Finite Group Databasebeta

← #29663808

Subgroups — #29663808

Order 15616.

690 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456404C211yesmaximal, characteristic, complemented
2#21455617C211yesmaximal, characteristic
3#21455611C211yesmaximal, characteristic
4C8×D4×D61C211yesmaximal, characteristic
5#21445271C211yesmaximal, characteristic, complemented
6#21445270C211yesmaximal, characteristic, complemented
7#21445267C211yesmaximal, characteristic, complemented
8#21445217C211yesmaximal, characteristic, complemented
9#21445216C211yesmaximal, characteristic, complemented
10#21445183C211yesmaximal, characteristic, complemented
11#21445182C211yesmaximal, characteristic, complemented
12#21444576C211yesmaximal, characteristic, complemented
13#21444508C211yesmaximal, characteristic, complemented
14#21444506C211yesmaximal, characteristic, complemented
15C61×#1683C211yesmaximal, characteristic, complemented
16C22×C8×D61—21no—
17#16840317C2211yescharacteristic
18#16840317C2211yescharacteristic
19C4×D4×D61C2211yescharacteristic
20#16840165C2211yescharacteristic, complemented
21#16840141C2211yescharacteristic, complemented
22#16840113C2211yescharacteristic, complemented
23#16840065—21no—
24#16840064—21no—
25#16840061—21no—
26#16840060—21no—
27#16840053C2211yescharacteristic
28#16840052C2211yescharacteristic
29#16840049C2211yescharacteristic
30#16840043C2211yescharacteristic
31#16840042C2211yescharacteristic
32#16840039C2211yescharacteristic
33#16840002—21no—
34#16840002—21no—
35#16839854—21no—
36#16839854—21no—
37C8×D244C2211yescharacteristic
38D4×#92418C2211yescharacteristic
39#16839827C2211yescharacteristic
40C8×C22⋊D61—21no—
41#16839825—21no—
42#16839803—21no—
43#16839801—21no—
44#16839784C2211yescharacteristic
45#16839784C2211yescharacteristic
46C4⋊C8×D61C2211yescharacteristic
47#16839765—21no—
48#16839765—21no—
49C22⋊C8×D61—21no—
50C4×C8×D61C2211yescharacteristic
51#16839515C2211yescharacteristic, complemented
52#16839514C2211yescharacteristic, complemented
53#16839514C2211yescharacteristic, complemented
54#16839511C2211yescharacteristic, complemented
55#16839510C2211yescharacteristic, complemented
56#16839510C2211yescharacteristic, complemented
57#16839509C2211yescharacteristic, complemented
58#16839508C2211yescharacteristic, complemented
59#16839507C2211yescharacteristic, complemented
60C61×#487C2211yescharacteristic, complemented
61C61×#476C2211yescharacteristic
62C61×#474C2211yescharacteristic
63C8×D4×C61C2211yescharacteristic
64C61×#335C2211yescharacteristic, complemented
65C61×#333C2211yescharacteristic, complemented
66C61×#332C2211yescharacteristic, complemented
67C2×D4×D61D411yescharacteristic
68C22×C4×D61—21no—
69C2×#11758202—21no—
70C2×#11758202—21no—
71C2×#11758189—21no—
72C2×#11758189—21no—
73C22×#92418—21no—
74C2×C8×D61—21no—
75C2×C8×D61—21no—
76C2×C8×D61—21no—
77C2×C8×D61—44no—
78C2×C8×D61D411yescharacteristic
79C2×C8×D61D411yescharacteristic
80#14519777C2311yescharacteristic
81#14519777C2311yescharacteristic
82#14519768—21no—
83#14519768—21no—
84#14519757—21no—
85#14519757—21no—
86#14519753—21no—
87#14519753—21no—
88Q8×Dic61—21no—
89Q8×Dic61—21no—
90#14519742—21no—
91#14519742—21no—
92C4×D244C2311yescharacteristic
93D4×Dic61D411yescharacteristic
94#14519739D411yescharacteristic
95C4×C22⋊D61—21no—
96#14519737—21no—
97#14519735—21no—
98#14519735—21no—
99C4⋊C4×D61D411yescharacteristic
