Finite Group Databasebeta

← #29669684

Subgroups — #29669684

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#21458803C211yesmaximal, characteristic, complemented
2#21457738C211yesmaximal, characteristic, complemented
3#21457632C211yesmaximal, characteristic
4#21445480C211yesmaximal, characteristic, complemented
5#21445366C211yesmaximal, characteristic, complemented
6#21445261C211yesmaximal, characteristic, complemented
7#21445253C211yesmaximal, characteristic, complemented
8#21445182C211yesmaximal, characteristic, complemented
9#21445102C211yesmaximal, characteristic, complemented
10#21445089C211yesmaximal, characteristic, complemented
11#21444998C211yesmaximal, characteristic, complemented
12#21444928C211yesmaximal, characteristic, complemented
13#21444646C211yesmaximal, characteristic, complemented
14SD16⋊#92418C211yesmaximal, characteristic, complemented
15C61×#3324C211yesmaximal, characteristic, complemented
16#16840485C2211yescharacteristic, complemented
17#16840416C2211yescharacteristic
18#16840158C2211yescharacteristic, complemented
19#16840144—21no—
20#16840141C2211yescharacteristic, complemented
21#16840113C2211yescharacteristic, complemented
22#16840108C2211yescharacteristic
23#16840108C2211yescharacteristic
24#16840106—21no—
25#16840073—21no—
26#16840042C2211yescharacteristic
27#16840042C2211yescharacteristic
28#16840002—21no—
29#16839996—21no—
30#16839984—21no—
31#16839981—21no—
32#16839972C2211yescharacteristic, complemented
33#16839967C2211yescharacteristic
34#16839964C2211yescharacteristic, complemented
35#16839954C2211yescharacteristic
36#16839921—21no—
37#16839896—21no—
38Q8×#92418C2211yescharacteristic, complemented
39C4×#11758202—21no—
40C4×#11758199C2211yescharacteristic
41C8×D244C2211yescharacteristic, complemented
42D4×#92418C2211yescharacteristic
43#16839755—21no—
44C2×#14519698—21no—
45#16839515C2211yescharacteristic, complemented
46#16839514C2211yescharacteristic, complemented
47#16839509C2211yescharacteristic, complemented
48#16839508C2211yescharacteristic, complemented
49#16839507C2211yescharacteristic, complemented
50#16839506C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic, complemented
52D4⋊#92418C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic
54#16839489C2211yescharacteristic, complemented
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×#11758224—21no—
68C2×#11758202—21no—
69C2×#11758202—21no—
70C2×#11758202—21no—
71C2×#11758200—21no—
72C2×#11758201—21no—
73C2×#11758199—21no—
74C2×#11758199D411yescharacteristic
75C22×#92418—21no—
76C2×C8×D61—21no—
77#14519777C2311yescharacteristic
78#14519777C2311yescharacteristic
79#14519772—41no—
80#14519765—21no—
81#14519761—41no—
82#14519757—21no—
83#14519753—21no—
84#14519751—21no—
85C4×Dic122—21no—
86C4×D244C2311yescharacteristic
87#14519735—21no—
88C2×C4⋊Dic61—21no—
89#14519710—21no—
90#14519710—21no—
91#14519710D411yescharacteristic
92#14519708—41no—
93C4⋊#92418—21no—
94C4⋊#92418C2311yescharacteristic
95C4⋊#92418C2311yescharacteristic
96C4⋊#92418C2311yescharacteristic
97#14519702—21no—
98#14519702—21no—
99#14519701—21no—
100#14519700—21no—
101#14519700—21no—
102#14519698—21no—
103#14519698—21no—
104#14519698—21no—
105#14519698—21no—
106#14519698—21no—
107#14519698—41no—
108#14519698D411yescharacteristic
109#14519697D411yescharacteristic
110C22⋊#92418—21no—
111#14519689—21no—
112C4×#92418—21no—
113C4×#92418C2311yescharacteristic
114C4×#92418C2311yescharacteristic
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×C22⋊D61—41no—
161C22×Dic61—41no—
162#11758224—42no—
163#11758224—81no—
164C2×D244C2×D411yescharacteristic
165C2×C4×D61—21no—
166C2×Dic122—21no—
167C2×Dic122—21no—
168SD16×C61—22no—
169SD16×C61—81no—
170M16×C61—81no—
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—41no—
178C4⋊C4×C61C2×D411yescharacteristic
179C4⋊C4×C61C2×D411yescharacteristic
