Finite Group Databasebeta

← #29662809

Subgroups — #29662809

Order 15616.

674 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456360C211yesmaximal, characteristic, complemented
2#21455826C211yesmaximal, characteristic
3#476×D61C211yesmaximal, characteristic
4#21454555C211yesmaximal, characteristic
5#21445342C211yesmaximal, characteristic, complemented
6#21445340C211yesmaximal, characteristic, complemented
7#21445316C211yesmaximal, characteristic, complemented
8#21445219C211yesmaximal, characteristic, complemented
9#21445217C211yesmaximal, characteristic, complemented
10#21445193C211yesmaximal, characteristic, complemented
11#21445191C211yesmaximal, characteristic, complemented
12#21444570C211yesmaximal, characteristic, complemented
13#21444508C211yesmaximal, characteristic, complemented
14#21444498C211yesmaximal, characteristic, complemented
15C61×#1691C211yesmaximal, characteristic, complemented
16C2×M16×D61—21no—
17#16840317C2211yescharacteristic
18C4⋊Q8×D61C2211yescharacteristic
19C4×D4×D61C2211yescharacteristic
20#16840169—21no—
21#16840169—21no—
22#16840167C2211yescharacteristic, complemented
23#16840165C2211yescharacteristic, complemented
24#16840158C2211yescharacteristic, complemented
25#16840091—21no—
26#16840053C2211yescharacteristic
27#16840053C2211yescharacteristic
28#16840052C2211yescharacteristic
29#16840052C2211yescharacteristic
30#16840051—21no—
31#16840050—21no—
32#16840049C2211yescharacteristic
33#16839964C2211yescharacteristic
34#16839911—21no—
35SD16×Dic61—21no—
36#16839847—21no—
37#16839840C2211yescharacteristic
38#16839836—21no—
39#16839831—21no—
40C8×D244C2211yescharacteristic
41#16839827C2211yescharacteristic
42#16839814C2211yescharacteristic
43#16839801—21no—
44#16839801—21no—
45#106×D61—21no—
46#16839784C2211yescharacteristic
47C4⋊C8×D61C2211yescharacteristic
48#102×D61—21no—
49#16839724—21no—
50C4×#11758189C2211yescharacteristic
51#16839514C2211yescharacteristic, complemented
52#16839513C2211yescharacteristic, complemented
53#16839512C2211yescharacteristic, complemented
54#16839511C2211yescharacteristic, complemented
55#16839510C2211yescharacteristic, complemented
56#16839510C2211yescharacteristic, complemented
57#16839510C2211yescharacteristic, complemented
58#16839510C2211yescharacteristic, complemented
59#16839508C2211yescharacteristic, complemented
60C61×#498C2211yescharacteristic, complemented
61C61×#476C2211yescharacteristic
62C61×#461C2211yescharacteristic
63C61×#436C2211yescharacteristic
64C61×#335C2211yescharacteristic, complemented
65C61×#333C2211yescharacteristic, complemented
66C61×#332C2211yescharacteristic, complemented
67C2×Q8×D61—21no—
68C2×D4×D61D411yescharacteristic
69C22×C4×D61—21no—
70C2×#11758201—21no—
71C2×#11758194—21no—
72C2×#11758189—21no—
73C2×#11758189D411yescharacteristic
74C2×#11758189D411yescharacteristic
75M16×D61—44no—
76C2×C8×D61—21no—
77C2×C8×D61—21no—
78C2×C8×D61—21no—
79#14519777C2311yescharacteristic
80#14519777C2311yescharacteristic
81#14519776—21no—
82#14519776—21no—
83#14519775—21no—
84#14519768—21no—
85#14519757—21no—
86#14519753—21no—
87Q8×Dic61—21no—
88#14519742—21no—
89C4×D244C2311yescharacteristic
90D4×Dic61D411yescharacteristic
91#14519739D411yescharacteristic
92C4×C22⋊D61—21no—
93#14519737—21no—
94#14519735—21no—
95C4⋊C4×D61—21no—