100C22⋊C4×D61—21no—
101C42×D61C2311yescharacteristic
102#14519711—21no—
103#14519710—21no—
104#14519710—21no—
105#14519710—21no—
106#14519710—21no—
107#14519709—21no—
108#14519708—21no—
109#14519708—21no—
110C4⋊#92418C2311yescharacteristic
111C4⋊#92418C2311yescharacteristic
112C4⋊#92418C2311yescharacteristic
113#14519706—21no—
114#14519706—21no—
115#14519706—21no—
116#14519702—21no—
117#14519702—21no—
118#14519699—21no—
119#14519699—21no—
120#14519698—21no—
121#14519698—21no—
122#14519697—21no—
123#14519697—21no—
124C22⋊#92418—21no—
125#14519689—21no—
126#14519689—21no—
127C4×#92418C2311yescharacteristic
128C8×Dic61D411yescharacteristic
129C8×Dic61D411yescharacteristic
130C22×C8×C61—21no—
131C4⋊Q8×C61C2311yescharacteristic
132C4⋊Q8×C61C2311yescharacteristic
133C4×D4×C61C2311yescharacteristic
134#107×C61—21no—
135#106×C61—21no—
136C4⋊C8×C61C2311yescharacteristic
137C4⋊C8×C61C2311yescharacteristic
138C4⋊C8×C61C2311yescharacteristic
139#103×C61—21no—
140#103×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×Q8×C61—21no—
148C2×D4×C61C2×D411yescharacteristic
149C22×C4×C61—21no—
150C2×C22⋊D61—21no—
151C22×Dic61—21no—
152D4×D61—22no—
153D4×D61Q1614yescomplemented
154C2×D244C2×D411yescharacteristic
155C2×C4×D61—21no—
156C2×C4×D61—21no—
157C2×C4×D61—21no—
158C2×C4×D61—42no—
159C2×C4×D61—42no—
160C2×C4×D61C2×D411yescharacteristic
161C2×C4×D61C2×D411yescharacteristic
162C2×C4×D61#4111yescharacteristic
163C2×Dic122—21no—
164C2×Dic122—21no—
165C2×C8×C61—21no—
166C2×C8×C61—21no—
167C2×C8×C61—21no—
168C2×C8×C61—42no—
169C2×C8×C61C2×D411yescharacteristic
170C2×C8×C61C2×D411yescharacteristic
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61C2×D411yescharacteristic
176C22⋊C4×C61—21no—
177C42×C61C2411yescharacteristic
178C22⋊Dic61—21no—
179#11758202—42no—
180#11758202—42no—
181#11758198—21no—
182#11758198—21no—
183#11758198—41no—
184#11758198—41no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61—41no—
190C4⋊Dic61—41no—
191C4⋊Dic61C2×D411yescharacteristic
192#11758196—21no—
193#11758196—41no—
194#11758196—41no—
195C4×Dic61—41no—
196C4×Dic61—41no—
197C4×Dic61C2×D411yescharacteristic
198C4×Dic61C2×D411yescharacteristic
199C4×Dic61#4111yescharacteristic
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—42no—
204C2×#92418C2×D411yescharacteristic
205C2×#92418C2×D411yescharacteristic
206#11758189—42no—
207#11758189—42no—
208C8×D61—22no—
209C8×D61—41no—
210C8×D61—42no—
211C8×D61—42no—
212C8×D61—42no—
213C23×C61—21no—
214C22×D61—21no—
215C22×D61—24no—
216C22×D61—42no—
217C22×D61—42no—
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—42no—
235C2×Dic61—41no—
236C2×Dic61—41no—
237C2×Dic61#12111yescharacteristic
238C2×Dic61#12111yescharacteristic
239C2×Dic61#12111yescharacteristic
240D244—21no—
241D244C2×Q1612yes—
242C4×D61—21no—
243C4×D61—21no—
244C4×D61—41no—
245C4×D61—42no—
246C4×D61—42no—
247C4×D61—41no—
248C4×D61—41no—
249C4×D61C2×Q1612yes—
250C4×D61#13512yes—
251Dic122—41no—
252Dic122—41no—
253Dic122—81no—
254C8×C61—21no—
255C8×C61—21no—
256C8×C61—41no—
257C8×C61—41no—
258C8×C61—41no—
259#92418—21no—
260#92418—41no—
261#92418—41no—
262#92418—41no—
263#92418AGL(1,ℤ/8)12yes—
264#19256—611nomaximal, Sylow
265C22×C61—22no—
266C22×C61—41no—
267C22×C61—41no—
268C22×C61C2×#12111yescharacteristic
269C2×D61—21no—
270C2×D61—21no—
271C2×D61—22no—
272C2×D61—48no—
273C2×D61—41no—
274C2×D61—41no—
275C2×D61#46712yes—