180C4⋊C4×C61C2×D411yescharacteristic
181C22⋊C4×C61—21no—
182C22⋊C4×C61—21no—
183C22⋊C4×C61—41no—
184C22⋊C4×C61—41no—
185C22⋊C4×C61—41no—
186C42×C61—21no—
187C42×C61—41no—
188C42×C61C2411yescharacteristic
189#11758202—42no—
190#11758202—81no—
191#11758202—81no—
192#11758201—81no—
193#11758200—81no—
194#11758199—41no—
195#11758199—81no—
196#11758198—21no—
197#11758198—41no—
198#11758198—41no—
199#11758198—41no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—41no—
205C4⋊Dic61C2×D411yescharacteristic
206#11758196—41no—
207C4×Dic61—41no—
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—
218C23×C61—41no—
219C23×C61—82no—
220C23×C61—82no—
221C23×C61—161no—
222C22×D61—21no—
223Q8×C61—21no—
224Q8×C61—21no—
225Q8×C61—41no—
226Q8×C61—41no—
227D4×C61—21no—
228D4×C61—21no—
229D4×C61—21no—
230D4×C61—41no—
231D4×C61—41no—
232D4×C61—41no—
233D4×C61—41no—
234D4×C61—81no—
235D4×C61—82no—
236D4×C61—82no—
237D4×C61#13512yes—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61C22×D411yescharacteristic
248C2×C4×C61C22×D411yescharacteristic
249C2×C4×C61H(5,𝔽2)11yescharacteristic
250C22⋊D61—82no—
251C2×Dic61—21no—
252C2×Dic61—41no—
253C2×Dic61—41no—
254C2×Dic61—82no—
255D244—21no—
256D244—21no—
257C4×D61—41no—
258C4×D61—41no—
259Dic122—41no—
260Dic122—41no—
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—22no—
274C22×C61—41no—
275C22×C61—41no—
276C22×C61—41no—
277C22×C61—41no—
278C22×C61—41no—
279C22×C61—81no—
280C22×C61—81no—
281C22×C61—81no—
282C22×C61—164no—
283C22×C61D4211yescharacteristic
284C2×D61—41no—
285C2×D61—41no—
286C2×D61—41no—
287C4×C61—21no—
288C4×C61—41no—
289C4×C61—41no—
290C4×C61—41no—
291C4×C61—41no—
292C4×C61—81no—
293C4×C61C2×#13511yescharacteristic
294C4×C61#57511yescharacteristic
295C4×C61#57611yescharacteristic
296C4×C61#57611yescharacteristic
297Dic61—41no—
298Dic61—81no—
299Dic61—81no—
300#3393—611no—
301#3324D6111yescharacteristic, complemented
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—42no—
316C2×C61—81no—
317C2×C61—161no—
318C2×C61#326811yescharacteristic
319C2×C61#328011yescharacteristic
320C2×C61#328911yescharacteristic
321D61—81no—
322#556—611no—
323#556C2×D6111yescharacteristic
324#547—611no—
325D42C2×D6111yescharacteristic
326#498—611no—
327#495—1221no—
328#494—611no—
329#487—611no—
330#486—611no—
331#486—611no—
332#486C2×D6111yescharacteristic
333#485—1221no—
334#482—21no—
335#480—1221no—
336#474—611no—
337#474—611no—
338#474C2×D6111yescharacteristic
339#468—21no—
340#467—1221no—
341#466—1221no—
342#464—1221no—
343#464—1221no—
344#462—611no—
345#461—611no—
346#461—611no—
347#460C2×D6111yescharacteristic
348#459—611no—
349#452—1221no—
350#450—21no—
351#447—1221no—
352C8×Q8—611no—
353C4×Q16—1221no—
354C4×SD16C2×D6111yescharacteristic
355C4×D8—611no—
356#436C2×D6111yescharacteristic
357C8×D4—611no—
358C8×D4—611no—
359#418—21no—
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—21no—
376C22×D4—21no—
377C2×Q16—1221no—
378C2×Q16—1221no—
379C2×Q16—1221no—
380C2×SD16—21no—
381C2×SD16—1221no—
382C2×SD16—1221no—
383C2×SD16C22⋊D6111yescharacteristic
384C2×D8—611no—
385C2×D8—1221no—
386C2×M16—21no—
387C22×C8—1221no—
388C22×C8—1221no—
389C4⋊Q8—611no—
390C4⋊Q8—611no—
391C4⋊Q8C22×D6111yescharacteristic
392#127C22×D6111yescharacteristic
393#126—41no—
394#126—2441no—
395#124—21no—
396#124—1221no—
397#123—2441no—
398C22⋊Q8—21no—
399C22⋊Q8—1221no—
400C22⋊Q8—1221no—
401#121—21no—
402#121—21no—
403#121—1221no—
404#120—41no—
405C4×Q8—21no—
406C4×Q8—1221no—
407C4×Q8C22×D6111yescharacteristic
408C4×D4—611no—
409C4×D4C22×D6111yescharacteristic
410C4×D4C22×D6111yescharacteristic
411#117—21no—
412#117—1221no—
413C2×C4⋊C4—1221no—
414#107—21no—
415#107—611no—
416#107—1221no—
417#107—1221no—
418#106—2441no—
419#106C22⋊D6111yescharacteristic
420C4⋊C8—611no—
421C4⋊C8—611no—
422C4⋊C8—611no—
423C4⋊C8—1221no—
424C4⋊C8C22×D6111yescharacteristic
