Finite Group Databasebeta

← #29665411

Subgroups — #29665411

Order 15616.

704 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456756C211yesmaximal, characteristic, complemented
2#21455600C211yesmaximal, characteristic, complemented
3#21455294C211yesmaximal, characteristic
4#21454926C211yesmaximal, characteristic, complemented
5#21445236C211yesmaximal, characteristic, complemented
6#21445234C211yesmaximal, characteristic, complemented
7#21445228C211yesmaximal, characteristic, complemented
8#21445226C211yesmaximal, characteristic, complemented
9#21445170C211yesmaximal, characteristic, complemented
10#21445164C211yesmaximal, characteristic, complemented
11#21445162C211yesmaximal, characteristic, complemented
12#21444588C211yesmaximal, characteristic, complemented
13#21444508C211yesmaximal, characteristic, complemented
14#331×D61C211yesmaximal, characteristic, complemented
15C61×#1665C211yesmaximal, characteristic, complemented
16C2×#14519801—21no—
17#16840317C2211yescharacteristic
18#127×D61C2211yescharacteristic
19C4×#11758224C2211yescharacteristic
20#16840136C2211yescharacteristic
21#16840109C2211yescharacteristic
22#16840108C2211yescharacteristic
23#16840068—21no—
24#16840060—21no—
25#16840058C2211yescharacteristic, complemented
26#16840052C2211yescharacteristic, complemented
27#16840044C2211yescharacteristic, complemented
28#16839963C2211yescharacteristic
29#16839961—21no—
30#16839960—21no—
31#16839960—21no—
32#16839959C2211yescharacteristic
33#16839957C2211yescharacteristic
34#16839955—21no—
35#16839911—21no—
36#16839911—21no—
37C8×Dic122C2211yescharacteristic, complemented
38#16839886C2211yescharacteristic, complemented
39D4×#92418C2211yescharacteristic, complemented
40C8×C22⋊D61—21no—
41#16839825—21no—
42#16839813C2211yescharacteristic
43#16839812—21no—
44#16839799—21no—
45#16839784C2211yescharacteristic
46C2×#14519706—21no—
47C4⋊C8×D61C2211yescharacteristic
48C4⋊C8×D61C2211yescharacteristic
49#16839765—21no—
50#16839755—21no—
51#16839513C2211yescharacteristic, complemented
52#16839512C2211yescharacteristic, complemented
53#16839512C2211yescharacteristic, complemented
54#16839510C2211yescharacteristic, complemented
55#16839510C2211yescharacteristic, complemented
56#16839508C2211yescharacteristic, complemented
57#16839506C2211yescharacteristic
58#16839505C2211yescharacteristic
59#16839505C2211yescharacteristic
60C61×#492C2211yescharacteristic
61C61×#474C2211yescharacteristic, complemented
62C61×#459C2211yescharacteristic
63C8×D4×C61C2211yescharacteristic, complemented
64C61×#334C2211yescharacteristic, complemented
65C61×#332C2211yescharacteristic, complemented
66C61×#331C2211yescharacteristic
67C2×#11758224D411yescharacteristic
68C2×D4×D61—21no—
69C2×D4×D61—21no—
70C2×D4×D61—21no—
71#14519801—44no—
72C2×#11758190—21no—
73C2×#11758200—21no—
74C2×#11758201—21no—
75C2×#11758201—21no—
76C2×#11758199—21no—
77C2×#11758189—21no—
78C22×#92418—21no—
79C2×C8×D61—21no—
80C2×C8×D61—21no—
81#14519777C2311yescharacteristic
82#14519774C2311yescharacteristic
83#14519773—21no—
84#14519773—21no—
85#14519773—21no—
86#14519768—21no—
87#14519757—21no—
88#14519753—21no—
89Q8×Dic61—21no—
90C4×Dic122C2311yescharacteristic
91#14519742—21no—
92#14519742D411yescharacteristic
93D4×Dic61D411yescharacteristic
94C4×C22⋊D61—21no—
95#14519737—21no—
96#14519735—21no—
97#14519735D411yescharacteristic
98#14519733—21no—
99C2×C4×Dic61—21no—
100C42×D61C2311yescharacteristic
101#14519711—21no—
