Finite Group Databasebeta

← #29676863

Subgroups — #29676863

Order 15616.

670 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458846C211yesmaximal, characteristic, complemented
2#21458052C211yesmaximal, characteristic, complemented
3#21457439C211yesmaximal, characteristic
4#21445423C211yesmaximal, characteristic, complemented
5#21445387C211yesmaximal, characteristic, complemented
6#21445377C211yesmaximal, characteristic, complemented
7#21445306C211yesmaximal, characteristic, complemented
8#21445301C211yesmaximal, characteristic, complemented
9#21445012C211yesmaximal, characteristic, complemented
10#21444997C211yesmaximal, characteristic, complemented
11#21444928C211yesmaximal, characteristic, complemented
12#21444915C211yesmaximal, characteristic, complemented
13#21444710C211yesmaximal, characteristic, complemented
14#21444709C211yesmaximal, characteristic, complemented
15C61×#3368C211yesmaximal, characteristic, complemented
16C2×#14519802—21no—
17#16840485C2211yescharacteristic
18D4×D244C2211yescharacteristic
19#16840154—21no—
20#16840141C2211yescharacteristic, complemented
21#16840138C2211yescharacteristic
22#16840110C2211yescharacteristic
23#16840109C2211yescharacteristic
24#16840108C2211yescharacteristic, complemented
25#16840088—21no—
26#16840042C2211yescharacteristic
27#16840013—21no—
28#16840011—21no—
29#16839984—21no—
30#16839972C2211yescharacteristic, complemented
31#16839969—21no—
32#16839964C2211yescharacteristic
33#16839962C2211yescharacteristic
34#16839962C2211yescharacteristic
35#16839954C2211yescharacteristic
36#16839911—21no—
37#16839907—21no—
38#16839895—21no—
39#16839893C2211yescharacteristic, complemented
40#16839867—21no—
41C4×#11758201C2211yescharacteristic
42#16839841C2211yescharacteristic
43#16839840C2211yescharacteristic
44#16839765—21no—
45#16839513C2211yescharacteristic, complemented
46#16839512C2211yescharacteristic, complemented
47#16839508C2211yescharacteristic, complemented
48#16839507C2211yescharacteristic, complemented
49#16839506C2211yescharacteristic, complemented
50#16839505C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic, complemented
52Q8⋊#92418C2211yescharacteristic, complemented
53D4⋊#92418C2211yescharacteristic
54#16839489C2211yescharacteristic
55C8×#92418C2211yescharacteristic, complemented
56C61×#557C2211yescharacteristic, complemented
57C61×#550C2211yescharacteristic
58C61×#490—21no—
59C61×#486C2211yescharacteristic, complemented
60C61×#478—21no—
61C61×#474C2211yescharacteristic
62C61×#460—21no—
63C61×#460C2211yescharacteristic, complemented
64C61×#446C2211yescharacteristic, complemented
65C61×#440—21no—
66C4×SD16×C61C2211yescharacteristic
67C22×D244—21no—
68C2×D4×D61—21no—
69#14519802—44no—
70C2×#11758202—21no—
71C2×#11758202—21no—
72C2×#11758202—21no—
73C2×#11758201—21no—
74C2×#11758201—21no—
75C2×#11758201—21no—
76C2×#11758201D411yescharacteristic
77C2×#11758199—21no—
78C2×#11758199—21no—
79C2×#11758199—21no—
80C2×#11758194—21no—
81C2×#11758189—21no—
82#14519777C2311yescharacteristic
83#14519774C2311yescharacteristic
84#14519772—41no—
85#14519765—21no—
86#14519753—21no—
87#14519752—21no—
88#14519751—21no—
89#14519747—41no—
90C4×Dic122—21no—
91C4×D244C2311yescharacteristic
92#14519735—21no—
93#14519710—21no—
94#14519708D411yescharacteristic
95C4⋊#92418—21no—
96C4⋊#92418C2311yescharacteristic
