Finite Group Databasebeta

← #29663626

Subgroups — #29663626

Order 15616.

636 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457162C211yesmaximal, characteristic, complemented
2#21456759C211yesmaximal, characteristic
3#21456190C211yesmaximal, characteristic
4#21454767C211yesmaximal, characteristic, complemented
5#21445390C211yesmaximal, characteristic, complemented
6#21445359C211yesmaximal, characteristic, complemented
7#21445358C211yesmaximal, characteristic, complemented
8#21445265C211yesmaximal, characteristic, complemented
9#21445260C211yesmaximal, characteristic, complemented
10#21445236C211yesmaximal, characteristic, complemented
11#21445233C211yesmaximal, characteristic, complemented
12#21444568C211yesmaximal, characteristic, complemented
13#21444475C211yesmaximal, characteristic, complemented
14#331×D61C211yesmaximal, characteristic
15C61×#1702C211yesmaximal, characteristic, complemented
16#16840317C2211yescharacteristic
17#16840316C2211yescharacteristic
18#127×D61C2211yescharacteristic
19#16840133—21no—
20#16840125—21no—
21#16840125—21no—
22#16840110C2211yescharacteristic
23#16840109C2211yescharacteristic
24#16840108C2211yescharacteristic
25#16840107—21no—
26#16840058C2211yescharacteristic, complemented
27#16840057C2211yescharacteristic, complemented
28#16840055—21no—
29#16840002—21no—
30#16840000—21no—
31#16839970—21no—
32#16839969C2211yescharacteristic, complemented
33#16839963C2211yescharacteristic
34#16839962C2211yescharacteristic
35#16839959C2211yescharacteristic
36#16839914—21no—
37#16839911—21no—
38#16839890—21no—
39#16839889—21no—
40#16839889C2211yescharacteristic, complemented
41Q8×#92418C2211yescharacteristic, complemented
42C8×Dic122C2211yescharacteristic, complemented
43#16839814C2211yescharacteristic
44#16839787—21no—
45C4⋊C8×D61C2211yescharacteristic
46C4⋊C8×D61C2211yescharacteristic
47#16839765—21no—
48#16839765—21no—
49#16839719—21no—
50C4×#11758189C2211yescharacteristic
51#16839515C2211yescharacteristic, complemented
52#16839514C2211yescharacteristic, complemented
53#16839512C2211yescharacteristic, complemented
54#16839510C2211yescharacteristic, complemented
55#16839510C2211yescharacteristic, complemented
56#16839508C2211yescharacteristic, complemented
57#16839506C2211yescharacteristic
58#16839505C2211yescharacteristic
59#16839505C2211yescharacteristic
60C61×#486C2211yescharacteristic
61C61×#462C2211yescharacteristic, complemented
62C61×#460C2211yescharacteristic
63C61×#446C2211yescharacteristic, complemented
64C61×#334C2211yescharacteristic, complemented
65C61×#332C2211yescharacteristic, complemented
66C61×#331C2211yescharacteristic
67C2×D4×D61—21no—
68C2×D4×D61—21no—
69C2×D4×D61—21no—
70C2×#11758202—21no—
71C2×#11758190—21no—
72C2×#11758200—21no—
73C2×#11758201—21no—
74C2×#11758189D411yescharacteristic
75C2×#11758189D411yescharacteristic
76C2×C8×D61—21no—
77C2×C8×D61—21no—
78#14519777C2311yescharacteristic
79#14519776D411yescharacteristic
80#14519775D411yescharacteristic
81#14519774C2311yescharacteristic
82#14519773—21no—
83#14519773—21no—
84#14519773—21no—
85#14519768—21no—
86#14519767—21no—
87#14519757—21no—
88#14519757D411yescharacteristic
89#14519755—21no—
90#14519753—21no—
91#14519753D411yescharacteristic
92Q8×Dic61—21no—
93C4×Dic122—21no—