276C4×C61—21no—
277C4×C61—41no—
278C4×C61—81no—
279C4×C61—81no—
280C4×C61C22×Q1611yescharacteristic
281C4×C61C2×#13511yescharacteristic
282C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
283C4×C61#57511yescharacteristic
284Dic61—21no—
285Dic61—21no—
286Dic61—41no—
287Dic61—81no—
288Dic61—81no—
289Dic61#46712yes—
290#3170—611no—
291#3077—611no—
292#3077—611no—
293C2×C8×D4—611no—
294#1683—611no—
295#1683—611no—
296#1683—611no—
297#1683D6111yescharacteristic, complemented
298#1675—611no—
299#1675—611no—
300#1669—611no—
301#1669—611no—
302#1567—611no—
303#1550—611no—
304#1550—611no—
305C2×C61—42no—
306C2×C61#303611yescharacteristic
307C2×C61C2×#46711yescharacteristic
308C2×C61#306711yescharacteristic
309D61—22no—
310D61—42no—
311C23×C8—1221no—
312#533—611no—
313#533—611no—
314C2×C4×D4—611no—
315#500—611no—
316#494—611no—
317#487—611no—
318#487C2×D6111yescharacteristic, complemented
319#478—1221no—
320#478—1221no—
321#478—1221no—
322#478—1221no—
323#476—611no—
324#476—611no—
325#476—611no—
326#476C2×D6111yescharacteristic
327#474—611no—
328#474—611no—
329#474—611no—
330#474C2×D6111yescharacteristic
331#467—1221no—
332#467—1221no—
333C4×Q16—1221no—
334C4×Q16—1221no—
335C8×D4—611no—
336C8×D4—611no—
337C8×D4—611no—
338C8×D4—1221no—
339C8×D4—1221no—
340C8×D4C2×D6111yescharacteristic
341#428—1221no—
342#428—1221no—
343#423—611no—
344#423—611no—
345C2×C4⋊C8—611no—
346#418—1221no—
347#418—1221no—
348C2×C22⋊C8—1221no—
349C2×C4×C8—611no—
350#335—611no—
351#335—611no—
352#335—611no—
353#335C2×D6111yescharacteristic, complemented
354#333—611no—
355#333—611no—
356#333—611no—
357#333C2×D6111yescharacteristic, complemented
358#332—611no—
359#332—611no—
360#332—611no—
361#332C2×D6111yescharacteristic, complemented
362C61#1925611yescharacteristic, complemented, Sylow
363C22×D4—611no—
364C23×C4—1221no—
365C2×Q16—1221no—
366C2×Q16—1221no—
367C2×M16—1221no—
368C2×M16—1221no—
369C22×C8—21no—
370C22×C8—611no—
371C22×C8—611no—
372C22×C8—1221no—
373C22×C8—1221no—
374C22×C8—1221no—
375C22×C8—1221no—
376C22×C8—2444no—
377C4⋊Q8—611no—
378C4⋊Q8—611no—
379C4⋊Q8C22×D6111yescharacteristic
380C4⋊Q8C22×D6111yescharacteristic
381#125—1221no—
382#125—1221no—
383C22⋊Q8—1221no—
384C22⋊Q8—1221no—
385C22⋊Q8—1221no—
386C22⋊Q8—1221no—
387C4×Q8—1221no—
388C4×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—
398#117—1221no—
399C2×C4⋊C4—611no—
400C2×C22⋊C4—1221no—
401C2×C42—611no—
402#107—21no—
403#107—1221no—
404#107—1221no—
405#107—1221no—
406#107—1221no—
407#107—1221no—
408#106—21no—
409#106—1221no—
410#106—1221no—
411#106—1221no—
412C4⋊C8—611no—
413C4⋊C8—611no—
414C4⋊C8—611no—
415C4⋊C8—1221no—
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—1221no—
435C22⋊C8—1221no—
436C22⋊C8—1221no—
437C4×C8—611no—
438C4×C8—611no—
439C4×C8—611no—
440C4×C8C22×D6111yescharacteristic
441C24—1221no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8—1221no—
445C2×Q8—1221no—
446C2×D4—614no—
447C2×D4—611no—
448C2×D4—1222no—
449C2×D4—1221no—
450C2×D4D4×D6111yescharacteristic
451C22×C4—21no—
452C22×C4—611no—
453C22×C4—611no—
454C22×C4—611no—
455C22×C4—1221no—
456C22×C4—1221no—
457C22×C4—1221no—
458C22×C4—1221no—
459C22×C4—2442no—
460C22×C4—2442no—
461Q16—2442no—
462Q16—2442no—
463M16—2442no—
464M16—2442no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—42no—
469C2×C8—611no—
470C2×C8—611no—
471C2×C8—1222no—
472C2×C8—1221no—
473C2×C8—1221no—
474C2×C8—1221no—
