Finite Group Databasebeta

← #10575044

Subgroups — #10575044

Order 512.

7835 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#62859C211yesmaximal, characteristic
2#62654C211yesmaximal, characteristic, complemented
3#60526C218yesmaximal, complemented
4#60524C218yesmaximal, complemented
5#60524C218yesmaximal, complemented
6#60522C218yesmaximal, complemented
7#60519C216yesmaximal, complemented
8#60517C216yesmaximal, complemented
9#60516C216yesmaximal, complemented
10C2×#3097C216yesmaximal, complemented
11#60109C211yesmaximal, characteristic, complemented
12#60108C211yesmaximal, characteristic, complemented
13#60088C211yesmaximal, characteristic, complemented
14C2×#2898C211yesmaximal, characteristic, complemented
15#60065C211yesmaximal, characteristic, complemented
16C2×#585C2211yescharacteristic
17C22×2−1+4C2211yescharacteristic
18C2×#568C2211yescharacteristic
19C22×#131C2211yescharacteristic
20#3500—24no—
21#3499—24no—
22#3489C2218yes—
23#3489C2218yes—
24#3486—22no—
25#3485—26no—
26#3449C2216yes—
27C2×#536C2216yes—
28C4×2−1+4—24no—
29C4×2−1+4—22no—
30C4×H(5,𝔽2)—26no—
31C4×H(5,𝔽2)—24no—
32#3428C2216yes—
33#3428C2211yescharacteristic
34#3428C2211yescharacteristic
35#3428C2218yescomplemented
36C2×#520C2216yes—
37C2×#520C2211yescharacteristic
38C2×#520C2218yescomplemented
39#3099C22112yescomplemented
40#3099C22112yescomplemented
41#3099C22124yescomplemented
42#3098C22112yescomplemented
43#3098C22112yescomplemented
44#3098C22124yescomplemented
45#3097C22112yescomplemented
46#3097C22124yescomplemented
47#3097C22124yescomplemented
48#3097C22124yescomplemented
49#3097C22112yescomplemented
50#3097C22124yescomplemented
51#3097C22124yescomplemented
52#3096C2214yescomplemented
53#3096C22112yescomplemented
54#3095C22124yescomplemented
55#3095C22124yescomplemented
56#3093C22124yescomplemented
57#3092C22124yescomplemented
58#3091C22124yescomplemented
59#3091C22124yescomplemented
60C2×#484C22112yescomplemented
61C2×#485C22112yescomplemented
62C2×#485C22112yescomplemented
63C2×#483C22112yescomplemented
64#2913C2214yescomplemented
65#2913C2218yescomplemented
66#2913C2214yescomplemented
67#2912C2214yescomplemented
68#2912C2218yescomplemented
69#2912C2214yescomplemented
70#2911C2216yescomplemented
71#2911C2216yescomplemented
72C2×#428C2216yescomplemented
73C2×#427C2216yescomplemented
74#2898C2218yescomplemented
75#2898C2218yescomplemented
76#2898C2218yescomplemented
77#2898C2218yescomplemented
78#2896C2216yescomplemented
79C2×#417C2213yescomplemented
80C2×#419C2216yescomplemented
81C2×#419C2213yescomplemented
82C2×#416C2213yescomplemented
83C22×#103C2213yescomplemented
84#2880C2218yescomplemented
85#2880C2218yescomplemented
86#2879C2216yes—
87C2×#408C2216yes—
88#585—28no—
89#585—28no—
90C2×2−1+4—24no—
91C2×2−1+4—28no—
92C2×2−1+4—28no—
93C2×2−1+4D412yes—
94C2×2−1+4D414yes—
95C2×2−1+4C2318yes—
96C2×H(5,𝔽2)D416yes—
97C2×H(5,𝔽2)D414yes—
98C22×#41—23no—
99C22×#41C2313yes—
