Finite Group Databasebeta

← #6359039

Subgroups — #6359039

Order 512.

1829 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#62243C212yesmaximal, complemented
2#61370C211yesmaximal, characteristic, complemented
3#61084C211yesmaximal, characteristic
4#19806C214yesmaximal, complemented
5#19803C214yesmaximal, complemented
6#19766C214yesmaximal, complemented
7#19763C214yesmaximal, complemented
8#19608C211yesmaximal, characteristic, complemented
9#19254C212yesmaximal, complemented
10#19205C211yesmaximal, characteristic, complemented
11#19194C211yesmaximal, characteristic, complemented
12#18664C212yesmaximal, complemented
13#18663C212yesmaximal, complemented
14#17976C211yesmaximal, characteristic, complemented
15C2×#1580C211yesmaximal, characteristic, complemented
16C2×#576—21no—
17#3501C2212yescomplemented
18#3496C2212yes—
19#3389—22no—
20#3384C2214yescomplemented
21#3380—22no—
22#3380C2212yescomplemented
23Q8×D8C2212yescomplemented
24#3335—22no—
25#3335C2214yescomplemented
26#3298—22no—
27#3298C2214yescomplemented
28#3295—22no—
29#3295C2214yescomplemented
30#3283C2212yes—
31D4×SD16—22no—
32D4×SD16C2214yescomplemented
33#3280—22no—
34D4×D8C2212yes—
35#3273—22no—
36#3272C2214yes—
37#3246C2212yes—
38#3246C2212yescomplemented
39#3187—24no—
40#3167C2211yescharacteristic
41#3167C2212yescomplemented
42#3167C2212yescomplemented
43#3164—21no—
44#3160C2211yescharacteristic, complemented
45#3147C2211yescharacteristic
46#3081—22no—
47#3077C2212yescomplemented
48#3061—21no—
49C2×#465—24no—
50#3036C2212yes—
51#3012—22no—
52#2960C2211yescharacteristic, complemented
53#2948—21no—
54C2×C4×D8C2211yescharacteristic
55#2931C2211yescharacteristic
56C2×C8×D4C2212yescomplemented
57#1716C2212yescomplemented
58#1715C2212yescomplemented
59#1714C2212yescomplemented
60#1714C2212yescomplemented
61#1695C2212yescomplemented
62#1693C2212yescomplemented
63#1690C2212yescomplemented
64#1688C2212yescomplemented
65#1679C2214yescomplemented
66#1677C2214yescomplemented
67#1671C2214yescomplemented
68#1671C2214yescomplemented
69#1666C2214yescomplemented
70#1665C2214yescomplemented
71#1654C2214yescomplemented
72#1651C2214yescomplemented
73#1643C2214yescomplemented
74#1639C2214yescomplemented
75#1635C2214yescomplemented
76#1629C2214yescomplemented
77#1627C2214yescomplemented
78#1622C2214yescomplemented
79#1583C2214yescomplemented
80#1582C2214yescomplemented
81#1580C2214yescomplemented
82#1580C2214yescomplemented
83#1566C2211yescharacteristic, complemented
84C2×#334C2211yescharacteristic, complemented
85#1550C2212yescomplemented
86#1547C2212yescomplemented
87#1493C2211yescharacteristic, complemented
88#1492C2211yescharacteristic
89C2×D4⋊C8C2212yescomplemented
90C2×#322C2211yescharacteristic, complemented
91C2×H(5,𝔽2)—22no—
92#576—48no—
93C2×AGL(1,ℤ/8)—22no—
94C2×AGL(1,ℤ/8)—24no—
95C2×#135—22no—
96C2×#135—21no—
97C2×#135—21no—
