Finite Group Databasebeta

← #29665198

Subgroups — #29665198

Order 15616.

654 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456768C211yesmaximal, characteristic, complemented
2#21455617C211yesmaximal, characteristic
3#474×D61C211yesmaximal, characteristic
4#21454840C211yesmaximal, characteristic
5#21445273C211yesmaximal, characteristic, complemented
6#21445272C211yesmaximal, characteristic, complemented
7#21445267C211yesmaximal, characteristic, complemented
8#21445185C211yesmaximal, characteristic, complemented
9#21445184C211yesmaximal, characteristic, complemented
10#21445183C211yesmaximal, characteristic, complemented
11#21445182C211yesmaximal, characteristic, complemented
12#21444592C211yesmaximal, characteristic, complemented
13#21444502C211yesmaximal, characteristic, complemented
14#21444473C211yesmaximal, characteristic, complemented
15C61×#1683C211yesmaximal, characteristic, complemented
16#16840317C2211yescharacteristic
17C4⋊Q8×D61C2211yescharacteristic
18C4×D4×D61C2211yescharacteristic
19#16840142C2211yescharacteristic, complemented
20#16840141C2211yescharacteristic, complemented
21#16840114C2211yescharacteristic, complemented
22#16840061—21no—
23#16840060—21no—
24#16840049C2211yescharacteristic
25#16840043C2211yescharacteristic
26#16840043C2211yescharacteristic
27#16840042C2211yescharacteristic
28#16840042C2211yescharacteristic
29#16840041—21no—
30#16840040—21no—
31#16840039C2211yescharacteristic
32#16839971—21no—
33#16839971—21no—
34#16839919—21no—
35#16839919—21no—
36C8×D244C2211yescharacteristic
37D4×#92418C2211yescharacteristic
38#16839827C2211yescharacteristic
39C8×C22⋊D61—21no—
40#16839825—21no—
41#16839814C2211yescharacteristic
42#16839813C2211yescharacteristic
43#16839801—21no—
44#107×D61—21no—
45#16839784C2211yescharacteristic
46C4⋊C8×D61C2211yescharacteristic
47#16839765—21no—
48#102×D61—21no—
49C2×#14519689—21no—
50C22⋊C8×D61—21no—
51#16839515C2211yescharacteristic, complemented
52#16839514C2211yescharacteristic, complemented
53#16839514C2211yescharacteristic, complemented
54#16839511C2211yescharacteristic, complemented
55#16839509C2211yescharacteristic, complemented
56#16839509C2211yescharacteristic, complemented
57#16839508C2211yescharacteristic, complemented
58#16839507C2211yescharacteristic, complemented
59#16839507C2211yescharacteristic, complemented
60C61×#487C2211yescharacteristic, complemented
61C61×#476C2211yescharacteristic
62C61×#474C2211yescharacteristic
63C8×D4×C61C2211yescharacteristic
64C61×#335C2211yescharacteristic, complemented
65C61×#333C2211yescharacteristic, complemented
66C61×#332C2211yescharacteristic, complemented
67C2×Q8×D61—21no—
68C2×D4×D61D411yescharacteristic
69C22×C4×D61—21no—
70C2×Dic244—21no—
71C2×Dic244—21no—
72C2×#11758202—21no—
73C2×#11758202—21no—
74C2×#11758189—21no—
75C22×#92418—21no—
76C2×C8×D61—21no—
77C2×C8×D61—21no—
78C2×C8×D61—21no—
79#14519777C2311yescharacteristic
80#14519777C2311yescharacteristic
81#14519776—21no—
82#14519776—21no—
83#14519775—21no—
84#14519768—21no—
85#14519757—21no—
86#14519753—21no—
87Q8×Dic61—21no—
88#14519742—21no—
89C4×D244C2311yescharacteristic
90D4×Dic61D411yescharacteristic
91#14519739D411yescharacteristic
92C4×C22⋊D61—21no—
93#14519737—21no—
94#14519735—21no—
95C4⋊C4×D61—21no—
96C4⋊C4×D61—21no—
97C4⋊C4×D61D411yescharacteristic
