Finite Group Databasebeta

← #29676870

Subgroups — #29676870

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#21458548C211yesmaximal, characteristic, complemented
2#21458044C211yesmaximal, characteristic, complemented
3#21457440C211yesmaximal, characteristic
4#21445426C211yesmaximal, characteristic, complemented
5#21445389C211yesmaximal, characteristic, complemented
6#21445382C211yesmaximal, characteristic, complemented
7#21445305C211yesmaximal, characteristic, complemented
8#21445303C211yesmaximal, characteristic, complemented
9#21445010C211yesmaximal, characteristic, complemented
10#21444996C211yesmaximal, characteristic, complemented
11#21444929C211yesmaximal, characteristic, complemented
12#21444915C211yesmaximal, characteristic, complemented
13#21444710C211yesmaximal, characteristic, complemented
14#21444709C211yesmaximal, characteristic, complemented
15C61×#3398C211yesmaximal, characteristic, complemented
16C2×#14519804—21no—
17#16840434C2211yescharacteristic
18D4×D244C2211yescharacteristic
19#16840139C2211yescharacteristic
20#16840128—21no—
21#16840110C2211yescharacteristic
22#16840110C2211yescharacteristic
23#16840110C2211yescharacteristic, complemented
24#16840110C2211yescharacteristic, complemented
25#16840086—21no—
26#16840059—21no—
27#16840039C2211yescharacteristic
28#16840039C2211yescharacteristic
29#16840016—21no—
30#16840010—21no—
31#16839962—21no—
32#16839962C2211yescharacteristic
33#16839962C2211yescharacteristic
34#16839962C2211yescharacteristic, complemented
35#16839957C2211yescharacteristic
36#16839912—21no—
37#16839910—21no—
38#16839897—21no—
39#16839893C2211yescharacteristic, complemented
40#16839860—21no—
41C4×#11758201C2211yescharacteristic
42#16839841C2211yescharacteristic
43#16839840C2211yescharacteristic
44#16839766—21no—
45#16839513C2211yescharacteristic, complemented
46#16839512C2211yescharacteristic, complemented
47#16839508C2211yescharacteristic, complemented
48#16839508C2211yescharacteristic, complemented
49#16839505C2211yescharacteristic
50#16839505C2211yescharacteristic, complemented
51Q8⋊#92418C2211yescharacteristic, complemented
52Q8⋊#92418C2211yescharacteristic, complemented
53D4⋊#92418C2211yescharacteristic
54#16839489C2211yescharacteristic
55C8×#92418C2211yescharacteristic, complemented
56C61×#557C2211yescharacteristic, complemented
57C61×#556C2211yescharacteristic
58C61×#497—21no—
59C61×#494C2211yescharacteristic, complemented
60C61×#464—21no—
61C61×#462—21no—
62C61×#462C2211yescharacteristic, complemented
63C61×#461C2211yescharacteristic
64C61×#446C2211yescharacteristic, complemented
65C61×#441—21no—
66C4×SD16×C61C2211yescharacteristic
67C2×#11758226—21no—
68C22×D244—21no—
69C2×D4×D61—21no—
70#14519804—44no—
71C2×#11758201—21no—
72C2×#11758201—21no—
73C2×#11758201—21no—
74C2×#11758201D411yescharacteristic
75C2×D488—21no—
76C2×D488—21no—
77C2×#11758199—21no—
78C2×#11758194—21no—
79C2×#11758189—21no—
80#14519774—21no—
81#14519774C2311yescharacteristic
82#14519774C2311yescharacteristic
83#14519752—21no—
84#14519752—21no—
85#14519752—41no—
86#14519751—21no—
87#14519747—41no—
88C4×D244—21no—
89C4×D244C2311yescharacteristic
90#14519710—21no—
91#14519710—21no—
