Finite Group Databasebeta

← #6487861

Subgroups — #6487861

Order 512.

1900 conjugacy classes of subgroups.

The subgroup lattice has 387 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35181C211yesmaximal, characteristic, complemented
2C2×#2349C211yesmaximal, characteristic
3#32959C214yesmaximal
4#29973C214yesmaximal, complemented
5#26125C218yesmaximal, complemented
6#26059C218yesmaximal, complemented
7#21986C212yesmaximal, complemented
8#21955C211yesmaximal, characteristic, complemented
9#20738C212yesmaximal
10C22×C4×D4C2211yescharacteristic
11C22×C4×D4C2212yes—
12C2×#478C2218yes—
13C2×#432C2212yes—
14#2851C2214yes—
15#2646—21no—
16#2646—21no—
17#2571C2214yescomplemented
18#2570—24no—
19#2552C2214yescomplemented
20#2349—24no—
21#2349C2218yes—
22#2317—22no—
23C2×#389C2212yes—
24C2×#385—22no—
25#2285C2212yes—
26C2×#382C2211yescharacteristic
27C2×#380—21no—
28C2×C4×C4⋊C4C2211yescharacteristic
29C22×#95—21no—
30#2101C2218yescomplemented
31#2088C2218yescomplemented
32#2086C2218yescomplemented
33#2059C2218yescomplemented
34#1949C2218yescomplemented
35#1940C2218yescomplemented
36#1864C2218yes—
37#1862C2218yes—
38#1853C2214yes—
39#1852C2214yes—
40#1847C2214yescomplemented
41#1836C2214yescomplemented
42#1824C2214yes—
43#1804C2218yes—
44#1802C2214yes—
45#1802C2214yes—
46#1797C2214yes—
47#1786C2214yes—
48#1762C2218yes—
49C23×D4C2311yescharacteristic
50C24×C4—21no—
51C24×C4—22no—
52C2×#125C2314yes—
53C2×C4×D4—22no—
54C2×C4×D4—24no—
55C2×C4×D4—24no—
56C2×C4×D4—48no—
57C2×C4×D4Q818yes—
58C2×C4×D4C2318yes—
59C2×C4×D4C2314yes—
60C2×#117—22no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4D412yes—
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2312yes—
65C22×C22⋊C4—21no—
66C22×C22⋊C4—22no—
67C22×C42C2312yes—
68C22×C42C2311yescharacteristic
69#478—416no—
70#432—28no—
71C2×#107C2318yes—
72C2×#106C2318yes—
73C2×C4⋊C8C2314yes—
74C2×C4⋊C8C2318yes—
75C2×#102C2318yes—
76C2×#102C2318yes—
77C2×C22⋊C8C2314yes—
78C2×#97C2314yes—
79#400—24no—
80#400—24no—
81#398—24no—
82#398—24no—
83#389—24no—
84#389—24no—
85#389—24no—
86#389C2318yes—
87#389C2314yes—
88#389C2314yes—
89#387—24no—
90#387—24no—
91#385—24no—
92#385—24no—
93#385—28no—
94#385—28no—
95#384—24no—
96#384C2312yes—
97#384C2312yes—
98#382—24no—
99#382—22no—
100#382—22no—
101#382C2314yes—
102#382C2314yes—
103#380—22no—
104#380—28no—
105#380C2314yes—
106C4×C4⋊C4—22no—
107C4×C4⋊C4D414yes—
108C4×C4⋊C4C2314yes—
109C4×C4⋊C4C2314yes—
110C2×#95—21no—
111C2×#95—22no—
112C2×#95—21no—
113C2×#95—28no—
114C2×#95D411yescharacteristic
115C2×#95D411yescharacteristic
116#340C2318yes—
117#340C2318yes—
118#340C2318yes—
119#340C2318yes—
120#336C2318yes—
121C25—21no—
122C22×D4—24no—
123C22×D4—24no—
124C22×D4—21no—
125C22×D4—22no—
126C22×D4C2×Q814yes—
127C22×D4C2412yes—
128C23×C4—21no—
129C23×C4—28no—
130C23×C4—24no—
131C23×C4—22no—
132C23×C4—28no—
133C23×C4—48no—
134C23×C4—48no—
135C23×C4—84no—
136C23×C4—84no—
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D412yes—
139C23×C4C2×D412yes—
140C23×C4C2×D411yescharacteristic
141C23×C4C2411yescharacteristic
142C22×C8C2418yes—
143#125—48no—
144C4×D4—216no—
145C4×D4—216no—
146C4×D4—28no—
147C4×D4—48no—
148C4×D4—416no—
149C4×D4C2×Q8116yes—
150#117—48no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—28no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—44no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4—84no—
166C2×C4⋊C4C2×D414yes—
167C2×C4⋊C4C2×D414yes—
168C2×C4⋊C4C2×Q814yes—
169C2×C4⋊C4C2414yes—
170C2×C4⋊C4C2414yes—
171C2×C4⋊C4C2414yes—
172C2×C4⋊C4C2412yes—
173C2×C22⋊C4—28no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—24no—
176C2×C22⋊C4—28no—
177C2×C22⋊C4—48no—
178C2×C42—24no—
179C2×C42—44no—
180C2×C42C2×D414yes—
181C2×C42C2×D412yes—
182C2×C42C2×Q814yes—
