Finite Group Databasebeta

← #29663287

Subgroups — #29663287

Order 15616.

646 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456969C211yesmaximal, characteristic, complemented
2#21456759C211yesmaximal, characteristic
3#486×D61C211yesmaximal, characteristic
4#21454750C211yesmaximal, characteristic, complemented
5#21445360C211yesmaximal, characteristic, complemented
6#21445359C211yesmaximal, characteristic, complemented
7#21445356C211yesmaximal, characteristic, complemented
8#21445238C211yesmaximal, characteristic, complemented
9#21445237C211yesmaximal, characteristic, complemented
10#21445234C211yesmaximal, characteristic, complemented
11#21445233C211yesmaximal, characteristic, complemented
12#21444566C211yesmaximal, characteristic, complemented
13#21444498C211yesmaximal, characteristic, complemented
14#21444485C211yesmaximal, characteristic
15C61×#1698C211yesmaximal, characteristic, complemented
16#16840316C2211yescharacteristic
17C4⋊Q8×D61C2211yescharacteristic
18#127×D61C2211yescharacteristic
19#16840166—21no—
20#16840133—21no—
21#16840111—21no—
22#16840110C2211yescharacteristic
23#16840109C2211yescharacteristic
24#16840108C2211yescharacteristic
25#16840107—21no—
26#16840107—21no—
27#16840107—21no—
28#16840099—21no—
29#16840096—21no—
30#16840058C2211yescharacteristic, complemented
31#16840057C2211yescharacteristic, complemented
32#16840054C2211yescharacteristic, complemented
33#16839963C2211yescharacteristic
34#16839962C2211yescharacteristic
35#16839959C2211yescharacteristic
36#16839914—21no—
37#16839911—21no—
38#16839892C2211yescharacteristic, complemented
39#16839890—21no—
40Q8×#92418C2211yescharacteristic, complemented
41#16839886—21no—
42#16839886C2211yescharacteristic, complemented
43#16839814C2211yescharacteristic
44#16839814C2211yescharacteristic
45#16839787—21no—
46#16839784C2211yescharacteristic
47#102×D61—21no—
48#102×D61—21no—
49#16839719—21no—
50C4×C8×D61C2211yescharacteristic
51#16839515C2211yescharacteristic, 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×#477C2211yescharacteristic, complemented
62C61×#460C2211yescharacteristic
63C61×#446C2211yescharacteristic, complemented
64C61×#335C2211yescharacteristic, complemented
65C61×#333C2211yescharacteristic, complemented
66C61×#331C2211yescharacteristic
67C2×Q8×D61—21no—
68C2×D4×D61—21no—
69C2×D4×D61—21no—
70C2×D4×D61—21no—
71C2×#11758190—21no—
72C2×#11758200—21no—
73C2×#11758201—21no—
74C2×#11758189—21no—
75C2×C8×D61D411yescharacteristic
76C2×C8×D61D411yescharacteristic
77#14519777C2311yescharacteristic
78#14519776—21no—
79#14519776—21no—
80#14519776D411yescharacteristic
81#14519775—21no—
82#14519775D411yescharacteristic
83#14519774C2311yescharacteristic
84#14519773—21no—
85#14519773—21no—
86#14519773—21no—
87#14519767—21no—
88#14519757D411yescharacteristic
89#14519755—21no—
90#14519753D411yescharacteristic
91C4×Dic122—21no—
92C4×Dic122C2311yescharacteristic
93#14519734—21no—
94C4⋊C4×D61—21no—
95C4⋊C4×D61—21no—
96C42×D61C2311yescharacteristic
97#14519711—21no—
98#14519710—21no—
99#14519710—21no—
100C4⋊#92418—21no—
101C4⋊#92418C2311yescharacteristic
102C4⋊#92418C2311yescharacteristic
103C4⋊#92418C2311yescharacteristic
104#14519706—21no—
105#14519706—21no—
106#14519706—21no—
107#14519702—21no—
108#14519701—21no—
109#14519700—21no—
110#14519699—21no—
111#14519699—21no—
112#14519699—21no—
113#14519698—21no—
114#14519698—21no—
115#14519698—21no—
116#14519697—21no—
117#14519697—21no—
118#14519697—21no—
119#14519689—21no—
120#14519689—21no—
121#14519687—21no—
122C4×#92418—21no—
123C4×#92418C2311yescharacteristic
124C8×Dic61—21no—
125C8×Dic61—21no—
126C8×Dic61—21no—
127C8×Dic61D411yescharacteristic
128C8×Dic61D411yescharacteristic
129C2×SD16×C61—21no—
130C4⋊Q8×C61C2311yescharacteristic
131#127×C61C2311yescharacteristic
132C4×Q8×C61C2311yescharacteristic
133#107×C61—21no—
134C4⋊C8×C61—21no—
135C4⋊C8×C61C2311yescharacteristic
136C4⋊C8×C61C2311yescharacteristic
137C4⋊C8×C61C2311yescharacteristic
138#103×C61—21no—
139#102×C61—21no—
140#102×C61—21no—
141#102×C61—21no—
142#97×C61—21no—
143C4×C8×C61C2311yescharacteristic
144C23×D61—41no—
145C2×Q8×C61—21no—
146C2×Q8×C61C2×D411yescharacteristic
147C2×D4×C61—21no—
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C2×C22⋊D61—41no—
151C2×C22⋊D61—41no—
