Finite Group Databasebeta

← #29665204

Subgroups — #29665204

Order 15616.

662 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#21456770C211yesmaximal, characteristic, complemented
2#21455920C211yesmaximal, characteristic, complemented
3#21455818C211yesmaximal, characteristic
4#21454840C211yesmaximal, characteristic, complemented
5#21445255C211yesmaximal, characteristic, complemented
6#21445254C211yesmaximal, characteristic, complemented
7#21445243C211yesmaximal, characteristic, complemented
8#21445242C211yesmaximal, characteristic, complemented
9#21445183C211yesmaximal, characteristic, complemented
10#21445182C211yesmaximal, characteristic, complemented
11#21445157C211yesmaximal, characteristic, complemented
12#21444596C211yesmaximal, characteristic, complemented
13#21444521C211yesmaximal, characteristic, complemented
14#21444482C211yesmaximal, characteristic, complemented
15C61×#1665C211yesmaximal, characteristic, complemented
16#16840317C2211yescharacteristic
17#16840310C2211yescharacteristic
18C4×D4×D61C2211yescharacteristic
19#16840141C2211yescharacteristic, complemented
20#16840113C2211yescharacteristic, complemented
21#16840108C2211yescharacteristic, complemented
22#16840099—21no—
23#16840074—21no—
24#16840043C2211yescharacteristic
25#16840042C2211yescharacteristic
26#16840039C2211yescharacteristic
27#16839982—21no—
28#16839982—21no—
29#16839968C2211yescharacteristic
30#16839967C2211yescharacteristic
31#16839954C2211yescharacteristic
32#16839921—21no—
33#16839921—21no—
34#16839914—21no—
35#16839894—21no—
36#16839861—21no—
37#16839859—21no—
38D8×Dic61—21no—
39#16839842—21no—
40C8×D244C2211yescharacteristic
41D4×#92418C2211yescharacteristic
42#16839827C2211yescharacteristic
43C8×C22⋊D61—21no—
44#16839825—21no—
45#16839818C2211yescharacteristic
46#16839814C2211yescharacteristic
47#16839814C2211yescharacteristic
48#16839813C2211yescharacteristic
49C2×#14519689—21no—
50C22⋊C8×D61—21no—
51#16839514C2211yescharacteristic, complemented
52#16839514C2211yescharacteristic, complemented
53#16839513C2211yescharacteristic, complemented
54#16839509C2211yescharacteristic, complemented
55#16839509C2211yescharacteristic, complemented
56#16839508C2211yescharacteristic, complemented
57#16839507C2211yescharacteristic, complemented
58#16839507C2211yescharacteristic, complemented
59#16839506C2211yescharacteristic, complemented
60C61×#492C2211yescharacteristic, complemented
61C61×#474C2211yescharacteristic
62C61×#459C2211yescharacteristic
63C8×D4×C61C2211yescharacteristic
64C61×#334C2211yescharacteristic, complemented
65C61×#332C2211yescharacteristic, complemented
66C61×#331C2211yescharacteristic, complemented
67C2×#11758224—21no—
68C2×D4×D61D411yescharacteristic
69C22×C4×D61—21no—
70C2×#11758202—21no—
71C2×#11758202—21no—
72C2×#11758190—21no—
73C2×#11758200—21no—
74C2×#11758199—21no—
75C22×#92418—21no—
76C2×C8×D61—21no—
77C2×C8×D61—21no—
78#14519777C2311yescharacteristic
79#14519777C2311yescharacteristic
80#14519768—21no—
81#14519764—21no—
82#14519757—21no—
83#14519753—21no—
84#14519748—21no—
85#14519748—21no—
86Q8×Dic61—21no—
87#14519742—21no—
88C4×D244C2311yescharacteristic
89D4×Dic61—21no—
90D4×Dic61—21no—
91D4×Dic61D411yescharacteristic
92#14519739D411yescharacteristic
93C4×C22⋊D61—21no—
94#14519737—21no—
95#14519735—21no—
96C4⋊C4×D61D411yescharacteristic
97C22⋊C4×D61—21no—
98C42×D61C2311yescharacteristic
99#14519711—21no—
100#14519710—21no—
101#14519710—21no—
102C4⋊#92418C2311yescharacteristic
103C4⋊#92418C2311yescharacteristic
104C4⋊#92418C2311yescharacteristic
105#14519706D411yescharacteristic
106#14519706D411yescharacteristic
107#14519702—21no—
108#14519702—21no—
109#14519702—21no—
110#14519700—21no—
111#14519699—21no—
112#14519698—21no—
113#14519698—21no—
114#14519698—21no—
115#14519698—21no—
116#14519697—21no—
117C22⋊#92418—21no—
118#14519689—21no—
119#14519689—21no—
120#14519689—21no—
121#14519689—21no—
122#14519689—21no—
123#14519689D411yescharacteristic
124#14519689D411yescharacteristic
125#14519687—21no—
126C4×#92418C2311yescharacteristic
127C8×Dic61—21no—
128C8×Dic61—21no—
129C2×SD16×C61—21no—
130C2×SD16×C61—21no—
131C2×D8×C61—21no—
132C22×C8×C61—21no—
