Finite Group Databasebeta

← #29676293

Subgroups — #29676293

Order 15616.

666 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458803C211yesmaximal, characteristic, complemented
2#21458405C211yesmaximal, characteristic, complemented
3#21457415C211yesmaximal, characteristic
4#21445492C211yesmaximal, characteristic, complemented
5#21445377C211yesmaximal, characteristic, complemented
6#21445332C211yesmaximal, characteristic, complemented
7#21445306C211yesmaximal, characteristic, complemented
8#21445165C211yesmaximal, characteristic, complemented
9#21445075C211yesmaximal, characteristic, complemented
10#21445053C211yesmaximal, characteristic, complemented
11#21444998C211yesmaximal, characteristic, complemented
12#21444928C211yesmaximal, characteristic, complemented
13#21444731C211yesmaximal, characteristic, complemented
14#21444663C211yesmaximal, characteristic, complemented
15C61×#3324C211yesmaximal, characteristic, complemented
16C2×#14519803—21no—
17#16840485C2211yescharacteristic, complemented
18#16840417C2211yescharacteristic
19#16840167C2211yescharacteristic, complemented
20#16840165C2211yescharacteristic, complemented
21#16840144—21no—
22#16840141C2211yescharacteristic, complemented
23#16840134C2211yescharacteristic
24#16840108C2211yescharacteristic
25#16840096—21no—
26#16840084—21no—
27#16840047C2211yescharacteristic, complemented
28#16840042C2211yescharacteristic
29#16840006—21no—
30#16840005—21no—
31#16839984—21no—
32#16839981—21no—
33#16839967C2211yescharacteristic
34#16839967C2211yescharacteristic
35#16839964C2211yescharacteristic, complemented
36#16839954C2211yescharacteristic
37#16839913—21no—
38#16839905—21no—
39Q8×#92418C2211yescharacteristic, complemented
40C4×#11758202—21no—
41C4×#11758199C2211yescharacteristic
42#16839841C2211yescharacteristic, complemented
43#16839840C2211yescharacteristic
44#16839801—21no—
45#16839513C2211yescharacteristic, complemented
46#16839512C2211yescharacteristic, complemented
47#16839508C2211yescharacteristic, complemented
48#16839507C2211yescharacteristic, complemented
49#16839507C2211yescharacteristic, complemented
50#16839506C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic, complemented
52D4⋊#92418C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic
54#16839489C2211yescharacteristic, complemented
55#16839484C2211yescharacteristic, complemented
56C61×#556C2211yescharacteristic
57D42×C61C2211yescharacteristic
58C61×#486C2211yescharacteristic
59C61×#482—21no—
60C61×#474C2211yescharacteristic
61C61×#468—21no—
62C61×#460C2211yescharacteristic
63C61×#450—21no—
64C4×SD16×C61C2211yescharacteristic
65C61×#436C2211yescharacteristic
66C61×#418—21no—
67C2×#11758222—21no—
68#14519803—44no—
69C2×#11758202—21no—
70C2×#11758202—21no—
71C2×#11758200—21no—
72C2×#11758200—21no—
73C2×#11758200—21no—
74C2×#11758200—21no—
75C2×#11758201—21no—
76C2×#11758199—21no—
77C2×#11758199D411yescharacteristic
78C2×#11758194—21no—
79C2×#11758189—21no—
80#14519777C2311yescharacteristic
81#14519777C2311yescharacteristic
82#14519772—41no—
83#14519765—21no—
84#14519759—41no—
85#14519756—21no—
86#14519753—21no—
87#14519748—21no—
88C4×Dic122—21no—
89C4×D244C2311yescharacteristic
90#14519735—21no—
91C4⋊C4×D61—21no—
92#14519711—21no—
93#14519711—21no—
94#14519710—21no—
95#14519710—41no—
96#14519710D411yescharacteristic
97#14519708—21no—
