Finite Group Databasebeta

← #29671482

Subgroups — #29671482

Order 15616.

666 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458063C211yesmaximal, characteristic, complemented
2#21457747C211yesmaximal, characteristic, complemented
3#21457626C211yesmaximal, characteristic, complemented
4#21445432C211yesmaximal, characteristic, complemented
5#21445363C211yesmaximal, characteristic, complemented
6#21445352C211yesmaximal, characteristic, complemented
7#21445268C211yesmaximal, characteristic, complemented
8#21445258C211yesmaximal, characteristic, complemented
9#21445045C211yesmaximal, characteristic, complemented
10#21444997C211yesmaximal, characteristic, complemented
11#21444970C211yesmaximal, characteristic, complemented
12#21444908C211yesmaximal, characteristic, complemented
13#21444702C211yesmaximal, characteristic, complemented
14D8×#92418C211yesmaximal, characteristic, complemented
15C61×#3330C211yesmaximal, 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#16840075—21no—
26#16840052C2211yescharacteristic
27#16840052C2211yescharacteristic, complemented
28#16840039C2211yescharacteristic
29#16840014—21no—
30#16839996—21no—
31#16839993—21no—
32#16839964C2211yescharacteristic
33#16839957C2211yescharacteristic
34#16839957C2211yescharacteristic
35#16839907—21no—
36#16839895—21no—
37#16839894—21no—
38C4×#11758199C2211yescharacteristic
39#16839841C2211yescharacteristic
40D4×#92418C2211yescharacteristic
41D4×#92418C2211yescharacteristic, complemented
42#16839765—21no—
43#16839758—21no—
44C2×#14519697—21no—
45#16839515C2211yescharacteristic, complemented
46#16839514C2211yescharacteristic, complemented
47#16839510C2211yescharacteristic, complemented
48#16839508C2211yescharacteristic, complemented
49#16839508C2211yescharacteristic, complemented
50#16839505C2211yescharacteristic
51D4⋊#92418C2211yescharacteristic
52D4⋊#92418C2211yescharacteristic, complemented
53D4⋊#92418C2211yescharacteristic, complemented
54#16839489C2211yescharacteristic
55C8×#92418C2211yescharacteristic, complemented
56C61×#547C2211yescharacteristic
57C61×#547C2211yescharacteristic, complemented
58C61×#487C2211yescharacteristic, complemented
59C61×#484—21no—
60C61×#474C2211yescharacteristic
61C61×#468—21no—
62C61×#461C2211yescharacteristic, complemented
63C61×#449—21no—
64C4×D8×C61C2211yescharacteristic
65C61×#436C2211yescharacteristic, complemented
66C61×#417—21no—
67C22×D244—21no—
68C2×D4×D61—21no—
69C2×#11758201—21no—
70C2×#11758199—21no—
71C2×#11758199—21no—
72C2×#11758199D411yescharacteristic
73C2×#11758189—21no—
74C22×#92418—21no—
75C22×#92418—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#14519710D411yescharacteristic
89#14519708—21no—
90#14519708—21no—
91C4⋊#92418C2311yescharacteristic
92C4⋊#92418C2311yescharacteristic
93C4⋊#92418C2311yescharacteristic
94#14519701—21no—
95#14519701—21no—
96#14519701—21no—
97#14519700—41no—
98#14519699—21no—
99#14519699—21no—
100#14519698—21no—
101#14519698—21no—
102#14519698D411yescharacteristic
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×#41×C61—21no—
116C2×#41×C61—21no—
117C2×SD16×C61—21no—
118C2×D8×C61D411yescharacteristic
119C2×M16×C61—21no—
120C4⋊Q8×C61C2311yescharacteristic
121C4⋊Q8×C61C2311yescharacteristic
122#123×C61—41no—
123#123×C61—41no—
124C22⋊Q8×C61—21no—
125C22⋊Q8×C61—21no—
126#121×C61—21no—
127#121×C61—21no—
128C4×D4×C61C2311yescharacteristic
129C4×D4×C61C2311yescharacteristic
130C4×D4×C61C2311yescharacteristic
131C2×C4⋊C4×C61—21no—
132C2×C4⋊C4×C61—21no—
133#107×C61—21no—
134#107×C61D411yescharacteristic
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
142C22⋊C8×C61—21no—
143C4×C8×C61C2311yescharacteristic
144C23×D61—41no—
145#41×C61—42no—
146#41×C61—42no—
147#41×C61—81no—
148#41×C61—81no—
149C2×Q8×C61—21no—
150C2×Q8×C61—21no—
151C2×D4×C61—41no—
152C2×D4×C61—41no—
153C2×D4×C61C2×D411yescharacteristic
154C2×D4×C61C2×D411yescharacteristic
155C2×D4×C61C2×D411yescharacteristic
156C22×C4×C61—21no—
157C22×C4×C61—21no—
158C22×C4×C61—21no—
159C22×C4×C61—41no—
160C22×C4×C61—41no—
161C2×C22⋊D61—41no—
162C2×C22⋊D61—41no—
163D4×D61—42no—
164D4×D61—81no—
165C2×D244—21no—
166C2×D244—21no—
167C2×D244—21no—
168C2×D244—42no—
