Finite Group Databasebeta

← #29676869

Subgroups — #29676869

Order 15616.

670 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458850C211yesmaximal, characteristic, complemented
2#21458546C211yesmaximal, characteristic, complemented
3#21457435C211yesmaximal, characteristic
4#21445423C211yesmaximal, characteristic, complemented
5#21445384C211yesmaximal, characteristic, complemented
6#21445380C211yesmaximal, characteristic, complemented
7#21445306C211yesmaximal, characteristic, complemented
8#21445301C211yesmaximal, characteristic, complemented
9#21445011C211yesmaximal, characteristic, complemented
10#21444998C211yesmaximal, characteristic, complemented
11#21444929C211yesmaximal, characteristic, complemented
12#21444914C211yesmaximal, characteristic, complemented
13#21444710C211yesmaximal, characteristic, complemented
14#21444707C211yesmaximal, characteristic, complemented
15C61×#3324C211yesmaximal, characteristic, complemented
16C2×#14519803—21no—
17#16840491C2211yescharacteristic, complemented
18#16840433C2211yescharacteristic
19#16840157—21no—
20#16840141C2211yescharacteristic, complemented
21#16840138C2211yescharacteristic
22#16840125—21no—
23#16840110C2211yescharacteristic, complemented
24#16840109C2211yescharacteristic, complemented
25#16840108C2211yescharacteristic
26#16840061—21no—
27#16840042C2211yescharacteristic
28#16840013—21no—
29#16839981—21no—
30#16839972—21no—
31#16839969C2211yescharacteristic, complemented
32#16839967C2211yescharacteristic
33#16839962C2211yescharacteristic, complemented
34#16839960—21no—
35#16839959C2211yescharacteristic
36#16839957C2211yescharacteristic
37#16839905—21no—
38#16839896—21no—
39#16839893C2211yescharacteristic, complemented
40#16839892C2211yescharacteristic, complemented
41#16839867—21no—
42#16839865—21no—
43C4×#11758200C2211yescharacteristic
44#16839840C2211yescharacteristic
45#16839513C2211yescharacteristic, complemented
46#16839512C2211yescharacteristic, complemented
47#16839508C2211yescharacteristic, complemented
48#16839507C2211yescharacteristic, complemented
49#16839506C2211yescharacteristic
50#16839505C2211yescharacteristic, complemented
51Q8⋊#92418C2211yescharacteristic, complemented
52#16839492C2211yescharacteristic, complemented
53D4⋊#92418C2211yescharacteristic
54D4⋊#92418C2211yescharacteristic
55C8×#92418C2211yescharacteristic, complemented
56C61×#556C2211yescharacteristic
57D42×C61C2211yescharacteristic
58C61×#486C2211yescharacteristic
59C61×#482—21no—
60C61×#474C2211yescharacteristic
61C61×#468—21no—
62C61×#460C2211yescharacteristic
63C61×#450—21no—
64C4×SD16×C61C2211yescharacteristic
65C61×#436C2211yescharacteristic
66C61×#418—21no—
67C2×#11758222—21no—
68#14519803—44no—
69C2×#11758202—21no—
70C2×#11758202—21no—
71C2×#11758202—21no—
72C2×#11758200—21no—
73C2×#11758200—21no—
74C2×#11758200D411yescharacteristic
75C2×#11758201—21no—
76C2×#11758199—21no—
77C2×#11758199—21no—
78C2×#11758199—21no—
79C2×#11758194—21no—
80#14519777—21no—
81#14519777C2311yescharacteristic
82#14519774C2311yescharacteristic
83#14519773—21no—
84#14519768—21no—
85#14519768—41no—
86#14519751—21no—
87#14519750—41no—
88Q8×Dic61—21no—
89C4×Dic122—21no—
90C4×Dic122C2311yescharacteristic
91#14519739—21no—
92#14519710—21no—
93#14519708D411yescharacteristic
94C4⋊#92418—21no—
95C4⋊#92418C2311yescharacteristic
96C4⋊#92418C2311yescharacteristic
