Finite Group Databasebeta

← #7569885

Subgroups — #7569885

Order 512.

2085 conjugacy classes of subgroups.

The subgroup lattice has 627 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59864C211yesmaximal, characteristic, complemented
2#36260C211yesmaximal, characteristic
3#32036C212yesmaximal, complemented
4#32029C212yesmaximal, complemented
5#28753C212yesmaximal, complemented
6#28612C214yesmaximal, complemented
7#26242C214yesmaximal, complemented
8#25897C214yesmaximal, complemented
9#24055C212yesmaximal, complemented
10#24033C212yesmaximal, complemented
11#23175C211yesmaximal, characteristic, complemented
12#22419C214yesmaximal, complemented
13#21613C212yesmaximal, complemented
14C2×#547C2211yescharacteristic
15C2×#546C2214yes—
16C2×D42C2211yescharacteristic
17C2×#489C2214yescomplemented
18C2×#481C2212yes—
19C2×#483C2212yescomplemented
20C2×#474C2212yes—
21C2×#435C2214yescomplemented
22C2×#417C2212yes—
23C22×#102C2212yes—
24#2858C2211yescharacteristic
25#2856—21no—
26#2804—21no—
27#2804—21no—
28#2580—24no—
29#2571—21no—
30#2563C2214yescomplemented
31#2542C2211yescharacteristic
32#2541C2211yescharacteristic
33#2540C2214yes—
34#2539—24no—
35#2501—21no—
36#2487C2212yes—
37#2487C2212yescomplemented
38#2433—21no—
39#2366—21no—
40D4×C4⋊C4C2214yescomplemented
41C4×C4⋊Q8C2212yescomplemented
42C4×#127C2211yescharacteristic
43C2×#390—21no—
44C42×D4C2211yescharacteristic
45#2043C2214yescomplemented
46#2041C2214yes—
47#2036C2214yescomplemented
48#2036C2214yescomplemented
49#2035C2214yescomplemented
50#2034C2214yes—
51#2033C2214yescomplemented
52#2033C2214yescomplemented
53#1988C2212yescomplemented
54#1983C2214yescomplemented
55#1983C2214yescomplemented
56#1959C2214yescomplemented
57#1955C2214yescomplemented
58#1948C2212yescomplemented
59#1921C2214yescomplemented
60#1895C2214yescomplemented
61#1894C2212yescomplemented
62#1893C2212yes—
63#1892C2214yescomplemented
64#1874C2214yescomplemented
65#1872C2214yescomplemented
66#1870C2212yescomplemented
67#1869C2212yes—
68#1864C2214yes—
69#1802C2212yes—
70#1788C2212yes—
71#1788C2214yes—
72#1786C2212yes—
73#1786C2214yes—
74#1776C2214yescomplemented
75#1767C2212yescomplemented
76#1762C2214yes—
77C22×#41C2312yes—
78C23×D4—21no—
79C23×D4C2312yes—
80C22×M16C2312yes—
81C23×C8C2312yes—
82#547—42no—
83#547—42no—
84#546—216no—
85#546—48no—
86D42—42no—
87D42—42no—
88C2×C4⋊Q8C2311yescharacteristic
89C2×C4⋊Q8C2312yes—
90C2×#127—21no—
91C2×#127D411yescharacteristic
92C2×#127C2311yescharacteristic
93C2×#124C2314yes—
94C2×#123—24no—
95C2×#123—22no—
96C2×C22⋊Q8C2312yes—
97C2×C22⋊Q8C2314yes—
98C2×#121—21no—
99C2×#121—21no—
100C2×#121—24no—
101C2×#121C2312yes—
102C2×#121C2314yes—
103C2×#120—24no—
104C2×#120—22no—
105C2×C4×Q8—22no—
106C2×C4×D4—24no—
107C2×C4×D4C2311yescharacteristic
108C2×C4×D4C2312yes—
109C2×C4×D4C2314yes—
110C2×C4×D4C2311yescharacteristic
111C2×C4×D4C2311yescharacteristic
112C22×C4⋊C4—21no—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C22⋊C4—24no—
116C22×C42—21no—
117#489—48no—
