Finite Group Databasebeta

← #243969

Subgroups — #243969

Order 512.

1108 conjugacy classes of subgroups.

The subgroup lattice has 449 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21982C211yesmaximal, characteristic, complemented
2#21803C211yesmaximal, characteristic
3#21203C211yesmaximal, characteristic
4C2×#1957C211yesmaximal, characteristic
5#20816C211yesmaximal, characteristic, complemented
6#20764C211yesmaximal, characteristic, complemented
7#10678C212yesmaximal, complemented
8#10677C212yesmaximal, complemented
9#10672C212yesmaximal, complemented
10#10671C212yesmaximal, complemented
11#7848C211yesmaximal, characteristic, complemented
12C22×#127C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#418—22no—
15C22×#102—22no—
16C2×C4×M16C2211yescharacteristic
17C22×C4×C8C2211yescharacteristic
18#2285C2211yescharacteristic
19#2285C2211yescharacteristic
20C2×#384C2211yescharacteristic
21#1982—22no—
22#1961—22no—
23#1957—22no—
24#1957C2212yes—
25#1957C2212yes—
26#1957C2212yes—
27#1957C2212yes—
28#1957C2212yes—
29#1957C2212yes—
30#1947—22no—
31#1852—22no—
32#1845C2212yescomplemented
33#1844C2211yescharacteristic, complemented
34#1844C2211yescharacteristic, complemented
35#1836—22no—
36#1834C2212yescomplemented
37#1834C2212yescomplemented
38#1834C2211yescharacteristic, complemented
39#1834C2211yescharacteristic, complemented
40#1834C2212yescomplemented
41#1807—22no—
42#1786—22no—
43C4×#106—22no—
44#1736—22no—
45#1273C2212yescomplemented
46#1273C2212yescomplemented
47C23×D4—21no—
48C23×D4—21no—
49C23×D4—21no—
50C22×M16D412yes—
51C23×C8D412yes—
52C2×#127—21no—
53C2×#127—21no—
54C2×#127—21no—
55C2×#127—22no—
56C2×#127—22no—
57C2×#127—22no—
58C2×#127Q814yes—
59C2×#127C2312yes—
60C2×#117—21no—
61C2×#117—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C42C2311yescharacteristic
66#431—22no—
67#431—22no—
68C2×#107—24no—
69C2×#106—22no—
70C2×#106—22no—
71C2×C4⋊C8D412yes—
72#418—48no—
73C2×#102—24no—
74C2×#102—24no—
75C2×#102—24no—
76C2×#102—24no—
77C2×#102—24no—
78C2×#102—24no—
79C2×#102—48no—
80C2×C22⋊C8D412yes—
81C4×M16—22no—
82C4×M16—22no—
83C2×#97D412yes—
84C2×C4×C8—22no—
85C2×C4×C8—22no—
86C2×C4×C8D412yes—
87C2×C4×C8C2311yescharacteristic
88C2×C4×C8C2312yes—
89C2×C4×C8C2312yes—
90C2×C4×C8C2311yescharacteristic
91#384—21no—
92#384—21no—
93#384—21no—
94#384C2311yescharacteristic
95#384C2311yescharacteristic
96#384C2311yescharacteristic
97#384C2312yes—
98#384C2311yescharacteristic
99#382—21no—
100#382—21no—
101#380—21no—
102#380—21no—
103#380—21no—
104#380—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109#340—22no—
110#340—22no—
111#340—24no—
112#340—24no—
113#340—24no—
114C25—41no—
115C22×D4—24no—
116C22×D4—24no—
117C22×D4—24no—
118C22×D4—22no—
119C22×D4—22no—
120C22×D4—22no—
121C22×D4—44no—
122C22×D4—44no—
123C22×D4—42no—
124C22×D4—42no—
125C22×D4—42no—
126C22×D4—41no—
127C22×D4—41no—