98C22×SD16—24no—
99C22×D8—22no—
100C22×D8D411yescharacteristic
101C2×#131—21no—
102C23×C8—22no—
103#554—22no—
104#554—24no—
105#554—42no—
106#554C2314yes—
107D4×Q8—22no—
108D4×Q8C2314yes—
109#547—22no—
110#547C2314yes—
111#546—24no—
112#546—42no—
113D42—22no—
114D42C2314yes—
115#544—22no—
116#543—42no—
117#537—22no—
118#534—42no—
119#533C2312yes—
120#533C2311yescharacteristic
121#532C2312yes—
122#532C2311yescharacteristic
123C2×C22⋊Q8—24no—
124C2×#121—24no—
125#518—22no—
126C2×C4×D4C2312yes—
127#501—24no—
128#500—22no—
129#500C2311yescharacteristic
130#500C2311yescharacteristic
131#499—22no—
132#499—21no—
133#499—21no—
134#498D411yescharacteristic
135#498C2312yes—
136#498C2312yes—
137#498C2312yes—
138#495—21no—
139#495—21no—
140#494C2311yescharacteristic
141#493C2311yescharacteristic
142#492D411yescharacteristic
143#492C2312yes—
144#492C2312yes—
145#492C2312yes—
146#487D411yescharacteristic
147#487C2312yes—
148#487C2312yes—
149#486D411yescharacteristic
150#486C2312yes—
151#486C2312yes—
152#483—24no—
153#483—24no—
154#481—24no—
155#481—24no—
156#478—24no—
157#478—24no—
158#478—24no—
159#476—22no—
160#476—24no—
161#476C2314yes—
162#476C2314yes—
163#475—22no—
164#475C2314yes—
165#474—24no—
166#474C2314yes—
167#468—24no—
168#467—22no—
169#466—22no—
170#465—24no—
171#465—22no—
172#465—22no—
173#465—24no—
174#465—24no—
175#465—24no—
176#465—24no—
177#465—24no—
178#465—42no—
179#465D414yes—
180#463—24no—
181#461—22no—
182#461C2314yes—
183#461C2314yes—
184#460—22no—
185#460C2314yes—
186#459C2314yes—
187#452—24no—
188#450—24no—
189#450—24no—
190#450—24no—
191#449—24no—
192#448—24no—
193#448—24no—
194#447—24no—
195#446—21no—
196#446C2312yes—
197C8×Q8—21no—
198C8×Q8C2312yes—
199#442—24no—
200C4×SD16—22no—
201C4×SD16—22no—
202C4×SD16—42no—
203C4×SD16C2314yes—
204C4×D8—22no—
205C4×D8—22no—
206C4×D8D412yes—
207C4×D8D412yes—
208C4×D8C2312yes—
209C4×D8C2312yes—
210#436—21no—
211#436C2312yes—
212C8×D4—22no—
213C8×D4—22no—
214C8×D4—21no—
215C8×D4C2314yes—
216C8×D4C2314yes—
217C8×D4C2312yes—
218#428—22no—
219#427—21no—
220#427—21no—
221C2×#107D411yescharacteristic
222C2×#106—24no—
223#424—21no—
224#423C2311yescharacteristic
225#423C2311yescharacteristic
226C2×C4⋊C8C2312yes—
227#418—22no—
228#417—22no—
229C2×#103—24no—
230C2×#102—24no—
231C2×#102D412yes—
232#408—21no—
233C2×C22⋊C8—22no—
234#405—21no—
235C2×C4×C8C2311yescharacteristic
236C2×C4×C8C2311yescharacteristic
237C2×C4×C8C2311yescharacteristic
238#334C2312yes—
239#334C2312yes—
240#334C2314yes—
241#333C2314yes—
242#332C2314yes—
243#332C2314yes—
244#331C2314yes—
245Q8⋊C8C2314yes—
246D4⋊C8C2314yes—
247D4⋊C8C2314yes—
248D4⋊C8C2314yes—
249#322C2312yes—
250#322C2312yes—
251H(5,𝔽2)—48no—
252H(5,𝔽2)—84no—
253C2×#41—21no—
254C2×#41—24no—
255C2×#41—22no—
256C2×#41—42no—
257C22×Q8—24no—
258C22×D4—24no—
259C22×D4—24no—
260C22×D4—42no—
261C22×D4—44no—
262C22×D4C2×D412yes—