100C22×#41C2316yes—
101C22×#41C2318yes—
102C22×#41C2311yescharacteristic
103C23×Q8—21no—
104C23×Q8C2313yes—
105#568—28no—
106C2×#131—28no—
107C2×#131C2314yes—
108C2×#131C2318yes—
109C2×#131C2314yes—
110C22×M16C2316yes—
111C22×M16C2313yes—
112C23×C8C2313yes—
113D4×Q8—28no—
114D4×Q8—28no—
115#548—28no—
116#548—28no—
117#547—224no—
118#547—224no—
119#546—224no—
120#546—224no—
121#537C23112yes—
122#537C23112yes—
123#536—212no—
124#536—212no—
125#536—26no—
126#536C23112yes—
127#536C23124yes—
128#535—26no—
129#535C2314yes—
130#534—212no—
131#532—212no—
132C2×C4⋊Q8—26no—
133C2×#124—26no—
134#525—212no—
135#525C23124yes—
136C2×#123—212no—
137C2×C22⋊Q8C23112yes—
138C2×C22⋊Q8C23112yes—
139#520—212no—
140#520—212no—
141#520—24no—
142#520—224no—
143#520—28no—
144#520C23112yes—
145#520C2314yes—
146#520C23124yes—
147#520C23112yes—
148#520C23112yes—
149#520C2318yes—
150#520C2314yes—
151#520C2314yes—
152#520C2314yes—
153#520C2314yes—
154#519—26no—
155#519—21no—
156#519—21no—
157#519—28no—
158#518—212no—
159#518—26no—
160#518—23no—
161#518—23no—
162#518—224no—
163C4×#41—212no—
164C4×#41—212no—
165C4×#41—24no—
166C4×#41—24no—
167C4×#41—224no—
168C4×#41—28no—
169C2×C4×Q8—26no—
170C2×C4×Q8—21no—
171C2×C4×Q8—28no—
172C2×C4×D4—212no—
173C2×C4×D4—26no—
174C2×C4×D4—23no—
175C2×C4×D4—224no—
176C2×#117C23112yes—
177C2×#117C23124yes—
178C2×#117C2313yes—
179C2×#117C2316yes—
180C2×#117C2316yes—
181C22×C4⋊C4C23112yes—
182C22×C4⋊C4C2313yes—
183#485C23124yes—
184#485C23124yes—
185#485C23148yes—
186#485C23148yes—
187#485C23124yes—
188#485C23124yes—
189#484C23124yes—
190#484C23148yes—
191#484C23124yes—
192#483C23124yes—
193#483C23148yes—
194#483C23124yes—
195#482C2318yes—
196#482C2318yes—
197#482C23124yes—
198#482C23124yes—
199#481C2318yes—
200#481C23124yes—
201#480C2318yes—
202#480C23124yes—
203#428C23112yes—
204#428C23124yes—
205#428C23112yes—
206#428C23112yes—
207#428C23112yes—
208#428C23112yes—
209#428C23112yes—
210#428C23112yes—
211#428C2314yes—
212#427C23112yes—
213#427C23112yes—
214#427C23124yes—
215#427C2314yes—
216#427C23112yes—
217C2×#107C2316yes—
218C2×#107C23112yes—
219C2×#107C2316yes—
220C2×#106C2316yes—
221C2×#106C23112yes—
222C2×#106C2316yes—
223#419C23124yes—
224#419C23124yes—
225#419C23112yes—
226#418C2314yes—
227#417C23112yes—
228#417C23124yes—
229#417C23112yes—
230#416C23112yes—
231#416C23124yes—
232#416C23112yes—
233C2×#103C23112yes—
234C2×#103C23112yes—
235C2×#103C23112yes—
236C2×#103C23112yes—
237C2×#103C2313yes—
238C2×#103C2316yes—
239C2×#103C2313yes—
240C2×#102C2311yescharacteristic
241C2×#102C2313yes—
242#408C2314yes—
243#408C23112yes—
244#408C23112yes—
245#408C23124yes—
246#408C23112yes—
247#407C23124yes—
248C2×C22⋊C8C23112yes—
249C2×C22⋊C8C23112yes—
2502−1+4—232no—
2512−1+4—216no—
2522−1+4—232no—
2532−1+4—232no—
2542−1+4—432no—
255H(5,𝔽2)—232no—
256H(5,𝔽2)—248no—
257C2×#41—232no—
258C2×#41—232no—
259C2×#41—224no—
260C2×#41—26no—
261C2×#41—224no—
262C2×#41—224no—
