Finite Group Databasebeta

← #29663962

Subgroups — #29663962

Order 15616.

722 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#486×D61C211yesmaximal, characteristic, complemented
2#21455631C211yesmaximal, characteristic, complemented
3#21455294C211yesmaximal, characteristic
4C8×#11758224C211yesmaximal, characteristic, complemented
5#21445238C211yesmaximal, characteristic, complemented
6#21445236C211yesmaximal, characteristic, complemented
7#21445234C211yesmaximal, characteristic, complemented
8#21445232C211yesmaximal, characteristic, complemented
9#21445230C211yesmaximal, characteristic, complemented
10#21445228C211yesmaximal, characteristic, complemented
11#21445226C211yesmaximal, characteristic, complemented
12#21444573C211yesmaximal, characteristic, complemented
13#21444498C211yesmaximal, characteristic, complemented
14#21444485C211yesmaximal, characteristic, complemented
15C61×#1677C211yesmaximal, characteristic, complemented
16C2×#14519801—21no—
17C4⋊Q8×D61C2211yescharacteristic
18#127×D61C2211yescharacteristic
19C4×#11758224C2211yescharacteristic
20#16840110C2211yescharacteristic
21#16840109C2211yescharacteristic
22#16840108C2211yescharacteristic
23#16840107—21no—
24#16840107—21no—
25#16840058C2211yescharacteristic, complemented
26#16840054C2211yescharacteristic, complemented
27#16840052C2211yescharacteristic, complemented
28#16840046—21no—
29#16840040—21no—
30#16839963C2211yescharacteristic
31#16839961—21no—
32#16839959C2211yescharacteristic
33#16839957C2211yescharacteristic
34#16839955—21no—
35C8×Dic122C2211yescharacteristic, complemented
36#16839886C2211yescharacteristic, complemented
37D4×#92418C2211yescharacteristic, complemented
38C8×C22⋊D61—21no—
39#16839825—21no—
40#16839814C2211yescharacteristic
41#16839812—21no—
42#16839801—21no—
43#16839784C2211yescharacteristic
44#16839784C2211yescharacteristic
45#16839765—21no—
46#16839760—21no—
47#102×D61—21no—
48#102×D61—21no—
49C2×C8×Dic61—21no—
50C4×C8×D61C2211yescharacteristic
51#16839513C2211yescharacteristic, complemented
52#16839512C2211yescharacteristic, complemented
53#16839512C2211yescharacteristic, complemented
54#16839511C2211yescharacteristic, complemented
55#16839510C2211yescharacteristic, complemented
56#16839510C2211yescharacteristic, complemented
57#16839506C2211yescharacteristic
58#16839505C2211yescharacteristic
59#16839505C2211yescharacteristic
60C61×#486C2211yescharacteristic
61C61×#476C2211yescharacteristic, complemented
62C61×#459C2211yescharacteristic
63C8×D4×C61C2211yescharacteristic, complemented
64C61×#334C2211yescharacteristic, complemented
65C61×#333C2211yescharacteristic, complemented
66C61×#331C2211yescharacteristic
67C2×#11758224D411yescharacteristic
68C2×Q8×D61—21no—
69C2×D4×D61—21no—
70C2×D4×D61—21no—
71C2×D4×D61—21no—
72#14519801—44no—
73C2×#11758190—21no—
74C2×#11758200—21no—
75C2×#11758199—21no—
76C2×#11758189—21no—
77C2×#11758189—21no—
78C22×#92418—21no—
79C2×C8×D61D411yescharacteristic
80C2×C8×D61D411yescharacteristic
81#14519777C2311yescharacteristic
82#14519776—21no—
83#14519776—21no—
84#14519775—21no—
85#14519774C2311yescharacteristic
86#14519773—21no—
87#14519773—21no—
88#14519773—21no—
89C4×Dic122C2311yescharacteristic
90#14519742D411yescharacteristic
91D4×Dic61D411yescharacteristic
92C4×C22⋊D61—21no—
93#14519737—21no—
94#14519735D411yescharacteristic
95#14519733—21no—
96C4⋊C4×D61—21no—
97C4⋊C4×D61—21no—
98C2×C4×Dic61—21no—
99C42×D61C2311yescharacteristic