94C4×Dic122C2311yescharacteristic
95#14519742—21no—
96#14519735—21no—
97#14519734—21no—
98C42×D61C2311yescharacteristic
99#14519711—21no—
100#14519710—21no—
101C4⋊#92418—21no—
102C4⋊#92418C2311yescharacteristic
103C4⋊#92418C2311yescharacteristic
104C4⋊#92418C2311yescharacteristic
105#14519706—21no—
106#14519706—21no—
107#14519706—21no—
108#14519706—21no—
109#14519702—21no—
110#14519701—21no—
111#14519701—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#14519689—21no—
122#14519687—21no—
123#14519687D411yescharacteristic
124#14519687D411yescharacteristic
125C4×#92418—21no—
126C4×#92418C2311yescharacteristic
127C8×Dic61—21no—
128C8×Dic61—21no—
129C2×Q16×C61—21no—
130C2×SD16×C61—21no—
131C4⋊Q8×C61C2311yescharacteristic
132#127×C61C2311yescharacteristic
133C4×Q8×C61C2311yescharacteristic
134C4⋊C8×C61—21no—
135C4⋊C8×C61C2311yescharacteristic
136C4⋊C8×C61C2311yescharacteristic
137C4⋊C8×C61C2311yescharacteristic
138#103×C61—21no—
139#102×C61—21no—
140#102×C61—21no—
141#102×C61—21no—
142#97×C61—21no—
143C4×C8×C61C2311yescharacteristic
144C23×D61—41no—
145C2×Q8×C61—21no—
146C2×Q8×C61C2×D411yescharacteristic
147C2×D4×C61—21no—
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C2×C22⋊D61—41no—
151C2×C22⋊D61—41no—
152C2×C22⋊D61—41no—
153D4×D61—42no—
154D4×D61—42no—
155D4×D61—81no—
156D4×D61—81no—
157D4×D61—82no—
158C2×D244—21no—
159C2×D244—21no—
160C2×D244—21no—
161C2×C4×D61C2×D411yescharacteristic
162C2×C4×D61C2×D411yescharacteristic
163C2×C4×D61#4111yescharacteristic
164C2×Dic122—21no—
165C2×Dic122—21no—
166C2×Dic122C2×D411yescharacteristic
167Q16×C61—81no—
168SD16×C61—81no—
169C2×C8×C61—21no—
170C2×C8×C61—21no—
171C2×C8×C61—21no—
172C2×C8×C61C2×D411yescharacteristic
173C2×C8×C61C2×D411yescharacteristic
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61C2×D411yescharacteristic
178C42×C61—21no—
179C42×C61C2411yescharacteristic
180#11758202—81no—
181#11758201—81no—
182#11758200—81no—
183#11758198—21no—
184#11758198—21no—
185#11758198—41no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61—41no—
190C4⋊Dic61C2×D411yescharacteristic
191#11758196—21no—
192#11758196—21no—
193#11758196—21no—
194#11758196—21no—
195#11758196—21no—
196#11758196—41no—
197C4×Dic61—21no—
198C4×Dic61—41no—
199C4×Dic61C2×D411yescharacteristic
200C4×Dic61C2×D411yescharacteristic
201C4×Dic61#4111yescharacteristic
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418C2×D411yescharacteristic
206C2×#92418C2×D411yescharacteristic
207#11758190—81no—
208#11758189—22no—
209#11758189—22no—
210C8×D61—42no—
211C8×D61—81no—
212C23×C61—41no—
213C22×D61—41no—
214C22×D61—82no—
215C22×D61—82no—
216C22×D61—82no—
217C22×D61#12111yescharacteristic
218Q8×C61—21no—
219Q8×C61—21no—
220Q8×C61—41no—
221Q8×C61—41no—
222D4×C61—41no—
223D4×C61—41no—
224D4×C61—41no—
225D4×C61—41no—
226D4×C61—81no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—41no—
230C2×C4×C61C22×D411yescharacteristic
231C2×C4×C61C22×D411yescharacteristic
232C2×C4×C61C2×#4111yescharacteristic
233C22⋊D61—82no—
234C22⋊D61—82no—
235C22⋊D61—82no—
236C2×Dic61—21no—
237C2×Dic61—21no—
238C2×Dic61—41no—
239C2×Dic61#12111yescharacteristic
240C2×Dic61#12111yescharacteristic
241C2×Dic61#12111yescharacteristic
242D244—41no—
243D244—41no—
244D244—41no—
245D244—41no—
246D244—81no—
