Finite Group Databasebeta

← #29676295

Subgroups — #29676295

Order 15616.

666 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458397C211yesmaximal, characteristic, complemented
2#21458044C211yesmaximal, characteristic, complemented
3#21457420C211yesmaximal, characteristic
4#21445491C211yesmaximal, characteristic, complemented
5#21445382C211yesmaximal, characteristic, complemented
6#21445330C211yesmaximal, characteristic, complemented
7#21445305C211yesmaximal, characteristic, complemented
8#21445167C211yesmaximal, characteristic, complemented
9#21445075C211yesmaximal, characteristic, complemented
10#21445052C211yesmaximal, characteristic, complemented
11#21444996C211yesmaximal, characteristic, complemented
12#21444929C211yesmaximal, characteristic, complemented
13#21444733C211yesmaximal, characteristic, complemented
14#21444662C211yesmaximal, characteristic, complemented
15C61×#3393C211yesmaximal, characteristic, complemented
16C2×#14519801—21no—
17#16840416C2211yescharacteristic
18D4×D244C2211yescharacteristic
19#16840165C2211yescharacteristic, complemented
20#16840165C2211yescharacteristic, complemented
21#16840135C2211yescharacteristic
22#16840110C2211yescharacteristic
23#16840110C2211yescharacteristic
24#16840095—21no—
25#16840086—21no—
26#16840086—21no—
27#16840047C2211yescharacteristic, complemented
28#16840039C2211yescharacteristic
29#16840039C2211yescharacteristic
30#16840010—21no—
31#16840010—21no—
32#16840009—21no—
33#16839967C2211yescharacteristic
34#16839962C2211yescharacteristic
35#16839957C2211yescharacteristic
36#16839914—21no—
37#16839912—21no—
38#16839908—21no—
39C4×#11758201C2211yescharacteristic
40#16839841C2211yescharacteristic
41#16839840C2211yescharacteristic
42#16839840C2211yescharacteristic, complemented
43#16839805—21no—
44#16839766—21no—
45#16839513C2211yescharacteristic, complemented
46#16839512C2211yescharacteristic, complemented
47#16839509C2211yescharacteristic, complemented
48#16839508C2211yescharacteristic, complemented
49#16839508C2211yescharacteristic, complemented
50#16839505C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic, complemented
52D4⋊#92418C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic, complemented
54#16839489C2211yescharacteristic
55#16839483C2211yescharacteristic, complemented
56C61×#556C2211yescharacteristic
57C61×#556C2211yescharacteristic, complemented
58C61×#495—21no—
59C61×#494C2211yescharacteristic, complemented
60C61×#464—21no—
61C61×#464—21no—
62C61×#461C2211yescharacteristic
63C61×#461C2211yescharacteristic, complemented
64C8×Q8×C61C2211yescharacteristic, complemented
65C4×Q16×C61—21no—
66C4×D8×C61C2211yescharacteristic
67C2×#11758224—21no—
68C22×D244—21no—
69C2×D4×D61—21no—
70#14519801—44no—
71C2×#11758190—21no—
72C2×#11758190—21no—
73C2×#11758200—21no—
74C2×#11758201—21no—
75C2×#11758201D411yescharacteristic
76C2×#11758199—21no—
77C2×#11758194—21no—
78C2×#11758194—21no—
79C2×#11758189—21no—
80#14519777C2311yescharacteristic
81#14519774C2311yescharacteristic
82#14519761—41no—
83#14519757—21no—
84#14519752—21no—
85#14519751—21no—
86#14519751—21no—
87#14519747—41no—
88C4×D244C2311yescharacteristic
89C2×C4⋊Dic61—21no—
90#14519710—21no—
91#14519710—21no—
92#14519710—21no—
93#14519710—21no—
94#14519710—41no—
95#14519708—21no—
96#14519708D411yescharacteristic
97C4⋊#92418C2311yescharacteristic