475C2×C8—2441no—
476C2×C8—2442no—
477C2×C8—2442no—
478C2×C8—2442no—
479C2×C8—2442no—
480C2×C8D4×D6111yescharacteristic
481C2×C8D4×D6111yescharacteristic
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—611no—
487C4⋊C4—1221no—
488C4⋊C4—1221no—
489C4⋊C4—1221no—
490C4⋊C4—1221no—
491C4⋊C4—1221no—
492C4⋊C4—2441no—
493C4⋊C4—2441no—
494C4⋊C4—2441no—
495C4⋊C4—2441no—
496C4⋊C4D4×D6111yescharacteristic
497C22⋊C4—21no—
498C22⋊C4—1221no—
499C22⋊C4—1221no—
500C22⋊C4—1221no—
501C22⋊C4—2441no—
502C22⋊C4—2441no—
503C42—611no—
504C42—611no—
505C42—611no—
506C42—2441no—
507C42—2441no—
508C42C23×D6111yescharacteristic
509C23—21no—
510C23—611no—
511C23—1224no—
512C23—1221no—
513C23—2442no—
514C23—2442no—
515Q8—41no—
516Q8—41no—
517Q8—81no—
518Q8—4881no—
519Q8—2441no—
520Q8—2441no—
521D4—21no—
522D4—612no—
523D4—1224no—
524D4—1221no—
525D4Q16×D6112yes—
526C2×C4—21no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—611no—
532C2×C4—612no—
533C2×C4—612no—
534C2×C4—611no—
535C2×C4—611no—
536C2×C4—1221no—
537C2×C4—1221no—
538C2×C4—1221no—
539C2×C4—2441no—
540C2×C4—2442no—
541C2×C4—2442no—
542C2×C4—2442no—
543C2×C4—2441no—
544C2×C4—2441no—
545C2×C4—2441no—
546C2×C4—2441no—
547C2×C4C2×D4×D6111yescharacteristic
548C2×C4C2×D4×D6111yescharacteristic
549C2×C4#41×D6111yescharacteristic
550C8—21no—
551C8—21no—
552C8—41no—
553C8—41no—
554C8—41no—
555C8—612no—
556C8—1221no—
557C8—2441no—
558C8—2441no—
559C8—2441no—
560C22—22no—
561C22—41no—
562C22—41no—
563C22—612no—
564C22—1221no—
565C22—1221no—
566C22—1222no—
567C22—2448no—
568C22—2441no—
569C22—2441no—
570C22#121×D6111yescharacteristic
571C4—21no—
572C4—41no—
573C4—81no—
574C4—81no—
575C4—612no—
576C4—1221no—
577C4—1221no—
578C4—4881no—
579C4—4881no—
580C4—2441no—
581C4C2×Q16×D6111yescharacteristic
582C4#135×D6111yescharacteristic
583C4AGL(1,ℤ/8)×D6111yescharacteristic
584C4#1684055911yescharacteristic
585C2—42no—
586C2—1222no—
587C2—2442no—
588C2#2145528111yescharacteristic
589C2#467×D6111yescharacteristic
590C2#2145553411yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1683C2
socleC22×C61C2×#121
prime radicalC2×D4×D61D4
hypercenter#1683D61
hypocenterC61#19256
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. #21456404C2semidirect
5. #21445271C2semidirect
6. #21445270C2semidirect
7. #21445267C2semidirect
8. #21445217C2semidirect
9. #21445216C2semidirect
10. #21445183C2semidirect
11. #21445182C2semidirect
12. #21444576C2semidirect
13. #21444508C2semidirect
14. #21444506C2semidirect
15. C61×#1683C2semidirect
20. #16840165C22semidirect
21. #16840141C22semidirect
22. #16840113C22semidirect
51. #16839515C22semidirect
52. #16839514C22semidirect
53. #16839514C22semidirect
54. #16839511C22semidirect
55. #16839510C22semidirect
56. #16839510C22semidirect
57. #16839509C22semidirect
58. #16839508C22semidirect
59. #16839507C22semidirect
60. C61×#487C22semidirect
64. C61×#335C22semidirect
65. C61×#333C22semidirect
66. C61×#332C22semidirect
153. D4×D61Q16semidirect
297. #1683D61semidirect
318. #487C2×D61semidirect
353. #335C2×D61semidirect
357. #333C2×D61semidirect
361. #332C2×D61semidirect
362. C61#19256semidirect

Maximal subgroups

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

subgroupindex
#214564042
#214556172
#214556112
C8×D4×D612
#214452712
#214452702
#214452672
#214452172
#214452162
#214451832
#214451822
#214445762
#214445082
#214445062
C61×#16832
#1925661