Finite Group Databasebeta

← #7487761

Subgroups — #7487761

Order 512.

2443 conjugacy classes of subgroups.

The subgroup lattice has 653 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59816C211yesmaximal, characteristic, complemented
2#39491C211yesmaximal, characteristic, complemented
3#31411C214yesmaximal, complemented
4#31270C212yesmaximal, complemented
5#30510C212yesmaximal, complemented
6#30417C214yesmaximal, complemented
7#30039C212yesmaximal, complemented
8#29364C212yesmaximal, complemented
9#24448C214yesmaximal, complemented
10#23976C212yesmaximal, complemented
11#23864C212yesmaximal, complemented
12#23860C214yesmaximal, complemented
13#23357C211yesmaximal, characteristic, complemented
14C22×AGL(1,ℤ/8)C2212yescomplemented
15C2×#555C2211yescharacteristic, complemented
16C2×#547C2211yescharacteristic, complemented
17C2×D42—21no—
18C2×D42C2214yes—
19C2×#468C2214yescomplemented
20C2×#449C2214yescomplemented
21C2×#450C2214yescomplemented
22C2×#450C2212yescomplemented
23C2×#447C2212yescomplemented
24C2×#418C2212yescomplemented
25#2862C2211yescharacteristic, complemented
26#2710—24no—
27#2679—21no—
28#2679—21no—
29#2619—21no—
30#2563C2211yescharacteristic, complemented
31#2542C2211yescharacteristic, complemented
32#2537—21no—
33#2537C2214yes—
34#2504—21no—
35#2500—21no—
36#2415C2214yescomplemented
37#2408—24no—
38#2408C2212yes—
39#2408C2212yes—
40#2407C2214yescomplemented
41#2317—21no—
42C4×#127C2212yes—
43C4×#127C2211yescharacteristic
44#2293C2211yescharacteristic, complemented
45#2099C2212yescomplemented
46#2093C2212yescomplemented
47#2076C2214yescomplemented
48#2072C2214yescomplemented
49#2041C2212yes—
50#2036C2214yescomplemented
51#2033C2214yescomplemented
52#2023C2214yescomplemented
53#2014C2214yes—
54#2013C2214yescomplemented
55#2011C2212yes—
56#2000C2214yescomplemented
57#2000C2214yescomplemented
58#1999C2214yescomplemented
59#1925C2214yescomplemented
60#1895C2214yescomplemented
61#1895C2212yescomplemented
62#1894C2212yescomplemented
63#1879C2214yescomplemented
64#1879C2214yescomplemented
65#1877C2214yescomplemented
66#1877C2214yescomplemented
67#1875C2214yescomplemented
68#1870C2212yescomplemented
69#1870C2212yescomplemented
70#1870C2214yescomplemented
71#1870C2214yescomplemented
72#1862C2214yescomplemented
73#1839C2212yescomplemented
74#1825C2212yescomplemented
75#1788C2212yescomplemented
76#1786C2214yescomplemented
77C22×#41C2312yes—
78C23×D4—24no—
79C23×D4C2312yes—
80C23×D4C2312yes—
81C2×AGL(1,ℤ/8)—44no—
82C2×AGL(1,ℤ/8)—48no—
83C22×SD16C2314yes—
84C22×SD16C2314yes—
85C22×D8C2314yes—
86C22×M16C2312yes—
87C22×M16C2312yes—
88#555—42no—
89#555—42no—
90#547—42no—
91#547—42no—
92D42—48no—
93D42—48no—
94D42—48no—
95C2×C4⋊Q8C2311yescharacteristic
96C2×C4⋊Q8C2311yescharacteristic
97C2×#127D411yescharacteristic
98C2×#127C2312yes—
99C2×#124—22no—
100C2×#123—22no—
101C2×C22⋊Q8C2312yes—
102C2×C22⋊Q8C2314yes—
103C2×#121—21no—
104C2×#121—21no—
105C2×#121—24no—
106C2×#121—24no—
107C2×#121C2314yes—
108C2×#121C2312yes—
109C2×#121C2314yes—
110C2×#120—22no—
111C2×#120—22no—
112C2×#120—24no—
113C2×#120—24no—
114C2×C4×D4—22no—
115C2×C4×D4—22no—
116C2×C4×D4C2311yescharacteristic