102#14519708—21no—
103#14519708—21no—
104C4⋊#92418C2311yescharacteristic
105C4⋊#92418C2311yescharacteristic
106C4⋊#92418C2311yescharacteristic
107#14519706—21no—
108#14519706—21no—
109#14519706—21no—
110#14519706—21no—
111#14519706—21no—
112#14519706D411yescharacteristic
113#14519706D411yescharacteristic
114#14519702—21no—
115#14519700—21no—
116#14519699—21no—
117#14519699—21no—
118#14519699—21no—
119#14519698—21no—
120#14519698—21no—
121#14519698—21no—
122#14519698—21no—
123#14519697—21no—
124C22⋊#92418—21no—
125#14519689D411yescharacteristic
126#14519689D411yescharacteristic
127C4×#92418C2311yescharacteristic
128C8×Dic61—21no—
129C8×Dic61—21no—
130C2×SD16×C61—21no—
131C2×SD16×C61—21no—
132C2×D8×C61—21no—
133C22×C8×C61—21no—
134C4⋊Q8×C61C2311yescharacteristic
135#127×C61C2311yescharacteristic
136C4×D4×C61C2311yescharacteristic
137#107×C61—21no—
138C4⋊C8×C61C2311yescharacteristic
139C4⋊C8×C61C2311yescharacteristic
140C4⋊C8×C61C2311yescharacteristic
141#102×C61—21no—
142#102×C61—21no—
143C22⋊C8×C61—21no—
144C4×C8×C61C2311yescharacteristic
145C23×D61—41no—
146C2×Q8×C61—21no—
147C2×D4×C61—21no—
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61C2×D411yescharacteristic
151C22×C4×C61—21no—
152C2×C22⋊D61—21no—
153C2×C22⋊D61—41no—
154C2×C22⋊D61—41no—
155C2×C22⋊D61—41no—
156C22×Dic61—21no—
157C22×Dic61—21no—
158#11758224—22no—
159#11758224SD1614yescomplemented
160D4×D61—42no—
161D4×D61—42no—
162D4×D61—81no—
163D4×D61—81no—
164D4×D61—82no—
165C2×D244—21no—
166C2×D244—21no—
167C2×D244—21no—
168C2×C4×D61C2×D411yescharacteristic
169C2×C4×D61C2×D411yescharacteristic
170C2×C4×D61#4111yescharacteristic
171C2×Dic122—21no—
172C2×Dic122C2×D411yescharacteristic
173SD16×C61—81no—
174SD16×C61—81no—
175D8×C61—42no—
176C2×C8×C61—21no—
177C2×C8×C61—21no—
178C2×C8×C61—21no—
179C2×C8×C61—81no—
180C2×C8×C61C2×D411yescharacteristic
181C2×C8×C61C2×D411yescharacteristic
182C4⋊C4×C61—21no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61C2×D411yescharacteristic
185C22⋊C4×C61—21no—
186C42×C61C2411yescharacteristic
187C22⋊Dic61—21no—
188#11758201—81no—
189#11758201—81no—
190#11758200—42no—
191#11758199—42no—
192#11758198—21no—
193#11758198—41no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—41no—
197C4⋊Dic61C2×D411yescharacteristic
198#11758196—21no—
199#11758196—21no—
200#11758196—41no—
201C4×Dic61—21no—
202C4×Dic61—21no—
203C4×Dic61—21no—
204C4×Dic61—41no—
205C4×Dic61C2×D411yescharacteristic
206C4×Dic61C2×D411yescharacteristic
207C4×Dic61#4111yescharacteristic
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—81no—
212C2×#92418C2×D411yescharacteristic
213C2×#92418C2×D411yescharacteristic
214#11758190—42no—
215#11758189—42no—
216C8×D61—42no—
217C8×D61—81no—
218C23×C61—21no—
219C23×C61—41no—
220C22×D61—41no—
221C22×D61—82no—
222C22×D61—82no—
223C22×D61—82no—
224C22×D61#12111yescharacteristic
225Q8×C61—41no—
226Q8×C61—41no—
227D4×C61—21no—
228D4×C61—41no—
229D4×C61—41no—
230D4×C61—41no—
231D4×C61—41no—
232D4×C61—81no—
233D4×C61C2×SD1612yes—
234C2×C4×C61—21no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61C22×D411yescharacteristic
239C2×C4×C61C22×D411yescharacteristic
240C2×C4×C61C2×#4111yescharacteristic
241C22⋊D61—24no—
242C22⋊D61—82no—
243C22⋊D61—82no—
244C22⋊D61—82no—
245C2×Dic61—21no—
246C2×Dic61—24no—
247C2×Dic61—21no—
248C2×Dic61—41no—
249C2×Dic61—81no—
250C2×Dic61#12111yescharacteristic
251C2×Dic61#12111yescharacteristic
252C2×Dic61#12111yescharacteristic
