Finite Group Databasebeta

← #29663816

Subgroups — #29663816

Order 15616.

716 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456377C211yesmaximal, characteristic, complemented
2#21455917C211yesmaximal, characteristic, complemented
3#21455909C211yesmaximal, characteristic
4C8×D4×D61C211yesmaximal, characteristic, complemented
5#21445271C211yesmaximal, characteristic, complemented
6#21445270C211yesmaximal, characteristic, complemented
7#21445253C211yesmaximal, characteristic, complemented
8#21445193C211yesmaximal, characteristic, complemented
9#21445192C211yesmaximal, characteristic, complemented
10#21445159C211yesmaximal, characteristic, complemented
11#21445158C211yesmaximal, characteristic, complemented
12#21444581C211yesmaximal, characteristic, complemented
13#21444516C211yesmaximal, characteristic, complemented
14#21444511C211yesmaximal, characteristic, complemented
15C61×#1681C211yesmaximal, characteristic, complemented
16C22×C8×D61—21no—
17#16840317C2211yescharacteristic
18#16840310C2211yescharacteristic
19C4×D4×D61C2211yescharacteristic
20#16840158C2211yescharacteristic, complemented
21#16840134C2211yescharacteristic, complemented
22#16840113C2211yescharacteristic, complemented
23#16840096—21no—
24#16840079—21no—
25#16840053C2211yescharacteristic
26#16840052C2211yescharacteristic
27#16840043C2211yescharacteristic
28#16840042C2211yescharacteristic
29#16839990—21no—
30#16839990—21no—
31#16839964C2211yescharacteristic
32#16839954C2211yescharacteristic
33#16839911—21no—
34#16839894—21no—
35#16839874—21no—
36#16839871—21no—
37#16839865—21no—
38#16839863—21no—
39#16839853—21no—
40#16839853—21no—
41C8×D244C2211yescharacteristic
42D4×#92418C2211yescharacteristic
43#16839827C2211yescharacteristic
44C8×C22⋊D61—21no—
45#16839825—21no—
46#16839818C2211yescharacteristic
47#16839818C2211yescharacteristic
48C4⋊C8×D61C2211yescharacteristic
49C22⋊C8×D61—21no—
50C4×C8×D61C2211yescharacteristic
51#16839515C2211yescharacteristic, complemented
52#16839512C2211yescharacteristic, complemented
53#16839512C2211yescharacteristic, complemented
54#16839510C2211yescharacteristic, complemented
55#16839510C2211yescharacteristic, complemented
56#16839509C2211yescharacteristic, complemented
57#16839508C2211yescharacteristic, complemented
58#16839507C2211yescharacteristic, complemented
59#16839506C2211yescharacteristic, complemented
60C61×#486C2211yescharacteristic, complemented
61C61×#461C2211yescharacteristic
62C61×#459C2211yescharacteristic
63C8×D4×C61C2211yescharacteristic
64C61×#335C2211yescharacteristic, complemented
65C61×#332C2211yescharacteristic, complemented
66C61×#331C2211yescharacteristic, complemented
67C2×#11758224—21no—
68C2×D4×D61D411yescharacteristic
69C22×C4×D61—21no—
70C2×#11758200—21no—
71C2×#11758200—21no—
72C2×#11758201—21no—
73C2×#11758199—21no—
74C22×#92418—21no—
75C2×C8×D61—21no—
76C2×C8×D61—21no—
77C2×C8×D61—21no—
78C2×C8×D61—44no—
79C2×C8×D61D411yescharacteristic
80C2×C8×D61D411yescharacteristic
81#14519777C2311yescharacteristic
82#14519777C2311yescharacteristic
83#14519768—21no—
84#14519764—21no—
85#14519757—21no—
86#14519753—21no—
87#14519748—21no—
88#14519748—21no—
89Q8×Dic61—21no—
90#14519742—21no—
91C4×D244C2311yescharacteristic
92D4×Dic61—21no—
93D4×Dic61—21no—
94D4×Dic61D411yescharacteristic
95#14519739D411yescharacteristic
