Finite Group Databasebeta

← #30612887

Subgroups — #30612887

Order 15616.

606 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#21462727C211yesmaximal, characteristic, complemented
2#21462682C211yesmaximal, characteristic
3#21461461C211yesmaximal, characteristic, complemented
4#21461460C211yesmaximal, characteristic, complemented
5#21461456C211yesmaximal, characteristic, complemented
6#21461454C211yesmaximal, characteristic, complemented
7#21461449C211yesmaximal, characteristic, complemented
8#21461447C211yesmaximal, characteristic, complemented
9#21461429C211yesmaximal, characteristic, complemented
10#21461427C211yesmaximal, characteristic, complemented
11#21461391C211yesmaximal, characteristic, complemented
12#21461159C211yesmaximal, characteristic
13#21461143C211yesmaximal, characteristic, complemented
14#21461037C211yesmaximal, characteristic, complemented
15#21455829C211yesmaximal, characteristic
16#16840813—21no—
17#16840813—21no—
18#16840812C2211yescharacteristic, complemented
19#16840812C2211yescharacteristic, complemented
20#16840811C2211yescharacteristic, complemented
21#16840811C2211yescharacteristic, complemented
22#16840797C2211yescharacteristic
23#16840797C2211yescharacteristic
24#16840796C2211yescharacteristic
25#16840796C2211yescharacteristic
26#16840795—21no—
27#16840795—21no—
28#16840787—21no—
29#16840787—21no—
30#16840784C2211yescharacteristic
31#16840781C2211yescharacteristic
32#16840780C2211yescharacteristic
33#16840734—21no—
34#16840733—21no—
35C4×#11758213C2211yescharacteristic
36#16840626C2211yescharacteristic, complemented
37#16840625C2211yescharacteristic, complemented
38#16840624C2211yescharacteristic, complemented
39#16840623C2211yescharacteristic, complemented
40#16840612C2211yescharacteristic, complemented
41#16840612C2211yescharacteristic, complemented
42#16840612C2211yescharacteristic, complemented
43#16840612C2211yescharacteristic, complemented
44#16840610C2211yescharacteristic
45#16840610C2211yescharacteristic
46D4⋊#92420C2211yescharacteristic
47D4⋊#92420C2211yescharacteristic
48#92420⋊C8C2211yescharacteristic, complemented
49#92420⋊C8C2211yescharacteristic, complemented
50#16840603C2211yescharacteristic, complemented
51C8×#92420C2211yescharacteristic, complemented
52#16840317C2211yescharacteristic
53C4×D4×D61C2211yescharacteristic
54#16840094C411yescharacteristic
55#16840094C411yescharacteristic
56#16840049C411yescharacteristic
57#16839968C411yescharacteristic
58#16839967C411yescharacteristic
59#16839914C411yescharacteristic
60#16839914C411yescharacteristic
61#16839853C2211yescharacteristic
62#16839853C2211yescharacteristic
63#16839849C2211yescharacteristic
64#16839849C2211yescharacteristic
65#16839814C2211yescharacteristic
66C61×#476C411yescharacteristic
67C2×#11758213—21no—
68C2×#11758213—21no—
69C2×#11758213—21no—
70C2×#11758217—21no—
71C2×#11758217—21no—
72#14519844—21no—
73#14519844—21no—
74#14519844D411yescharacteristic
75#14519844D411yescharacteristic