263C2×#41—212no—
264C2×#41—212no—
265C2×#41—424no—
266C2×#41C2×D4112yes—
267C2×#41C2×D4112yes—
268C2×#41C2×D4124yes—
269C2×#41C2×D418yes—
270C2×#41C2×D414yes—
271C2×#41#4118yes—
272C2×#41#4118yes—
273C2×#41C24112yes—
274C2×#41C2414yes—
275C2×#41C24112yes—
276C2×#41C2414yes—
277C22×Q8—28no—
278C22×Q8—26no—
279C22×Q8—23no—
280C22×Q8—212no—
281C22×Q8—212no—
282C22×Q8—48no—
283C22×Q8C2×D416yes—
284C22×Q8C2×D418yes—
285C22×Q8C2×D411yescharacteristic
286C22×Q8C24112yes—
287C22×Q8C2413yes—
288C22×D4—23no—
289C22×D4C2×D4112yes—
290C22×D4C2×D416yes—
291C22×D4C2×D4124yes—
292C22×D4C2×D413yes—
293C22×D4C2411yescharacteristic
294C23×C4—23no—
295C23×C4—212no—
296C23×C4—23no—
297C23×C4—23no—
298C23×C4C24112yes—
299C23×C4C2413yes—
300#131—216no—
301#131—216no—
302#131—232no—
303C2×M16—212no—
304C2×M16—224no—
305C2×M16C2414yes—
306C2×M16C24112yes—
307C2×M16C24112yes—
308C22×C8—212no—
309C22×C8C2411yescharacteristic
310C22×C8C2413yes—
311C22×C8C2416yes—
312C22×C8C2416yes—
313C22×C8C24112yes—
314C4⋊Q8—212no—
315C4⋊Q8—224no—
316#127—24no—
317#124—24no—
318#124—212no—
319#124—224no—
320#123—224no—
321#123—248no—
322#123—224no—
323C22⋊Q8—212no—
324C22⋊Q8—212no—
325C22⋊Q8—224no—
326C22⋊Q8—224no—
327C22⋊Q8C24124yes—
328C22⋊Q8C24124yes—
329#121—212no—
330#121—212no—
331#121—224no—
332#121—224no—
333#121C2418yes—
334#121C2418yes—
335#120—224no—
336#120—28no—
337C4×Q8—224no—
338C4×Q8—212no—
339C4×Q8—212no—
340C4×Q8—28no—
341C4×Q8—24no—
342C4×Q8—212no—
343C4×Q8—24no—
344C4×Q8—24no—
345C4×Q8—24no—
346C4×Q8—24no—
347C4×D4—224no—
348C4×D4—248no—
349C4×D4—224no—
350C4×D4—224no—
351C4×D4—212no—
352C4×D4—212no—
353C4×D4—224no—
354C4×D4—212no—
355C4×D4—28no—
356C4×D4—224no—
357C4×D4—212no—
358C4×D4—24no—
359C4×D4—212no—
360C4×D4—212no—
361C4×D4—212no—
362C4×D4—224no—
363C4×D4—224no—
364#117—224no—
365#117—224no—
366#117—224no—
367#117—26no—
368#117—26no—
369#117—26no—
370#117—26no—
371#117—26no—
372#117C24112yes—
373#117C24112yes—
374#117C2414yes—
375#117C24124yes—
376#117C2418yes—
377#117C2414yes—
378#117C24112yes—
379C2×C4⋊C4—212no—
380C2×C4⋊C4—224no—
381C2×C4⋊C4—26no—
382C2×C4⋊C4—23no—
383C2×C4⋊C4—212no—
384C2×C4⋊C4—212no—
385C2×C4⋊C4C2413yes—
386C2×C4⋊C4C2416yes—
387C2×C4⋊C4C24112yes—
388C2×C4⋊C4C24124yes—
389C2×C4⋊C4C2411yescharacteristic
390C2×C4⋊C4C2416yes—
391C2×C4⋊C4C2418yes—
392C2×C4⋊C4C2416yes—
393C2×C4⋊C4C2416yes—
394C2×C22⋊C4—212no—
395C2×C22⋊C4—224no—
396C2×C22⋊C4—23no—
397C2×C22⋊C4—26no—
398C2×C22⋊C4—26no—
399C2×C22⋊C4—212no—
400C2×C22⋊C4—212no—
401C2×C42—212no—
402C2×C42—224no—
403C2×C42—26no—
404C2×C42—23no—
405C2×C42—26no—
406#107C2414yes—
407#107C2414yes—
408#107C24124yes—
409#107C24112yes—
410#107C24112yes—
411#107C2418yes—
412#106C2414yes—
413#106C2414yes—
414#106C24124yes—
415#106C24112yes—
416#106C24112yes—
417#106C2418yes—
418#103C2416yes—
419#103C2416yes—
420#103C24124yes—
421#103C24124yes—
422#103C24124yes—
423#103C24112yes—
424#102C2412yes—
425#102C2412yes—
