Finite Group Databasebeta

← #29663305

Subgroups — #29663305

Order 15616.

658 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#21456766C211yesmaximal, characteristic, complemented
2#21456759C211yesmaximal, characteristic
3#486×D61C211yesmaximal, characteristic
4#21454750C211yesmaximal, characteristic, complemented
5#21445388C211yesmaximal, characteristic, complemented
6#21445387C211yesmaximal, characteristic, complemented
7#21445384C211yesmaximal, characteristic, complemented
8#21445266C211yesmaximal, characteristic, complemented
9#21445265C211yesmaximal, characteristic, complemented
10#21445262C211yesmaximal, characteristic, complemented
11#21445261C211yesmaximal, characteristic, complemented
12#21444564C211yesmaximal, characteristic, complemented
13#21444502C211yesmaximal, characteristic, complemented
14#21444485C211yesmaximal, characteristic
15C61×#1702C211yesmaximal, characteristic, complemented
16#16840316C2211yescharacteristic
17C4⋊Q8×D61C2211yescharacteristic
18#127×D61C2211yescharacteristic
19#16840140—21no—
20#16840133—21no—
21#16840111—21no—
22#16840110C2211yescharacteristic
23#16840109C2211yescharacteristic
24#16840108C2211yescharacteristic
25#16840107—21no—
26#16840107—21no—
27#16840107—21no—
28#16839973C2211yescharacteristic, complemented
29#16839972C2211yescharacteristic, complemented
30#16839969C2211yescharacteristic, complemented
31#16839963C2211yescharacteristic
32#16839962C2211yescharacteristic
33#16839959C2211yescharacteristic
34#16839919—21no—
35#16839916—21no—
36#16839914—21no—
37#16839911—21no—
38#16839892C2211yescharacteristic, complemented
39#16839890—21no—
40Q8×#92418C2211yescharacteristic, complemented
41#16839886—21no—
42#16839886C2211yescharacteristic, complemented
43#16839814C2211yescharacteristic
44#16839814C2211yescharacteristic
45#16839787—21no—
46#16839784C2211yescharacteristic
47#102×D61—21no—
48#102×D61—21no—
49#16839719—21no—
50C4×C8×D61C2211yescharacteristic
51#16839514C2211yescharacteristic, complemented
52#16839514C2211yescharacteristic, complemented
53#16839513C2211yescharacteristic, complemented
54#16839509C2211yescharacteristic, complemented
55#16839508C2211yescharacteristic, complemented
56#16839507C2211yescharacteristic, complemented
57#16839506C2211yescharacteristic
58#16839505C2211yescharacteristic
59#16839505C2211yescharacteristic
60C61×#486C2211yescharacteristic
61C61×#462C2211yescharacteristic, complemented
62C61×#460C2211yescharacteristic
63C61×#446C2211yescharacteristic, complemented
64C61×#334C2211yescharacteristic, complemented
65C61×#332C2211yescharacteristic, complemented
66C61×#331C2211yescharacteristic
67C2×Q8×D61—21no—
68C2×D4×D61—21no—
69C2×D4×D61—21no—
70C2×D4×D61—21no—
71C2×Dic244—21no—
72C2×#11758202—21no—
73C2×#11758202—21no—
74C2×#11758190—21no—
75C2×#11758200—21no—
76C2×#11758201—21no—
77C2×#11758189—21no—
78C2×C8×D61D411yescharacteristic
79C2×C8×D61D411yescharacteristic
80#14519777C2311yescharacteristic
81#14519776—21no—
82#14519776—21no—
83#14519776D411yescharacteristic
84#14519775—21no—
85#14519775D411yescharacteristic
86#14519774C2311yescharacteristic
87#14519773—21no—
88#14519773—21no—
89#14519773—21no—
90#14519767—21no—
91#14519757D411yescharacteristic
92#14519755—21no—
93#14519753D411yescharacteristic
94C4×Dic122—21no—
95C4×Dic122C2311yescharacteristic
96#14519734—21no—
97C4⋊C4×D61—21no—
