Finite Group Databasebeta

← #6481945

Subgroups — #6481945

Order 512.

1600 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35181C211yesmaximal, characteristic, complemented
2C2×#2357C211yesmaximal, characteristic
3#33150C214yesmaximal, complemented
4#29956C214yesmaximal
5#26103C218yesmaximal, complemented
6#26037C218yesmaximal, complemented
7#21986C212yesmaximal, complemented
8#21955C211yesmaximal, characteristic, complemented
9#20738C212yesmaximal
10C22×C4×D4C2211yescharacteristic
11C2×#432C2212yes—
12#2851C2214yes—
13#2646—21no—
14#2646—21no—
15#2571C2214yescomplemented
16#2570—24no—
17#2552C2214yescomplemented
18#2357—24no—
19#2357C2218yes—
20#2317—22no—
21C2×#390C2212yes—
22C2×#388—22no—
23#2285C2212yes—
24C2×#382C2211yescharacteristic
25C2×#381—21no—
26C2×C4×C4⋊C4C2212yes—
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C22⋊C4—21no—
29#2092C2218yes—
30#2067C2218yes—
31#2065C2218yes—
32#2025C2218yes—
33#1989C2218yescomplemented
34#1949C2218yescomplemented
35#1949C2218yescomplemented
36#1940C2218yescomplemented
37#1920C2218yescomplemented
38#1918C2218yescomplemented
39#1853C2214yes—
40#1852C2214yes—
41#1847C2214yescomplemented
42#1836C2214yescomplemented
43#1824C2214yes—
44#1802C2214yes—
45#1802C2214yes—
46#1797C2214yes—
47#1786C2214yes—
48#1762C2218yes—
49C23×D4C2311yescharacteristic
50C24×C4—21no—
51C2×#125C2314yes—
52C2×C4×D4—22no—
53C2×C4×D4—24no—
54C2×C4×D4Q818yes—
55C2×C4×D4C2314yes—
56C2×#117—22no—
57C22×C4⋊C4D412yes—
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2312yes—
60C22×C22⋊C4—21no—
61C22×C22⋊C4—22no—
62C22×C22⋊C4—22no—
63C22×C42—22no—
64C22×C42C2312yes—
65C22×C42C2311yescharacteristic
66#432—28no—
67C2×#107C2318yes—
68C2×#106C2318yes—
69C2×C4⋊C8C2314yes—
70C2×C4⋊C8C2318yes—
71C2×#102C2318yes—
72C2×#102C2318yes—
73C2×C22⋊C8C2314yes—
74C2×#97C2314yes—
75#400—24no—
76#400—24no—
77#398—24no—
78#398—24no—
79#390—24no—
80#390C2318yes—
81#389—24no—
82#389—24no—
83#389C2314yes—
84#389C2314yes—
85#388—28no—
86#388—28no—
87#387—24no—
88#387—24no—
89#385—24no—
90#385—24no—
91#384—24no—
92#384C2312yes—
93#384C2312yes—
94#382—24no—
95#382—22no—
96#382—22no—
97#382C2314yes—
98#382C2314yes—
99#381—28no—
100#380—22no—
101#380C2314yes—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—24no—
104C4×C4⋊C4D414yes—
105C4×C4⋊C4C2314yes—
106C4×C4⋊C4C2318yes—
107C4×C4⋊C4C2314yes—
108C4×C22⋊C4—28no—
109C2×#95—21no—
110C2×#95—22no—
111C2×#95—21no—
112C2×#95D411yescharacteristic
113C2×#95D411yescharacteristic
114#340C2318yes—
115#340C2318yes—
116#340C2318yes—
117#340C2318yes—
118#336C2318yes—
119C25—21no—
120C22×D4—21no—
121C22×D4—22no—
122C22×D4—42no—
123C22×D4—42no—
124C22×D4C2×Q814yes—
125C22×D4C2412yes—
126C23×C4—21no—
127C23×C4—28no—
128C23×C4—24no—
129C23×C4—22no—
130C23×C4—84no—
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D412yes—
133C23×C4C2×D412yes—
134C23×C4C2×D411yescharacteristic
135C23×C4C2411yescharacteristic
136C22×C8C2418yes—
137#125—48no—
138C4×D4—216no—
139C4×D4—28no—
140C4×D4C2×Q8116yes—
141#117—48no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—44no—
154C2×C4⋊C4—44no—
155C2×C4⋊C4C2×D414yes—
156C2×C4⋊C4C2×D414yes—
157C2×C4⋊C4C2×Q814yes—
158C2×C4⋊C4C2414yes—
159C2×C4⋊C4C2414yes—
160C2×C4⋊C4C2414yes—
161C2×C4⋊C4C2412yes—
162C2×C22⋊C4—28no—
163C2×C22⋊C4—24no—
164C2×C22⋊C4—24no—
165C2×C22⋊C4—28no—
166C2×C22⋊C4—28no—
167C2×C22⋊C4—48no—
168C2×C22⋊C4—84no—
169C2×C42—24no—
170C2×C42—28no—
171C2×C42—44no—
172C2×C42—84no—
173C2×C42C2×D414yes—
174C2×C42C2×D412yes—
175C2×C42C2×Q814yes—
176C2×C42#4114yes—
177C2×C42C2414yes—
178C2×C42C2412yes—
179#107—48no—
180#106—48no—
181C4⋊C8—28no—
182C4⋊C8—48no—
