Finite Group Databasebeta

← #29669691

Subgroups — #29669691

Order 15616.

648 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458044C211yesmaximal, characteristic, complemented
2#21457730C211yesmaximal, characteristic, complemented
3#21457634C211yesmaximal, characteristic
4#21445482C211yesmaximal, characteristic, complemented
5#21445368C211yesmaximal, characteristic, complemented
6#21445263C211yesmaximal, characteristic, complemented
7#21445258C211yesmaximal, characteristic, complemented
8#21445181C211yesmaximal, characteristic, complemented
9#21445101C211yesmaximal, characteristic, complemented
10#21445089C211yesmaximal, characteristic, complemented
11#21444996C211yesmaximal, characteristic, complemented
12#21444929C211yesmaximal, characteristic, complemented
13#21444649C211yesmaximal, characteristic, complemented
14#21444629C211yesmaximal, characteristic, complemented
15C61×#3393C211yesmaximal, characteristic, complemented
16#16840417C2211yescharacteristic
17D4×D244C2211yescharacteristic
18#16840158C2211yescharacteristic, complemented
19#16840114C2211yescharacteristic, complemented
20#16840110C2211yescharacteristic
21#16840110C2211yescharacteristic
22#16840109C2211yescharacteristic
23#16840103—21no—
24#16840086—21no—
25#16840077—21no—
26#16840042C2211yescharacteristic
27#16840039C2211yescharacteristic
28#16840039C2211yescharacteristic
29#16840010—21no—
30#16840001—21no—
31#16839997—21no—
32#16839972C2211yescharacteristic, complemented
33#16839962C2211yescharacteristic
34#16839957C2211yescharacteristic
35#16839922—21no—
36#16839912—21no—
37#16839894—21no—
38C4×#11758201C2211yescharacteristic
39#16839840C2211yescharacteristic
40C8×D244C2211yescharacteristic
41D4×#92418C2211yescharacteristic, complemented
42#16839766—21no—
43#16839756—21no—
44#102×D61—21no—
45#16839515C2211yescharacteristic, complemented
46#16839514C2211yescharacteristic, complemented
47#16839508C2211yescharacteristic, complemented
48#16839508C2211yescharacteristic, complemented
49#16839507C2211yescharacteristic, complemented
50#16839505C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic, complemented
52D4⋊#92418C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic, complemented
54#16839489C2211yescharacteristic
55#16839484C2211yescharacteristic, complemented
56C61×#556C2211yescharacteristic
57C61×#556C2211yescharacteristic, complemented
58C61×#495—21no—
59C61×#494C2211yescharacteristic, complemented
60C61×#464—21no—
61C61×#464—21no—
62C61×#461C2211yescharacteristic
63C61×#461C2211yescharacteristic, complemented
64C8×Q8×C61C2211yescharacteristic, complemented
65C4×Q16×C61—21no—
66C4×D8×C61C2211yescharacteristic
67C2×#11758222—21no—
68C22×D244—21no—
69C2×D4×D61—21no—
70C2×#11758202—21no—
71C2×#11758201—21no—
72C2×#11758201D411yescharacteristic
73C2×#11758199—21no—
74C2×#11758194—21no—
75C22×#92418—21no—
76C2×C8×D61—21no—
77#14519777C2311yescharacteristic
78#14519774C2311yescharacteristic
79#14519759—41no—
80#14519756—21no—
81#14519752—21no—
82#14519751—21no—
83#14519748—21no—
84#14519747—41no—
85C4×D244C2311yescharacteristic
86C4⋊C4×D61—21no—
87#14519710—21no—
88#14519710—21no—
89#14519710—21no—
90#14519708—41no—
91#14519708D411yescharacteristic
92C4⋊#92418C2311yescharacteristic