98C4⋊#92418C2311yescharacteristic
99C4⋊#92418C2311yescharacteristic
100#14519701—21no—
101#14519701D411yescharacteristic
102#14519700—21no—
103#14519697—21no—
104#14519697—21no—
105#14519697—21no—
106#14519697—21no—
107#14519697—21no—
108#14519697—21no—
109#14519697—41no—
110#14519697D411yescharacteristic
111C22⋊#92418—21no—
112C22⋊#92418—21no—
113#14519689—21no—
114C4×#92418C2311yescharacteristic
115C4×#92418C2311yescharacteristic
116C2×Q16×C61—21no—
117C2×Q16×C61—21no—
118C2×SD16×C61—21no—
119C2×SD16×C61—21no—
120C2×D8×C61D411yescharacteristic
121C4⋊Q8×C61C2311yescharacteristic
122C4⋊Q8×C61C2311yescharacteristic
123#126×C61—41no—
124#126×C61—41no—
125#124×C61—21no—
126#124×C61—21no—
127C22⋊Q8×C61—21no—
128C22⋊Q8×C61—21no—
129C4×Q8×C61—21no—
130C4×Q8×C61—21no—
131C4×Q8×C61C2311yescharacteristic
132C4×D4×C61C2311yescharacteristic
133C4×D4×C61C2311yescharacteristic
134#117×C61—21no—
135#117×C61—21no—
136#107×C61D411yescharacteristic
137C4⋊C8×C61—21no—
138C4⋊C8×C61C2311yescharacteristic
139#103×C61—21no—
140#103×C61—21no—
141#102×C61D411yescharacteristic
142#102×C61D411yescharacteristic
143C4×C8×C61—21no—
144C4×C8×C61C2311yescharacteristic
145C23×D61—41no—
146C2×Q8×C61—21no—
147C2×Q8×C61—21no—
148C2×Q8×C61C2×D411yescharacteristic
149C2×D4×C61C2×D411yescharacteristic
150C2×D4×C61C2×D411yescharacteristic
151C22×C4×C61—21no—
152C22×C4×C61—21no—
153C2×C22⋊D61—41no—
154C2×C22⋊D61—41no—
155C22×Dic61—41no—
156#11758224—42no—
157#11758224—81no—
158D4×D61—42no—
159D4×D61—81no—
160C2×D244—21no—
161C2×D244—21no—
162C2×D244—21no—
163C2×D244—42no—
164C2×D244—41no—
165C2×D244—81no—
166C2×D244C2×D411yescharacteristic
167C2×C4×D61—21no—
168C2×Dic122—21no—
169Q16×C61—42no—
170Q16×C61—81no—
171SD16×C61—81no—
172SD16×C61—81no—
173D8×C61—22no—
174C2×C8×C61—21no—
175C2×C8×C61C2×D411yescharacteristic
176C2×C8×C61C2×D411yescharacteristic
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—41no—
183C4⋊C4×C61—41no—
184C4⋊C4×C61C2×D411yescharacteristic
185C4⋊C4×C61C2×D411yescharacteristic
186C4⋊C4×C61C2×D411yescharacteristic
187C22⋊C4×C61—21no—
188C22⋊C4×C61—21no—
189C22⋊C4×C61—41no—
190C22⋊C4×C61—41no—
191C22⋊C4×C61—41no—
192C22⋊C4×C61—41no—
193C42×C61—21no—
194C42×C61—41no—
195C42×C61—41no—
196C42×C61C2411yescharacteristic
197#11758201—41no—
198#11758201—81no—
199#11758200—42no—
200#11758199—42no—
201#11758198—21no—
202#11758198—41no—
203#11758198—41no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61—41no—
207C4⋊Dic61C2×D411yescharacteristic
208#11758194—42no—
209#11758194—81no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418C2×D411yescharacteristic
215C2×#92418C2×D411yescharacteristic
216#11758190—42no—
217#11758190—81no—
218#11758189—42no—
219C23×C61—21no—
220C23×C61—21no—
221C22×D61—21no—
222C22×D61—41no—
223C22×D61—82no—
224C22×D61—82no—
225C22×D61—161no—
226Q8×C61—21no—
227Q8×C61—21no—
228Q8×C61—41no—
229Q8×C61—41no—
230Q8×C61—41no—
231Q8×C61—41no—
232D4×C61—21no—
233D4×C61—21no—
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—41no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61C22×D411yescharacteristic
247C2×C4×C61C22×D411yescharacteristic
248C2×C4×C61H(5,𝔽2)11yescharacteristic
249C22⋊D61—82no—
250C22⋊D61—82no—
251C2×Dic61—21no—
252C2×Dic61—41no—
253C2×Dic61—82no—
254D244—21no—
255D244—21no—
256D244—41no—
257D244—41no—
258D244—41no—
259D244—41no—