98C22⋊C4×D61—21no—
99C42×D61C2311yescharacteristic
100#14519711—21no—
101#14519711—21no—
102#14519710—21no—
103#14519710—21no—
104#14519710—21no—
105#14519708—21no—
106C4⋊#92418C2311yescharacteristic
107C4⋊#92418C2311yescharacteristic
108C4⋊#92418C2311yescharacteristic
109#14519706—21no—
110#14519706—21no—
111#14519706D411yescharacteristic
112#14519706D411yescharacteristic
113#14519700—21no—
114#14519700—21no—
115#14519699—21no—
116#14519699—21no—
117#14519698—21no—
118#14519698—21no—
119#14519697—21no—
120#14519697—21no—
121C22⋊#92418—21no—
122#14519689—21no—
123#14519689—21no—
124#14519689—21no—
125#14519689D411yescharacteristic
126#14519689D411yescharacteristic
127C4×#92418C2311yescharacteristic
128C8×Dic61—21no—
129C22×C8×C61—21no—
130C4⋊Q8×C61C2311yescharacteristic
131C4⋊Q8×C61C2311yescharacteristic
132C4×D4×C61C2311yescharacteristic
133#107×C61—21no—
134#106×C61—21no—
135C4⋊C8×C61C2311yescharacteristic
136C4⋊C8×C61C2311yescharacteristic
137C4⋊C8×C61C2311yescharacteristic
138#103×C61—21no—
139#103×C61—21no—
140#102×C61—21no—
141#102×C61—21no—
142C22⋊C8×C61—21no—
143C4×C8×C61C2311yescharacteristic
144C23×D61—21no—
145C2×Q8×C61—21no—
146C2×Q8×C61—21no—
147C2×D4×C61C2×D411yescharacteristic
148C22×C4×C61—21no—
149C2×C22⋊D61—21no—
150C22×Dic61—21no—
151Q8×D61—81no—
152Q8×D61—81no—
153D4×D61—22no—
154D4×D61Q1614yescomplemented
155C2×D244C2×D411yescharacteristic
156C2×C4×D61—21no—
157C2×C4×D61—21no—
158C2×C4×D61—21no—
159C2×C4×D61—41no—
160C2×C4×D61—81no—
161C2×C4×D61—81no—
162C2×C4×D61C2×D411yescharacteristic
163C2×C4×D61C2×D411yescharacteristic
164C2×C4×D61#4111yescharacteristic
165C2×Dic122—21no—
166C2×Dic122—21no—
167C2×Dic122—41no—
168C2×C8×C61—21no—
169C2×C8×C61—21no—
170C2×C8×C61—21no—
171C2×C8×C61—81no—
172C2×C8×C61C2×D411yescharacteristic
173C2×C8×C61C2×D411yescharacteristic
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61C2×D411yescharacteristic
179C22⋊C4×C61—21no—
180C42×C61C2411yescharacteristic
181C22⋊Dic61—21no—
182#11758202—81no—
183#11758202—81no—
184#11758198—21no—
185#11758198—21no—
186#11758198—41no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—41no—
192C4⋊Dic61C2×D411yescharacteristic
193#11758196—21no—
194#11758196—41no—
195#11758196—41no—
196#11758196—41no—
197C4×Dic61—41no—
198C4×Dic61C2×D411yescharacteristic
199C4×Dic61C2×D411yescharacteristic
200C4×Dic61#4111yescharacteristic
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—81no—
205C2×#92418C2×D411yescharacteristic
206C2×#92418C2×D411yescharacteristic
207Dic244—81no—
208Dic244—81no—
209#11758189—42no—
210C8×D61—42no—
211C8×D61—81no—
212C8×D61—81no—
213C23×C61—21no—
214C22×D61—21no—
215C22×D61—24no—
216C22×D61—81no—
217C22×D61—81no—
218C22×D61#12111yescharacteristic
219Q8×C61—41no—
220Q8×C61—41no—
221Q8×C61—81no—
222D4×C61—21no—
223D4×C61C2×Q1612yes—
224C2×C4×C61—21no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61C22×D411yescharacteristic
230C2×C4×C61C22×D411yescharacteristic
231C2×C4×C61C2×#4111yescharacteristic
232C22⋊D61—24no—
233C2×Dic61—21no—
234C2×Dic61—41no—
235C2×Dic61—41no—
236C2×Dic61—81no—
237C2×Dic61#12111yescharacteristic
238C2×Dic61#12111yescharacteristic
239C2×Dic61#12111yescharacteristic
240D244—21no—
241D244C2×Q1612yes—
242C4×D61—21no—
243C4×D61—21no—
244C4×D61—41no—
245C4×D61—41no—
246C4×D61—41no—
