Finite Group Databasebeta

← #29676419

Subgroups — #29676419

Order 15616.

658 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458837C211yesmaximal, characteristic, complemented
2#21458050C211yesmaximal, characteristic, complemented
3#21457763C211yesmaximal, characteristic, complemented
4#21445460C211yesmaximal, characteristic, complemented
5#21445356C211yesmaximal, characteristic, complemented
6#21445354C211yesmaximal, characteristic, complemented
7#21445284C211yesmaximal, characteristic, complemented
8#21445254C211yesmaximal, characteristic, complemented
9#21445047C211yesmaximal, characteristic, complemented
10#21444989C211yesmaximal, characteristic, complemented
11#21444883C211yesmaximal, characteristic, complemented
12#21444873C211yesmaximal, characteristic, complemented
13#21444721C211yesmaximal, characteristic, complemented
14#21444649C211yesmaximal, characteristic, complemented
15C61×#3325C211yesmaximal, characteristic, complemented
16#16840482C2211yescharacteristic
17#16840433C2211yescharacteristic, complemented
18#16840150—21no—
19#16840138C2211yescharacteristic
20#16840135C2211yescharacteristic, complemented
21#16840112C2211yescharacteristic, complemented
22#16840110C2211yescharacteristic, complemented
23#16840108C2211yescharacteristic
24#16840086—21no—
25#16840083—21no—
26#16840054C2211yescharacteristic, complemented
27#16840042C2211yescharacteristic
28#16840006—21no—
29#16839996—21no—
30#16839976—21no—
31#16839967C2211yescharacteristic, complemented
32#16839961—21no—
33#16839959C2211yescharacteristic
34#16839957C2211yescharacteristic
35#16839957C2211yescharacteristic
36#16839913—21no—
37#16839905—21no—
38#16839892C2211yescharacteristic, complemented
39#16839861—21no—
40C4×#11758200C2211yescharacteristic
41#16839840C2211yescharacteristic
42D4×#92418C2211yescharacteristic, complemented
43#16839767—21no—
44#16839756—21no—
45#16839515C2211yescharacteristic, complemented
46#16839514C2211yescharacteristic, complemented
47#16839511C2211yescharacteristic, complemented
48#16839509C2211yescharacteristic, complemented
49#16839506C2211yescharacteristic
50#16839505C2211yescharacteristic, complemented
51#16839492C2211yescharacteristic, complemented
52D4⋊#92418C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic
54D4⋊#92418C2211yescharacteristic, complemented
55#16839484C2211yescharacteristic, complemented
56C61×#555C2211yescharacteristic
57D42×C61C2211yescharacteristic
58C61×#486C2211yescharacteristic
59C61×#482—21no—
60C61×#476C2211yescharacteristic
61C61×#469—21no—
62C61×#459C2211yescharacteristic
63C61×#447—21no—
64C4×D8×C61C2211yescharacteristic
65C61×#436C2211yescharacteristic
66C61×#418—21no—
67C2×#11758222—21no—
68C2×#11758190—21no—
69C2×#11758190—21no—
70C2×#11758200—21no—
71C2×#11758200—21no—
72C2×#11758200D411yescharacteristic
73C2×#11758199—21no—
74C2×#11758199—21no—
75C2×#11758199—41no—
76C2×#11758194—21no—
77C22×#92418—21no—
78#14519777C2311yescharacteristic
79#14519774C2311yescharacteristic
80#14519773—21no—
81#14519770—41no—
82#14519764—21no—
83#14519756—21no—
84#14519751—21no—
85#14519750—41no—
86C4×Dic122C2311yescharacteristic
87#14519742—21no—
88#14519739—21no—
89#14519736—21no—
90#14519710—21no—
91#14519710—21no—
92#14519708D411yescharacteristic
93C4⋊#92418C2311yescharacteristic
94C4⋊#92418C2311yescharacteristic
95C4⋊#92418C2311yescharacteristic
96#14519706—21no—
97#14519701—41no—
98#14519700—21no—
99#14519700—21no—
100#14519700D411yescharacteristic
101#14519698—21no—
102#14519698—21no—
103#14519698—21no—
104#14519698—21no—
105#14519698D411yescharacteristic
106#14519697—21no—
107#14519697—21no—
108#14519697—21no—
109#14519697—21no—
110C22⋊#92418—21no—
111C22⋊#92418—21no—
112#14519687—21no—
113C4×#92418C2311yescharacteristic
114C4×#92418C2311yescharacteristic
115C22×D4×C61—21no—
116C22×D4×C61—21no—
117C2×D8×C61—21no—
118C2×D8×C61D411yescharacteristic
119C2×M16×C61—21no—
120C4⋊Q8×C61C2311yescharacteristic
121#127×C61C2311yescharacteristic
122#124×C61—41no—
123#121×C61—21no—
124#121×C61—21no—
125#121×C61—21no—
126#121×C61—21no—
127#120×C61—41no—
128C4×D4×C61C2311yescharacteristic
129C4×D4×C61C2311yescharacteristic
130C4×D4×C61C2311yescharacteristic
131#117×C61—21no—
132#117×C61—21no—
133#107×C61D411yescharacteristic
134#106×C61—21no—
135C4⋊C8×C61C2311yescharacteristic
136#102×C61—21no—
137#102×C61—21no—