96C4×C22⋊D61—21no—
97#14519737—21no—
98#14519735—21no—
99C4⋊C4×D61D411yescharacteristic
100C22⋊C4×D61—21no—
101C42×D61C2311yescharacteristic
102#14519711—21no—
103#14519710—21no—
104#14519709—21no—
105#14519708—21no—
106#14519708—21no—
107#14519708—21no—
108C4⋊#92418C2311yescharacteristic
109C4⋊#92418C2311yescharacteristic
110C4⋊#92418C2311yescharacteristic
111#14519706—21no—
112#14519702—21no—
113#14519702—21no—
114#14519701—21no—
115#14519699—21no—
116#14519699—21no—
117#14519698—21no—
118#14519698—21no—
119#14519698—21no—
120C22⋊#92418—21no—
121#14519689—21no—
122#14519689—21no—
123#14519689—21no—
124#14519689—21no—
125#14519687—21no—
126#14519687—21no—
127C4×#92418C2311yescharacteristic
128C8×Dic61D411yescharacteristic
129C8×Dic61D411yescharacteristic
130C2×SD16×C61—21no—
131C2×D8×C61—21no—
132C22×C8×C61—21no—
133C4⋊Q8×C61C2311yescharacteristic
134#127×C61C2311yescharacteristic
135C4×D4×C61C2311yescharacteristic
136C4⋊C8×C61C2311yescharacteristic
137C4⋊C8×C61C2311yescharacteristic
138C4⋊C8×C61C2311yescharacteristic
139#103×C61—21no—
140#102×C61—21no—
141#102×C61—21no—
142#102×C61—21no—
143C22⋊C8×C61—21no—
144C4×C8×C61C2311yescharacteristic
145C23×D61—21no—
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—
154C22×Dic61—21no—
155C22×Dic61—41no—
156#11758224—82no—
157D4×D61—22no—
158D4×D61—22no—
159C2×D244C2×D411yescharacteristic
160C2×C4×D61—21no—
161C2×C4×D61—21no—
162C2×C4×D61—21no—
163C2×C4×D61—42no—
164C2×C4×D61—42no—
165C2×C4×D61C2×D411yescharacteristic
166C2×C4×D61C2×D411yescharacteristic
167C2×C4×D61#4111yescharacteristic
168C2×Dic122—21no—
169C2×Dic122—21no—
170SD16×C61—42no—
171D8×C61—42no—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C2×C8×C61—42no—
176C2×C8×C61C2×D411yescharacteristic
177C2×C8×C61C2×D411yescharacteristic
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61C2×D411yescharacteristic
181C22⋊C4×C61—21no—
182C42×C61C2411yescharacteristic
183C22⋊Dic61—21no—
184C22⋊Dic61—41no—
185C22⋊Dic61—41no—
186#11758201—42no—
187#11758200—42no—
188#11758200—42no—
189#11758199—42no—
190#11758198—21no—
191#11758198—21no—
192#11758198—41no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—41no—
198C4⋊Dic61C2×D411yescharacteristic
199#11758196—21no—
200#11758196—41no—
201C4×Dic61—41no—
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
211C8×D61—22no—
212C8×D61—41no—
213C8×D61—42no—
214C8×D61—42no—
215C8×D61—42no—
216C23×C61—21no—
217C23×C61—41no—
218C22×D61—24no—
219C22×D61—21no—
220C22×D61—42no—
221C22×D61—42no—
222C22×D61#12111yescharacteristic
223Q8×C61—41no—
224Q8×C61—41no—
225D4×C61—21no—
226D4×C61—41no—
227D4×C61—41no—
228D4×C61—41no—
229D4×C61—41no—
230D4×C61—81no—
231D4×C61#13512yes—
232C2×C4×C61—21no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61C22×D411yescharacteristic
237C2×C4×C61C22×D411yescharacteristic
238C2×C4×C61C2×#4111yescharacteristic
239C22⋊D61—24no—
240C22⋊D61—82no—
241C2×Dic61—21no—
242C2×Dic61—42no—
243C2×Dic61—41no—
244C2×Dic61—41no—
245C2×Dic61—82no—
246C2×Dic61#12111yescharacteristic
247C2×Dic61#12111yescharacteristic
248C2×Dic61#12111yescharacteristic
249D244—21no—
250D244#13512yes—
251C4×D61—22no—
252C4×D61—21no—
253C4×D61—21no—
254C4×D61—21no—
255C4×D61—21no—
256C4×D61—42no—
257C4×D61—42no—
258C4×D61—41no—
259C4×D61—41no—
260Dic122—41no—
261Dic122—41no—
262Dic122—81no—
263C8×C61—21no—