253D244—41no—
254D244—41no—
255D244—41no—
256D244—41no—
257D244—81no—
258C4×D61—21no—
259C4×D61—21no—
260C4×D61—41no—
261C4×D61C2×SD1612yes—
262C4×D61#13512yes—
263Dic122—21no—
264Dic122—81no—
265Dic122C2×SD1612yes—
266C8×C61—21no—
267C8×C61—21no—
268C8×C61—41no—
269C8×C61—41no—
270C8×C61—41no—
271#92418—21no—
272#92418—21no—
273#92418—41no—
274#92418—41no—
275#92418—41no—
276#19409—611nomaximal, Sylow
277C22×C61—22no—
278C22×C61—41no—
279C22×C61—41no—
280C22×C61—81no—
281C22×C61—81no—
282C22×C61—81no—
283C22×C61C2×#12111yescharacteristic
284C2×D61—21no—
285C2×D61—21no—
286C2×D61—81no—
287C2×D61—81no—
288C2×D61—81no—
289C2×D61—164no—
290C2×D61#46512yes—
291C4×C61—21no—
292C4×C61—41no—
293C4×C61—81no—
294C4×C61C22×SD1611yescharacteristic
295C4×C61C2×#13511yescharacteristic
296C4×C61#57611yescharacteristic
297C4×C61#57711yescharacteristic
298Dic61—21no—
299Dic61—22no—
300Dic61—21no—
301Dic61—41no—
302Dic61—81no—
303Dic61#46512yes—
304#3208—611no—
305#3076—611no—
306#3038—611no—
307#2988—611no—
308#1678—611no—
309#1678—611no—
310#1677—611no—
311#1677—611no—
312#1666—611no—
313#1666—611no—
314#1665—611no—
315#1665D6111yescharacteristic, complemented
316#1570—611no—
317#1550—611no—
318C2×#331—611no—
319C2×C61—42no—
320C2×C61—161no—
321C2×C61#304511yescharacteristic
322C2×C61C2×#46511yescharacteristic
323C2×C61#306811yescharacteristic
324D61—22no—
325D61—161no—
326C2×AGL(1,ℤ/8)—1221no—
327#533—611no—
328C2×#127—611no—
329C4×#41—611no—
330#492—611no—
331#492C2×D6111yescharacteristic
332#486—611no—
333#486—611no—
334#479—1221no—
335#478—1221no—
336#477—611no—
337#476—611no—
338#475—611no—
339#474C2×D6111yescharacteristic, complemented
340#460—611no—
341#460—611no—
342#460—1221no—
343#460—1221no—
344#460—1221no—
345#459—611no—
346#459—1221no—
347#459C2×D6111yescharacteristic
348#450—1221no—
349#450—1221no—
350C8×Q8—611no—
351C8×Q8—611no—
352C8×D4—611no—
353C8×D4—1221no—
354C8×D4—1221no—
355C8×D4C2×D6111yescharacteristic, complemented
356#431—611no—
357#431—1221no—
358#427—1221no—
359#423—611no—
360C2×C4⋊C8—611no—
361C2×C4⋊C8—611no—
362C2×C4⋊C8—1221no—
363#418—1221no—
364#416—1221no—
365#334—611no—
366#334—611no—
367#334—611no—
368#334C2×D6111yescharacteristic, complemented
369#333—611no—
370#333—611no—
371#332—611no—
372#332C2×D6111yescharacteristic, complemented
373#331—611no—
374#331—611no—
375#331—611no—
376#331C2×D6111yescharacteristic
377C61#1940911yescharacteristic, complemented, Sylow
378C2×#41—611no—
379C22×D4—1221no—
380C22×D4—1221no—
381C22×D4—1221no—
382AGL(1,ℤ/8)—2444no—
383C2×SD16—21no—
384C2×SD16—21no—
385C2×SD16—1221no—
386C2×SD16—1221no—
387C2×SD16—1221no—
388C2×SD16—1221no—
389C2×D8—21no—
390C2×D8—1221no—
391C2×M16—1221no—
392C22×C8—21no—
393C22×C8—1221no—
394C22×C8—1221no—
395C22×C8—1221no—
396C4⋊Q8—611no—
397C4⋊Q8C22×D6111yescharacteristic
398#127—611no—
399#127—1221no—
400#127—1221no—
401#127—1221no—
402#127C22×D6111yescharacteristic
403#125—1221no—
404C22⋊Q8—1221no—
405C22⋊Q8—1221no—
406C4×Q8—611no—
407C4×Q8—611no—
408C4×Q8—1221no—
409C4×Q8—1221no—
410C4×D4—611no—
411C4×D4—1221no—
412C4×D4—1221no—
413C4×D4C22×D6111yescharacteristic
414#117—611no—
415#117—1221no—
416#117—1221no—
417C2×C42—611no—
418C2×C42—1221no—
419#107—21no—
420#107—1221no—
421#106—1221no—
422#106—1221no—
423C4⋊C8—611no—
424C4⋊C8—611no—
425C4⋊C8—611no—
426C4⋊C8—611no—
427C4⋊C8—611no—
428C4⋊C8—1221no—
429C4⋊C8—1221no—
430C4⋊C8—1221no—
431C4⋊C8—1221no—
432C4⋊C8—1221no—
