Finite Group Databasebeta

← #29663964

Subgroups — #29663964

Order 15616.

722 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#486×D61C211yesmaximal, characteristic, complemented
2#21455633C211yesmaximal, characteristic, complemented
3#21455296C211yesmaximal, characteristic
4C8×#11758226C211yesmaximal, characteristic, complemented
5#21445237C211yesmaximal, characteristic, complemented
6#21445235C211yesmaximal, characteristic, complemented
7#21445233C211yesmaximal, characteristic, complemented
8#21445231C211yesmaximal, characteristic, complemented
9#21445229C211yesmaximal, characteristic, complemented
10#21445227C211yesmaximal, characteristic, complemented
11#21445225C211yesmaximal, characteristic, complemented
12#21444573C211yesmaximal, characteristic, complemented
13#21444498C211yesmaximal, characteristic, complemented
14#21444485C211yesmaximal, characteristic, complemented
15C61×#1678C211yesmaximal, characteristic, complemented
16C2×#14519804—21no—
17C4⋊Q8×D61C2211yescharacteristic
18#127×D61C2211yescharacteristic
19C4×#11758226C2211yescharacteristic
20#16840110C2211yescharacteristic
21#16840109C2211yescharacteristic
22#16840108C2211yescharacteristic
23#16840107—21no—
24#16840107—21no—
25#16840057C2211yescharacteristic, complemented
26#16840053C2211yescharacteristic, complemented
27#16840049C2211yescharacteristic, complemented
28#16840045—21no—
29#16840041—21no—
30#16839962C2211yescharacteristic
31#16839960—21no—
32#16839958C2211yescharacteristic
33#16839956—21no—
34#16839954C2211yescharacteristic
35Q8×#92418C2211yescharacteristic, complemented
36C8×D244—21no—
37C8×D244C2211yescharacteristic, complemented
38#16839827—21no—
39#16839827C2211yescharacteristic, complemented
40#16839814—21no—
41#16839814C2211yescharacteristic
42#16839803—21no—
43#16839784C2211yescharacteristic
44#16839784C2211yescharacteristic
45#16839765—21no—
46#16839762—21no—
47#102×D61—21no—
48#102×D61—21no—
49C4×C8×D61—21no—
50C4×C8×D61C2211yescharacteristic
51#16839513C2211yescharacteristic, complemented
52#16839512C2211yescharacteristic, complemented
53#16839512C2211yescharacteristic, complemented
54#16839511C2211yescharacteristic, complemented
55#16839510C2211yescharacteristic, complemented
56#16839510C2211yescharacteristic, complemented
57#16839506C2211yescharacteristic
58#16839505C2211yescharacteristic
59#16839505C2211yescharacteristic
60C61×#486C2211yescharacteristic
61C61×#477C2211yescharacteristic, complemented
62C61×#460C2211yescharacteristic
63C8×Q8×C61C2211yescharacteristic, complemented
64C61×#334C2211yescharacteristic, complemented
65C61×#333C2211yescharacteristic, complemented
66C61×#331C2211yescharacteristic
67C2×#11758226D411yescharacteristic
68C2×Q8×D61—21no—
69C2×D4×D61—21no—
70C2×D4×D61—21no—
71C2×D4×D61—21no—
72#14519804—44no—
73C2×#11758201—21no—
74C2×D488—21no—
75C2×#11758199—21no—
76C2×#11758189—21no—
77C2×#11758189—21no—
78C2×C8×D61—21no—
79C2×C8×D61—21no—
80C2×C8×D61D411yescharacteristic
81C2×C8×D61D411yescharacteristic
82#14519777C2311yescharacteristic
83#14519776—21no—
84#14519776—21no—
85#14519775—21no—
86#14519774C2311yescharacteristic
87#14519773—21no—
88#14519773—21no—
89#14519773—21no—
90Q8×Dic61D411yescharacteristic
91C4×D244—21no—
92C4×D244C2311yescharacteristic
93#14519739—21no—
94#14519739D411yescharacteristic
95#14519735—21no—
96#14519735D411yescharacteristic
97C4⋊C4×D61—21no—
98C4⋊C4×D61—21no—
99C42×D61—21no—
