Finite Group Databasebeta

← #29669515

Subgroups — #29669515

Order 15616.

668 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458731C211yesmaximal, characteristic, complemented
2SD16×Dic122C211yesmaximal, characteristic, complemented
3#21457645C211yesmaximal, characteristic
4D4×#11758190C211yesmaximal, characteristic
5#21445441C211yesmaximal, characteristic, complemented
6#21445291C211yesmaximal, characteristic, complemented
7#21445236C211yesmaximal, characteristic, complemented
8#21445218C211yesmaximal, characteristic, complemented
9#21445195C211yesmaximal, characteristic, complemented
10#21445029C211yesmaximal, characteristic, complemented
11#21444987C211yesmaximal, characteristic, complemented
12#21444937C211yesmaximal, characteristic
13#21444899C211yesmaximal, characteristic, complemented
14#21444633C211yesmaximal, characteristic, complemented
15C61×#1582C211yesmaximal, characteristic, complemented
16C22×#11758190—21no—
17Q8×Dic122C2211yescharacteristic, complemented
18#16840472C2211yescharacteristic
19D4×Dic122C2211yescharacteristic
20D4×D244C2211yescharacteristic
21#16840171—21no—
22#16840167C2211yescharacteristic, complemented
23#16840160C2211yescharacteristic, complemented
24#16840159—21no—
25#16840135C2211yescharacteristic
26#16840135C2211yescharacteristic
27#16840113C2211yescharacteristic, complemented
28#16840109C2211yescharacteristic
29#16840082—21no—
30#16840070—21no—
31#16840058C2211yescharacteristic, complemented
32#16840053C2211yescharacteristic
33#16840052C2211yescharacteristic
34#16840052C2211yescharacteristic
35#16840050—21no—
36#16840047C2211yescharacteristic, complemented
37#16840045—21no—
38#16840043—21no—
39#16839994—21no—
40#16839991—21no—
41#16839968C2211yescharacteristic
42#16839963C2211yescharacteristic
43#16839962C2211yescharacteristic
44#16839915—21no—
45#16839911—21no—
46#16839903—21no—
47C8×Dic122C2211yescharacteristic, complemented
48#16839864—21no—
49C4×#11758190C2211yescharacteristic
50SD16×Dic61—21no—
51C8×D244C2211yescharacteristic
52#102×D61—21no—
53#16839515C2211yescharacteristic, complemented
54#16839510C2211yescharacteristic, complemented
55#16839510C2211yescharacteristic, complemented
56#16839509C2211yescharacteristic, complemented
57#16839505C2211yescharacteristic
58#16839494C2211yescharacteristic, complemented
59#16839491C2211yescharacteristic
60C61×#446C2211yescharacteristic, complemented
61C4×SD16×C61C2211yescharacteristic
62C8×D4×C61C2211yescharacteristic
63C61×#334C2211yescharacteristic, complemented
64C61×Q8⋊C8C2211yescharacteristic, complemented
65C61×D4⋊C8C2211yescharacteristic
66C61×#322C2211yescharacteristic, complemented
67C22×Dic122—21no—
68C22×D244—21no—
69C2×D4×D61—21no—
70C2×#11758190—21no—
71C2×#11758190—21no—
72C2×#11758190—21no—
73C2×#11758190—21no—
74C2×#11758190—44no—
75C2×#11758190—41no—
76C2×#11758190D411yescharacteristic
77C2×#11758201—21no—
78C2×C8×D61—21no—
79#14519777—21no—
80#14519777C2311yescharacteristic
81#14519777C2311yescharacteristic
82#14519777C2311yescharacteristic
83#14519776—21no—
84#14519776—21no—
85#14519776—41no—
86#14519774C2311yescharacteristic
87#14519769—21no—
88#14519756—21no—
89#14519755—41no—
90#14519754—41no—
91#14519753—21no—
92#14519752—21no—
93#14519751—21no—
94#14519747—41no—
95Q8×Dic61—21no—
96C4×Dic122—21no—
97C4×Dic122C2311yescharacteristic