97C4⋊#92418C2311yescharacteristic
98#14519706—21no—
99#14519701—21no—
100#14519700—21no—
101#14519700D411yescharacteristic
102#14519699—21no—
103#14519699—21no—
104#14519698—21no—
105#14519698—21no—
106#14519698—41no—
107#14519698D411yescharacteristic
108#14519697—21no—
109#14519697—21no—
110#14519697—41no—
111C22⋊#92418—21no—
112#14519688—21no—
113#14519687—21no—
114C4×#92418C2311yescharacteristic
115C4×#92418C2311yescharacteristic
116C22×D4×C61—21no—
117C22×D4×C61—21no—
118C2×SD16×C61—21no—
119C2×SD16×C61D411yescharacteristic
120C2×M16×C61—21no—
121C4⋊Q8×C61C2311yescharacteristic
122#127×C61C2311yescharacteristic
123#126×C61—41no—
124#124×C61—21no—
125C22⋊Q8×C61—21no—
126#121×C61—21no—
127#121×C61—21no—
128#120×C61—41no—
129C4×Q8×C61—21no—
130C4×Q8×C61C2311yescharacteristic
131C4×D4×C61C2311yescharacteristic
132C4×D4×C61C2311yescharacteristic
133#117×C61—21no—
134#107×C61—21no—
135#106×C61D411yescharacteristic
136C4⋊C8×C61C2311yescharacteristic
137#103×C61D411yescharacteristic
138#102×C61—21no—
139#102×C61—21no—
140#102×C61—21no—
141#102×C61—21no—
142#102×C61D411yescharacteristic
143C22⋊C8×C61—21no—
144C4×C8×C61C2311yescharacteristic
145C24×C61—41no—
146C2×Q8×C61—21no—
147C2×Q8×C61C2×D411yescharacteristic
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61—21no—
151C2×D4×C61—41no—
152C2×D4×C61—41no—
153C2×D4×C61—42no—
154C2×D4×C61—42no—
155C2×D4×C61—81no—
156C2×D4×C61—81no—
157C2×D4×C61C2×D411yescharacteristic
158C2×D4×C61C2×D411yescharacteristic
159C22×C4×C61—21no—
160C22×C4×C61—21no—
161C2×C22⋊D61—41no—
162C2×C22⋊D61—41no—
163#11758222—42no—
164#11758222—81no—
165C2×D244—21no—
166C2×D244—21no—
167C2×D244—21no—
168C2×C4×D61—41no—
169C2×Dic122—21no—
170C2×Dic122C2×D411yescharacteristic
171SD16×C61—41no—
172SD16×C61—81no—
173M16×C61—81no—
174C2×C8×C61—21no—
175C2×C8×C61C2×D411yescharacteristic
176C2×C8×C61C2×D411yescharacteristic
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—41no—
181C4⋊C4×C61C2×D411yescharacteristic
182C4⋊C4×C61C2×D411yescharacteristic
183C4⋊C4×C61C2×D411yescharacteristic
184C22⋊C4×C61—21no—
185C22⋊C4×C61—21no—
186C22⋊C4×C61—41no—
187C22⋊C4×C61—41no—
188C22⋊C4×C61—41no—
189C42×C61—21no—
190C42×C61—41no—
191C42×C61C2411yescharacteristic
192#11758202—42no—
193#11758202—81no—
194#11758202—81no—
195#11758201—42no—
196#11758200—22no—
197#11758200—42no—
198#11758200—81no—
199#11758199—42no—
200#11758199—42no—
201#11758199—81no—
202#11758198—41no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—41no—
206C4⋊Dic61—41no—
207C4⋊Dic61C2×D411yescharacteristic
208#11758196—21no—
209#11758196—41no—
210C4×Dic61—21no—
211C4×Dic61—41no—
212#11758194—42no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418—21no—
217C2×#92418C2×D411yescharacteristic
218C2×#92418C2×D411yescharacteristic
219C23×C61—21no—
220C23×C61—21no—
221C23×C61—41no—
222C23×C61—82no—
223C23×C61—82no—
224C23×C61—161no—
225C22×D61—41no—
226Q8×C61—21no—
227Q8×C61—21no—
228Q8×C61—41no—
229Q8×C61—41no—
230D4×C61—21no—
231D4×C61—21no—
232D4×C61—21no—
233D4×C61—21no—
234D4×C61—41no—
235D4×C61—41no—
236D4×C61—41no—
237D4×C61—41no—
238D4×C61—81no—
239D4×C61—82no—
240D4×C61—82no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61C22×D411yescharacteristic
251C2×C4×C61C22×D411yescharacteristic
252C2×C4×C61H(5,𝔽2)11yescharacteristic
253C22⋊D61—82no—
254C22⋊D61—161no—
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—41no—
