Finite Group Databasebeta

← #29670410

Subgroups — #29670410

Order 15616.

642 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#21458811C211yesmaximal, characteristic, complemented
2#21458052C211yesmaximal, characteristic, complemented
3#21457637C211yesmaximal, characteristic, complemented
4#21445455C211yesmaximal, characteristic, complemented
5#21445349C211yesmaximal, characteristic, complemented
6#21445263C211yesmaximal, characteristic, complemented
7#21445237C211yesmaximal, characteristic, complemented
8#21445158C211yesmaximal, characteristic, complemented
9#21445033C211yesmaximal, characteristic, complemented
10#21444990C211yesmaximal, characteristic, complemented
11#21444878C211yesmaximal, characteristic, complemented
12#21444872C211yesmaximal, characteristic
13#21444662C211yesmaximal, characteristic, complemented
14SD16⋊#92418C211yesmaximal, characteristic, complemented
15C61×#3361C211yesmaximal, characteristic, complemented
16#16840485C2211yescharacteristic
17D4×D244C2211yescharacteristic
18#16840144—21no—
19#16840138C2211yescharacteristic
20#16840134C2211yescharacteristic, complemented
21#16840112C2211yescharacteristic, complemented
22#16840110C2211yescharacteristic
23#16840109C2211yescharacteristic
24#16840088—21no—
25#16840080—21no—
26#16840049C2211yescharacteristic
27#16840042C2211yescharacteristic
28#16840011—21no—
29#16839993—21no—
30#16839981—21no—
31#16839977—21no—
32#16839972C2211yescharacteristic, complemented
33#16839962C2211yescharacteristic
34#16839962C2211yescharacteristic
35#16839954C2211yescharacteristic
36#16839912—21no—
37#16839911—21no—
38Q8×#92418C2211yescharacteristic, complemented
39C4×#11758201C2211yescharacteristic
40C4×#11758200—21no—
41#16839841C2211yescharacteristic
42D4×#92418C2211yescharacteristic
43#16839765—21no—
44C2×#14519697—21no—
45#16839515C2211yescharacteristic, complemented
46#16839512C2211yescharacteristic, complemented
47#16839511C2211yescharacteristic, complemented
48#16839507C2211yescharacteristic, complemented
49#16839506C2211yescharacteristic, complemented
50#16839505C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic, complemented
52Q8⋊#92418C2211yescharacteristic, complemented
53D4⋊#92418C2211yescharacteristic
54#16839489C2211yescharacteristic
55#16839483C2211yescharacteristic, complemented
56C61×#559C2211yescharacteristic, complemented
57C61×#554C2211yescharacteristic
58C61×#488—21no—
59C61×#486C2211yescharacteristic, complemented
60C61×#478—21no—
61C61×#477C2211yescharacteristic, complemented
62C61×#463—21no—
63C61×#459C2211yescharacteristic
64C8×Q8×C61C2211yescharacteristic, complemented
65C4×SD16×C61C2211yescharacteristic
66C4×D8×C61—21no—
67C22×D244—21no—
68C2×D4×D61—21no—
69C2×#11758202—21no—
70C2×#11758200—21no—
71C2×#11758200—21no—
72C2×#11758201—21no—