169C2×D244—41no—
170C2×D244—81no—
171C2×D244C2×D411yescharacteristic
172C2×C4×D61—21no—
173C2×Dic122—21no—
174SD16×C61—81no—
175D8×C61—22no—
176M16×C61—81no—
177C2×C8×C61—21no—
178C2×C8×C61C2×D411yescharacteristic
179C2×C8×C61C2×D411yescharacteristic
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—41no—
185C4⋊C4×C61—41no—
186C4⋊C4×C61C2×D411yescharacteristic
187C4⋊C4×C61C2×D411yescharacteristic
188C4⋊C4×C61C2×D411yescharacteristic
189C22⋊C4×C61—21no—
190C22⋊C4×C61—21no—
191C22⋊C4×C61—21no—
192C22⋊C4×C61—41no—
193C22⋊C4×C61—41no—
194C42×C61C2411yescharacteristic
195C22⋊Dic61—41no—
196#11758201—81no—
197#11758199—41no—
198#11758199—81no—
199#11758199—81no—
200#11758198—21no—
201#11758198—41no—
202#11758198—41no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61C2×D411yescharacteristic
206C4×Dic61—41no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—81no—
212C2×#92418—81no—
213C2×#92418C2×D411yescharacteristic
214C2×#92418C2×D411yescharacteristic
215#11758189—81no—
216C23×C61—21no—
217C23×C61—21no—
218C23×C61—21no—
219C22×D61—21no—
220C22×D61—41no—
221C22×D61—82no—
222C22×D61—82no—
223C22×D61—161no—
224Q8×C61—41no—
225Q8×C61—41no—
226Q8×C61—41no—
227Q8×C61—41no—
228D4×C61—21no—
229D4×C61—21no—
230D4×C61—21no—
231D4×C61—21no—
232D4×C61—21no—
233D4×C61—82no—
234D4×C61—82no—
235D4×C61#13512yes—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—82no—
248C2×C4×C61—82no—
249C2×C4×C61C22×D411yescharacteristic
250C2×C4×C61C22×D411yescharacteristic
251C2×C4×C61H(5,𝔽2)11yescharacteristic
252C22⋊D61—82no—
253C22⋊D61—161no—
254C2×Dic61—21no—
255C2×Dic61—41no—
256D244—21no—
257D244—21no—
258D244—41no—
259D244—41no—
260D244—41no—
261D244—41no—
262D244—81no—
263D244—82no—
264C4×D61—41no—
265C4×D61—41no—
266Dic122—81no—
267C8×C61—21no—
268C8×C61—21no—
269C8×C61—41no—
270#92418—21no—
271#92418—21no—
272#92418—41no—
273#92418—41no—
274#92418—41no—
275#92418—41no—
276#19899—611nomaximal, Sylow
277C22×C61—22no—
278C22×C61—41no—
279C22×C61—41no—
280C22×C61—41no—
281C22×C61—41no—
282C22×C61—41no—
283C22×C61—41no—
284C22×C61—41no—
285C22×C61—41no—
286C22×C61D4211yescharacteristic
287C2×D61—41no—
288C2×D61—41no—
289C2×D61—41no—
290C2×D61—81no—
291C2×D61—81no—
292C2×D61—81no—
293C2×D61—164no—
294C4×C61—21no—
295C4×C61—41no—
296C4×C61—41no—
297C4×C61—41no—
298C4×C61—81no—
299C4×C61—81no—
300C4×C61C2×#13511yescharacteristic
301C4×C61#57511yescharacteristic
302C4×C61#57611yescharacteristic
303C4×C61#57611yescharacteristic
304Dic61—41no—
305Dic61—81no—
306#3330D6111yescharacteristic, complemented
307#3325—611no—
308#3303—611no—
309#3294—611no—
310#1711—611no—
311#1699—611no—
312#1697—611no—
313#1683—611no—
314#1681—611no—
315#1646—611no—
316#1639—611no—
317#1635—611no—
318#1625—611no—
319#1589—611no—
320C8×D8—611no—
321C2×C61—42no—
322C2×C61—81no—
323C2×C61—81no—
324C2×C61#327311yescharacteristic
325C2×C61#328011yescharacteristic
326C2×C61#329011yescharacteristic
327D61—81no—
328D61—161no—
329#555—611no—
330#547C2×D6111yescharacteristic
331#547C2×D6111yescharacteristic, complemented
332D42—611no—
333#487—611no—
334#487—611no—
335#487C2×D6111yescharacteristic, complemented
336#486—611no—
337#486—611no—
338#486—611no—
339#485—1221no—
340#484—21no—
341#482—1221no—
342#480—1221no—
343#476—611no—
344#476—611no—
345#474—611no—
346#474C2×D6111yescharacteristic
347#469—1221no—
348#468—21no—
349#466—1221no—
350#465—1221no—
351#461—611no—
352#461C2×D6111yescharacteristic, complemented
353#459—611no—
354#459—611no—
355#449—21no—
356#449—1221no—
357#447—1221no—
358#447—1221no—
359C4×D8—611no—
360C4×D8C2×D6111yescharacteristic
361#436—611no—
362#436C2×D6111yescharacteristic, complemented
363C8×D4—611no—
364C8×D4—611no—
365#418—1221no—
366#417—21no—
367#416—1221no—
368C2×#102—1221no—
369#335—611no—
370#335—611no—
371#333—611no—
372#332—611no—
373#332—611no—
374#331—611no—
375D4⋊C8—611no—
376D4⋊C8—611no—
377D4⋊C8—611no—
378D4⋊C8—611no—
379C82—611no—
380C61#1989911yescharacteristic, complemented, Sylow
381C2×#41—21no—
382C2×#41—21no—
383C22×D4—1221no—
384C22×D4—1221no—
385C2×SD16—21no—
386C2×SD16—1221no—
387C2×D8—611no—
388C2×D8—1221no—
389C2×D8—1221no—
390C2×D8C22⋊D6111yescharacteristic