426#102C24112yes—
427#102C2418yes—
428#102C2418yes—
429C22⋊C8C2418yes—
430C22⋊C8C2418yes—
431C22⋊C8C24124yes—
432C22⋊C8C24124yes—
433C24—21no—
434C24C22×D4112yes—
435C24C22×D413yes—
436#41—232no—
437#41—296no—
438#41—248no—
439#41—248no—
440#41—28no—
441#41—224no—
442#41—224no—
443#41—28no—
444#41—224no—
445#41—224no—
446#41—496no—
447#41—424no—
448#41#123132yes—
449#41C2×#41116yes—
450#41C2×#41116yes—
451C2×Q8—232no—
452C2×Q8—224no—
453C2×Q8—212no—
454C2×Q8—23no—
455C2×Q8—26no—
456C2×Q8—26no—
457C2×Q8—224no—
458C2×Q8—224no—
459C2×Q8—432no—
460C2×Q8—412no—
461C2×Q8—424no—
462C2×Q8#12318yes—
463C2×Q8C22×D4112yes—
464C2×Q8C22×D414yes—
465C2×Q8C22×D414yes—
466C2×Q8C2516yes—
467C2×D4—248no—
468C2×D4—296no—
469C2×D4—212no—
470C2×D4—28no—
471C2×D4—28no—
472C2×D4—224no—
473C2×D4—21no—
474C2×D4—22no—
475C2×D4—22no—
476C2×D4—412no—
477C2×D4#123124yes—
478C2×D4C22×D418yes—
479C2×D4C22×D4124yes—
480C2×D4C22×D4112yes—
481C2×D4C22×D414yes—
482C2×D4C22×D4112yes—
483C2×D4C2512yes—
484C22×C4—224no—
485C22×C4—224no—
486C22×C4—212no—
487C22×C4—23no—
488C22×C4—28no—
489C22×C4—21no—
490C22×C4—26no—
491C22×C4—26no—
492C22×C4—26no—
493C22×C4—224no—
494C22×C4—23no—
495C22×C4—212no—
496C22×C4—26no—
497C22×C4—212no—
498C22×C4—412no—
499C22×C4—412no—
500C22×C4—163no—
501C22×C4—1612no—
502C22×C4C22×D4112yes—
503C22×C4C22×D4124yes—
504C22×C4C22×D416yes—
505C22×C4C22×D413yes—
506C22×C4C2×#41112yes—
507C22×C4C2×#41112yes—
508C22×C4C2518yes—
509C22×C4C2516yes—
510C22×C4C2511yescharacteristic
511M16—28no—
512M16—224no—
513M16—224no—
514C2×C8—22no—
515C2×C8—26no—
516C2×C8—212no—
517C2×C8—212no—
518C2×C8—224no—
519C2×C8C2511yescharacteristic
520C2×C8C2518yes—
521C2×C8C2516yes—
522C2×C8C2511yescharacteristic
523C4⋊C4—224no—
524C4⋊C4—23no—
525C4⋊C4—26no—
526C4⋊C4—212no—
527C4⋊C4—224no—
528C4⋊C4—224no—
529C4⋊C4—28no—
530C4⋊C4—21no—
531C4⋊C4—26no—
532C4⋊C4—26no—
533C4⋊C4—26no—
534C4⋊C4C2516yes—
535C4⋊C4C2514yes—
536C4⋊C4C2512yes—
537C4⋊C4C2514yes—
538C4⋊C4C2514yes—
539C4⋊C4C2514yes—
540C22⋊C4—224no—
541C22⋊C4—28no—
542C22⋊C4—24no—
543C22⋊C4—212no—
544C22⋊C4—212no—
545C22⋊C4—24no—
546C22⋊C4—224no—
547C22⋊C4—224no—
548C22⋊C4—212no—
549C22⋊C4—28no—
550C22⋊C4—28no—
551C42—224no—
552C42—28no—
553C42—24no—
554C42—212no—
555C42—212no—
556C42—24no—
557C42—212no—
558C23—224no—
559C23—22no—
560C23—43no—
561C23—43no—
562C23—412no—
563C23—412no—
564C23—42no—
565C23—81no—
566C23—81no—
567C23C2×#123112yes—
568C23C23×D418yes—
569C23C23×D411yescharacteristic
570C23C23×D416yes—
571Q8—224no—
572Q8—28no—
573Q8—26no—
574Q8—26no—
575Q8—424no—
576Q8—46no—
577Q8C2×#123116yes—
578D4—248no—
579D4—216no—
580D4—22no—
581D4—22no—
582D4—224no—
583D4—28no—
584D4—424no—
585D4—42no—
586D4—88no—
587D4—88no—
588D4C2×#123148yes—
589C2×C4—224no—
590C2×C4—28no—
591C2×C4—28no—
592C2×C4—248no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—26no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—26no—
600C2×C4—46no—
601C2×C4—424no—