98C4×D244—21no—
99C4×D244C2311yescharacteristic
100D4×Dic61—21no—
101C4⋊C4×D61—21no—
102#14519711—21no—
103#14519711—21no—
104#14519711—21no—
105#14519709—21no—
106#14519709—21no—
107#14519709—21no—
108#14519709D411yescharacteristic
109#14519708—21no—
110#14519708—21no—
111#14519708—21no—
112C4⋊#92418C2311yescharacteristic
113C4⋊#92418C2311yescharacteristic
114#14519706—21no—
115#14519702—21no—
116#14519702—21no—
117#14519702—41no—
118#14519702D411yescharacteristic
119#14519701—21no—
120#14519699—21no—
121#14519699—21no—
122#14519699—21no—
123#14519699—21no—
124#14519699D411yescharacteristic
125#14519698—21no—
126#14519698—21no—
127#14519697—21no—
128#14519689—21no—
129C8×Dic61—21no—
130C2×SD16×C61D411yescharacteristic
131C22×C8×C61—21no—
132C4×Q8×C61C2311yescharacteristic
133C4×D4×C61C2311yescharacteristic
134#106×C61D411yescharacteristic
135C4⋊C8×C61—21no—
136C4⋊C8×C61C2311yescharacteristic
137C4⋊C8×C61C2311yescharacteristic
138#103×C61D411yescharacteristic
139#102×C61D411yescharacteristic
140C22⋊C8×C61—21no—
141#97×C61—21no—
142C4×C8×C61C2311yescharacteristic
143C4×C8×C61C2311yescharacteristic
144C4×C8×C61C2311yescharacteristic
145C23×D61—41no—
146C2×Q8×C61C2×D411yescharacteristic
147C2×D4×C61C2×D411yescharacteristic
148C22×C4×C61—21no—
149C2×C22⋊D61—41no—
150C22×Dic61—41no—
151D4×D61—42no—
152D4×D61—81no—
153C2×D244—21no—
154C2×D244—21no—
155C2×D244—21no—
156C2×D244—42no—
157C2×D244—41no—
158C2×D244—81no—
159C2×D244C2×D411yescharacteristic
160C2×C4×D61—21no—
161C2×C4×D61—41no—
162C2×Dic122—21no—
163C2×Dic122—21no—
164C2×Dic122—21no—
165C2×Dic122—42no—
166C2×Dic122—41no—
167C2×Dic122—41no—
168C2×Dic122—81no—
169C2×Dic122C2×D411yescharacteristic
170SD16×C61—22no—
171C2×C8×C61—21no—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61—42no—
175C2×C8×C61C2×D411yescharacteristic
176C2×C8×C61C2×D411yescharacteristic
177C2×C8×C61C2×D411yescharacteristic
178C2×C8×C61C2×D411yescharacteristic
179C4⋊C4×C61—21no—
180C4⋊C4×C61C2×D411yescharacteristic
181C4⋊C4×C61C2×D411yescharacteristic
182C22⋊C4×C61—21no—
183C42×C61—21no—
184C42×C61C2411yescharacteristic
185C22⋊Dic61—41no—
186#11758201—81no—
187#11758198—21no—
188#11758198—41no—
189#11758198—41no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—41no—
197C4⋊Dic61C2×D411yescharacteristic
198C4⋊Dic61C2×D411yescharacteristic
199#11758196—21no—
200#11758196—41no—
201#11758196—41no—
202#11758196—41no—
203#11758196—41no—
204C4×Dic61—21no—
205C4×Dic61—41no—
206C2×#92418—21no—
207C2×#92418—21no—
208#11758190—42no—
209#11758190—42no—
210#11758190—42no—
211#11758190—81no—
212#11758190—82no—
213#11758190SD1614yescomplemented
214C8×D61—81no—
215C23×C61—21no—
216C22×D61—21no—
217C22×D61—41no—
218C22×D61—82no—
219C22×D61—82no—
220C22×D61—161no—
221Q8×C61—21no—
222Q8×C61—21no—
223D4×C61—21no—
224D4×C61#13712yes—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61C22×D411yescharacteristic
231C2×C4×C61C22×D411yescharacteristic
232C2×C4×C61H(5,𝔽2)11yescharacteristic
233C22⋊D61—82no—
234C2×Dic61—21no—
235C2×Dic61—21no—
236C2×Dic61—21no—
237C2×Dic61—41no—
238C2×Dic61—41no—
239C2×Dic61—41no—
240C2×Dic61—82no—
241D244—21no—
242D244—41no—
243D244—41no—
244D244—41no—
245D244—41no—
246D244—81no—
247D244—82no—
248D244C2×SD1612yes—
249C4×D61—41no—
250C4×D61—41no—
251C4×D61—161no—
252Dic122—21no—