260D244—81no—
261D244—82no—
262C4×D61—41no—
263C4×D61—41no—
264Dic122—41no—
265Dic122—41no—
266C8×C61—21no—
267C8×C61—41no—
268C8×C61AGL(1,ℤ/8)12yes—
269#92418—21no—
270#92418—21no—
271#92418—41no—
272#92418—41no—
273#92418—41no—
274#92418—41no—
275#20250—611nomaximal, Sylow
276C22×C61—41no—
277C22×C61—41no—
278C22×C61—41no—
279C22×C61—41no—
280C22×C61—41no—
281C22×C61—41no—
282C22×C61D4211yescharacteristic
283C2×D61—41no—
284C2×D61—41no—
285C2×D61—41no—
286C2×D61—81no—
287C2×D61—81no—
288C2×D61—81no—
289C2×D61—164no—
290C4×C61—21no—
291C4×C61—41no—
292C4×C61—41no—
293C4×C61—41no—
294C4×C61—41no—
295C4×C61—81no—
296C4×C61—81no—
297C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
298C4×C61#57511yescharacteristic
299C4×C61#57611yescharacteristic
300C4×C61#57711yescharacteristic
301Dic61—41no—
302Dic61—81no—
303#3393D6111yescharacteristic, complemented
304#3353—611no—
305#3324—611no—
306#3272—611no—
307#1722—611no—
308#1701—611no—
309#1693—611no—
310#1689—611no—
311#1666—611no—
312#1650—611no—
313#1647—611no—
314#1639—611no—
315#1628—611no—
316#1594—611no—
317#1583—611no—
318C2×C61—81no—
319C2×C61—81no—
320C2×C61#326711yescharacteristic
321C2×C61#327411yescharacteristic
322C2×C61#328911yescharacteristic
323D61—81no—
324D61—161no—
325C2×AGL(1,ℤ/8)—1221no—
326#556C2×D6111yescharacteristic
327#556C2×D6111yescharacteristic, complemented
328#547—611no—
329D42—611no—
330#500—611no—
331#500—611no—
332#495—21no—
333#494C2×D6111yescharacteristic, complemented
334#492—611no—
335#486—611no—
336#486—611no—
337#484—1221no—
338#482—1221no—
339#482—1221no—
340#475—611no—
341#474—611no—
342#474—611no—
343#468—1221no—
344#468—1221no—
345#468—1221no—
346#464—21no—
347#464—21no—
348#461—611no—
349#461C2×D6111yescharacteristic
350#461C2×D6111yescharacteristic, complemented
351#460—611no—
352#459—611no—
353#450—1221no—
354#450—1221no—
355#449—1221no—
356C8×Q8C2×D6111yescharacteristic, complemented
357C4×Q16—21no—
358C4×SD16—611no—
359C4×D8C2×D6111yescharacteristic
360#436—611no—
361#436—611no—
362#436—611no—
363#428—1221no—
364#418—1221no—
365#334—611no—
366#334—611no—
367#332—611no—
368#332—611no—
369#332—611no—
370#331—611no—
371Q8⋊C8—611no—
372D4⋊C8—611no—
373D4⋊C8—611no—
374D4⋊C8—611no—
375#322—611no—
376C61#2025011yescharacteristic, complemented, Sylow
377C2×#41—1221no—
378C22×D4—1221no—
379C22×D4—1221no—
380AGL(1,ℤ/8)—2444no—
381C2×Q16—21no—
382C2×Q16—21no—
383C2×SD16—21no—
384C2×SD16—21no—
385C2×SD16—611no—
386C2×SD16—1221no—
387C2×SD16—1221no—
388C2×SD16—1221no—
389C2×SD16—1221no—
390C2×D8—1221no—
391C2×D8C22⋊D6111yescharacteristic
392C2×M16—1221no—
393C2×M16—1221no—
394C2×M16—1221no—
395C4⋊Q8—611no—
396C4⋊Q8C22×D6111yescharacteristic
397C4⋊Q8C22×D6111yescharacteristic
398#127—611no—
399#126—41no—
400#126—41no—
401#124—21no—
402#124—21no—
403#123—2441no—
404C22⋊Q8—21no—
405C22⋊Q8—21no—
406C22⋊Q8—1221no—
407#121—1221no—
408#121—1221no—
409#121—1221no—
410#120—2441no—
411C4×Q8—21no—
412C4×Q8—21no—
413C4×Q8C22×D6111yescharacteristic
414C4×D4—611no—
415C4×D4C22×D6111yescharacteristic
416C4×D4C22×D6111yescharacteristic
417#117—21no—
418#117—21no—
419C2×C4⋊C4—1221no—
420#107—1221no—
421#107—1221no—
422#107—1221no—
423#107—1221no—
424#107—2441no—
425#107C22⋊D6111yescharacteristic
426#106—611no—
427#106—1221no—
428C4⋊C8—21no—
429C4⋊C8—611no—
430C4⋊C8—611no—
431C4⋊C8—611no—