100C42×D61C2311yescharacteristic
101#14519710—21no—
102#14519709—21no—
103#14519708—21no—
104C4⋊#92418—21no—
105C4⋊#92418C2311yescharacteristic
106C4⋊#92418C2311yescharacteristic
107C4⋊#92418C2311yescharacteristic
108#14519706—21no—
109#14519706—21no—
110#14519701—21no—
111#14519699—21no—
112#14519699—21no—
113#14519699—21no—
114#14519699—21no—
115#14519698—21no—
116#14519698—21no—
117#14519698—21no—
118#14519697—21no—
119#14519697—21no—
120#14519697—21no—
121#14519697—21no—
122#14519689—21no—
123#14519689—21no—
124#14519689—21no—
125C4×#92418—21no—
126C4×#92418C2311yescharacteristic
127C8×Dic61—21no—
128C8×Dic61D411yescharacteristic
129C8×Dic61D411yescharacteristic
130C2×SD16×C61—21no—
131C4⋊Q8×C61C2311yescharacteristic
132#127×C61C2311yescharacteristic
133C4×Q8×C61C2311yescharacteristic
134#107×C61—21no—
135C4⋊C8×C61—21no—
136C4⋊C8×C61C2311yescharacteristic
137C4⋊C8×C61C2311yescharacteristic
138C4⋊C8×C61C2311yescharacteristic
139#103×C61—21no—
140#102×C61—21no—
141#102×C61—21no—
142#102×C61—21no—
143C4×C8×C61—21no—
144C4×C8×C61C2311yescharacteristic
145C23×D61—41no—
146C2×Q8×C61—21no—
147C2×Q8×C61C2×D411yescharacteristic
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61—21no—
151C2×C22⋊D61—41no—
152C2×C22⋊D61—41no—
153C2×C22⋊D61—41no—
154#11758226—22no—
155#11758226SD1614yes—
156Q8×D61—82no—
157D4×D61—42no—
158D4×D61—42no—
159D4×D61—42no—
160D4×D61—42no—
161D4×D61—82no—
162C2×D244—21no—
163C2×D244—21no—
164C2×D244—21no—
165C2×D244—21no—
166C2×D244C2×D411yescharacteristic
167C2×C4×D61—21no—
168C2×C4×D61—21no—
169C2×C4×D61—21no—
170C2×C4×D61—41no—
171C2×C4×D61C2×D411yescharacteristic
172C2×C4×D61C2×D411yescharacteristic
173C2×C4×D61#4111yescharacteristic
174C2×Dic122—21no—
175C2×Dic122—41no—
176SD16×C61—42no—
177C2×C8×C61—21no—
178C2×C8×C61—21no—
179C2×C8×C61—21no—
180C2×C8×C61C2×D411yescharacteristic
181C2×C8×C61C2×D411yescharacteristic
182C4⋊C4×C61—21no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61C2×D411yescharacteristic
186C42×C61—21no—
187C42×C61C2411yescharacteristic
188#11758201—42no—
189#11758199—42no—
190#11758198—21no—
191#11758198—21no—
192#11758198—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61C2×D411yescharacteristic
197#11758196—41no—
198#11758196—41no—
199C4×Dic61—21no—
200C4×Dic61—21no—
201C4×Dic61C2×D411yescharacteristic
202C4×Dic61C2×D411yescharacteristic
203C4×Dic61#4111yescharacteristic
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418C2×D411yescharacteristic
208C2×#92418C2×D411yescharacteristic
209D488—42no—
210#11758189—42no—
211#11758189—42no—
212C8×D61—41no—
213C8×D61—41no—
214C8×D61—42no—
215C8×D61—42no—
216C23×C61—41no—
217C22×D61—21no—
218C22×D61—41no—
219C22×D61—82no—
220C22×D61—82no—
221C22×D61—82no—
222C22×D61#12111yescharacteristic
223Q8×C61—21no—
224Q8×C61—81no—
225Q8×C61C2×SD1612yes—
226D4×C61—41no—
227D4×C61—41no—
228D4×C61—41no—
229D4×C61—41no—
230D4×C61—81no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—41no—
234C2×C4×C61C22×D411yescharacteristic
235C2×C4×C61C22×D411yescharacteristic
236C2×C4×C61C2×#4111yescharacteristic
237C22⋊D61—82no—
238C22⋊D61—82no—
239C22⋊D61—82no—
240C2×Dic61—21no—
241C2×Dic61—41no—
242C2×Dic61#12111yescharacteristic
243C2×Dic61#12111yescharacteristic
244C2×Dic61#12111yescharacteristic
245D244—24no—
246D244—21no—
247D244—41no—
248D244—41no—
249D244—41no—
250D244—41no—
251D244—81no—
252D244C2×SD1612yes—
253C4×D61—21no—
254C4×D61—21no—
255C4×D61—24no—