117C2×C4×D4C2314yes—
118C2×C4×D4C2311yescharacteristic
119C2×C4×D4C2314yes—
120C2×C4×D4C2311yescharacteristic
121C2×#117—21no—
122C2×#117—21no—
123C2×#117C2312yes—
124C22×C4⋊C4—21no—
125#468—48no—
126#450—44no—
127#450—48no—
128#449—48no—
129#447—44no—
130C2×#107C2314yes—
131#418—44no—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#102C2314yes—
135C2×#102C2314yes—
136C2×#102C2314yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×#102C2314yes—
140C2×C22⋊C8C2314yes—
141C2×C22⋊C8C2312yes—
142C2×C22⋊C8C2314yes—
143C2×C22⋊C8C2312yes—
144#399—22no—
145#397—24no—
146#394—24no—
147#394—22no—
148#394—22no—
149#394—24no—
150#394—24no—
151#394—24no—
152#393—22no—
153#392—24no—
154#392—22no—
155#392—22no—
156#391—22no—
157#391C2314yes—
158#390—21no—
159#390—21no—
160#390—22no—
161#390—22no—
162#390D414yes—
163#390D411yescharacteristic
164#390D411yescharacteristic
165#390C2314yes—
166#390C2312yes—
167#389—24no—
168#389C2312yes—
169#388—24no—
170#388—22no—
171#386—24no—
172#385—24no—
173#384—21no—
174#384D411yescharacteristic
175#384C2312yes—
176#382—21no—
177#382—21no—
178#382—21no—
179#380—24no—
180#380C2314yes—
181#380C2312yes—
182#380C2312yes—
183#379—22no—
184#379—22no—
185C4×C4⋊C4C2311yescharacteristic
186C4×C22⋊C4—21no—
187C43C2311yescharacteristic
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#336C2314yes—
193#336C2312yes—
194#336C2312yes—
195#336C2314yes—
196C25—22no—
197C25—22no—
198C2×#41—24no—
199C2×#41—42no—
200C2×#41—44no—
201C2×#41—44no—
202C22×Q8—21no—
203C22×Q8C2412yes—
204C22×D4—28no—
205C22×D4—28no—
206C22×D4—24no—
207C22×D4—24no—
208C22×D4—22no—
209C22×D4—22no—
210C22×D4—22no—
211C22×D4—22no—
212C22×D4—22no—
213C22×D4—22no—
214C22×D4—42no—
215C22×D4—42no—
216C22×D4—44no—
217C22×D4—44no—
218C22×D4—42no—
219C22×D4—42no—
220C22×D4—416no—
221C22×D4—416no—
222C22×D4—88no—
223C22×D4C2×D411yescharacteristic
224C22×D4C2×D411yescharacteristic
225C22×D4C2×D414yes—
226C22×D4C2414yes—
227C22×D4C2411yescharacteristic
228C23×C4—21no—
229C23×C4—24no—
230C23×C4—22no—
231C23×C4C2412yes—
232C23×C4C2412yes—
233AGL(1,ℤ/8)—432no—
234C2×SD16—28no—
235C2×SD16—28no—
236C2×SD16—44no—
237C2×SD16—44no—
238C2×SD16—44no—
239C2×SD16—44no—
240C2×D8—28no—
241C2×D8—28no—
242C2×D8—44no—
243C2×M16—42no—
244C2×M16—44no—
245C2×M16—42no—
246C2×M16—44no—
247C22×C8C2414yes—
248C22×C8C2414yes—
249C4⋊Q8—41no—
250C4⋊Q8—41no—
251C4⋊Q8—41no—
252C4⋊Q8—41no—
253#127—28no—
254#127—24no—
255#124—48no—
256#123—48no—
257C22⋊Q8—24no—
258C22⋊Q8—42no—
259C22⋊Q8—44no—
260C22⋊Q8—44no—
261#121—28no—
262#121—28no—
263#121—24no—
264#121—44no—
265#121—44no—
266#121—42no—
267#121—44no—
268#121—44no—
269#121—416no—
270#121—416no—
271#120—416no—
272#120—416no—
273#120—48no—
274#120—48no—
275C4×D4—22no—
276C4×D4—22no—
277C4×D4—44no—
278C4×D4—44no—
279C4×D4—44no—
280C4×D4—44no—
281C4×D4—41no—
282C4×D4—41no—
283C4×D4—41no—
284C4×D4—41no—
285C4×D4—48no—
286C4×D4—84no—
287#117—24no—
288#117—42no—
289#117—44no—
290#117—44no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4C2×D411yescharacteristic
301C2×C4⋊C4C2×D411yescharacteristic
302C2×C4⋊C4C2×D414yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2414yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2411yescharacteristic