432C4⋊C8C22×D6111yescharacteristic
433#103—21no—
434#103—21no—
435#103—611no—
436#103—1221no—
437#103—1221no—
438#102—611no—
439#102—1221no—
440#102—1221no—
441#102—1221no—
442#102—1221no—
443#102—1221no—
444#102—1221no—
445#102—2441no—
446#102C22⋊D6111yescharacteristic
447#102C22⋊D6111yescharacteristic
448C22⋊C8—1221no—
449C22⋊C8—1221no—
450C22⋊C8—1221no—
451C4×C8—21no—
452C4×C8—611no—
453C4×C8—611no—
454C4×C8C22×D6111yescharacteristic
455C24—2441no—
456#41—4881no—
457#41—2442no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—1221no—
461C2×Q8D4×D6111yescharacteristic
462C2×D4—611no—
463C2×D4—1221no—
464C2×D4—1221no—
465C2×D4—1221no—
466C2×D4—4881no—
467C2×D4—4881no—
468C2×D4—2441no—
469C2×D4—2442no—
470C2×D4—2441no—
471C2×D4—2442no—
472C2×D4—2441no—
473C2×D4C2×C22⋊D6111yescharacteristic
474C2×D4C2×C22⋊D6111yescharacteristic
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—1221no—
478C22×C4—2441no—
479Q16—42no—
480Q16—81no—
481SD16—81no—
482SD16—81no—
483SD16—4881no—
484SD16—4881no—
485SD16—2441no—
486SD16—2442no—
487SD16—2442no—
488D8—22no—
489D8—2442no—
490M16—4881no—
491M16—2442no—
492M16—2442no—
493C2×C8—21no—
494C2×C8—611no—
495C2×C8—611no—
496C2×C8—1221no—
497C2×C8—1221no—
498C2×C8—1221no—
499C2×C8—1221no—
500C2×C8C2×C22⋊D6111yescharacteristic
501C2×C8C2×C22⋊D6111yescharacteristic
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—41no—
508C4⋊C4—41no—
509C4⋊C4—611no—
510C4⋊C4—1221no—
511C4⋊C4—1221no—
512C4⋊C4—2441no—
513C4⋊C4D4×D6111yescharacteristic
514C4⋊C4C2×C22⋊D6111yescharacteristic
515C4⋊C4C2×C22⋊D6111yescharacteristic
516C22⋊C4—21no—
517C22⋊C4—21no—
518C22⋊C4—41no—
519C22⋊C4—41no—
520C22⋊C4—41no—
521C22⋊C4—41no—
522C22⋊C4—1221no—
523C22⋊C4—2441no—
524C22⋊C4—2441no—
525C42—21no—
526C42—41no—
527C42—41no—
528C42C23×D6111yescharacteristic
529C23—21no—
530C23—21no—
531C23—1221no—
532C23—9761no—
533C23—4882no—
534C23—4882no—
535C23—2441no—
536Q8—21no—
537Q8—21no—
538Q8—41no—
539Q8—41no—
540Q8—41no—
541Q8—41no—
542Q8—2441no—
543Q8—2441no—
544D4—21no—
545D4—21no—
546D4—21no—
547D4—21no—
548D4—1221no—
549D4—1221no—
550D4—4881no—
551D4—4882no—
552D4—4882no—
553D4—4882no—
554D4—2441no—
555D4—2441no—
556D4—2441no—
557D4—2441no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—1221no—
569C2×C4—4882no—
570C2×C4—2441no—
571C2×C4—2441no—
572C2×C4—2441no—
573C2×C4C2×D4×D6111yescharacteristic
574C2×C4C22×C22⋊D6111yescharacteristic
575C2×C4#1451982211yescharacteristic
576C8—21no—
577C8—41no—
578C8—1221no—
579C8—1221no—
580C8—2441no—
581C8—2441no—
582C8—2441no—
583C8—2441no—
584C8#1451980212yes—
585C22—41no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22—9764no—
592C22—4881no—
593C22—4881no—
594C22—4881no—
595C22—2441no—
596C22—2441no—
597C22—2441no—
598C22D4×C22⋊D6111yescharacteristic
599C4—21no—
600C4—41no—
601C4—41no—
602C4—41no—
603C4—41no—
604C4—81no—
605C4—81no—
606C4—4881no—
607C4—2441no—
608C4C2×#1451980211yescharacteristic
609C4#1684055911yescharacteristic
610C4#1684056511yescharacteristic
611C4#1684056911yescharacteristic
612C2—81no—
613C2—81no—
614C2—9761no—
615C2—4881no—
616C2#2145734411yescharacteristic
617C2#2145743911yescharacteristic
618C2#2145759011yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3393C2
socleC22×C61D42
prime radical#21457420C2
hypercenter#3393D61
hypocenterC61#20250
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3393

