Finite Group Databasebeta

← #29676555

Subgroups — #29676555

Order 15616.

616 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458850C211yesmaximal, characteristic, complemented
2#21458502C211yesmaximal, characteristic, complemented
3#21458050C211yesmaximal, characteristic
4#21445480C211yesmaximal, characteristic, complemented
5#21445385C211yesmaximal, characteristic, complemented
6#21445366C211yesmaximal, characteristic, complemented
7#21445306C211yesmaximal, characteristic, complemented
8#21445260C211yesmaximal, characteristic, complemented
9#21445089C211yesmaximal, characteristic, complemented
10#21445088C211yesmaximal, characteristic, complemented
11#21444998C211yesmaximal, characteristic
12#21444914C211yesmaximal, characteristic, complemented
13#21444722C211yesmaximal, characteristic, complemented
14#21444660C211yesmaximal, characteristic, complemented
15C61×#3324C211yesmaximal, characteristic
16#16840491C2211yescharacteristic, complemented
17#16840482C2211yescharacteristic
18#16840158C2211yescharacteristic, complemented
19#16840157—21no—
20#16840141C2211yescharacteristic, complemented
21#16840115—21no—
22#16840113C2211yescharacteristic, complemented
23#16840108C2211yescharacteristic
24#16840108C2211yescharacteristic
25#16840083—21no—
26#16840072—21no—
27#16840042C2211yescharacteristic
28#16840042C2211yescharacteristic
29#16840006—21no—
30#16839982—21no—
31#16839981—21no—
32#16839972—21no—
33#16839972C2211yescharacteristic, complemented
34#16839969C2211yescharacteristic, complemented
35#16839967C2211yescharacteristic
36#16839959C2211yescharacteristic
37#16839913—21no—
38#16839893C2211yescharacteristic, complemented
39C8×Dic122C2211yescharacteristic, complemented
40#16839867—21no—
41#16839854—21no—
42C4×#11758200C2211yescharacteristic
43#16839840C2211yescharacteristic
44#16839767—21no—
45#16839515C2211yescharacteristic, complemented
46#16839512C2211yescharacteristic, complemented
47#16839509C2211yescharacteristic, complemented
48#16839508C2211yescharacteristic, complemented
49#16839507C2211yescharacteristic, complemented
50#16839506C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic, complemented
52#16839492C2211yescharacteristic, complemented
53D4⋊#92418C2211yescharacteristic
54D4⋊#92418C2211yescharacteristic
55#16839483C2211yescharacteristic, 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×#11758202—21no—
68C2×#11758202—21no—
69C2×#11758202—21no—
70C2×#11758202—21no—
71C2×#11758200—21no—
72C2×#11758200—21no—
73C2×#11758200D411yescharacteristic
74C2×#11758194—21no—
75#14519777—21no—
76#14519777C2311yescharacteristic
77#14519777C2311yescharacteristic
78#14519770—41no—
79#14519768—21no—
80#14519768—41no—
81#14519764—21no—
82#14519756—21no—
83Q8×Dic61—21no—
84C4×Dic122—21no—
85C4×Dic122C2311yescharacteristic
86#14519742—21no—
87#14519736—21no—
88#14519710—21no—
89#14519710—21no—
90#14519708—41no—
91#14519708D411yescharacteristic
92C4⋊#92418—21no—
93C4⋊#92418C2311yescharacteristic
94C4⋊#92418C2311yescharacteristic
95C4⋊#92418C2311yescharacteristic
96#14519706—21no—
97#14519702—21no—
98#14519702—21no—
99#14519701—21no—
100#14519700—21no—
101#14519700—21no—
102#14519700D411yescharacteristic
103#14519698—21no—
104#14519698—21no—
105#14519698—21no—
106#14519698—21no—
107#14519698—21no—
108#14519698—41no—
109#14519698D411yescharacteristic
110C22⋊#92418—21no—
111#14519688—21no—