98C4⋊#92418—21no—
99C4⋊#92418C2311yescharacteristic
100C4⋊#92418C2311yescharacteristic
101C4⋊#92418C2311yescharacteristic
102#14519701—21no—
103#14519701—21no—
104#14519700—21no—
105#14519700—21no—
106#14519698—21no—
107#14519698—21no—
108#14519698—41no—
109#14519698D411yescharacteristic
110#14519697D411yescharacteristic
111C22⋊#92418—21no—
112#14519689—21no—
113C4×#92418—21no—
114C4×#92418C2311yescharacteristic
115C4×#92418C2311yescharacteristic
116C22×D4×C61—21no—
117C22×D4×C61—21no—
118C2×SD16×C61—21no—
119C2×SD16×C61D411yescharacteristic
120C2×M16×C61—21no—
121C4⋊Q8×C61C2311yescharacteristic
122#127×C61C2311yescharacteristic
123#126×C61—41no—
124#124×C61—21no—
125C22⋊Q8×C61—21no—
126#121×C61—21no—
127#121×C61—21no—
128#120×C61—41no—
129C4×Q8×C61—21no—
130C4×Q8×C61C2311yescharacteristic
131C4×D4×C61C2311yescharacteristic
132C4×D4×C61C2311yescharacteristic
133#117×C61—21no—
134#107×C61—21no—
135#106×C61D411yescharacteristic
136C4⋊C8×C61C2311yescharacteristic
137#103×C61D411yescharacteristic
138#102×C61—21no—
139#102×C61—21no—
140#102×C61—21no—
141#102×C61—21no—
142#102×C61D411yescharacteristic
143C22⋊C8×C61—21no—
144C4×C8×C61C2311yescharacteristic
145C24×C61—41no—
146C2×Q8×C61—21no—
147C2×Q8×C61C2×D411yescharacteristic
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61—21no—
151C2×D4×C61—41no—
152C2×D4×C61—41no—
153C2×D4×C61—42no—
154C2×D4×C61—42no—
155C2×D4×C61—81no—
156C2×D4×C61—81no—
157C2×D4×C61C2×D411yescharacteristic
158C2×D4×C61C2×D411yescharacteristic
159C22×C4×C61—21no—
160C22×C4×C61—21no—
161C2×C22⋊D61—41no—
162#11758222—42no—
163#11758222—81no—
164C2×D244C2×D411yescharacteristic
165C2×C4×D61—21no—
166C2×C4×D61—41no—
167C2×Dic122—21no—
168C2×Dic122—21no—
169SD16×C61—41no—
170SD16×C61—81no—
171M16×C61—81no—
172C2×C8×C61—21no—
173C2×C8×C61C2×D411yescharacteristic
174C2×C8×C61C2×D411yescharacteristic
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—41no—
179C4⋊C4×C61C2×D411yescharacteristic
180C4⋊C4×C61C2×D411yescharacteristic
181C4⋊C4×C61C2×D411yescharacteristic
182C22⋊C4×C61—21no—
183C22⋊C4×C61—21no—
184C22⋊C4×C61—41no—
185C22⋊C4×C61—41no—
186C22⋊C4×C61—41no—
187C42×C61—21no—
188C42×C61—41no—
189C42×C61C2411yescharacteristic
190C22⋊Dic61—41no—
191#11758202—42no—
192#11758202—81no—
193#11758201—81no—
194#11758200—42no—
195#11758200—42no—
196#11758200—81no—
197#11758200—81no—
198#11758199—22no—
199#11758199—42no—
200#11758198—21no—
201#11758198—41no—
202#11758198—41no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61C2×D411yescharacteristic
208#11758196—41no—
209#11758196—41no—
210C4×Dic61—41no—
211#11758194—42no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418C2×D411yescharacteristic
217C2×#92418C2×D411yescharacteristic
218#11758189—42no—
219C23×C61—21no—
220C23×C61—21no—
221C23×C61—41no—
222C23×C61—82no—
223C23×C61—82no—
224C23×C61—161no—
225C22×D61—21no—
226Q8×C61—21no—
227Q8×C61—21no—
228Q8×C61—41no—
229Q8×C61—41no—
230D4×C61—21no—
231D4×C61—21no—
232D4×C61—21no—
233D4×C61—21no—
234D4×C61—41no—
235D4×C61—41no—
236D4×C61—41no—
237D4×C61—41no—
238D4×C61—81no—
239D4×C61—82no—
240D4×C61—82no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61C22×D411yescharacteristic
251C2×C4×C61C22×D411yescharacteristic
252C2×C4×C61H(5,𝔽2)11yescharacteristic
253C22⋊D61—82no—
254C2×Dic61—21no—
255C2×Dic61—41no—
256C2×Dic61—41no—
257D244—21no—