93C4⋊#92418C2311yescharacteristic
94C4⋊#92418C2311yescharacteristic
95#14519701D411yescharacteristic
96#14519700—21no—
97#14519700—21no—
98#14519700—21no—
99#14519699—21no—
100#14519699—21no—
101#14519698—21no—
102#14519697—21no—
103#14519697—21no—
104#14519697—21no—
105#14519697—21no—
106#14519697—21no—
107#14519697—21no—
108#14519697—41no—
109#14519697D411yescharacteristic
110C22⋊#92418—21no—
111C22⋊#92418—21no—
112#14519689—21no—
113C4×#92418C2311yescharacteristic
114C4×#92418C2311yescharacteristic
115C2×Q16×C61—21no—
116C2×Q16×C61—21no—
117C2×SD16×C61—21no—
118C2×SD16×C61—21no—
119C2×D8×C61D411yescharacteristic
120C4⋊Q8×C61C2311yescharacteristic
121C4⋊Q8×C61C2311yescharacteristic
122#126×C61—41no—
123#126×C61—41no—
124#124×C61—21no—
125#124×C61—21no—
126C22⋊Q8×C61—21no—
127C22⋊Q8×C61—21no—
128C4×Q8×C61—21no—
129C4×Q8×C61—21no—
130C4×Q8×C61C2311yescharacteristic
131C4×D4×C61C2311yescharacteristic
132C4×D4×C61C2311yescharacteristic
133#117×C61—21no—
134#117×C61—21no—
135#107×C61D411yescharacteristic
136C4⋊C8×C61—21no—
137C4⋊C8×C61C2311yescharacteristic
138#103×C61—21no—
139#103×C61—21no—
140#102×C61D411yescharacteristic
141#102×C61D411yescharacteristic
142C4×C8×C61—21no—
143C4×C8×C61C2311yescharacteristic
144C23×D61—41no—
145C2×Q8×C61—21no—
146C2×Q8×C61—21no—
147C2×Q8×C61C2×D411yescharacteristic
148C2×D4×C61C2×D411yescharacteristic
149C2×D4×C61C2×D411yescharacteristic
150C22×C4×C61—21no—
151C22×C4×C61—21no—
152C2×C22⋊D61—41no—
153C2×C22⋊D61—41no—
154D4×D61—42no—
155D4×D61—81no—
156#11758222—42no—
157#11758222—81no—
158C2×D244—21no—
159C2×D244—21no—
160C2×D244—21no—
161C2×D244—42no—
162C2×D244—41no—
163C2×D244—81no—
164C2×D244C2×D411yescharacteristic
165C2×C4×D61—21no—
166C2×C4×D61—41no—
167C2×Dic122—21no—
168Q16×C61—42no—
169Q16×C61—81no—
170SD16×C61—81no—
171SD16×C61—81no—
172D8×C61—41no—
173C2×C8×C61—21no—
174C2×C8×C61C2×D411yescharacteristic
175C2×C8×C61C2×D411yescharacteristic
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—41no—
182C4⋊C4×C61—41no—
183C4⋊C4×C61C2×D411yescharacteristic
184C4⋊C4×C61C2×D411yescharacteristic
185C4⋊C4×C61C2×D411yescharacteristic
186C22⋊C4×C61—21no—
187C22⋊C4×C61—21no—
188C22⋊C4×C61—41no—
189C22⋊C4×C61—41no—
190C22⋊C4×C61—41no—
191C22⋊C4×C61—41no—
192C42×C61—21no—
193C42×C61—41no—
194C42×C61—41no—
195C42×C61C2411yescharacteristic
196C22⋊Dic61—41no—
197#11758202—81no—
198#11758201—22no—
199#11758201—81no—
200#11758199—81no—
201#11758198—21no—
202#11758198—41no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61C2×D411yescharacteristic
206#11758196—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
215C8×D61—81no—
216C23×C61—21no—
217C23×C61—21no—
218C22×D61—21no—
219C22×D61—41no—
220C22×D61—82no—
221C22×D61—82no—
222C22×D61—161no—
223Q8×C61—21no—
224Q8×C61—21no—
225Q8×C61—41no—
226Q8×C61—41no—
227Q8×C61—41no—
228Q8×C61—41no—
229D4×C61—21no—
230D4×C61—21no—
231D4×C61—21no—
232D4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—41no—
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
246C22⋊D61—82no—
247C22⋊D61—82no—
248C2×Dic61—21no—
249C2×Dic61—41no—
250D244—21no—
251D244—41no—
252D244—41no—
253D244—41no—
254D244—41no—
255D244—81no—
256D244—82no—
257D244#13512yes—
258C4×D61—41no—
259C4×D61—41no—
260C4×D61—82no—
261Dic122—41no—