118#483—44no—
119#481—44no—
120#474—28no—
121#435—48no—
122C2×#107C2314yes—
123C2×#106C2314yes—
124C2×C4⋊C8C2312yes—
125C2×C4⋊C8C2312yes—
126#417—44no—
127C2×#103C2314yes—
128C2×#103C2314yes—
129C2×#103C2314yes—
130C2×#103C2314yes—
131C2×#102—44no—
132C2×#102C2314yes—
133C2×#102C2314yes—
134C2×#102C2314yes—
135C2×#102C2314yes—
136C2×#102C2314yes—
137C2×#102C2314yes—
138C2×C22⋊C8C2312yes—
139C2×C22⋊C8C2312yes—
140C2×C22⋊C8C2314yes—
141C2×C22⋊C8C2314yes—
142C2×#97C2314yes—
143#400—22no—
144#400—22no—
145#399—24no—
146#399—22no—
147#399—22no—
148#399—22no—
149#393—24no—
150#393—24no—
151#392—24no—
152#392—22no—
153#391—22no—
154#390—24no—
155#390—21no—
156#390—21no—
157#390—24no—
158#390—24no—
159#390—21no—
160#390—21no—
161#390—22no—
162#390D414yes—
163#390D411yescharacteristic
164#390D411yescharacteristic
165#389—24no—
166#389—22no—
167#389C2314yes—
168#386—22no—
169#386—22no—
170#385—22no—
171#385—22no—
172#385—24no—
173#385—24no—
174#384—21no—
175#384D411yescharacteristic
176#384C2312yes—
177#382—24no—
178#380C2312yes—
179#380C2312yes—
180#380C2314yes—
181#380C2314yes—
182C4×C4⋊C4—22no—
183C4×C4⋊C4C2314yes—
184C4×C4⋊C4C2311yescharacteristic
185C4×C4⋊C4C2312yes—
186C4×C22⋊C4—21no—
187C4×C22⋊C4—22no—
188C43C2311yescharacteristic
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#336C2314yes—
194C25—22no—
195C2×#41—28no—
196C2×#41—24no—
197C2×#41—44no—
198C2×#41—42no—
199C22×Q8—22no—
200C22×Q8C2412yes—
201C22×D4—28no—
202C22×D4—24no—
203C22×D4—22no—
204C22×D4—22no—
205C22×D4—24no—
206C22×D4—22no—
207C22×D4—21no—
208C22×D4—21no—
209C22×D4—42no—
210C22×D4—44no—
211C22×D4—44no—
212C22×D4—44no—
213C22×D4—82no—
214C22×D4C2×D414yes—
215C22×D4C2×D411yescharacteristic
216C22×D4C2×D411yescharacteristic
217C22×D4C2412yes—
218C22×D4C2411yescharacteristic
219C23×C4—21no—
220C23×C4—22no—
221C23×C4—22no—
222C23×C4—22no—
223C23×C4—22no—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C2×M16—42no—
227C2×M16—44no—
228C22×C8—42no—
229C22×C8—44no—
230C22×C8C2414yes—
231C22×C8C2414yes—
232C4⋊Q8—24no—
233C4⋊Q8—24no—
234C4⋊Q8—41no—
235C4⋊Q8—41no—
236#127—24no—
237#127—41no—
238#127—41no—
239#127—82no—
240#124—28no—
241#124—28no—
242#123—416no—
243#123—48no—
244C22⋊Q8—28no—
245C22⋊Q8—24no—
246C22⋊Q8—44no—
247C22⋊Q8—42no—
248#121—28no—
249#121—24no—
250#121—416no—
251#121—42no—
252#121—44no—
253#121—44no—
254#121—44no—
255#120—416no—
256#120—48no—
257C4×Q8—84no—
258C4×D4—22no—
259C4×D4—22no—
260C4×D4—28no—
261C4×D4—24no—
262C4×D4—28no—
263C4×D4—41no—
264C4×D4—41no—
265C4×D4—41no—
266C4×D4—41no—
267C4×D4—88no—
268C4×D4#4118yes—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—42no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4—42no—
282C2×C4⋊C4—42no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4C2×D414yes—
285C2×C4⋊C4C2×D411yescharacteristic
286C2×C4⋊C4C2×D411yescharacteristic
287C2×C4⋊C4C2414yes—
288C2×C4⋊C4C2414yes—
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2411yescharacteristic
291C2×C22⋊C4—21no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—416no—
