Finite Group Databasebeta

← #29676848

Subgroups — #29676848

Order 15616.

634 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458113C211yesmaximal, characteristic, complemented
2#21458063C211yesmaximal, characteristic
3#21458044C211yesmaximal, characteristic, complemented
4#21445432C211yesmaximal, characteristic, complemented
5#21445392C211yesmaximal, characteristic, complemented
6#21445382C211yesmaximal, characteristic, complemented
7#21445363C211yesmaximal, characteristic, complemented
8#21445359C211yesmaximal, characteristic, complemented
9#21445045C211yesmaximal, characteristic, complemented
10#21444997C211yesmaximal, characteristic, complemented
11#21444991C211yesmaximal, characteristic, complemented
12#21444922C211yesmaximal, characteristic, complemented
13#21444709C211yesmaximal, characteristic, complemented
14#21444704C211yesmaximal, characteristic, complemented
15C61×#3367C211yesmaximal, characteristic, complemented
16#16840477C2211yescharacteristic
17D4×D244C2211yescharacteristic
18#16840114C2211yescharacteristic, complemented
19#16840112C2211yescharacteristic, complemented
20#16840110C2211yescharacteristic
21#16840110C2211yescharacteristic
22#16840109C2211yescharacteristic
23#16840100—21no—
24#16840089—21no—
25#16840086—21no—
26#16840057C2211yescharacteristic, complemented
27#16840052C2211yescharacteristic
28#16840039C2211yescharacteristic
29#16840014—21no—
30#16840010—21no—
31#16840010—21no—
32#16839964C2211yescharacteristic
33#16839962C2211yescharacteristic
34#16839957C2211yescharacteristic
35#16839912—21no—
36#16839901—21no—
37#16839894—21no—
38C4×#11758201C2211yescharacteristic
39#16839841C2211yescharacteristic
40#16839840C2211yescharacteristic
41#16839840C2211yescharacteristic, complemented
42#16839771—21no—
43#16839766—21no—
44#16839765—21no—
45#16839515C2211yescharacteristic, complemented
46#16839512C2211yescharacteristic, complemented
47#16839510C2211yescharacteristic, complemented
48#16839508C2211yescharacteristic, complemented
49#16839508C2211yescharacteristic, complemented
50#16839505C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic, complemented
52D4⋊#92418C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic, complemented
54#16839489C2211yescharacteristic
55C8×#92418C2211yescharacteristic, complemented
56C61×#556C2211yescharacteristic
57D4×Q8×C61C2211yescharacteristic, complemented
58C61×#489—21no—
59C61×#487C2211yescharacteristic, complemented
60C61×#477—21no—
61C61×#474C2211yescharacteristic
62C61×#463—21no—
63C61×#461C2211yescharacteristic, complemented
64C61×#446C2211yescharacteristic, complemented
65C61×#440—21no—
66C4×D8×C61C2211yescharacteristic
67C22×D244—21no—
68C2×D4×D61—21no—
69C2×#11758201—21no—
70C2×#11758201—21no—
71C2×#11758201D411yescharacteristic
72C2×#11758199—21no—
73C2×#11758194—21no—
74C2×#11758194—21no—
75C2×#11758189—21no—
76#14519777C2311yescharacteristic
77#14519774C2311yescharacteristic
78#14519763—41no—
79#14519752—21no—
80#14519751—21no—
81#14519751—21no—
82#14519748—21no—
83#14519747—41no—
84C4×D244C2311yescharacteristic
85#14519736—21no—
86#14519735—21no—
87#14519710—21no—
88#14519710—21no—
89#14519708—21no—
90#14519708D411yescharacteristic
91C4⋊#92418C2311yescharacteristic
92C4⋊#92418C2311yescharacteristic
93C4⋊#92418C2311yescharacteristic
94#14519701—21no—
95#14519701—21no—
96#14519701—21no—
97#14519701—21no—
98#14519701D411yescharacteristic
99#14519700—41no—
100#14519699—21no—
101#14519699—21no—
102#14519698—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
115C22×Q8×C61—21no—
116C2×SD16×C61—21no—
117C2×D8×C61D411yescharacteristic
118C4⋊Q8×C61—21no—
119C4⋊Q8×C61C2311yescharacteristic
120C4⋊Q8×C61C2311yescharacteristic
121#126×C61—41no—
122#124×C61—21no—
123C22⋊Q8×C61—21no—
124C22⋊Q8×C61—21no—
125C22⋊Q8×C61—41no—
126C4×Q8×C61—21no—
127C4×Q8×C61C2311yescharacteristic
128C4×D4×C61—21no—
129C4×D4×C61C2311yescharacteristic
130C4×D4×C61C2311yescharacteristic
131#117×C61—21no—
132#107×C61—21no—
133#107×C61D411yescharacteristic
134C4⋊C8×C61—21no—
135C4⋊C8×C61C2311yescharacteristic
136#103×C61—21no—
137#103×C61—21no—
138#103×C61—21no—
139#102×C61—21no—
140#102×C61D411yescharacteristic
141#102×C61D411yescharacteristic
142#97×C61—21no—
143C4×C8×C61C2311yescharacteristic
144C23×D61—41no—