96C4⋊C4×D61—21no—
97C4⋊C4×D61D411yescharacteristic
98C22⋊C4×D61—21no—
99C42×D61C2311yescharacteristic
100#14519711—21no—
101#14519711—21no—
102#14519710—21no—
103#14519710—21no—
104#14519709—21no—
105#14519709—21no—
106#14519708—21no—
107#14519708—21no—
108#14519708—21no—
109#14519708—21no—
110#14519708—21no—
111C4⋊#92418C2311yescharacteristic
112C4⋊#92418C2311yescharacteristic
113C4⋊#92418C2311yescharacteristic
114#14519706—21no—
115#14519706—21no—
116#14519701—21no—
117#14519699—21no—
118#14519699—21no—
119#14519698—21no—
120#14519698—21no—
121#14519697—21no—
122C22⋊#92418—21no—
123#14519689—21no—
124#14519689—21no—
125#14519689—21no—
126#14519687D411yescharacteristic
127#14519687D411yescharacteristic
128C4×#92418C2311yescharacteristic
129C8×Dic61—21no—
130C2×SD16×C61—21no—
131C2×M16×C61—21no—
132C4⋊Q8×C61C2311yescharacteristic
133C4⋊Q8×C61C2311yescharacteristic
134C4×D4×C61C2311yescharacteristic
135#106×C61—21no—
136#106×C61—21no—
137#106×C61—21no—
138C4⋊C8×C61C2311yescharacteristic
139C4⋊C8×C61C2311yescharacteristic
140C4⋊C8×C61C2311yescharacteristic
141#103×C61—21no—
142#102×C61—21no—
143C22⋊C8×C61—21no—
144C4×C8×C61C2311yescharacteristic
145C23×D61—21no—
146C2×Q8×C61—21no—
147C2×Q8×C61—21no—
148C2×D4×C61C2×D411yescharacteristic
149C22×C4×C61—21no—
150C2×C22⋊D61—21no—
151C22×Dic61—21no—
152Q8×D61—82no—
153D4×D61—22no—
154D4×D61—41no—
155C2×D244C2×D411yescharacteristic
156C2×C4×D61—21no—
157C2×C4×D61—21no—
158C2×C4×D61—21no—
159C2×C4×D61—42no—
160C2×C4×D61—42no—
161C2×C4×D61—41no—
162C2×C4×D61C2×D411yescharacteristic
163C2×C4×D61C2×D411yescharacteristic
164C2×C4×D61#4111yescharacteristic
165C2×Dic122—21no—
166C2×Dic122—21no—
167C2×Dic122—41no—
168SD16×C61—42no—
169M16×C61—42no—
170C2×C8×C61—21no—
171C2×C8×C61—21no—
172C2×C8×C61—21no—
173C2×C8×C61C2×D411yescharacteristic
174C2×C8×C61C2×D411yescharacteristic
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61C2×D411yescharacteristic
180C22⋊C4×C61—21no—
181C42×C61C2411yescharacteristic
182C22⋊Dic61—21no—
183#11758201—42no—
184#11758198—21no—
185#11758198—21no—
186#11758198—41no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—41no—
192C4⋊Dic61C2×D411yescharacteristic
193#11758196—21no—
194#11758196—41no—
195#11758196—41no—
196#11758196—41no—
197C4×Dic61—41no—
198C4×Dic61C2×D411yescharacteristic
199C4×Dic61C2×D411yescharacteristic
200C4×Dic61#4111yescharacteristic
201#11758194—42no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418C2×D411yescharacteristic
206C2×#92418C2×D411yescharacteristic
207#11758189—22no—
208#11758189—42no—
209#11758189SD1614yescomplemented
210C8×D61—42no—
211C8×D61—42no—
212C8×D61—42no—
213C23×C61—21no—
214C22×D61—21no—
215C22×D61—42no—
216C22×D61—42no—
217C22×D61—81no—
218C22×D61#12111yescharacteristic
219Q8×C61—41no—
220Q8×C61—41no—
221Q8×C61—81no—
222D4×C61—21no—
223D4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61C22×D411yescharacteristic
230C2×C4×C61C22×D411yescharacteristic
231C2×C4×C61C2×#4111yescharacteristic
232C22⋊D61—42no—
233C2×Dic61—21no—
234C2×Dic61—42no—
235C2×Dic61—41no—
236C2×Dic61—41no—
237C2×Dic61#12111yescharacteristic
238C2×Dic61#12111yescharacteristic
239C2×Dic61#12111yescharacteristic
240D244—21no—
241D244#13712yes—
242C4×D61—21no—
243C4×D61—21no—
244C4×D61—41no—
245C4×D61—42no—