391C2×M16—21no—
392C2×M16—1221no—
393C22×C8—1221no—
394C22×C8—1221no—
395C4⋊Q8—611no—
396C4⋊Q8C22×D6111yescharacteristic
397C4⋊Q8C22×D6111yescharacteristic
398#127—611no—
399#124—2441no—
400#123—41no—
401#123—41no—
402C22⋊Q8—21no—
403C22⋊Q8—21no—
404#121—21no—
405#121—21no—
406#121—1221no—
407#121—1221no—
408#121—1221no—
409#121—1221no—
410#120—2441no—
411C4×D4—611no—
412C4×D4C22×D6111yescharacteristic
413C4×D4C22×D6111yescharacteristic
414C4×D4C22×D6111yescharacteristic
415#117—1221no—
416#117—1221no—
417C2×C4⋊C4—21no—
418C2×C4⋊C4—21no—
419#107—21no—
420#107—611no—
421#107—1221no—
422#107C22⋊D6111yescharacteristic
423#106—1221no—
424#106—1221no—
425C4⋊C8—611no—
426C4⋊C8—611no—
427C4⋊C8—611no—
428C4⋊C8C22×D6111yescharacteristic
429#103—21no—
430#103—21no—
431#103—21no—
432#103—1221no—
433#103—1221no—
434#103—1221no—
435#103—2441no—
436#102—21no—
437#102—611no—
438#102—611no—
439#102—1221no—
440#102—1221no—
441#102—1221no—
442#102—1221no—
443#102—1221no—
444#102—1221no—
445#102—1221no—
446#102—1221no—
447#102—1221no—
448#102—2441no—
449#102C22⋊D6111yescharacteristic
450#102C22⋊D6111yescharacteristic
451C22⋊C8—21no—
452C22⋊C8—1221no—
453C22⋊C8—1221no—
454C22⋊C8—1221no—
455C4×C8—611no—
456C4×C8—611no—
457C4×C8C22×D6111yescharacteristic
458C24—2441no—
459#41—42no—
460#41—42no—
461#41—81no—
462#41—81no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—1221no—
466C2×D4—41no—
467C2×D4—41no—
468C2×D4—611no—
469C2×D4—1221no—
470C2×D4—1221no—
471C2×D4—1221no—
472C2×D4—4881no—
473C2×D4—4881no—
474C2×D4—2441no—
475C2×D4—2442no—
476C2×D4—2441no—
477C2×D4—2441no—
478C2×D4—2442no—
479C2×D4D4×D6111yescharacteristic
480C2×D4C2×C22⋊D6111yescharacteristic
481C2×D4C2×C22⋊D6111yescharacteristic
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—41no—
486C22×C4—41no—
487C22×C4—1221no—
488SD16—81no—
489SD16—4881no—
490D8—22no—
491D8—4881no—
492D8—4881no—
493D8—2441no—
494M16—81no—
495M16—4881no—
496C2×C8—21no—
497C2×C8—611no—
498C2×C8—611no—
499C2×C8—1221no—
500C2×C8—1221no—
501C2×C8—1221no—
502C2×C8—1221no—
503C2×C8—4881no—
504C2×C8—4881no—
505C2×C8C2×C22⋊D6111yescharacteristic
506C2×C8C2×C22⋊D6111yescharacteristic
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—41no—
512C4⋊C4—41no—
513C4⋊C4—611no—
514C4⋊C4—1221no—
515C4⋊C4—1221no—
516C4⋊C4D4×D6111yescharacteristic
517C4⋊C4C2×C22⋊D6111yescharacteristic
518C4⋊C4C2×C22⋊D6111yescharacteristic
519C22⋊C4—21no—
520C22⋊C4—21no—
521C22⋊C4—21no—
522C22⋊C4—41no—
523C22⋊C4—41no—
524C22⋊C4—1221no—
525C22⋊C4—2441no—
526C22⋊C4—2441no—
527C22⋊C4—2441no—
528C42—2441no—
529C42C23×D6111yescharacteristic
530C23—21no—
531C23—21no—
532C23—21no—
533C23—1221no—
534C23—9761no—
535C23—4882no—
536C23—4882no—
537C23—2441no—
538Q8—41no—
539Q8—41no—
540Q8—41no—
541Q8—41no—
542Q8—4881no—
543D4—21no—
544D4—21no—
545D4—21no—
546D4—21no—
547D4—21no—
548D4—82no—
549D4—82no—
550D4—1221no—
551D4—1221no—
552D4—9761no—
553D4—4881no—
554D4—4882no—
555D4—4882no—
556D4—2441no—
557D4—2441no—
558D4—2441no—
559D4—2441no—
560D4#1451979712yes—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—82no—
573C2×C4—82no—
574C2×C4—1221no—
575C2×C4—2441no—
576C2×C4—2441no—
577C2×C4—2441no—
578C2×C4C2×D4×D6111yescharacteristic
579C2×C4C22×C22⋊D6111yescharacteristic
580C2×C4#1451982211yescharacteristic
581C8—21no—
582C8—21no—
583C8—41no—
584C8—1221no—
585C8—1221no—
586C8—2441no—
587C8—2441no—
588C8—2441no—
589C8—2441no—
590C22—22no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—9764no—
600C22—4881no—
601C22—4881no—
602C22—4881no—
603C22—2441no—
604C22—2441no—
605C22—2441no—
606C22D4×C22⋊D6111yescharacteristic
607C4—21no—
608C4—41no—
609C4—41no—
610C4—41no—
611C4—81no—
612C4—81no—
613C4—4881no—
614C4—2441no—
615C4C2×#1451979711yescharacteristic
616C4#1684055811yescharacteristic
617C4#1684056511yescharacteristic
618C4#1684056611yescharacteristic
619C2—42no—
620C2—81no—
621C2—81no—
622C2—9761no—
623C2—4881no—
624C2#2145742411yescharacteristic
625C2#2145750011yescharacteristic
626C2#2145760511yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3330C2
socleC22×C61D42
hypercenter#3330D61
hypocenterC61#19899
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3330