98C4⋊C4×D61—21no—
99C42×D61C2311yescharacteristic
100C4⋊#92418—21no—
101C4⋊#92418C2311yescharacteristic
102C4⋊#92418C2311yescharacteristic
103C4⋊#92418C2311yescharacteristic
104#14519706—21no—
105#14519706—21no—
106#14519706—21no—
107#14519702—21no—
108#14519701—21no—
109#14519700—21no—
110#14519699—21no—
111#14519699—21no—
112#14519699—21no—
113#14519698—21no—
114#14519698—21no—
115#14519698—21no—
116#14519697—21no—
117#14519697—21no—
118#14519697—21no—
119#14519689—21no—
120#14519689—21no—
121#14519687—21no—
122C4×#92418—21no—
123C4×#92418C2311yescharacteristic
124C8×Dic61—21no—
125C8×Dic61—21no—
126C8×Dic61—21no—
127C8×Dic61D411yescharacteristic
128C8×Dic61D411yescharacteristic
129C2×Q16×C61—21no—
130C2×SD16×C61—21no—
131C4⋊Q8×C61C2311yescharacteristic
132#127×C61C2311yescharacteristic
133C4×Q8×C61C2311yescharacteristic
134C4⋊C8×C61—21no—
135C4⋊C8×C61C2311yescharacteristic
136C4⋊C8×C61C2311yescharacteristic
137C4⋊C8×C61C2311yescharacteristic
138#103×C61—21no—
139#102×C61—21no—
140#102×C61—21no—
141#102×C61—21no—
142#97×C61—21no—
143C4×C8×C61C2311yescharacteristic
144C23×D61—41no—
145C2×Q8×C61—21no—
146C2×Q8×C61C2×D411yescharacteristic
147C2×D4×C61—21no—
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C2×C22⋊D61—41no—
151C2×C22⋊D61—41no—
152C2×C22⋊D61—41no—
153Q8×D61—81no—
154Q8×D61—81no—
155D4×D61—42no—
156D4×D61—42no—
157D4×D61—81no—
158D4×D61—81no—
159D4×D61—82no—
160C2×D244—21no—
161C2×D244—21no—
162C2×D244—21no—
163C2×C4×D61—41no—
164C2×C4×D61C2×D411yescharacteristic
165C2×C4×D61C2×D411yescharacteristic
166C2×C4×D61#4111yescharacteristic
167C2×Dic122—21no—
168C2×Dic122—21no—
169C2×Dic122—41no—
170C2×Dic122C2×D411yescharacteristic
171Q16×C61—81no—
172SD16×C61—81no—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61C2×D411yescharacteristic
177C2×C8×C61C2×D411yescharacteristic
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61C2×D411yescharacteristic
182C42×C61—21no—
183C42×C61C2411yescharacteristic
184#11758202—81no—
185#11758202—81no—
186#11758201—81no—
187#11758200—81no—
188#11758198—21no—
189#11758198—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61C2×D411yescharacteristic
194#11758196—21no—
195#11758196—21no—
196#11758196—21no—
197#11758196—21no—
198#11758196—21no—
199#11758196—41no—
200#11758196—41no—
201C4×Dic61—21no—
202C4×Dic61C2×D411yescharacteristic
203C4×Dic61C2×D411yescharacteristic
204C4×Dic61#4111yescharacteristic
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418C2×D411yescharacteristic
209C2×#92418C2×D411yescharacteristic
210Dic244—81no—
211#11758190—81no—
212#11758189—42no—
213C8×D61—22no—
214C8×D61—22no—
215C23×C61—41no—
216C22×D61—41no—
217C22×D61—82no—
218C22×D61—82no—
219C22×D61—82no—
220C22×D61#12111yescharacteristic
221Q8×C61—21no—
222Q8×C61—21no—
223Q8×C61—41no—
224Q8×C61—41no—
225D4×C61—41no—
226D4×C61—41no—
227D4×C61—41no—
228D4×C61—41no—
229D4×C61—81no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—41no—
233C2×C4×C61C22×D411yescharacteristic
234C2×C4×C61C22×D411yescharacteristic
235C2×C4×C61C2×#4111yescharacteristic
236C22⋊D61—82no—
237C22⋊D61—82no—
238C22⋊D61—82no—
239C2×Dic61—21no—
240C2×Dic61—21no—
241C2×Dic61—41no—
242C2×Dic61#12111yescharacteristic
243C2×Dic61#12111yescharacteristic
244C2×Dic61#12111yescharacteristic
245D244—41no—
246D244—41no—
247D244—41no—
248D244—41no—
249D244—81no—