145C2×Q8×C61—21no—
146C2×Q8×C61—21no—
147C2×Q8×C61—41no—
148C2×Q8×C61—42no—
149C2×Q8×C61—81no—
150C2×Q8×C61C2×D411yescharacteristic
151C2×D4×C61C2×D411yescharacteristic
152C2×D4×C61C2×D411yescharacteristic
153C22×C4×C61—21no—
154C22×C4×C61—21no—
155C22×C4×C61—41no—
156C2×C22⋊D61—41no—
157C2×C22⋊D61—41no—
158D4×D61—42no—
159D4×D61—81no—
160C2×D244—21no—
161C2×D244—21no—
162C2×D244—21no—
163C2×D244—42no—
164C2×D244—41no—
165C2×D244—81no—
166C2×D244C2×D411yescharacteristic
167C2×C4×D61—21no—
168C2×Dic122—21no—
169SD16×C61—42no—
170D8×C61—41no—
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×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—41no—
180C4⋊C4×C61—41no—
181C4⋊C4×C61C2×D411yescharacteristic
182C4⋊C4×C61C2×D411yescharacteristic
183C4⋊C4×C61C2×D411yescharacteristic
184C22⋊C4×C61—21no—
185C22⋊C4×C61—21no—
186C22⋊C4×C61—41no—
187C22⋊C4×C61—41no—
188C22⋊C4×C61—41no—
189C42×C61—21no—
190C42×C61—41no—
191C42×C61C2411yescharacteristic
192C22⋊Dic61—41no—
193#11758201—41no—
194#11758201—81no—
195#11758201—81no—
196#11758199—81no—
197#11758198—21no—
198#11758198—41no—
199#11758198—41no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61C2×D411yescharacteristic
203C4×Dic61—41no—
204#11758194—81no—
205#11758194—81no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418C2×D411yescharacteristic
211C2×#92418C2×D411yescharacteristic
212#11758189—81no—
213C23×C61—21no—
214C23×C61—21no—
215C22×D61—21no—
216C22×D61—41no—
217C22×D61—82no—
218C22×D61—82no—
219C22×D61—161no—
220Q8×C61—21no—
221Q8×C61—21no—
222Q8×C61—41no—
223Q8×C61—41no—
224Q8×C61—41no—
225Q8×C61—41no—
226Q8×C61—82no—
227D4×C61—21no—
228D4×C61—21no—
229D4×C61—21no—
230D4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—82no—
242C2×C4×C61C22×D411yescharacteristic
243C2×C4×C61C22×D411yescharacteristic
244C2×C4×C61H(5,𝔽2)11yescharacteristic
245C22⋊D61—82no—
246C22⋊D61—161no—
247C2×Dic61—21no—
248C2×Dic61—41no—
249D244—21no—
250D244—21no—
251D244—41no—
252D244—41no—
253D244—41no—
254D244—41no—
255D244—81no—
256D244—82no—
257C4×D61—41no—
258C4×D61—41no—
259Dic122—81no—
260C8×C61—21no—
261C8×C61—21no—
262C8×C61—41no—
263#92418—21no—
264#92418—21no—
265#92418—41no—
266#92418—41no—
267#92418—41no—
268#92418—41no—
269#20287—611nomaximal, Sylow
270C22×C61—41no—
271C22×C61—41no—
272C22×C61—41no—
273C22×C61—41no—
274C22×C61—41no—
275C22×C61—41no—
276C22×C61D4211yescharacteristic
277C2×D61—41no—
278C2×D61—41no—
279C2×D61—41no—
280C2×D61—81no—
281C2×D61—81no—
282C2×D61—81no—
283C2×D61—164no—
284C4×C61—21no—
285C4×C61—41no—
286C4×C61—41no—
287C4×C61—41no—
288C4×C61—41no—
289C4×C61—81no—
290C4×C61—81no—
291C4×C61#57511yescharacteristic
292C4×C61#57511yescharacteristic
293C4×C61#57611yescharacteristic
294C4×C61#57711yescharacteristic
295Dic61—41no—
296Dic61—81no—
297#3367D6111yescharacteristic, complemented
298#3329—611no—
299#3325—611no—
300#3324—611no—
301#1711—611no—
302#1703—611no—
303#1701—611no—
304#1699—611no—
305#1698—611no—
306#1646—611no—
307#1639—611no—
308#1638—611no—
309#1627—611no—
310#1590—611no—
311#1589—611no—
312C2×C61—81no—
313C2×C61—81no—
314C2×C61#327311yescharacteristic
315C2×C61#327511yescharacteristic
316C2×C61#329011yescharacteristic
317D61—81no—
318D61—161no—
319#556C2×D6111yescharacteristic
320#555—611no—
321D4×Q8C2×D6111yescharacteristic, complemented
322D42—611no—
323#489—21no—
324#487—611no—
325#487—611no—
326#487C2×D6111yescharacteristic, complemented
327#486—611no—
328#486—611no—
329#486—611no—
330#485—1221no—
331#482—1221no—
332#482—1221no—
333#477—21no—
334#477—611no—
335#476—611no—
336#474—611no—
337#474C2×D6111yescharacteristic
338#469—1221no—
339#468—1221no—
340#468—1221no—
341#463—21no—
342#461—611no—
343#461C2×D6111yescharacteristic, complemented
344#460—611no—
345#459—611no—
346#450—1221no—
347#448—1221no—
348#447—1221no—
349#446C2×D6111yescharacteristic, complemented
350#440—21no—
351C4×SD16—611no—
352C4×D8C2×D6111yescharacteristic
353#436—611no—
354#436—611no—
355#436—611no—
356#419—1221no—
357#418—1221no—
358#418—1221no—
359#335—611no—
360#334—611no—
361#333—611no—
362#332—611no—
363#332—611no—
364#331—611no—
365Q8⋊C8—611no—
366D4⋊C8—611no—
367D4⋊C8—611no—
368D4⋊C8—611no—
369C82—611no—
370C61#2028711yescharacteristic, complemented, Sylow