258D244—21no—
259C4×D61—41no—
260C4×D61—41no—
261C4×D61—82no—
262Dic122—41no—
263Dic122—41no—
264Dic122—41no—
265Dic122—41no—
266C8×C61—21no—
267C8×C61—21no—
268C8×C61—41no—
269#92418—21no—
270#92418—41no—
271#92418—41no—
272#92418—41no—
273#92418—41no—
274#92418AGL(1,ℤ/8)12yes—
275#20250—611nomaximal, Sylow
276C22×C61—41no—
277C22×C61—41no—
278C22×C61—41no—
279C22×C61—41no—
280C22×C61—41no—
281C22×C61—41no—
282C22×C61—81no—
283C22×C61—81no—
284C22×C61—81no—
285C22×C61—164no—
286C22×C61D4211yescharacteristic
287C2×D61—41no—
288C2×D61—41no—
289C2×D61—41no—
290C4×C61—21no—
291C4×C61—41no—
292C4×C61—41no—
293C4×C61—41no—
294C4×C61—41no—
295C4×C61—81no—
296C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
297C4×C61#57511yescharacteristic
298C4×C61#57611yescharacteristic
299C4×C61#57711yescharacteristic
300Dic61—41no—
301Dic61—81no—
302Dic61—81no—
303#3393—611no—
304#3353—611no—
305#3324D6111yescharacteristic, complemented
306#3272—611no—
307#1722—611no—
308#1701—611no—
309#1693—611no—
310#1689—611no—
311#1666—611no—
312#1650—611no—
313#1647—611no—
314#1639—611no—
315#1628—611no—
316#1594—611no—
317#1583—611no—
318C2×C61—81no—
319C2×C61—81no—
320C2×C61—161no—
321C2×C61#326711yescharacteristic
322C2×C61#327411yescharacteristic
323C2×C61#328911yescharacteristic
324D61—81no—
325C2×AGL(1,ℤ/8)—1221no—
326#556—611no—
327#556C2×D6111yescharacteristic
328#547—611no—
329D42C2×D6111yescharacteristic
330#500—611no—
331#500—611no—
332#495—1221no—
333#494—611no—
334#492—611no—
335#486—611no—
336#486C2×D6111yescharacteristic
337#484—1221no—
338#482—21no—
339#482—1221no—
340#475—611no—
341#474—611no—
342#474C2×D6111yescharacteristic
343#468—21no—
344#468—1221no—
345#468—1221no—
346#464—1221no—
347#464—1221no—
348#461—611no—
349#461—611no—
350#461—611no—
351#460C2×D6111yescharacteristic
352#459—611no—
353#450—21no—
354#450—1221no—
355#449—1221no—
356C8×Q8—611no—
357C4×Q16—1221no—
358C4×SD16C2×D6111yescharacteristic
359C4×D8—611no—
360#436—611no—
361#436—611no—
362#436C2×D6111yescharacteristic
363#428—1221no—
364#418—21no—
365#334—611no—
366#334—611no—
367#332—611no—
368#332—611no—
369#332—611no—
370#331—611no—
371Q8⋊C8—611no—
372D4⋊C8—611no—
373D4⋊C8—611no—
374D4⋊C8—611no—
375#322—611no—
376C61#2025011yescharacteristic, complemented, Sylow
377C2×#41—1221no—
378C22×D4—21no—
379C22×D4—21no—
380AGL(1,ℤ/8)—2444no—
381C2×Q16—1221no—
382C2×Q16—1221no—
383C2×SD16—21no—
384C2×SD16—1221no—
385C2×SD16—1221no—
386C2×SD16—1221no—
387C2×SD16—1221no—
388C2×SD16—1221no—
389C2×SD16C22⋊D6111yescharacteristic
390C2×D8—611no—
391C2×D8—1221no—
392C2×M16—21no—
393C2×M16—1221no—
394C2×M16—1221no—
395C4⋊Q8—611no—
396C4⋊Q8—611no—
397C4⋊Q8C22×D6111yescharacteristic
398#127C22×D6111yescharacteristic
399#126—41no—
400#126—2441no—
401#124—21no—
402#124—1221no—
403#123—2441no—
404C22⋊Q8—21no—
405C22⋊Q8—1221no—
406C22⋊Q8—1221no—
407#121—21no—
408#121—21no—
409#121—1221no—
410#120—41no—
411C4×Q8—21no—
412C4×Q8—1221no—
413C4×Q8C22×D6111yescharacteristic
414C4×D4—611no—
415C4×D4C22×D6111yescharacteristic
416C4×D4C22×D6111yescharacteristic
417#117—21no—
418#117—1221no—
419C2×C4⋊C4—1221no—
420#107—21no—
421#107—611no—
422#107—1221no—
423#107—1221no—
424#107—1221no—
425#107—2441no—
426#106—1221no—
427#106C22⋊D6111yescharacteristic
428C4⋊C8—611no—
429C4⋊C8—611no—
430C4⋊C8—611no—
431C4⋊C8—1221no—
432C4⋊C8C22×D6111yescharacteristic