Complements

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

NH ≅ G/Ntype
1. #21458063C2semidirect
2. #21457747C2semidirect
3. #21457626C2semidirect
4. #21445432C2semidirect
5. #21445363C2semidirect
6. #21445352C2semidirect
7. #21445268C2semidirect
8. #21445258C2semidirect
9. #21445045C2semidirect
10. #21444997C2semidirect
11. #21444970C2semidirect
12. #21444908C2semidirect
13. #21444702C2semidirect
14. D8×#92418C2semidirect
15. C61×#3330C2semidirect
18. #16840114C22semidirect
19. #16840112C22semidirect
27. #16840052C22semidirect
41. D4×#92418C22semidirect
45. #16839515C22semidirect
46. #16839514C22semidirect
47. #16839510C22semidirect
48. #16839508C22semidirect
49. #16839508C22semidirect
52. D4⋊#92418C22semidirect
53. D4⋊#92418C22semidirect
55. C8×#92418C22semidirect
57. C61×#547C22semidirect
58. C61×#487C22semidirect
62. C61×#461C22semidirect
65. C61×#436C22semidirect
306. #3330D61semidirect
331. #547C2×D61semidirect
335. #487C2×D61semidirect
352. #461C2×D61semidirect
362. #436C2×D61semidirect
380. C61#19899semidirect

Maximal subgroups

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

subgroupindex
#214580632
#214577472
#214576262
#214454322
#214453632
#214453522
#214452682
#214452582
#214450452
#214449972
#214449702
#214449082
#214447022
D8×#924182
C61×#33302
#1989961