256C4×D61—22no—
257C4×D61—41no—
258C4×D61—41no—
259C4×D61—42no—
260C4×D61—82no—
261C4×D61C2×SD1612yes—
262C4×D61#13512yes—
263Dic122—82no—
264Dic122—81no—
265C8×C61—21no—
266C8×C61—21no—
267C8×C61—41no—
268C8×C61—41no—
269C8×C61—41no—
270#92418—21no—
271#92418—21no—
272#92418—41no—
273#92418—41no—
274#92418—41no—
275#19274—611nomaximal, Sylow
276C22×C61—81no—
277C22×C61—81no—
278C22×C61—81no—
279C22×C61C2×#12111yescharacteristic
280C2×D61—21no—
281C2×D61—21no—
282C2×D61—22no—
283C2×D61—41no—
284C2×D61—41no—
285C2×D61—81no—
286C2×D61—81no—
287C2×D61—81no—
288C2×D61—164no—
289C2×D61#46512yes—
290C4×C61—22no—
291C4×C61—21no—
292C4×C61—41no—
293C4×C61—81no—
294C4×C61C22×SD1611yescharacteristic
295C4×C61C2×#13511yescharacteristic
296C4×C61#57511yescharacteristic
297C4×C61#57511yescharacteristic
298Dic61—21no—
299Dic61—41no—
300Dic61—81no—
301Dic61#46512yes—
302Dic61#47212yes—
303C2×#486—611no—
304#3079—611no—
305#3038—611no—
306C8×#41—611no—
307#1678—611no—
308#1678—611no—
309#1678—611no—
310#1678D6111yescharacteristic, complemented
311#1677—611no—
312#1677—611no—
313#1677—611no—
314#1677—611no—
315#1566—611no—
316#1548—611no—
317#1545—611no—
318C2×C61—161no—
319C2×C61#304011yescharacteristic
320C2×C61C2×#46511yescharacteristic
321C2×C61C2×#47211yescharacteristic
322D61—22no—
323D61—42no—
324D61—161no—
325C2×AGL(1,ℤ/8)—1221no—
326C2×C4⋊Q8—611no—
327C2×#127—611no—
328C4×#41—611no—
329#486—611no—
330#486—611no—
331#486—611no—
332#486—1221no—
333#486—1221no—
334#486C2×D6111yescharacteristic
335#477—611no—
336#477C2×D6111yescharacteristic, complemented
337#476—611no—
338#476—611no—
339#475—1221no—
340#474—1221no—
341#460—611no—
342#460—1221no—
343#460C2×D6111yescharacteristic
344#459—611no—
345#459—611no—
346#459—1221no—
347C8×Q8—611no—
348C8×Q8C2×D6111yescharacteristic, complemented
349C8×D4—611no—
350C8×D4—611no—
351C8×D4—1221no—
352C8×D4—1221no—
353#431—611no—
354#431—1221no—
355#428—1221no—
356#423—611no—
357#423—611no—
358#418—1221no—
359#417—1221no—
360C2×#102—1221no—
361C2×#102—1221no—
362C2×C4×C8—611no—
363C2×C4×C8—1221no—
364#334—611no—
365#334—611no—
366#334—611no—
367#334C2×D6111yescharacteristic, complemented
368#333—611no—
369#333—611no—
370#333—611no—
371#333C2×D6111yescharacteristic, complemented
372#331—611no—
373#331—611no—
374#331—611no—
375#331C2×D6111yescharacteristic
376C61#1927411yescharacteristic, complemented, Sylow
377C2×#41—611no—
378C22×Q8—1221no—
379C22×D4—1221no—
380C22×D4—1221no—
381C22×D4—1221no—
382AGL(1,ℤ/8)—2444no—
383C2×SD16—21no—
384C2×SD16—1221no—
385C2×D8—1221no—
386C2×D8—1221no—
387C2×M16—1221no—
388C2×M16—1221no—
389C22×C8—611no—
390C22×C8—611no—
391C22×C8—1221no—
392C22×C8—1221no—
393C4⋊Q8—611no—
394C4⋊Q8—1221no—
395C4⋊Q8—1221no—
396C4⋊Q8—1221no—
397C4⋊Q8C22×D6111yescharacteristic
398#127—611no—
399#127—1221no—
400#127—1221no—
401#127—1221no—
402#127C22×D6111yescharacteristic
403C4×Q8—611no—
404C4×Q8C22×D6111yescharacteristic
405C4×D4—611no—
406C4×D4—611no—
407C4×D4—1221no—
408C4×D4—1221no—
409#117—611no—
410#117—1221no—
411C2×C4⋊C4—1221no—
412C2×C4⋊C4—1221no—
413C2×C42—611no—
414C2×C42—1221no—
415#107—21no—
416#107—1221no—
417#106—1221no—
418#106—1221no—
419C4⋊C8—21no—
420C4⋊C8—611no—
421C4⋊C8—611no—
422C4⋊C8—611no—
423C4⋊C8—1221no—
424C4⋊C8—1221no—
425C4⋊C8—1221no—
426C4⋊C8C22×D6111yescharacteristic
427C4⋊C8C22×D6111yescharacteristic
428C4⋊C8C22×D6111yescharacteristic
429#103—21no—
430#103—1221no—
