Finite Group Databasebeta

← #29663206

Subgroups — #29663206

Order 15616.

710 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#21456962C211yesmaximal, characteristic, complemented
2#21456759C211yesmaximal, characteristic
3#492×D61C211yesmaximal, characteristic
4#21454748C211yesmaximal, characteristic, complemented
5#21445294C211yesmaximal, characteristic, complemented
6#21445293C211yesmaximal, characteristic, complemented
7#21445290C211yesmaximal, characteristic, complemented
8#21445289C211yesmaximal, characteristic, complemented
9#21445168C211yesmaximal, characteristic, complemented
10#21445167C211yesmaximal, characteristic, complemented
11#21445164C211yesmaximal, characteristic, complemented
12#21444564C211yesmaximal, characteristic, complemented
13#21444489C211yesmaximal, characteristic, complemented
14#21444485C211yesmaximal, characteristic
15C61×#1666C211yesmaximal, characteristic, complemented
16C2×SD16×D61—21no—
17C2×SD16×D61—21no—
18#16840316C2211yescharacteristic
19C4⋊Q8×D61C2211yescharacteristic
20#127×D61C2211yescharacteristic
21#16840159—21no—
22#16840136C2211yescharacteristic
23#16840135C2211yescharacteristic
24#16840134C2211yescharacteristic
25#16840133—21no—
26#16840133—21no—
27#16840133—21no—
28#16840111—21no—
29#16840107—21no—
30#16840094—21no—
31#16840091—21no—
32#16840048C2211yescharacteristic, complemented
33#16840047C2211yescharacteristic, complemented
34#16840044C2211yescharacteristic, complemented
35#16839963C2211yescharacteristic
36#16839962C2211yescharacteristic
37#16839959C2211yescharacteristic
38#16839914—21no—
39#16839911—21no—
40#16839893C2211yescharacteristic, complemented
41#16839891—21no—
42#16839889C2211yescharacteristic, complemented
43C8×Dic122—21no—
44C8×Dic122C2211yescharacteristic, complemented
45#16839814C2211yescharacteristic
46#16839814C2211yescharacteristic
47#16839787—21no—
48#16839784C2211yescharacteristic
49#16839716—21no—
50C4×C8×D61C2211yescharacteristic
51#16839514C2211yescharacteristic, complemented
52#16839514C2211yescharacteristic, complemented
53#16839513C2211yescharacteristic, complemented
54#16839509C2211yescharacteristic, complemented
55#16839508C2211yescharacteristic, complemented
56#16839507C2211yescharacteristic, complemented
57#16839506C2211yescharacteristic
58#16839505C2211yescharacteristic
59#16839505C2211yescharacteristic
60C61×#492C2211yescharacteristic
61C61×#475C2211yescharacteristic, complemented
62C61×#460C2211yescharacteristic
63C8×Q8×C61C2211yescharacteristic, complemented
64C61×#334C2211yescharacteristic, complemented
65C61×#332C2211yescharacteristic, complemented
66C61×#331C2211yescharacteristic
67C2×Q8×D61—21no—
68C2×D4×D61—21no—
69C2×D4×D61—21no—
70C2×D4×D61—21no—
71C2×#11758190—21no—
72C2×#11758190—21no—
73C2×#11758190—21no—
74C2×#11758200—21no—
75C2×#11758200—21no—
76C2×#11758200—21no—
77C2×#11758201—21no—
78C2×#11758201—21no—
79C2×#11758201—21no—
80SD16×D61—44no—
81SD16×D61—44no—
82C2×#11758189—21no—
83C2×C8×D61D411yescharacteristic
84C2×C8×D61D411yescharacteristic
85#14519777C2311yescharacteristic
86#14519776—21no—
87#14519776—21no—
88#14519776D411yescharacteristic
89#14519775—21no—
90#14519775D411yescharacteristic
91#14519774C2311yescharacteristic
92#14519773—21no—
93#14519773—21no—
94#14519773—21no—
95#14519767—21no—
96#14519757D411yescharacteristic
97#14519755—21no—
98#14519753D411yescharacteristic
99C4×Dic122—21no—
100C4×Dic122C2311yescharacteristic
101#14519734—21no—