247C4×D61—81no—
248C4×D61—81no—
249C4×D61—82no—
250C4×D61C2×Q1612yes—
251C4×D61#13512yes—
252Dic122—41no—
253Dic122—41no—
254Dic122—41no—
255Dic122—41no—
256Dic122—82no—
257C8×C61—21no—
258C8×C61—21no—
259C8×C61—41no—
260C8×C61—41no—
261C8×C61—41no—
262#92418—21no—
263#92418—21no—
264#92418—41no—
265#92418—41no—
266#92418—41no—
267#19387—611nomaximal, Sylow
268C22×C61—22no—
269C22×C61—41no—
270C22×C61—41no—
271C22×C61C2×#12111yescharacteristic
272C2×D61—21no—
273C2×D61—21no—
274C2×D61—22no—
275C2×D61—41no—
276C2×D61—41no—
277C2×D61—84no—
278C2×D61#46712yes—
279C4×C61—21no—
280C4×C61—41no—
281C4×C61—81no—
282C4×C61—81no—
283C4×C61C22×Q1611yescharacteristic
284C4×C61C2×#13511yescharacteristic
285C4×C61#57611yescharacteristic
286C4×C61#57611yescharacteristic
287Dic61—21no—
288Dic61—21no—
289Dic61—41no—
290Dic61—81no—
291Dic61—81no—
292Dic61#46712yes—
293#3209—611no—
294#3077—611no—
295C2×#474—611no—
296#2972—611no—
297#1683—611no—
298#1683—611no—
299#1683—611no—
300#1683D6111yescharacteristic, complemented
301#1669—611no—
302#1669—611no—
303#1669—611no—
304#1669—611no—
305#1571—611no—
306#1549—611no—
307#1542—611no—
308C2×C61—42no—
309C2×C61#304111yescharacteristic
310C2×C61C2×#46711yescharacteristic
311C2×C61#306911yescharacteristic
312D61—22no—
313D61—42no—
314#533—611no—
315C2×C4⋊Q8—611no—
316C2×C4×D4—611no—
317#494—611no—
318#494—611no—
319#487—611no—
320#487C2×D6111yescharacteristic, complemented
321#478—1221no—
322#478—1221no—
323#476—611no—
324#476C2×D6111yescharacteristic
325#474—611no—
326#474—611no—
327#474—611no—
328#474—611no—
329#474—611no—
330#474—1221no—
331#474—1221no—
332#474C2×D6111yescharacteristic
333#462—1221no—
334#462—1221no—
335#451—1221no—
336#451—1221no—
337C8×D4—611no—
338C8×D4—611no—
339C8×D4—611no—
340C8×D4—1221no—
341C8×D4—1221no—
342C8×D4C2×D6111yescharacteristic
343#431—611no—
344#431—611no—
345#428—1221no—
346C2×#107—1221no—
347#423—611no—
348C2×C4⋊C8—611no—
349#418—1221no—
350C2×#102—1221no—
351C2×C22⋊C8—1221no—
352C2×C22⋊C8—1221no—
353#335—611no—
354#335—611no—
355#335—611no—
356#335C2×D6111yescharacteristic, complemented
357#333—611no—
358#333C2×D6111yescharacteristic, complemented
359#332—611no—
360#332—611no—
361#332—611no—
362#332—611no—
363#332—611no—
364#332C2×D6111yescharacteristic, complemented
365C61#1938711yescharacteristic, complemented, Sylow
366C22×Q8—1221no—
367C22×D4—611no—
368C23×C4—1221no—
369C2×Q16—1221no—
370C2×Q16—1221no—
371C2×Q16—1221no—
372C2×Q16—1221no—
373C2×M16—1221no—
374C22×C8—21no—
375C22×C8—1221no—
376C22×C8—1221no—
377C22×C8—1221no—
378C22×C8—1221no—
379C4⋊Q8—611no—
380C4⋊Q8—611no—
381C4⋊Q8—1221no—
382C4⋊Q8—1221no—
383C4⋊Q8—1221no—
384C4⋊Q8C22×D6111yescharacteristic
385C4⋊Q8C22×D6111yescharacteristic
386#125—1221no—
387C22⋊Q8—1221no—
388C22⋊Q8—1221no—
389C4×Q8—1221no—
390C4×Q8—1221no—
391C4×D4—611no—
392C4×D4—611no—
393C4×D4—611no—
394C4×D4—1221no—
395C4×D4—1221no—
396C4×D4C22×D6111yescharacteristic
397#117—1221no—
398C2×C4⋊C4—611no—
399C2×C4⋊C4—1221no—
400C2×C4⋊C4—1221no—
401C2×C22⋊C4—1221no—
402C2×C42—611no—
403#107—21no—
404#107—1221no—
405#107—1221no—
406#107—1221no—
407#107—1221no—
408#107—1221no—
409#106—21no—
410#106—1221no—
411C4⋊C8—611no—
412C4⋊C8—611no—
413C4⋊C8—611no—
414C4⋊C8—611no—
415C4⋊C8—611no—
416C4⋊C8—1221no—
417C4⋊C8—1221no—
418C4⋊C8C22×D6111yescharacteristic