262Dic122—41no—
263C8×C61—21no—
264C8×C61—21no—
265C8×C61—41no—
266#92418—21no—
267#92418—21no—
268#92418—41no—
269#92418—41no—
270#92418—41no—
271#92418—41no—
272#19777—611nomaximal, Sylow
273C22×C61—41no—
274C22×C61—41no—
275C22×C61—41no—
276C22×C61—41no—
277C22×C61—41no—
278C22×C61—41no—
279C22×C61D4211yescharacteristic
280C2×D61—22no—
281C2×D61—41no—
282C2×D61—41no—
283C2×D61—81no—
284C2×D61—81no—
285C2×D61—81no—
286C2×D61—164no—
287C4×C61—21no—
288C4×C61—41no—
289C4×C61—41no—
290C4×C61—41no—
291C4×C61—41no—
292C4×C61—81no—
293C4×C61—81no—
294C4×C61C2×#13511yescharacteristic
295C4×C61#57511yescharacteristic
296C4×C61#57611yescharacteristic
297C4×C61#57611yescharacteristic
298Dic61—41no—
299Dic61—81no—
300#3393D6111yescharacteristic, complemented
301#3324—611no—
302#3302—611no—
303#3294—611no—
304#1720—611no—
305#1699—611no—
306#1682—611no—
307#1681—611no—
308#1669—611no—
309#1654—611no—
310#1652—611no—
311#1639—611no—
312#1628—611no—
313#1581—611no—
314SD16⋊C8—611no—
315C2×C61—81no—
316C2×C61—81no—
317C2×C61#326811yescharacteristic
318C2×C61#328011yescharacteristic
319C2×C61#328911yescharacteristic
320D61—42no—
321D61—161no—
322#556C2×D6111yescharacteristic
323#556C2×D6111yescharacteristic, complemented
324#547—611no—
325D42—611no—
326#498—611no—
327#495—21no—
328#494C2×D6111yescharacteristic, complemented
329#487—611no—
330#486—611no—
331#486—611no—
332#486—611no—
333#485—1221no—
334#482—1221no—
335#480—1221no—
336#474—611no—
337#474—611no—
338#474—611no—
339#468—1221no—
340#467—1221no—
341#466—1221no—
342#464—21no—
343#464—21no—
344#462—611no—
345#461C2×D6111yescharacteristic
346#461C2×D6111yescharacteristic, complemented
347#460—611no—
348#459—611no—
349#452—1221no—
350#450—1221no—
351#447—1221no—
352C8×Q8C2×D6111yescharacteristic, complemented
353C4×Q16—21no—
354C4×SD16—611no—
355C4×D8C2×D6111yescharacteristic
356#436—611no—
357C8×D4—611no—
358C8×D4—611no—
359#418—1221no—
360#416—1221no—
361C2×#102—1221no—
362#335—611no—
363#335—611no—
364#332—611no—
365#332—611no—
366#332—611no—
367#331—611no—
368Q8⋊C8—611no—
369D4⋊C8—611no—
370D4⋊C8—611no—
371D4⋊C8—611no—
372#322—611no—
373C61#1977711yescharacteristic, complemented, Sylow
374C2×#41—1221no—
375C22×D4—1221no—
376C22×D4—1221no—
377C2×Q16—21no—
378C2×Q16—21no—
379C2×Q16—1221no—
380C2×SD16—21no—
381C2×SD16—21no—
382C2×SD16—611no—
383C2×SD16—1221no—
384C2×D8—1221no—
385C2×D8C22⋊D6111yescharacteristic
386C2×M16—1221no—
387C22×C8—1221no—
388C22×C8—1221no—
389C4⋊Q8—611no—
390C4⋊Q8C22×D6111yescharacteristic
391C4⋊Q8C22×D6111yescharacteristic
392#127—611no—
393#126—41no—
394#126—41no—
395#124—21no—
396#124—21no—
397#123—2441no—
398C22⋊Q8—21no—
399C22⋊Q8—21no—
400C22⋊Q8—1221no—
401#121—1221no—
402#121—1221no—
403#121—1221no—
404#120—2441no—
405C4×Q8—21no—
406C4×Q8—21no—
407C4×Q8C22×D6111yescharacteristic
408C4×D4—611no—
409C4×D4C22×D6111yescharacteristic
410C4×D4C22×D6111yescharacteristic
411#117—21no—
412#117—21no—
413C2×C4⋊C4—1221no—
414#107—1221no—
415#107—1221no—
416#107—1221no—
417#107C22⋊D6111yescharacteristic
418#106—611no—
419#106—2441no—
420C4⋊C8—21no—
421C4⋊C8—611no—
422C4⋊C8—611no—
423C4⋊C8—611no—
424C4⋊C8C22×D6111yescharacteristic
425#103—21no—
426#103—21no—
427#103—611no—
428#103—1221no—
429#103—1221no—
430#103—1221no—
