Finite Group Databasebeta

← #29662798

Subgroups — #29662798

Order 15616.

706 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456360C211yesmaximal, characteristic, complemented
2#21455829C211yesmaximal, characteristic
3#461×D61C211yesmaximal, characteristic
4#21454555C211yesmaximal, characteristic
5#21445332C211yesmaximal, characteristic, complemented
6#21445330C211yesmaximal, characteristic, complemented
7#21445320C211yesmaximal, characteristic, complemented
8#21445209C211yesmaximal, characteristic, complemented
9#21445207C211yesmaximal, characteristic, complemented
10#21445197C211yesmaximal, characteristic, complemented
11#21445191C211yesmaximal, characteristic, complemented
12#21444570C211yesmaximal, characteristic, complemented
13#21444506C211yesmaximal, characteristic, complemented
14#21444502C211yesmaximal, characteristic, complemented
15C61×#1691C211yesmaximal, characteristic, complemented
16C2×SD16×D61—21no—
17C2×M16×D61—21no—
18#16840317C2211yescharacteristic
19C4⋊Q8×D61C2211yescharacteristic
20C4×D4×D61C2211yescharacteristic
21#16840169—21no—
22#16840169—21no—
23#16840167C2211yescharacteristic, complemented
24#16840165C2211yescharacteristic, complemented
25#16840158C2211yescharacteristic, complemented
26#16840094—21no—
27#16840049C2211yescharacteristic
28#16839968C2211yescharacteristic
29#16839968C2211yescharacteristic
30#16839967C2211yescharacteristic
31#16839967C2211yescharacteristic
32#16839966—21no—
33#16839965—21no—
34#16839964C2211yescharacteristic
35#16839914—21no—
36#16839853—21no—
37#16839849—21no—
38#16839840C2211yescharacteristic
39#16839836—21no—
40#16839831—21no—
41C8×D244C2211yescharacteristic
42#16839827C2211yescharacteristic
43#16839814C2211yescharacteristic
44#16839801—21no—
45#16839801—21no—
46#16839784C2211yescharacteristic
47C4⋊C8×D61C2211yescharacteristic
48#103×D61—21no—
49#16839724—21no—
50C4×#11758189C2211yescharacteristic
51#16839514C2211yescharacteristic, complemented
52#16839513C2211yescharacteristic, complemented
53#16839512C2211yescharacteristic, complemented
54#16839511C2211yescharacteristic, complemented
55#16839509C2211yescharacteristic, complemented
56#16839509C2211yescharacteristic, complemented
57#16839508C2211yescharacteristic, complemented
58#16839507C2211yescharacteristic, complemented
59#16839507C2211yescharacteristic, complemented
60C61×#498C2211yescharacteristic, complemented
61C61×#476C2211yescharacteristic
62C61×#461C2211yescharacteristic
63C61×#436C2211yescharacteristic
64C61×#335C2211yescharacteristic, complemented
65C61×#333C2211yescharacteristic, complemented
66C61×#332C2211yescharacteristic, complemented
67C2×Q8×D61—21no—
68C2×D4×D61D411yescharacteristic
69C22×C4×D61—21no—
70C2×#11758190—21no—
71C2×#11758190—21no—
72C2×#11758200—21no—
73C2×#11758200—21no—
74C2×#11758201—21no—
75SD16×D61—44no—
76C2×#11758194—21no—
77C2×#11758189—21no—
78C2×#11758189D411yescharacteristic
79C2×#11758189D411yescharacteristic
80M16×D61—44no—
81C2×C8×D61—21no—
82C2×C8×D61—21no—
83C2×C8×D61—21no—
84#14519777C2311yescharacteristic
85#14519777C2311yescharacteristic
86#14519776—21no—
87#14519776—21no—
88#14519775—21no—
89#14519768—21no—
90#14519757—21no—
91#14519753—21no—
92Q8×Dic61—21no—
93#14519742—21no—
94C4×D244C2311yescharacteristic
95D4×Dic61D411yescharacteristic
96#14519739D411yescharacteristic
97C4×C22⋊D61—21no—
98#14519737—21no—
99#14519735—21no—