112C4×#92418C2311yescharacteristic
113C4×#92418C2311yescharacteristic
114C8×Dic61—21no—
115C22×D4×C61—21no—
116C22×D4×C61—21no—
117C2×SD16×C61—21no—
118C2×SD16×C61D411yescharacteristic
119C2×M16×C61—21no—
120C4⋊Q8×C61C2311yescharacteristic
121#127×C61C2311yescharacteristic
122#126×C61—41no—
123#124×C61—21no—
124C22⋊Q8×C61—21no—
125#121×C61—21no—
126#121×C61—21no—
127#120×C61—41no—
128C4×Q8×C61—21no—
129C4×Q8×C61C2311yescharacteristic
130C4×D4×C61C2311yescharacteristic
131C4×D4×C61C2311yescharacteristic
132#117×C61—21no—
133#107×C61—21no—
134#106×C61D411yescharacteristic
135C4⋊C8×C61C2311yescharacteristic
136#103×C61D411yescharacteristic
137#102×C61—21no—
138#102×C61—21no—
139#102×C61—21no—
140#102×C61—21no—
141#102×C61D411yescharacteristic
142C22⋊C8×C61—21no—
143C4×C8×C61C2311yescharacteristic
144C24×C61—41no—
145C2×Q8×C61—21no—
146C2×Q8×C61C2×D411yescharacteristic
147C2×D4×C61—21no—
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61—41no—
151C2×D4×C61—41no—
152C2×D4×C61—42no—
153C2×D4×C61—42no—
154C2×D4×C61—81no—
155C2×D4×C61—81no—
156C2×D4×C61C2×D411yescharacteristic
157C2×D4×C61C2×D411yescharacteristic
158C22×C4×C61—21no—
159C22×C4×C61—21no—
160C2×Dic122—21no—
161C2×Dic122—21no—
162C2×Dic122C2×D411yescharacteristic
163SD16×C61—41no—
164SD16×C61—81no—
165M16×C61—81no—
166C2×C8×C61—21no—
167C2×C8×C61C2×D411yescharacteristic
168C2×C8×C61C2×D411yescharacteristic
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—41no—
173C4⋊C4×C61C2×D411yescharacteristic
174C4⋊C4×C61C2×D411yescharacteristic
175C4⋊C4×C61C2×D411yescharacteristic
176C22⋊C4×C61—21no—
177C22⋊C4×C61—21no—
178C22⋊C4×C61—41no—
179C22⋊C4×C61—41no—
180C22⋊C4×C61—41no—
181C42×C61—21no—
182C42×C61—41no—
183C42×C61C2411yescharacteristic
184C22⋊Dic61—41no—
185C22⋊Dic61—41no—
186#11758202—42no—
187#11758202—42no—
188#11758202—81no—
189#11758202—81no—
190#11758200—41no—
191#11758200—81no—
192#11758200—81no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—41no—
198C4⋊Dic61—41no—
199C4⋊Dic61C2×D411yescharacteristic
200#11758196—21no—
201#11758196—41no—
202#11758196—41no—
203C4×Dic61—21no—
204C4×Dic61—41no—
205C4×Dic61—41no—
206#11758194—81no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418C2×D411yescharacteristic
212C2×#92418C2×D411yescharacteristic
213C23×C61—21no—
214C23×C61—21no—
215C23×C61—41no—
216C23×C61—82no—
217C23×C61—82no—
218C23×C61—161no—
219Q8×C61—21no—
220Q8×C61—21no—
221Q8×C61—41no—
222Q8×C61—41no—
223D4×C61—21no—
224D4×C61—21no—
225D4×C61—21no—
226D4×C61—21no—
227D4×C61—41no—
228D4×C61—41no—
229D4×C61—41no—
230D4×C61—41no—
231D4×C61—81no—
232D4×C61—82no—
233D4×C61—82no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61C22×D411yescharacteristic
244C2×C4×C61C22×D411yescharacteristic
245C2×C4×C61H(5,𝔽2)11yescharacteristic
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—41no—
249C2×Dic61—41no—
250Dic122—21no—
251Dic122—21no—
252Dic122—41no—
253Dic122—41no—
254Dic122—41no—
255Dic122—41no—
256C8×C61—21no—
257C8×C61—21no—
258C8×C61—41no—
259#92418—21no—
260#92418—21no—
261#92418—41no—
262#92418—41no—
263#92418—41no—
264#92418—41no—
265#20267—611nomaximal, Sylow
266C22×C61—41no—
267C22×C61—41no—
268C22×C61—41no—
269C22×C61—41no—
270C22×C61—41no—
271C22×C61—41no—
272C22×C61—81no—
273C22×C61—81no—
274C22×C61—81no—
275C22×C61—164no—
276C22×C61D4211yescharacteristic
277C4×C61—21no—
278C4×C61—41no—
279C4×C61—41no—
280C4×C61—41no—
