Finite Group Databasebeta

← #6532004

Subgroups — #6532004

Order 512.

1656 conjugacy classes of subgroups.

The subgroup lattice has 371 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2580C211yesmaximal, characteristic
2#34187C211yesmaximal, characteristic, complemented
3#27617C214yesmaximal, complemented
4#27588C214yesmaximal
5#25069C218yesmaximal, complemented
6#25003C218yesmaximal, complemented
7#21986C212yesmaximal, complemented
8#21822C211yesmaximal, characteristic, complemented
9#20738C212yesmaximal
10C22×C4×Q8C2211yescharacteristic
11C22×C4×D4C2211yescharacteristic
12C2×#432C2212yes—
13#2580—24no—
14#2580C2218yes—
15C2×#399—21no—
16C2×#393—21no—
17#2356—24no—
18#2355C2214yescomplemented
19#2355C2214yescomplemented
20#2313—22no—
21#2310—21no—
22C2×#390C2212yes—
23C2×#389C2212yes—
24C2×#385—22no—
25#2285C2212yes—
26#2283C2214yes—
27C2×#382C2211yescharacteristic
28#2274—21no—
29#2101C2218yescomplemented
30#2092C2218yes—
31#2085C2218yescomplemented
32#2073C2218yes—
33#1985C2218yescomplemented
34#1949C2218yescomplemented
35#1926C2218yes—
36#1925C2218yes—
37#1919C2218yescomplemented
38#1918C2218yescomplemented
39#1853C2214yes—
40#1852C2214yes—
41#1847C2214yescomplemented
42#1836C2214yescomplemented
43#1824C2214yes—
44#1817C2214yes—
45#1802C2214yes—
46#1802C2214yes—
47#1786C2214yes—
48#1762C2218yes—
49C23×Q8D411yescharacteristic
50C23×D4C2311yescharacteristic
51C24×C4—21no—
52C2×C4×Q8—28no—
53C2×C4×Q8D414yes—
54C2×C4×Q8C2314yes—
55C2×C4×D4—22no—
56C2×C4×D4—28no—
57C2×C4×D4C2314yes—
58C2×#117—22no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4D412yes—
61C22×C4⋊C4D411yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2312yes—
64C22×C22⋊C4—21no—
65C22×C22⋊C4—22no—
66C22×C22⋊C4—22no—
67C22×C42C2312yes—
68C22×C42C2311yescharacteristic
69#432—28no—
70C2×#107C2318yes—
71C2×#106C2318yes—
72C2×C4⋊C8C2314yes—
73C2×#102C2318yes—
74C2×#102C2318yes—
75C2×C22⋊C8C2314yes—
76C2×#97C2314yes—
77#399—28no—
78#393—28no—
79#391—24no—
80#391—24no—
81#390—24no—
82#390C2318yes—
83#389—24no—
84#389C2318yes—
85#387—24no—
86#387—24no—
87#387—24no—
88#387—24no—
89#385—28no—
90#385—28no—
91#384C2312yes—
92#384C2312yes—
93#383—24no—
94#382—22no—
95#382—22no—
96#382C2314yes—
97#382C2314yes—
98#382C2314yes—
99#381—24no—
100#380—22no—
101#380C2314yes—
102C4×C4⋊C4—24no—
103C4×C4⋊C4—24no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4C2314yes—
106C4×C4⋊C4C2314yes—
107C4×C4⋊C4C2314yes—
108C4×C22⋊C4—24no—
109#376—24no—
110C2×#95—21no—
111C2×#95—22no—
112C2×#95—21no—
113#340C2318yes—
114#340C2318yes—
115#340C2318yes—
116#340C2318yes—
117#336C2318yes—
118#336C2318yes—
119C25—21no—
120C22×Q8—24no—
121C22×Q8—44no—
122C22×Q8C2×D414yes—
123C22×D4—24no—
124C22×D4—21no—
125C22×D4—24no—
126C22×D4—42no—
127C22×D4C2412yes—
128C23×C4—22no—
129C23×C4—21no—
130C23×C4—24no—
131C23×C4—44no—
132C23×C4—84no—
133C23×C4C2×D412yes—
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D412yes—
136C23×C4C2×D411yescharacteristic
137C23×C4C2411yescharacteristic
138C22×C8C2418yes—
139C4×Q8—216no—
140C4×Q8—28no—
141C4×Q8—28no—
142C4×D4—28no—
143C4×D4—28no—
144C4×D4—48no—
145#117—48no—
146C2×C4⋊C4—28no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—28no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—44no—
157C2×C4⋊C4—84no—
158C2×C4⋊C4C2×D414yes—
159C2×C4⋊C4C2×D414yes—
160C2×C4⋊C4C2×D414yes—
161C2×C4⋊C4C2412yes—
162C2×C4⋊C4C2414yes—
163C2×C4⋊C4C2414yes—
164C2×C4⋊C4C2414yes—
165C2×C22⋊C4—24no—
166C2×C22⋊C4—24no—
167C2×C22⋊C4—28no—
168C2×C22⋊C4—28no—
