Finite Group Databasebeta

← #6528520

Subgroups — #6528520

Order 512.

1748 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35167C211yesmaximal, characteristic, complemented
2C2×#2563C211yesmaximal, characteristic
3#29109C214yesmaximal
4#27969C214yesmaximal, complemented
5#27821C218yesmaximal, complemented
6#27735C218yesmaximal, complemented
7#21834C212yesmaximal, complemented
8#20732C212yesmaximal
9#20536C211yesmaximal, characteristic, complemented
10C2×#544—21no—
11C22×C4⋊Q8C2211yescharacteristic
12C22×C4×D4C2211yescharacteristic
13C2×#431C2212yes—
14#2857C2214yes—
15#2659—22no—
16#2563—24no—
17#2563C2218yes—
18#2552C2214yescomplemented
19#2551C2214yescomplemented
20#2550—24no—
21C2×#393—22no—
22#2316—21no—
23C2×#390C2212yes—
24#2285C2212yes—
25C2×#382C2212yes—
26C2×#380—21no—
27C2×#380—21no—
28C2×C4×C4⋊C4C2211yescharacteristic
29#2101C22116yescomplemented
30#2071C2218yes—
31#2023C2218yes—
32#1993C2218yescomplemented
33#1990C2214yescomplemented
34#1990C2214yescomplemented
35#1987C2218yescomplemented
36#1936C22116yes—
37#1836C2218yescomplemented
38#1824C2218yes—
39#1823C2214yes—
40#1802C2214yes—
41#1802C2214yes—
42#1786C2214yes—
43#1770C2218yescomplemented
44C4×#102C2218yes—
45#1749C2214yes—
46C23×Q8—21no—
47C23×D4C2311yescharacteristic
48C24×C4—21no—
49#544—48no—
50C2×C4⋊Q8—24no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—28no—
53C2×C4⋊Q8C2314yes—
54C2×C4⋊Q8C2314yes—
55C2×#125—24no—
56C2×C22⋊Q8—24no—
57C2×C4×D4—22no—
58C2×C4×D4—28no—
59C2×C4×D4C2314yes—
60C2×#117—22no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4D411yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2312yes—
66C22×C22⋊C4—21no—
67C22×C22⋊C4—22no—
68C22×C22⋊C4—22no—
69C22×C42C2312yes—
70C22×C42C2311yescharacteristic
71#431—28no—
72C2×C4⋊C8C2314yes—
73C2×#102C2318yes—
74C2×#102C2318yes—
75C2×C22⋊C8C2314yes—
76C2×#97C2318yes—
77C2×C4×C8C2314yes—
78#398—28no—
79#395—28no—
80#393—216no—
81#391—24no—
82#390—24no—
83#390C2318yes—
84#389—24no—
85#389—24no—
86#389—24no—
87#389C2318yes—
88#384C2312yes—
89#384C2312yes—
90#384C2314yes—
91#382—24no—
92#382—22no—
93#382C2318yes—
94#380—24no—
95#380—22no—
96#380—28no—
97#380—28no—
98#380C2314yes—
99C4×C4⋊C4—22no—
100C4×C4⋊C4D414yes—
101C4×C4⋊C4C2314yes—
102C4×C4⋊C4C2314yes—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95D412yes—
106#340C2318yes—
107#340C2318yes—
108#340C23116yes—
109#340C23116yes—
110#336C2318yes—
111C25—21no—
112C22×Q8—24no—
113C22×Q8—24no—
114C22×Q8—44no—
115C22×Q8—82no—
116C22×Q8—82no—
117C22×D4—24no—
118C22×D4—21no—
119C22×D4—24no—
120C22×D4—42no—
121C22×D4C2412yes—
122C23×C4—22no—
123C23×C4—21no—
124C23×C4—24no—
125C23×C4—44no—
126C23×C4—44no—
127C23×C4—44no—
128C23×C4C2×D412yes—
129C23×C4C2×D412yes—
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C23×C4C2411yescharacteristic
133C22×C8C2418yes—
134C4⋊Q8—28no—
135C4⋊Q8—28no—
136C4⋊Q8—48no—
137C4⋊Q8—416no—
138C4⋊Q8—48no—
139#125—416no—
140C22⋊Q8—416no—
141C4×D4—28no—
142C4×D4—28no—
143C4×D4—48no—
144#117—48no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—28no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—28no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—44no—
161C2×C4⋊C4C2×D414yes—
162C2×C4⋊C4C2×D414yes—
163C2×C4⋊C4C2412yes—
164C2×C4⋊C4C2414yes—
165C2×C4⋊C4C2414yes—
166C2×C4⋊C4C2414yes—
167C2×C22⋊C4—24no—
168C2×C22⋊C4—24no—
169C2×C22⋊C4—28no—
170C2×C22⋊C4—28no—
171C2×C22⋊C4—44no—
172C2×C22⋊C4—84no—
173C2×C22⋊C4—84no—
174C2×C42—24no—
175C2×C42—44no—
176C2×C42C2×D414yes—
177C2×C42C2×D412yes—
178C2×C42#4118yes—
179C2×C42C2414yes—
180C2×C42C2412yes—
181C4⋊C8—28no—
182#102—216no—
183#102—48no—
184C22⋊C8—28no—
185#97—48no—
186C4×C8—28no—
187#95—28no—
188#95—28no—
189#95—24no—
190#95—24no—
191#95—28no—
192#95C2×D418yes—
193C24—24no—
194C24—22no—
195C24—42no—
196C24—42no—