431#102—21no—
432#102—21no—
433#102—21no—
434#102—1221no—
435#102—1221no—
436#102—1221no—
437#102—1221no—
438#102—1221no—
439#102—1221no—
440#102—1221no—
441#102—1221no—
442#102—1221no—
443#102—1221no—
444#102—1221no—
445C22⋊C8—1221no—
446C22⋊C8—1221no—
447C22⋊C8—1221no—
448C4×C8—21no—
449C4×C8—611no—
450C4×C8—611no—
451C4×C8—611no—
452C4×C8—1221no—
453C4×C8—1221no—
454C4×C8C22×D6111yescharacteristic
455C24—2441no—
456#41—614no—
457#41—1222no—
458C2×Q8—21no—
459C2×Q8—1221no—
460C2×Q8—4882no—
461C2×Q8—2441no—
462C2×Q8D4×D6111yescharacteristic
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—611no—
467C2×D4—1221no—
468C2×D4—1221no—
469C2×D4—1221no—
470C2×D4—1221no—
471C2×D4—4882no—
472C2×D4—2442no—
473C2×D4—2442no—
474C2×D4—2441no—
475C2×D4—2441no—
476C2×D4—2441no—
477C2×D4—2442no—
478C2×D4—2442no—
479C22×C4—611no—
480C22×C4—611no—
481C22×C4—611no—
482C22×C4—1221no—
483C22×C4—1221no—
484C22×C4—1221no—
485C22×C4—2441no—
486SD16—42no—
487SD16—2442no—
488D8—2442no—
489D8—2442no—
490M16—2442no—
491M16—2442no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—611no—
496C2×C8—611no—
497C2×C8—1221no—
498C2×C8—1221no—
499C2×C8—1221no—
500C2×C8—2441no—
501C2×C8—2441no—
502C2×C8—2442no—
503C2×C8—2442no—
504C2×C8D4×D6111yescharacteristic
505C2×C8D4×D6111yescharacteristic
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—611no—
510C4⋊C4—1221no—
511C4⋊C4—1221no—
512C4⋊C4—1221no—
513C4⋊C4—2441no—
514C4⋊C4—2441no—
515C4⋊C4D4×D6111yescharacteristic
516C22⋊C4—1221no—
517C22⋊C4—1221no—
518C22⋊C4—1221no—
519C42—21no—
520C42—611no—
521C42—611no—
522C42—611no—
523C42—1221no—
524C42—1221no—
525C42C23×D6111yescharacteristic
526C23—41no—
527C23—611no—
528C23—1221no—
529C23—4882no—
530C23—4882no—
531C23—4882no—
532C23—2441no—
533Q8—21no—
534Q8—81no—
535Q8—4881no—
536Q8—4882no—
537Q8SD16×D6112yes—
538D4—41no—
539D4—41no—
540D4—41no—
541D4—41no—
542D4—81no—
543D4—612no—
544D4—1224no—
545D4—1221no—
546D4—4881no—
547D4—4882no—
548D4—4882no—
549D4—4882no—
550D4—2441no—
551D4—2441no—
552D4—2441no—
553D4—2441no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—41no—
557C2×C4—611no—
558C2×C4—611no—
559C2×C4—612no—
560C2×C4—612no—
561C2×C4—611no—
562C2×C4—1222no—
563C2×C4—1224no—
564C2×C4—1221no—
565C2×C4—1221no—
566C2×C4—1221no—
567C2×C4—4882no—
568C2×C4—2442no—
569C2×C4—2441no—
570C2×C4—2441no—
571C2×C4—2441no—
572C2×C4C2×D4×D6111yescharacteristic
573C2×C4C2×D4×D6111yescharacteristic
574C2×C4#41×D6111yescharacteristic
575C8—21no—
576C8—21no—
577C8—41no—
578C8—41no—
579C8—41no—
580C8—1221no—
581C8—1221no—
582C8—2441no—
583C8—2441no—
584C8—2441no—
585C22—81no—
586C22—81no—
587C22—81no—
588C22—612no—
589C22—1221no—
590C22—1221no—
591C22—1222no—
592C22—9764no—
593C22—4881no—
594C22—4881no—
595C22—4881no—
596C22—2441no—
597C22—2441no—
598C22#121×D6111yescharacteristic
599C4—22no—
600C4—21no—
601C4—41no—
602C4—81no—
603C4—612no—
604C4—612no—
605C4—1221no—
606C4—4881no—
607C4—2441no—
608C4C2×SD16×D6111yescharacteristic
609C4#135×D6111yescharacteristic
610C4#1684055911yescharacteristic
611C4#1684056111yescharacteristic
612C2—161no—
613C2—1222no—
614C2—9761no—
615C2—2442no—
616C2#2145531011yescharacteristic
617C2#465×D6111yescharacteristic
618C2#472×D6111yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1678C2
socleC22×C61C2×#121
prime radical#21455296C2
hypercenter#1678D61
hypocenterC61#19274
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1678