419C4⋊C8C22×D6111yescharacteristic
420C4⋊C8C22×D6111yescharacteristic
421#103—21no—
422#103—21no—
423#103—1221no—
424#103—1221no—
425#102—21no—
426#102—21no—
427#102—1221no—
428#102—1221no—
429#102—1221no—
430#102—1221no—
431#102—1221no—
432#102—1221no—
433C22⋊C8—21no—
434C22⋊C8—611no—
435C22⋊C8—611no—
436C22⋊C8—1221no—
437C22⋊C8—1221no—
438C22⋊C8—1221no—
439C22⋊C8—1221no—
440C4×C8—611no—
441C4×C8—1221no—
442C4×C8C22×D6111yescharacteristic
443C24—1221no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—1221no—
447C2×Q8—1221no—
448C2×Q8—4881no—
449C2×Q8—4881no—
450C2×Q8—2441no—
451C2×D4—614no—
452C2×D4—611no—
453C2×D4—1222no—
454C2×D4—1221no—
455C2×D4D4×D6111yescharacteristic
456C22×C4—21no—
457C22×C4—611no—
458C22×C4—611no—
459C22×C4—611no—
460C22×C4—1221no—
461C22×C4—1221no—
462C22×C4—1221no—
463C22×C4—1221no—
464C22×C4—4881no—
465C22×C4—4881no—
466C22×C4—2441no—
467Q16—4881no—
468Q16—4881no—
469Q16—4881no—
470Q16—4881no—
471M16—2442no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—81no—
476C2×C8—611no—
477C2×C8—611no—
478C2×C8—1221no—
479C2×C8—1221no—
480C2×C8—1221no—
481C2×C8—4881no—
482C2×C8—4881no—
483C2×C8—4881no—
484C2×C8—2442no—
485C2×C8D4×D6111yescharacteristic
486C2×C8D4×D6111yescharacteristic
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—611no—
492C4⋊C4—1221no—
493C4⋊C4—1221no—
494C4⋊C4—1221no—
495C4⋊C4—1221no—
496C4⋊C4—1221no—
497C4⋊C4—2441no—
498C4⋊C4—2441no—
499C4⋊C4—2441no—
500C4⋊C4—2441no—
501C4⋊C4D4×D6111yescharacteristic
502C22⋊C4—21no—
503C22⋊C4—1221no—
504C22⋊C4—1221no—
505C22⋊C4—1221no—
506C22⋊C4—2441no—
507C42—611no—
508C42—611no—
509C42—611no—
510C42—2441no—
511C42C23×D6111yescharacteristic
512C23—21no—
513C23—611no—
514C23—1224no—
515C23—1221no—
516C23—4881no—
517C23—4881no—
518Q8—41no—
519Q8—41no—
520Q8—81no—
521Q8—4882no—
522Q8—2441no—
523Q8—2441no—
524Q8—2441no—
525Q8—2441no—
526D4—21no—
527D4—612no—
528D4—1224no—
529D4—1221no—
530D4Q16×D6112yes—
531C2×C4—21no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—611no—
537C2×C4—612no—
538C2×C4—612no—
539C2×C4—611no—
540C2×C4—611no—
541C2×C4—1221no—
542C2×C4—1221no—
543C2×C4—1221no—
544C2×C4—4881no—
545C2×C4—4881no—
546C2×C4—4882no—
547C2×C4—4881no—
548C2×C4—2441no—
549C2×C4—2441no—
550C2×C4—2441no—
551C2×C4—2441no—
552C2×C4—2441no—
553C2×C4C2×D4×D6111yescharacteristic
554C2×C4C2×D4×D6111yescharacteristic
555C2×C4#41×D6111yescharacteristic
556C8—21no—
557C8—21no—
558C8—41no—
559C8—41no—
560C8—41no—
561C8—1221no—
562C8—1221no—
563C8—2441no—
564C8—2441no—
565C8—2441no—
566C22—22no—
567C22—41no—
568C22—41no—
569C22—612no—
570C22—1221no—
571C22—1221no—
572C22—1222no—
573C22—4884no—
574C22—2441no—
575C22—2441no—
576C22#121×D6111yescharacteristic
577C4—21no—
578C4—41no—
579C4—81no—
580C4—81no—
581C4—612no—
582C4—1221no—
583C4—1221no—
584C4—4881no—
585C4—4881no—
586C4—2441no—
587C4C2×Q16×D6111yescharacteristic
588C4#135×D6111yescharacteristic
589C4#1684056411yescharacteristic
590C4#1684056611yescharacteristic
591C2—42no—
592C2—1222no—
593C2—2442no—
594C2#2145532011yescharacteristic
595C2#467×D6111yescharacteristic
596C2#2145555111yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1683C2
socleC22×C61C2×#121
prime radicalC2×D4×D61D4
hypercenter#1683D61
hypocenterC61#19387
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1683