253Dic122—41no—
254Dic122—41no—
255Dic122—82no—
256Dic122—81no—
257Dic122—81no—
258Dic122—161no—
259Dic122C2×SD1612yes—
260C8×C61—21no—
261C8×C61—21no—
262C8×C61—21no—
263C8×C61—41no—
264C8×C61—41no—
265C8×C61—41no—
266C8×C61C2×SD1612yes—
267#92418—41no—
268#92418—41no—
269#19765—611nomaximal, Sylow
270C22×C61—22no—
271C22×C61—41no—
272C22×C61—41no—
273C22×C61D4211yescharacteristic
274C2×D61—22no—
275C2×D61—41no—
276C2×D61—41no—
277C2×D61—81no—
278C2×D61—81no—
279C2×D61—81no—
280C2×D61—164no—
281C4×C61—21no—
282C4×C61—41no—
283C4×C61—41no—
284C4×C61—41no—
285C4×C61C22×SD1611yescharacteristic
286C4×C61C2×#13711yescharacteristic
287C4×C61#57711yescharacteristic
288C4×C61#57711yescharacteristic
289Dic61—22no—
290Dic61—41no—
291Dic61—41no—
292Dic61—81no—
293Dic61—81no—
294Dic61—81no—
295#3385—611no—
296Q8×SD16—611no—
297#3295—611no—
298D4×SD16—611no—
299#1714—611no—
300#1686—611no—
301#1678—611no—
302#1675—611no—
303#1671—611no—
304#1643—611no—
305#1637—611no—
306#1629—611no—
307#1623—611no—
308#1582D6111yescharacteristic, complemented
309SD16⋊C8—611no—
310C2×C61—42no—
311C2×C61#326611yescharacteristic
312C2×C61D4×SD1611yescharacteristic
313C2×C61#328811yescharacteristic
314D61—42no—
315D61—161no—
316C22×SD16—1221no—
317Q82—611no—
318#554—611no—
319D4×Q8—611no—
320D42—611no—
321#501—1221no—
322#500—611no—
323#498—611no—
324#498—1221no—
325#492—611no—
326#492—611no—
327#487—611no—
328#486—611no—
329#481—1221no—
330#479—1221no—
331#477—611no—
332#476—611no—
333#476—611no—
334#476—611no—
335#476—1221no—
336#475—611no—
337#475—1221no—
338#474—1221no—
339#465—1221no—
340#465—1221no—
341#461—611no—
342#460—611no—
343#460—611no—
344#450—1221no—
345#450—1221no—
346#448—1221no—
347#446C2×D6111yescharacteristic, complemented
348C8×Q8—611no—
349#440—1221no—
350C4×SD16—611no—
351C4×SD16—1221no—
352C4×SD16C2×D6111yescharacteristic
353C8×D4—611no—
354C8×D4C2×D6111yescharacteristic
355C2×#102—1221no—
356#335—611no—
357#334C2×D6111yescharacteristic, complemented
358#333—611no—
359#333—611no—
360#332—611no—
361#331—611no—
362Q8⋊C8—611no—
363Q8⋊C8C2×D6111yescharacteristic, complemented
364D4⋊C8—611no—
365D4⋊C8C2×D6111yescharacteristic
366#322C2×D6111yescharacteristic, complemented
367C61#1976511yescharacteristic, complemented, Sylow
368C22×Q8—1221no—
369C22×D4—1221no—
370C22×D4—1221no—
371C2×SD16—611no—
372C2×SD16—1221no—
373C2×SD16—1221no—
374C2×SD16—1221no—
375C2×SD16—1221no—
376C2×SD16—1221no—
377C2×SD16—2441no—
378C2×SD16—2444no—
379C2×SD16D24411yescharacteristic
380C22×C8—21no—
381C22×C8—1221no—
382C4⋊Q8—611no—
383C4⋊Q8—611no—
384C4⋊Q8—611no—
385C4⋊Q8—1221no—
386C4⋊Q8—1221no—
387C4⋊Q8—1221no—
388C4⋊Q8—2441no—
389#127—611no—
390#125—1221no—
391C22⋊Q8—1221no—
392C22⋊Q8—1221no—
393C22⋊Q8—2441no—
394C22⋊Q8—2441no—
395#121—1221no—
396#121—1221no—
397#120—2441no—
398C4×Q8—611no—
399C4×Q8—1221no—
400C4×Q8—1221no—
401C4×Q8C22×D6111yescharacteristic
402C4×D4—611no—
403C4×D4—1221no—
404C4×D4—1221no—
405C4×D4C22×D6111yescharacteristic
406C2×C4⋊C4—1221no—
407#107—1221no—
408#107—1221no—
409#107—1221no—
410#106—611no—
411#106—1221no—
412#106—1221no—
413#106—1221no—
414#106—1221no—
415#106—1221no—
416#106—1221no—
417#106D24411yescharacteristic
418C4⋊C8—21no—
419C4⋊C8—611no—
420C4⋊C8—611no—
421C4⋊C8—1221no—
422C4⋊C8C22×D6111yescharacteristic
423C4⋊C8C22×D6111yescharacteristic