433#103—1221no—
434#103—1221no—
435#103—1221no—
436#103—1221no—
437#103C22⋊D6111yescharacteristic
438#102—21no—
439#102—21no—
440#102—21no—
441#102—21no—
442#102—611no—
443#102—611no—
444#102—1221no—
445#102—1221no—
446#102—2441no—
447#102C22⋊D6111yescharacteristic
448C22⋊C8—21no—
449C22⋊C8—1221no—
450C22⋊C8—1221no—
451C4×C8—611no—
452C4×C8—611no—
453C4×C8—1221no—
454C4×C8C22×D6111yescharacteristic
455C24—41no—
456#41—4881no—
457#41—2442no—
458C2×Q8—21no—
459C2×Q8—1221no—
460C2×Q8—1221no—
461C2×Q8C2×C22⋊D6111yescharacteristic
462C2×D4—21no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—41no—
466C2×D4—41no—
467C2×D4—42no—
468C2×D4—42no—
469C2×D4—81no—
470C2×D4—81no—
471C2×D4—611no—
472C2×D4—2441no—
473C2×D4D4×D6111yescharacteristic
474C2×D4C2×C22⋊D6111yescharacteristic
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—1221no—
478C22×C4—2441no—
479Q16—4881no—
480Q16—2442no—
481SD16—41no—
482SD16—81no—
483SD16—4881no—
484SD16—4881no—
485SD16—4881no—
486SD16—2442no—
487SD16—2442no—
488D8—1222no—
489D8—2442no—
490M16—81no—
491M16—2442no—
492M16—2442no—
493C2×C8—21no—
494C2×C8—611no—
495C2×C8—611no—
496C2×C8—1221no—
497C2×C8—1221no—
498C2×C8—1221no—
499C2×C8—1221no—
500C2×C8C2×C22⋊D6111yescharacteristic
501C2×C8C2×C22⋊D6111yescharacteristic
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—41no—
506C4⋊C4—611no—
507C4⋊C4—1221no—
508C4⋊C4—1221no—
509C4⋊C4—1221no—
510C4⋊C4—1221no—
511C4⋊C4—2441no—
512C4⋊C4—2441no—
513C4⋊C4D4×D6111yescharacteristic
514C4⋊C4C2×C22⋊D6111yescharacteristic
515C4⋊C4C2×C22⋊D6111yescharacteristic
516C22⋊C4—21no—
517C22⋊C4—21no—
518C22⋊C4—41no—
519C22⋊C4—41no—
520C22⋊C4—41no—
521C22⋊C4—1221no—
522C22⋊C4—2441no—
523C22⋊C4—2441no—
524C22⋊C4—2441no—
525C42—21no—
526C42—41no—
527C42—2441no—
528C42C23×D6111yescharacteristic
529C23—21no—
530C23—21no—
531C23—41no—
532C23—82no—
533C23—82no—
534C23—161no—
535C23—1221no—
536Q8—21no—
537Q8—21no—
538Q8—41no—
539Q8—41no—
540Q8—2441no—
541Q8—2441no—
542Q8—2441no—
543Q8—2441no—
544D4—21no—
545D4—21no—
546D4—21no—
547D4—21no—
548D4—41no—
549D4—41no—
550D4—41no—
551D4—41no—
552D4—81no—
553D4—82no—
554D4—82no—
555D4—1221no—
556D4—1221no—
557D4—4882no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—1221no—
568C2×C4—4882no—
569C2×C4—2441no—
570C2×C4—2441no—
571C2×C4—2441no—
572C2×C4—2441no—
573C2×C4C2×D4×D6111yescharacteristic
574C2×C4C22×C22⋊D6111yescharacteristic
575C2×C4#1451982211yescharacteristic
576C8—21no—
577C8—21no—
578C8—41no—
579C8—612no—
580C8—1221no—
581C8—2441no—
582C8—2441no—
583C8—2441no—
584C8—2441no—
585C22—41no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22—81no—
592C22—81no—
593C22—81no—
594C22—164no—
595C22—2441no—
596C22—2441no—
597C22—2441no—
598C22D4×C22⋊D6111yescharacteristic
599C4—21no—
600C4—41no—
601C4—41no—
602C4—41no—
603C4—41no—
604C4—81no—
605C4—4881no—
606C4—4881no—
607C4—2441no—
608C4AGL(1,ℤ/8)×D6111yescharacteristic
609C4#1684056011yescharacteristic
610C4#1684056611yescharacteristic
611C4#1684057311yescharacteristic
612C2—81no—
613C2—81no—
614C2—161no—
615C2—4881no—
616C2#2145734011yescharacteristic
617C2#2145743311yescharacteristic
618C2#2145759211yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3324C2
socleC22×C61D42
prime radical#21457415C2
hypercenter#3324D61
hypocenterC61#20250
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3324