250C4×D61—21no—
251C4×D61—21no—
252C4×D61—22no—
253C4×D61—82no—
254C4×D61#13512yes—
255C4×D61#13712yes—
256Dic122—21no—
257Dic122—21no—
258Dic122—42no—
259Dic122—41no—
260Dic122—41no—
261Dic122—82no—
262C8×C61—21no—
263C8×C61—21no—
264C8×C61—41no—
265C8×C61—41no—
266C8×C61—41no—
267#92418—21no—
268#92418—21no—
269#92418—41no—
270#92418—41no—
271#92418—41no—
272#19214—611nomaximal, Sylow
273C22×C61—81no—
274C22×C61—81no—
275C22×C61—81no—
276C22×C61C2×#12111yescharacteristic
277C2×D61—21no—
278C2×D61—21no—
279C2×D61—81no—
280C2×D61—81no—
281C2×D61—81no—
282C2×D61—164no—
283C2×D61#46412yes—
284C4×C61—21no—
285C4×C61—41no—
286C4×C61—41no—
287C4×C61—81no—
288C4×C61C2×#13511yescharacteristic
289C4×C61C2×#13711yescharacteristic
290C4×C61#57611yescharacteristic
291C4×C61#57611yescharacteristic
292Dic61—21no—
293Dic61—41no—
294Dic61—41no—
295Dic61—81no—
296Dic61#46412yes—
297Dic61#47112yes—
298#3209—611no—
299#3208—611no—
300C2×#486—611no—
301#2960—611no—
302#1702—611no—
303#1702—611no—
304#1702—611no—
305#1702D6111yescharacteristic, complemented
306#1682—611no—
307#1682—611no—
308#1682—611no—
309#1682—611no—
310#1564—611no—
311#1549—611no—
312#1545—611no—
313C2×C61—161no—
314C2×C61C2×#46411yescharacteristic
315C2×C61#305811yescharacteristic
316C2×C61C2×#47111yescharacteristic
317D61—22no—
318D61—161no—
319#533—611no—
320C2×C4⋊Q8—611no—
321C2×#127—611no—
322#494—1221no—
323#492—1221no—
324#487—1221no—
325#486—611no—
326#486—611no—
327#486—611no—
328#486—1221no—
329#486—1221no—
330#486—1221no—
331#486C2×D6111yescharacteristic
332#462—611no—
333#462—611no—
334#462—611no—
335#462C2×D6111yescharacteristic, complemented
336#460—611no—
337#460—611no—
338#460—611no—
339#460C2×D6111yescharacteristic
340#451—1221no—
341#451—1221no—
342#450—1221no—
343#450—1221no—
344#446—611no—
345#446—1221no—
346#446C2×D6111yescharacteristic, complemented
347C8×Q8—611no—
348C8×Q8—611no—
349C8×Q8—1221no—
350#431—611no—
351#431—611no—
352#424—1221no—
353#423—611no—
354C2×#102—1221no—
355C2×#102—1221no—
356#405—1221no—
357C2×C4×C8—611no—
358#335—611no—
359#335—611no—
360#334—611no—
361#334C2×D6111yescharacteristic, complemented
362#332—611no—
363#332—611no—
364#332—611no—
365#332C2×D6111yescharacteristic, complemented
366#331—611no—
367#331—611no—
368#331—611no—
369#331C2×D6111yescharacteristic
370C61#1921411yescharacteristic, complemented, Sylow
371C22×Q8—1221no—
372C22×D4—1221no—
373C22×D4—1221no—
374C22×D4—1221no—
375C2×Q16—21no—
376C2×Q16—1221no—
377C2×Q16—1221no—
378C2×Q16—1221no—
379C2×SD16—21no—
380C2×SD16—1221no—
381C2×SD16—1221no—
382C2×SD16—1221no—
383C2×M16—1221no—
384C22×C8—611no—
385C22×C8—611no—
386C4⋊Q8—611no—
387C4⋊Q8—611no—
388C4⋊Q8—611no—
389C4⋊Q8—1221no—
390C4⋊Q8—1221no—
391C4⋊Q8—1221no—
392C4⋊Q8C22×D6111yescharacteristic
393#127—611no—
394#127—1221no—
395#127—1221no—
396#127—1221no—
397#127C22×D6111yescharacteristic
398#125—1221no—
399C22⋊Q8—611no—
400C22⋊Q8—611no—
401C22⋊Q8—1221no—
402C4×Q8—611no—
403C4×Q8—1221no—
404C4×Q8C22×D6111yescharacteristic
405#117—1221no—
406C2×C4⋊C4—1221no—
407C2×C4⋊C4—1221no—
408C2×C42—611no—
409C4⋊C8—21no—
410C4⋊C8—611no—
411C4⋊C8—611no—
412C4⋊C8—611no—
413C4⋊C8—1221no—
414C4⋊C8—1221no—
415C4⋊C8—1221no—
416C4⋊C8—1221no—
417C4⋊C8C22×D6111yescharacteristic
418C4⋊C8C22×D6111yescharacteristic
419C4⋊C8C22×D6111yescharacteristic
420#103—21no—
421#103—1221no—