73C2×#11758201—21no—
74C2×#11758201D411yescharacteristic
75C2×#11758199—21no—
76C2×#11758199—41no—
77C2×#11758189—21no—
78C22×#92418—21no—
79#14519777C2311yescharacteristic
80#14519774C2311yescharacteristic
81#14519772—41no—
82#14519765—21no—
83#14519753—21no—
84#14519752—21no—
85#14519751—21no—
86#14519747—41no—
87C4×Dic122—21no—
88C4×D244C2311yescharacteristic
89#14519735—21no—
90#14519710—21no—
91#14519708—21no—
92#14519708D411yescharacteristic
93C4⋊#92418—21no—
94C4⋊#92418C2311yescharacteristic
95C4⋊#92418C2311yescharacteristic
96C4⋊#92418C2311yescharacteristic
97#14519701—41no—
98#14519701D411yescharacteristic
99#14519700—21no—
100#14519699—21no—
101#14519699—21no—
102#14519698—21no—
103#14519698—21no—
104#14519697—21no—
105#14519697—21no—
106#14519697—21no—
107#14519697—21no—
108#14519697—21no—
109#14519697D411yescharacteristic
110C22⋊#92418—21no—
111#14519689—21no—
112C4×#92418—21no—
113C4×#92418C2311yescharacteristic
114C4×#92418C2311yescharacteristic
115C2×SD16×C61D411yescharacteristic
116C2×D8×C61—21no—
117C4⋊Q8×C61C2311yescharacteristic
118#127×C61C2311yescharacteristic
119#125×C61—21no—
120#124×C61—21no—
121#124×C61—21no—
122#123×C61—41no—
123C22⋊Q8×C61—21no—
124C22⋊Q8×C61—41no—
125C4×Q8×C61C2311yescharacteristic
126C4×Q8×C61C2311yescharacteristic
127C4×D4×C61—21no—
128C4×D4×C61—21no—
129C4×D4×C61—21no—
130C4×D4×C61C2311yescharacteristic
131C2×C4⋊C4×C61—21no—
132#107×C61—21no—
133#106×C61D411yescharacteristic
134C4⋊C8×C61—21no—
135C4⋊C8×C61C2311yescharacteristic
136#103×C61—21no—
137#103×C61D411yescharacteristic
138#102×C61—21no—
139#102×C61—21no—
140#102×C61—21no—
141#102×C61D411yescharacteristic
142C4×C8×C61—21no—
143C4×C8×C61C2311yescharacteristic
144C23×D61—41no—
145C2×Q8×C61—21no—
146C2×Q8×C61C2×D411yescharacteristic
147C2×Q8×C61C2×D411yescharacteristic
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61—21no—
151C2×D4×C61C2×D411yescharacteristic
152C22×C4×C61—21no—
153C22×C4×C61—41no—
154C22×C4×C61—41no—
155C2×C22⋊D61—41no—
156D4×D61—42no—
157D4×D61—81no—
158C2×D244—21no—
159C2×D244—21no—
160C2×D244—21no—
161C2×D244—42no—
162C2×D244—41no—
163C2×D244—81no—
164C2×D244C2×D411yescharacteristic
165C2×C4×D61—21no—
166C2×Dic122—21no—
167SD16×C61—22no—
168D8×C61—42no—
169C2×C8×C61—21no—
170C2×C8×C61C2×D411yescharacteristic
171C2×C8×C61C2×D411yescharacteristic
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—41no—
177C4⋊C4×C61—41no—
178C4⋊C4×C61C2×D411yescharacteristic
179C4⋊C4×C61C2×D411yescharacteristic
180C4⋊C4×C61C2×D411yescharacteristic
181C22⋊C4×C61—21no—
182C22⋊C4×C61—41no—
183C22⋊C4×C61—41no—
184C22⋊C4×C61—41no—
185C22⋊C4×C61—41no—
186C42×C61—21no—
187C42×C61—21no—
188C42×C61C2411yescharacteristic
189#11758202—81no—
190#11758201—41no—
191#11758201—81no—
192#11758201—81no—
193#11758200—42no—
