218yescomplemented
43#1936C2218yes—
44#1934C2218yes—
45#1836C2214yescomplemented
46#1835C2214yescomplemented
47#1824C2214yes—
48#1824C2214yes—
49#1823C2214yes—
50#1802C2214yes—
51#1802C2214yes—
52#1786C2214yes—
53#1771C2218yescomplemented
54#1762C2218yes—
55#1749C2212yes—
56C4×C22⋊C8C2212yes—
57C23×D4C2311yescharacteristic
58C24×C4—21no—
59C2×#125—22no—
60C2×#125—28no—
61C2×#125C2314yes—
62C2×#125C2312yes—
63C2×#125C2312yes—
64C2×C4×D4—22no—
65C2×C4×D4—28no—
66C2×C4×D4C2314yes—
67C2×#117—22no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4D412yes—
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C22⋊C4—21no—
76C22×C22⋊C4—22no—
77C22×C22⋊C4—22no—
78C22×C42C2311yescharacteristic
79C22×C42C2311yescharacteristic
80C22×C42C2311yescharacteristic
81#432—28no—
82C2×C4⋊C8C2314yes—
83C2×#102C2318yes—
84C2×#102C2318yes—
85C2×C22⋊C8C2314yes—
86C2×#97C2314yes—
87C2×#97C2314yes—
88C2×C4×C8C2314yes—
89#400—24no—
90#400—24no—
91#398—24no—
92#398—24no—
93#396—28no—
94#396—28no—
95#390—22no—
96#390—22no—
97#390C2314yes—
98#390C2314yes—
99#389—24no—
100#389—24no—
101#389C2314yes—
102#389C2314yes—
103#387—24no—
104#387—24no—
105#385—24no—
106#385—24no—
107#385—28no—
108#385—28no—
109#384—24no—
110#384—22no—
111#384C2314yes—
112#384C2312yes—
113#384C2312yes—
114#382—24no—
115#382—22no—
116#382—22no—
117#382C2314yes—
118#382C2312yes—
119#382C2312yes—
120#380—22no—
121#380C2314yes—
122C4×C4⋊C4—22no—
123C4×C4⋊C4D414yes—
124C4×C4⋊C4C2314yes—
125C4×C4⋊C4C2314yes—
126C2×#95—22no—
127C2×#95D412yes—
128#340C2318yes—
129#340C2318yes—
130#340C2318yes—
131#340C2318yes—
132#340C2318yes—
133#340C2318yes—
134#336C2314yes—
135#336C2314yes—
136C25—21no—
137C22×D4—24no—
138C22×D4—21no—
139C22×D4—24no—
140C22×D4—42no—
141C22×D4C2412yes—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—24no—
146C23×C4—44no—
147C23×C4—84no—
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D412yes—
150C23×C4C2×D412yes—
151C23×C4C2×D411yescharacteristic
152C23×C4C2411yescharacteristic
153C22×C8C2418yes—
154#125—28no—
155#125—28no—
156#125—44no—
157#125—44no—
158#125—48no—
159C4×D4—28no—
160C4×D4—28no—
161C4×D4—48no—
162#117—48no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—28no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—84no—
184C2×C4⋊C4C2×D414yes—
185C2×C4⋊C4C2×D414yes—
186C2×C4⋊C4C2412yes—
187C2×C4⋊C4C2412yes—
188C2×C4⋊C4C2412yes—
189C2×C4⋊C4C2414yes—
190C2×C4⋊C4C2414yes—
191C2×C22⋊C4—24no—
192C2×C22⋊C4—24no—
193C2×C22⋊C4—28no—
194C2×C22⋊C4—28no—
195C2×C22⋊C4—48no—
196C2×C22⋊C4—44no—
197C2×C22⋊C4—84no—
198C2×C42—24no—
199C2×C42—42no—
200C2×C42—42no—
201C2×C42C2×D412yes—
202C2×C42C2×D412yes—
203C2×C42C2×D412yes—
204C2×C42#4118yes—
205C2×C42C2412yes—
206C2×C42C2412yes—
207C2×C42C2412yes—
208C4⋊C8—28no—
209#102—216no—
210#102—48no—
211C22⋊C8—28no—
212#97—28no—
213#97—44no—
214C4×C8—44no—
215#95—28no—
216#95—24no—
217#95—24no—
218#95—24no—
219#95—24no—
220#95—24no—
221#95—24no—
222#95C2×D414yes—
223#95C2×D414yes—
224C24—24no—
225C24—22no—
226C24—44no—
227C24—42no—
228C24—42no—
229C24—44no—
230C24C22×D411yescharacteristic