Complements

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

NH ≅ G/Ntype
1. #486×D61C2semidirect
2. #21455633C2semidirect
4. C8×#11758226C2semidirect
5. #21445237C2semidirect
6. #21445235C2semidirect
7. #21445233C2semidirect
8. #21445231C2semidirect
9. #21445229C2semidirect
10. #21445227C2semidirect
11. #21445225C2semidirect
12. #21444573C2semidirect
13. #21444498C2semidirect
14. #21444485C2semidirect
15. C61×#1678C2semidirect
25. #16840057C22semidirect
26. #16840053C22semidirect
27. #16840049C22semidirect
35. Q8×#92418C22semidirect
37. C8×D244C22semidirect
39. #16839827C22semidirect
51. #16839513C22semidirect
52. #16839512C22semidirect
53. #16839512C22semidirect
54. #16839511C22semidirect
55. #16839510C22semidirect
56. #16839510C22semidirect
61. C61×#477C22semidirect
63. C8×Q8×C61C22semidirect
64. C61×#334C22semidirect
65. C61×#333C22semidirect
310. #1678D61semidirect
336. #477C2×D61semidirect
348. C8×Q8C2×D61semidirect
367. #334C2×D61semidirect
371. #333C2×D61semidirect
376. C61#19274semidirect

Maximal subgroups

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

subgroupindex
#486×D612
#214556332
#214552962
C8×#117582262
#214452372
#214452352
#214452332
#214452312
#214452292
#214452272
#214452252
#214445732
#214444982
#214444852
C61×#16782
#1927461