100C4⋊C4×D61—21no—
101C4⋊C4×D61—21no—
102C4⋊C4×D61D411yescharacteristic
103C22⋊C4×D61—21no—
104C42×D61C2311yescharacteristic
105#14519711—21no—
106#14519711—21no—
107#14519710—21no—
108#14519710—21no—
109#14519708—21no—
110#14519708—21no—
111#14519708—21no—
112C4⋊#92418C2311yescharacteristic
113C4⋊#92418C2311yescharacteristic
114C4⋊#92418C2311yescharacteristic
115#14519706—21no—
116#14519706—21no—
117#14519702—21no—
118#14519702—21no—
119#14519701—21no—
120#14519700—21no—
121#14519700—21no—
122#14519697—21no—
123C22⋊#92418—21no—
124#14519689—21no—
125#14519689—21no—
126#14519689—21no—
127#14519687D411yescharacteristic
128#14519687D411yescharacteristic
129C4×#92418C2311yescharacteristic
130C8×Dic61—21no—
131C2×SD16×C61—21no—
132C2×M16×C61—21no—
133C4⋊Q8×C61C2311yescharacteristic
134C4⋊Q8×C61C2311yescharacteristic
135C4×D4×C61C2311yescharacteristic
136#106×C61—21no—
137#106×C61—21no—
138#106×C61—21no—
139C4⋊C8×C61C2311yescharacteristic
140C4⋊C8×C61C2311yescharacteristic
141C4⋊C8×C61C2311yescharacteristic
142#103×C61—21no—
143#102×C61—21no—
144C22⋊C8×C61—21no—
145C4×C8×C61C2311yescharacteristic
146C23×D61—21no—
147C2×Q8×C61—21no—
148C2×Q8×C61—21no—
149C2×D4×C61C2×D411yescharacteristic
150C22×C4×C61—21no—
151C2×C22⋊D61—21no—
152C22×Dic61—21no—
153Q8×D61—42no—
154Q8×D61—42no—
155D4×D61—22no—
156D4×D61—41no—
157C2×D244C2×D411yescharacteristic
158C2×C4×D61—21no—
159C2×C4×D61—21no—
160C2×C4×D61—21no—
161C2×C4×D61—42no—
162C2×C4×D61—42no—
163C2×C4×D61—41no—
164C2×C4×D61C2×D411yescharacteristic
165C2×C4×D61C2×D411yescharacteristic
166C2×C4×D61#4111yescharacteristic
167C2×Dic122—21no—
168C2×Dic122—21no—
169C2×Dic122—41no—
170SD16×C61—42no—
171M16×C61—42no—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C2×C8×C61C2×D411yescharacteristic
176C2×C8×C61C2×D411yescharacteristic
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61C2×D411yescharacteristic
182C22⋊C4×C61—21no—
183C42×C61C2411yescharacteristic
184C22⋊Dic61—21no—
185#11758201—42no—
186#11758200—42no—
187#11758200—42no—
188#11758198—21no—
189#11758198—21no—
190#11758198—41no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—41no—
196C4⋊Dic61C2×D411yescharacteristic
197#11758196—21no—
198#11758196—41no—
199#11758196—41no—
200#11758196—41no—
201C4×Dic61—41no—
202C4×Dic61C2×D411yescharacteristic
203C4×Dic61C2×D411yescharacteristic
204C4×Dic61#4111yescharacteristic
205#11758194—42no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418C2×D411yescharacteristic
210C2×#92418C2×D411yescharacteristic
211#11758190—42no—
212#11758190—42no—
213#11758189—22no—
214#11758189—42no—
215#11758189SD1614yescomplemented
216C8×D61—42no—
217C8×D61—42no—
218C8×D61—42no—
219C23×C61—21no—
220C22×D61—21no—
221C22×D61—42no—
222C22×D61—42no—
223C22×D61—81no—
224C22×D61#12111yescharacteristic
225Q8×C61—41no—
226Q8×C61—41no—
227Q8×C61—81no—
228D4×C61—21no—
229D4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61C22×D411yescharacteristic
236C2×C4×C61C22×D411yescharacteristic
237C2×C4×C61C2×#4111yescharacteristic
238C22⋊D61—42no—
239C2×Dic61—21no—
240C2×Dic61—42no—
241C2×Dic61—41no—
242C2×Dic61—41no—
243C2×Dic61#12111yescharacteristic
244C2×Dic61#12111yescharacteristic
245C2×Dic61#12111yescharacteristic
246D244—21no—
247D244AGL(1,ℤ/8)12yes—
248C4×D61—21no—
249C4×D61—21no—
250C4×D61—41no—
251C4×D61—42no—
252C4×D61—41no—
253C4×D61—41no—
254C4×D61—42no—
255C4×D61—82no—