307C2×C22⋊C4—21no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4C2×D414yes—
323C2×C22⋊C4C2×D414yes—
324C2×C22⋊C4C2×D412yes—
325C2×C22⋊C4C2×D412yes—
326C2×C42—22no—
327C2×C42—21no—
328C2×C42C2×D412yes—
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42C2412yes—
332C2×C42C2411yescharacteristic
333#107—44no—
334#103—44no—
335#103—44no—
336#102—28no—
337#102—28no—
338#102—44no—
339#102—44no—
340#102—44no—
341#102—44no—
342C22⋊C8—42no—
343C22⋊C8—44no—
344C22⋊C8—42no—
345C22⋊C8—44no—
346#95—22no—
347#95—21no—
348#95—22no—
349#95—22no—
350#95—24no—
351#95—24no—
352#95—24no—
353C24—21no—
354C24—24no—
355C24—48no—
356C24—48no—
357C24—44no—
358C24—44no—
359C24—82no—
360C24—82no—
361C24—84no—
362C24—84no—
363C24—82no—
364C24—82no—
365C24C22×D412yes—
366C24C22×D412yes—
367#41—48no—
368#41—84no—
369#41—88no—
370C2×Q8—24no—
371C2×Q8—22no—
372C2×Q8—22no—
373C2×Q8—24no—
374C2×Q8—42no—
375C2×Q8—82no—
376C2×D4—28no—
377C2×D4—28no—
378C2×D4—28no—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—24no—
382C2×D4—24no—
383C2×D4—24no—
384C2×D4—24no—
385C2×D4—24no—
386C2×D4—21no—
387C2×D4—24no—
388C2×D4—44no—
389C2×D4—44no—
390C2×D4—44no—
391C2×D4—41no—
392C2×D4—41no—
393C2×D4—41no—
394C2×D4—41no—
395C2×D4—48no—
396C2×D4—48no—
397C2×D4—48no—
398C2×D4—48no—
399C2×D4—48no—
400C2×D4—48no—
401C2×D4—44no—
402C2×D4—48no—
403C2×D4—48no—
404C2×D4—44no—
405C2×D4—44no—
406C2×D4—82no—
407C2×D4—82no—
408C2×D4—82no—
409C2×D4—82no—
410C2×D4—88no—
411C2×D4—88no—
412C2×D4—84no—
413C2×D4—84no—
414C2×D4—84no—
415C2×D4—84no—
416C2×D4—84no—
417C2×D4—84no—
418C2×D4—864no—
419C2×D4H(5,𝔽2)18yes—
420C2×D4H(5,𝔽2)12yes—
421C22×C4—24no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—42no—
436C22×C4—44no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4—42no—
440C22×C4—48no—
441C22×C4—84no—
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D412yes—
447C22×C4C22×D411yescharacteristic
448C22×C4C2511yescharacteristic
449SD16—416no—
450SD16—88no—
451D8—416no—
452M16—48no—
453M16—84no—
454C2×C8—24no—
455C2×C8—24no—
456C2×C8—24no—
457C2×C8—24no—
458C2×C8—24no—
459C2×C8—24no—
460C4⋊C4—21no—
461C4⋊C4—24no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—41no—
467C4⋊C4—41no—
468C4⋊C4—44no—
469C4⋊C4—44no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—44no—
474C4⋊C4—44no—
475C4⋊C4—82no—
476C4⋊C4—84no—
477C4⋊C4H(5,𝔽2)12yes—
478C22⋊C4—24no—
479C22⋊C4—44no—
480C22⋊C4—44no—
481C22⋊C4—42no—
482C22⋊C4—42no—
483C22⋊C4—44no—
484C22⋊C4—44no—
485C22⋊C4—48no—
486C22⋊C4—48no—
487C22⋊C4—44no—
488C22⋊C4—48no—
489C22⋊C4—44no—
490C22⋊C4—44no—
491C22⋊C4—44no—
492C22⋊C4—48no—
493C22⋊C4—82no—
494C22⋊C4—84no—
495C22⋊C4—84no—
496C22⋊C4—84no—
497C42—21no—
498C42—21no—
499C42—24no—
500C42—41no—
501C42—42no—
502C42—44no—
503C42—42no—
504C42AGL(1,ℤ/8)14yescomplemented
505C23—24no—
506C23—24no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—28no—
513C23—24no—
514C23—24no—
515C23—22no—
516C23—22no—
517C23—44no—
518C23—42no—
519C23—44no—
520C23—41no—
521C23—44no—
522C23—44no—
523C23—42no—
524C23—44no—
525C23—41no—
526C23—44no—
527C23—88no—
528C23—88no—
529C23—816no—
530C23—816no—