133C4⋊Q8×C61C2311yescharacteristic
134#127×C61C2311yescharacteristic
135C4×D4×C61C2311yescharacteristic
136#107×C61—21no—
137C4⋊C8×C61C2311yescharacteristic
138C4⋊C8×C61C2311yescharacteristic
139C4⋊C8×C61C2311yescharacteristic
140#102×C61—21no—
141#102×C61—21no—
142C22⋊C8×C61—21no—
143C4×C8×C61C2311yescharacteristic
144C23×D61—21no—
145C2×Q8×C61—21no—
146C2×D4×C61—21no—
147C2×D4×C61—21no—
148C2×D4×C61—21no—
149C2×D4×C61C2×D411yescharacteristic
150C22×C4×C61—21no—
151C2×C22⋊D61—21no—
152C2×C22⋊D61—41no—
153C22×Dic61—21no—
154C22×Dic61—41no—
155#11758224—81no—
156#11758224—81no—
157D4×D61—22no—
158D4×D61—22no—
159C2×D244C2×D411yescharacteristic
160C2×C4×D61—21no—
161C2×C4×D61—21no—
162C2×C4×D61—21no—
163C2×C4×D61—81no—
164C2×C4×D61—81no—
165C2×C4×D61C2×D411yescharacteristic
166C2×C4×D61C2×D411yescharacteristic
167C2×C4×D61#4111yescharacteristic
168C2×Dic122—21no—
169C2×Dic122—21no—
170SD16×C61—81no—
171SD16×C61—81no—
172D8×C61—42no—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61—81no—
177C2×C8×C61C2×D411yescharacteristic
178C2×C8×C61C2×D411yescharacteristic
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61C2×D411yescharacteristic
182C22⋊C4×C61—21no—
183C42×C61C2411yescharacteristic
184C22⋊Dic61—21no—
185C22⋊Dic61—41no—
186C22⋊Dic61—41no—
187#11758202—81no—
188#11758202—81no—
189#11758200—42no—
190#11758199—42no—
191#11758198—21no—
192#11758198—21no—
193#11758198—41no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—41no—
199C4⋊Dic61C2×D411yescharacteristic
200#11758196—21no—
201#11758196—41no—
202C4×Dic61—41no—
203C4×Dic61C2×D411yescharacteristic
204C4×Dic61C2×D411yescharacteristic
205C4×Dic61#4111yescharacteristic
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—81no—
210C2×#92418C2×D411yescharacteristic
211C2×#92418C2×D411yescharacteristic
212#11758190—42no—
213C8×D61—81no—
214C8×D61—81no—
215C23×C61—21no—
216C23×C61—41no—
217C22×D61—21no—
218C22×D61—24no—
219C22×D61—81no—
220C22×D61—81no—
221C22×D61#12111yescharacteristic
222Q8×C61—41no—
223Q8×C61—41no—
224D4×C61—21no—
225D4×C61—41no—
226D4×C61—41no—
227D4×C61—41no—
228D4×C61—41no—
229D4×C61—81no—
230D4×C61#13512yes—
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—
240C2×Dic61—21no—
241C2×Dic61—41no—
242C2×Dic61—41no—
243C2×Dic61—81no—
244C2×Dic61—82no—
245C2×Dic61#12111yescharacteristic
246C2×Dic61#12111yescharacteristic
247C2×Dic61#12111yescharacteristic
248D244—21no—
249D244#13512yes—
250C4×D61—21no—
251C4×D61—21no—
252C4×D61—21no—
253C4×D61—21no—
254C4×D61—41no—
255C4×D61—41no—
256C4×D61—41no—
257C4×D61—81no—
258C4×D61—81no—
259Dic122—41no—
260Dic122—41no—
261Dic122—41no—
262Dic122—41no—
263C8×C61—21no—
264C8×C61—21no—
265C8×C61—41no—
266C8×C61—41no—
267C8×C61—41no—
268#92418—21no—
269#92418—21no—
270#92418—41no—
271#92418—41no—
272#92418—41no—
273#19388—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—21no—
285C2×D61—41no—
286C2×D61—41no—
287C2×D61—84no—
288C4×C61—21no—
289C4×C61—41no—
290C4×C61—81no—
291C4×C61C2×#13511yescharacteristic
292C4×C61C2×#13511yescharacteristic
293C4×C61#57611yescharacteristic
294C4×C61#57611yescharacteristic
295Dic61—21no—
296Dic61—21no—
297Dic61—21no—
298Dic61—41no—
299Dic61—81no—
300Dic61—81no—
301#3210—611no—
302#3110—611no—
303#3100—611no—
304#2972—611no—
305#1681—611no—
306#1681—611no—
307#1679—611no—
308#1679—611no—
309#1669—611no—
310#1669—611no—
311#1665—611no—
312#1665D6111yescharacteristic, complemented
313#1572—611no—
314#1553—611no—
315#1544—611no—
316C2×C61—42no—
317C2×C61—161no—
318C2×C61#304111yescharacteristic
319C2×C61#305011yescharacteristic
320C2×C61#306911yescharacteristic
321D61—41no—
322D61—42no—
323#533—611no—
324#532—611no—
325C2×C4×D4—611no—
326#494—611no—
327#492C2×D6111yescharacteristic, complemented
328#487—611no—
329#486—611no—
330#484—1221no—
331#480—1221no—
332#474—611no—
333#474—611no—
334#474—611no—
335#474C2×D6111yescharacteristic
336#464—1221no—
337#464—1221no—
338#461—611no—
339#461—611no—
340#459—611no—
341#459C2×D6111yescharacteristic
342#452—1221no—
343#452—1221no—
344#450—1221no—
345#447—1221no—
346#440—1221no—
347#440—1221no—
348C4×D8—1221no—
349C4×D8—1221no—
350C8×D4—611no—
351C8×D4—611no—