197C24—42no—
198C24—42no—
199C24—42no—
200C24—42no—
201C24C22×D411yescharacteristic
202C2×Q8—48no—
203C2×Q8—48no—
204C2×Q8—84no—
205C2×Q8—84no—
206C2×Q8—84no—
207C2×Q8—84no—
208C2×Q8—816no—
209C2×D4—24no—
210C2×D4—24no—
211C2×D4—22no—
212C2×D4—24no—
213C2×D4—416no—
214C2×D4—42no—
215C2×D4—42no—
216C2×D42−1+414yes—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—24no—
220C22×C4—24no—
221C22×C4—22no—
222C22×C4—24no—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—44no—
232C22×C4—44no—
233C22×C4—42no—
234C22×C4—44no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—48no—
238C22×C4—48no—
239C22×C4—48no—
240C22×C4—816no—
241C22×C4#12414yes—
242C22×C4#12414yes—
243C22×C4AGL(1,ℤ/8)14yes—
244C22×C4#13714yes—
245C22×C4C22×D414yes—
246C22×C4C22×D414yes—
247C22×C4C22×D411yescharacteristic
248C22×C4C22×D412yes—
249C22×C4C22×D412yes—
250C22×C4C2×#4114yes—
251C22×C4C2511yescharacteristic
252C2×C8—28no—
253C2×C8—28no—
254C2×C8—28no—
255C4⋊C4—28no—
256C4⋊C4—28no—
257C4⋊C4—24no—
258C4⋊C4—28no—
259C4⋊C4—28no—
260C4⋊C4—28no—
261C4⋊C4—48no—
262C4⋊C4—416no—
263C4⋊C4—42no—
264C4⋊C4—44no—
265C4⋊C4—48no—
266C4⋊C4—44no—
267C4⋊C4—44no—
268C4⋊C4—82no—
269C4⋊C4—82no—
270C4⋊C4—84no—
271C4⋊C4—84no—
272C22⋊C4—48no—
273C22⋊C4—48no—
274C22⋊C4—88no—
275C22⋊C4—88no—
276C42—28no—
277C42—44no—
278C42—48no—
279C42#12418yes—
280C42C2×#4114yes—
281C42C2×#4114yes—
282C23—21no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—24no—
287C23—48no—
288C23—48no—
289C23—48no—
290C23—42no—
291C23—42no—
292C23—42no—
293C23—42no—
294C23—42no—
295C23—42no—
296C23—84no—
297C23—84no—
298C23—84no—
299C23—84no—
300C23C2×#12412yes—
301C23C2×AGL(1,ℤ/8)12yes—
302C23C2×#13712yes—
303C23C23×D411yescharacteristic
304Q8—816no—
305Q8—168no—
306D4—28no—
307D4—48no—
308D4—44no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—28no—
314C2×C4—24no—
315C2×C4—24no—
316C2×C4—24no—
317C2×C4—24no—
318C2×C4—24no—
319C2×C4—24no—
320C2×C4—44no—
321C2×C4—44no—
322C2×C4—44no—
323C2×C4—44no—
324C2×C4—44no—
325C2×C4—42no—
326C2×C4—44no—
327C2×C4—44no—
328C2×C4—44no—
329C2×C4—44no—
330C2×C4—44no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—84no—
334C2×C4—84no—
335C2×C4—88no—
336C2×C4—84no—
337C2×C4—816no—
338C2×C4#489116yes—
339C2×C4C2×#12412yes—
340C2×C4C2×#12414yes—
341C2×C4C2×#12414yes—
342C2×C4C2×AGL(1,ℤ/8)14yes—
343C2×C4C2×#13714yes—
344C2×C4C22×#4112yes—
345C2×C4C2×2−1+412yes—
346C2×C4#58512yes—
347C8—48no—
348C22—22no—
349C22—22no—
350C22—22no—
351C22—22no—
352C22—22no—
353C22—22no—
354C22—44no—
355C22—44no—
356C22—44no—
357C22—816no—
358C22—82no—
359C22—82no—
360C22—82no—
361C22—82no—
362C22C2×#48914yes—
363C22C22×#12411yescharacteristic
364C22#348511yescharacteristic
365C22#348611yescharacteristic
366C22C22×AGL(1,ℤ/8)11yescharacteristic
367C22C22×#13711yescharacteristic
368C22#358511yescharacteristic
369C22#358511yescharacteristic
370C4—22no—
371C4—24no—
372C4—44no—
373C4—44no—
374C4—84no—
375C4—84no—
376C4—82no—
377C4—84no—
378C4C2×#48918yes—
379C4#349514yes—
380C2—24no—
381C2—84no—
382C2#3564411yescharacteristic
383C2C22×#48911yescharacteristic
384C2#6061911yescharacteristic
385C2#6172011yescharacteristic
386C2#6172111yescharacteristic
387C2#6182511yescharacteristic
388C2#6182611yescharacteristic
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. #35167C2semidirect
4. #27969C2semidirect
5. #27821C2semidirect
6. #27735C2semidirect
7. #21834C2semidirect
9. #20536C2semidirect
18. #2552C22semidirect
19. #2551C22semidirect
29. #2101C22semidirect
32. #1993C22semidirect
33. #1990C22semidirect
34. #1990C22semidirect
35. #1987C22semidirect
37. #1836C22semidirect
43. #1770C22semidirect

Maximal subgroups

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

subgroupindex
#351672
C2×#25632
#291092
#279692
#278212
#277352
#218342
#207322
#205362