152C2×C22⋊D61—41no—
153Q8×D61—82no—
154D4×D61—42no—
155D4×D61—42no—
156D4×D61—81no—
157D4×D61—81no—
158D4×D61—82no—
159C2×D244—21no—
160C2×D244—21no—
161C2×D244—21no—
162C2×C4×D61—41no—
163C2×C4×D61C2×D411yescharacteristic
164C2×C4×D61C2×D411yescharacteristic
165C2×C4×D61#4111yescharacteristic
166C2×Dic122—21no—
167C2×Dic122—21no—
168C2×Dic122—41no—
169C2×Dic122C2×D411yescharacteristic
170SD16×C61—81no—
171C2×C8×C61—21no—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61C2×D411yescharacteristic
175C2×C8×C61C2×D411yescharacteristic
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61C2×D411yescharacteristic
180C42×C61—21no—
181C42×C61C2411yescharacteristic
182#11758201—81no—
183#11758200—81no—
184#11758198—21no—
185#11758198—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61C2×D411yescharacteristic
190#11758196—21no—
191#11758196—21no—
192#11758196—21no—
193#11758196—21no—
194#11758196—21no—
195#11758196—41no—
196#11758196—41no—
197C4×Dic61—21no—
198C4×Dic61C2×D411yescharacteristic
199C4×Dic61C2×D411yescharacteristic
200C4×Dic61#4111yescharacteristic
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418C2×D411yescharacteristic
205C2×#92418C2×D411yescharacteristic
206#11758190—81no—
207#11758189—42no—
208C8×D61—22no—
209C8×D61—22no—
210C23×C61—41no—
211C22×D61—41no—
212C22×D61—82no—
213C22×D61—82no—
214C22×D61—82no—
215C22×D61#12111yescharacteristic
216Q8×C61—21no—
217Q8×C61—21no—
218Q8×C61—81no—
219D4×C61—41no—
220D4×C61—41no—
221D4×C61—41no—
222D4×C61—41no—
223D4×C61—81no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—41no—
227C2×C4×C61C22×D411yescharacteristic
228C2×C4×C61C22×D411yescharacteristic
229C2×C4×C61C2×#4111yescharacteristic
230C22⋊D61—82no—
231C22⋊D61—82no—
232C22⋊D61—82no—
233C2×Dic61—21no—
234C2×Dic61—21no—
235C2×Dic61—41no—
236C2×Dic61#12111yescharacteristic
237C2×Dic61#12111yescharacteristic
238C2×Dic61#12111yescharacteristic
239D244—41no—
240D244—41no—
241D244—41no—
242D244—41no—
243D244—81no—
244C4×D61—21no—
245C4×D61—21no—
246C4×D61—22no—
247C4×D61—82no—
248C4×D61#13512yes—
249C4×D61#13712yes—
250Dic122—21no—
251Dic122—21no—
252Dic122—42no—
253Dic122—82no—
254Dic122—81no—
255C8×C61—21no—
256C8×C61—21no—
257C8×C61—41no—
258C8×C61—41no—
259C8×C61—41no—
260#92418—21no—
261#92418—21no—
262#92418—41no—
263#92418—41no—
264#92418—41no—
265#19212—611nomaximal, Sylow
266C22×C61—81no—
267C22×C61—81no—
268C22×C61—81no—
269C22×C61C2×#12111yescharacteristic
270C2×D61—21no—
271C2×D61—21no—
272C2×D61—81no—
273C2×D61—81no—
274C2×D61—81no—
275C2×D61—164no—
276C2×D61#46412yes—
277C4×C61—21no—
278C4×C61—41no—
279C4×C61—41no—
280C4×C61—81no—
281C4×C61C2×#13511yescharacteristic
282C4×C61C2×#13711yescharacteristic
283C4×C61#57711yescharacteristic
284C4×C61#57711yescharacteristic
285Dic61—21no—
286Dic61—41no—
287Dic61—41no—
288Dic61—81no—
289Dic61#46412yes—
290Dic61#47212yes—
291#3230—611no—
292#3208—611no—
293C2×#486—611no—
294#2960—611no—
295#1698—611no—
296#1698—611no—
297#1698—611no—
298#1698D6111yescharacteristic, complemented
299#1678—611no—
300#1678—611no—
301#1678—611no—
302#1678—611no—
303#1564—611no—
304#1548—611no—
305#1545—611no—
306C2×C61—161no—
307C2×C61C2×#46411yescharacteristic
308C2×C61#305711yescharacteristic
309C2×C61C2×#47211yescharacteristic
310D61—22no—
311D61—161no—
312#533—611no—
313C2×C4⋊Q8—611no—
314C2×#127—611no—
315#500—1221no—
316#492—1221no—
317#487—1221no—
318#486—611no—
319#486—611no—
320#486—611no—
321#486—1221no—
322#486—1221no—
323#486—1221no—
324#486C2×D6111yescharacteristic
325#484—1221no—
326#484—1221no—
327#477—611no—
328#477—611no—
329#477—611no—
330#477C2×D6111yescharacteristic, complemented
331#460—611no—
332#460—611no—
333#460—611no—
334#460C2×D6111yescharacteristic
335#450—1221no—
336#450—1221no—
337#446—611no—
338#446—1221no—
339#446C2×D6111yescharacteristic, complemented
340C8×Q8—611no—
341C8×Q8—611no—
342C8×Q8—1221no—
343#431—611no—
344#431—611no—
345#424—1221no—
346#423—611no—
347C2×#102—1221no—
348C2×#102—1221no—
349#405—1221no—
350C2×C4×C8—611no—
351#335—611no—
352#335C2×D6111yescharacteristic, complemented
353#334—611no—
354#334—611no—
355#333—611no—
356#333—611no—
357#333—611no—
358#333C2×D6111yescharacteristic, complemented
359#331—611no—
360#331—611no—
361#331—611no—
362#331C2×D6111yescharacteristic