431#102—611no—
432#102—1221no—
433#102—1221no—
434#102—1221no—
435#102—1221no—
436#102—1221no—
437#102—1221no—
438#102—1221no—
439#102—1221no—
440#102—1221no—
441#102—2441no—
442#102C22⋊D6111yescharacteristic
443#102C22⋊D6111yescharacteristic
444C22⋊C8—1221no—
445C22⋊C8—1221no—
446C22⋊C8—1221no—
447C4×C8—21no—
448C4×C8—611no—
449C4×C8—611no—
450C4×C8C22×D6111yescharacteristic
451C24—2441no—
452#41—4881no—
453#41—2442no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—1221no—
457C2×Q8D4×D6111yescharacteristic
458C2×D4—611no—
459C2×D4—1221no—
460C2×D4—1221no—
461C2×D4—1221no—
462C2×D4—4881no—
463C2×D4—4881no—
464C2×D4—2441no—
465C2×D4—2442no—
466C2×D4—2441no—
467C2×D4—2442no—
468C2×D4—2441no—
469C2×D4C2×C22⋊D6111yescharacteristic
470C2×D4C2×C22⋊D6111yescharacteristic
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—1221no—
474C22×C4—2441no—
475Q16—42no—
476Q16—81no—
477Q16—4881no—
478SD16—81no—
479SD16—81no—
480SD16—1222no—
481SD16—4881no—
482D8—41no—
483D8—4881no—
484M16—4881no—
485C2×C8—21no—
486C2×C8—611no—
487C2×C8—611no—
488C2×C8—1221no—
489C2×C8—1221no—
490C2×C8—1221no—
491C2×C8—1221no—
492C2×C8—4881no—
493C2×C8—4881no—
494C2×C8C2×C22⋊D6111yescharacteristic
495C2×C8C2×C22⋊D6111yescharacteristic
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—41no—
502C4⋊C4—41no—
503C4⋊C4—611no—
504C4⋊C4—1221no—
505C4⋊C4—1221no—
506C4⋊C4—2441no—
507C4⋊C4D4×D6111yescharacteristic
508C4⋊C4C2×C22⋊D6111yescharacteristic
509C4⋊C4C2×C22⋊D6111yescharacteristic
510C22⋊C4—21no—
511C22⋊C4—21no—
512C22⋊C4—41no—
513C22⋊C4—41no—
514C22⋊C4—41no—
515C22⋊C4—41no—
516C22⋊C4—1221no—
517C22⋊C4—2441no—
518C22⋊C4—2441no—
519C42—21no—
520C42—41no—
521C42—41no—
522C42C23×D6111yescharacteristic
523C23—21no—
524C23—21no—
525C23—1221no—
526C23—9761no—
527C23—4882no—
528C23—4882no—
529C23—2441no—
530Q8—21no—
531Q8—21no—
532Q8—41no—
533Q8—41no—
534Q8—41no—
535Q8—41no—
536Q8—2441no—
537Q8—2441no—
538D4—21no—
539D4—21no—
540D4—21no—
541D4—21no—
542D4—612no—
543D4—1221no—
544D4—4881no—
545D4—4882no—
546D4—4882no—
547D4—4882no—
548D4—2441no—
549D4—2441no—
550D4—2441no—
551D4—2441no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—1221no—
563C2×C4—4882no—
564C2×C4—2441no—
565C2×C4—2441no—
566C2×C4—2441no—
567C2×C4C2×D4×D6111yescharacteristic
568C2×C4C22×C22⋊D6111yescharacteristic
569C2×C4#1451982211yescharacteristic
570C8—21no—
571C8—21no—
572C8—41no—
573C8—1221no—
574C8—1221no—
575C8—2441no—
576C8—2441no—
577C8—2441no—
578C8—2441no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22—1222no—
586C22—9764no—
587C22—4881no—
588C22—4881no—
589C22—4881no—
590C22—2441no—
591C22—2441no—
592C22D4×C22⋊D6111yescharacteristic
593C4—21no—
594C4—41no—
595C4—41no—
596C4—41no—
597C4—41no—
598C4—81no—
599C4—81no—
600C4—4881no—
601C4—2441no—
602C4#135×D6111yescharacteristic
603C4#1684056011yescharacteristic
604C4#1684056411yescharacteristic
605C4#1684056511yescharacteristic
606C2—81no—
607C2—81no—
608C2—9761no—
609C2—2442no—
610C2#2145736011yescharacteristic
611C2#2145749811yescharacteristic
612C2#2145759011yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3393C2
socleC22×C61D42
prime radical#21457634C2
hypercenter#3393D61
hypocenterC61#19777
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3393