258D244—41no—
259D244—41no—
260D244—41no—
261D244—41no—
262D244—81no—
263C4×D61—82no—
264Dic122—21no—
265Dic122—21no—
266Dic122—41no—
267Dic122—41no—
268C8×C61—21no—
269C8×C61—21no—
270C8×C61—41no—
271#92418—21no—
272#92418—41no—
273#92418—41no—
274#92418—41no—
275#92418—41no—
276#92418AGL(1,ℤ/8)12yes—
277#20288—611nomaximal, Sylow
278C22×C61—41no—
279C22×C61—41no—
280C22×C61—41no—
281C22×C61—41no—
282C22×C61—41no—
283C22×C61—41no—
284C22×C61—81no—
285C22×C61—81no—
286C22×C61—81no—
287C22×C61—164no—
288C22×C61D4211yescharacteristic
289C2×D61—81no—
290C2×D61—81no—
291C2×D61—81no—
292C4×C61—21no—
293C4×C61—41no—
294C4×C61—41no—
295C4×C61—41no—
296C4×C61—41no—
297C4×C61—81no—
298C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
299C4×C61#57511yescharacteristic
300C4×C61#57611yescharacteristic
301C4×C61#57711yescharacteristic
302Dic61—41no—
303Dic61—41no—
304Dic61—81no—
305#3398—611no—
306#3368—611no—
307#3324D6111yescharacteristic, complemented
308#3274—611no—
309#1709—611no—
310#1702—611no—
311#1701—611no—
312#1689—611no—
313#1688—611no—
314#1641—611no—
315#1639—611no—
316#1628—611no—
317#1626—611no—
318#1590—611no—
319#1590—611no—
320C2×C61—81no—
321C2×C61—81no—
322C2×C61—161no—
323C2×C61#327411yescharacteristic
324C2×C61#327511yescharacteristic
325C2×C61#329011yescharacteristic
326D61—161no—
327C2×AGL(1,ℤ/8)—1221no—
328#557—611no—
329#556C2×D6111yescharacteristic
330#550—611no—
331D42C2×D6111yescharacteristic
332#497—1221no—
333#494—611no—
334#493—611no—
335#490—1221no—
336#486—611no—
337#486—611no—
338#486—611no—
339#486C2×D6111yescharacteristic
340#482—21no—
341#478—1221no—
342#474—611no—
343#474C2×D6111yescharacteristic
344#469—1221no—
345#468—21no—
346#464—1221no—
347#462—611no—
348#462—1221no—
349#461—611no—
350#460—611no—
351#460—611no—
352#460—1221no—
353#460C2×D6111yescharacteristic
354#459—611no—
355#450—21no—
356#449—1221no—
357#447—1221no—
358#446—611no—
359#446—611no—
360#441—1221no—
361#440—1221no—
362C4×SD16—611no—
363C4×SD16C2×D6111yescharacteristic
364#436—611no—
365#436C2×D6111yescharacteristic
366#418—21no—
367#334—611no—
368#334—611no—
369#332—611no—
370#332—611no—
371#331—611no—
372#331—611no—
373Q8⋊C8—611no—
374Q8⋊C8—611no—
375D4⋊C8—611no—
376D4⋊C8—611no—
377C82—611no—
378C61#2028811yescharacteristic, complemented, Sylow
379C2×#41—1221no—
380C22×D4—21no—
381C22×D4—21no—
382AGL(1,ℤ/8)—2444no—
383C2×Q16—1221no—
384C2×Q16—1221no—
385C2×Q16—1221no—
386C2×SD16—21no—
387C2×SD16—611no—
388C2×SD16—1221no—
389C2×SD16—1221no—
390C2×SD16—1221no—
391C2×SD16C22⋊D6111yescharacteristic
392C2×D8—1221no—
393C2×D8—1221no—
394C2×D8—1221no—
395C2×M16—21no—
396C2×M16—1221no—
397C4⋊Q8—611no—
398C4⋊Q8—1221no—
399C4⋊Q8C22×D6111yescharacteristic
400#127—611no—
401#127—1221no—
402#127C22×D6111yescharacteristic
403#126—41no—
404#125—1221no—
405#125—2441no—
406#124—21no—
407C22⋊Q8—21no—
408#121—21no—
409#121—21no—
410#121—1221no—
411#121—2441no—
412#120—41no—
413C4×Q8—21no—
414C4×Q8—611no—
415C4×Q8—1221no—
416C4×Q8—1221no—
417C4×Q8C22×D6111yescharacteristic
418C4×D4—1221no—
419C4×D4C22×D6111yescharacteristic
420C4×D4C22×D6111yescharacteristic
421#117—21no—
422#107—21no—
423#107—1221no—
424#106—611no—
425#106C22⋊D6111yescharacteristic
426C4⋊C8—611no—
427C4⋊C8—611no—
428C4⋊C8—611no—
429C4⋊C8—1221no—
430C4⋊C8—1221no—
431C4⋊C8C22×D6111yescharacteristic