602C2×C4—424no—
603C2×C4—86no—
604C2×C4—86no—
605C2×C4—81no—
606C2×C4—88no—
607C2×C4—86no—
608C2×C4—812no—
609C2×C4—86no—
610C2×C4—83no—
611C2×C4—824no—
612C2×C4C2×#123112yes—
613C2×C4C2×#123124yes—
614C2×C4C23×D418yes—
615C2×C4C23×D411yescharacteristic
616C2×C4C23×D416yes—
617C2×C4C22×#4112yes—
618C2×C4C22×#41112yes—
619C2×C4C22×#41112yes—
620C2×C4C22×#4112yes—
621C2×C4C2611yescharacteristic
622C8—21no—
623C8—21no—
624C8—26no—
625C8—28no—
626C22—26no—
627C22—21no—
628C22—26no—
629C22—21no—
630C22—28no—
631C22—28no—
632C22—28no—
633C22—412no—
634C22—42no—
635C22—42no—
636C22—42no—
637C22—824no—
638C22—84no—
639C22C22×#123112yes—
640C22C22×#12312yes—
641C22C24×D411yescharacteristic
642C4—28no—
643C4—46no—
644C4—44no—
645C4—44no—
646C4—44no—
647C4—42no—
648C4—44no—
649C4—46no—
650C4C22×#123112yes—
651C4C22×#12312yes—
652C4C23×#4111yescharacteristic
653C4C23×#4111yescharacteristic
654C2—22no—
655C2—212no—
656C2—48no—
657C2—82no—
658C2C23×#12311yescharacteristic
659C2#6309211yescharacteristic
660C2#6309311yescharacteristic
subgroupG/H
center, socleC22C24×D4
commutator, Frattini, cosocleC2×C4C26
prime radical#62859C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×#585 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #62654C2semidirect
3. #60526C2semidirect
4. #60524C2semidirect
5. #60524C2semidirect
6. #60522C2semidirect
7. #60519C2semidirect
8. #60517C2semidirect
9. #60516C2semidirect
10. C2×#3097C2semidirect
11. #60109C2semidirect
12. #60108C2semidirect
13. #60088C2semidirect
14. C2×#2898C2semidirect
15. #60065C2semidirect
35. #3428C22semidirect
38. C2×#520C22semidirect
39. #3099C22semidirect
40. #3099C22semidirect
41. #3099C22semidirect
42. #3098C22semidirect
43. #3098C22semidirect
44. #3098C22semidirect
45. #3097C22semidirect
46. #3097C22semidirect
47. #3097C22semidirect
48. #3097C22semidirect
49. #3097C22semidirect
50. #3097C22semidirect
51. #3097C22semidirect
52. #3096C22semidirect
53. #3096C22semidirect
54. #3095C22semidirect
55. #3095C22semidirect
56. #3093C22semidirect
57. #3092C22semidirect
58. #3091C22semidirect
59. #3091C22semidirect
60. C2×#484C22semidirect
61. C2×#485C22semidirect
62. C2×#485C22semidirect
63. C2×#483C22semidirect
64. #2913C22semidirect
65. #2913C22semidirect
66. #2913C22semidirect
67. #2912C22semidirect
68. #2912C22semidirect
69. #2912C22semidirect
70. #2911C22semidirect
71. #2911C22semidirect
72. C2×#428C22semidirect
73. C2×#427C22semidirect
74. #2898C22semidirect
75. #2898C22semidirect
76. #2898C22semidirect
77. #2898C22semidirect
78. #2896C22semidirect
79. C2×#417C22semidirect
80. C2×#419C22semidirect
81. C2×#419C22semidirect
82. C2×#416C22semidirect
83. C22×#103C22semidirect
84. #2880C22semidirect
85. #2880C22semidirect

Maximal subgroups

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

subgroupindex
#628592
#626542
#605262
#605242
#605222
#605192
#605172
#605162
C2×#30972
#601092
#601082
#600882
C2×#28982
#600652