Finite Group Databasebeta

← #7487729

Subgroups — #7487729

Order 512.

2291 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59817C211yesmaximal, characteristic, complemented
2#39362C211yesmaximal, characteristic
3#31386C214yesmaximal, complemented
4#31262C212yesmaximal, complemented
5#30506C212yesmaximal, complemented
6#30404C214yesmaximal, complemented
7#30041C212yesmaximal, complemented
8#29368C212yesmaximal, complemented
9#24429C214yesmaximal, complemented
10#23961C212yesmaximal, complemented
11#23855C214yesmaximal, complemented
12#23849C212yesmaximal, complemented
13#23355C211yesmaximal, characteristic, complemented
14C23×SD16C2212yes—
15C2×D4×Q8C2214yescomplemented
16C2×#547C2211yescharacteristic
17C2×D42C2211yescharacteristic
18C2×#535—21no—
19C2×#465C2214yescomplemented
20C2×#448C2212yes—
21C2×#448C2212yes—
22C2×#448C2214yescomplemented
23C2×#450C2214yes—
24C22×#103C2212yescomplemented
25#2858C2211yescharacteristic
26#2683—21no—
27#2683—21no—
28#2678—24no—
29#2620—21no—
30#2551—21no—
31#2547C2211yescharacteristic
32#2541C2211yescharacteristic
33#2538C2214yescomplemented
34#2452—24no—
35#2435—21no—
36#2431—21no—
37#2416C2214yes—
38#2409C2212yes—
39#2409C2212yescomplemented
40#2408C2214yes—
41#2366—21no—
42C4×C4⋊Q8C2212yescomplemented
43C4×#127C2211yescharacteristic
44#2293C2211yescharacteristic
45#2100C2212yescomplemented
46#2094C2212yes—
47#2077C2214yescomplemented
48#2071C2214yescomplemented
49#2043C2212yescomplemented
50#2035C2214yescomplemented
51#2034C2214yescomplemented
52#2024C2214yescomplemented
53#2014C2214yes—
54#2013C2212yes—
55#2013C2214yescomplemented
56#2000C2214yes—
57#2000C2214yes—
58#2000C2214yescomplemented
59#1927C2214yescomplemented
60#1893C2212yes—
61#1893C2212yes—
62#1892C2214yescomplemented
63#1880C2214yes—
64#1880C2214yescomplemented
65#1880C2214yescomplemented
66#1877C2214yes—
67#1872C2214yescomplemented
68#1869C2212yes—
69#1869C2212yescomplemented
70#1869C2214yescomplemented
71#1869C2214yescomplemented
72#1865C2214yescomplemented
73#1839C2212yescomplemented
74#1823C2212yescomplemented
75#1787C2214yescomplemented
76#1786C2212yes—
77C22×#41—21no—
78C23×Q8C2312yes—
79C23×Q8C2312yes—
80C23×D4—21no—
81C23×D4C2312yes—
82C22×SD16—44no—
83C22×SD16—48no—
84C22×SD16C2314yes—
85C22×SD16C2314yes—
86C22×SD16C2314yes—
87C23×C8C2312yes—
88C23×C8C2312yes—
89D4×Q8—48no—
90D4×Q8—48no—
91#547—42no—
92#547—42no—
93D42—42no—
94D42—42no—
95#535—48no—
96C2×C4⋊Q8—24no—
97C2×C4⋊Q8C2312yes—
98C2×C4⋊Q8C2311yescharacteristic
99C2×#127—21no—
100C2×#127D411yescharacteristic
101C2×#127C2311yescharacteristic
102C2×#124—21no—
103C2×#124—21no—
104C2×#124—21no—
105C2×#123—22no—
106C2×C22⋊Q8—21no—
107C2×C22⋊Q8—24no—
108C2×C22⋊Q8—24no—
109C2×C22⋊Q8C2314yes—
110C2×C22⋊Q8C2314yes—
111C2×#121—24no—
112C2×#121—21no—
113C2×#121C2312yes—
114C2×#121C2312yes—
115C2×#121C2314yes—
116C2×#120—24no—
117C2×#120—22no—
118C2×C4×Q8C2314yes—
119C2×C4×D4—24no—
120C2×C4×D4C2311yescharacteristic