281C4×C61—41no—
282C4×C61—81no—
283C4×C61#57511yescharacteristic
284C4×C61#57511yescharacteristic
285C4×C61#57611yescharacteristic
286C4×C61#57711yescharacteristic
287Dic61—41no—
288Dic61—41no—
289Dic61—81no—
290Dic61—81no—
291#3398—611no—
292#3364—611no—
293#3324—611no—
294#3324D6111yescharacteristic
295#1720—611no—
296#1702—611no—
297#1699—611no—
298#1689—611no—
299#1682—611no—
300#1652—611no—
301#1652—611no—
302#1639—611no—
303#1626—611no—
304#1592—611no—
305#1583—611no—
306C2×C61—81no—
307C2×C61—81no—
308C2×C61—161no—
309C2×C61#326811yescharacteristic
310C2×C61#327511yescharacteristic
311C2×C61#328911yescharacteristic
312#557—611no—
313#556—611no—
314#556C2×D6111yescharacteristic
315D42C2×D6111yescharacteristic
316#498—611no—
317#497—1221no—
318#494—611no—
319#488—1221no—
320#487—611no—
321#486—611no—
322#486—611no—
323#486C2×D6111yescharacteristic
324#482—21no—
325#482—1221no—
326#479—1221no—
327#474—611no—
328#474—611no—
329#474C2×D6111yescharacteristic
330#468—21no—
331#468—1221no—
332#464—1221no—
333#464—1221no—
334#462—611no—
335#462—611no—
336#462—1221no—
337#461—611no—
338#460—611no—
339#460C2×D6111yescharacteristic
340#450—21no—
341#450—1221no—
342#446—611no—
343C8×Q8—611no—
344#441—1221no—
345C4×Q16—1221no—
346C4×SD16—611no—
347C4×SD16C2×D6111yescharacteristic
348#436—611no—
349#436C2×D6111yescharacteristic
350#418—21no—
351#418—1221no—
352#335—611no—
353#334—611no—
354#332—611no—
355#332—611no—
356#332—611no—
357#331—611no—
358Q8⋊C8—611no—
359Q8⋊C8—611no—
360D4⋊C8—611no—
361D4⋊C8—611no—
362#322—611no—
363C61#2026711yescharacteristic, complemented, Sylow
364C22×D4—21no—
365C22×D4—21no—
366C2×Q16—1221no—
367C2×Q16—1221no—
368C2×Q16—1221no—
369C2×Q16—1221no—
370C2×SD16—21no—
371C2×SD16—611no—
372C2×SD16—1221no—
373C2×SD16—1221no—
374C2×SD16C22⋊D6111yescharacteristic
375C2×M16—21no—
376C2×M16—1221no—
377C4⋊Q8—611no—
378C4⋊Q8—611no—
379C4⋊Q8—1221no—
380C4⋊Q8C22×D6111yescharacteristic
381#127C22×D6111yescharacteristic
382#126—41no—
383#126—2441no—
384#125—1221no—
385#125—2441no—
386#124—21no—
387#124—1221no—
388C22⋊Q8—21no—
389C22⋊Q8—1221no—
390#121—21no—
391#121—21no—
392#120—41no—
393C4×Q8—21no—
394C4×Q8—611no—
395C4×Q8—1221no—
396C4×Q8—1221no—
397C4×Q8—1221no—
398C4×Q8C22×D6111yescharacteristic
399C4×D4C22×D6111yescharacteristic
400C4×D4C22×D6111yescharacteristic
401#117—21no—
402#117—1221no—
403#107—21no—
404#107—1221no—
405#107—1221no—
406#106—611no—
407#106—2441no—
408#106C22⋊D6111yescharacteristic
409C4⋊C8—611no—
410C4⋊C8—611no—
411C4⋊C8—611no—
412C4⋊C8—1221no—
413C4⋊C8—1221no—
414C4⋊C8C22×D6111yescharacteristic
415#103—611no—
416#103—1221no—
417#103—1221no—
418#103—1221no—
419#103—1221no—
420#103—1221no—
421#103C22⋊D6111yescharacteristic
422#102—21no—
423#102—21no—
424#102—21no—
425#102—21no—
426#102—611no—
427#102—1221no—
428#102—1221no—
429#102—1221no—
430#102—1221no—
431#102—1221no—
432#102—2441no—
433#102C22⋊D6111yescharacteristic
434C22⋊C8—21no—
435C22⋊C8—1221no—
436#97—1221no—
437C4×C8—611no—
438C4×C8—611no—
439C4×C8—1221no—
440C4×C8C22×D6111yescharacteristic
441C24—41no—
442C2×Q8—21no—
443C2×Q8—611no—
444C2×Q8—1221no—
445C2×Q8—1221no—
446C2×Q8C2×C22⋊D6111yescharacteristic
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—41no—
451C2×D4—41no—
452C2×D4—42no—
453C2×D4—42no—
454C2×D4—81no—
455C2×D4—81no—
456C2×D4D4×D6111yescharacteristic
457C2×D4C2×C22⋊D6111yescharacteristic