264C8×C61—21no—
265C8×C61—41no—
266C8×C61—41no—
267C8×C61—41no—
268#92418—21no—
269#92418—41no—
270#92418—41no—
271#92418—41no—
272#92418AGL(1,ℤ/8)12yes—
273#19257—611nomaximal, Sylow
274C22×C61—22no—
275C22×C61—41no—
276C22×C61—41no—
277C22×C61—81no—
278C22×C61—81no—
279C22×C61—81no—
280C22×C61C2×#12111yescharacteristic
281C2×D61—21no—
282C2×D61—21no—
283C2×D61—22no—
284C2×D61—48no—
285C2×D61—41no—
286C2×D61—41no—
287C2×D61#47112yes—
288C4×C61—21no—
289C4×C61—41no—
290C4×C61—81no—
291C4×C61C2×#13511yescharacteristic
292C4×C61C2×#13511yescharacteristic
293C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
294C4×C61#57511yescharacteristic
295Dic61—21no—
296Dic61—21no—
297Dic61—21no—
298Dic61—41no—
299Dic61—81no—
300Dic61—81no—
301#3167—611no—
302#3110—611no—
303#3109—611no—
304C2×C8×D4—611no—
305#1683—611no—
306#1683—611no—
307#1681—611no—
308#1681D6111yescharacteristic, complemented
309#1671—611no—
310#1671—611no—
311#1665—611no—
312#1665—611no—
313#1568—611no—
314#1552—611no—
315#1551—611no—
316C2×C61—42no—
317C2×C61—161no—
318C2×C61#303611yescharacteristic
319C2×C61#305011yescharacteristic
320C2×C61C2×#47111yescharacteristic
321D61—22no—
322D61—42no—
323C23×C8—1221no—
324#533—611no—
325#532—611no—
326C2×C4×D4—611no—
327#498—611no—
328#492—611no—
329#487—611no—
330#486C2×D6111yescharacteristic, complemented
331#484—1221no—
332#481—1221no—
333#476—611no—
334#476—611no—
335#474—611no—
336#474—611no—
337#465—1221no—
338#465—1221no—
339#461—611no—
340#461C2×D6111yescharacteristic
341#459—611no—
342#459C2×D6111yescharacteristic
343#450—1221no—
344#447—1221no—
345#442—1221no—
346#442—1221no—
347#440—1221no—
348#440—1221no—
349C4×SD16—1221no—
350C4×SD16—1221no—
351C8×D4—611no—
352C8×D4—611no—
353C8×D4—611no—
354C8×D4—1221no—
355C8×D4—1221no—
356C8×D4C2×D6111yescharacteristic
357#432—611no—
358#432—611no—
359C2×C4⋊C8—611no—
360C2×C22⋊C8—1221no—
361C2×C4×C8—611no—
362#335—611no—
363#335C2×D6111yescharacteristic, complemented
364#334—611no—
365#334—611no—
366#333—611no—
367#333—611no—
368#332—611no—
369#332—611no—
370#332—611no—
371#332C2×D6111yescharacteristic, complemented
372#331—611no—
373#331C2×D6111yescharacteristic, complemented
374C61#1925711yescharacteristic, complemented, Sylow
375C2×#41—1221no—
376C22×D4—611no—
377C23×C4—1221no—
378C2×SD16—21no—
379C2×SD16—1221no—
380C2×SD16—1221no—
381C2×SD16—1221no—
382C2×D8—21no—
383C2×D8—1221no—
384C22×C8—21no—
385C22×C8—611no—
386C22×C8—611no—
387C22×C8—1221no—
388C22×C8—1221no—
389C22×C8—1221no—
390C22×C8—1221no—
391C22×C8—2444no—
392C4⋊Q8—611no—
393C4⋊Q8—611no—
394C4⋊Q8C22×D6111yescharacteristic
395#127C22×D6111yescharacteristic
396#125—1221no—
397#124—1221no—
398C22⋊Q8—1221no—
399C22⋊Q8—1221no—
400#121—1221no—
401#121—1221no—
402C4×Q8—1221no—
403C4×Q8—1221no—
404C4×D4—611no—
405C4×D4—611no—
406C4×D4—611no—
407C4×D4—1221no—
408C4×D4—1221no—
409C4×D4—1221no—
410C4×D4—1221no—
411C4×D4C22×D6111yescharacteristic
412#117—1221no—
413C2×C4⋊C4—611no—
414C2×C22⋊C4—1221no—
415C2×C42—611no—
416#107—1221no—
417#107—1221no—
418#106—1221no—
419#106—1221no—
420#106—1221no—
421#106—1221no—
422C4⋊C8—611no—
423C4⋊C8—611no—
424C4⋊C8—611no—
425C4⋊C8—1221no—
426C4⋊C8C22×D6111yescharacteristic
427C4⋊C8C22×D6111yescharacteristic
428C4⋊C8C22×D6111yescharacteristic
429#103—21no—
430#103—1221no—
431#103—1221no—
432#103—1221no—
433#102—21no—