247C4×D61—21no—
248C4×D61—21no—
249C4×D61—41no—
250C4×D61#13512yes—
251C4×D61#13712yes—
252Dic122—21no—
253Dic122—21no—
254Dic122—81no—
255Dic122—81no—
256C8×C61—21no—
257C8×C61—21no—
258C8×C61—41no—
259C8×C61—41no—
260C8×C61—41no—
261#92418—21no—
262#92418—21no—
263#92418—41no—
264#92418—41no—
265#92418—41no—
266#19242—611nomaximal, Sylow
267C22×C61—81no—
268C22×C61—81no—
269C22×C61—81no—
270C22×C61C2×#12111yescharacteristic
271C2×D61—21no—
272C2×D61—21no—
273C2×D61—81no—
274C2×D61—81no—
275C2×D61—81no—
276C2×D61—164no—
277C2×D61#46412yes—
278C4×C61—21no—
279C4×C61—41no—
280C4×C61—41no—
281C4×C61—81no—
282C4×C61C2×#13511yescharacteristic
283C4×C61C2×#13711yescharacteristic
284C4×C61#57611yescharacteristic
285C4×C61#57711yescharacteristic
286Dic61—21no—
287Dic61—21no—
288Dic61—41no—
289Dic61—41no—
290Dic61—81no—
291Dic61#46412yes—
292#3249—611no—
293#3208—611no—
294#3138—611no—
295#2962—611no—
296#1702—611no—
297#1702D6111yescharacteristic, complemented
298#1698—611no—
299#1698—611no—
300#1682—611no—
301#1682—611no—
302#1678—611no—
303#1678—611no—
304#1565—611no—
305#1542—611no—
306C2×#331—611no—
307C2×C61—161no—
308C2×C61C2×#46411yescharacteristic
309C2×C61#306211yescharacteristic
310C2×C61#306811yescharacteristic
311D61—22no—
312D61—161no—
313#533—611no—
314#533—611no—
315C2×#127—611no—
316#492—1221no—
317#490—1221no—
318#490—1221no—
319#486—611no—
320#486—611no—
321#486—611no—
322#486—1221no—
323#486C2×D6111yescharacteristic
324#477—611no—
325#477—611no—
326#477—1221no—
327#467—1221no—
328#467—1221no—
329#462—611no—
330#462—1221no—
331#462C2×D6111yescharacteristic, complemented
332#460—611no—
333#460—611no—
334#460—611no—
335#460C2×D6111yescharacteristic
336#450—1221no—
337#450—1221no—
338#446—611no—
339#446—1221no—
340#446—1221no—
341#446C2×D6111yescharacteristic, complemented
342C8×Q8—611no—
343C8×Q8—611no—
344#431—611no—
345#424—1221no—
346C2×C4⋊C8—611no—
347C2×C4⋊C8—611no—
348#418—1221no—
349#418—1221no—
350#405—1221no—
351C4×M16—611no—
352#335—611no—
353#335—611no—
354#334—611no—
355#334C2×D6111yescharacteristic, complemented
356#333—611no—
357#333—611no—
358#332—611no—
359#332C2×D6111yescharacteristic, complemented
360#331—611no—
361#331—611no—
362#331—611no—
363#331C2×D6111yescharacteristic
364C61#1924211yescharacteristic, complemented, Sylow
365C22×D4—1221no—
366C22×D4—1221no—
367C22×D4—1221no—
368C2×Q16—21no—
369C2×Q16—1221no—
370C2×SD16—21no—
371C2×SD16—1221no—
372C2×SD16—1221no—
373C2×SD16—1221no—
374C2×M16—611no—
375C2×M16—611no—
376C22×C8—1221no—
377C22×C8—1221no—
378C4⋊Q8—611no—
379C4⋊Q8—611no—
380C4⋊Q8—611no—
381C4⋊Q8C22×D6111yescharacteristic
382#127—611no—
383#127—1221no—
384#127—1221no—
385#127—1221no—
386#127C22×D6111yescharacteristic
387#125—1221no—
388#125—1221no—
389C22⋊Q8—611no—
390C22⋊Q8—611no—
391C22⋊Q8—1221no—
392C22⋊Q8—1221no—
393C22⋊Q8—1221no—
394C4×Q8—611no—
395C4×Q8—1221no—
396C4×Q8—1221no—
397C4×Q8—1221no—
398C4×Q8C22×D6111yescharacteristic
399#117—1221no—
400#117—1221no—
401C2×C42—611no—
402#107—1221no—
403#107—1221no—
404C4⋊C8—21no—
405C4⋊C8—611no—
406C4⋊C8—611no—
407C4⋊C8—611no—
408C4⋊C8—1221no—
409C4⋊C8—1221no—
410C4⋊C8—1221no—
411C4⋊C8—1221no—
412C4⋊C8—1221no—
413C4⋊C8C22×D6111yescharacteristic
414C4⋊C8C22×D6111yescharacteristic