Complements

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

NH ≅ G/Ntype
1. #21458397C2semidirect
2. #21458044C2semidirect
4. #21445491C2semidirect
5. #21445382C2semidirect
6. #21445330C2semidirect
7. #21445305C2semidirect
8. #21445167C2semidirect
9. #21445075C2semidirect
10. #21445052C2semidirect
11. #21444996C2semidirect
12. #21444929C2semidirect
13. #21444733C2semidirect
14. #21444662C2semidirect
15. C61×#3393C2semidirect
19. #16840165C22semidirect
20. #16840165C22semidirect
27. #16840047C22semidirect
42. #16839840C22semidirect
45. #16839513C22semidirect
46. #16839512C22semidirect
47. #16839509C22semidirect
48. #16839508C22semidirect
49. #16839508C22semidirect
51. Q8⋊#92418C22semidirect
53. D4⋊#92418C22semidirect
55. #16839483C22semidirect
57. C61×#556C22semidirect
59. C61×#494C22semidirect
63. C61×#461C22semidirect
64. C8×Q8×C61C22semidirect
303. #3393D61semidirect
327. #556C2×D61semidirect
333. #494C2×D61semidirect
350. #461C2×D61semidirect
356. C8×Q8C2×D61semidirect
376. C61#20250semidirect

Maximal subgroups

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

subgroupindex
#214583972
#214580442
#214574202
#214454912
#214453822
#214453302
#214453052
#214451672
#214450752
#214450522
#214449962
#214449292
#214447332
#214446622
C61×#33932
#2025061