194#11758200—81no—
195#11758199—81no—
196#11758199—82no—
197#11758198—21no—
198#11758198—41no—
199#11758198—41no—
200#11758198—41no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61C2×D411yescharacteristic
204#11758196—41no—
205C4×Dic61—41no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—81no—
211C2×#92418C2×D411yescharacteristic
212C2×#92418C2×D411yescharacteristic
213#11758189—81no—
214C23×C61—21no—
215C23×C61—41no—
216C22×D61—21no—
217C22×D61—41no—
218C22×D61—82no—
219C22×D61—82no—
220C22×D61—161no—
221Q8×C61—21no—
222Q8×C61—21no—
223Q8×C61—21no—
224Q8×C61—21no—
225Q8×C61—81no—
226D4×C61—21no—
227D4×C61—41no—
228D4×C61—41no—
229D4×C61—41no—
230D4×C61—41no—
231D4×C61—81no—
232D4×C61#13512yes—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—161no—
242C2×C4×C61—161no—
243C2×C4×C61C22×D411yescharacteristic
244C2×C4×C61C22×D411yescharacteristic
245C2×C4×C61H(5,𝔽2)11yescharacteristic
246C22⋊D61—82no—
247C2×Dic61—21no—
248C2×Dic61—41no—
249D244—21no—
250D244—21no—
251D244—41no—
252D244—41no—
253D244—41no—
254D244—41no—
255D244—81no—
256D244—82no—
257C4×D61—41no—
258C4×D61—41no—
259Dic122—41no—
260Dic122—41no—
261C8×C61—21no—
262C8×C61—21no—
263C8×C61—41no—
264#92418—21no—
265#92418—21no—
266#92418—41no—
267#92418—41no—
268#92418—41no—
269#92418—41no—
270#19825—611nomaximal, Sylow
271C22×C61—22no—
272C22×C61—41no—
273C22×C61—41no—
274C22×C61—81no—
275C22×C61—81no—
276C22×C61—81no—
277C22×C61D4211yescharacteristic
278C2×D61—41no—
279C2×D61—41no—
280C2×D61—41no—
281C2×D61—81no—
282C2×D61—81no—
283C2×D61—81no—
284C2×D61—164no—
285C4×C61—21no—
286C4×C61—41no—
287C4×C61—41no—
288C4×C61—41no—
289C4×C61—41no—
290C4×C61—41no—
291C4×C61—81no—
292C4×C61C2×#13511yescharacteristic
293C4×C61#57511yescharacteristic
294C4×C61#57611yescharacteristic
295C4×C61#57711yescharacteristic
296Dic61—41no—
297Dic61—81no—
298#3394—611no—
299#3361D6111yescharacteristic, complemented
300#3324—611no—
301#3295—611no—
302#1717—611no—
303#1697—611no—
304#1682—611no—
305#1678—611no—
306#1665—611no—
307#1644—611no—
308#1638—611no—
309#1620—611no—
310#1619—611no—
311#1583—611no—
312SD16⋊C8—611no—
313C2×C61—42no—
314C2×C61—161no—
315C2×C61#326811yescharacteristic
316C2×C61#328211yescharacteristic
317C2×C61#328811yescharacteristic
318D61—81no—
319D61—161no—
320#559C2×D6111yescharacteristic, complemented
321#556—611no—
322#554C2×D6111yescharacteristic
323D42—611no—
324#495—1221no—
325#493—611no—
326#492—611no—
327#488—21no—
328#487—611no—
329#486—611no—
330#486—611no—
331#486C2×D6111yescharacteristic, complemented
332#482—1221no—
333#481—1221no—
334#478—21no—
335#477C2×D6111yescharacteristic, complemented
336#476—611no—
337#474—611no—
338#468—1221no—
339#465—1221no—
340#464—1221no—
341#463—21no—
342#463—1221no—
343#462—611no—
344#460—611no—