Complements

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

NH ≅ G/Ntype
1. #21456768C2semidirect
5. #21445273C2semidirect
6. #21445272C2semidirect
7. #21445267C2semidirect
8. #21445185C2semidirect
9. #21445184C2semidirect
10. #21445183C2semidirect
11. #21445182C2semidirect
12. #21444592C2semidirect
13. #21444502C2semidirect
14. #21444473C2semidirect
15. C61×#1683C2semidirect
19. #16840142C22semidirect
20. #16840141C22semidirect
21. #16840114C22semidirect
51. #16839515C22semidirect
52. #16839514C22semidirect
53. #16839514C22semidirect
54. #16839511C22semidirect
55. #16839509C22semidirect
56. #16839509C22semidirect
57. #16839508C22semidirect
58. #16839507C22semidirect
59. #16839507C22semidirect
60. C61×#487C22semidirect
64. C61×#335C22semidirect
65. C61×#333C22semidirect
66. C61×#332C22semidirect
154. D4×D61Q16semidirect
300. #1683D61semidirect
320. #487C2×D61semidirect
356. #335C2×D61semidirect
358. #333C2×D61semidirect
364. #332C2×D61semidirect
365. C61#19387semidirect

Maximal subgroups

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

subgroupindex
#214567682
#214556172
#474×D612
#214548402
#214452732
#214452722
#214452672
#214451852
#214451842
#214451832
#214451822
#214445922
#214445022
#214444732
C61×#16832
#1938761