424#103—611no—
425#103—1221no—
426#103—1221no—
427#103—1221no—
428#103—2441no—
429#103D24411yescharacteristic
430#102—611no—
431#102—1221no—
432#102—1221no—
433#102—1221no—
434#102—1221no—
435#102—1221no—
436#102—1221no—
437#102—1221no—
438#102D24411yescharacteristic
439C22⋊C8—21no—
440C22⋊C8—1221no—
441#97—21no—
442C4×C8—1221no—
443C4×C8C22×D6111yescharacteristic
444C4×C8C22×D6111yescharacteristic
445C4×C8C22×D6111yescharacteristic
446C24—2441no—
447C2×Q8—611no—
448C2×Q8—1221no—
449C2×Q8—1221no—
450C2×Q8—1221no—
451C2×Q8—4881no—
452C2×Q8—2441no—
453C2×Q8—2441no—
454C2×Q8—2442no—
455C2×Q8C2×D24411yescharacteristic
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×D4C2×D24411yescharacteristic
467C22×C4—21no—
468C22×C4—1221no—
469C22×C4—2441no—
470C22×C4—2441no—
471SD16—22no—
472SD16—614no—
473SD16—4881no—
474SD16—4882no—
475SD16—4881no—
476SD16—2442no—
477SD16—2442no—
478SD16—2442no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—42no—
483C2×C8—1221no—
484C2×C8—1221no—
485C2×C8—4881no—
486C2×C8C2×D24411yescharacteristic
487C2×C8C2×D24411yescharacteristic
488C2×C8D4×D6111yescharacteristic
489C2×C8D4×D6111yescharacteristic
490C4⋊C4—21no—
491C4⋊C4—611no—
492C4⋊C4—611no—
493C4⋊C4—1221no—
494C4⋊C4—1221no—
495C4⋊C4—1221no—
496C4⋊C4—1221no—
497C4⋊C4—1221no—
498C4⋊C4—1221no—
499C4⋊C4—1221no—
500C4⋊C4—2441no—
501C4⋊C4—2441no—
502C4⋊C4—2441no—
503C4⋊C4—2441no—
504C4⋊C4—2441no—
505C4⋊C4C2×D24411yescharacteristic
506C4⋊C4C2×D24411yescharacteristic
507C22⋊C4—21no—
508C22⋊C4—1221no—
509C22⋊C4—2441no—
510C22⋊C4—2441no—
511C22⋊C4—2441no—
512C42—21no—
513C42—1221no—
514C42—2441no—
515C42C23×D6111yescharacteristic
516C23—21no—
517C23—1221no—
518C23—9761no—
519C23—4882no—
520C23—4882no—
521C23—2441no—
522Q8—21no—
523Q8—21no—
524Q8—612no—
525Q8—1221no—
526Q8—9761no—
527Q8—4881no—
528Q8—4881no—
529Q8—4882no—
530Q8—2441no—
531Q8—2441no—
532D4—21no—
533D4—612no—
534D4—1221no—
535D4—4881no—
536D4—4882no—
537D4—4882no—
538D4—2441no—
539D4—2441no—
540D4—2441no—
541D4—2441no—
542D4#1451981012yes—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—1221no—
549C2×C4—1221no—
550C2×C4—1221no—
551C2×C4—9761no—
552C2×C4—4882no—
553C2×C4—2441no—
554C2×C4—2441no—
555C2×C4—2441no—
556C2×C4—2441no—
557C2×C4—2441no—
558C2×C4C2×D4×D6111yescharacteristic
559C2×C4C22×D24411yescharacteristic
560C2×C4#1451982311yescharacteristic
561C8—21no—
562C8—21no—
563C8—21no—
564C8—41no—
565C8—41no—
566C8—41no—
567C8—2441no—
568C8—2441no—
569C8SD16×D6112yes—
570C22—22no—
571C22—41no—
572C22—41no—
573C22—1222no—
574C22—9764no—
575C22—4881no—
576C22—4881no—
577C22—4881no—
578C22—2441no—
579C22—2441no—
580C22D4×D24411yescharacteristic
581C4—21no—
582C4—41no—
583C4—41no—
584C4—41no—
585C4—1222no—
586C4—4881no—
587C4—4881no—
588C4—4881no—
589C4—2441no—
590C4—2441no—
591C4C2×SD16×D6111yescharacteristic
592C4C2×#1451981011yescharacteristic
593C4#1684056911yescharacteristic
594C4#1684057411yescharacteristic
595C2—42no—
596C2—9761no—
597C2—2442no—
598C2#2145733411yescharacteristic
599C2SD16×D24411yescharacteristic
600C2#2145758611yescharacteristic
subgroupG/H
centerC22D4×D244
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1582C2
socleC22×C61D42
prime radicalD4×D244C22
hypercenter#1582D61
hypocenterC61#19765
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1582