345#460—611no—
346#459—611no—
347#459C2×D6111yescharacteristic
348#450—1221no—
349#450—1221no—
350C8×Q8—611no—
351C8×Q8C2×D6111yescharacteristic, complemented
352C4×SD16—611no—
353C4×SD16—1221no—
354C4×SD16C2×D6111yescharacteristic
355C4×D8—21no—
356#436—611no—
357C8×D4—611no—
358#418—1221no—
359C2×#102—1221no—
360#335—611no—
361#334—611no—
362#333—611no—
363#332—611no—
364#331—611no—
365#331—611no—
366Q8⋊C8—611no—
367Q8⋊C8—611no—
368D4⋊C8—611no—
369D4⋊C8—611no—
370#322—611no—
371C61#1982511yescharacteristic, complemented, Sylow
372C22×D4—1221no—
373C22×D4—1221no—
374C2×Q16—1221no—
375C2×SD16—611no—
376C2×SD16—1221no—
377C2×SD16—1221no—
378C2×SD16—1221no—
379C2×SD16—1221no—
380C2×SD16C22⋊D6111yescharacteristic
381C2×D8—21no—
382C2×D8—1221no—
383C2×D8—2441no—
384C2×M16—1221no—
385C22×C8—1221no—
386C4⋊Q8—611no—
387C4⋊Q8C22×D6111yescharacteristic
388#127—611no—
389#127C22×D6111yescharacteristic
390#126—2441no—
391#125—21no—
392#124—21no—
393#124—21no—
394#124—1221no—
395#123—41no—
396C22⋊Q8—21no—
397C22⋊Q8—41no—
398C22⋊Q8—1221no—
399#121—1221no—
400#121—1221no—
401#120—2441no—
402C4×Q8—1221no—
403C4×Q8C22×D6111yescharacteristic
404C4×Q8C22×D6111yescharacteristic
405C4×D4—21no—
406C4×D4—21no—
407C4×D4—21no—
408C4×D4—611no—
409C4×D4C22×D6111yescharacteristic
410#117—1221no—
411C2×C4⋊C4—21no—
412#107—21no—
413#107—1221no—
414#106—611no—
415#106—1221no—
416#106C22⋊D6111yescharacteristic
417C4⋊C8—21no—
418C4⋊C8—611no—
419C4⋊C8—611no—
420C4⋊C8—611no—
421C4⋊C8—1221no—
422C4⋊C8C22×D6111yescharacteristic
423#103—21no—
424#103—611no—
425#103—1221no—
426#103—2441no—
427#103C22⋊D6111yescharacteristic
428#102—21no—
429#102—21no—
430#102—21no—
431#102—611no—
432#102—1221no—
433#102—1221no—
434#102—1221no—
435#102—1221no—
436#102—1221no—
437#102—1221no—
438#102—1221no—
439#102—1221no—
440#102—1221no—
441#102C22⋊D6111yescharacteristic
442C22⋊C8—1221no—
443C22⋊C8—1221no—
444C4×C8—21no—
445C4×C8—611no—
446C4×C8—611no—
447C4×C8—1221no—
448C4×C8C22×D6111yescharacteristic
449C24—2441no—
450C2×Q8—21no—
451C2×Q8—1221no—
452C2×Q8D4×D6111yescharacteristic
453C2×Q8C2×C22⋊D6111yescharacteristic
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—611no—
458C2×D4—1221no—
459C2×D4—1221no—
460C2×D4—1221no—
461C2×D4—4881no—
462C2×D4—4881no—
463C2×D4—2441no—
464C2×D4—2442no—
465C2×D4—2441no—
466C2×D4—2442no—
467C2×D4C2×C22⋊D6111yescharacteristic
468C22×C4—21no—
469C22×C4—41no—
470C22×C4—41no—
471C22×C4—1221no—
472Q16—4881no—
473SD16—22no—
474SD16—4881no—
475SD16—4881no—
476SD16—4881no—
477SD16—2442no—
478SD16—2441no—
479D8—42no—
480D8—4881no—
481D8—4882no—
482M16—4881no—
483C2×C8—21no—
484C2×C8—611no—
485C2×C8—611no—
486C2×C8—1221no—
487C2×C8—1221no—
488C2×C8—1221no—
489C2×C8—1221no—
490C2×C8—4881no—
491C2×C8C2×C22⋊D6111yescharacteristic
