Finite Group Databasebeta

← #29669889

Subgroups — #29669889

Order 15616.

626 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458698C211yesmaximal, characteristic, complemented
2#21458604C211yesmaximal, characteristic, complemented
3#21457643C211yesmaximal, characteristic
4#21457643C211yesmaximal, characteristic
5#21445478C211yesmaximal, characteristic, complemented
6#21445358C211yesmaximal, characteristic, complemented
7#21445268C211yesmaximal, characteristic, complemented
8#21445234C211yesmaximal, characteristic, complemented
9#21445192C211yesmaximal, characteristic, complemented
10#21444955C211yesmaximal, characteristic, complemented
11#21444955C211yesmaximal, characteristic, complemented
12#21444912C211yesmaximal, characteristic, complemented
13#21444652C211yesmaximal, characteristic, complemented
14#21444628C211yesmaximal, characteristic, complemented
15C61×#1639C211yesmaximal, characteristic
16#16840487C2211yescharacteristic, complemented
17#16840469C2211yescharacteristic
18#16840469C2211yescharacteristic
19#16840163—21no—
20#16840158C2211yescharacteristic, complemented
21#16840142C2211yescharacteristic, complemented
22#16840129—21no—
23#16840112C2211yescharacteristic, complemented
24#16840108C2211yescharacteristic
25#16840108C2211yescharacteristic
26#16840078—21no—
27#16840078—21no—
28#16840071—21no—
29#16840068—21no—
30#16840060—21no—
31#16840058C2211yescharacteristic, complemented
32#16840058C2211yescharacteristic, complemented
33#16840052C2211yescharacteristic
34#16840052C2211yescharacteristic
35#16839989—21no—
36#16839989—21no—
37#16839965—21no—
38#16839959C2211yescharacteristic
39#16839959C2211yescharacteristic
40#16839913—21no—
41#16839913—21no—
42#16839889C2211yescharacteristic, complemented
43#16839886C2211yescharacteristic, complemented
44#16839863—21no—
45#16839853—21no—
46C4×#11758200C2211yescharacteristic
47C4×#11758200C2211yescharacteristic
48D4×#92418C2211yescharacteristic
49D4×#92418C2211yescharacteristic
50C2×#14519698—21no—
51C2×#14519698—21no—
52#16839514C2211yescharacteristic, complemented
53#16839512C2211yescharacteristic, complemented
54#16839510C2211yescharacteristic, complemented
55#16839510C2211yescharacteristic, complemented
56#16839508C2211yescharacteristic, complemented
57#16839494C2211yescharacteristic, complemented
58#16839494C2211yescharacteristic, complemented
59#16839484C2211yescharacteristic, complemented
60D42×C61C2211yescharacteristic
61C61×#486C2211yescharacteristic
62C61×#474C2211yescharacteristic
63C61×#461C2211yescharacteristic
64C61×#331C2211yescharacteristic
65C61×D4⋊C8C2211yescharacteristic
66C61×D4⋊C8C2211yescharacteristic
67C2×Dic244—41no—
68C2×#11758200—21no—
69C2×#11758200—21no—
70C2×#11758200D411yescharacteristic
71C2×#11758200D411yescharacteristic
72C22×#92418—21no—
73C22×#92418—21no—
74#14519777C2311yescharacteristic
75#14519777C2311yescharacteristic
76#14519775—21no—
77#14519768—21no—
78#14519768—21no—
79#14519768—21no—
80#14519767—41no—
81#14519756—21no—
82#14519756—21no—
83#14519754—41no—
84#14519754—41no—
85Q8×Dic61—21no—
86C4×Dic122C2311yescharacteristic
87C4×Dic122C2311yescharacteristic
88#14519742—21no—
89D4×Dic61—21no—
90D4×Dic61—21no—
91C2×C4⋊Dic61—21no—
92C2×C4⋊Dic61—21no—
93#14519711—21no—
94#14519711—21no—
95#14519708—21no—
96#14519708—21no—
97#14519708—21no—
98#14519708—21no—
99#14519708D411yescharacteristic
100#14519708D411yescharacteristic
101C4⋊#92418C2311yescharacteristic
102C4⋊#92418C2311yescharacteristic
103#14519706—21no—