Complements

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

NH ≅ G/Ntype
1. #21458731C2semidirect
2. SD16×Dic122C2semidirect
5. #21445441C2semidirect
6. #21445291C2semidirect
7. #21445236C2semidirect
8. #21445218C2semidirect
9. #21445195C2semidirect
10. #21445029C2semidirect
11. #21444987C2semidirect
13. #21444899C2semidirect
14. #21444633C2semidirect
15. C61×#1582C2semidirect
17. Q8×Dic122C22semidirect
22. #16840167C22semidirect
23. #16840160C22semidirect
27. #16840113C22semidirect
31. #16840058C22semidirect
36. #16840047C22semidirect
47. C8×Dic122C22semidirect
53. #16839515C22semidirect
54. #16839510C22semidirect
55. #16839510C22semidirect
56. #16839509C22semidirect
58. #16839494C22semidirect
60. C61×#446C22semidirect
63. C61×#334C22semidirect
64. C61×Q8⋊C8C22semidirect
66. C61×#322C22semidirect
213. #11758190SD16semidirect
308. #1582D61semidirect
347. #446C2×D61semidirect
357. #334C2×D61semidirect
363. Q8⋊C8C2×D61semidirect
366. #322C2×D61semidirect
367. C61#19765semidirect

Maximal subgroups

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

subgroupindex
#214587312
SD16×Dic1222
#214576452
D4×#117581902
#214454412
#214452912
#214452362
#214452182
#214451952
#214450292
#214449872
#214449372
#214448992
#214446332
C61×#15822
#1976561