231C2×D4—24no—
232C2×D4—24no—
233C2×D4—22no—
234C2×D4—24no—
235C2×D4—416no—
236C2×D4—42no—
237C2×D4—42no—
238C2×D42−1+414yes—
239C22×C4—24no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—44no—
255C22×C4—44no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—44no—
260C22×C4—48no—
261C22×C4—84no—
262C22×C4—84no—
263C22×C4—816no—
264C22×C4#12414yes—
265C22×C4#12414yes—
266C22×C4C22×D414yes—
267C22×C4C22×D414yes—
268C22×C4C22×D411yescharacteristic
269C22×C4C22×D412yes—
270C22×C4C22×D412yes—
271C22×C4C2×#4114yes—
272C22×C4C2511yescharacteristic
273C2×C8—28no—
274C2×C8—28no—
275C2×C8—28no—
276C4⋊C4—24no—
277C4⋊C4—24no—
278C4⋊C4—24no—
279C4⋊C4—28no—
280C4⋊C4—28no—
281C4⋊C4—44no—
282C4⋊C4—44no—
283C4⋊C4—44no—
284C4⋊C4—44no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—48no—
289C4⋊C4—48no—
290C4⋊C4—82no—
291C4⋊C4—82no—
292C4⋊C4—84no—
293C4⋊C4—84no—
294C4⋊C4—88no—
295C4⋊C4—84no—
296C4⋊C4—84no—
297C22⋊C4—416no—
298C22⋊C4—48no—
299C22⋊C4—48no—
300C22⋊C4—88no—
301C42—24no—
302C42—24no—
303C42—42no—
304C42—42no—
305C42—48no—
306C42#12418yes—
307C42C2×#4114yes—
308C42C2×#4114yes—
309C23—22no—
310C23—21no—
311C23—21no—
312C23—24no—
313C23—48no—
314C23—416no—
315C23—44no—
316C23—42no—
317C23—42no—
318C23—44no—
319C23—84no—
320C23—88no—
321C23—84no—
322C23C2×#12412yes—
323C23#57514yes—
324C23C23×D411yescharacteristic
325D4—28no—
326D4—48no—
327D4—44no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—28no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—28no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—44no—
343C2×C4—48no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—816no—
357C2×C4—84no—
358C2×C4—84no—
359C2×C4—84no—
360C2×C4—84no—
361C2×C4C2×#12412yes—
362C2×C4C2×#12414yes—
363C2×C4C2×#12414yes—
364C2×C4C22×#4112yes—
365C2×C4C2×2−1+412yes—
366C2×C4#58512yes—
367C8—48no—
368C22—24no—
369C22—22no—
370C22—24no—
371C22—22no—
372C22—44no—
373C22—48no—
374C22—816no—
375C22—84no—
376C22—82no—
377C22—82no—
378C22#314214yes—
379C22C22×#12411yescharacteristic
380C22#348511yescharacteristic
381C22#348611yescharacteristic
382C22C2×#57512yes—
383C22#358512yes—
384C4—22no—
385C4—24no—
386C4—24no—
387C4—44no—
388C4—44no—
389C4—84no—
390C4—84no—
391C4—82no—
392C4—82no—
393C4—82no—
394C4#349514yes—
395C2—24no—
396C2—84no—
397C2#3564411yescharacteristic
398C2C2×#314211yescharacteristic
399C2#6061911yescharacteristic
400C2#6171312yes—
401C2#6183112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35181C2semidirect
5. #27965C2semidirect
6. #27964C2semidirect
7. #27829C2semidirect
8. #27743C2semidirect
9. #21838C2semidirect
11. #20534C2semidirect
18. #2571C22semidirect
22. #2552C22semidirect
32. #2101C22semidirect
33. #2099C22semidirect
38. #1992C22semidirect
39. #1992C22semidirect
40. #1992C22semidirect
41. #1991C22semidirect
42. #1989C22semidirect
45. #1836C22semidirect
46. #1835C22semidirect
53. #1771C22semidirect

Maximal subgroups

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

subgroupindex
#351812
C2×#25662
#291102
#291042
#279652
#279642
#278292
#277432
#218382
#207382
#205342