Finite Group Databasebeta

← #10583754

Subgroups — #10583754

Order 512.

6576 conjugacy classes of subgroups.

The subgroup lattice has 665 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#63040C211yesmaximal, characteristic, complemented
2#63039C211yesmaximal, characteristic, complemented
3#62529C213yesmaximal, complemented
4#62525C213yesmaximal, complemented
5#62342C219yesmaximal, complemented
6#62150C213yesmaximal, complemented
7#62149C213yesmaximal, complemented
8#62001C213yesmaximal, complemented
9#61999C213yesmaximal, complemented
10#61788C219yesmaximal, complemented
11#61668C219yesmaximal, complemented
12#61436C213yesmaximal, complemented
13#61434C213yesmaximal, complemented
14#61215C219yesmaximal, complemented
15#60308C211yesmaximal, characteristic, complemented
16#3549—29no—
17#3549C2219yes—
18#3548C2219yescomplemented
19#3547—29no—
20#3517—23no—
21#3511C2219yescomplemented
22#3503C2213yescomplemented
23#3499C2213yescomplemented
24#3498—23no—
25#3498—23no—
26#3498C2219yes—
27#3498C2213yes—
28#3498C2213yes—
29#3497—23no—
30#3483—21no—
31#3483C2213yes—
32#3483C2213yes—
33#3483C2211yescharacteristic
34#3483C2213yescomplemented
35#3482C2213yescomplemented
36Q8×#41—21no—
37#3392C2213yescomplemented
38#3391C2213yescomplemented
39Q8×Q16C2213yescomplemented
40Q8×SD16C2213yescomplemented
41#3363C2211yescharacteristic, complemented
42#3362C2216yescomplemented
43#3362C2213yescomplemented
44#3362C2211yescharacteristic, complemented
45#3358C2216yescomplemented
46#3358C2213yescomplemented
47#3357C22118yescomplemented
48#3354C22118yescomplemented
49#3341C2219yescomplemented
50#3340C2219yescomplemented
51#3333C22118yescomplemented
52#3332C22118yescomplemented
53#3325C22118yescomplemented
54#3324C22118yescomplemented
55#3305C2219yescomplemented
56#3305C2219yescomplemented
57#3304C2219yescomplemented
58#3303C2219yescomplemented
59#3295C22118yescomplemented
60#3295C22118yescomplemented
61#3294C22118yescomplemented
62#3294C22118yescomplemented
63#3287C2219yescomplemented
64#3284C2219yescomplemented
65#3274C22118yescomplemented
66#3272C22118yescomplemented
67#3254C2219yescomplemented
68#3253C2219yescomplemented
69#3227C2219yescomplemented
70#3226C2219yescomplemented
71#3208C2219yescomplemented
72#3205C2219yescomplemented
73#3198C22136yescomplemented
74#3169C2219yescomplemented
75#3167C2219yescomplemented
76C2×#486C2219yescomplemented
77#3107C2219yescomplemented
78#3106C2219yescomplemented
79#3106C2219yescomplemented
80#3106C2219yescomplemented
81#3096C22118yescomplemented
82#3079C2219yescomplemented
83#3062C22118yescomplemented
84#3038C2219yescomplemented
85#3012C22118yescomplemented
86#2991C2213yescomplemented
87#2991C2213yescomplemented
88#2979C2219yescomplemented
89C8×#41C2213yescomplemented
90C8×#41C2213yescomplemented
91C4×#135C2219yescomplemented
92#2931C2219yescomplemented
93#2897C2216yescomplemented
94C2×2−1+4—23no—
95C2×H(5,𝔽2)—23no—
96C2×H(5,𝔽2)—23no—