363C61#1921211yescharacteristic, complemented, Sylow
364C22×Q8—1221no—
365C22×D4—1221no—
366C22×D4—1221no—
367C22×D4—1221no—
368C2×SD16—21no—
369C2×SD16—1221no—
370C2×SD16—1221no—
371C2×SD16—1221no—
372C2×M16—1221no—
373C22×C8—611no—
374C22×C8—611no—
375C4⋊Q8—611no—
376C4⋊Q8—611no—
377C4⋊Q8—611no—
378C4⋊Q8—1221no—
379C4⋊Q8—1221no—
380C4⋊Q8—1221no—
381C4⋊Q8C22×D6111yescharacteristic
382#127—611no—
383#127—1221no—
384#127—1221no—
385#127—1221no—
386#127C22×D6111yescharacteristic
387#125—1221no—
388C22⋊Q8—611no—
389C22⋊Q8—611no—
390C22⋊Q8—1221no—
391C4×Q8—611no—
392C4×Q8—1221no—
393C4×Q8C22×D6111yescharacteristic
394#117—1221no—
395C2×C4⋊C4—1221no—
396C2×C4⋊C4—1221no—
397C2×C42—611no—
398#107—21no—
399#107—1221no—
400#107—1221no—
401#107—1221no—
402C4⋊C8—21no—
403C4⋊C8—611no—
404C4⋊C8—611no—
405C4⋊C8—611no—
406C4⋊C8—1221no—
407C4⋊C8—1221no—
408C4⋊C8—1221no—
409C4⋊C8—1221no—
410C4⋊C8C22×D6111yescharacteristic
411C4⋊C8C22×D6111yescharacteristic
412C4⋊C8C22×D6111yescharacteristic
413#103—21no—
414#103—1221no—
415#103—1221no—
416#103—1221no—
417#102—21no—
418#102—21no—
419#102—21no—
420#102—1221no—
421#102—1221no—
422#102—1221no—
423#102—1221no—
424#102—1221no—
425#102—1221no—
426#102—1221no—
427#102—1221no—
428#102—1221no—
429C22⋊C8—1221no—
430C22⋊C8—1221no—
431#97—21no—
432#97—1221no—
433C4×C8—611no—
434C4×C8—611no—
435C4×C8—611no—
436C4×C8—1221no—
437C4×C8—1221no—
438C4×C8—1221no—
439C4×C8—1221no—
440C4×C8C22×D6111yescharacteristic
441C24—2441no—
442C2×Q8—21no—
443C2×Q8—611no—
444C2×Q8—1221no—
445C2×Q8—1221no—
446C2×Q8—4882no—
447C2×Q8—2441no—
448C2×Q8D4×D6111yescharacteristic
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—1221no—
453C2×D4—1221no—
454C2×D4—1221no—
455C2×D4—4881no—
456C2×D4—4881no—
457C2×D4—4882no—
458C2×D4—2442no—
459C2×D4—2442no—
460C2×D4—2441no—
461C2×D4—2441no—
462C2×D4—2441no—
463C22×C4—611no—
464C22×C4—611no—
465C22×C4—611no—
466C22×C4—2441no—
467SD16—81no—
468SD16—4881no—
469SD16—4881no—
470SD16—4881no—
471M16—2442no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—611no—
476C2×C8—611no—
477C2×C8—1222no—
478C2×C8—1222no—
479C2×C8—1221no—
480C2×C8—1221no—
481C2×C8—1221no—
482C2×C8D4×D6111yescharacteristic
483C2×C8D4×D6111yescharacteristic
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—611no—
488C4⋊C4—1221no—
489C4⋊C4—1221no—
490C4⋊C4—1221no—
491C4⋊C4—1221no—
492C4⋊C4—1221no—
493C4⋊C4—1221no—
494C4⋊C4—1221no—
495C4⋊C4—1221no—
496C4⋊C4—2441no—
497C4⋊C4—2441no—
498C4⋊C4D4×D6111yescharacteristic
499C22⋊C4—1221no—
500C22⋊C4—1221no—
501C42—21no—
502C42—611no—
503C42—611no—
504C42—611no—
505C42—1221no—
506C42C23×D6111yescharacteristic
507C23—41no—
508C23—611no—
509C23—4882no—
510C23—4882no—
511C23—4882no—
512C23—2441no—
513Q8—21no—
514Q8—21no—
515Q8—81no—
516Q8—1221no—
517Q8—1221no—
518Q8—4881no—
519Q8—4882no—
520Q8—2442no—
521D4—41no—
522D4—41no—
523D4—41no—
524D4—41no—
525D4—81no—
526D4—4881no—
527D4—4882no—
528D4—4882no—
529D4—4882no—
530D4—2441no—
531D4—2441no—
532D4—2441no—
533D4—2441no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—41no—
537C2×C4—611no—
538C2×C4—611no—
539C2×C4—612no—
540C2×C4—612no—
541C2×C4—611no—
542C2×C4—1222no—
543C2×C4—1221no—
544C2×C4—1221no—
545C2×C4—1221no—
546C2×C4—1221no—
547C2×C4—4882no—
548C2×C4—2441no—
549C2×C4C2×D4×D6111yescharacteristic
550C2×C4C2×D4×D6111yescharacteristic
551C2×C4#41×D6111yescharacteristic
552C8—21no—
553C8—21no—
554C8—41no—
555C8—41no—
556C8—41no—
557C8—1221no—
558C8—1221no—
559C8—2441no—
560C8—2441no—
561C8—2441no—
562C22—81no—
563C22—81no—
564C22—81no—
565C22—612no—
566C22—1221no—
567C22—1221no—
568C22—9764no—
569C22—4881no—
570C22—4881no—
571C22—4881no—
572C22#121×D6111yescharacteristic
573C4—21no—
574C4—41no—
575C4—41no—
576C4—81no—
577C4—612no—
578C4—612no—
579C4—1221no—
580C4—4881no—
581C4—2441no—
582C4—2441no—
583C4#135×D6111yescharacteristic
584C4#137×D6111yescharacteristic
585C4#1684057111yescharacteristic
586C4#1684057211yescharacteristic
587C2—161no—
588C2—1222no—
589C2—9761no—
590C2#464×D6111yescharacteristic
591C2#2145545611yescharacteristic
592C2#472×D6111yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1698C2
socleC22×C61C2×#121
prime radical#127×D61C22
hypercenter#1698D61
hypocenterC61#19212
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1698