76C4⋊#92420—21no—
77C4⋊#92420—21no—
78C4⋊#92420C2311yescharacteristic
79C4⋊#92420C2311yescharacteristic
80C4⋊#92420C2311yescharacteristic
81C4⋊#92420C2311yescharacteristic
82#14519834—21no—
83#14519834—21no—
84#14519834—21no—
85#14519834—21no—
86#14519834D411yescharacteristic
87#14519834D411yescharacteristic
88C22⋊#92420—21no—
89C22⋊#92420—21no—
90#14519832—21no—
91#14519832—21no—
92#14519832—21no—
93#14519832—21no—
94C4×#92420—21no—
95C4×#92420—21no—
96C4×#92420—21no—
97C4×#92420C2311yescharacteristic
98C4×#92420C2311yescharacteristic
99C4×#92420C2311yescharacteristic
100C4×#92420C2311yescharacteristic
101C2×D4×D61C2×C411yescharacteristic
102C22×C4×D61—21no—
103C2×#11758190C2×C411yescharacteristic
104C2×#11758190C2×C411yescharacteristic
105C2×#11758200C2×C411yescharacteristic
106C2×#11758200C2×C411yescharacteristic
107#14519777C2×C411yescharacteristic
108#14519768—21no—
109#14519757C2×C411yescharacteristic
110#14519757C2×C411yescharacteristic
111#14519753—21no—
112Q8×Dic61—21no—
113#14519742C2311yescharacteristic
114#14519742C2311yescharacteristic
115C4×D244C2×C411yescharacteristic
116D4×Dic61C2311yescharacteristic
117#14519739C2311yescharacteristic
118C4×C22⋊D61—21no—
119#14519737—21no—
120#14519735—21no—
121C4⋊C4×D61C2×C411yescharacteristic
122C22⋊C4×D61—21no—
123C42×D61C2311yescharacteristic
124#14519708C2×C411yescharacteristic
125#14519708C2×C411yescharacteristic
126C4⋊#92418C2×C411yescharacteristic
127#14519702C2×C411yescharacteristic
128#14519702C2×C411yescharacteristic
129#14519700C2×C411yescharacteristic
130#14519700C2×C411yescharacteristic
131#14519697C2×C411yescharacteristic
132#14519697C2×C411yescharacteristic
133#14519689C2×C411yescharacteristic
134#14519689C2×C411yescharacteristic
135C8×Dic61C2311yescharacteristic
136C8×Dic61C2311yescharacteristic
137C4⋊Q8×C61C2×C411yescharacteristic
138C4×D4×C61C2×C411yescharacteristic
139#106×C61C2×C411yescharacteristic
140#106×C61C2×C411yescharacteristic
141C4⋊C8×C61C2×C411yescharacteristic
142#102×C61C2×C411yescharacteristic
143#102×C61C2×C411yescharacteristic
144C23×D61—21no—
145C2×Q8×C61—21no—
146C2×D4×C61C22×C411yescharacteristic
147C22×C4×C61—21no—
148C2×C22⋊D61—21no—
149C22×Dic61—21no—
150D4×D61—41no—
151D4×D61—41no—
152C2×D244C22×C411yescharacteristic
153C2×C4×D61—21no—
154C2×C4×D61—21no—
155C2×C4×D61—21no—
156C2×C4×D61—81no—
157C2×C4×D61—81no—
158C2×C4×D61C22×C411yescharacteristic
159C2×C4×D61C2×D411yescharacteristic
160C2×C4×D61#4111yescharacteristic
161C2×Dic122C22×C411yescharacteristic
162C2×Dic122C22×C411yescharacteristic
163#11758217—81no—
164#11758217—81no—
165C2×#92420—21no—
166C2×#92420—21no—
167C2×#92420—21no—
168C2×#92420—21no—
169C2×#92420—21no—
170C2×#92420—21no—
171C2×#92420C2×D411yescharacteristic
172C2×#92420C2×D411yescharacteristic
173C2×#92420C2×D411yescharacteristic
174C2×#92420C2×D411yescharacteristic
175#11758213—42no—
176#11758213—81no—
177#11758213—81no—
178C2×C8×C61C22×C411yescharacteristic