102C4⋊C4×D61—21no—
103C4⋊C4×D61—21no—
104C42×D61C2311yescharacteristic
105#14519709—21no—
106#14519708—21no—
107#14519708—21no—
108C4⋊#92418—21no—
109C4⋊#92418C2311yescharacteristic
110C4⋊#92418C2311yescharacteristic
111C4⋊#92418C2311yescharacteristic
112#14519706—21no—
113#14519706—21no—
114#14519706—21no—
115#14519702—21no—
116#14519701—21no—
117#14519700—21no—
118#14519699—21no—
119#14519698—21no—
120#14519697—21no—
121#14519689—21no—
122#14519689—21no—
123#14519688—21no—
124#14519687—21no—
125C4×#92418C2311yescharacteristic
126C8×Dic61—21no—
127C8×Dic61—21no—
128C8×Dic61—21no—
129C8×Dic61D411yescharacteristic
130C8×Dic61D411yescharacteristic
131C2×SD16×C61—21no—
132C2×SD16×C61—21no—
133C2×SD16×C61—21no—
134C4⋊Q8×C61C2311yescharacteristic
135#127×C61C2311yescharacteristic
136C4×Q8×C61C2311yescharacteristic
137#106×C61—21no—
138C4⋊C8×C61—21no—
139C4⋊C8×C61C2311yescharacteristic
140C4⋊C8×C61C2311yescharacteristic
141C4⋊C8×C61C2311yescharacteristic
142#103×C61—21no—
143#102×C61—21no—
144C4×C8×C61—21no—
145C4×C8×C61C2311yescharacteristic
146C23×D61—41no—
147C2×Q8×C61—21no—
148C2×Q8×C61C2×D411yescharacteristic
149C2×D4×C61—21no—
150C2×D4×C61—21no—
151C2×D4×C61—21no—
152C2×C22⋊D61—41no—
153C2×C22⋊D61—41no—
154C2×C22⋊D61—41no—
155Q8×D61—42no—
156Q8×D61—42no—
157D4×D61—42no—
158D4×D61—42no—
159D4×D61—81no—
160D4×D61—81no—
161D4×D61—82no—
162C2×D244—21no—
163C2×D244—21no—
164C2×D244—21no—
165C2×C4×D61—41no—
166C2×C4×D61C2×D411yescharacteristic
167C2×C4×D61C2×D411yescharacteristic
168C2×C4×D61#4111yescharacteristic
169C2×Dic122—21no—
170C2×Dic122—21no—
171C2×Dic122—41no—
172C2×Dic122C2×D411yescharacteristic
173SD16×C61—42no—
174SD16×C61—42no—
175SD16×C61—81no—
176C2×C8×C61—21no—
177C2×C8×C61—21no—
178C2×C8×C61—21no—
179C2×C8×C61C2×D411yescharacteristic
180C2×C8×C61C2×D411yescharacteristic
181C4⋊C4×C61—21no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61C2×D411yescharacteristic
185C42×C61—21no—
186C42×C61C2411yescharacteristic
187#11758201—42no—
188#11758201—42no—
189#11758201—81no—
190#11758200—42no—
191#11758200—42no—
192#11758200—81no—
193#11758198—21no—
194#11758198—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61C2×D411yescharacteristic
199#11758196—21no—
200#11758196—21no—
201#11758196—21no—
202#11758196—21no—
203#11758196—21no—
204#11758196—41no—
205#11758196—41no—
206C4×Dic61—21no—
207C4×Dic61C2×D411yescharacteristic
208C4×Dic61C2×D411yescharacteristic
209C4×Dic61#4111yescharacteristic
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418C2×D411yescharacteristic
214C2×#92418C2×D411yescharacteristic
215#11758190—42no—
216#11758190—42no—
217#11758190—81no—
218#11758189—42no—
219C8×D61—22no—
220C8×D61SD1614yescomplemented
221C23×C61—41no—
222C22×D61—41no—
223C22×D61—82no—
224C22×D61—82no—
225C22×D61—82no—
226C22×D61#12111yescharacteristic
227Q8×C61—21no—
228Q8×C61—21no—
229Q8×C61—41no—
230Q8×C61—41no—
231D4×C61—41no—
232D4×C61—41no—
233D4×C61—41no—
234D4×C61—41no—
235D4×C61—81no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—41no—
239C2×C4×C61C22×D411yescharacteristic
240C2×C4×C61C22×D411yescharacteristic
241C2×C4×C61C2×#4111yescharacteristic
242C22⋊D61—82no—
243C22⋊D61—82no—
244C22⋊D61—82no—
245C2×Dic61—21no—
246C2×Dic61—21no—
247C2×Dic61—41no—
248C2×Dic61#12111yescharacteristic
249C2×Dic61#12111yescharacteristic
250C2×Dic61#12111yescharacteristic
251D244—41no—
252D244—41no—
253D244—41no—
254D244—41no—
255D244—81no—