92#14519708D411yescharacteristic
93C4⋊#92418—21no—
94C4⋊#92418C2311yescharacteristic
95C4⋊#92418C2311yescharacteristic
96C4⋊#92418C2311yescharacteristic
97#14519701D411yescharacteristic
98#14519697—21no—
99#14519697—21no—
100#14519697—21no—
101#14519697—21no—
102#14519697—21no—
103#14519697—21no—
104#14519697—21no—
105#14519697—21no—
106#14519697—21no—
107#14519697—21no—
108#14519697—41no—
109#14519697—41no—
110#14519697D411yescharacteristic
111C22⋊#92418—21no—
112#14519689—21no—
113#14519688—21no—
114C4×#92418C2311yescharacteristic
115C4×#92418C2311yescharacteristic
116C2×Q16×C61—21no—
117C2×Q16×C61—21no—
118C2×Q16×C61—21no—
119C2×SD16×C61—21no—
120C2×SD16×C61D411yescharacteristic
121C4⋊Q8×C61—21no—
122C4⋊Q8×C61C2311yescharacteristic
123C4⋊Q8×C61C2311yescharacteristic
124#126×C61—41no—
125#125×C61—21no—
126#125×C61—41no—
127#124×C61—21no—
128C22⋊Q8×C61—21no—
129C4×Q8×C61—21no—
130C4×Q8×C61—21no—
131C4×Q8×C61—21no—
132C4×Q8×C61C2311yescharacteristic
133C4×Q8×C61C2311yescharacteristic
134C4×D4×C61C2311yescharacteristic
135#117×C61—21no—
136#106×C61D411yescharacteristic
137C4⋊C8×C61—21no—
138C4⋊C8×C61C2311yescharacteristic
139#103×C61—21no—
140#103×C61—21no—
141#103×C61D411yescharacteristic
142#102×C61D411yescharacteristic
143#97×C61—21no—
144C4×C8×C61C2311yescharacteristic
145C23×D61—41no—
146C2×Q8×C61—21no—
147C2×Q8×C61—21no—
148C2×Q8×C61C2×D411yescharacteristic
149C2×Q8×C61C2×D411yescharacteristic
150C2×D4×C61C2×D411yescharacteristic
151C22×C4×C61—21no—
152C2×C22⋊D61—41no—
153#11758226—42no—
154#11758226—81no—
155D4×D61—42no—
156D4×D61—81no—
157C2×D244—21no—
158C2×D244—21no—
159C2×D244—21no—
160C2×D244—21no—
161C2×D244—21no—
162C2×D244—21no—
163C2×D244—42no—
164C2×D244—41no—
165C2×D244—41no—
166C2×D244—41no—
167C2×D244—81no—
168C2×D244C2×D411yescharacteristic
169C2×C4×D61—21no—
170C2×C4×D61—41no—
171Q16×C61—42no—
172Q16×C61—81no—
173Q16×C61—81no—
174SD16×C61—22no—
175SD16×C61—81no—
176C2×C8×C61—21no—
177C2×C8×C61C2×D411yescharacteristic
178C2×C8×C61C2×D411yescharacteristic
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—41no—
186C4⋊C4×C61—41no—
187C4⋊C4×C61—41no—
188C4⋊C4×C61—41no—
189C4⋊C4×C61C2×D411yescharacteristic
190C4⋊C4×C61C2×D411yescharacteristic
191C4⋊C4×C61C2×D411yescharacteristic
192C22⋊C4×C61—21no—
193C22⋊C4×C61—41no—
194C22⋊C4×C61—41no—
195C42×C61—21no—
196C42×C61—21no—
197C42×C61—41no—
198C42×C61—41no—
199C42×C61C2411yescharacteristic
200#11758201—41no—
201#11758201—42no—
202#11758201—42no—
203#11758201—81no—
204#11758199—42no—
205#11758198—21no—
206#11758198—41no—
207#11758198—41no—
208C4⋊Dic61C2×D411yescharacteristic
209#11758194—81no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418C2×D411yescharacteristic
215C2×#92418C2×D411yescharacteristic
216D488—42no—
217D488—81no—
218#11758189—42no—
219C23×C61—21no—
220C22×D61—21no—
221C22×D61—41no—
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—
232Q8×C61—41no—
233Q8×C61—41no—
234D4×C61—21no—
235D4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61C22×D411yescharacteristic
246C2×C4×C61C22×D411yescharacteristic
247C2×C4×C61H(5,𝔽2)11yescharacteristic
248C22⋊D61—82no—
249C2×Dic61—21no—
250D244—21no—
251D244—21no—
252D244—41no—
253D244—41no—
254D244—41no—