97C4⋊#92418C2311yescharacteristic
98C4⋊#92418C2311yescharacteristic
99#14519701—21no—
100#14519701D411yescharacteristic
101#14519700—21no—
102#14519699—21no—
103#14519699—21no—
104#14519698—21no—
105#14519698—21no—
106#14519698—41no—
107#14519697—21no—
108#14519697—21no—
109#14519697—41no—
110#14519697D411yescharacteristic
111C22⋊#92418—21no—
112#14519689—21no—
113#14519688—21no—
114C4×#92418C2311yescharacteristic
115C4×#92418C2311yescharacteristic
116C2×#41×C61—21no—
117C2×SD16×C61—21no—
118C2×SD16×C61D411yescharacteristic
119C4⋊Q8×C61—21no—
120C4⋊Q8×C61C2311yescharacteristic
121#127×C61—21no—
122#127×C61C2311yescharacteristic
123#125×C61—21no—
124#125×C61—41no—
125#121×C61—21no—
126#121×C61—41no—
127C4×Q8×C61—21no—
128C4×Q8×C61—21no—
129C4×Q8×C61C2311yescharacteristic
130C4×Q8×C61C2311yescharacteristic
131C4×D4×C61—21no—
132C4×D4×C61C2311yescharacteristic
133#107×C61—21no—
134#106×C61D411yescharacteristic
135C4⋊C8×C61—21no—
136C4⋊C8×C61C2311yescharacteristic
137#103×C61D411yescharacteristic
138#102×C61—21no—
139#102×C61—21no—
140#102×C61—21no—
141#102×C61—21no—
142#102×C61D411yescharacteristic
143#97×C61—21no—
144C4×C8×C61C2311yescharacteristic
145C23×D61—41no—
146#41×C61—42no—
147#41×C61—81no—
148C2×Q8×C61—21no—
149C2×Q8×C61C2×D411yescharacteristic
150C2×Q8×C61C2×D411yescharacteristic
151C2×D4×C61—21no—
152C2×D4×C61—21no—
153C2×D4×C61—21no—
154C2×D4×C61—41no—
155C2×D4×C61—41no—
156C2×D4×C61C2×D411yescharacteristic
157C22×C4×C61—21no—
158C22×C4×C61—41no—
159C2×C22⋊D61—41no—
160D4×D61—42no—
161D4×D61—81no—
162C2×D244—21no—
163C2×D244—21no—
164C2×D244—21no—
165C2×D244—42no—
166C2×D244—41no—
167C2×D244—81no—
168C2×D244C2×D411yescharacteristic
169C2×C4×D61—21no—
170C2×Dic122—21no—
171SD16×C61—41no—
172SD16×C61—42no—
173C2×C8×C61—21no—
174C2×C8×C61C2×D411yescharacteristic
175C2×C8×C61C2×D411yescharacteristic
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—41no—
181C4⋊C4×C61—41no—
182C4⋊C4×C61—41no—
183C4⋊C4×C61C2×D411yescharacteristic
184C4⋊C4×C61C2×D411yescharacteristic
185C4⋊C4×C61C2×D411yescharacteristic
186C22⋊C4×C61—21no—
187C22⋊C4×C61—41no—
188C42×C61—21no—
189C42×C61—21no—
190C42×C61—41no—
191C42×C61C2411yescharacteristic
192#11758202—42no—
193#11758202—81no—
194#11758202—81no—
195#11758201—22no—
196#11758201—42no—
197#11758201—81no—
198#11758201—81no—
199#11758199—42no—
200#11758199—42no—
201#11758199—81no—
202#11758198—21no—
203#11758198—41no—
204#11758198—41no—
205#11758198—41no—
206C4⋊Dic61—21no—
207C4⋊Dic61—21no—
208C4⋊Dic61C2×D411yescharacteristic
209#11758196—41no—
210C4×Dic61—41no—
211#11758194—42no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418C2×D411yescharacteristic
217C2×#92418C2×D411yescharacteristic
218#11758189—81no—
219C23×C61—21no—
220C23×C61—41no—
221C22×D61—21no—
222C22×D61—41no—
223C22×D61—82no—
224C22×D61—82no—
225C22×D61—161no—
226Q8×C61—21no—
227Q8×C61—21no—
228Q8×C61—21no—
229Q8×C61—21no—
230Q8×C61—41no—
231Q8×C61—41no—
232D4×C61—21no—
233D4×C61—21no—
234D4×C61—41no—
235D4×C61—41no—
236D4×C61—41no—
237D4×C61—41no—
238D4×C61—81no—
239D4×C61—82no—
240D4×C61—161no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—82no—
250C2×C4×C61C22×D411yescharacteristic
251C2×C4×C61C22×D411yescharacteristic
252C2×C4×C61H(5,𝔽2)11yescharacteristic
253C22⋊D61—82no—
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—41no—