422#103—1221no—
423#103—1221no—
424#102—21no—
425#102—21no—
426#102—21no—
427#102—1221no—
428#102—1221no—
429#102—1221no—
430#102—1221no—
431#102—1221no—
432#102—1221no—
433#102—1221no—
434#102—1221no—
435#102—1221no—
436C22⋊C8—1221no—
437C22⋊C8—1221no—
438#97—21no—
439#97—1221no—
440C4×C8—611no—
441C4×C8—611no—
442C4×C8—611no—
443C4×C8—1221no—
444C4×C8—1221no—
445C4×C8—1221no—
446C4×C8—1221no—
447C4×C8C22×D6111yescharacteristic
448C24—2441no—
449C2×Q8—21no—
450C2×Q8—611no—
451C2×Q8—1221no—
452C2×Q8—1221no—
453C2×Q8—4881no—
454C2×Q8—4881no—
455C2×Q8—2441no—
456C2×Q8D4×D6111yescharacteristic
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—1221no—
461C2×D4—1221no—
462C2×D4—1221no—
463C2×D4—4881no—
464C2×D4—4881no—
465C2×D4—4882no—
466C2×D4—2442no—
467C2×D4—2442no—
468C2×D4—2441no—
469C2×D4—2441no—
470C2×D4—2441no—
471C22×C4—611no—
472C22×C4—611no—
473C22×C4—611no—
474C22×C4—2441no—
475Q16—81no—
476Q16—4881no—
477Q16—4881no—
478Q16—4881no—
479SD16—81no—
480SD16—4881no—
481SD16—4881no—
482SD16—4881no—
483M16—2442no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—611no—
488C2×C8—611no—
489C2×C8—1222no—
490C2×C8—1222no—
491C2×C8—1221no—
492C2×C8—1221no—
493C2×C8—1221no—
494C2×C8D4×D6111yescharacteristic
495C2×C8D4×D6111yescharacteristic
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—611no—
500C4⋊C4—1221no—
501C4⋊C4—1221no—
502C4⋊C4—1221no—
503C4⋊C4—1221no—
504C4⋊C4—1221no—
505C4⋊C4—1221no—
506C4⋊C4—1221no—
507C4⋊C4—1221no—
508C4⋊C4—2441no—
509C4⋊C4—2441no—
510C4⋊C4D4×D6111yescharacteristic
511C22⋊C4—1221no—
512C22⋊C4—1221no—
513C42—21no—
514C42—611no—
515C42—611no—
516C42—611no—
517C42—1221no—
518C42C23×D6111yescharacteristic
519C23—41no—
520C23—611no—
521C23—4882no—
522C23—4882no—
523C23—4882no—
524C23—2441no—
525Q8—21no—
526Q8—21no—
527Q8—41no—
528Q8—41no—
529Q8—1221no—
530Q8—1221no—
531Q8—4882no—
532Q8—2442no—
533Q8—2441no—
534Q8—2441no—
535D4—41no—
536D4—41no—
537D4—41no—
538D4—41no—
539D4—81no—
540D4—4881no—
541D4—4882no—
542D4—4882no—
543D4—4882no—
544D4—2441no—
545D4—2441no—
546D4—2441no—
547D4—2441no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—41no—
551C2×C4—611no—
552C2×C4—611no—
553C2×C4—612no—
554C2×C4—612no—
555C2×C4—611no—
556C2×C4—1222no—
557C2×C4—1221no—
558C2×C4—1221no—
559C2×C4—1221no—
560C2×C4—1221no—
561C2×C4—4882no—
562C2×C4—2441no—
563C2×C4C2×D4×D6111yescharacteristic
564C2×C4C2×D4×D6111yescharacteristic
565C2×C4#41×D6111yescharacteristic
566C8—21no—
567C8—21no—
568C8—41no—
569C8—41no—
570C8—41no—
571C8—1221no—
572C8—1221no—
573C8—2441no—
574C8—2441no—
575C8—2441no—
576C22—81no—
577C22—81no—
578C22—81no—
579C22—612no—
580C22—1221no—
581C22—1221no—
582C22—9764no—
583C22—4881no—
584C22—4881no—
585C22—4881no—
586C22#121×D6111yescharacteristic
587C4—21no—
588C4—41no—
589C4—41no—
590C4—81no—
591C4—612no—
592C4—612no—
593C4—1221no—
594C4—4881no—
595C4—2441no—
596C4—2441no—
597C4#135×D6111yescharacteristic
598C4#137×D6111yescharacteristic
599C4#1684056611yescharacteristic
600C4#1684056611yescharacteristic
601C2—161no—
602C2—1222no—
603C2—9761no—
604C2#464×D6111yescharacteristic
605C2#2145546511yescharacteristic
606C2#471×D6111yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1702C2
socleC22×C61C2×#121
prime radical#127×D61C22
hypercenter#1702D61
hypocenterC61#19214
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1702