371C22×Q8—21no—
372C22×D4—1221no—
373C22×D4—1221no—
374C2×SD16—21no—
375C2×SD16—611no—
376C2×SD16—1221no—
377C2×SD16—1221no—
378C2×D8—1221no—
379C2×D8C22⋊D6111yescharacteristic
380C2×M16—1221no—
381C2×M16—1221no—
382C2×M16—1221no—
383C4⋊Q8—21no—
384C4⋊Q8—611no—
385C4⋊Q8C22×D6111yescharacteristic
386C4⋊Q8C22×D6111yescharacteristic
387#127—611no—
388#126—41no—
389#124—21no—
390#124—2441no—
391C22⋊Q8—21no—
392C22⋊Q8—21no—
393C22⋊Q8—41no—
394#121—1221no—
395#121—1221no—
396#121—1221no—
397#121—1221no—
398#120—2441no—
399C4×Q8—21no—
400C4×Q8C22×D6111yescharacteristic
401C4×D4—21no—
402C4×D4—611no—
403C4×D4C22×D6111yescharacteristic
404C4×D4C22×D6111yescharacteristic
405#117—21no—
406#117—1221no—
407#117—1221no—
408#107—21no—
409#107—1221no—
410#107—1221no—
411#107C22⋊D6111yescharacteristic
412#106—611no—
413#106—1221no—
414C4⋊C8—21no—
415C4⋊C8—611no—
416C4⋊C8—611no—
417C4⋊C8—611no—
418C4⋊C8C22×D6111yescharacteristic
419#103—21no—
420#103—21no—
421#103—21no—
422#103—611no—
423#103—1221no—
424#103—1221no—
425#103—1221no—
426#103—1221no—
427#103—2441no—
428#102—21no—
429#102—611no—
430#102—1221no—
431#102—1221no—
432#102—1221no—
433#102—1221no—
434#102—1221no—
435#102—1221no—
436#102—1221no—
437#102—1221no—
438#102—2441no—
439#102C22⋊D6111yescharacteristic
440#102C22⋊D6111yescharacteristic
441C22⋊C8—1221no—
442C22⋊C8—1221no—
443C22⋊C8—1221no—
444#97—21no—
445C4×C8—611no—
446C4×C8—611no—
447C4×C8C22×D6111yescharacteristic
448C24—2441no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—41no—
452C2×Q8—42no—
453C2×Q8—81no—
454C2×Q8—1221no—
455C2×Q8D4×D6111yescharacteristic
456C2×D4—611no—
457C2×D4—1221no—
458C2×D4—1221no—
459C2×D4—1221no—
460C2×D4—4881no—
461C2×D4—4881no—
462C2×D4—2441no—
463C2×D4—2442no—
464C2×D4—2441no—
465C2×D4—2442no—
466C2×D4—2441no—
467C2×D4C2×C22⋊D6111yescharacteristic
468C2×D4C2×C22⋊D6111yescharacteristic
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—41no—
472C22×C4—1221no—
473SD16—42no—
474SD16—4881no—
475SD16—4881no—
476SD16—2441no—
477D8—41no—
478D8—4881no—
479M16—4881no—
480M16—4881no—
481M16—4881no—
482C2×C8—21no—
483C2×C8—611no—
484C2×C8—611no—
485C2×C8—1221no—
486C2×C8—1221no—
487C2×C8—1221no—
488C2×C8—1221no—
489C2×C8C2×C22⋊D6111yescharacteristic
490C2×C8C2×C22⋊D6111yescharacteristic
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—41no—
497C4⋊C4—41no—
498C4⋊C4—611no—
499C4⋊C4—1221no—
500C4⋊C4—1221no—
501C4⋊C4D4×D6111yescharacteristic
502C4⋊C4C2×C22⋊D6111yescharacteristic
503C4⋊C4C2×C22⋊D6111yescharacteristic
504C22⋊C4—21no—
505C22⋊C4—21no—
506C22⋊C4—41no—
507C22⋊C4—41no—
508C22⋊C4—41no—
509C22⋊C4—1221no—
510C22⋊C4—2441no—
511C22⋊C4—2441no—
512C22⋊C4—2441no—
513C42—21no—
514C42—41no—
515C42—2441no—
516C42C23×D6111yescharacteristic
517C23—21no—
518C23—21no—
519C23—1221no—
520C23—9761no—
521C23—4882no—
522C23—4882no—
523C23—2441no—
524Q8—21no—
525Q8—21no—
526Q8—41no—
527Q8—41no—
528Q8—41no—
529Q8—41no—
530Q8—82no—
531Q8—4881no—
532D4—21no—
533D4—21no—
534D4—21no—
535D4—21no—
536D4—1221no—
537D4—1221no—
538D4—9761no—
539D4—4881no—
540D4—4882no—
541D4—4882no—
542D4—2441no—
543D4—2441no—
544D4—2441no—
545D4—2441no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—82no—
557C2×C4—1221no—
558C2×C4—2441no—
559C2×C4—2441no—
560C2×C4—2441no—
561C2×C4C2×D4×D6111yescharacteristic
562C2×C4C22×C22⋊D6111yescharacteristic
563C2×C4#1451982211yescharacteristic
564C8—21no—
565C8—21no—
566C8—41no—
567C8—1221no—
568C8—1221no—
569C8—2441no—
570C8—2441no—
571C8—2441no—
572C8—2441no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—9764no—
580C22—4881no—
581C22—4881no—
582C22—4881no—
583C22—2441no—
584C22—2441no—
585C22—2441no—
586C22D4×C22⋊D6111yescharacteristic
587C4—21no—
588C4—41no—
589C4—41no—
590C4—41no—
591C4—41no—
592C4—81no—
593C4—81no—
594C4—4881no—
595C4—2441no—
596C4#1684055811yescharacteristic
597C4#1684056011yescharacteristic
598C4#1684056611yescharacteristic
599C4#1684057011yescharacteristic
600C2—81no—
601C2—81no—
602C2—9761no—
603C2—4881no—
604C2#2145742411yescharacteristic
605C2#2145745111yescharacteristic
606C2#2145760511yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3367C2
socleC22×C61D42
prime radical#21458063C2
hypercenter#3367D61
hypocenterC61#20287
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3367