Complements

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

NH ≅ G/Ntype
1. #21458803C2semidirect
2. #21458405C2semidirect
4. #21445492C2semidirect
5. #21445377C2semidirect
6. #21445332C2semidirect
7. #21445306C2semidirect
8. #21445165C2semidirect
9. #21445075C2semidirect
10. #21445053C2semidirect
11. #21444998C2semidirect
12. #21444928C2semidirect
13. #21444731C2semidirect
14. #21444663C2semidirect
15. C61×#3324C2semidirect
17. #16840485C22semidirect
19. #16840167C22semidirect
20. #16840165C22semidirect
22. #16840141C22semidirect
27. #16840047C22semidirect
35. #16839964C22semidirect
39. Q8×#92418C22semidirect
42. #16839841C22semidirect
45. #16839513C22semidirect
46. #16839512C22semidirect
47. #16839508C22semidirect
48. #16839507C22semidirect
49. #16839507C22semidirect
51. Q8⋊#92418C22semidirect
54. #16839489C22semidirect
55. #16839484C22semidirect
305. #3324D61semidirect
376. C61#20250semidirect

Maximal subgroups

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

subgroupindex
#214588032
#214584052
#214574152
#214454922
#214453772
#214453322
#214453062
#214451652
#214450752
#214450532
#214449982
#214449282
#214447312
#214446632
C61×#33242
#2025061