121C2×C4×D4C2311yescharacteristic
122C2×C4×D4C2314yes—
123C2×C4×D4C2311yescharacteristic
124C2×#117—21no—
125C22×C4⋊C4C2312yes—
126#465—48no—
127#450—48no—
128#448—44no—
129#448—48no—
130#448—44no—
131C2×#106C2314yes—
132C2×#103—44no—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#102C2314yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×C22⋊C8C2314yes—
142C2×C22⋊C8C2312yes—
143C2×C22⋊C8C2312yes—
144C2×C22⋊C8C2314yes—
145#397—22no—
146#396—24no—
147#396—24no—
148#396—24no—
149#395—22no—
150#395—22no—
151#394—24no—
152#394—24no—
153#393—22no—
154#393—22no—
155#392—24no—
156#392—22no—
157#391—21no—
158#391C2314yes—
159#390—24no—
160#390—21no—
161#390—22no—
162#390D414yes—
163#390D411yescharacteristic
164#390D411yescharacteristic
165#390C2314yes—
166#390C2312yes—
167#389—24no—
168#389—22no—
169#389—22no—
170#389C2312yes—
171#388—22no—
172#386—22no—
173#386—22no—
174#386—24no—
175#385—22no—
176#385—22no—
177#384D411yescharacteristic
178#384C2312yes—
179#380C2312yes—
180#380C2314yes—
181#380C2312yes—
182#379—24no—
183C4×C4⋊C4—22no—
184C4×C4⋊C4—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—
197C2×#41—44no—
198C2×#41—44no—
199C2×#41—82no—
200C22×Q8—21no—
201C22×Q8—22no—
202C22×Q8—22no—
203C22×Q8—28no—
204C22×Q8—28no—
205C22×Q8—44no—
206C22×Q8—44no—
207C22×Q8—42no—
208C22×Q8—42no—
209C22×Q8—42no—
210C22×Q8—42no—
211C22×Q8C2414yes—
212C22×D4—24no—
213C22×D4—22no—
214C22×D4—22no—
215C22×D4—24no—
216C22×D4—21no—
217C22×D4—21no—
218C22×D4—22no—
219C22×D4—44no—
220C22×D4—42no—
221C22×D4—44no—
222C22×D4—44no—
223C22×D4—44no—
224C22×D4—82no—
225C22×D4C2×D411yescharacteristic
226C22×D4C2×D414yes—
227C22×D4C2×D411yescharacteristic
228C22×D4C2412yes—
229C22×D4C2411yescharacteristic
230C23×C4—21no—
231C23×C4—22no—
232C23×C4—22no—
233C23×C4—22no—
234C23×C4C2412yes—
235C23×C4C2412yes—
236C2×SD16—28no—
237C2×SD16—28no—
238C2×SD16—28no—
239C2×SD16—28no—
240C2×SD16—44no—
241C2×SD16—44no—
242C2×SD16—44no—
243C2×SD16—44no—
244C2×SD16—44no—
245C2×SD16—432no—
246C22×C8—44no—
247C22×C8—42no—
248C22×C8—42no—
249C22×C8—44no—
250C22×C8C2414yes—
251C22×C8C2414yes—
252C4⋊Q8—28no—
253C4⋊Q8—41no—
254C4⋊Q8—41no—
255C4⋊Q8—416no—
256#127—41no—
257#127—41no—
258#127—44no—
259#127SD1618yescomplemented
260#124—44no—
261#124—44no—
262#124—82no—
263#123—48no—
264C22⋊Q8—28no—
265C22⋊Q8—28no—
266C22⋊Q8—44no—
267C22⋊Q8—44no—
268C22⋊Q8—416no—
269C22⋊Q8—416no—
270C22⋊Q8—44no—
271#121—24no—
272#121—24no—
273#121—42no—
274#121—42no—
275#121—44no—
276#121—416no—
277#121—44no—
278#121—44no—
279#120—48no—
280#120—416no—
281C4×Q8—44no—
282C4×Q8—44no—
283C4×D4—22no—
284C4×D4—22no—
285C4×D4—41no—
286C4×D4—41no—
287C4×D4—41no—
288C4×D4—41no—
289C4×D4—44no—
290C4×D4—44no—
291C4×D4—416no—
292#117—44no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—42no—
305C2×C4⋊C4C2×D411yescharacteristic
306C2×C4⋊C4C2×D411yescharacteristic
307C2×C4⋊C4C2×D414yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2414yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2411yescharacteristic