183#102—48no—
184#102—48no—
185C22⋊C8—28no—
186#97—28no—
187#95—24no—
188#95—28no—
189#95—24no—
190#95—24no—
191#95—24no—
192#95—24no—
193#95—24no—
194#95—24no—
195#95—24no—
196#95C2×D414yes—
197#95C2×D414yes—
198C24—24no—
199C24—22no—
200C24—44no—
201C24—42no—
202C24—42no—
203C24—44no—
204C24C22×D411yescharacteristic
205C2×D4—24no—
206C2×D4—22no—
207C2×D4—24no—
208C2×D4—24no—
209C2×D4—48no—
210C2×D4—42no—
211C2×D4—42no—
212C2×D4—48no—
213C2×D4C22×Q814yes—
214C22×C4—24no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—24no—
218C22×C4—24no—
219C22×C4—22no—
220C22×C4—24no—
221C22×C4—216no—
222C22×C4—24no—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—44no—
231C22×C4—44no—
232C22×C4—44no—
233C22×C4—44no—
234C22×C4—84no—
235C22×C4—84no—
236C22×C4—816no—
237C22×C4C22⋊Q814yes—
238C22×C4C22⋊Q814yes—
239C22×C4C22×D414yes—
240C22×C4C22×D414yes—
241C22×C4C22×D412yes—
242C22×C4C22×D411yescharacteristic
243C22×C4C22×D412yes—
244C22×C4C22×Q812yes—
245C22×C4C2×#4112yes—
246C22×C4C2511yescharacteristic
247C2×C8—28no—
248C2×C8—28no—
249C2×C8—28no—
250C4⋊C4—24no—
251C4⋊C4—22no—
252C4⋊C4—28no—
253C4⋊C4—28no—
254C4⋊C4—28no—
255C4⋊C4—44no—
256C4⋊C4—44no—
257C4⋊C4—44no—
258C4⋊C4—44no—
259C4⋊C4—44no—
260C4⋊C4—82no—
261C4⋊C4—82no—
262C4⋊C4—84no—
263C4⋊C4—84no—
264C4⋊C4—84no—
265C4⋊C4—84no—
266C4⋊C4C22×Q814yes—
267C22⋊C4—216no—
268C22⋊C4—416no—
269C22⋊C4—48no—
270C22⋊C4—88no—
271C42—28no—
272C42—44no—
273C42—48no—
274C42—88no—
275C42C22⋊Q818yes—
276C42C22×Q814yes—
277C42C2×#4114yes—
278C23—22no—
279C23—21no—
280C23—21no—
281C23—24no—
282C23—48no—
283C23—48no—
284C23—44no—
285C23—48no—
286C23—42no—
287C23—42no—
288C23—44no—
289C23—88no—
290C23—88no—
291C23C2×C22⋊Q812yes—
292C23#57514yes—
293C23C23×D411yescharacteristic
294D4—28no—
295D4—48no—
296D4—44no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—28no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—24no—
305C2×C4—24no—
306C2×C4—24no—
307C2×C4—24no—
308C2×C4—24no—
309C2×C4—24no—
310C2×C4—24no—
311C2×C4—24no—
312C2×C4—24no—
313C2×C4—44no—
314C2×C4—48no—
315C2×C4—48no—
316C2×C4—44no—
317C2×C4—44no—
318C2×C4—44no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—44no—
322C2×C4—44no—
323C2×C4—44no—
324C2×C4—44no—
325C2×C4—44no—
326C2×C4—816no—
327C2×C4—84no—
328C2×C4—88no—
329C2×C4C2×C22⋊Q812yes—
330C2×C4C2×C22⋊Q814yes—
331C2×C4C2×C22⋊Q814yes—
332C2×C4C23×Q811yescharacteristic
333C2×C4C22×#4111yescharacteristic
334C2×C4#58512yes—
335C2×C4#58512yes—
336C8—48no—
337C22—24no—
338C22—22no—
339C22—22no—
340C22—24no—
341C22—44no—
342C22—44no—
343C22—44no—
344C22—816no—
345C22—84no—
346C22—84no—
347C22#308014yes—
348C22C22×C22⋊Q811yescharacteristic
349C22#348511yescharacteristic
350C22#348711yescharacteristic
351C22C2×#57512yes—
352C22#358512yes—
353C4—24no—
354C4—24no—
355C4—22no—
356C4—44no—
357C4—44no—
358C4—44no—
359C4—84no—
360C4—84no—
361C4—84no—
362C4#345812yes—
363C4#349312yes—
364C2—24no—
365C2—84no—
366C2#3515111yescharacteristic
367C2C2×#308011yescharacteristic
368C2#6049011yescharacteristic
369C2#6171312yes—
370C2#6189512yes—
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
3. #33150C2semidirect
5. #26103C2semidirect
6. #26037C2semidirect
7. #21986C2semidirect
8. #21955C2semidirect
15. #2571C22semidirect
17. #2552C22semidirect
33. #1989C22semidirect
34. #1949C22semidirect
35. #1949C22semidirect
36. #1940C22semidirect
37. #1920C22semidirect
38. #1918C22semidirect
41. #1847C22semidirect
42. #1836C22semidirect

Maximal subgroups

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

subgroupindex
#351812
C2×#23572
#331502
#299562
#261032
#260372
#219862
#219552
#207382