104#14519706—21no—
105#14519702—21no—
106#14519702—21no—
107#14519701—21no—
108#14519701—21no—
109#14519700D411yescharacteristic
110#14519700D411yescharacteristic
111#14519698—21no—
112#14519698—21no—
113#14519698—21no—
114#14519698—21no—
115#14519698—21no—
116#14519698—21no—
117#14519698—21no—
118#14519698—21no—
119#14519698D411yescharacteristic
120#14519698D411yescharacteristic
121C22⋊#92418—21no—
122C22⋊#92418—21no—
123#14519687—21no—
124C4×#92418C2311yescharacteristic
125C4×#92418C2311yescharacteristic
126C8×Dic61—21no—
127C22×D4×C61—21no—
128C22×D4×C61—21no—
129C2×SD16×C61—21no—
130C4⋊Q8×C61C2311yescharacteristic
131#127×C61C2311yescharacteristic
132#121×C61—21no—
133#121×C61—21no—
134#120×C61—41no—
135C4×D4×C61C2311yescharacteristic
136C4×D4×C61C2311yescharacteristic
137#107×C61—21no—
138C4⋊C8×C61C2311yescharacteristic
139C4⋊C8×C61C2311yescharacteristic
140#103×C61—21no—
141#102×C61—21no—
142#102×C61—41no—
143C4×C8×C61C2311yescharacteristic
144C24×C61—41no—
145C2×Q8×C61—21no—
146C2×D4×C61—21no—
147C2×D4×C61—21no—
148C2×D4×C61—21no—
149C2×D4×C61—41no—
150C2×D4×C61—41no—
151C2×D4×C61—42no—
152C2×D4×C61—42no—
153C2×D4×C61—81no—
154C2×D4×C61—81no—
155C2×D4×C61C2×D411yescharacteristic
156C2×D4×C61C2×D411yescharacteristic
157C22×C4×C61—21no—
158C22×C4×C61—21no—
159C22×Dic61—41no—
160C22×Dic61—41no—
161C2×Dic122—21no—
162C2×Dic122—21no—
163C2×Dic122C2×D411yescharacteristic
164C2×Dic122C2×D411yescharacteristic
165SD16×C61—42no—
166C2×C8×C61—21no—
167C2×C8×C61—21no—
168C2×C8×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61C2×D411yescharacteristic
172C4⋊C4×C61C2×D411yescharacteristic
173C22⋊C4×C61—21no—
174C22⋊C4×C61—21no—
175C22⋊C4×C61—41no—
176C42×C61C2411yescharacteristic
177C22⋊Dic61—41no—
178C22⋊Dic61—41no—
179#11758200—22no—
180#11758200—22no—
181#11758200—81no—
182#11758200—81no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—41no—
188C4⋊Dic61—41no—
189C4⋊Dic61—41no—
190C4⋊Dic61C2×D411yescharacteristic
191C4⋊Dic61C2×D411yescharacteristic
192#11758196—21no—
193#11758196—21no—
194#11758196—41no—
195#11758196—41no—
196#11758196—41no—
197#11758196—41no—
198C4×Dic61—21no—
199C4×Dic61—21no—
200C4×Dic61—41no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—81no—
204C2×#92418—81no—
205C2×#92418C2×D411yescharacteristic
206C2×#92418C2×D411yescharacteristic
207C2×#92418C2×D411yescharacteristic
208C2×#92418C2×D411yescharacteristic
209Dic244—82no—
210C23×C61—21no—
211C23×C61—21no—
212C23×C61—41no—
213C23×C61—82no—
214C23×C61—82no—
215C23×C61—161no—
216Q8×C61—41no—
217Q8×C61—41no—
218D4×C61—21no—
219D4×C61—21no—
220D4×C61—41no—
221D4×C61—41no—
222D4×C61—41no—
223D4×C61—41no—
224D4×C61—81no—
225D4×C61—82no—
226D4×C61—82no—
227D4×C61#13512yes—
228D4×C61#13712yes—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61C22×D411yescharacteristic
237C2×C4×C61C22×D411yescharacteristic
238C2×C4×C61H(5,𝔽2)11yescharacteristic
239C2×Dic61—21no—
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—21no—
243C2×Dic61—41no—
244C2×Dic61—41no—
245C2×Dic61—161no—
246C2×Dic61—161no—
247Dic122—21no—
248Dic122—21no—
249Dic122—21no—
250Dic122—21no—
251Dic122—81no—
252Dic122—81no—
253C8×C61—41no—
254C8×C61—41no—
255C8×C61—41no—
256#92418—21no—
257#92418—21no—
258#92418—21no—
259#92418—21no—
260#92418—41no—
261#92418—41no—
262#19790—611nomaximal, Sylow
263C22×C61—22no—
264C22×C61—22no—
265C22×C61—41no—
266C22×C61—41no—
267C22×C61—41no—
268C22×C61—41no—
269C22×C61—81no—
270C22×C61—81no—
271C22×C61—81no—
272C22×C61—164no—