263C23×C4—22no—
264AGL(1,ℤ/8)—48no—
265AGL(1,ℤ/8)—416no—
266#135—48no—
267#135—44no—
268#135—82no—
269C2×Q16—21no—
270C2×Q16—21no—
271C2×SD16—24no—
272C2×SD16—24no—
273C2×SD16—24no—
274C2×SD16—22no—
275C2×SD16—22no—
276C2×SD16—22no—
277C2×SD16—22no—
278C2×SD16—42no—
279C2×SD16—88no—
280C2×SD16C2×D414yes—
281C2×D8—24no—
282C2×D8—21no—
283C2×D8—21no—
284C2×D8—48no—
285C2×D8SD1618yescomplemented
286C2×D8C2×D412yes—
287C2×D8C2×D412yes—
288#131—44no—
289C2×M16—21no—
290C2×M16—21no—
291C2×M16—22no—
292C22×C8—22no—
293C22×C8—21no—
294C22×C8—24no—
295C22×C8—22no—
296C22×C8—24no—
297C22×C8—84no—
298C22×C8C2×D411yescharacteristic
299C22×C8C2×D412yes—
300C22×C8C2×D411yescharacteristic
301C4⋊Q8—24no—
302C4⋊Q8—42no—
303C4⋊Q8C2×D411yescharacteristic
304C4⋊Q8C2×D411yescharacteristic
305C4⋊Q8C2×D411yescharacteristic
306C4⋊Q8C2412yes—
307C4⋊Q8C2412yes—
308C4⋊Q8C2412yes—
309#127C2×D411yescharacteristic
310#127C2412yes—
311#125—24no—
312#125—22no—
313#125—21no—
314#125—42no—
315#124—22no—
316#124—21no—
317#123—44no—
318#123—44no—
319C22⋊Q8—21no—
320C22⋊Q8—24no—
321C22⋊Q8—22no—
322C22⋊Q8—22no—
323C22⋊Q8—24no—
324C22⋊Q8—24no—
325C22⋊Q8—44no—
326C22⋊Q8—44no—
327C22⋊Q8—44no—
328C22⋊Q8—44no—
329C22⋊Q8—44no—
330C22⋊Q8—44no—
331C22⋊Q8—42no—
332C22⋊Q8C2×D412yes—
333#121—21no—
334#121—24no—
335#121—22no—
336#121—22no—
337#121—24no—
338#121—24no—
339#121—44no—
340#121—42no—
341#121—44no—
342#121C2×D412yes—
343#120—44no—
344#120—44no—
345C4×Q8—21no—
346C4×Q8—22no—
347C4×Q8—22no—
348C4×Q8C2412yes—
349C4×D4—24no—
350C4×D4—24no—
351C4×D4—22no—
352C4×D4—21no—
353C4×D4—22no—
354C4×D4—22no—
355C4×D4—22no—
356C4×D4—42no—
357C4×D4—42no—
358C4×D4C2×D414yes—
359C4×D4C2414yes—
360C4×D4C2412yes—
361#117—21no—
362#117—22no—
363C2×C4⋊C4—24no—
364C2×C4⋊C4—24no—
365C2×C4⋊C4—44no—
366C2×C4⋊C4C2×D412yes—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—44no—
369C2×C22⋊C4—44no—
370C2×C42C2411yescharacteristic
371#107—21no—
372#107—21no—
373#107—21no—
374#107—21no—
375#107—24no—
376#107C2×D412yes—
377#107C2×D412yes—
378#106—22no—
379#106—22no—
380#106—22no—
381#106—22no—
382#106—24no—
383#106—24no—
384#106—24no—
385#106—42no—
386#106C2×D414yes—
387C4⋊C8—22no—
388C4⋊C8—22no—
389C4⋊C8—22no—
390C4⋊C8—21no—
391C4⋊C8—21no—
392C4⋊C8C2412yes—
393C4⋊C8C2414yes—
394C4⋊C8C2412yes—
395#103—24no—
396#103—22no—
397#103—22no—
398#103—24no—
399#103—24no—
400#103—42no—
401#103C2×D414yes—
402#102—24no—
403#102—24no—
404#102—24no—
405#102—22no—
406#102—22no—
407#102—24no—
408#102—24no—
409#102—42no—
410#102C2×D414yes—
411#102C2×D414yes—
412#102C2×D414yes—
413C22⋊C8—24no—
414C22⋊C8—24no—
415C22⋊C8—22no—
416C22⋊C8—22no—
417#97—22no—
418C4×C8—22no—
419C4×C8—21no—
420C4×C8C2×D411yescharacteristic
421C4×C8C2×D411yescharacteristic
422C4×C8C2×D411yescharacteristic
423C4×C8C2×D411yescharacteristic
424C4×C8C2412yes—
425C4×C8C2412yes—
426C4×C8C2411yescharacteristic