128C22×D4—41no—
129C22×D4—44no—
130C22×D4—44no—
131C22×D4—82no—
132C22×D4—82no—
133C23×C4—41no—
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C2×M16—24no—
138C2×M16—24no—
139C2×M16—24no—
140C22×C8—24no—
141C22×C8—24no—
142C22×C8—42no—
143C22×C8C2×D412yes—
144C22×C8C2×D414yes—
145C22×C8C2×D414yes—
146C22×C8C2×D412yes—
147#127—24no—
148#127—24no—
149#127—24no—
150#127—22no—
151#127—22no—
152#127—22no—
153#127—42no—
154#127—42no—
155#127—42no—
156#127C2×Q814yes—
157#117—44no—
158#117—44no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—82no—
180C2×C22⋊C4—41no—
181C2×C22⋊C4—41no—
182C2×C42—41no—
183C2×C42—41no—
184C2×C42C2×D411yescharacteristic
185C2×C42C2×D411yescharacteristic
186C2×C42C2×D411yescharacteristic
187C2×C42C2×Q812yes—
188C2×C42#4112yes—
189C2×C42#4112yes—
190C2×C42#4112yes—
191C2×C42C2411yescharacteristic
192#107—84no—
193#106—44no—
194#106—44no—
195C4⋊C8—24no—
196#102—48no—
197#102—48no—
198#102—48no—
199#102—48no—
200#102—84no—
201#102—84no—
202C22⋊C8—24no—
203#97—24no—
204C4×C8—24no—
205C4×C8—21no—
206C4×C8—21no—
207C4×C8—21no—
208C4×C8—21no—
209C4×C8—22no—
210C4×C8—22no—
211C4×C8—22no—
212C4×C8—22no—
213#95—41no—
214#95—41no—
215#95—41no—
216#95—41no—
217C24—42no—
218C24—44no—
219C24—82no—
220C24—82no—
221C24—82no—
222C24—82no—
223C24—82no—
224C24—82no—
225C24#12711yescharacteristic
226C2×D4—24no—
227C2×D4—24no—
228C2×D4—24no—
229C2×D4—416no—
230C2×D4—42no—
231C2×D4—42no—
232C2×D4—42no—
233C2×D4—44no—
234C2×D4—44no—
235C2×D4—44no—
236C2×D4—44no—
237C2×D4—44no—
238C2×D4—44no—
239C2×D4—44no—
240C2×D4—44no—
241C2×D4—44no—
242C2×D4—416no—
243C2×D4—88no—
244C2×D4—82no—
245C2×D4—82no—
246C2×D4—88no—
247C2×D4—82no—
248C2×D4—82no—
249C2×D4—88no—
250C2×D4—88no—
251C2×D4—82no—
252C2×D4—82no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—42no—
262C22×C4—82no—
263C22×C4—82no—
264C22×C4—82no—
265C22×C4C22⋊Q812yes—
266C22×C4C22⋊Q812yes—
267C22×C4C22⋊Q812yes—
268C22×C4#12312yes—
269C22×C4#12312yes—
270C22×C4#12312yes—
271C22×C4#12711yescharacteristic
272C22×C4#12711yescharacteristic
273C22×C4#12711yescharacteristic
274C22×C4#13514yes—
275C22×C4AGL(1,ℤ/8)14yes—
276C22×C4C22×D411yescharacteristic
277C22×C4C22×D411yescharacteristic
278C22×C4C22×D411yescharacteristic
279M16—48no—
280C2×C8—22no—
281C2×C8—22no—
282C2×C8—22no—
283C2×C8—24no—
284C2×C8—24no—
285C2×C8—24no—
286C2×C8—22no—
287C2×C8—24no—
288C2×C8—22no—
289C2×C8—24no—
290C2×C8—24no—
291C2×C8—48no—
292C2×C8AGL(1,ℤ/8)14yes—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—44no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—44no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—44no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—82no—
309C4⋊C4—82no—
310C4⋊C4—82no—
311C22⋊C4—82no—
312C22⋊C4—82no—
313C42—82no—
314C42—82no—
315C42C22⋊Q812yes—
316C42C22⋊Q812yes—
317C42C22⋊Q812yes—
318C42#12312yes—
319C42#12312yes—