179C2×C8×C61C22×C411yescharacteristic
180C4⋊C4×C61—21no—
181C4⋊C4×C61C22×C411yescharacteristic
182C4⋊C4×C61C22×C411yescharacteristic
183C4⋊C4×C61C22×C411yescharacteristic
184C22⋊C4×C61—21no—
185C42×C61C22×C411yescharacteristic
186C22⋊Dic61—21no—
187#11758200—41no—
188#11758200—41no—
189#11758198—21no—
190#11758198—21no—
191#11758198—21no—
192#11758198—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61C22×C411yescharacteristic
198#11758196—21no—
199#11758196—21no—
200#11758196—21no—
201C4×Dic61—21no—
202C4×Dic61—21no—
203C4×Dic61C2×D411yescharacteristic
204C4×Dic61#4111yescharacteristic
205C4×Dic61C2411yescharacteristic
206C2×#92418C22×C411yescharacteristic
207C2×#92418C22×C411yescharacteristic
208#11758190—41no—
209#11758190—41no—
210C23×C61—21no—
211C22×D61—21no—
212C22×D61—42no—
213C22×D61—81no—
214C22×D61—81no—
215C22×D61C4×D411yescharacteristic
216Q8×C61—81no—
217D4×C61—21no—
218D4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—81no—
223C2×C4×C61C4×D411yescharacteristic
224C2×C4×C61C4×D411yescharacteristic
225C2×C4×C61C23×C411yescharacteristic
226C22⋊D61—42no—
227C2×Dic61—21no—
228C2×Dic61—21no—
229C2×Dic61—21no—
230C2×Dic61—41no—
231C2×Dic61—41no—
232C2×Dic61C4×D411yescharacteristic
233C2×Dic61C22×D411yescharacteristic
234C2×Dic61C2×#4111yescharacteristic
235D244—21no—
236D244—21no—
237C4×D61—21no—
238C4×D61—21no—
239C4×D61—21no—
240C4×D61—21no—
241C4×D61—41no—
242C4×D61—41no—
243C4×D61—41no—
244C4×D61—81no—
245C4×D61—81no—
246Dic122—21no—
247Dic122—21no—
248Dic122—21no—
249Dic122—21no—
250#92420—22no—
251#92420—21no—
252#92420—21no—
253#92420—21no—
254#92420—21no—
255#92420—41no—
256#92420—41no—
257#92420—41no—
258#92420—41no—
259#92420—41no—
260C8×C61—21no—
261C8×C61—21no—
262#92418—21no—
263#92418—21no—
264#17952—611nomaximal, Sylow
265C22×C61—41no—
266C22×C61—41no—
267C22×C61—41no—
268C22×C61C2×C4×D411yescharacteristic
269C2×D61—21no—
270C2×D61—21no—
271C2×D61—21no—
272C2×D61—41no—
273C2×D61—41no—
274C2×D61—41no—
275C2×D61—84no—
276C4×C61—21no—
277C4×C61—21no—
278C4×C61—41no—
279C4×C61—41no—
280C4×C61—41no—
281C4×C61#56811yescharacteristic
282C4×C61#56811yescharacteristic
283Dic61—21no—
284Dic61—21no—
285Dic61—41no—
286Dic61—41no—
287Dic61—41no—
288Dic61#57711yescharacteristic
289Dic61#57711yescharacteristic
290#3101—611no—
291#2985—611no—
292#2975—611no—
293#1592—611no—
294#1592—611no—
295#1590—611no—
296#1590—611no—
297#1588—611no—
298#1588—611no—
299#1583—611no—
300#1583—611no—
301#1572—611no—
302#1490—611no—
303#1485—611no—
304#1453—611no—
305C2×C61—81no—
306C2×C61#293011yescharacteristic
307C2×C61#294911yescharacteristic
308C2×C61#295611yescharacteristic
309D61—41no—
310D61—81no—
311#533—611no—
312C2×C4×D4—611no—
313#483—611no—
314#483—611no—
315#476—611no—
316#476#692311yescharacteristic
317#461—611no—
318#461—611no—
319#450—611no—
320#450—611no—
321#446—611no—
322#446—611no—
323#446—1221no—
324#446—1221no—
325C8×Q8—611no—
326C8×Q8—611no—
327C4×SD16—611no—