432#103—611no—
433#103—1221no—
434#103—1221no—
435#103C22⋊D6111yescharacteristic
436#102—21no—
437#102—21no—
438#102—21no—
439#102—21no—
440#102—611no—
441#102—1221no—
442#102—1221no—
443#102—1221no—
444#102—1221no—
445#102—1221no—
446#102—1221no—
447#102—2441no—
448#102—2441no—
449#102C22⋊D6111yescharacteristic
450C22⋊C8—21no—
451C22⋊C8—1221no—
452#97—1221no—
453#97—1221no—
454C4×C8—611no—
455C4×C8—611no—
456C4×C8C22×D6111yescharacteristic
457C24—41no—
458#41—4881no—
459#41—2442no—
460C2×Q8—21no—
461C2×Q8—611no—
462C2×Q8—1221no—
463C2×Q8C2×C22⋊D6111yescharacteristic
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—41no—
468C2×D4—41no—
469C2×D4—42no—
470C2×D4—42no—
471C2×D4—81no—
472C2×D4—81no—
473C2×D4—1221no—
474C2×D4—1221no—
475C2×D4—1221no—
476C2×D4—2441no—
477C2×D4—2441no—
478C2×D4D4×D6111yescharacteristic
479C2×D4C2×C22⋊D6111yescharacteristic
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—2441no—
483Q16—4881no—
484Q16—4881no—
485Q16—2442no—
486SD16—41no—
487SD16—81no—
488SD16—1222no—
489SD16—4881no—
490SD16—2442no—
491SD16—2442no—
492D8—4881no—
493D8—2442no—
494D8—2442no—
495M16—81no—
496M16—2442no—
497C2×C8—21no—
498C2×C8—611no—
499C2×C8—611no—
500C2×C8—1221no—
501C2×C8—1221no—
502C2×C8—1221no—
503C2×C8—1221no—
504C2×C8C2×C22⋊D6111yescharacteristic
505C2×C8C2×C22⋊D6111yescharacteristic
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—41no—
510C4⋊C4—611no—
511C4⋊C4—1221no—
512C4⋊C4—1221no—
513C4⋊C4—1221no—
514C4⋊C4—2441no—
515C4⋊C4—2441no—
516C4⋊C4—2441no—
517C4⋊C4D4×D6111yescharacteristic
518C4⋊C4C2×C22⋊D6111yescharacteristic
519C4⋊C4C2×C22⋊D6111yescharacteristic
520C22⋊C4—21no—
521C22⋊C4—21no—
522C22⋊C4—41no—
523C22⋊C4—41no—
524C22⋊C4—41no—
525C22⋊C4—2441no—
526C42—21no—
527C42—41no—
528C42—1221no—
529C42—2441no—
530C42C23×D6111yescharacteristic
531C23—21no—
532C23—21no—
533C23—41no—
534C23—82no—
535C23—82no—
536C23—161no—
537C23—2441no—
538Q8—21no—
539Q8—21no—
540Q8—41no—
541Q8—41no—
542Q8—1221no—
543Q8—1221no—
544Q8—2441no—
545Q8—2441no—
546D4—21no—
547D4—21no—
548D4—21no—
549D4—21no—
550D4—41no—
551D4—41no—
552D4—41no—
553D4—41no—
554D4—81no—
555D4—82no—
556D4—82no—
557D4—9761no—
558D4—4881no—
559D4—4882no—
560D4—2441no—
561D4—2441no—
562D4—2441no—
563D4—2441no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—1221no—
574C2×C4—1221no—
575C2×C4—4882no—
576C2×C4—2441no—
577C2×C4C2×D4×D6111yescharacteristic
578C2×C4C22×C22⋊D6111yescharacteristic
579C2×C4#1451982211yescharacteristic
580C8—21no—
581C8—21no—
582C8—41no—
583C8—612no—
584C8—1221no—
585C8—2441no—
586C8—2441no—
587C8—2441no—
588C8—2441no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—81no—
596C22—81no—
597C22—81no—
598C22—164no—
599C22—4881no—
600C22—4881no—
601C22—4881no—
602C22D4×C22⋊D6111yescharacteristic
603C4—21no—
604C4—41no—
605C4—41no—
606C4—41no—
607C4—41no—
608C4—81no—
609C4—4881no—
610C4—2441no—
611C4—2441no—
612C4AGL(1,ℤ/8)×D6111yescharacteristic
613C4#1684056011yescharacteristic
614C4#1684056611yescharacteristic
615C4#1684057311yescharacteristic
616C2—81no—
617C2—81no—
618C2—161no—
619C2—9761no—
620C2#2145743311yescharacteristic
621C2#2145744611yescharacteristic
622C2#2145759911yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3324C2
socleC22×C61D42
prime radical#21457435C2
hypercenter#3324D61
hypocenterC61#20288
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3324