100#14519711—21no—
101#14519710—21no—
102#14519708—21no—
103C4⋊#92418C2311yescharacteristic
104C4⋊#92418C2311yescharacteristic
105C4⋊#92418C2311yescharacteristic
106#14519706—21no—
107#14519706—21no—
108#14519706—21no—
109#14519706—21no—
110#14519702—21no—
111#14519700—21no—
112#14519699—21no—
113#14519699—21no—
114#14519699—21no—
115#14519698—21no—
116#14519698—21no—
117#14519698—21no—
118#14519698—21no—
119#14519697—21no—
120#14519697—21no—
121#14519697—21no—
122C22⋊#92418—21no—
123#14519689—21no—
124C4×#92418C2311yescharacteristic
125C8×Dic61—21no—
126C8×Dic61—21no—
127C8×Dic61—21no—
128C8×Dic61D411yescharacteristic
129C8×Dic61D411yescharacteristic
130C2×D8×C61—21no—
131C22×C8×C61—21no—
132C4⋊Q8×C61C2311yescharacteristic
133#127×C61C2311yescharacteristic
134C4×D4×C61C2311yescharacteristic
135#106×C61—21no—
136C4⋊C8×C61C2311yescharacteristic
137C4⋊C8×C61C2311yescharacteristic
138C4⋊C8×C61C2311yescharacteristic
139#102×C61—21no—
140#102×C61—21no—
141#102×C61—21no—
142#102×C61—21no—
143C22⋊C8×C61—21no—
144C4×C8×C61C2311yescharacteristic
145C23×D61—41no—
146C2×Q8×C61—21no—
147C2×D4×C61—21no—
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61C2×D411yescharacteristic
151C22×C4×C61—21no—
152C2×C22⋊D61—21no—
153C2×C22⋊D61—41no—
154C2×C22⋊D61—41no—
155C2×C22⋊D61—41no—
156C22×Dic61—21no—
157C22×Dic61—21no—
158Q8×D61—82no—
159#11758224—22no—
160#11758224SD1614yes—
161D4×D61—42no—
162D4×D61—42no—
163D4×D61—42no—
164D4×D61—42no—
165D4×D61—82no—
166C2×D244—21no—
167C2×D244—21no—
168C2×D244—21no—
169C2×C4×D61—41no—
170C2×C4×D61C2×D411yescharacteristic
171C2×C4×D61C2×D411yescharacteristic
172C2×C4×D61#4111yescharacteristic
173C2×Dic122—21no—
174C2×Dic122—41no—
175C2×Dic122C2×D411yescharacteristic
176D8×C61—42no—
177C2×C8×C61—21no—
178C2×C8×C61—21no—
179C2×C8×C61—21no—
180C2×C8×C61—42no—
181C2×C8×C61C2×D411yescharacteristic
182C2×C8×C61C2×D411yescharacteristic
183C4⋊C4×C61—21no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61C2×D411yescharacteristic
186C22⋊C4×C61—21no—
187C42×C61C2411yescharacteristic
188C22⋊Dic61—21no—
189#11758200—42no—
190#11758199—42no—
191#11758198—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61C2×D411yescharacteristic
195#11758196—21no—
196#11758196—21no—
197#11758196—41no—
198#11758196—41no—
199C4×Dic61—21no—
200C4×Dic61—21no—
201C4×Dic61—21no—
202C4×Dic61C2×D411yescharacteristic
203C4×Dic61C2×D411yescharacteristic
204C4×Dic61#4111yescharacteristic
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—42no—
209C2×#92418C2×D411yescharacteristic
210C2×#92418C2×D411yescharacteristic
211#11758190—42no—
212#11758189—42no—
213#11758189—42no—
214C8×D61—41no—
215C8×D61—41no—
216C23×C61—21no—
217C23×C61—41no—
218C22×D61—41no—
219C22×D61—82no—
220C22×D61—82no—
221C22×D61—82no—
222C22×D61#12111yescharacteristic
223Q8×C61—81no—
224D4×C61—21no—
225D4×C61—41no—
226D4×C61—41no—
227D4×C61—41no—
228D4×C61—41no—
229D4×C61—81no—
230D4×C61C2×SD1612yes—
231C2×C4×C61—21no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61C22×D411yescharacteristic
236C2×C4×C61C22×D411yescharacteristic
237C2×C4×C61C2×#4111yescharacteristic
238C22⋊D61—24no—
239C22⋊D61—82no—
240C22⋊D61—82no—
241C22⋊D61—82no—
242C2×Dic61—21no—
243C2×Dic61—24no—
244C2×Dic61—21no—
245C2×Dic61—42no—
246C2×Dic61—41no—
247C2×Dic61#12111yescharacteristic
248C2×Dic61#12111yescharacteristic
249C2×Dic61#12111yescharacteristic
250D244—41no—
251D244—41no—
252D244—41no—