246C4×D61—41no—
247C4×D61—41no—
248C4×D61—42no—
249C4×D61—82no—
250C4×D61C2×SD1612yes—
251C4×D61#13712yes—
252Dic122—82no—
253Dic122—81no—
254Dic122—81no—
255C8×C61—21no—
256C8×C61—41no—
257C8×C61—41no—
258C8×C61—41no—
259C8×C61C2×SD1612yes—
260#92418—21no—
261#92418—41no—
262#92418—41no—
263#92418—41no—
264#92418C2×SD1612yes—
265#19165—611nomaximal, Sylow
266C22×C61—41no—
267C22×C61—41no—
268C22×C61—41no—
269C22×C61C2×#12111yescharacteristic
270C2×D61—21no—
271C2×D61—21no—
272C2×D61—22no—
273C2×D61—41no—
274C2×D61—41no—
275C2×D61—84no—
276C2×D61#46112yes—
277C4×C61—21no—
278C4×C61—41no—
279C4×C61—81no—
280C4×C61—81no—
281C4×C61C22×SD1611yescharacteristic
282C4×C61C2×#13711yescharacteristic
283C4×C61C2×#13711yescharacteristic
284C4×C61#57511yescharacteristic
285Dic61—21no—
286Dic61—21no—
287Dic61—41no—
288Dic61—81no—
289Dic61—81no—
290Dic61#46112yes—
291#3165—611no—
292#3101—611no—
293C2×#476—611no—
294D4×M16—611no—
295#1695—611no—
296#1695—611no—
297#1691—611no—
298#1691D6111yescharacteristic, complemented
299#1675—611no—
300#1675—611no—
301#1671—611no—
302#1671—611no—
303#1565—611no—
304#1550—611no—
305#1548—611no—
306C2×C61—81no—
307C2×C61C2×#46111yescharacteristic
308C2×C61#305611yescharacteristic
309C2×C61#306811yescharacteristic
310D61—22no—
311D61—42no—
312C22×M16—1221no—
313#533—611no—
314C2×C4⋊Q8—611no—
315C2×C4×D4—611no—
316#501—1221no—
317#501—1221no—
318#500—611no—
319#500—611no—
320#498—611no—
321#498C2×D6111yescharacteristic, complemented
322#483—1221no—
323#476—611no—
324#476—611no—
325#476—611no—
326#476—611no—
327#476—611no—
328#476—1221no—
329#476—1221no—
330#476C2×D6111yescharacteristic
331#461—611no—
332#461C2×D6111yescharacteristic
333#450—1221no—
334C4×SD16—1221no—
335C4×SD16—1221no—
336#436—611no—
337#436C2×D6111yescharacteristic
338#435—1221no—
339#435—1221no—
340C8×D4—611no—
341C8×D4—611no—
342#431—611no—
343#428—1221no—
344#428—1221no—
345C2×#106—1221no—
346#423—611no—
347C2×C4⋊C8—611no—
348C2×#102—1221no—
349#407—1221no—
350C4×M16—611no—
351#335—611no—
352#335C2×D6111yescharacteristic, complemented
353#334—611no—
354#334—611no—
355#333—611no—
356#333—611no—
357#333—611no—
358#333—611no—
359#333—611no—
360#333C2×D6111yescharacteristic, complemented
361#332—611no—
362#332C2×D6111yescharacteristic, complemented
363C61#1916511yescharacteristic, complemented, Sylow
364C22×Q8—1221no—
365C22×D4—611no—
366C23×C4—1221no—
367C2×SD16—21no—
368C2×SD16—1221no—
369C2×M16—21no—
370C2×M16—611no—
371C2×M16—611no—
372C2×M16—1221no—
373C2×M16—1221no—
374C2×M16—2444no—
375C22×C8—1221no—
376C22×C8—1221no—
377C22×C8—1221no—
378C4⋊Q8—611no—
379C4⋊Q8—611no—
380C4⋊Q8—1221no—
381C4⋊Q8—1221no—
382C4⋊Q8—1221no—
383C4⋊Q8C22×D6111yescharacteristic
384C4⋊Q8C22×D6111yescharacteristic
385#125—1221no—
386C22⋊Q8—1221no—
387C22⋊Q8—1221no—
388C4×Q8—1221no—
389C4×Q8—1221no—
390C4×D4—611no—
391C4×D4—611no—
392C4×D4—611no—
393C4×D4—1221no—
394C4×D4—1221no—
395C4×D4C22×D6111yescharacteristic
396#117—1221no—
397C2×C4⋊C4—611no—
398C2×C4⋊C4—1221no—
399C2×C4⋊C4—1221no—
400C2×C22⋊C4—1221no—
401C2×C42—611no—
402#107—1221no—
403#107—1221no—
404#107—1221no—
405#107—1221no—
406#106—21no—
407#106—21no—
408#106—21no—
409#106—1221no—
410#106—1221no—
411#106—1221no—
412#106—1221no—
413#106—1221no—
414#106—1221no—
415#106—1221no—
416C4⋊C8—611no—