169C2×C22⋊C4—48no—
170C2×C22⋊C4—44no—
171C2×C22⋊C4—84no—
172C2×C42—28no—
173C2×C42—24no—
174C2×C42—24no—
175C2×C42—24no—
176C2×C42C2×D414yes—
177C2×C42C2×D412yes—
178C2×C42#4118yes—
179C2×C42C2414yes—
180C2×C42C2412yes—
181#107—48no—
182#106—48no—
183C4⋊C8—28no—
184#102—216no—
185#102—48no—
186C22⋊C8—28no—
187#97—28no—
188#95—24no—
189#95—24no—
190#95—28no—
191#95—24no—
192#95—24no—
193#95—24no—
194#95—24no—
195#95—24no—
196#95—24no—
197#95—24no—
198#95—24no—
199C24—24no—
200C24—22no—
201C24—44no—
202C24—42no—
203C24—42no—
204C24—44no—
205C24C22×D411yescharacteristic
206C2×Q8—28no—
207C2×Q8—44no—
208C2×Q8—44no—
209C2×Q8—416no—
210C2×D4—24no—
211C2×D4—24no—
212C2×D4—22no—
213C2×D4—24no—
214C2×D4—416no—
215C2×D4—42no—
216C2×D4—42no—
217C2×D42−1+414yes—
218C22×C4—24no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—24no—
226C22×C4—24no—
227C22×C4—24no—
228C22×C4—22no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—44no—
233C22×C4—44no—
234C22×C4—44no—
235C22×C4—42no—
236C22×C4—44no—
237C22×C4—48no—
238C22×C4—84no—
239C22×C4—84no—
240C22×C4—816no—
241C22×C4#12314yes—
242C22×C4#12314yes—
243C22×C4C22×D414yes—
244C22×C4C22×D414yes—
245C22×C4C22×D411yescharacteristic
246C22×C4C22×D412yes—
247C22×C4C22×D412yes—
248C22×C4C2×#4114yes—
249C22×C4C2511yescharacteristic
250C2×C8—28no—
251C2×C8—28no—
252C2×C8—28no—
253C4⋊C4—28no—
254C4⋊C4—28no—
255C4⋊C4—28no—
256C4⋊C4—24no—
257C4⋊C4—28no—
258C4⋊C4—28no—
259C4⋊C4—28no—
260C4⋊C4—44no—
261C4⋊C4—44no—
262C4⋊C4—44no—
263C4⋊C4—42no—
264C4⋊C4—82no—
265C4⋊C4—82no—
266C4⋊C4—88no—
267C22⋊C4—416no—
268C22⋊C4—48no—
269C22⋊C4—48no—
270C22⋊C4—88no—
271C42—28no—
272C42—28no—
273C42—48no—
274C42—84no—
275C42—84no—
276C42#12318yes—
277C42C2×#4114yes—
278C42C2×#4114yes—
279C23—22no—
280C23—21no—
281C23—21no—
282C23—24no—
283C23—48no—
284C23—416no—
285C23—44no—
286C23—42no—
287C23—42no—
288C23—44no—
289C23—84no—
290C23—88no—
291C23—84no—
292C23C2×#12312yes—
293C23#57514yes—
294C23C23×D411yescharacteristic
295Q8—416no—
296D4—28no—
297D4—48no—
298D4—44no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—28no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—28no—
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—44no—
316C2×C4—44no—
317C2×C4—44no—
318C2×C4—44no—
319C2×C4—42no—
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×C4—84no—
330C2×C4C2×#12312yes—
331C2×C4C2×#12314yes—
332C2×C4C2×#12314yes—
333C2×C4C22×#4112yes—
334C2×C4C2×2−1+412yes—
335C2×C4#58512yes—
336C8—48no—
337C22—22no—
338C22—22no—
339C22—24no—
340C22—24no—
341C22—44no—
342C22—48no—
343C22—816no—
344C22—82no—
345C22—82no—
346C22—84no—
347C22#309314yes—
348C22C22×#12311yescharacteristic
349C22#348611yescharacteristic
350C22#348711yescharacteristic
351C22C2×#57512yes—
352C22#358512yes—
353C4—22no—
354C4—24no—
355C4—24no—
356C4—44no—
357C4—44no—
358C4—82no—
359C4—84no—
360C4—84no—
361C4—84no—
362C4#349514yes—
363C2—24no—
364C2—84no—
365C2#3567211yescharacteristic
366C2C2×#309311yescharacteristic
367C2#6051811yescharacteristic
368C2#6183112yes—
369C2#6189512yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34187C2semidirect
3. #27617C2semidirect
5. #25069C2semidirect
6. #25003C2semidirect
7. #21986C2semidirect
8. #21822C2semidirect
18. #2355C22semidirect
19. #2355C22semidirect
29. #2101C22semidirect
31. #2085C22semidirect
33. #1985C22semidirect
34. #1949C22semidirect
37. #1919C22semidirect
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
C2×#25802
#341872
#276172
#275882
#250692
#250032
#219862
#218222
#207382