328C4×SD16—611no—
329C4×SD16—611no—
330C4×SD16—611no—
331#436—611no—
332#436—611no—
333#436—611no—
334#436—611no—
335#436—1221no—
336#436—1221no—
337#433—1221no—
338#433—1221no—
339#432—611no—
340#431—611no—
341#431—611no—
342#431—611no—
343#407—1221no—
344#407—1221no—
345C4×M16—611no—
346#335—611no—
347#335—611no—
348#334—611no—
349#334—611no—
350Q8⋊C8—611no—
351Q8⋊C8—611no—
352Q8⋊C8—611no—
353Q8⋊C8—611no—
354D4⋊C8—611no—
355D4⋊C8—611no—
356D4⋊C8—611no—
357D4⋊C8—611no—
358#322—611no—
359#322—611no—
360#322—611no—
361C82—611no—
362C61#1795211yescharacteristic, complemented, Sylow
363C22×D4—611no—
364C23×C4—1221no—
365C2×SD16—611no—
366C2×SD16—611no—
367C2×SD16—611no—
368C2×SD16—611no—
369C2×M16—1221no—
370C2×M16—1221no—
371C2×M16—1221no—
372C2×M16—1221no—
373C2×M16—1221no—
374C4⋊Q8—611no—
375C4⋊Q8C2×#692311yescharacteristic
376#125—1221no—
377C22⋊Q8—611no—
378C22⋊Q8—611no—
379C22⋊Q8—1221no—
380C4×Q8—611no—
381C4×Q8—611no—
382C4×Q8—1221no—
383C4×D4—611no—
384C4×D4—611no—
385C4×D4—611no—
386C4×D4—1221no—
387C4×D4—1221no—
388C4×D4C2×#692311yescharacteristic
389#117—1221no—
390C2×C4⋊C4—611no—
391C2×C22⋊C4—1221no—
392C2×C42—611no—
393#106—611no—
394#106—611no—
395#106C2×#692311yescharacteristic
396#106C2×#692311yescharacteristic
397C4⋊C8—611no—
398C4⋊C8—611no—
399C4⋊C8—611no—
400C4⋊C8—611no—
401C4⋊C8—611no—
402C4⋊C8—611no—
403C4⋊C8—611no—
404C4⋊C8—1221no—
405C4⋊C8—1221no—
406C4⋊C8—1221no—
407C4⋊C8—1221no—
408C4⋊C8C2×#692311yescharacteristic
409#103—611no—
410#103—611no—
411#103—611no—
412#103—611no—
413#102—611no—
414#102—611no—
415#102C2×#692311yescharacteristic
416#102C2×#692311yescharacteristic
417C22⋊C8—611no—
418C22⋊C8—611no—
419C22⋊C8—611no—
420C22⋊C8—611no—
421C22⋊C8—1221no—
422C22⋊C8—1221no—
423C22⋊C8—1221no—
424C22⋊C8—1221no—
425C22⋊C8—1221no—
426C22⋊C8—1221no—
427#97—1221no—
428#97—1221no—
429#97—1221no—
430#97—1221no—
431C4×C8—611no—
432C4×C8—611no—
433C4×C8—611no—
434C4×C8—611no—
435C4×C8—611no—
436C4×C8—611no—
437C4×C8—1221no—
438C4×C8—1221no—
439C4×C8—1221no—
440C24—1221no—
441C2×Q8—21no—
442C2×Q8—611no—
443C2×Q8—611no—
444C2×D4—611no—
445C2×D4—1221no—
446C2×D4—2441no—
447C2×D4—2441no—
448C2×D4C22×#692311yescharacteristic
449C22×C4—21no—
450C22×C4—611no—
451C22×C4—611no—
452C22×C4—611no—
453C22×C4—1221no—
454C22×C4—1221no—
455C22×C4—1221no—
456C22×C4—1221no—
457C22×C4—4881no—
458C22×C4—4881no—
459SD16—2441no—
460SD16—2441no—
461SD16—2441no—
462SD16—2441no—
463M16—4881no—
464M16—4881no—
465M16—4881no—
466M16—4881no—
467M16—2442no—
468C2×C8—611no—
469C2×C8—611no—
470C2×C8—611no—
471C2×C8—611no—
472C2×C8—611no—
473C2×C8—611no—
474C2×C8—1221no—
475C2×C8—1221no—
476C2×C8—1221no—
477C2×C8—1221no—
478C2×C8—1221no—
479C2×C8—1221no—
480C2×C8C22×#692311yescharacteristic
481C2×C8C22×#692311yescharacteristic
482C4⋊C4—21no—
483C4⋊C4—611no—
484C4⋊C4—1221no—
485C4⋊C4—1221no—
486C4⋊C4—1221no—
487C4⋊C4—1221no—
488C4⋊C4—1221no—
489C4⋊C4—1221no—
490C4⋊C4—1221no—
491C4⋊C4C22×#692311yescharacteristic