427C4×C8C2411yescharacteristic
428C24—22no—
429C24—44no—
430#41—42no—
431#41—42no—
432#41—48no—
433#41—44no—
434#41—44no—
435#41—84no—
436#41—88no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—22no—
440C2×Q8—22no—
441C2×Q8—22no—
442C2×Q8—44no—
443C2×Q8—84no—
444C2×Q8—84no—
445C2×Q8C22×D412yes—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—24no—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—22no—
454C2×D4—44no—
455C2×D4—44no—
456C2×D4—48no—
457C2×D4—44no—
458C2×D4—42no—
459C2×D4—84no—
460C2×D4—84no—
461C2×D4—84no—
462C2×D4—88no—
463C2×D4—816no—
464C2×D4C2×SD1618yescomplemented
465C2×D4C22×D412yes—
466C2×D4C22×D414yes—
467C22×C4—22no—
468C22×C4—24no—
469C22×C4—22no—
470C22×C4—24no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—42no—
475C22×C4—82no—
476C22×C4—82no—
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4H(5,𝔽2)11yescharacteristic
480Q16—42no—
481Q16—81no—
482SD16—44no—
483SD16—44no—
484SD16—48no—
485SD16—44no—
486SD16—82no—
487SD16—82no—
488SD16—84no—
489SD16—84no—
490SD16—84no—
491D8—48no—
492D8—42no—
493D8—81no—
494D8C2×SD1618yescomplemented
495D8C2×SD1618yescomplemented
496M16—42no—
497M16—42no—
498M16—44no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—24no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—42no—
505C2×C8—48no—
506C2×C8—44no—
507C2×C8—84no—
508C2×C8—82no—
509C2×C8—81no—
510C2×C8C2×SD1614yescomplemented
511C2×C8C22×D412yes—
512C2×C8C22×D412yes—
513C2×C8C22×D411yescharacteristic
514C2×C8C22×D411yescharacteristic
515C2×C8C22×D411yescharacteristic
516C2×C8C22×D411yescharacteristic
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—44no—
531C4⋊C4—44no—
532C4⋊C4—44no—
533C4⋊C4—44no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4C22×D412yes—
537C4⋊C4C22×D414yes—
538C4⋊C4C22×D412yes—
539C22⋊C4—22no—
540C22⋊C4—24no—
541C22⋊C4—22no—
542C22⋊C4—24no—
543C22⋊C4—22no—
544C22⋊C4—44no—
545C22⋊C4—44no—
546C22⋊C4—44no—
547C22⋊C4—44no—
548C22⋊C4—42no—
549C22⋊C4—42no—
550C22⋊C4—42no—
551C42—22no—
552C42—42no—
553C42C22×D411yescharacteristic
554C42C22×D411yescharacteristic
555C42H(5,𝔽2)11yescharacteristic
556C42C2511yescharacteristic
557C23—24no—
558C23—28no—
559C23—22no—
560C23—42no—
561C23—82no—
562C23—82no—
563C23—88no—
564C23—164no—
565C23—164no—
566C23D4211yescharacteristic
567Q8—24no—
568Q8—42no—
569Q8—42no—
570Q8—42no—
571Q8—42no—
572Q8—42no—
573Q8—81no—
574Q8—82no—
575Q8—164no—
576D4—24no—
577D4—28no—
578D4—24no—
579D4—44no—
580D4—42no—
581D4—42no—
582D4—42no—
583D4—42no—
584D4—42no—
585D4—81no—
586D4—82no—
587D4—88no—
588D4—88no—
589D4—84no—
590D4—164no—
591D4C22×SD1618yes—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—24no—
596C2×C4—22no—
597C2×C4—41no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—84no—
606C2×C4—82no—
607C2×C4—88no—
608C2×C4—84no—
609C2×C4—164no—
610C2×C4—164no—
611C2×C4D4211yescharacteristic