352C8×D4—611no—
353C8×D4—1221no—
354C8×D4—1221no—
355C8×D4C2×D6111yescharacteristic
356#432—611no—
357#431—611no—
358#431—611no—
359#431—611no—
360C2×C22⋊C8—1221no—
361C2×C22⋊C8—1221no—
362#335—611no—
363#335—611no—
364#334—611no—
365#334C2×D6111yescharacteristic, complemented
366#332—611no—
367#332—611no—
368#332—611no—
369#332—611no—
370#332—611no—
371#332C2×D6111yescharacteristic, complemented
372#331—611no—
373#331C2×D6111yescharacteristic, complemented
374C61#1938811yescharacteristic, complemented, Sylow
375C2×#41—1221no—
376C22×D4—611no—
377C23×C4—1221no—
378C2×Q16—1221no—
379C2×Q16—1221no—
380C2×SD16—21no—
381C2×SD16—21no—
382C2×SD16—1221no—
383C2×SD16—1221no—
384C2×D8—21no—
385C2×D8—1221no—
386C22×C8—21no—
387C22×C8—1221no—
388C22×C8—1221no—
389C22×C8—1221no—
390C4⋊Q8—611no—
391C4⋊Q8—611no—
392C4⋊Q8C22×D6111yescharacteristic
393#127C22×D6111yescharacteristic
394#125—1221no—
395#124—1221no—
396C22⋊Q8—1221no—
397C22⋊Q8—1221no—
398#121—1221no—
399#121—1221no—
400C4×Q8—1221no—
401C4×Q8—1221no—
402C4×D4—611no—
403C4×D4—611no—
404C4×D4—611no—
405C4×D4—1221no—
406C4×D4—1221no—
407C4×D4—1221no—
408C4×D4—1221no—
409C4×D4C22×D6111yescharacteristic
410#117—1221no—
411C2×C4⋊C4—611no—
412C2×C22⋊C4—1221no—
413C2×C42—611no—
414#107—21no—
415#107—1221no—
416#107—1221no—
417#107—1221no—
418C4⋊C8—611no—
419C4⋊C8—611no—
420C4⋊C8—611no—
421C4⋊C8—611no—
422C4⋊C8—611no—
423C4⋊C8C22×D6111yescharacteristic
424C4⋊C8C22×D6111yescharacteristic
425C4⋊C8C22×D6111yescharacteristic
426#103—1221no—
427#103—1221no—
428#103—1221no—
429#103—1221no—
430#102—21no—
431#102—21no—
432#102—1221no—
433#102—1221no—
434#102—1221no—
435#102—1221no—
436#102—1221no—
437#102—1221no—
438C22⋊C8—21no—
439C22⋊C8—611no—
440C22⋊C8—611no—
441C22⋊C8—1221no—
442C22⋊C8—1221no—
443C22⋊C8—1221no—
444C22⋊C8—1221no—
445C22⋊C8—1221no—
446C22⋊C8—1221no—
447#97—1221no—
448C4×C8—611no—
449C4×C8—1221no—
450C4×C8—1221no—
451C4×C8C22×D6111yescharacteristic
452C24—1221no—
453#41—4881no—
454#41—4881no—
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—4881no—
476C22×C4—4881no—
477C22×C4—2441no—
478Q16—4881no—
479Q16—4881no—
480SD16—81no—
481SD16—81no—
482SD16—2442no—
483SD16—2442no—
484D8—42no—
485D8—2442no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—81no—
490C2×C8—611no—
491C2×C8—611no—
492C2×C8—1221no—
493C2×C8—1221no—
494C2×C8—1221no—
495C2×C8—4881no—
496C2×C8—4881no—
497C2×C8—4881no—
498C2×C8D4×D6111yescharacteristic
499C2×C8D4×D6111yescharacteristic
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—611no—
503C4⋊C4—1221no—
504C4⋊C4—1221no—
505C4⋊C4—1221no—
506C4⋊C4—1221no—
507C4⋊C4—1221no—
508C4⋊C4—2441no—
509C4⋊C4—2441no—
510C4⋊C4D4×D6111yescharacteristic
511C22⋊C4—21no—
512C22⋊C4—1221no—
513C22⋊C4—1221no—
514C22⋊C4—1221no—
515C22⋊C4—2441no—
516C22⋊C4—2441no—
517C22⋊C4—2441no—
518C42—611no—
519C42—611no—
520C42—611no—
521C42—2441no—
522C42C23×D6111yescharacteristic
523C23—21no—
524C23—41no—
525C23—611no—
526C23—1224no—
527C23—1221no—
528C23—4881no—
529C23—4881no—
530Q8—41no—
531Q8—41no—
532Q8—2441no—
533Q8—2441no—
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#1451979812yes—
547C2×C4—21no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—611no—
552C2×C4—611no—
553C2×C4—611no—
554C2×C4—1221no—
555C2×C4—1221no—
556C2×C4—1221no—
557C2×C4—1221no—
558C2×C4—1221no—
559C2×C4—4881no—
560C2×C4—4881no—
561C2×C4—4882no—
562C2×C4—4881no—
563C2×C4—2441no—
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—1221no—
577C8—1221no—
578C8—2441no—
579C8—2441no—
580C8—2441no—
581C22—22no—
582C22—41no—
583C22—41no—
584C22—81no—
585C22—81no—
586C22—81no—
587C22—1221no—
588C22—1222no—
589C22—1221no—
590C22—1221no—
591C22—4884no—
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×#1451979811yescharacteristic
606C4#1684056311yescharacteristic
607C4#1684056611yescharacteristic
608C2—42no—
609C2—161no—
610C2—2441no—
611C2—2442no—
612C2#2145531711yescharacteristic
613C2#2145539511yescharacteristic
614C2#2145555111yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1665C2
socleC22×C61C2×#121
prime radical#21455818C2
hypercenter#1665D61
hypocenterC61#19388
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1665