267Dic122—41no—
268C8×C61—21no—
269C8×C61—21no—
270C8×C61—41no—
271#92418—21no—
272#92418—41no—
273#92418—41no—
274#92418—41no—
275#92418—41no—
276#92418AGL(1,ℤ/8)12yes—
277#20288—611nomaximal, Sylow
278C22×C61—41no—
279C22×C61—41no—
280C22×C61—41no—
281C22×C61—81no—
282C22×C61—81no—
283C22×C61—81no—
284C22×C61D4211yescharacteristic
285C2×D61—41no—
286C2×D61—41no—
287C2×D61—41no—
288C2×D61—81no—
289C2×D61—81no—
290C2×D61—81no—
291C2×D61—164no—
292C4×C61—21no—
293C4×C61—41no—
294C4×C61—41no—
295C4×C61—41no—
296C4×C61—41no—
297C4×C61—41no—
298C4×C61—81no—
299C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
300C4×C61#57511yescharacteristic
301C4×C61#57611yescharacteristic
302C4×C61#57711yescharacteristic
303Dic61—41no—
304Dic61—81no—
305#3398—611no—
306#3368D6111yescharacteristic, complemented
307#3324—611no—
308#3274—611no—
309#1709—611no—
310#1702—611no—
311#1701—611no—
312#1689—611no—
313#1688—611no—
314#1641—611no—
315#1639—611no—
316#1628—611no—
317#1626—611no—
318#1590—611no—
319#1590—611no—
320C2×C61—81no—
321C2×C61—161no—
322C2×C61#327411yescharacteristic
323C2×C61#327511yescharacteristic
324C2×C61#329011yescharacteristic
325D61—81no—
326D61—161no—
327C2×AGL(1,ℤ/8)—1221no—
328#557C2×D6111yescharacteristic, complemented
329#556—611no—
330#550C2×D6111yescharacteristic
331D42—611no—
332#497—1221no—
333#494—611no—
334#493—611no—
335#490—21no—
336#486—611no—
337#486—611no—
338#486—611no—
339#486C2×D6111yescharacteristic, complemented
340#482—1221no—
341#478—21no—
342#474—611no—
343#474C2×D6111yescharacteristic
344#469—1221no—
345#468—1221no—
346#464—1221no—
347#462—611no—
348#462—1221no—
349#461—611no—
350#460—21no—
351#460—611no—
352#460—611no—
353#460C2×D6111yescharacteristic, complemented
354#459—611no—
355#450—1221no—
356#449—1221no—
357#447—1221no—
358#446—611no—
359#446C2×D6111yescharacteristic, complemented
360#441—1221no—
361#440—21no—
362C4×SD16—611no—
363C4×SD16C2×D6111yescharacteristic
364#436—611no—
365#436—611no—
366#418—1221no—
367#334—611no—
368#334—611no—
369#332—611no—
370#332—611no—
371#331—611no—
372#331—611no—
373Q8⋊C8—611no—
374Q8⋊C8—611no—
375D4⋊C8—611no—
376D4⋊C8—611no—
377C82—611no—
378C61#2028811yescharacteristic, complemented, Sylow
379C2×#41—21no—
380C22×D4—1221no—
381C22×D4—1221no—
382AGL(1,ℤ/8)—2444no—
383C2×Q16—1221no—
384C2×Q16—1221no—
385C2×Q16—1221no—
386C2×SD16—21no—
387C2×SD16—611no—
388C2×SD16—1221no—
389C2×SD16—1221no—
390C2×SD16—1221no—
391C2×SD16C22⋊D6111yescharacteristic
392C2×D8—1221no—
393C2×D8—1221no—
394C2×D8—1221no—
395C2×M16—1221no—
396C2×M16—1221no—
397C4⋊Q8—21no—
398C4⋊Q8—611no—
399C4⋊Q8C22×D6111yescharacteristic
400#127—21no—
401#127—611no—
402#127C22×D6111yescharacteristic
403#126—2441no—
404#125—21no—
405#125—41no—
406#124—1221no—
407C22⋊Q8—1221no—
408#121—21no—
409#121—41no—
410#121—1221no—
411#121—1221no—
412#120—2441no—
413C4×Q8—21no—
414C4×Q8—21no—
415C4×Q8—1221no—
416C4×Q8C22×D6111yescharacteristic
417C4×Q8C22×D6111yescharacteristic
418C4×D4—21no—
419C4×D4—611no—
420C4×D4C22×D6111yescharacteristic
421#117—1221no—
422#107—21no—
423#107—1221no—
424#106—611no—
425#106C22⋊D6111yescharacteristic
426C4⋊C8—21no—
427C4⋊C8—611no—
428C4⋊C8—611no—
429C4⋊C8—611no—
430C4⋊C8—1221no—
431C4⋊C8C22×D6111yescharacteristic
432#103—611no—
433#103—1221no—
434#103—1221no—
435#103C22⋊D6111yescharacteristic