253D244—41no—
254D244—81no—
255C4×D61—21no—
256C4×D61—21no—
257C4×D61—22no—
258C4×D61—82no—
259C4×D61C2×SD1612yes—
260C4×D61#13512yes—
261Dic122—21no—
262Dic122—82no—
263Dic122—81no—
264Dic122C2×SD1612yes—
265C8×C61—21no—
266C8×C61—21no—
267C8×C61—41no—
268C8×C61—41no—
269C8×C61—41no—
270#92418—21no—
271#92418—21no—
272#92418—41no—
273#92418—41no—
274#92418—41no—
275#19274—611nomaximal, Sylow
276C22×C61—22no—
277C22×C61—41no—
278C22×C61—41no—
279C22×C61—81no—
280C22×C61—81no—
281C22×C61—81no—
282C22×C61C2×#12111yescharacteristic
283C2×D61—21no—
284C2×D61—21no—
285C2×D61—81no—
286C2×D61—81no—
287C2×D61—81no—
288C2×D61—164no—
289C2×D61#46512yes—
290C4×C61—21no—
291C4×C61—41no—
292C4×C61—81no—
293C4×C61C22×SD1611yescharacteristic
294C4×C61C2×#13511yescharacteristic
295C4×C61#57511yescharacteristic
296C4×C61#57511yescharacteristic
297Dic61—21no—
298Dic61—22no—
299Dic61—41no—
300Dic61—81no—
301Dic61#46512yes—
302Dic61#47212yes—
303C2×#486—611no—
304#3079—611no—
305#3038—611no—
306C8×#41—611no—
307#1678—611no—
308#1678—611no—
309#1678—611no—
310#1678—611no—
311#1677—611no—
312#1677—611no—
313#1677—611no—
314#1677D6111yescharacteristic, complemented
315#1566—611no—
316#1548—611no—
317#1545—611no—
318C2×C61—42no—
319C2×C61—161no—
320C2×C61#304011yescharacteristic
321C2×C61C2×#46511yescharacteristic
322C2×C61C2×#47211yescharacteristic
323D61—22no—
324D61—161no—
325C2×AGL(1,ℤ/8)—1221no—
326C2×C4⋊Q8—611no—
327C2×#127—611no—
328C4×#41—611no—
329#486—611no—
330#486—611no—
331#486—611no—
332#486—1221no—
333#486—1221no—
334#486C2×D6111yescharacteristic
335#477—611no—
336#477—611no—
337#476—611no—
338#476C2×D6111yescharacteristic, complemented
339#475—1221no—
340#474—1221no—
341#460—611no—
342#460—611no—
343#460—1221no—
344#459—611no—
345#459—1221no—
346#459C2×D6111yescharacteristic
347C8×Q8—611no—
348C8×Q8—611no—
349C8×D4—611no—
350C8×D4—1221no—
351C8×D4—1221no—
352C8×D4C2×D6111yescharacteristic, complemented
353#431—611no—
354#431—1221no—
355#428—1221no—
356#423—611no—
357#423—611no—
358#418—1221no—
359#417—1221no—
360C2×#102—1221no—
361C2×#102—1221no—
362C2×C4×C8—611no—
363C2×C4×C8—1221no—
364#334—611no—
365#334—611no—
366#334—611no—
367#334C2×D6111yescharacteristic, complemented
368#333—611no—
369#333—611no—
370#333—611no—
371#333C2×D6111yescharacteristic, complemented
372#331—611no—
373#331—611no—
374#331—611no—
375#331C2×D6111yescharacteristic
376C61#1927411yescharacteristic, complemented, Sylow
377C2×#41—611no—
378C22×Q8—1221no—
379C22×D4—1221no—
380C22×D4—1221no—
381C22×D4—1221no—
382AGL(1,ℤ/8)—2444no—
383C2×SD16—1221no—
384C2×SD16—1221no—
385C2×D8—21no—
386C2×D8—1221no—
387C2×M16—1221no—
388C2×M16—1221no—
389C22×C8—21no—
390C22×C8—611no—
391C22×C8—611no—
392C22×C8—1221no—
393C4⋊Q8—611no—
394C4⋊Q8—1221no—
395C4⋊Q8—1221no—
396C4⋊Q8—1221no—
397C4⋊Q8C22×D6111yescharacteristic
398#127—611no—
399#127—1221no—
400#127—1221no—
401#127—1221no—
402#127C22×D6111yescharacteristic
403C4×Q8—611no—
404C4×Q8—611no—
405C4×D4—611no—
406C4×D4—1221no—
407C4×D4—1221no—
408C4×D4C22×D6111yescharacteristic
409#117—611no—
410#117—1221no—
411C2×C4⋊C4—1221no—
412C2×C4⋊C4—1221no—
413C2×C42—611no—
414C2×C42—1221no—
415#107—1221no—
416#107—1221no—
417#106—21no—
418#106—1221no—
419C4⋊C8—611no—
420C4⋊C8—611no—
421C4⋊C8—611no—
422C4⋊C8—1221no—
423C4⋊C8—1221no—
424C4⋊C8—1221no—
425C4⋊C8—1221no—
426C4⋊C8C22×D6111yescharacteristic
427C4⋊C8C22×D6111yescharacteristic
428C4⋊C8C22×D6111yescharacteristic