256C4×D61C2×SD1612yes—
257C4×D61#13712yes—
258Dic122—41no—
259Dic122—41no—
260Dic122—41no—
261Dic122—41no—
262Dic122—82no—
263C8×C61—21no—
264C8×C61—41no—
265C8×C61—41no—
266C8×C61—41no—
267C8×C61C2×SD1612yescomplemented
268#92418—21no—
269#92418—41no—
270#92418—41no—
271#92418—41no—
272#92418C2×SD1612yescomplemented
273#19164—611nomaximal, Sylow
274C22×C61—41no—
275C22×C61—41no—
276C22×C61—41no—
277C22×C61C2×#12111yescharacteristic
278C2×D61—21no—
279C2×D61—21no—
280C2×D61—22no—
281C2×D61—41no—
282C2×D61—41no—
283C2×D61—84no—
284C2×D61#46112yes—
285C4×C61—21no—
286C4×C61—41no—
287C4×C61—81no—
288C4×C61—81no—
289C4×C61C22×SD1611yescharacteristic
290C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
291C4×C61C2×#13711yescharacteristic
292C4×C61#57511yescharacteristic
293Dic61—21no—
294Dic61—21no—
295Dic61—41no—
296Dic61—81no—
297Dic61—81no—
298Dic61#46112yes—
299#3165—611no—
300#3101—611no—
301C2×#461—611no—
302D4×M16—611no—
303#1693—611no—
304#1693—611no—
305#1691—611no—
306#1691D6111yescharacteristic, complemented
307#1673—611no—
308#1673—611no—
309#1671—611no—
310#1671—611no—
311#1565—611no—
312#1550—611no—
313#1549—611no—
314C2×C61—81no—
315C2×C61C2×#46111yescharacteristic
316C2×C61#305511yescharacteristic
317C2×C61#306911yescharacteristic
318D61—22no—
319D61—42no—
320C22×SD16—1221no—
321C22×M16—1221no—
322#533—611no—
323C2×C4⋊Q8—611no—
324C2×C4×D4—611no—
325#501—1221no—
326#501—1221no—
327#500—611no—
328#500—611no—
329#498—611no—
330#498C2×D6111yescharacteristic, complemented
331#483—1221no—
332#476—611no—
333#476C2×D6111yescharacteristic
334#461—611no—
335#461—611no—
336#461—611no—
337#461—611no—
338#461—611no—
339#461—1221no—
340#461—1221no—
341#461C2×D6111yescharacteristic
342#450—1221no—
343C4×SD16—1221no—
344C4×SD16—1221no—
345#436—611no—
346#436C2×D6111yescharacteristic
347#435—1221no—
348#435—1221no—
349C8×D4—611no—
350C8×D4—611no—
351#431—611no—
352#428—1221no—
353#428—1221no—
354#423—611no—
355C2×C4⋊C8—611no—
356C2×#103—1221no—
357#407—1221no—
358C4×M16—611no—
359#335—611no—
360#335C2×D6111yescharacteristic, complemented
361#334—611no—
362#334—611no—
363#333—611no—
364#333C2×D6111yescharacteristic, complemented
365#332—611no—
366#332—611no—
367#332—611no—
368#332—611no—
369#332—611no—
370#332C2×D6111yescharacteristic, complemented
371C61#1916411yescharacteristic, complemented, Sylow
372C22×Q8—1221no—
373C22×D4—611no—
374C23×C4—1221no—
375C2×SD16—21no—
376C2×SD16—1221no—
377C2×SD16—1221no—
378C2×SD16—1221no—
379C2×SD16—1221no—
380C2×SD16—1221no—
381C2×SD16—2444no—
382C2×M16—21no—
383C2×M16—611no—
384C2×M16—611no—
385C2×M16—1221no—
386C2×M16—1221no—
387C2×M16—2444no—
388C22×C8—1221no—
389C22×C8—1221no—
390C22×C8—1221no—
391C4⋊Q8—611no—
392C4⋊Q8—611no—
393C4⋊Q8—1221no—
394C4⋊Q8—1221no—
395C4⋊Q8—1221no—
396C4⋊Q8C22×D6111yescharacteristic
397C4⋊Q8C22×D6111yescharacteristic
398#125—1221no—
399C22⋊Q8—1221no—
400C22⋊Q8—1221no—
401C4×Q8—1221no—
402C4×Q8—1221no—
403C4×D4—611no—
404C4×D4—611no—
405C4×D4—611no—
406C4×D4—1221no—
407C4×D4—1221no—
408C4×D4C22×D6111yescharacteristic
409#117—1221no—
410C2×C4⋊C4—611no—
411C2×C4⋊C4—1221no—
412C2×C4⋊C4—1221no—
413C2×C22⋊C4—1221no—
414C2×C42—611no—
415#107—1221no—
416#107—1221no—
417#107—1221no—
418#107—1221no—
419#106—21no—
420#106—21no—
421#106—21no—
422#106—1221no—
423#106—1221no—
424#106—1221no—
425C4⋊C8—611no—
426C4⋊C8—611no—