417C4⋊C8—611no—
418C4⋊C8—611no—
419C4⋊C8—1221no—
420C4⋊C8—1221no—
421C4⋊C8C22×D6111yescharacteristic
422C4⋊C8C22×D6111yescharacteristic
423C4⋊C8C22×D6111yescharacteristic
424#103—21no—
425#103—1221no—
426#102—21no—
427#102—1221no—
428#102—1221no—
429#102—1221no—
430#102—1221no—
431#102—1221no—
432C22⋊C8—21no—
433C22⋊C8—1221no—
434C22⋊C8—1221no—
435C22⋊C8—1221no—
436C22⋊C8—1221no—
437#97—611no—
438#97—611no—
439C4×C8—611no—
440C4×C8—1221no—
441C4×C8C22×D6111yescharacteristic
442C24—1221no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—1221no—
446C2×Q8—1221no—
447C2×Q8—4882no—
448C2×Q8—2441no—
449C2×D4—611no—
450C2×D4—1221no—
451C2×D4—1222no—
452C2×D4—2441no—
453C2×D4D4×D6111yescharacteristic
454C22×C4—21no—
455C22×C4—611no—
456C22×C4—611no—
457C22×C4—611no—
458C22×C4—1221no—
459C22×C4—1221no—
460C22×C4—1221no—
461C22×C4—1221no—
462C22×C4—2442no—
463C22×C4—2442no—
464C22×C4—2441no—
465SD16—42no—
466SD16—2442no—
467M16—42no—
468M16—614no—
469M16—1222no—
470M16—2442no—
471M16—2442no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—611no—
476C2×C8—611no—
477C2×C8—1221no—
478C2×C8—1221no—
479C2×C8—1221no—
480C2×C8—2442no—
481C2×C8—2442no—
482C2×C8—2442no—
483C2×C8D4×D6111yescharacteristic
484C2×C8D4×D6111yescharacteristic
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—611no—
490C4⋊C4—1221no—
491C4⋊C4—1221no—
492C4⋊C4—1221no—
493C4⋊C4—1221no—
494C4⋊C4—1221no—
495C4⋊C4—2441no—
496C4⋊C4—2441no—
497C4⋊C4—2441no—
498C4⋊C4—2441no—
499C4⋊C4D4×D6111yescharacteristic
500C22⋊C4—21no—
501C22⋊C4—1221no—
502C22⋊C4—1221no—
503C22⋊C4—1221no—
504C22⋊C4—2441no—
505C42—611no—
506C42—611no—
507C42—611no—
508C42—2441no—
509C42C23×D6111yescharacteristic
510C23—21no—
511C23—611no—
512C23—1221no—
513C23—4881no—
514C23—2442no—
515C23—2442no—
516Q8—41no—
517Q8—41no—
518Q8—81no—
519Q8—4881no—
520Q8—4881no—
521Q8—4882no—
522D4—21no—
523D4—21no—
524D4—612no—
525D4—1221no—
526D4—2442no—
527C2×C4—21no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—611no—
533C2×C4—612no—
534C2×C4—612no—
535C2×C4—611no—
536C2×C4—611no—
537C2×C4—1221no—
538C2×C4—1221no—
539C2×C4—1221no—
540C2×C4—4882no—
541C2×C4—2441no—
542C2×C4—2442no—
543C2×C4—2442no—
544C2×C4—2442no—
545C2×C4—2441no—
546C2×C4—2441no—
547C2×C4—2441no—
548C2×C4—2441no—
549C2×C4C2×D4×D6111yescharacteristic
550C2×C4C2×D4×D6111yescharacteristic
551C2×C4#41×D6111yescharacteristic
552C8—21no—
553C8—41no—
554C8—41no—
555C8—41no—
556C8—612no—
557C8—1221no—
558C8—2441no—
559C8—2441no—
560C8—2441no—
561C8SD16×D6112yes—
562C22—41no—
563C22—41no—
564C22—41no—
565C22—612no—
566C22—1221no—
567C22—1221no—
568C22—1222no—
569C22—4884no—
570C22—2441no—
571C22—2441no—
572C22#121×D6111yescharacteristic
573C4—21no—
574C4—41no—
575C4—81no—
576C4—81no—
577C4—612no—
578C4—1221no—
579C4—1221no—
580C4—4881no—
581C4—4881no—
582C4—2441no—
583C4C2×SD16×D6111yescharacteristic
584C4#137×D6111yescharacteristic
585C4#137×D6111yescharacteristic
586C4#1684055811yescharacteristic
587C2—81no—
588C2—1222no—
589C2—2442no—
590C2#461×D6111yescharacteristic
591C2#2145544411yescharacteristic
592C2#2145554211yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1691C2
socleC22×C61C2×#121
prime radicalC2×D4×D61D4
hypercenter#1691D61
hypocenterC61#19165
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1691