434#102—21no—
435#102—21no—
436#102—1221no—
437#102—1221no—
438#102—1221no—
439#102—1221no—
440#102—1221no—
441C22⋊C8—21no—
442C22⋊C8—1221no—
443C22⋊C8—1221no—
444C22⋊C8—1221no—
445C22⋊C8—1221no—
446C22⋊C8—1221no—
447#97—1221no—
448#97—1221no—
449C4×C8—611no—
450C4×C8—611no—
451C4×C8—611no—
452C4×C8C22×D6111yescharacteristic
453C24—1221no—
454#41—4882no—
455C2×Q8—21no—
456C2×Q8—1221no—
457C2×Q8—1221no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—611no—
462C2×D4—1221no—
463C2×D4—1222no—
464C2×D4—1222no—
465C2×D4—2441no—
466C2×D4D4×D6111yescharacteristic
467C22×C4—21no—
468C22×C4—611no—
469C22×C4—611no—
470C22×C4—611no—
471C22×C4—1221no—
472C22×C4—1221no—
473C22×C4—1221no—
474C22×C4—1221no—
475C22×C4—2442no—
476C22×C4—2442no—
477C22×C4—2441no—
478SD16—42no—
479SD16—2442no—
480SD16—2442no—
481SD16—2442no—
482D8—42no—
483D8—2442no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—42no—
488C2×C8—611no—
489C2×C8—611no—
490C2×C8—1222no—
491C2×C8—1221no—
492C2×C8—1221no—
493C2×C8—1221no—
494C2×C8—2441no—
495C2×C8—2442no—
496C2×C8—2442no—
497C2×C8—2442no—
498C2×C8—2442no—
499C2×C8D4×D6111yescharacteristic
500C2×C8D4×D6111yescharacteristic
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—611no—
504C4⋊C4—1221no—
505C4⋊C4—1221no—
506C4⋊C4—1221no—
507C4⋊C4—1221no—
508C4⋊C4—1221no—
509C4⋊C4—2441no—
510C4⋊C4—2441no—
511C4⋊C4D4×D6111yescharacteristic
512C22⋊C4—21no—
513C22⋊C4—1221no—
514C22⋊C4—1221no—
515C22⋊C4—1221no—
516C22⋊C4—2441no—
517C22⋊C4—2441no—
518C22⋊C4—2441no—
519C42—611no—
520C42—611no—
521C42—611no—
522C42—2441no—
523C42C23×D6111yescharacteristic
524C23—21no—
525C23—41no—
526C23—611no—
527C23—1224no—
528C23—1221no—
529C23—2442no—
530C23—2442no—
531Q8—41no—
532Q8—41no—
533Q8—4881no—
534Q8—2441no—
535Q8—2441no—
536D4—21no—
537D4—41no—
538D4—41no—
539D4—41no—
540D4—41no—
541D4—81no—
542D4—612no—
543D4—1224no—
544D4—1221no—
545D4—4882no—
546D4#1451979612yes—
547C2×C4—21no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—611no—
552C2×C4—611no—
553C2×C4—611no—
554C2×C4—1222no—
555C2×C4—1221no—
556C2×C4—1221no—
557C2×C4—1221no—
558C2×C4—1221no—
559C2×C4—1221no—
560C2×C4—4882no—
561C2×C4—2442no—
562C2×C4—2442no—
563C2×C4—2442no—
564C2×C4—2441no—
565C2×C4—2441no—
566C2×C4—2441no—
567C2×C4—2441no—
568C2×C4C2×D4×D6111yescharacteristic
569C2×C4C2×D4×D6111yescharacteristic
570C2×C4#41×D6111yescharacteristic
571C8—21no—
572C8—21no—
573C8—41no—
574C8—41no—
575C8—41no—
576C8—612no—
577C8—1221no—
578C8—2441no—
579C8—2441no—
580C8—2441no—
581C22—22no—
582C22—41no—
583C22—41no—
584C22—81no—
585C22—81no—
586C22—81no—
587C22—612no—
588C22—1221no—
589C22—1221no—
590C22—1222no—
591C22—2448no—
592C22—2441no—
593C22—2441no—
594C22#121×D6111yescharacteristic
595C4—21no—
596C4—41no—
597C4—81no—
598C4—1221no—
599C4—1221no—
600C4—1221no—
601C4—4881no—
602C4—4881no—
603C4—2441no—
604C4#135×D6111yescharacteristic
605C4C2×#1451979611yescharacteristic
606C4AGL(1,ℤ/8)×D6111yescharacteristic
607C4#1684055911yescharacteristic
608C2—42no—
609C2—161no—
610C2—1222no—
611C2—2442no—
612C2#2145528111yescharacteristic
613C2#2145539811yescharacteristic
614C2#471×D6111yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1681C2
socleC22×C61C2×#121
prime radical#21455909C2
hypercenter#1681D61
hypocenterC61#19257
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1681