429#103—1221no—
430#103—1221no—
431#102—21no—
432#102—21no—
433#102—21no—
434#102—21no—
435#102—1221no—
436#102—1221no—
437#102—1221no—
438#102—1221no—
439#102—1221no—
440#102—1221no—
441#102—1221no—
442#102—1221no—
443#102—1221no—
444#102—1221no—
445C22⋊C8—21no—
446C22⋊C8—1221no—
447C22⋊C8—1221no—
448C4×C8—611no—
449C4×C8—611no—
450C4×C8—611no—
451C4×C8—1221no—
452C4×C8—1221no—
453C4×C8—1221no—
454C4×C8C22×D6111yescharacteristic
455C24—2441no—
456#41—614no—
457#41—1222no—
458C2×Q8—21no—
459C2×Q8—611no—
460C2×Q8—1221no—
461C2×Q8—4882no—
462C2×Q8—2441no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—1221no—
467C2×D4—1221no—
468C2×D4—1221no—
469C2×D4—1221no—
470C2×D4—4882no—
471C2×D4—2442no—
472C2×D4—2442no—
473C2×D4—2441no—
474C2×D4—2441no—
475C2×D4—2441no—
476C2×D4—2442no—
477C2×D4—2442no—
478C2×D4D4×D6111yescharacteristic
479C22×C4—21no—
480C22×C4—611no—
481C22×C4—611no—
482C22×C4—611no—
483C22×C4—1221no—
484C22×C4—1221no—
485C22×C4—2441no—
486SD16—2442no—
487SD16—2442no—
488D8—42no—
489D8—2442no—
490M16—2442no—
491M16—2442no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—42no—
496C2×C8—611no—
497C2×C8—611no—
498C2×C8—1221no—
499C2×C8—1221no—
500C2×C8—1221no—
501C2×C8—2441no—
502C2×C8—2441no—
503C2×C8—2442no—
504C2×C8D4×D6111yescharacteristic
505C2×C8D4×D6111yescharacteristic
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—611no—
509C4⋊C4—1221no—
510C4⋊C4—1221no—
511C4⋊C4—1221no—
512C4⋊C4—1221no—
513C4⋊C4—2441no—
514C4⋊C4—2441no—
515C4⋊C4D4×D6111yescharacteristic
516C22⋊C4—21no—
517C22⋊C4—1221no—
518C22⋊C4—1221no—
519C42—611no—
520C42—611no—
521C42—611no—
522C42—1221no—
523C42—1221no—
524C42—1221no—
525C42C23×D6111yescharacteristic
526C23—21no—
527C23—41no—
528C23—611no—
529C23—4882no—
530C23—4882no—
531C23—4882no—
532C23—2441no—
533Q8—81no—
534Q8—612no—
535Q8—1221no—
536Q8—4881no—
537Q8—4882no—
538D4—21no—
539D4—41no—
540D4—41no—
541D4—41no—
542D4—41no—
543D4—81no—
544D4—1224no—
545D4—4881no—
546D4—4882no—
547D4—4882no—
548D4—4882no—
549D4—2441no—
550D4—2441no—
551D4—2441no—
552D4—2441no—
553D4SD16×D6112yes—
554C2×C4—21no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—611no—
559C2×C4—611no—
560C2×C4—612no—
561C2×C4—612no—
562C2×C4—611no—
563C2×C4—1222no—
564C2×C4—1224no—
565C2×C4—1221no—
566C2×C4—1221no—
567C2×C4—1221no—
568C2×C4—1221no—
569C2×C4—4882no—
570C2×C4—2442no—
571C2×C4—2441no—
572C2×C4C2×D4×D6111yescharacteristic
573C2×C4C2×D4×D6111yescharacteristic
574C2×C4#41×D6111yescharacteristic
575C8—21no—
576C8—21no—
577C8—41no—
578C8—41no—
579C8—41no—
580C8—1221no—
581C8—1221no—
582C8—2441no—
583C8—2441no—
584C8—2441no—
585C22—22no—
586C22—41no—
587C22—41no—
588C22—81no—
589C22—81no—
590C22—81no—
591C22—612no—
592C22—1221no—
593C22—1221no—
594C22—9764no—
595C22—4881no—
596C22—4881no—
597C22—4881no—
598C22#121×D6111yescharacteristic
599C4—21no—
600C4—41no—
601C4—81no—
602C4—612no—
603C4—612no—
604C4—1222no—
605C4—1221no—
606C4—4881no—
607C4—2441no—
608C4C2×SD16×D6111yescharacteristic
609C4#135×D6111yescharacteristic
610C4#1684055811yescharacteristic
611C4#1684055911yescharacteristic
612C2—42no—
613C2—161no—
614C2—1222no—
615C2—9761no—
616C2#2145530811yescharacteristic
617C2#465×D6111yescharacteristic
618C2#472×D6111yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1677C2
socleC22×C61C2×#121
prime radical#21455294C2
hypercenter#1677D61
hypocenterC61#19274
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1677