433C4⋊C8C22×D6111yescharacteristic
434C4⋊C8C22×D6111yescharacteristic
435C4⋊C8C22×D6111yescharacteristic
436#103—1221no—
437#103—1221no—
438#102—21no—
439#102—21no—
440#102—1221no—
441#102—1221no—
442#102—1221no—
443#102—1221no—
444#102—1221no—
445#102—1221no—
446#102—1221no—
447#102—1221no—
448C22⋊C8—21no—
449C22⋊C8—611no—
450C22⋊C8—611no—
451C22⋊C8—1221no—
452C4×C8—611no—
453C4×C8—1221no—
454C4×C8—1221no—
455C4×C8C22×D6111yescharacteristic
456C24—2441no—
457#41—614no—
458#41—1222no—
459C2×Q8—21no—
460C2×Q8—611no—
461C2×Q8—1221no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—1221no—
466C2×D4—1221no—
467C2×D4—1221no—
468C2×D4—1221no—
469C2×D4—4881no—
470C2×D4—4881no—
471C2×D4—4882no—
472C2×D4—2442no—
473C2×D4—2442no—
474C2×D4—2441no—
475C2×D4—2441no—
476C2×D4—2441no—
477C2×D4D4×D6111yescharacteristic
478C22×C4—21no—
479C22×C4—611no—
480C22×C4—611no—
481C22×C4—611no—
482C22×C4—1221no—
483C22×C4—1221no—
484SD16—81no—
485SD16—81no—
486SD16—4881no—
487SD16—4881no—
488SD16—2442no—
489SD16—2442no—
490D8—42no—
491D8—2442no—
492M16—2442no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—81no—
497C2×C8—611no—
498C2×C8—611no—
499C2×C8—1221no—
500C2×C8—1221no—
501C2×C8—1221no—
502C2×C8—4881no—
503C2×C8—4881no—
504C2×C8—2442no—
505C2×C8D4×D6111yescharacteristic
506C2×C8D4×D6111yescharacteristic
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—611no—
510C4⋊C4—1221no—
511C4⋊C4—1221no—
512C4⋊C4—1221no—
513C4⋊C4—1221no—
514C4⋊C4—2441no—
515C4⋊C4—2441no—
516C4⋊C4D4×D6111yescharacteristic
517C22⋊C4—21no—
518C22⋊C4—1221no—
519C22⋊C4—1221no—
520C22⋊C4—2441no—
521C42—611no—
522C42—611no—
523C42—611no—
524C42—1221no—
525C42—1221no—
526C42—1221no—
527C42—2441no—
528C42C23×D6111yescharacteristic
529C23—21no—
530C23—41no—
531C23—611no—
532C23—4882no—
533C23—4882no—
534C23—4882no—
535C23—2441no—
536Q8—41no—
537Q8—41no—
538Q8—612no—
539Q8—1221no—
540Q8—4881no—
541D4—21no—
542D4—41no—
543D4—41no—
544D4—41no—
545D4—41no—
546D4—81no—
547D4—1224no—
548D4—4881no—
549D4—4882no—
550D4—4882no—
551D4—4882no—
552D4—2441no—
553D4—2441no—
554D4—2441no—
555D4—2441no—
556D4SD16×D6112yes—
557C2×C4—21no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—611no—
562C2×C4—612no—
563C2×C4—612no—
564C2×C4—611no—
565C2×C4—611no—
566C2×C4—1224no—
567C2×C4—1221no—
568C2×C4—1221no—
569C2×C4—1221no—
570C2×C4—1221no—
571C2×C4—4881no—
572C2×C4—2441no—
573C2×C4—2441no—
574C2×C4C2×D4×D6111yescharacteristic
575C2×C4C2×D4×D6111yescharacteristic
576C2×C4#41×D6111yescharacteristic
577C8—21no—
578C8—21no—
579C8—41no—
580C8—41no—
581C8—41no—
582C8—1221no—
583C8—1221no—
584C8—2441no—
585C8—2441no—
586C8—2441no—
587C22—22no—
588C22—41no—
589C22—41no—
590C22—81no—
591C22—81no—
592C22—81no—
593C22—612no—
594C22—1221no—
595C22—1221no—
596C22—9764no—
597C22—4881no—
598C22—4881no—
599C22—4881no—
600C22#121×D6111yescharacteristic
601C4—21no—
602C4—41no—
603C4—81no—
604C4—612no—
605C4—1222no—
606C4—1221no—
607C4—1221no—
608C4—4881no—
609C4—2441no—
610C4C2×SD16×D6111yescharacteristic
611C4#135×D6111yescharacteristic
612C4#1684056311yescharacteristic
613C4#1684057211yescharacteristic
614C2—42no—
615C2—161no—
616C2—1222no—
617C2—9761no—
618C2#2145535311yescharacteristic
619C2#465×D6111yescharacteristic
620C2#2145554211yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1665C2
socleC22×C61C2×#121
prime radical#21455294C2
hypercenter#1665D61
hypocenterC61#19409
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1665