304C2×C22⋊C4C2×D414yes—
305C2×C22⋊C4C2×D414yes—
306C2×C22⋊C4C2×D412yes—
307C2×C22⋊C4C2×D412yes—
308C2×C42—24no—
309C2×C42—21no—
310C2×C42—22no—
311C2×C42—82no—
312C2×C42C2×D411yescharacteristic
313C2×C42C2×D412yes—
314C2×C42C2×D411yescharacteristic
315C2×C42C2411yescharacteristic
316C2×C42C2412yes—
317#107—28no—
318#106—28no—
319C4⋊C8—24no—
320C4⋊C8—24no—
321#103—44no—
322#103—44no—
323#103—44no—
324#103—44no—
325#102—28no—
326#102—28no—
327#102—44no—
328#102—44no—
329#102—44no—
330#102—44no—
331C22⋊C8—44no—
332C22⋊C8—44no—
333C22⋊C8—42no—
334C22⋊C8—42no—
335#97—28no—
336#95—21no—
337#95—22no—
338#95—24no—
339#95—24no—
340#95—22no—
341#95—22no—
342#95—24no—
343#95—24no—
344C24—21no—
345C24—22no—
346C24—48no—
347C24—44no—
348C24—44no—
349C24—82no—
350C24—84no—
351C24C22×D412yes—
352C24C22×D412yes—
353#41—48no—
354#41—88no—
355#41—84no—
356C2×Q8—24no—
357C2×Q8—22no—
358C2×Q8—24no—
359C2×Q8—22no—
360C2×Q8—44no—
361C2×Q8—82no—
362C2×Q8—82no—
363C2×D4—28no—
364C2×D4—28no—
365C2×D4—22no—
366C2×D4—24no—
367C2×D4—21no—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—24no—
371C2×D4—22no—
372C2×D4—41no—
373C2×D4—44no—
374C2×D4—41no—
375C2×D4—41no—
376C2×D4—41no—
377C2×D4—44no—
378C2×D4—44no—
379C2×D4—48no—
380C2×D4—42no—
381C2×D4—42no—
382C2×D4—48no—
383C2×D4—42no—
384C2×D4—48no—
385C2×D4—48no—
386C2×D4—44no—
387C2×D4—44no—
388C2×D4—42no—
389C2×D4—82no—
390C2×D4—88no—
391C2×D4—84no—
392C2×D4—816no—
393C2×D4—82no—
394C2×D4—84no—
395C2×D4—84no—
396C2×D4#13714yes—
397C2×D4C2×#4114yes—
398C2×D4H(5,𝔽2)12yes—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—28no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—42no—
415C22×C4—48no—
416C22×C4—44no—
417C22×C4—42no—
418C22×C4—82no—
419C22×C4—82no—
420C22×C4—82no—
421C22×C4—82no—
422C22×C4—84no—
423C22×C4—84no—
424C22×C4C22×D412yes—
425C22×C4C22×D412yes—
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D412yes—
428C22×C4C22×D412yes—
429C22×C4C22×D411yescharacteristic
430C22×C4C2511yescharacteristic
431M16—84no—
432C2×C8—24no—
433C2×C8—24no—
434C2×C8—24no—
435C2×C8—24no—
436C2×C8—24no—
437C2×C8—24no—
438C2×C8—84no—
439C4⋊C4—28no—
440C4⋊C4—22no—
441C4⋊C4—21no—
442C4⋊C4—24no—
443C4⋊C4—24no—
444C4⋊C4—24no—
445C4⋊C4—24no—
446C4⋊C4—41no—
447C4⋊C4—41no—
448C4⋊C4—44no—
449C4⋊C4—48no—
450C4⋊C4—42no—
451C4⋊C4—44no—
452C4⋊C4—42no—
453C4⋊C4—44no—
454C4⋊C4—82no—
455C4⋊C4—82no—
456C4⋊C4—82no—
457C4⋊C4—84no—
458C4⋊C4C2×#4114yes—
459C4⋊C4H(5,𝔽2)12yes—
460C22⋊C4—24no—
461C22⋊C4—28no—
462C22⋊C4—28no—
463C22⋊C4—28no—
464C22⋊C4—42no—
465C22⋊C4—42no—
466C22⋊C4—48no—
467C22⋊C4—44no—
468C22⋊C4—48no—
469C22⋊C4—44no—
470C22⋊C4—48no—
471C22⋊C4—48no—
472C22⋊C4—82no—
473C22⋊C4—82no—
474C22⋊C4—84no—
475C22⋊C4—84no—
476C42—22no—
477C42—21no—
478C42—21no—
479C42—41no—
480C42—42no—
481C42—42no—
482C42—48no—
483C42—82no—
484C42C2×#4114yes—
485C42H(5,𝔽2)14yes—
486C23—24no—
487C23—22no—
488C23—24no—
489C23—24no—
490C23—22no—
491C23—22no—
492C23—24no—
493C23—22no—
494C23—24no—