Complements

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

NH ≅ G/Ntype
1. #21456770C2semidirect
2. #21455920C2semidirect
4. #21454840C2semidirect
5. #21445255C2semidirect
6. #21445254C2semidirect
7. #21445243C2semidirect
8. #21445242C2semidirect
9. #21445183C2semidirect
10. #21445182C2semidirect
11. #21445157C2semidirect
12. #21444596C2semidirect
13. #21444521C2semidirect
14. #21444482C2semidirect
15. C61×#1665C2semidirect
19. #16840141C22semidirect
20. #16840113C22semidirect
21. #16840108C22semidirect
51. #16839514C22semidirect
52. #16839514C22semidirect
53. #16839513C22semidirect
54. #16839509C22semidirect
55. #16839509C22semidirect
56. #16839508C22semidirect
57. #16839507C22semidirect
58. #16839507C22semidirect
59. #16839506C22semidirect
60. C61×#492C22semidirect
64. C61×#334C22semidirect
65. C61×#332C22semidirect
66. C61×#331C22semidirect
312. #1665D61semidirect
327. #492C2×D61semidirect
365. #334C2×D61semidirect
371. #332C2×D61semidirect
373. #331C2×D61semidirect
374. C61#19388semidirect

Maximal subgroups

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

subgroupindex
#214567702
#214559202
#214558182
#214548402
#214452552
#214452542
#214452432
#214452422
#214451832
#214451822
#214451572
#214445962
#214445212
#214444822
C61×#16652
#1938861