Complements

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

NH ≅ G/Ntype
1. #486×D61C2semidirect
2. #21455631C2semidirect
4. C8×#11758224C2semidirect
5. #21445238C2semidirect
6. #21445236C2semidirect
7. #21445234C2semidirect
8. #21445232C2semidirect
9. #21445230C2semidirect
10. #21445228C2semidirect
11. #21445226C2semidirect
12. #21444573C2semidirect
13. #21444498C2semidirect
14. #21444485C2semidirect
15. C61×#1677C2semidirect
25. #16840058C22semidirect
26. #16840054C22semidirect
27. #16840052C22semidirect
35. C8×Dic122C22semidirect
36. #16839886C22semidirect
37. D4×#92418C22semidirect
51. #16839513C22semidirect
52. #16839512C22semidirect
53. #16839512C22semidirect
54. #16839511C22semidirect
55. #16839510C22semidirect
56. #16839510C22semidirect
61. C61×#476C22semidirect
63. C8×D4×C61C22semidirect
64. C61×#334C22semidirect
65. C61×#333C22semidirect
314. #1677D61semidirect
338. #476C2×D61semidirect
352. C8×D4C2×D61semidirect
367. #334C2×D61semidirect
371. #333C2×D61semidirect
376. C61#19274semidirect

Maximal subgroups

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

subgroupindex
#486×D612
#214556312
#214552942
C8×#117582242
#214452382
#214452362
#214452342
#214452322
#214452302
#214452282
#214452262
#214445732
#214444982
#214444852
C61×#16772
#1927461