97C2×H(5,𝔽2)C2316yes—
98C2×AGL(1,ℤ/8)C23118yes—
99C2×#135C2319yes—
100C2×#131C2316yes—
101#562—218no—
102H(3,𝔽4)—218no—
103Q82D411yescharacteristic
104Q82C2313yes—
105Q82C2311yescharacteristic
106Q82C2311yescharacteristic
107#557—23no—
108#557C2311yescharacteristic
109#556C2319yes—
110#555—29no—
111#555—29no—
112#555—29no—
113#555—29no—
114#555D419yes—
115#555C2319yes—
116#555C2319yes—
117#555C2319yes—
118#554—29no—
119#554C23118yes—
120#553—218no—
121#552—29no—
122#550—23no—
123#550—23no—
124#550—23no—
125#550—29no—
126#550—23no—
127#550—23no—
128#550D413yes—
129#550D413yes—
130#550C2319yes—
131#550C2319yes—
132#550C2319yes—
133#550C2313yes—
134#550C2313yes—
135#550C2311yescharacteristic
136D4×Q8—23no—
137D4×Q8—29no—
138D4×Q8C2313yes—
139#548—23no—
140#548—218no—
141#547—29no—
142#547C2319yes—
143#547C23118yes—
144D42—29no—
145D42—218no—
146D42—218no—
147D42—29no—
148D42—29no—
149D42C23118yes—
150D42C23118yes—
151#544—29no—
152#542—218no—
153#540—218no—
154#537—218no—
155#537C23118yes—
156#536—29no—
157#535—218no—
158#535—218no—
159#535—218no—
160#535—29no—
161#535—29no—
162#535C23118yes—
163#533—29no—
164#533C2319yes—
165#532—29no—
166#532—29no—
167#532—29no—
168#532—29no—
169#532—29no—
170#532C2319yes—
171#532C2319yes—
172#532C2319yes—
173#532C2319yes—
174C2×C4⋊Q8C2319yes—
175C2×#127—29no—
176C2×#127C2319yes—
177#529—29no—
178#520—23no—
179#520C2316yes—
180C4×#41—23no—
181C4×#41—23no—
182C4×#41—23no—
183C4×#41C2319yes—
184C4×#41C2319yes—
185C4×#41C2313yes—
186C4×#41C2313yes—
187C2×C4×Q8—23no—
188#500C2319yes—
189#498C2319yes—
190#493C2319yes—
191#492C2319yes—
192#487C2313yes—
193#486C23118yes—
194#486C23118yes—
195#486C2319yes—
196#486C2319yes—
197#486C2313yes—
198#485C23118yes—
199#482C23118yes—
200#482C23118yes—
201#482C23118yes—
202#481C23136yes—
203#480C23136yes—
204#477C2319yes—
205#477C2313yes—
206#477C2319yes—
207#476C23118yes—
208#476C2313yes—
209#476C2319yes—
210#476C2319yes—
211#475C2319yes—
212#474C23118yes—
213#474C2319yes—
214#469C23136yes—
215#468C23136yes—
216#466C23118yes—
217#466C23118yes—
218#465C23118yes—
219#465C23118yes—
220#462C2313yes—
221#461C2319yes—
222#460C23118yes—
223#460C2313yes—
224#460C2319yes—
225#460C2319yes—
226#459C23118yes—
227#459C2319yes—
228#459C2319yes—
229#459C2319yes—
230#452C23118yes—
231#450C23136yes—
232#449C23118yes—
233#449C23118yes—
234#449C23118yes—
235#447C23136yes—
236C8×Q8C2311yescharacteristic
237C8×Q8C2313yes—
238C8×Q8C2311yescharacteristic
239C8×Q8C2313yes—
240C4×Q16C2319yes—
241C4×Q16C2313yes—
242C4×SD16C2319yes—
243C4×SD16C2319yes—
244C4×SD16C23118yes—
245C4×SD16C2319yes—
246C4×SD16C2313yes—
247C4×D8C2319yes—
248C4×D8C23118yes—
249C4×D8C2319yes—
250#436C23118yes—
251#436C23118yes—
252C8×D4C2319yes—
253C8×D4C2319yes—
254C8×D4C2319yes—
255C8×D4C2319yes—
256#431C2319yes—
257#431C2319yes—
258#428C23118yes—
259#427C2319yes—
260#423C2319yes—
261#418C23118yes—
262#417C23118yes—
263#416C2319yes—
264#416C2319yes—
265C2×#102C23118yes—