531C23—168no—
532C23—168no—
533C23—164no—
534C23—164no—
535C23—164no—
536C23—164no—
537C23C23×D411yescharacteristic
538Q8—42no—
539Q8—42no—
540Q8—44no—
541Q8—162no—
542D4—24no—
543D4—28no—
544D4—28no—
545D4—416no—
546D4—416no—
547D4—44no—
548D4—44no—
549D4—42no—
550D4—82no—
551D4—82no—
552D4—88no—
553D4—88no—
554D4—88no—
555D4—816no—
556D4—816no—
557D4—88no—
558D4—164no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—48no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—816no—
593C2×C4—88no—
594C2×C4C2×AGL(1,ℤ/8)12yes—
595C2×C4C2×AGL(1,ℤ/8)14yes—
596C2×C4C2×H(5,𝔽2)11yescharacteristic
597C2×C4C2×H(5,𝔽2)14yes—
598C2×C4C2×H(5,𝔽2)11yescharacteristic
599C8—44no—
600C8—44no—
601C22—42no—
602C22—42no—
603C22—42no—
604C22—48no—
605C22—44no—
606C22—44no—
607C22—48no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—48no—
612C22—44no—
613C22—44no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22—84no—
618C22—82no—
619C22—84no—
620C22—82no—
621C22—84no—
622C22—84no—
623C22—1616no—
624C22—1616no—
625C22#348412yes—
626C22#348411yescharacteristic
627C22C22×AGL(1,ℤ/8)11yescharacteristic
628C22#358512yes—
629C22#358611yescharacteristic
630C4—24no—
631C4—22no—
632C4—42no—
633C4—42no—
634C4—42no—
635C4—42no—
636C4—41no—
637C4—82no—
638C4—82no—
639C4—84no—
640C4—82no—
641C4#353212yes—
642C4#353712yes—
643C2—82no—
644C2—82no—
645C2—82no—
646C2—88no—
647C2#5200411yescharacteristic
648C2#6158212yes—
649C2#6158511yescharacteristic
650C2#6167012yes—
651C2#6167311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
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. #59816C2semidirect
2. #39491C2semidirect
3. #31411C2semidirect
4. #31270C2semidirect
5. #30510C2semidirect
6. #30417C2semidirect
7. #30039C2semidirect
8. #29364C2semidirect
9. #24448C2semidirect
10. #23976C2semidirect
11. #23864C2semidirect
12. #23860C2semidirect
13. #23357C2semidirect
14. C22×AGL(1,ℤ/8)C22semidirect
15. C2×#555C22semidirect
16. C2×#547C22semidirect
19. C2×#468C22semidirect
20. C2×#449C22semidirect
21. C2×#450C22semidirect
22. C2×#450C22semidirect
23. C2×#447C22semidirect
24. C2×#418C22semidirect
25. #2862C22semidirect
30. #2563C22semidirect
31. #2542C22semidirect
36. #2415C22semidirect
40. #2407C22semidirect
44. #2293C22semidirect
45. #2099C22semidirect
46. #2093C22semidirect
47. #2076C22semidirect
48. #2072C22semidirect
50. #2036C22semidirect
51. #2033C22semidirect
52. #2023C22semidirect
54. #2013C22semidirect
56. #2000C22semidirect
57. #2000C22semidirect
58. #1999C22semidirect
59. #1925C22semidirect
60. #1895C22semidirect
61. #1895C22semidirect
62. #1894C22semidirect
63. #1879C22semidirect
64. #1879C22semidirect
65. #1877C22semidirect
66. #1877C22semidirect
67. #1875C22semidirect
68. #1870C22semidirect
69. #1870C22semidirect
70. #1870C22semidirect
71. #1870C22semidirect
72. #1862C22semidirect
73. #1839C22semidirect
74. #1825C22semidirect
75. #1788C22semidirect
76. #1786C22semidirect
504. C42AGL(1,ℤ/8)semidirect

Maximal subgroups

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

subgroupindex
#598162
#394912
#314112
#312702
#305102
#304172
#300392
#293642
#244482
#239762
#238642
#238602
#233572