138#102×C61—21no—
139#102×C61—21no—
140#102×C61D411yescharacteristic
141#102×C61D411yescharacteristic
142C22⋊C8×C61—21no—
143C4×C8×C61C2311yescharacteristic
144C24×C61—41no—
145C2×Q8×C61—21no—
146C2×D4×C61—21no—
147C2×D4×C61—21no—
148C2×D4×C61—21no—
149C2×D4×C61—41no—
150C2×D4×C61—41no—
151C2×D4×C61—42no—
152C2×D4×C61—42no—
153C2×D4×C61—41no—
154C2×D4×C61—81no—
155C2×D4×C61—81no—
156C2×D4×C61C2×D411yescharacteristic
157C2×D4×C61C2×D411yescharacteristic
158C2×D4×C61C2×D411yescharacteristic
159C22×C4×C61—21no—
160C22×C4×C61—21no—
161C22×C4×C61—21no—
162C2×C22⋊D61—41no—
163C2×C22⋊D61—41no—
164#11758222—42no—
165#11758222—81no—
166C2×D244—21no—
167C2×D244—21no—
168C2×D244—21no—
169C2×C4×D61—41no—
170C2×Dic122—21no—
171C2×Dic122C2×D411yescharacteristic
172D8×C61—41no—
173D8×C61—81no—
174M16×C61—81no—
175C2×C8×C61—21no—
176C2×C8×C61C2×D411yescharacteristic
177C2×C8×C61C2×D411yescharacteristic
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61C2×D411yescharacteristic
181C4⋊C4×C61C2×D411yescharacteristic
182C4⋊C4×C61C2×D411yescharacteristic
183C22⋊C4×C61—21no—
184C22⋊C4×C61—21no—
185C22⋊C4×C61—21no—
186C22⋊C4×C61—41no—
187C22⋊C4×C61—41no—
188C22⋊C4×C61—41no—
189C42×C61—41no—
190C42×C61C2411yescharacteristic
191C22⋊Dic61—41no—
192C22⋊Dic61—41no—
193#11758200—41no—
194#11758200—81no—
195#11758200—81no—
196#11758199—81no—
197#11758199—81no—
198#11758199—82no—
199#11758198—41no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61C2×D411yescharacteristic
203#11758196—21no—
204#11758196—41no—
205C4×Dic61—21no—
206C4×Dic61—41no—
207#11758194—81no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—81no—
213C2×#92418C2×D411yescharacteristic
214C2×#92418C2×D411yescharacteristic
215#11758190—42no—
216#11758190—81no—
217C23×C61—21no—
218C23×C61—21no—
219C23×C61—21no—
220C23×C61—41no—
221C23×C61—82no—
222C23×C61—82no—
223C23×C61—161no—
224C22×D61—41no—
225Q8×C61—81no—
226D4×C61—21no—
227D4×C61—21no—
228D4×C61—21no—
229D4×C61—21no—
230D4×C61—21no—
231D4×C61—21no—
232D4×C61—41no—
233D4×C61—41no—
234D4×C61—41no—
235D4×C61—41no—
236D4×C61—81no—
237D4×C61—82no—
238D4×C61—82no—
239D4×C61—161no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
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—21no—
273#92418—41no—
274#92418—41no—
275#92418—41no—
276#92418—41no—
277#20258—611nomaximal, Sylow
278C22×C61—41no—
279C22×C61—41no—
280C22×C61—41no—
281C22×C61—41no—
282C22×C61—41no—
283C22×C61—41no—
284C22×C61—41no—
285C22×C61—41no—
286C22×C61—41no—
287C22×C61—81no—
288C22×C61—81no—
289C22×C61—81no—
290C22×C61—164no—
291C22×C61D4211yescharacteristic
292C2×D61—81no—
293C2×D61—81no—
294C2×D61—81no—
295C4×C61—21no—
296C4×C61—41no—
297C4×C61—41no—
298C4×C61—41no—
299C4×C61—81no—
300C4×C61#57511yescharacteristic
301C4×C61#57511yescharacteristic
302C4×C61#57611yescharacteristic
303C4×C61#57611yescharacteristic
304Dic61—41no—
305Dic61—41no—
306Dic61—81no—
307#3397—611no—
308#3325D6111yescharacteristic, complemented
309#3324—611no—
310#3305—611no—
311#1717—611no—
312#1698—611no—
313#1697—611no—
314#1685—611no—
315#1681—611no—
316#1646—611no—
317#1638—611no—
318#1621—611no—
319#1619—611no—
320#1592—611no—
321#1581—611no—
322C2×C61—81no—
323C2×C61—81no—
324C2×C61—81no—
325C2×C61—161no—
326C2×C61#326811yescharacteristic
327C2×C61#327311yescharacteristic
328C2×C61#328811yescharacteristic
329D61—161no—
330#556—611no—
331#555C2×D6111yescharacteristic
332#550—611no—
333D42C2×D6111yescharacteristic
334#496—1221no—
335#493—611no—
336#492—611no—
337#487—611no—
338#486—611no—
339#486—611no—
340#486C2×D6111yescharacteristic
341#482—21no—
342#482—1221no—
343#482—1221no—
344#477—611no—
345#476C2×D6111yescharacteristic
346#474—611no—
347#469—21no—
348#468—1221no—
349#466—1221no—
350#463—1221no—
351#461—611no—
352#460—611no—
353#460—1221no—
354#459—611no—
355#459—611no—
356#459C2×D6111yescharacteristic
357#450—1221no—
358#449—1221no—
359#447—21no—
360#446—611no—
361#440—1221no—
362C4×SD16—611no—
363C4×D8C2×D6111yescharacteristic
364#436—611no—
365#436C2×D6111yescharacteristic
366C8×D4—611no—
367#418—21no—
368#418—1221no—
369#416—1221no—
370#335—611no—
371#335—611no—
372#333—611no—
373#332—611no—
374#331—611no—
375#331—611no—
376Q8⋊C8—611no—
377D4⋊C8—611no—
378D4⋊C8—611no—
379D4⋊C8—611no—
380#322—611no—
381C61#2025811yescharacteristic, complemented, Sylow
382C2×#41—1221no—