183C2×C42#4114yes—
184C2×C42C2414yes—
185C2×C42C2412yes—
186#107—48no—
187#106—48no—
188C4⋊C8—28no—
189C4⋊C8—48no—
190#102—216no—
191#102—216no—
192C22⋊C8—28no—
193#97—28no—
194#95—24no—
195#95—28no—
196#95—24no—
197#95—24no—
198#95—24no—
199#95—24no—
200#95—24no—
201#95—24no—
202#95—24no—
203#95C2×D414yes—
204#95C2×D414yes—
205C24—24no—
206C24—24no—
207C24—24no—
208C24—22no—
209C24—44no—
210C24—44no—
211C24C22×D411yescharacteristic
212C2×D4—24no—
213C2×D4—24no—
214C2×D4—216no—
215C2×D4—216no—
216C2×D4—24no—
217C2×D4—22no—
218C2×D4—22no—
219C2×D4—22no—
220C2×D4C22×Q814yes—
221C2×D4H(5,𝔽2)14yes—
222C2×D42−1+414yes—
223C22×C4—24no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—24no—
227C22×C4—24no—
228C22×C4—22no—
229C22×C4—24no—
230C22×C4—216no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—416no—
240C22×C4—416no—
241C22×C4—44no—
242C22×C4—44no—
243C22×C4—44no—
244C22×C4—44no—
245C22×C4—84no—
246C22×C4—84no—
247C22×C4—88no—
248C22×C4—816no—
249C22×C4—832no—
250C22×C4C22⋊Q814yes—
251C22×C4C22⋊Q814yes—
252C22×C4C22×D414yes—
253C22×C4C22×D414yes—
254C22×C4C22×D412yes—
255C22×C4C22×D411yescharacteristic
256C22×C4C22×D412yes—
257C22×C4C22×Q812yes—
258C22×C4C2×#4112yes—
259C22×C4C2511yescharacteristic
260C2×C8—28no—
261C2×C8—28no—
262C2×C8—28no—
263C4⋊C4—24no—
264C4⋊C4—22no—
265C4⋊C4—28no—
266C4⋊C4—28no—
267C4⋊C4—28no—
268C4⋊C4—44no—
269C4⋊C4—44no—
270C4⋊C4—44no—
271C4⋊C4—44no—
272C4⋊C4—44no—
273C4⋊C4—82no—
274C4⋊C4—82no—
275C4⋊C4—84no—
276C4⋊C4—84no—
277C4⋊C4—88no—
278C4⋊C4—84no—
279C4⋊C4—84no—
280C4⋊C4C22×Q814yes—
281C22⋊C4—216no—
282C22⋊C4—416no—
283C22⋊C4—48no—
284C42—28no—
285C42—44no—
286C42—48no—
287C42C22⋊Q818yes—
288C42C22×Q814yes—
289C42C2×#4114yes—
290C23—22no—
291C23—21no—
292C23—21no—
293C23—216no—
294C23—24no—
295C23—24no—
296C23—24no—
297C23—48no—
298C23—416no—
299C23—416no—
300C23—44no—
301C23—48no—
302C23—44no—
303C23C2×C22⋊Q812yes—
304C23#57514yes—
305C23C23×D411yescharacteristic
306D4—216no—
307D4—24no—
308D4—24no—
309D4—24no—
310D4#54418yes—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—28no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—24no—
319C2×C4—24no—
320C2×C4—24no—
321C2×C4—24no—
322C2×C4—24no—
323C2×C4—24no—
324C2×C4—24no—
325C2×C4—24no—
326C2×C4—24no—
327C2×C4—44no—
328C2×C4—48no—
329C2×C4—48no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—816no—
341C2×C4—84no—
342C2×C4—832no—
343C2×C4—88no—
344C2×C4C2×C22⋊Q812yes—
345C2×C4C2×C22⋊Q814yes—
346C2×C4C2×C22⋊Q814yes—
347C2×C4C23×Q811yescharacteristic
348C2×C4C22×#4111yescharacteristic
349C2×C4C2×H(5,𝔽2)12yes—
350C2×C4C2×2−1+412yes—
351C8—48no—
352C22—22no—
353C22—24no—
354C22—24no—
355C22—22no—
356C22—28no—
357C22—432no—
358C22—44no—
359C22—44no—
360C22—48no—
361C22—48no—
362C22#308014yes—
363C22C22×C22⋊Q811yescharacteristic
364C22#348411yescharacteristic
365C22#348611yescharacteristic
366C22C2×#57512yes—
367C22#358512yes—
368C4—24no—
369C4—24no—
370C4—22no—
371C4—44no—
372C4—44no—
373C4—44no—
374C4—84no—
375C4—84no—
376C4—84no—
377C4C2×#54412yes—
378C4#349212yes—
379C2—24no—
380C2—48no—
381C2#3522411yescharacteristic
382C2C2×#308011yescharacteristic
383C2#6049011yescharacteristic
384C2#6157612yes—
385C2#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
4. #29973C2semidirect
5. #26125C2semidirect
6. #26059C2semidirect
7. #21986C2semidirect
8. #21955C2semidirect
17. #2571C22semidirect
19. #2552C22semidirect
30. #2101C22semidirect
31. #2088C22semidirect
32. #2086C22semidirect
33. #2059C22semidirect
34. #1949C22semidirect
35. #1940C22semidirect
40. #1847C22semidirect
41. #1836C22semidirect

Maximal subgroups

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

subgroupindex
#351812
C2×#23492
#329592
#299732
#261252
#260592
#219862
#219552
#207382