312C2×C22⋊C4—21no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4C2×D412yes—
326C2×C22⋊C4C2×D414yes—
327C2×C22⋊C4C2×D414yes—
328C2×C22⋊C4C2×D412yes—
329C2×C42—21no—
330C2×C42—24no—
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D412yes—
333C2×C42C2×D411yescharacteristic
334C2×C42C2411yescharacteristic
335C2×C42C2412yes—
336#106—44no—
337#103—28no—
338#103—28no—
339#103—44no—
340#103—44no—
341#103—44no—
342#102—44no—
343#102—44no—
344#102—44no—
345C22⋊C8—44no—
346C22⋊C8—42no—
347C22⋊C8—42no—
348C22⋊C8—44no—
349#95—22no—
350#95—24no—
351#95—24no—
352#95—21no—
353#95—24no—
354#95—22no—
355#95—22no—
356C24—21no—
357C24—22no—
358C24—44no—
359C24—44no—
360C24—84no—
361C24—82no—
362C24—84no—
363C24C22×D412yes—
364C24C22×D412yes—
365#41—816no—
366C2×Q8—24no—
367C2×Q8—24no—
368C2×Q8—24no—
369C2×Q8—24no—
370C2×Q8—24no—
371C2×Q8—48no—
372C2×Q8—48no—
373C2×Q8—42no—
374C2×Q8—42no—
375C2×Q8—42no—
376C2×Q8—48no—
377C2×Q8—48no—
378C2×Q8—82no—
379C2×Q8—82no—
380C2×Q8—88no—
381C2×Q8—88no—
382C2×Q8—84no—
383C2×Q8—84no—
384C2×Q8H(5,𝔽2)18yes—
385C2×D4—24no—
386C2×D4—21no—
387C2×D4—24no—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—24no—
391C2×D4—44no—
392C2×D4—41no—
393C2×D4—41no—
394C2×D4—41no—
395C2×D4—44no—
396C2×D4—41no—
397C2×D4—44no—
398C2×D4—44no—
399C2×D4—42no—
400C2×D4—42no—
401C2×D4—42no—
402C2×D4—48no—
403C2×D4—44no—
404C2×D4—44no—
405C2×D4—44no—
406C2×D4—48no—
407C2×D4—42no—
408C2×D4—84no—
409C2×D4—82no—
410C2×D4—84no—
411C2×D4—88no—
412C2×D4—816no—
413C2×D4—82no—
414C2×D4—82no—
415C2×D4—84no—
416C2×D4—84no—
417C2×D4C2×SD1614yes—
418C2×D4C2×SD16116yescomplemented
419C2×D4C2×SD1614yescomplemented
420C2×D4H(5,𝔽2)12yes—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—42no—
436C22×C4—44no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4—42no—
440C22×C4—44no—
441C22×C4—82no—
442C22×C4—82no—
443C22×C4—82no—
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D412yes—
450C22×C4C2511yescharacteristic
451SD16—416no—
452SD16—416no—
453SD16—88no—
454C2×C8—24no—
455C2×C8—24no—
456C2×C8—24no—
457C2×C8—24no—
458C2×C8—24no—
459C2×C8—24no—
460C2×C8—48no—
461C2×C8—84no—
462C4⋊C4—28no—
463C4⋊C4—28no—
464C4⋊C4—21no—
465C4⋊C4—24no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—41no—
471C4⋊C4—41no—
472C4⋊C4—44no—
473C4⋊C4—44no—
474C4⋊C4—48no—
475C4⋊C4—48no—
476C4⋊C4—44no—
477C4⋊C4—48no—
478C4⋊C4—42no—
479C4⋊C4—44no—
480C4⋊C4—82no—
481C4⋊C4H(5,𝔽2)12yes—
482C22⋊C4—24no—
483C22⋊C4—42no—
484C22⋊C4—44no—
485C22⋊C4—44no—
486C22⋊C4—42no—
487C22⋊C4—44no—
488C22⋊C4—48no—
489C22⋊C4—48no—
490C22⋊C4—44no—
491C22⋊C4—48no—
492C22⋊C4—44no—
493C22⋊C4—48no—
494C22⋊C4—82no—
495C22⋊C4—84no—
496C22⋊C4—84no—
497C22⋊C4—82no—
498C22⋊C4—82no—
499C42—21no—
500C42—21no—
501C42—24no—
502C42—42no—
503C42—41no—
504C42—48no—
505C42—42no—
506C42C2×SD1614yescomplemented
507C23—22no—
508C23—22no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—41no—