Complements

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

NH ≅ G/Ntype
1. #21456766C2semidirect
4. #21454750C2semidirect
5. #21445388C2semidirect
6. #21445387C2semidirect
7. #21445384C2semidirect
8. #21445266C2semidirect
9. #21445265C2semidirect
10. #21445262C2semidirect
11. #21445261C2semidirect
12. #21444564C2semidirect
13. #21444502C2semidirect
15. C61×#1702C2semidirect
28. #16839973C22semidirect
29. #16839972C22semidirect
30. #16839969C22semidirect
38. #16839892C22semidirect
40. Q8×#92418C22semidirect
42. #16839886C22semidirect
51. #16839514C22semidirect
52. #16839514C22semidirect
53. #16839513C22semidirect
54. #16839509C22semidirect
55. #16839508C22semidirect
56. #16839507C22semidirect
61. C61×#462C22semidirect
63. C61×#446C22semidirect
64. C61×#334C22semidirect
65. C61×#332C22semidirect
305. #1702D61semidirect
335. #462C2×D61semidirect
346. #446C2×D61semidirect
361. #334C2×D61semidirect
365. #332C2×D61semidirect
370. C61#19214semidirect

Maximal subgroups

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

subgroupindex
#214567662
#214567592
#486×D612
#214547502
#214453882
#214453872
#214453842
#214452662
#214452652
#214452622
#214452612
#214445642
#214445022
#214444852
C61×#17022
#1921461