256C4×D61—21no—
257C4×D61—21no—
258C4×D61—22no—
259C4×D61—82no—
260C4×D61C2×SD1612yes—
261C4×D61AGL(1,ℤ/8)12yes—
262Dic122—21no—
263Dic122—21no—
264Dic122—42no—
265Dic122—41no—
266Dic122—41no—
267Dic122—82no—
268C8×C61—21no—
269C8×C61—41no—
270C8×C61—41no—
271C8×C61—41no—
272C8×C61C2×SD1612yes—
273#92418—21no—
274#92418—41no—
275#92418—41no—
276#92418—41no—
277#92418C2×SD1612yes—
278#19202—611nomaximal, Sylow
279C22×C61—81no—
280C22×C61—81no—
281C22×C61—81no—
282C22×C61C2×#12111yescharacteristic
283C2×D61—21no—
284C2×D61—21no—
285C2×D61—81no—
286C2×D61—81no—
287C2×D61—81no—
288C2×D61—164no—
289C2×D61#46012yes—
290C4×C61—21no—
291C4×C61—41no—
292C4×C61—41no—
293C4×C61—81no—
294C4×C61C22×SD1611yescharacteristic
295C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
296C4×C61#57711yescharacteristic
297C4×C61#57711yescharacteristic
298Dic61—21no—
299Dic61—41no—
300Dic61—41no—
301Dic61—81no—
302Dic61#46012yes—
303Dic61#47112yes—
304#3229—611no—
305#3208—611no—
306C2×#492—611no—
307#2960—611no—
308#1686—611no—
309#1686—611no—
310#1686—611no—
311#1686—611no—
312#1666—611no—
313#1666—611no—
314#1666—611no—
315#1666D6111yescharacteristic, complemented
316#1564—611no—
317#1546—611no—
318#1545—611no—
319C2×C61—161no—
320C2×C61C2×#46011yescharacteristic
321C2×C61#305711yescharacteristic
322C2×C61C2×#47111yescharacteristic
323D61—22no—
324D61—161no—
325C22×SD16—1221no—
326C22×SD16—1221no—
327#533—611no—
328C2×C4⋊Q8—611no—
329C2×#127—611no—
330#498—1221no—
331#492—611no—
332#492—611no—
333#492—611no—
334#492—1221no—
335#492—1221no—
336#492—1221no—
337#492C2×D6111yescharacteristic
338#487—1221no—
339#486—1221no—
340#483—1221no—
341#483—1221no—
342#475—611no—
343#475—611no—
344#475—611no—
345#475C2×D6111yescharacteristic, complemented
346#460—611no—
347#460—611no—
348#460—611no—
349#460C2×D6111yescharacteristic
350#450—1221no—
351#450—1221no—
352#446—611no—
353#446—611no—
354#446—1221no—
355C8×Q8—611no—
356C8×Q8—1221no—
357C8×Q8C2×D6111yescharacteristic, complemented
358#431—611no—
359#431—611no—
360#424—1221no—
361#423—611no—
362#405—1221no—
363C2×C4×C8—611no—
364#335—611no—
365#335—611no—
366#334—611no—
367#334C2×D6111yescharacteristic, complemented
368#332—611no—
369#332—611no—
370#332—611no—
371#332C2×D6111yescharacteristic, complemented
372#331—611no—
373#331—611no—
374#331—611no—
375#331C2×D6111yescharacteristic
376C61#1920211yescharacteristic, complemented, Sylow
377C22×Q8—1221no—
378C22×D4—1221no—
379C22×D4—1221no—
380C22×D4—1221no—
381C2×SD16—21no—
382C2×SD16—21no—
383C2×SD16—21no—
384C2×SD16—1221no—
385C2×SD16—1221no—
386C2×SD16—1221no—
387C2×SD16—1221no—
388C2×SD16—1221no—
389C2×SD16—1221no—
390C2×SD16—1221no—
391C2×SD16—1221no—
392C2×SD16—1221no—
393C2×SD16—2444no—
394C2×SD16—2444no—
395C2×M16—1221no—
396C22×C8—611no—
397C22×C8—611no—
398C4⋊Q8—611no—
399C4⋊Q8—611no—
400C4⋊Q8—611no—
401C4⋊Q8—1221no—
402C4⋊Q8—1221no—
403C4⋊Q8—1221no—
404C4⋊Q8C22×D6111yescharacteristic
405#127—611no—
406#127—1221no—
407#127—1221no—
408#127—1221no—
409#127C22×D6111yescharacteristic
410#125—1221no—
411C22⋊Q8—611no—
412C22⋊Q8—611no—
413C22⋊Q8—1221no—
414C4×Q8—611no—
415C4×Q8—1221no—
416C4×Q8C22×D6111yescharacteristic
417#117—1221no—
418C2×C4⋊C4—1221no—
419C2×C4⋊C4—1221no—
420C2×C42—611no—
421#106—21no—
422#106—1221no—
423#106—1221no—
424#106—1221no—
425C4⋊C8—21no—
426C4⋊C8—611no—
427C4⋊C8—611no—
428C4⋊C8—611no—
429C4⋊C8—1221no—
430C4⋊C8—1221no—
431C4⋊C8—1221no—
432C4⋊C8—1221no—