320C42#12312yes—
321C42C22×Q811yescharacteristic
322C42C2×#4111yescharacteristic
323C42C2×#4111yescharacteristic
324C42C2×#4111yescharacteristic
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—44no—
330C23—88no—
331C23—88no—
332C23—88no—
333C23—82no—
334C23—82no—
335C23—82no—
336C23—82no—
337C23—82no—
338C23—82no—
339C23—164no—
340C23—164no—
341C23—164no—
342C23—164no—
343C23#39912yes—
344C23#49512yes—
345C23#49612yes—
346C23C2×#12711yescharacteristic
347D4—48no—
348D4—48no—
349D4—48no—
350D4—88no—
351D4—88no—
352D4—84no—
353D4—84no—
354D4—84no—
355D4—88no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—24no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—82no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—164no—
380C2×C4#39912yes—
381C2×C4#39912yes—
382C2×C4#39912yes—
383C2×C4#47818yes—
384C2×C4#48218yes—
385C2×C4#49512yes—
386C2×C4#49612yes—
387C2×C4C2×C22⋊Q811yescharacteristic
388C2×C4C2×C22⋊Q811yescharacteristic
389C2×C4C2×C22⋊Q811yescharacteristic
390C2×C4C2×#12311yescharacteristic
391C2×C4C2×#12311yescharacteristic
392C2×C4C2×#12311yescharacteristic
393C2×C4C2×#13512yes—
394C2×C4C2×AGL(1,ℤ/8)12yes—
395C2×C4C2×AGL(1,ℤ/8)12yes—
396C2×C4#57512yes—
397C2×C4#57612yes—
398C2×C4#57712yes—
399C8—24no—
400C8—44no—
401C8—42no—
402C8—44no—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22—22no—
409C22—84no—
410C22—84no—
411C22—84no—
412C22—1616no—
413C22—162no—
414C22—162no—
415C22—162no—
416C22—162no—
417C22#209214yes—
418C22C2×#39911yescharacteristic
419C22C2×#49511yescharacteristic
420C22C2×#49611yescharacteristic
421C22#315111yescharacteristic
422C22#315311yescharacteristic
423C22#315411yescharacteristic
424C22#315511yescharacteristic
425C4—22no—
426C4—22no—
427C4—161no—
428C4—161no—
429C4—161no—
430C4—162no—
431C4—161no—
432C4#209214yes—
433C4C2×#47812yes—
434C4#307712yes—
435C4#308112yes—
436C4C2×#48212yes—
437C4#309412yes—
438C4#309612yes—
439C2—24no—
440C2—164no—
441C2#1962511yescharacteristic
442C2#1962711yescharacteristic
443C2#1967611yescharacteristic
444C2#1967811yescharacteristic
445C2C2×#209211yescharacteristic
446C2#2373111yescharacteristic
447C2#2373511yescharacteristic
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×#127C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #21982C2semidirect
5. #20816C2semidirect
6. #20764C2semidirect
7. #10678C2semidirect
8. #10677C2semidirect
9. #10672C2semidirect
10. #10671C2semidirect
11. #7848C2semidirect
32. #1845C22semidirect
33. #1844C22semidirect
34. #1844C22semidirect
36. #1834C22semidirect
37. #1834C22semidirect
38. #1834C22semidirect
39. #1834C22semidirect
40. #1834C22semidirect
45. #1273C22semidirect
46. #1273C22semidirect

Maximal subgroups

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

subgroupindex
#219822
#218032
#212032
C2×#19572
#208162
#207642
#106782
#106772
#106722
#106712
#78482