383C22×D4—21no—
384C22×D4—21no—
385C2×SD16—611no—
386C2×SD16—1221no—
387C2×SD16—1221no—
388C2×SD16—1221no—
389C2×SD16—1221no—
390C2×D8—21no—
391C2×D8—1221no—
392C2×D8—1221no—
393C2×D8—2441no—
394C2×D8C22⋊D6111yescharacteristic
395C2×M16—21no—
396C2×M16—1221no—
397C22×C8—1221no—
398C4⋊Q8—611no—
399C4⋊Q8C22×D6111yescharacteristic
400#127—611no—
401#127—1221no—
402#127C22×D6111yescharacteristic
403#126—2441no—
404#124—41no—
405#124—1221no—
406C22⋊Q8—1221no—
407#121—21no—
408#121—21no—
409#121—21no—
410#121—21no—
411#121—1221no—
412#121—2441no—
413#120—41no—
414C4×Q8—611no—
415C4×Q8—1221no—
416C4×D4—1221no—
417C4×D4C22×D6111yescharacteristic
418C4×D4C22×D6111yescharacteristic
419C4×D4C22×D6111yescharacteristic
420#117—21no—
421#117—21no—
422#117—1221no—
423#107—1221no—
424#107—1221no—
425#107C22⋊D6111yescharacteristic
426#106—21no—
427#106—611no—
428C4⋊C8—611no—
429C4⋊C8—611no—
430C4⋊C8—611no—
431C4⋊C8—1221no—
432C4⋊C8C22×D6111yescharacteristic
433#103—611no—
434#103—1221no—
435#103—1221no—
436#103—2441no—
437#102—21no—
438#102—21no—
439#102—21no—
440#102—21no—
441#102—611no—
442#102—1221no—
443#102—1221no—
444#102—1221no—
445#102—1221no—
446#102—1221no—
447#102—1221no—
448#102—1221no—
449#102—1221no—
450#102C22⋊D6111yescharacteristic
451#102C22⋊D6111yescharacteristic
452C22⋊C8—21no—
453C22⋊C8—1221no—
454C22⋊C8—1221no—
455#97—1221no—
456C4×C8—611no—
457C4×C8—611no—
458C4×C8C22×D6111yescharacteristic
459C24—41no—
460#41—4881no—
461#41—2442no—
462C2×Q8—21no—
463C2×Q8—611no—
464C2×Q8—1221no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—41no—
469C2×D4—41no—
470C2×D4—42no—
471C2×D4—42no—
472C2×D4—41no—
473C2×D4—81no—
474C2×D4—81no—
475C2×D4—1221no—
476C2×D4—1221no—
477C2×D4—1221no—
478C2×D4—2441no—
479C2×D4—2441no—
480C2×D4D4×D6111yescharacteristic
481C2×D4C2×C22⋊D6111yescharacteristic
482C2×D4C2×C22⋊D6111yescharacteristic
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—2441no—
487SD16—4881no—
488SD16—4881no—
489SD16—4881no—
490SD16—2442no—
491SD16—2441no—
492D8—41no—
493D8—81no—
494D8—4881no—
495D8—4881no—
496D8—4882no—
497M16—81no—
498M16—4881no—
499C2×C8—21no—
500C2×C8—611no—
501C2×C8—611no—
502C2×C8—1221no—
503C2×C8—1221no—
504C2×C8—1221no—
505C2×C8—1221no—
506C2×C8—4881no—
507C2×C8C2×C22⋊D6111yescharacteristic
508C2×C8C2×C22⋊D6111yescharacteristic
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—611no—
512C4⋊C4—1221no—
513C4⋊C4—1221no—
514C4⋊C4—1221no—
515C4⋊C4—2441no—
516C4⋊C4D4×D6111yescharacteristic
517C4⋊C4C2×C22⋊D6111yescharacteristic
518C4⋊C4C2×C22⋊D6111yescharacteristic
519C22⋊C4—21no—
520C22⋊C4—21no—
521C22⋊C4—21no—
522C22⋊C4—41no—
523C22⋊C4—41no—
524C22⋊C4—41no—
525C22⋊C4—2441no—
526C22⋊C4—2441no—
527C22⋊C4—2441no—
528C42—41no—
529C42—1221no—
530C42—2441no—
531C42C23×D6111yescharacteristic
532C23—21no—
533C23—21no—
534C23—21no—
535C23—41no—
536C23—82no—
537C23—82no—
538C23—161no—
539C23—2441no—
540Q8—81no—
541Q8—1221no—
542Q8—1221no—
543Q8—2441no—
544Q8—2441no—
545D4—21no—
546D4—21no—
547D4—21no—
548D4—21no—
549D4—21no—
550D4—21no—
551D4—41no—
552D4—41no—
553D4—41no—
554D4—41no—
555D4—81no—
556D4—82no—
557D4—82no—
558D4—161no—
559D4—9761no—
560D4—4881no—
561D4—4882no—
562D4—2441no—
563D4—2441no—
564D4—2441no—
565D4—2441no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—1221no—
577C2×C4—1221no—
578C2×C4—4882no—
579C2×C4—2441no—
580C2×C4C2×D4×D6111yescharacteristic
581C2×C4C22×C22⋊D6111yescharacteristic
582C2×C4#1451982211yescharacteristic
583C8—21no—
584C8—21no—
585C8—41no—
586C8—1221no—
587C8—1221no—
588C8—2441no—
589C8—2441no—
590C8—2441no—
591C8—2441no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—41no—
600C22—41no—
601C22—81no—
602C22—81no—
603C22—81no—
604C22—164no—
605C22—4881no—
606C22—4881no—
607C22—4881no—
608C22D4×C22⋊D6111yescharacteristic
609C4—21no—
610C4—41no—
611C4—41no—
612C4—41no—
613C4—81no—
614C4—4881no—
615C4—2441no—
616C4—2441no—
617C4#1684055811yescharacteristic
618C4#1684056011yescharacteristic
619C4#1684056511yescharacteristic
620C4#1684056611yescharacteristic
621C2—81no—
622C2—81no—
623C2—81no—
624C2—161no—
625C2—9761no—
626C2#2145735511yescharacteristic
627C2#2145742411yescharacteristic
628C2#2145758311yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3325C2
socleC22×C61D42
hypercenter#3325D61
hypocenterC61#20258
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3325