266C2×#102C23118yes—
267#408C23118yes—
268C2×C4×C8C2319yes—
2692−1+4—812no—
270H(5,𝔽2)—412no—
271H(5,𝔽2)—86no—
272H(5,𝔽2)—812no—
273H(5,𝔽2)—812no—
274C2×#41—21no—
275C2×#41—23no—
276C2×#41—23no—
277C2×#41—26no—
278C2×#41—218no—
279C2×#41—29no—
280C2×#41—29no—
281C2×#41—29no—
282C2×#41C2×D413yes—
283C2×#41C2×D413yes—
284C2×#41C2419yes—
285C2×#41C2419yes—
286C2×#41C2416yes—
287C22×Q8—23no—
288C22×Q8—29no—
289C22×D4—218no—
290C22×D4—218no—
291C22×D4—218no—
292C22×D4—29no—
293C22×D4—29no—
294C22×D4C24118yes—
295AGL(1,ℤ/8)—818no—
296#135—418no—
297C2×Q16C2413yes—
298C2×SD16C2413yes—
299C2×SD16C2419yes—
300C2×SD16C24118yes—
301C2×D8C2419yes—
302C2×D8C24118yes—
303#131—86no—
304C2×M16C24118yes—
305C22×C8C2419yes—
306C22×C8C2419yes—
307C4⋊Q8—23no—
308C4⋊Q8—23no—
309C4⋊Q8—29no—
310C4⋊Q8—29no—
311C4⋊Q8—29no—
312C4⋊Q8C2×D419yes—
313C4⋊Q8C2413yes—
314C4⋊Q8C2419yes—
315C4⋊Q8C2419yes—
316C4⋊Q8C2413yes—
317C4⋊Q8C2413yes—
318#127—29no—
319#127—29no—
320#127—218no—
321#127—29no—
322#127—29no—
323#127—29no—
324#127C2×D419yes—
325#127C2419yes—
326#127C2419yes—
327#127C2413yes—
328#126—218no—
329#125—23no—
330#125—218no—
331#125—29no—
332#124—218no—
333#124—29no—
334#124—218no—
335#124—218no—
336#124—218no—
337#124—29no—
338#124—29no—
339#124—29no—
340#124C2×D4118yes—
341#123—218no—
342#123—236no—
343C22⋊Q8—218no—
344C22⋊Q8—29no—
345C22⋊Q8—29no—
346C22⋊Q8—218no—
347C22⋊Q8—236no—
348C22⋊Q8C24118yes—
349#121—29no—
350#121—29no—
351#121—29no—
352#121—29no—
353#121—218no—
354#121—236no—
355#121—29no—
356#121—29no—
357#121—29no—
358#121—23no—
359#121—29no—
360#121—29no—
361#121—218no—
362#121—218no—
363#121—218no—
364#121—218no—
365#121—218no—
366#121C2×D4118yes—
367#121C2×D4118yes—
368#121C24118yes—
369#121C24118yes—
370#121C24118yes—
371#120—236no—
372#120—236no—
373#120—236no—
374C4×Q8—21no—
375C4×Q8—21no—
376C4×Q8—21no—
377C4×Q8—23no—
378C4×Q8—26no—
379C4×Q8—29no—
380C4×Q8—29no—
381C4×Q8C2×D413yes—
382C4×Q8C2×D413yes—
383C4×Q8C2413yes—
384C4×Q8C2413yes—
385C4×Q8C2413yes—
386C4×Q8C2419yes—
387C4×Q8C2411yescharacteristic
388C4×Q8C2411yescharacteristic
389C4×D4—218no—
390C4×D4—218no—
391C4×D4—29no—
392C4×D4—29no—
393C4×D4—29no—
394C4×D4—29no—
395C4×D4—29no—
396C4×D4—29no—
397C4×D4—29no—
398C4×D4—29no—
399C4×D4—29no—
400C4×D4—29no—
401C4×D4C2×D419yes—
402C4×D4C2×D419yes—
403C4×D4C2413yes—
404C4×D4C2419yes—
405C4×D4C2419yes—
406C4×D4C2419yes—
407C4×D4C2419yes—
408C4×D4C2419yes—
409#117—29no—
410#117—29no—
411#117—29no—
412#117—29no—
413#117—29no—
414#117—29no—
415#117C2×D419yes—
416#117C2×D419yes—
417#117C2419yes—
418#117C2419yes—
419C2×C4⋊C4—29no—
420C2×C4⋊C4—29no—
421C2×C4⋊C4C24118yes—
422C2×C42—29no—
423C2×C42—29no—
424C2×C42C2419yes—
425#107C2413yes—
426#107C2419yes—
427#107C24118yes—
428#106C2413yes—
429#106C2419yes—
430#106C24118yes—
431C4⋊C8C2419yes—
432C4⋊C8C2419yes—
433C4⋊C8C2413yes—
434C4⋊C8C2413yes—
435#103C2411yescharacteristic
436#103C2413yes—
437#103C2419yes—
438#103C2419yes—