255D244—41no—
256D244—41no—
257D244—41no—
258D244—41no—
259D244—41no—
260D244—81no—
261D244—81no—
262D244—82no—
263D244—82no—
264D244—161no—
265C4×D61—41no—
266C4×D61—41no—
267C4×D61—82no—
268C8×C61—21no—
269C8×C61—41no—
270C8×C61AGL(1,ℤ/8)12yes—
271#92418—21no—
272#92418—21no—
273#92418—41no—
274#92418—41no—
275#92418—41no—
276#92418—41no—
277#20288—611nomaximal, Sylow
278C22×C61—41no—
279C22×C61—41no—
280C22×C61—41no—
281C22×C61D4211yescharacteristic
282C2×D61—41no—
283C2×D61—41no—
284C2×D61—41no—
285C2×D61—81no—
286C2×D61—81no—
287C2×D61—81no—
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×C61—81no—
300C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
301C4×C61#57511yescharacteristic
302C4×C61#57611yescharacteristic
303C4×C61#57711yescharacteristic
304Dic61—41no—
305#3398D6111yescharacteristic, complemented
306#3368—611no—
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#327411yescharacteristic
322C2×C61#327511yescharacteristic
323C2×C61#329011yescharacteristic
324D61—81no—
325D61—161no—
326D61—161no—
327C2×AGL(1,ℤ/8)—1221no—
328#557C2×D6111yescharacteristic, complemented
329#556C2×D6111yescharacteristic
330#550—611no—
331D42—611no—
332#497—21no—
333#494C2×D6111yescharacteristic, complemented
334#493—611no—
335#490—1221no—
336#486—611no—
337#486—611no—
338#486—611no—
339#486—611no—
340#482—1221no—
341#478—1221no—
342#474—611no—
343#474—611no—
344#469—1221no—
345#468—1221no—
346#464—21no—
347#462—21no—
348#462C2×D6111yescharacteristic, complemented
349#461C2×D6111yescharacteristic
350#460—611no—
351#460—611no—
352#460—611no—
353#460—1221no—
354#459—611no—
355#450—1221no—
356#449—1221no—
357#447—1221no—
358#446—611no—
359#446C2×D6111yescharacteristic, complemented
360#441—21no—
361#440—1221no—
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—1221no—
380C22×D4—1221no—
381C22×D4—1221no—
382AGL(1,ℤ/8)—2444no—
383C2×Q16—21no—
384C2×Q16—21no—
385C2×Q16—21no—
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⋊Q8C22×D6111yescharacteristic
399C4⋊Q8C22×D6111yescharacteristic
400#127—611no—
401#127—611no—
402#127—1221no—
403#126—41no—
404#125—21no—
405#125—41no—
406#124—21no—
407C22⋊Q8—21no—
408#121—1221no—
409#121—1221no—
410#121—1221no—
411#121—2441no—
412#120—2441no—
413C4×Q8—21no—
414C4×Q8—21no—
415C4×Q8—21no—
416C4×Q8C22×D6111yescharacteristic
417C4×Q8C22×D6111yescharacteristic
418C4×D4—611no—
419C4×D4—1221no—
420C4×D4C22×D6111yescharacteristic
421#117—21no—
422#107—1221no—
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—21no—
433#103—21no—
434#103—611no—
435#103C22⋊D6111yescharacteristic
436#102—611no—
437#102—1221no—
438#102—1221no—
439#102—1221no—
440#102—1221no—
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—4881no—
459#41—2442no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8D4×D6111yescharacteristic
463C2×Q8C2×C22⋊D6111yescharacteristic
464C2×D4—611no—
465C2×D4—1221no—
466C2×D4—1221no—
467C2×D4—1221no—
468C2×D4—1221no—
469C2×D4—1221no—
470C2×D4—1221no—
471C2×D4—4881no—
472C2×D4—4881no—
473C2×D4—2441no—
474C2×D4—2441no—
475C2×D4—2441no—
476C2×D4—2442no—
477C2×D4—2441no—
478C2×D4—2442no—
479C2×D4C2×C22⋊D6111yescharacteristic
480C22×C4—21no—
481C22×C4—1221no—