415C4⋊C8C22×D6111yescharacteristic
416#103—21no—
417#103—1221no—
418#103—1221no—
419#103—1221no—
420#102—21no—
421#102—21no—
422#102—21no—
423#102—1221no—
424#102—1221no—
425#102—1221no—
426#102—1221no—
427#102—1221no—
428#102—1221no—
429#102—1221no—
430#102—1221no—
431#102—1221no—
432C22⋊C8—1221no—
433#97—21no—
434#97—611no—
435#97—611no—
436#97—1221no—
437C4×C8—611no—
438C4×C8—1221no—
439C4×C8—1221no—
440C4×C8—1221no—
441C4×C8C22×D6111yescharacteristic
442C24—2441no—
443C2×Q8—21no—
444C2×Q8—611no—
445C2×Q8—1221no—
446C2×Q8—1221no—
447C2×Q8D4×D6111yescharacteristic
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—1221no—
452C2×D4—1221no—
453C2×D4—1221no—
454C2×D4—4881no—
455C2×D4—4881no—
456C2×D4—4882no—
457C2×D4—2442no—
458C2×D4—2442no—
459C2×D4—2441no—
460C2×D4—2441no—
461C2×D4—2441no—
462C22×C4—611no—
463C22×C4—611no—
464C22×C4—611no—
465Q16—81no—
466Q16—4881no—
467SD16—81no—
468SD16—4881no—
469SD16—4881no—
470SD16—4881no—
471M16—1222no—
472M16—1222no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—611no—
477C2×C8—611no—
478C2×C8—1221no—
479C2×C8—1221no—
480C2×C8—1221no—
481C2×C8—4881no—
482C2×C8—2442no—
483C2×C8D4×D6111yescharacteristic
484C2×C8D4×D6111yescharacteristic
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—611no—
489C4⋊C4—1221no—
490C4⋊C4—1221no—
491C4⋊C4—1221no—
492C4⋊C4—1221no—
493C4⋊C4—1221no—
494C4⋊C4—1221no—
495C4⋊C4—1221no—
496C4⋊C4—1221no—
497C4⋊C4—2441no—
498C4⋊C4—2441no—
499C4⋊C4D4×D6111yescharacteristic
500C22⋊C4—1221no—
501C22⋊C4—1221no—
502C22⋊C4—2441no—
503C42—21no—
504C42—611no—
505C42—611no—
506C42—611no—
507C42—1221no—
508C42—2441no—
509C42C23×D6111yescharacteristic
510C23—41no—
511C23—611no—
512C23—4882no—
513C23—4882no—
514C23—4882no—
515C23—2441no—
516Q8—21no—
517Q8—21no—
518Q8—41no—
519Q8—41no—
520Q8—1221no—
521Q8—1221no—
522Q8—4881no—
523Q8—4881no—
524D4—41no—
525D4—41no—
526D4—41no—
527D4—41no—
528D4—81no—
529D4—4881no—
530D4—4882no—
531D4—4882no—
532D4—4882no—
533D4—2441no—
534D4—2441no—
535D4—2441no—
536D4—2441no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—41no—
540C2×C4—611no—
541C2×C4—612no—
542C2×C4—612no—
543C2×C4—611no—
544C2×C4—611no—
545C2×C4—1221no—
546C2×C4—1221no—
547C2×C4—1221no—
548C2×C4—1221no—
549C2×C4—2441no—
550C2×C4—2441no—
551C2×C4C2×D4×D6111yescharacteristic
552C2×C4C2×D4×D6111yescharacteristic
553C2×C4#41×D6111yescharacteristic
554C8—21no—
555C8—21no—
556C8—41no—
557C8—41no—
558C8—41no—
559C8—1221no—
560C8—1221no—
561C8—2441no—
562C8—2441no—
563C8—2441no—
564C22—81no—
565C22—81no—
566C22—81no—
567C22—612no—
568C22—1221no—
569C22—1221no—
570C22—9764no—
571C22—4881no—
572C22—4881no—
573C22—4881no—
574C22#121×D6111yescharacteristic
575C4—21no—
576C4—41no—
577C4—41no—
578C4—81no—
579C4—612no—
580C4—1221no—
581C4—1221no—
582C4—4881no—
583C4—2441no—
584C4—2441no—
585C4#135×D6111yescharacteristic
586C4#137×D6111yescharacteristic
587C4#1684056611yescharacteristic
588C4#1684057211yescharacteristic
589C2—161no—
590C2—1222no—
591C2—9761no—
592C2#464×D6111yescharacteristic
593C2#2145549611yescharacteristic
594C2#2145554211yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1702C2
socleC22×C61C2×#121
prime radical#127×D61C22
hypercenter#1702D61
hypocenterC61#19242
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1702