439#103C2416yes—
440#102C2419yes—
441#102C24118yes—
442#102C24118yes—
443#102C24136yes—
444#102C2413yes—
445#102C2419yes—
446#102C2419yes—
447#102C2416yes—
448C22⋊C8C24118yes—
449C22⋊C8C24118yes—
450C4×C8C2413yes—
451C4×C8C2413yes—
452C4×C8C2419yes—
453C4×C8C2419yes—
454C24—218no—
455#41—212no—
456#41—212no—
457#41—218no—
458#41—46no—
459#41—46no—
460#41—49no—
461#41—49no—
462#41—49no—
463#41—412no—
464#41—44no—
465#41—412no—
466#41—818no—
467#41—818no—
468#41—818no—
469#41—812no—
470#41—836no—
471C2×Q8—29no—
472C2×Q8—29no—
473C2×Q8—23no—
474C2×Q8—23no—
475C2×Q8—412no—
476C2×Q8—818no—
477C2×Q8C22×D416yes—
478C2×Q8C22×D411yescharacteristic
479C2×Q8C22×D411yescharacteristic
480C2×Q8C2513yes—
481C2×Q8C2511yescharacteristic
482C2×D4—29no—
483C2×D4—29no—
484C2×D4—218no—
485C2×D4—218no—
486C2×D4—218no—
487C2×D4—23no—
488C2×D4—29no—
489C2×D4—29no—
490C2×D4—218no—
491C2×D4—29no—
492C2×D4—29no—
493C2×D4—23no—
494C2×D4—418no—
495C2×D4—418no—
496C2×D4—818no—
497C2×D4—818no—
498C2×D4—836no—
499C2×D4—836no—
500C2×D4—836no—
501C2×D4C22×D419yes—
502C2×D4C22×D419yes—
503C2×D4C22×D4118yes—
504C2×D4C2519yes—
505C2×D4C2513yes—
506C22×C4—218no—
507C22×C4—23no—
508C22×C4—29no—
509C22×C4—29no—
510C22×C4—29no—
511C22×C4—29no—
512C22×C4—29no—
513C22×C4C22×D419yes—
514C22×C4C22×D419yes—
515C22×C4C2519yes—
516Q16—26no—
517SD16—26no—
518SD16—49no—
519SD16—418no—
520D8—49no—
521D8—418no—
522M16—418no—
523C2×C8—49no—
524C2×C8—49no—
525C2×C8C2511yescharacteristic
526C2×C8C2513yes—
527C2×C8C2513yes—
528C2×C8C2519yes—
529C4⋊C4—23no—
530C4⋊C4—29no—
531C4⋊C4—29no—
532C4⋊C4—218no—
533C4⋊C4—218no—
534C4⋊C4—23no—
535C4⋊C4—23no—
536C4⋊C4—23no—
537C4⋊C4—23no—
538C4⋊C4—23no—
539C4⋊C4—23no—
540C4⋊C4—29no—
541C4⋊C4C22×D419yes—
542C4⋊C4C22×D419yes—
543C4⋊C4C22×D413yes—
544C4⋊C4C22×D413yes—
545C4⋊C4C2519yes—
546C4⋊C4C2511yescharacteristic
547C4⋊C4C2513yes—
548C4⋊C4C2513yes—
549C22⋊C4—218no—
550C22⋊C4—29no—
551C22⋊C4—29no—
552C22⋊C4—23no—
553C22⋊C4—218no—
554C22⋊C4—218no—
555C22⋊C4—218no—
556C22⋊C4—29no—
557C22⋊C4—29no—
558C22⋊C4—29no—
559C22⋊C4C22×D4118yes—
560C22⋊C4C22×D4118yes—
561C42—23no—
562C42—23no—
563C42—23no—
564C42—29no—
565C42—29no—
566C42—29no—
567C42C22×D419yes—
568C42C22×D419yes—
569C42C2513yes—
570C42C2513yes—
571C23—23no—
572C23—29no—
573C23—818no—
574C23—818no—
575C23—818no—
576C23C23×D419yes—
577Q8—21no—
578Q8—23no—
579Q8—23no—
580Q8—46no—
581Q8—418no—
582Q8—89no—
583Q8—83no—
584Q8—83no—
585Q8#57614yes—
586Q8#57714yes—
587Q8#58512yes—
588D4—23no—
589D4—29no—
590D4—23no—
591D4—29no—
592D4—218no—
593D4—218no—
594D4—418no—
595D4—418no—
596D4—418no—
597D4—418no—
598D4—46no—
599D4—818no—
600D4—89no—
601D4—89no—
602D4—83no—
603D4—818no—
604D4—818no—
605D4—89no—
606D4—818no—
607C2×C4—21no—
608C2×C4—29no—
609C2×C4—29no—
610C2×C4—218no—
611C2×C4—218no—
612C2×C4—21no—
613C2×C4—23no—
614C2×C4—23no—
615C2×C4—23no—
616C2×C4—29no—
617C2×C4—418no—
618C2×C4—418no—
619C2×C4—418no—
620C2×C4—418no—