612C2×C4D4211yescharacteristic
613C2×C4D4211yescharacteristic
614C2×C4C22×SD1612yes—
615C2×C4#57712yes—
616C2×C4C23×D411yescharacteristic
617C2×C4C23×D411yescharacteristic
618C2×C4C2×H(5,𝔽2)11yescharacteristic
619C8—22no—
620C8—22no—
621C8—21no—
622C8—21no—
623C8—42no—
624C8—44no—
625C8—42no—
626C8—42no—
627C8C22×SD1612yes—
628C8C22×SD1612yes—
629C22—21no—
630C22—21no—
631C22—28no—
632C22—44no—
633C22—44no—
634C22—42no—
635C22—42no—
636C22—42no—
637C22—88no—
638C22—82no—
639C22—82no—
640C22—82no—
641C22—328no—
642C22D4×SD1612yes—
643C22C2×D4211yescharacteristic
644C4—21no—
645C4—21no—
646C4—21no—
647C4—42no—
648C4—44no—
649C4—42no—
650C4—42no—
651C4—82no—
652C4—82no—
653C4—82no—
654C4D4×SD1612yes—
655C4C23×SD1611yescharacteristic
656C4C2×#57711yescharacteristic
657C4#358512yes—
658C2—22no—
659C2—48no—
660C2—82no—
661C2—162no—
662C2C2×D4×SD1611yescharacteristic
663C2#6107911yescharacteristic
664C2#6110011yescharacteristic
subgroupG/H
center, socleC22C2×D42
commutator, Frattini, cosocleC42C25
prime radical#61084C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #62243C2semidirect
2. #61370C2semidirect
4. #19806C2semidirect
5. #19803C2semidirect
6. #19766C2semidirect
7. #19763C2semidirect
8. #19608C2semidirect
9. #19254C2semidirect
10. #19205C2semidirect
11. #19194C2semidirect
12. #18664C2semidirect
13. #18663C2semidirect
14. #17976C2semidirect
15. C2×#1580C2semidirect
17. #3501C22semidirect
20. #3384C22semidirect
22. #3380C22semidirect
23. Q8×D8C22semidirect
25. #3335C22semidirect
27. #3298C22semidirect
29. #3295C22semidirect
32. D4×SD16C22semidirect
38. #3246C22semidirect
41. #3167C22semidirect
42. #3167C22semidirect
44. #3160C22semidirect
47. #3077C22semidirect
52. #2960C22semidirect
56. C2×C8×D4C22semidirect
57. #1716C22semidirect
58. #1715C22semidirect
59. #1714C22semidirect
60. #1714C22semidirect
61. #1695C22semidirect
62. #1693C22semidirect
63. #1690C22semidirect
64. #1688C22semidirect
65. #1679C22semidirect
66. #1677C22semidirect
67. #1671C22semidirect
68. #1671C22semidirect
69. #1666C22semidirect
70. #1665C22semidirect
71. #1654C22semidirect
72. #1651C22semidirect
73. #1643C22semidirect
74. #1639C22semidirect
75. #1635C22semidirect
76. #1629C22semidirect
77. #1627C22semidirect
78. #1622C22semidirect
79. #1583C22semidirect
80. #1582C22semidirect
81. #1580C22semidirect
82. #1580C22semidirect
83. #1566C22semidirect
84. C2×#334C22semidirect
85. #1550C22semidirect
86. #1547C22semidirect
87. #1493C22semidirect
89. C2×D4⋊C8C22semidirect
90. C2×#322C22semidirect
285. C2×D8SD16semidirect
464. C2×D4C2×SD16semidirect
494. D8C2×SD16semidirect
495. D8C2×SD16semidirect
510. C2×C8C2×SD16semidirect

Maximal subgroups

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

subgroupindex
#622432
#613702
#610842
#198062
#198032
#197662
#197632
#196082
#192542
#192052
#191942
#186642
#186632
#179762
C2×#15802