492C4⋊C4C22×#692311yescharacteristic
493C4⋊C4C22×#692311yescharacteristic
494C22⋊C4—21no—
495C22⋊C4—1221no—
496C22⋊C4—1221no—
497C22⋊C4—1221no—
498C22⋊C4—1221no—
499C22⋊C4—1221no—
500C42—611no—
501C42—611no—
502C42—611no—
503C42—1221no—
504C42—1221no—
505C42C22×#692311yescharacteristic
506C23—21no—
507C23—611no—
508C23—1221no—
509C23—4881no—
510C23—4881no—
511C23—2442no—
512Q8—81no—
513Q8—1221no—
514Q8—1221no—
515Q8—1221no—
516Q8—1221no—
517D4—21no—
518D4—21no—
519D4—1221no—
520D4—1221no—
521D4—2442no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—81no—
526C2×C4—611no—
527C2×C4—611no—
528C2×C4—611no—
529C2×C4—1221no—
530C2×C4—1221no—
531C2×C4—1221no—
532C2×C4—1221no—
533C2×C4—1221no—
534C2×C4—1221no—
535C2×C4—1221no—
536C2×C4—4881no—
537C2×C4—4881no—
538C2×C4—2441no—
539C2×C4—2441no—
540C2×C4—2441no—
541C2×C4—2441no—
542C2×C4—2441no—
543C2×C4D4×#692311yescharacteristic
544C2×C4D4×#692311yescharacteristic
545C2×C4C23×#692311yescharacteristic
546C8—21no—
547C8—21no—
548C8—1221no—
549C8—1221no—
550C8—1221no—
551C8—1221no—
552C8—1222no—
553C8—1221no—
554C8—1221no—
555C8—2441no—
556C8—2441no—
557C8—2441no—
558C8—2441no—
559C8—2441no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22—1221no—
564C22—1221no—
565C22—1221no—
566C22—4884no—
567C22—2441no—
568C22—2441no—
569C22—2441no—
570C22C2×D4×#692311yescharacteristic
571C4—21no—
572C4—21no—
573C4—41no—
574C4—41no—
575C4—41no—
576C4—611no—
577C4—611no—
578C4—1221no—
579C4—1221no—
580C4—2441no—
581C4—2441no—
582C4—2441no—
583C4#1684084111yescharacteristic
584C4#1684084211yescharacteristic
585C2—81no—
586C2—4881no—
587C2—2441no—
588C2#2146253911yescharacteristic
589C2#2146257911yescharacteristic
590C2#2146259311yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#476C4
socleC22×C61C2×C4×D4
prime radicalC2×D4×D61C2×C4
cosocleC4×Dic61C24
hypercenter#476#6923
hypocenterC61#17952
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
1. #21462727C2semidirect
3. #21461461C2semidirect
4. #21461460C2semidirect
5. #21461456C2semidirect
6. #21461454C2semidirect
7. #21461449C2semidirect
8. #21461447C2semidirect
9. #21461429C2semidirect
10. #21461427C2semidirect
11. #21461391C2semidirect
13. #21461143C2semidirect
14. #21461037C2semidirect
18. #16840812C22semidirect
19. #16840812C22semidirect
20. #16840811C22semidirect
21. #16840811C22semidirect
36. #16840626C22semidirect
37. #16840625C22semidirect
38. #16840624C22semidirect
39. #16840623C22semidirect
40. #16840612C22semidirect
41. #16840612C22semidirect
42. #16840612C22semidirect
43. #16840612C22semidirect
48. #92420⋊C8C22semidirect
49. #92420⋊C8C22semidirect
50. #16840603C22semidirect
51. C8×#92420C22semidirect
362. C61#17952semidirect

Maximal subgroups

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

subgroupindex
#214627272
#214626822
#214614612
#214614602
#214614562
#214614542
#214614492
#214614472
#214614292
#214614272
#214613912
#214611592
#214611432
#214610372
#214558292
#1795261