273C22×C61D4211yescharacteristic
274C4×C61—21no—
275C4×C61—41no—
276C4×C61—41no—
277C4×C61—81no—
278C4×C61C2×#13511yescharacteristic
279C4×C61C2×#13711yescharacteristic
280C4×C61#57711yescharacteristic
281C4×C61#57711yescharacteristic
282Dic61—41no—
283Dic61—41no—
284Dic61—41no—
285Dic61—41no—
286Dic61—81no—
287Dic61—81no—
288#3381—611no—
289#3372—611no—
290#3295—611no—
291#3295—611no—
292#1720—611no—
293#1698—611no—
294#1683—611no—
295#1678—611no—
296#1671—611no—
297#1639D6111yescharacteristic
298#1632—611no—
299#1632—611no—
300#1626—611no—
301#1582—611no—
302SD16⋊C8—611no—
303C2×C61—42no—
304C2×C61—42no—
305C2×C61—161no—
306C2×C61#326611yescharacteristic
307C2×C61#328211yescharacteristic
308C2×C61#328811yescharacteristic
309#557—611no—
310#554—611no—
311#554—611no—
312D42C2×D6111yescharacteristic
313#499—1221no—
314#498—611no—
315#494—611no—
316#491—1221no—
317#487—611no—
318#486—611no—
319#486—611no—
320#486C2×D6111yescharacteristic
321#481—1221no—
322#481—1221no—
323#479—1221no—
324#479—1221no—
325#478—1221no—
326#477—611no—
327#477—611no—
328#476—611no—
329#476—611no—
330#474C2×D6111yescharacteristic
331#465—1221no—
332#465—1221no—
333#461—1221no—
334#461C2×D6111yescharacteristic
335#460—611no—
336#460—611no—
337#450—1221no—
338#450—1221no—
339#446—611no—
340C8×Q8—611no—
341#440—1221no—
342C4×SD16—611no—
343C4×SD16—611no—
344C4×SD16—1221no—
345C8×D4—611no—
346C8×D4—611no—
347C2×#102—1221no—
348C2×#102—1221no—
349#335—611no—
350#334—611no—
351#333—611no—
352#333—611no—
353#332—611no—
354#331C2×D6111yescharacteristic
355Q8⋊C8—611no—
356Q8⋊C8—611no—
357D4⋊C8C2×D6111yescharacteristic
358D4⋊C8C2×D6111yescharacteristic
359#322—611no—
360C61#1979011yescharacteristic, complemented, Sylow
361C22×D4—21no—
362C22×D4—21no—
363C2×Q16—2441no—
364C2×SD16—21no—
365C2×SD16—611no—
366C2×SD16—611no—
367C2×SD16—1221no—
368C2×SD16—1221no—
369C22×C8—1221no—
370C22×C8—1221no—
371C4⋊Q8—611no—
372C4⋊Q8—611no—
373C4⋊Q8—1221no—
374C4⋊Q8C22×D6111yescharacteristic
375#127C22×D6111yescharacteristic
376#125—1221no—
377#125—1221no—
378#125—1221no—
379#125—2441no—
380C22⋊Q8—1221no—
381C22⋊Q8—1221no—
382C22⋊Q8—2441no—
383C22⋊Q8—2441no—
384#121—21no—
385#121—21no—
386#120—41no—
387C4×Q8—611no—
388C4×Q8—611no—
389C4×Q8—1221no—
390C4×Q8—1221no—
391C4×D4—1221no—
392C4×D4—1221no—
393C4×D4C22×D6111yescharacteristic
394C4×D4C22×D6111yescharacteristic
395C2×C4⋊C4—1221no—
396C2×C4⋊C4—1221no—
397#107—21no—
398#107—1221no—
399#107—1221no—
400#106—611no—
401#106—611no—
402#106—1221no—
403#106—1221no—
404#106—1221no—
405#106—1221no—
406C4⋊C8—611no—
407C4⋊C8—611no—
408C4⋊C8—1221no—
409C4⋊C8—1221no—
410C4⋊C8C22×D6111yescharacteristic
411C4⋊C8C22×D6111yescharacteristic
412#103—21no—
413#103—611no—
414#103—611no—
415#103—1221no—
416#103—1221no—
417#103—1221no—
418#103—1221no—
419#102—21no—
420#102—41no—
421#102—611no—
422#102—611no—
423#102—1221no—
424#102—1221no—
425#102—1221no—
426#102—1221no—
427#102—1221no—
428#102—1221no—
429#102—1221no—
430#102—1221no—
431C22⋊C8—1221no—
432C22⋊C8—1221no—
433#97—1221no—
434C4×C8—611no—
435C4×C8—611no—
436C4×C8—1221no—
437C4×C8C22×D6111yescharacteristic
438C24—41no—
439C2×Q8—21no—
440C2×Q8—611no—
441C2×Q8—611no—
442C2×Q8—1221no—
443C2×Q8—1221no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—41no—
448C2×D4—41no—
449C2×D4—42no—
450C2×D4—42no—
451C2×D4—81no—
452C2×D4—81no—
453C2×D4D4×D6111yescharacteristic
454C2×D4D4×D6111yescharacteristic
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—2441no—