Complements

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

NH ≅ G/Ntype
1. #21457162C2semidirect
4. #21454767C2semidirect
5. #21445390C2semidirect
6. #21445359C2semidirect
7. #21445358C2semidirect
8. #21445265C2semidirect
9. #21445260C2semidirect
10. #21445236C2semidirect
11. #21445233C2semidirect
12. #21444568C2semidirect
13. #21444475C2semidirect
15. C61×#1702C2semidirect
26. #16840058C22semidirect
27. #16840057C22semidirect
32. #16839969C22semidirect
40. #16839889C22semidirect
41. Q8×#92418C22semidirect
42. C8×Dic122C22semidirect
51. #16839515C22semidirect
52. #16839514C22semidirect
53. #16839512C22semidirect
54. #16839510C22semidirect
55. #16839510C22semidirect
56. #16839508C22semidirect
61. C61×#462C22semidirect
63. C61×#446C22semidirect
64. C61×#334C22semidirect
65. C61×#332C22semidirect
297. #1702D61semidirect
331. #462C2×D61semidirect
341. #446C2×D61semidirect
355. #334C2×D61semidirect
359. #332C2×D61semidirect
364. C61#19242semidirect

Maximal subgroups

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

subgroupindex
#214571622
#214567592
#214561902
#214547672
#214453902
#214453592
#214453582
#214452652
#214452602
#214452362
#214452332
#214445682
#214444752
#331×D612
C61×#17022
#1924261