621C2×C4—46no—
622C2×C4—89no—
623C2×C4—818no—
624C2×C4C23×D419yes—
625C2×C4C23×D413yes—
626C2×C4C23×D413yes—
627C2×C4C23×D413yes—
628C2×C4C23×D413yes—
629C2×C4C2611yescharacteristic
630C8—23no—
631C8—23no—
632C8—21no—
633C8—29no—
634C22—29no—
635C22—29no—
636C22—29no—
637C22—49no—
638C22—43no—
639C22—43no—
640C22—49no—
641C22—49no—
642C22—43no—
643C22—3218no—
644C22C24×D411yescharacteristic
645C4—29no—
646C4—22no—
647C4—23no—
648C4—23no—
649C4—43no—
650C4—43no—
651C4—49no—
652C4—41no—
653C4—43no—
654C4C2×#57616yes—
655C4C2×#57716yes—
656C4C2×#58511yescharacteristic
657C4H(7,𝔽2)11yescharacteristic
658C2—49no—
659C2—83no—
660C2—89no—
661C2#6298711yescharacteristic
662C2C22×#57611yescharacteristic
663C2C22×#57711yescharacteristic
subgroupG/H
center, socleC22C24×D4
commutator, Frattini, cosocleC2×C4C26
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ #3483 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #63040C2semidirect
2. #63039C2semidirect
3. #62529C2semidirect
4. #62525C2semidirect
5. #62342C2semidirect
6. #62150C2semidirect
7. #62149C2semidirect
8. #62001C2semidirect
9. #61999C2semidirect
10. #61788C2semidirect
11. #61668C2semidirect
12. #61436C2semidirect
13. #61434C2semidirect
14. #61215C2semidirect
15. #60308C2semidirect
18. #3548C22semidirect
21. #3511C22semidirect
22. #3503C22semidirect
23. #3499C22semidirect
34. #3483C22semidirect
35. #3482C22semidirect
37. #3392C22semidirect
38. #3391C22semidirect
39. Q8×Q16C22semidirect
40. Q8×SD16C22semidirect
41. #3363C22semidirect
42. #3362C22semidirect
43. #3362C22semidirect
44. #3362C22semidirect
45. #3358C22semidirect
46. #3358C22semidirect
47. #3357C22semidirect
48. #3354C22semidirect
49. #3341C22semidirect
50. #3340C22semidirect
51. #3333C22semidirect
52. #3332C22semidirect
53. #3325C22semidirect
54. #3324C22semidirect
55. #3305C22semidirect
56. #3305C22semidirect
57. #3304C22semidirect
58. #3303C22semidirect
59. #3295C22semidirect
60. #3295C22semidirect
61. #3294C22semidirect
62. #3294C22semidirect
63. #3287C22semidirect
64. #3284C22semidirect
65. #3274C22semidirect
66. #3272C22semidirect
67. #3254C22semidirect
68. #3253C22semidirect
69. #3227C22semidirect
70. #3226C22semidirect
71. #3208C22semidirect
72. #3205C22semidirect
73. #3198C22semidirect
74. #3169C22semidirect
75. #3167C22semidirect
76. C2×#486C22semidirect
77. #3107C22semidirect
78. #3106C22semidirect
79. #3106C22semidirect
80. #3106C22semidirect
81. #3096C22semidirect
82. #3079C22semidirect
83. #3062C22semidirect
84. #3038C22semidirect
85. #3012C22semidirect
86. #2991C22semidirect
87. #2991C22semidirect
88. #2979C22semidirect
89. C8×#41C22semidirect
90. C8×#41C22semidirect
91. C4×#135C22semidirect
92. #2931C22semidirect
93. #2897C22semidirect

Maximal subgroups

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

subgroupindex
#630402
#630392
#625292
#625252
#623422
#621502
#621492
#620012
#619992
#617882
#616682
#614362
#614342
#612152
#603082