Complements

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

NH ≅ G/Ntype
1. #21456377C2semidirect
2. #21455917C2semidirect
4. C8×D4×D61C2semidirect
5. #21445271C2semidirect
6. #21445270C2semidirect
7. #21445253C2semidirect
8. #21445193C2semidirect
9. #21445192C2semidirect
10. #21445159C2semidirect
11. #21445158C2semidirect
12. #21444581C2semidirect
13. #21444516C2semidirect
14. #21444511C2semidirect
15. C61×#1681C2semidirect
20. #16840158C22semidirect
21. #16840134C22semidirect
22. #16840113C22semidirect
51. #16839515C22semidirect
52. #16839512C22semidirect
53. #16839512C22semidirect
54. #16839510C22semidirect
55. #16839510C22semidirect
56. #16839509C22semidirect
57. #16839508C22semidirect
58. #16839507C22semidirect
59. #16839506C22semidirect
60. C61×#486C22semidirect
64. C61×#335C22semidirect
65. C61×#332C22semidirect
66. C61×#331C22semidirect
308. #1681D61semidirect
330. #486C2×D61semidirect
363. #335C2×D61semidirect
371. #332C2×D61semidirect
373. #331C2×D61semidirect
374. C61#19257semidirect

Maximal subgroups

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

subgroupindex
#214563772
#214559172
#214559092
C8×D4×D612
#214452712
#214452702
#214452532
#214451932
#214451922
#214451592
#214451582
#214445812
#214445162
#214445112
C61×#16812
#1925761