458C22×C4—2441no—
459Q16—4882no—
460SD16—42no—
461SD16—1222no—
462SD16—1222no—
463SD16—4881no—
464SD16—4881no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—611no—
469C2×C8—611no—
470C2×C8—611no—
471C2×C8—611no—
472C2×C8—1221no—
473C2×C8—1221no—
474C2×C8—4881no—
475C2×C8—4881no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—611no—
479C4⋊C4—611no—
480C4⋊C4—1221no—
481C4⋊C4—1221no—
482C4⋊C4—1221no—
483C4⋊C4—1221no—
484C4⋊C4—1221no—
485C4⋊C4—1221no—
486C4⋊C4—2441no—
487C4⋊C4—2441no—
488C4⋊C4—2441no—
489C4⋊C4—2441no—
490C4⋊C4—2441no—
491C4⋊C4—2441no—
492C4⋊C4—2441no—
493C4⋊C4D4×D6111yescharacteristic
494C4⋊C4D4×D6111yescharacteristic
495C22⋊C4—21no—
496C22⋊C4—21no—
497C22⋊C4—41no—
498C22⋊C4—2441no—
499C22⋊C4—2441no—
500C42—1221no—
501C42—1221no—
502C42—2441no—
503C42C23×D6111yescharacteristic
504C23—21no—
505C23—21no—
506C23—41no—
507C23—82no—
508C23—82no—
509C23—161no—
510Q8—41no—
511Q8—41no—
512Q8—1221no—
513Q8—1221no—
514Q8—1221no—
515Q8—1221no—
516Q8—4881no—
517Q8—4881no—
518D4—21no—
519D4—21no—
520D4—41no—
521D4—41no—
522D4—41no—
523D4—41no—
524D4—81no—
525D4—82no—
526D4—82no—
527D4#1451979612yes—
528D4#1451980612yes—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—1221no—
537C2×C4—1221no—
538C2×C4—1221no—
539C2×C4—1221no—
540C2×C4—9761no—
541C2×C4—9761no—
542C2×C4—2441no—
543C2×C4—2441no—
544C2×C4C2×D4×D6111yescharacteristic
545C2×C4C2×D4×D6111yescharacteristic
546C2×C4#1451982211yescharacteristic
547C8—41no—
548C8—41no—
549C8—41no—
550C8—1221no—
551C8—1221no—
552C8—1221no—
553C8—1221no—
554C8—2441no—
555C8—2441no—
556C22—22no—
557C22—22no—
558C22—41no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22—81no—
563C22—81no—
564C22—81no—
565C22—164no—
566C22#1684039411yescharacteristic
567C4—21no—
568C4—41no—
569C4—41no—
570C4—81no—
571C4—4881no—
572C4—4881no—
573C4—2441no—
574C4—2441no—
575C4—2441no—
576C4—2441no—
577C4C2×#1451979611yescharacteristic
578C4C2×#1451980611yescharacteristic
579C4#1684057211yescharacteristic
580C4#1684057211yescharacteristic
581C2—42no—
582C2—42no—
583C2—161no—
584C2#2145733311yescharacteristic
585C2#2145753011yescharacteristic
586C2#2145758811yescharacteristic
subgroupG/H
centerC22#16840394
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1639C2
socleC22×C61D42
prime radicalD42×C61C22
hypercenter#1639D61
hypocenterC61#19790
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1639

Complements

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

NH ≅ G/Ntype
1. #21458698C2semidirect
2. #21458604C2semidirect
5. #21445478C2semidirect
6. #21445358C2semidirect
7. #21445268C2semidirect
8. #21445234C2semidirect
9. #21445192C2semidirect
10. #21444955C2semidirect
11. #21444955C2semidirect
12. #21444912C2semidirect
13. #21444652C2semidirect
14. #21444628C2semidirect
16. #16840487C22semidirect
20. #16840158C22semidirect
21. #16840142C22semidirect
23. #16840112C22semidirect
31. #16840058C22semidirect
32. #16840058C22semidirect
42. #16839889C22semidirect
43. #16839886C22semidirect
52. #16839514C22semidirect
53. #16839512C22semidirect
54. #16839510C22semidirect
55. #16839510C22semidirect
56. #16839508C22semidirect
57. #16839494C22semidirect
58. #16839494C22semidirect
59. #16839484C22semidirect
360. C61#19790semidirect

Maximal subgroups

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

subgroupindex
#214586982
#214586042
#214576432
#214454782
#214453582
#214452682
#214452342
#214451922
#214449552
#214449122
#214446522
#214446282
C61×#16392
#1979061