Complements

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

NH ≅ G/Ntype
1. #21456360C2semidirect
5. #21445342C2semidirect
6. #21445340C2semidirect
7. #21445316C2semidirect
8. #21445219C2semidirect
9. #21445217C2semidirect
10. #21445193C2semidirect
11. #21445191C2semidirect
12. #21444570C2semidirect
13. #21444508C2semidirect
14. #21444498C2semidirect
15. C61×#1691C2semidirect
22. #16840167C22semidirect
23. #16840165C22semidirect
24. #16840158C22semidirect
51. #16839514C22semidirect
52. #16839513C22semidirect
53. #16839512C22semidirect
54. #16839511C22semidirect
55. #16839510C22semidirect
56. #16839510C22semidirect
57. #16839510C22semidirect
58. #16839510C22semidirect
59. #16839508C22semidirect
60. C61×#498C22semidirect
64. C61×#335C22semidirect
65. C61×#333C22semidirect
66. C61×#332C22semidirect
209. #11758189SD16semidirect
298. #1691D61semidirect
321. #498C2×D61semidirect
352. #335C2×D61semidirect
360. #333C2×D61semidirect
362. #332C2×D61semidirect
363. C61#19165semidirect

Maximal subgroups

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

subgroupindex
#214563602
#214558262
#476×D612
#214545552
#214453422
#214453402
#214453162
#214452192
#214452172
#214451932
#214451912
#214445702
#214445082
#214444982
C61×#16912
#1916561