Complements

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

NH ≅ G/Ntype
1. #21458837C2semidirect
2. #21458050C2semidirect
3. #21457763C2semidirect
4. #21445460C2semidirect
5. #21445356C2semidirect
6. #21445354C2semidirect
7. #21445284C2semidirect
8. #21445254C2semidirect
9. #21445047C2semidirect
10. #21444989C2semidirect
11. #21444883C2semidirect
12. #21444873C2semidirect
13. #21444721C2semidirect
14. #21444649C2semidirect
15. C61×#3325C2semidirect
17. #16840433C22semidirect
20. #16840135C22semidirect
21. #16840112C22semidirect
22. #16840110C22semidirect
26. #16840054C22semidirect
31. #16839967C22semidirect
38. #16839892C22semidirect
42. D4×#92418C22semidirect
45. #16839515C22semidirect
46. #16839514C22semidirect
47. #16839511C22semidirect
48. #16839509C22semidirect
50. #16839505C22semidirect
51. #16839492C22semidirect
54. D4⋊#92418C22semidirect
55. #16839484C22semidirect
308. #3325D61semidirect
381. C61#20258semidirect

Maximal subgroups

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

subgroupindex
#214588372
#214580502
#214577632
#214454602
#214453562
#214453542
#214452842
#214452542
#214450472
#214449892
#214448832
#214448732
#214447212
#214446492
C61×#33252
#2025861