495C23—22no—
496C23—22no—
497C23—41no—
498C23—42no—
499C23—41no—
500C23—42no—
501C23—44no—
502C23—44no—
503C23—42no—
504C23—42no—
505C23—88no—
506C23—88no—
507C23—816no—
508C23—168no—
509C23—164no—
510C23C2×#13714yes—
511C23C23×D411yescharacteristic
512Q8—42no—
513Q8—44no—
514Q8—42no—
515Q8—88no—
516D4—24no—
517D4—22no—
518D4—44no—
519D4—416no—
520D4—42no—
521D4—44no—
522D4—42no—
523D4—82no—
524D4—84no—
525D4—88no—
526D4—88no—
527D4—88no—
528D4—168no—
529D4—162no—
530D4#55914yes—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—48no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—41no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—88no—
565C2×C4—88no—
566C2×C4—164no—
567C2×C4—164no—
568C2×C4#54614yes—
569C2×C4#54614yes—
570C2×C4C2×#13712yes—
571C2×C4C22×#4112yes—
572C2×C4C2×H(5,𝔽2)11yescharacteristic
573C2×C4C2×H(5,𝔽2)11yescharacteristic
574C2×C4C2×H(5,𝔽2)12yes—
575C8—44no—
576C8—44no—
577C22—24no—
578C22—24no—
579C22—28no—
580C22—42no—
581C22—44no—
582C22—44no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—44no—
587C22—42no—
588C22—44no—
589C22—42no—
590C22—42no—
591C22—44no—
592C22—82no—
593C22—84no—
594C22—82no—
595C22—82no—
596C22—84no—
597C22—1616no—
598C22C2×#54612yes—
599C22#348411yescharacteristic
600C22C22×#13711yescharacteristic
601C22#358511yescharacteristic
602C22#358512yes—
603C4—24no—
604C4—22no—
605C4—24no—
606C4—42no—
607C4—42no—
608C4—44no—
609C4—42no—
610C4—41no—
611C4—82no—
612C4—84no—
613C4—84no—
614C4C2×#55911yescharacteristic
615C4#350812yes—
616C4#354311yescharacteristic
617C2—44no—
618C2—44no—
619C2—82no—
620C2—162no—
621C2#5914311yescharacteristic
622C2#6116212yes—
623C2#6121812yes—
624C2#6157011yescharacteristic
625C2#6167311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36260C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#127 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #59864C2semidirect
3. #32036C2semidirect
4. #32029C2semidirect
5. #28753C2semidirect
6. #28612C2semidirect
7. #26242C2semidirect
8. #25897C2semidirect
9. #24055C2semidirect
10. #24033C2semidirect
11. #23175C2semidirect
12. #22419C2semidirect
13. #21613C2semidirect
17. C2×#489C22semidirect
19. C2×#483C22semidirect
21. C2×#435C22semidirect
30. #2563C22semidirect
37. #2487C22semidirect
40. D4×C4⋊C4C22semidirect
41. C4×C4⋊Q8C22semidirect
45. #2043C22semidirect
47. #2036C22semidirect
48. #2036C22semidirect
49. #2035C22semidirect
51. #2033C22semidirect
52. #2033C22semidirect
53. #1988C22semidirect
54. #1983C22semidirect
55. #1983C22semidirect
56. #1959C22semidirect
57. #1955C22semidirect
58. #1948C22semidirect
59. #1921C22semidirect
60. #1895C22semidirect
61. #1894C22semidirect
63. #1892C22semidirect
64. #1874C22semidirect
65. #1872C22semidirect
66. #1870C22semidirect
74. #1776C22semidirect
75. #1767C22semidirect

Maximal subgroups

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

subgroupindex
#598642
#362602
#320362
#320292
#287532
#286122
#262422
#258972
#240552
#240332
#231752
#224192
#216132