517C23—42no—
518C23—42no—
519C23—41no—
520C23—42no—
521C23—44no—
522C23—42no—
523C23—44no—
524C23—42no—
525C23—88no—
526C23—88no—
527C23—168no—
528C23—164no—
529C23—168no—
530C23C22×SD1614yes—
531C23C22×SD1614yes—
532C23C23×D411yescharacteristic
533Q8—28no—
534Q8—28no—
535Q8—44no—
536Q8—44no—
537Q8—84no—
538Q8—88no—
539Q8—88no—
540D4—24no—
541D4—44no—
542D4—416no—
543D4—44no—
544D4—42no—
545D4—42no—
546D4—42no—
547D4—44no—
548D4—84no—
549D4—88no—
550D4—88no—
551D4—88no—
552D4—162no—
553D4—168no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—28no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—48no—
570C2×C4—41no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—88no—
590C2×C4—88no—
591C2×C4—88no—
592C2×C4C22×SD1612yes—
593C2×C4C22×SD1614yes—
594C2×C4C2×H(5,𝔽2)11yescharacteristic
595C2×C4C2×H(5,𝔽2)14yes—
596C2×C4C2×H(5,𝔽2)11yescharacteristic
597C8—44no—
598C8—44no—
599C22—24no—
600C22—24no—
601C22—28no—
602C22—28no—
603C22—42no—
604C22—42no—
605C22—44no—
606C22—44no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—82no—
613C22—82no—
614C22—82no—
615C22—82no—
616C22—84no—
617C22—84no—
618C22—82no—
619C22—1616no—
620C22#348411yescharacteristic
621C22#348412yes—
622C22C23×SD1611yescharacteristic
623C22#358511yescharacteristic
624C22#358612yes—
625C4—22no—
626C4—24no—
627C4—42no—
628C4—42no—
629C4—48no—
630C4—42no—
631C4—42no—
632C4—41no—
633C4—82no—
634C4—82no—
635C4—84no—
636C4#353212yes—
637C4#353712yes—
638C2—44no—
639C2—44no—
640C2—82no—
641C2—162no—
642C2#5200411yescharacteristic
643C2#6156011yescharacteristic
644C2#6156212yes—
645C2#6167112yes—
646C2#6167411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#39362C2
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. #59817C2semidirect
3. #31386C2semidirect
4. #31262C2semidirect
5. #30506C2semidirect
6. #30404C2semidirect
7. #30041C2semidirect
8. #29368C2semidirect
9. #24429C2semidirect
10. #23961C2semidirect
11. #23855C2semidirect
12. #23849C2semidirect
13. #23355C2semidirect
15. C2×D4×Q8C22semidirect
19. C2×#465C22semidirect
22. C2×#448C22semidirect
24. C22×#103C22semidirect
33. #2538C22semidirect
39. #2409C22semidirect
42. C4×C4⋊Q8C22semidirect
45. #2100C22semidirect
47. #2077C22semidirect
48. #2071C22semidirect
49. #2043C22semidirect
50. #2035C22semidirect
51. #2034C22semidirect
52. #2024C22semidirect
55. #2013C22semidirect
58. #2000C22semidirect
59. #1927C22semidirect
62. #1892C22semidirect
64. #1880C22semidirect
65. #1880C22semidirect
67. #1872C22semidirect
69. #1869C22semidirect
70. #1869C22semidirect
71. #1869C22semidirect
72. #1865C22semidirect
73. #1839C22semidirect
74. #1823C22semidirect
75. #1787C22semidirect
259. #127SD16semidirect
418. C2×D4C2×SD16semidirect
419. C2×D4C2×SD16semidirect
506. C42C2×SD16semidirect

Maximal subgroups

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

subgroupindex
#598172
#393622
#313862
#312622
#305062
#304042
#300412
#293682
#244292
#239612
#238552
#238492
#233552