436#102—21no—
437#102—21no—
438#102—21no—
439#102—21no—
440#102—611no—
441#102—1221no—
442#102—1221no—
443#102—1221no—
444#102—1221no—
445#102—1221no—
446#102—1221no—
447#102—2441no—
448#102—2441no—
449#102C22⋊D6111yescharacteristic
450C22⋊C8—1221no—
451C22⋊C8—1221no—
452#97—21no—
453#97—1221no—
454C4×C8—611no—
455C4×C8—611no—
456C4×C8C22×D6111yescharacteristic
457C24—2441no—
458#41—42no—
459#41—81no—
460C2×Q8—21no—
461C2×Q8—1221no—
462C2×Q8D4×D6111yescharacteristic
463C2×Q8C2×C22⋊D6111yescharacteristic
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—41no—
468C2×D4—41no—
469C2×D4—611no—
470C2×D4—1221no—
471C2×D4—1221no—
472C2×D4—1221no—
473C2×D4—4881no—
474C2×D4—4881no—
475C2×D4—2441no—
476C2×D4—2442no—
477C2×D4—2441no—
478C2×D4—2442no—
479C2×D4C2×C22⋊D6111yescharacteristic
480C22×C4—21no—
481C22×C4—41no—
482C22×C4—1221no—
483Q16—4881no—
484Q16—4881no—
485Q16—2442no—
486SD16—41no—
487SD16—42no—
488SD16—1222no—
489SD16—4881no—
490SD16—4881no—
491SD16—2442no—
492D8—4881no—
493D8—2442no—
494D8—2442no—
495M16—4881no—
496M16—2442no—
497C2×C8—21no—
498C2×C8—611no—
499C2×C8—611no—
500C2×C8—1221no—
501C2×C8—1221no—
502C2×C8—1221no—
503C2×C8—1221no—
504C2×C8C2×C22⋊D6111yescharacteristic
505C2×C8C2×C22⋊D6111yescharacteristic
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—41no—
511C4⋊C4—41no—
512C4⋊C4—41no—
513C4⋊C4—611no—
514C4⋊C4—1221no—
515C4⋊C4—1221no—
516C4⋊C4—2441no—
517C4⋊C4D4×D6111yescharacteristic
518C4⋊C4C2×C22⋊D6111yescharacteristic
519C4⋊C4C2×C22⋊D6111yescharacteristic
520C22⋊C4—21no—
521C22⋊C4—41no—
522C22⋊C4—1221no—
523C22⋊C4—2441no—
524C22⋊C4—2441no—
525C22⋊C4—2441no—
526C42—21no—
527C42—21no—
528C42—41no—
529C42—2441no—
530C42C23×D6111yescharacteristic
531C23—21no—
532C23—41no—
533C23—1221no—
534C23—9761no—
535C23—4882no—
536C23—4882no—
537C23—2441no—
538Q8—21no—
539Q8—21no—
540Q8—21no—
541Q8—21no—
542Q8—41no—
543Q8—41no—
544Q8—2441no—
545Q8—2441no—
546D4—21no—
547D4—21no—
548D4—41no—
549D4—41no—
550D4—41no—
551D4—41no—
552D4—81no—
553D4—82no—
554D4—161no—
555D4—1221no—
556D4—1221no—
557D4—4881no—
558D4—4882no—
559D4—4882no—
560D4—2441no—
561D4—2441no—
562D4—2441no—
563D4—2441no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—82no—
573C2×C4—1221no—
574C2×C4—2441no—
575C2×C4—2441no—
576C2×C4—2441no—
577C2×C4C2×D4×D6111yescharacteristic
578C2×C4C22×C22⋊D6111yescharacteristic
579C2×C4#1451982211yescharacteristic
580C8—21no—
581C8—21no—
582C8—41no—
583C8—612no—
584C8—1221no—
585C8—2441no—
586C8—2441no—
587C8—2441no—
588C8—2441no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—81no—
593C22—81no—
594C22—81no—
595C22—9764no—
596C22—4881no—
597C22—4881no—
598C22—4881no—
599C22—2441no—
600C22—2441no—
601C22—2441no—
602C22D4×C22⋊D6111yescharacteristic
603C4—21no—
604C4—41no—
605C4—41no—
606C4—41no—
607C4—41no—
608C4—41no—
609C4—81no—
610C4—4881no—
611C4—2441no—
612C4AGL(1,ℤ/8)×D6111yescharacteristic
613C4#1684056011yescharacteristic
614C4#1684056611yescharacteristic
615C4#1684057011yescharacteristic
616C2—81no—
617C2—161no—
618C2—9761no—
619C2—4881no—
620C2#2145743311yescharacteristic
621C2#2145745111yescharacteristic
622C2#2145759911yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3368C2
socleC22×C61D42
prime radical#21457439C2
hypercenter#3368D61
hypocenterC61#20288
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3368