Complements

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

NH ≅ G/Ntype
1. #21456969C2semidirect
4. #21454750C2semidirect
5. #21445360C2semidirect
6. #21445359C2semidirect
7. #21445356C2semidirect
8. #21445238C2semidirect
9. #21445237C2semidirect
10. #21445234C2semidirect
11. #21445233C2semidirect
12. #21444566C2semidirect
13. #21444498C2semidirect
15. C61×#1698C2semidirect
30. #16840058C22semidirect
31. #16840057C22semidirect
32. #16840054C22semidirect
38. #16839892C22semidirect
40. Q8×#92418C22semidirect
42. #16839886C22semidirect
51. #16839515C22semidirect
52. #16839512C22semidirect
53. #16839512C22semidirect
54. #16839511C22semidirect
55. #16839510C22semidirect
56. #16839510C22semidirect
61. C61×#477C22semidirect
63. C61×#446C22semidirect
64. C61×#335C22semidirect
65. C61×#333C22semidirect
298. #1698D61semidirect
330. #477C2×D61semidirect
339. #446C2×D61semidirect
352. #335C2×D61semidirect
358. #333C2×D61semidirect
363. C61#19212semidirect

Maximal subgroups

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

subgroupindex
#214569692
#214567592
#486×D612
#214547502
#214453602
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