482C22×C4—2441no—
483Q16—42no—
484Q16—81no—
485Q16—81no—
486SD16—22no—
487SD16—81no—
488SD16—4881no—
489SD16—2441no—
490SD16—2442no—
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—21no—
511C4⋊C4—21no—
512C4⋊C4—41no—
513C4⋊C4—41no—
514C4⋊C4—41no—
515C4⋊C4—41no—
516C4⋊C4—611no—
517C4⋊C4D4×D6111yescharacteristic
518C4⋊C4C2×C22⋊D6111yescharacteristic
519C4⋊C4C2×C22⋊D6111yescharacteristic
520C22⋊C4—21no—
521C22⋊C4—41no—
522C22⋊C4—41no—
523C22⋊C4—1221no—
524C22⋊C4—2441no—
525C22⋊C4—2441no—
526C42—21no—
527C42—21no—
528C42—41no—
529C42—41no—
530C42C23×D6111yescharacteristic
531C23—21no—
532C23—1221no—
533C23—9761no—
534C23—4882no—
535C23—4882no—
536C23—2441no—
537C23—2441no—
538Q8—21no—
539Q8—21no—
540Q8—21no—
541Q8—21no—
542Q8—41no—
543Q8—41no—
544Q8—41no—
545Q8—41no—
546D4—21no—
547D4—21no—
548D4—1221no—
549D4—1221no—
550D4—9761no—
551D4—4881no—
552D4—4881no—
553D4—4882no—
554D4—4882no—
555D4—4882no—
556D4—2441no—
557D4—2441no—
558D4—2441no—
559D4—2441no—
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—41no—
573C2×C4—1221no—
574C2×C4—4882no—
575C2×C4—2441no—
576C2×C4—2441no—
577C2×C4C2×D4×D6111yescharacteristic
578C2×C4C22×C22⋊D6111yescharacteristic
579C2×C4#1451982211yescharacteristic
580C8—21no—
581C8—41no—
582C8—1221no—
583C8—1221no—
584C8—2441no—
585C8—2441no—
586C8—2441no—
587C8—2441no—
588C8#1451980212yes—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—9764no—
593C22—4881no—
594C22—4881no—
595C22—4881no—
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—81no—
611C4—2441no—
612C4C2×#1451980211yescharacteristic
613C4#1684056111yescharacteristic
614C4#1684056511yescharacteristic
615C4#1684057111yescharacteristic
616C2—81no—
617C2—9761no—
618C2—9761no—
619C2—4881no—
620C2#2145743911yescharacteristic
621C2#2145744411yescharacteristic
622C2#2145760511yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3398C2
socleC22×C61D42
prime radical#21457440C2
hypercenter#3398D61
hypocenterC61#20288
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3398

Complements

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

NH ≅ G/Ntype
1. #21458548C2semidirect
2. #21458044C2semidirect
4. #21445426C2semidirect
5. #21445389C2semidirect
6. #21445382C2semidirect
7. #21445305C2semidirect
8. #21445303C2semidirect
9. #21445010C2semidirect
10. #21444996C2semidirect
11. #21444929C2semidirect
12. #21444915C2semidirect
13. #21444710C2semidirect
14. #21444709C2semidirect
15. C61×#3398C2semidirect
23. #16840110C22semidirect
24. #16840110C22semidirect
34. #16839962C22semidirect
39. #16839893C22semidirect
45. #16839513C22semidirect
46. #16839512C22semidirect
47. #16839508C22semidirect
48. #16839508C22semidirect
50. #16839505C22semidirect
51. Q8⋊#92418C22semidirect
52. Q8⋊#92418C22semidirect
55. C8×#92418C22semidirect
56. C61×#557C22semidirect
59. C61×#494C22semidirect
62. C61×#462C22semidirect
64. C61×#446C22semidirect
305. #3398D61semidirect
328. #557C2×D61semidirect
333. #494C2×D61semidirect
348. #462C2×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
#214585482
#214580442
#214574402
#214454262
#214453892
#214453822
#214453052
#214453032
#214450102
#214449962
#214449292
#214449152
#214447102
#214447092
C61×#33982
#2028861