458C22×C4—21no—
459C22×C4—21no—
460Q16—4881no—
461Q16—4881no—
462Q16—2442no—
463Q16—2442no—
464SD16—41no—
465SD16—81no—
466SD16—4881no—
467SD16—4881no—
468SD16—2441no—
469M16—81no—
470M16—4881no—
471C2×C8—21no—
472C2×C8—611no—
473C2×C8—611no—
474C2×C8—1221no—
475C2×C8—1221no—
476C2×C8—1221no—
477C2×C8—1221no—
478C2×C8C2×C22⋊D6111yescharacteristic
479C2×C8C2×C22⋊D6111yescharacteristic
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—41no—
484C4⋊C4—611no—
485C4⋊C4—1221no—
486C4⋊C4—1221no—
487C4⋊C4—1221no—
488C4⋊C4—1221no—
489C4⋊C4—1221no—
490C4⋊C4—2441no—
491C4⋊C4—2441no—
492C4⋊C4—2441no—
493C4⋊C4—2441no—
494C4⋊C4D4×D6111yescharacteristic
495C4⋊C4C2×C22⋊D6111yescharacteristic
496C4⋊C4C2×C22⋊D6111yescharacteristic
497C22⋊C4—21no—
498C22⋊C4—21no—
499C22⋊C4—41no—
500C22⋊C4—41no—
501C22⋊C4—41no—
502C22⋊C4—2441no—
503C22⋊C4—2441no—
504C42—21no—
505C42—41no—
506C42—1221no—
507C42—2441no—
508C42—2441no—
509C42C23×D6111yescharacteristic
510C23—21no—
511C23—21no—
512C23—41no—
513C23—82no—
514C23—82no—
515C23—161no—
516Q8—21no—
517Q8—21no—
518Q8—41no—
519Q8—41no—
520Q8—1221no—
521Q8—1221no—
522Q8—2441no—
523Q8—2441no—
524Q8—2441no—
525Q8—2441no—
526D4—21no—
527D4—21no—
528D4—21no—
529D4—21no—
530D4—41no—
531D4—41no—
532D4—41no—
533D4—41no—
534D4—81no—
535D4—82no—
536D4—82no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—1221no—
547C2×C4—1221no—
548C2×C4—2441no—
549C2×C4—2441no—
550C2×C4C2×D4×D6111yescharacteristic
551C2×C4C22×C22⋊D6111yescharacteristic
552C2×C4#1451982211yescharacteristic
553C8—21no—
554C8—21no—
555C8—41no—
556C8—1221no—
557C8—1221no—
558C8—2441no—
559C8—2441no—
560C8—2441no—
561C8—2441no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22—41no—
568C22—81no—
569C22—81no—
570C22—81no—
571C22—164no—
572C22D4×C22⋊D6111yescharacteristic
573C4—21no—
574C4—41no—
575C4—41no—
576C4—41no—
577C4—41no—
578C4—81no—
579C4—4881no—
580C4—4881no—
581C4—2441no—
582C4—2441no—
583C4#1684055811yescharacteristic
584C4#1684056011yescharacteristic
585C4#1684056611yescharacteristic
586C4#1684057311yescharacteristic
587C2—81no—
588C2—81no—
589C2—161no—
590C2#2145735311yescharacteristic
591C2#2145744611yescharacteristic
592C2#2145759211yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3324C2
socleC22×C61D42
prime radicalD42×C61C22
hypercenter#3324D61
hypocenterC61#20267
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. #21458502C2semidirect
4. #21445480C2semidirect
5. #21445385C2semidirect
6. #21445366C2semidirect
7. #21445306C2semidirect
8. #21445260C2semidirect
9. #21445089C2semidirect
10. #21445088C2semidirect
12. #21444914C2semidirect
13. #21444722C2semidirect
14. #21444660C2semidirect
16. #16840491C22semidirect
18. #16840158C22semidirect
20. #16840141C22semidirect
22. #16840113C22semidirect
33. #16839972C22semidirect
34. #16839969C22semidirect
38. #16839893C22semidirect
39. C8×Dic122C22semidirect
45. #16839515C22semidirect
46. #16839512C22semidirect
47. #16839509C22semidirect
48. #16839508C22semidirect
49. #16839507C22semidirect
51. Q8⋊#92418C22semidirect
52. #16839492C22semidirect
55. #16839483C22semidirect
363. C61#20267semidirect

Maximal subgroups

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

subgroupindex
#214588502
#214585022
#214580502
#214454802
#214453852
#214453662
#214453062
#214452602
#214450892
#214450882
#214449982
#214449142
#214447222
#214446602
C61×#33242
#2026761