Complements

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

NH ≅ G/Ntype
1. #21456756C2semidirect
2. #21455600C2semidirect
4. #21454926C2semidirect
5. #21445236C2semidirect
6. #21445234C2semidirect
7. #21445228C2semidirect
8. #21445226C2semidirect
9. #21445170C2semidirect
10. #21445164C2semidirect
11. #21445162C2semidirect
12. #21444588C2semidirect
13. #21444508C2semidirect
14. #331×D61C2semidirect
15. C61×#1665C2semidirect
25. #16840058C22semidirect
26. #16840052C22semidirect
27. #16840044C22semidirect
37. C8×Dic122C22semidirect
38. #16839886C22semidirect
39. D4×#92418C22semidirect
51. #16839513C22semidirect
52. #16839512C22semidirect
53. #16839512C22semidirect
54. #16839510C22semidirect
55. #16839510C22semidirect
56. #16839508C22semidirect
61. C61×#474C22semidirect
63. C8×D4×C61C22semidirect
64. C61×#334C22semidirect
65. C61×#332C22semidirect
159. #11758224SD16semidirect
315. #1665D61semidirect
339. #474C2×D61semidirect
355. C8×D4C2×D61semidirect
368. #334C2×D61semidirect
372. #332C2×D61semidirect
377. C61#19409semidirect

Maximal subgroups

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

subgroupindex
#214567562
#214556002
#214552942
#214549262
#214452362
#214452342
#214452282
#214452262
#214451702
#214451642
#214451622
#214445882
#214445082
#331×D612
C61×#16652
#1940961