Complements

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

NH ≅ G/Ntype
1. #21458044C2semidirect
2. #21457730C2semidirect
4. #21445482C2semidirect
5. #21445368C2semidirect
6. #21445263C2semidirect
7. #21445258C2semidirect
8. #21445181C2semidirect
9. #21445101C2semidirect
10. #21445089C2semidirect
11. #21444996C2semidirect
12. #21444929C2semidirect
13. #21444649C2semidirect
14. #21444629C2semidirect
15. C61×#3393C2semidirect
18. #16840158C22semidirect
19. #16840114C22semidirect
32. #16839972C22semidirect
41. D4×#92418C22semidirect
45. #16839515C22semidirect
46. #16839514C22semidirect
47. #16839508C22semidirect
48. #16839508C22semidirect
49. #16839507C22semidirect
51. Q8⋊#92418C22semidirect
53. D4⋊#92418C22semidirect
55. #16839484C22semidirect
57. C61×#556C22semidirect
59. C61×#494C22semidirect
63. C61×#461C22semidirect
64. C8×Q8×C61C22semidirect
300. #3393D61semidirect
323. #556C2×D61semidirect
328. #494C2×D61semidirect
346. #461C2×D61semidirect
352. C8×Q8C2×D61semidirect
373. C61#19777semidirect

Maximal subgroups

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

subgroupindex
#214580442
#214577302
#214576342
#214454822
#214453682
#214452632
#214452582
#214451812
#214451012
#214450892
#214449962
#214449292
#214446492
#214446292
C61×#33932
#1977761