Complements

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

NH ≅ G/Ntype
1. #21458113C2semidirect
3. #21458044C2semidirect
4. #21445432C2semidirect
5. #21445392C2semidirect
6. #21445382C2semidirect
7. #21445363C2semidirect
8. #21445359C2semidirect
9. #21445045C2semidirect
10. #21444997C2semidirect
11. #21444991C2semidirect
12. #21444922C2semidirect
13. #21444709C2semidirect
14. #21444704C2semidirect
15. C61×#3367C2semidirect
18. #16840114C22semidirect
19. #16840112C22semidirect
26. #16840057C22semidirect
41. #16839840C22semidirect
45. #16839515C22semidirect
46. #16839512C22semidirect
47. #16839510C22semidirect
48. #16839508C22semidirect
49. #16839508C22semidirect
51. Q8⋊#92418C22semidirect
53. D4⋊#92418C22semidirect
55. C8×#92418C22semidirect
57. D4×Q8×C61C22semidirect
59. C61×#487C22semidirect
63. C61×#461C22semidirect
64. C61×#446C22semidirect
297. #3367D61semidirect
321. D4×Q8C2×D61semidirect
326. #487C2×D61semidirect
343. #461C2×D61semidirect
349. #446C2×D61semidirect
370. C61#20287semidirect

Maximal subgroups

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

subgroupindex
#214581132
#214580632
#214580442
#214454322
#214453922
#214453822
#214453632
#214453592
#214450452
#214449972
#214449912
#214449222
#214447092
#214447042
C61×#33672
#2028761