425#103—1221no—
426#103—1221no—
427#103—1221no—
428#103—1221no—
429#103—1221no—
430#103C22⋊D6111yescharacteristic
431#102—21no—
432#102—21no—
433#102—21no—
434#102—21no—
435#102—611no—
436#102—611no—
437#102—1221no—
438#102—1221no—
439#102—1221no—
440#102—1221no—
441#102—1221no—
442#102—2441no—
443#102C22⋊D6111yescharacteristic
444C22⋊C8—21no—
445C22⋊C8—1221no—
446C22⋊C8—1221no—
447C4×C8—611no—
448C4×C8—611no—
449C4×C8—1221no—
450C4×C8C22×D6111yescharacteristic
451C24—41no—
452#41—4881no—
453#41—2442no—
454C2×Q8—21no—
455C2×Q8—1221no—
456C2×Q8—1221no—
457C2×Q8C2×C22⋊D6111yescharacteristic
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—41no—
462C2×D4—41no—
463C2×D4—42no—
464C2×D4—42no—
465C2×D4—81no—
466C2×D4—81no—
467C2×D4—611no—
468C2×D4—2441no—
469C2×D4D4×D6111yescharacteristic
470C2×D4C2×C22⋊D6111yescharacteristic
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—1221no—
474C22×C4—2441no—
475Q16—4881no—
476Q16—4881no—
477Q16—2442no—
478SD16—22no—
479SD16—81no—
480SD16—4881no—
481SD16—4881no—
482D8—4881no—
483D8—2441no—
484M16—81no—
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—41no—
500C4⋊C4—611no—
501C4⋊C4—1221no—
502C4⋊C4—1221no—
503C4⋊C4—1221no—
504C4⋊C4—1221no—
505C4⋊C4—2441no—
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—1221no—
516C22⋊C4—2441no—
517C22⋊C4—2441no—
518C22⋊C4—2441no—
519C42—21no—
520C42—41no—
521C42—2441no—
522C42C23×D6111yescharacteristic
523C23—21no—
524C23—21no—
525C23—41no—
526C23—82no—
527C23—82no—
528C23—161no—
529C23—1221no—
530Q8—21no—
531Q8—21no—
532Q8—41no—
533Q8—41no—
534Q8—2441no—
535Q8—2441no—
536Q8—2441no—
537Q8—2441no—
538D4—21no—
539D4—21no—
540D4—21no—
541D4—41no—
542D4—41no—
543D4—41no—
544D4—41no—
545D4—81no—
546D4—82no—
547D4—82no—
548D4—1221no—
549D4—1221no—
550D4—4882no—
551D4#1451979712yes—
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—1221no—
562C2×C4—4882no—
563C2×C4—2441no—
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—22no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22—81no—
586C22—81no—
587C22—81no—
588C22—164no—
589C22—2441no—
590C22—2441no—
591C22—2441no—
592C22D4×C22⋊D6111yescharacteristic
593C4—21no—
594C4—41no—
595C4—41no—
596C4—41no—
597C4—41no—
598C4—81no—
599C4—4881no—
600C4—4881no—
601C4—2441no—
602C4C2×#1451979711yescharacteristic
603C4#1684055811yescharacteristic
604C4#1684056511yescharacteristic
605C4#1684056611yescharacteristic
606C2—42no—
607C2—81no—
608C2—161no—
609C2—4881no—
610C2#2145735311yescharacteristic
611C2#2145750011yescharacteristic
612C2#2145759211yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3324C2
socleC22×C61D42
prime radical#21457632C2
hypercenter#3324D61
hypocenterC61#19777
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. #21458803C2semidirect
2. #21457738C2semidirect
4. #21445480C2semidirect
5. #21445366C2semidirect
6. #21445261C2semidirect
7. #21445253C2semidirect
8. #21445182C2semidirect
9. #21445102C2semidirect
10. #21445089C2semidirect
11. #21444998C2semidirect
12. #21444928C2semidirect
13. #21444646C2semidirect
14. SD16⋊#92418C2semidirect
15. C61×#3324C2semidirect
16. #16840485C22semidirect
18. #16840158C22semidirect
20. #16840141C22semidirect
21. #16840113C22semidirect
32. #16839972C22semidirect
34. #16839964C22semidirect
38. Q8×#92418C22semidirect
41. C8×D244C22semidirect
45. #16839515C22semidirect
46. #16839514C22semidirect
47. #16839509C22semidirect
48. #16839508C22semidirect
49. #16839507C22semidirect
51. Q8⋊#92418C22semidirect
54. #16839489C22semidirect
55. #16839483C22semidirect
301. #3324D61semidirect
373. C61#19777semidirect

Maximal subgroups

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

subgroupindex
#214588032
#214577382
#214576322
#214454802
#214453662
#214452612
#214452532
#214451822
#214451022
#214450892
#214449982
#214449282
#214446462
SD16⋊#924182
C61×#33242
#1977761