Complements

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

NH ≅ G/Ntype
1. #21458850C2semidirect
2. #21458546C2semidirect
4. #21445423C2semidirect
5. #21445384C2semidirect
6. #21445380C2semidirect
7. #21445306C2semidirect
8. #21445301C2semidirect
9. #21445011C2semidirect
10. #21444998C2semidirect
11. #21444929C2semidirect
12. #21444914C2semidirect
13. #21444710C2semidirect
14. #21444707C2semidirect
15. C61×#3324C2semidirect
17. #16840491C22semidirect
20. #16840141C22semidirect
23. #16840110C22semidirect
24. #16840109C22semidirect
31. #16839969C22semidirect
33. #16839962C22semidirect
39. #16839893C22semidirect
40. #16839892C22semidirect
45. #16839513C22semidirect
46. #16839512C22semidirect
47. #16839508C22semidirect
48. #16839507C22semidirect
50. #16839505C22semidirect
51. Q8⋊#92418C22semidirect
52. #16839492C22semidirect
55. C8×#92418C22semidirect
307. #3324D61semidirect
378. C61#20288semidirect

Maximal subgroups

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

subgroupindex
#214588502
#214585462
#214574352
#214454232
#214453842
#214453802
#214453062
#214453012
#214450112
#214449982
#214449292
#214449142
#214447102
#214447072
C61×#33242
#2028861