427C4⋊C8—611no—
428C4⋊C8—1221no—
429C4⋊C8—1221no—
430C4⋊C8C22×D6111yescharacteristic
431C4⋊C8C22×D6111yescharacteristic
432C4⋊C8C22×D6111yescharacteristic
433#103—21no—
434#103—1221no—
435#103—1221no—
436#103—1221no—
437#103—1221no—
438#103—1221no—
439#102—21no—
440#102—1221no—
441C22⋊C8—21no—
442C22⋊C8—1221no—
443C22⋊C8—1221no—
444C22⋊C8—1221no—
445C22⋊C8—1221no—
446#97—611no—
447#97—611no—
448C4×C8—611no—
449C4×C8—1221no—
450C4×C8C22×D6111yescharacteristic
451C24—1221no—
452C2×Q8—21no—
453C2×Q8—21no—
454C2×Q8—1221no—
455C2×Q8—1221no—
456C2×Q8—2441no—
457C2×Q8—2442no—
458C2×Q8—2442no—
459C2×D4—611no—
460C2×D4—1221no—
461C2×D4—1222no—
462C2×D4—2441no—
463C2×D4D4×D6111yescharacteristic
464C22×C4—21no—
465C22×C4—611no—
466C22×C4—611no—
467C22×C4—611no—
468C22×C4—1221no—
469C22×C4—1221no—
470C22×C4—1221no—
471C22×C4—1221no—
472C22×C4—2442no—
473C22×C4—2442no—
474C22×C4—2441no—
475SD16—42no—
476SD16—2442no—
477SD16—2442no—
478SD16—2442no—
479SD16—2442no—
480SD16—2442no—
481M16—42no—
482M16—614no—
483M16—1222no—
484M16—2442no—
485M16—2442no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—611no—
490C2×C8—611no—
491C2×C8—1221no—
492C2×C8—1221no—
493C2×C8—1221no—
494C2×C8—2442no—
495C2×C8—2442no—
496C2×C8—2442no—
497C2×C8D4×D6111yescharacteristic
498C2×C8D4×D6111yescharacteristic
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—611no—
504C4⋊C4—1221no—
505C4⋊C4—1221no—
506C4⋊C4—1221no—
507C4⋊C4—1221no—
508C4⋊C4—1221no—
509C4⋊C4—2441no—
510C4⋊C4—2441no—
511C4⋊C4—2441no—
512C4⋊C4—2441no—
513C4⋊C4D4×D6111yescharacteristic
514C22⋊C4—21no—
515C22⋊C4—1221no—
516C22⋊C4—1221no—
517C22⋊C4—1221no—
518C22⋊C4—2441no—
519C42—611no—
520C42—611no—
521C42—611no—
522C42—2441no—
523C42C23×D6111yescharacteristic
524C23—21no—
525C23—611no—
526C23—1221no—
527C23—4881no—
528C23—2442no—
529C23—2442no—
530Q8—41no—
531Q8—41no—
532Q8—81no—
533Q8—4882no—
534Q8—2441no—
535Q8—2441no—
536Q8—2441no—
537Q8—2441no—
538D4—21no—
539D4—21no—
540D4—612no—
541D4—1221no—
542D4—2442no—
543C2×C4—21no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—611no—
549C2×C4—612no—
550C2×C4—612no—
551C2×C4—611no—
552C2×C4—611no—
553C2×C4—1221no—
554C2×C4—1221no—
555C2×C4—1221no—
556C2×C4—4882no—
557C2×C4—2441no—
558C2×C4—2442no—
559C2×C4—2442no—
560C2×C4—2442no—
561C2×C4—2441no—
562C2×C4—2441no—
563C2×C4—2441no—
564C2×C4—2441no—
565C2×C4C2×D4×D6111yescharacteristic
566C2×C4C2×D4×D6111yescharacteristic
567C2×C4#41×D6111yescharacteristic
568C8—21no—
569C8—41no—
570C8—41no—
571C8—41no—
572C8—612no—
573C8—1221no—
574C8—2441no—
575C8—2441no—
576C8—2441no—
577C8SD16×D6112yescomplemented
578C22—41no—
579C22—41no—
580C22—41no—
581C22—612no—
582C22—1221no—
583C22—1221no—
584C22—1222no—
585C22—4884no—
586C22—2441no—
587C22—2441no—
588C22#121×D6111yescharacteristic
589C4—21no—
590C4—41no—
591C4—81no—
592C4—81no—
593C4—612no—
594C4—1221no—
595C4—1221no—
596C4—4881no—
597C4—4881no—
598C4—2441no—
599C4C2×SD16×D6111yescharacteristic
600C4AGL(1,ℤ/8)×D6111yescharacteristic
601C4#137×D6111yescharacteristic
602C4#1684056111yescharacteristic
603C2—81no—
604C2—1222no—
605C2—2442no—
606C2#461×D6111yescharacteristic
607C2#2145543311yescharacteristic
608C2#2145555111yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1691C2
socleC22×C61C2×#121
prime radicalC2×D4×D61D4
hypercenter#1691D61
hypocenterC61#19164
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1691