Complements

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

NH ≅ G/Ntype
1. #21458846C2semidirect
2. #21458052C2semidirect
4. #21445423C2semidirect
5. #21445387C2semidirect
6. #21445377C2semidirect
7. #21445306C2semidirect
8. #21445301C2semidirect
9. #21445012C2semidirect
10. #21444997C2semidirect
11. #21444928C2semidirect
12. #21444915C2semidirect
13. #21444710C2semidirect
14. #21444709C2semidirect
15. C61×#3368C2semidirect
20. #16840141C22semidirect
24. #16840108C22semidirect
30. #16839972C22semidirect
39. #16839893C22semidirect
45. #16839513C22semidirect
46. #16839512C22semidirect
47. #16839508C22semidirect
48. #16839507C22semidirect
49. #16839506C22semidirect
51. Q8⋊#92418C22semidirect
52. Q8⋊#92418C22semidirect
55. C8×#92418C22semidirect
56. C61×#557C22semidirect
59. C61×#486C22semidirect
63. C61×#460C22semidirect
64. C61×#446C22semidirect
306. #3368D61semidirect
328. #557C2×D61semidirect
339. #486C2×D61semidirect
353. #460C2×D61semidirect
359. #446C2×D61semidirect
378. C61#20288semidirect

Maximal subgroups

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

subgroupindex
#214588462
#214580522
#214574392
#214454232
#214453872
#214453772
#214453062
#214453012
#214450122
#214449972
#214449282
#214449152
#214447102
#214447092
C61×#33682
#2028861