433C4⋊C8C22×D6111yescharacteristic
434C4⋊C8C22×D6111yescharacteristic
435C4⋊C8C22×D6111yescharacteristic
436#103—21no—
437#103—1221no—
438#103—1221no—
439#103—1221no—
440#102—21no—
441#102—1221no—
442#102—1221no—
443#102—1221no—
444C22⋊C8—1221no—
445C22⋊C8—1221no—
446#97—1221no—
447#97—1221no—
448C4×C8—21no—
449C4×C8—611no—
450C4×C8—611no—
451C4×C8—611no—
452C4×C8—1221no—
453C4×C8—1221no—
454C4×C8—1221no—
455C4×C8C22×D6111yescharacteristic
456C24—2441no—
457C2×Q8—21no—
458C2×Q8—611no—
459C2×Q8—1221no—
460C2×Q8—1221no—
461C2×Q8—2441no—
462C2×Q8—2442no—
463C2×Q8—2442no—
464C2×Q8D4×D6111yescharacteristic
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—1221no—
469C2×D4—1221no—
470C2×D4—1221no—
471C2×D4—4881no—
472C2×D4—4881no—
473C2×D4—4882no—
474C2×D4—2442no—
475C2×D4—2442no—
476C2×D4—2441no—
477C2×D4—2441no—
478C2×D4—2441no—
479C22×C4—611no—
480C22×C4—611no—
481C22×C4—611no—
482C22×C4—2441no—
483SD16—42no—
484SD16—42no—
485SD16—81no—
486SD16—4881no—
487SD16—4881no—
488SD16—4881no—
489SD16—2442no—
490SD16—2442no—
491SD16—2442no—
492SD16—2442no—
493SD16—2442no—
494SD16—2442no—
495M16—2442no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—611no—
500C2×C8—614no—
501C2×C8—611no—
502C2×C8—1222no—
503C2×C8—1221no—
504C2×C8—1221no—
505C2×C8—1221no—
506C2×C8D4×D6111yescharacteristic
507C2×C8D4×D6111yescharacteristic
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—611no—
512C4⋊C4—1221no—
513C4⋊C4—1221no—
514C4⋊C4—1221no—
515C4⋊C4—1221no—
516C4⋊C4—1221no—
517C4⋊C4—1221no—
518C4⋊C4—1221no—
519C4⋊C4—1221no—
520C4⋊C4—2441no—
521C4⋊C4—2441no—
522C4⋊C4D4×D6111yescharacteristic
523C22⋊C4—1221no—
524C22⋊C4—1221no—
525C42—21no—
526C42—611no—
527C42—611no—
528C42—611no—
529C42—1221no—
530C42C23×D6111yescharacteristic
531C23—41no—
532C23—611no—
533C23—4882no—
534C23—4882no—
535C23—4882no—
536C23—2441no—
537Q8—21no—
538Q8—21no—
539Q8—41no—
540Q8—41no—
541Q8—1221no—
542Q8—1221no—
543Q8—4882no—
544Q8—2442no—
545Q8—2441no—
546Q8—2441no—
547D4—41no—
548D4—41no—
549D4—41no—
550D4—41no—
551D4—81no—
552D4—4881no—
553D4—4882no—
554D4—4882no—
555D4—4882no—
556D4—2441no—
557D4—2441no—
558D4—2441no—
559D4—2441no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—41no—
563C2×C4—611no—
564C2×C4—611no—
565C2×C4—612no—
566C2×C4—612no—
567C2×C4—611no—
568C2×C4—1222no—
569C2×C4—1221no—
570C2×C4—1221no—
571C2×C4—1221no—
572C2×C4—1221no—
573C2×C4—4882no—
574C2×C4—2441no—
575C2×C4C2×D4×D6111yescharacteristic
576C2×C4C2×D4×D6111yescharacteristic
577C2×C4#41×D6111yescharacteristic
578C8—21no—
579C8—41no—
580C8—41no—
581C8—41no—
582C8—612no—
583C8—1221no—
584C8—2441no—
585C8—2441no—
586C8—2441no—
587C8SD16×D6112yes—
588C22—81no—
589C22—81no—
590C22—81no—
591C22—612no—
592C22—1221no—
593C22—1221no—
594C22—9764no—
595C22—4881no—
596C22—4881no—
597C22—4881no—
598C22#121×D6111yescharacteristic
599C4—21no—
600C4—41no—
601C4—41no—
602C4—81no—
603C4—612no—
604C4—612no—
605C4—1221no—
606C4—4881no—
607C4—2441no—
608C4—2441no—
609C4C2×SD16×D6111yescharacteristic
610C4AGL(1,ℤ/8)×D6111yescharacteristic
611C4#1684056811yescharacteristic
612C4#1684056911yescharacteristic
613C2—161no—
614C2—1222no—
615C2—9761no—
616C2#460×D6111yescharacteristic
617C2#2145545311yescharacteristic
618C2#471×D6111yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1666C2
socleC22×C61C2×#121
prime radical#127×D61C22
hypercenter#1666D61
hypocenterC61#19202
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1666