Complements

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

NH ≅ G/Ntype
1. #21456360C2semidirect
5. #21445332C2semidirect
6. #21445330C2semidirect
7. #21445320C2semidirect
8. #21445209C2semidirect
9. #21445207C2semidirect
10. #21445197C2semidirect
11. #21445191C2semidirect
12. #21444570C2semidirect
13. #21444506C2semidirect
14. #21444502C2semidirect
15. C61×#1691C2semidirect
23. #16840167C22semidirect
24. #16840165C22semidirect
25. #16840158C22semidirect
51. #16839514C22semidirect
52. #16839513C22semidirect
53. #16839512C22semidirect
54. #16839511C22semidirect
55. #16839509C22semidirect
56. #16839509C22semidirect
57. #16839508C22semidirect
58. #16839507C22semidirect
59. #16839507C22semidirect
60. C61×#498C22semidirect
64. C61×#335C22semidirect
65. C61×#333C22semidirect
66. C61×#332C22semidirect
215. #11758189SD16semidirect
267. C8×C61C2×SD16semidirect
272. #92418C2×SD16semidirect
306. #1691D61semidirect
330. #498C2×D61semidirect
360. #335C2×D61semidirect
364. #333C2×D61semidirect
370. #332C2×D61semidirect
371. C61#19164semidirect
577. C8SD16×D61semidirect

Maximal subgroups

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

subgroupindex
#214563602
#214558292
#461×D612
#214545552
#214453322
#214453302
#214453202
#214452092
#214452072
#214451972
#214451912
#214445702
#214445062
#214445022
C61×#16912
#1916461