Complements

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

NH ≅ G/Ntype
1. #21456962C2semidirect
4. #21454748C2semidirect
5. #21445294C2semidirect
6. #21445293C2semidirect
7. #21445290C2semidirect
8. #21445289C2semidirect
9. #21445168C2semidirect
10. #21445167C2semidirect
11. #21445164C2semidirect
12. #21444564C2semidirect
13. #21444489C2semidirect
15. C61×#1666C2semidirect
32. #16840048C22semidirect
33. #16840047C22semidirect
34. #16840044C22semidirect
40. #16839893C22semidirect
42. #16839889C22semidirect
44. C8×Dic122C22semidirect
51. #16839514C22semidirect
52. #16839514C22semidirect
53. #16839513C22semidirect
54. #16839509C22semidirect
55. #16839508C22semidirect
56. #16839507C22semidirect
61. C61×#475C22semidirect
63. C8×Q8×C61C22semidirect
64. C61×#334C22semidirect
65. C61×#332C22semidirect
220. C8×D61SD16semidirect
315. #1666D61semidirect
345. #475C2×D61semidirect
357. C8×Q8C2×D61semidirect
367. #334C2×D61semidirect
371. #332C2×D61semidirect
376. C61#19202semidirect

Maximal subgroups

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

subgroupindex
#214569622
#214567592
#492×D612
#214547482
#214452942
#214452932
#214452902
#214452892
#214451682
#214451672
#214451642
#214445642
#214444892
#214444852
C61×#16662
#1920261