492C2×C8C2×C22⋊D6111yescharacteristic
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—41no—
498C4⋊C4—41no—
499C4⋊C4—611no—
500C4⋊C4—1221no—
501C4⋊C4—1221no—
502C4⋊C4—2441no—
503C4⋊C4D4×D6111yescharacteristic
504C4⋊C4C2×C22⋊D6111yescharacteristic
505C4⋊C4C2×C22⋊D6111yescharacteristic
506C22⋊C4—21no—
507C22⋊C4—41no—
508C22⋊C4—41no—
509C22⋊C4—41no—
510C22⋊C4—41no—
511C22⋊C4—1221no—
512C22⋊C4—2441no—
513C22⋊C4—2441no—
514C22⋊C4—2441no—
515C42—21no—
516C42—21no—
517C42—2441no—
518C42C23×D6111yescharacteristic
519C23—21no—
520C23—41no—
521C23—1221no—
522C23—9761no—
523C23—4882no—
524C23—4882no—
525C23—2441no—
526Q8—21no—
527Q8—21no—
528Q8—21no—
529Q8—21no—
530Q8—81no—
531Q8—2441no—
532Q8—2441no—
533D4—21no—
534D4—41no—
535D4—41no—
536D4—41no—
537D4—41no—
538D4—81no—
539D4—1221no—
540D4—1221no—
541D4—4881no—
542D4—4882no—
543D4—4882no—
544D4—2441no—
545D4—2441no—
546D4—2441no—
547D4—2441no—
548D4#1451979712yes—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—161no—
558C2×C4—161no—
559C2×C4—1221no—
560C2×C4—2441no—
561C2×C4—2441no—
562C2×C4—2441no—
563C2×C4C2×D4×D6111yescharacteristic
564C2×C4C22×C22⋊D6111yescharacteristic
565C2×C4#1451982211yescharacteristic
566C8—21no—
567C8—21no—
568C8—41no—
569C8—1221no—
570C8—1221no—
571C8—2441no—
572C8—2441no—
573C8—2441no—
574C8—2441no—
575C22—22no—
576C22—41no—
577C22—41no—
578C22—81no—
579C22—81no—
580C22—81no—
581C22—9764no—
582C22—4881no—
583C22—4881no—
584C22—4881no—
585C22—2441no—
586C22—2441no—
587C22—2441no—
588C22D4×C22⋊D6111yescharacteristic
589C4—21no—
590C4—41no—
591C4—41no—
592C4—41no—
593C4—41no—
594C4—41no—
595C4—81no—
596C4—4881no—
597C4—2441no—
598C4C2×#1451979711yescharacteristic
599C4#1684055911yescharacteristic
600C4#1684056411yescharacteristic
601C4#1684057011yescharacteristic
602C2—42no—
603C2—161no—
604C2—9761no—
605C2—4881no—
606C2#2145735911yescharacteristic
607C2#2145752711yescharacteristic
608C2#2145758311yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3361C2
socleC22×C61D42
prime radical#21444872C2
hypercenter#3361D61
hypocenterC61#19825
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3361

Complements

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

NH ≅ G/Ntype
1. #21458811C2semidirect
2. #21458052C2semidirect
3. #21457637C2semidirect
4. #21445455C2semidirect
5. #21445349C2semidirect
6. #21445263C2semidirect
7. #21445237C2semidirect
8. #21445158C2semidirect
9. #21445033C2semidirect
10. #21444990C2semidirect
11. #21444878C2semidirect
13. #21444662C2semidirect
14. SD16⋊#92418C2semidirect
15. C61×#3361C2semidirect
20. #16840134C22semidirect
21. #16840112C22semidirect
32. #16839972C22semidirect
38. Q8×#92418C22semidirect
45. #16839515C22semidirect
46. #16839512C22semidirect
47. #16839511C22semidirect
48. #16839507C22semidirect
49. #16839506C22semidirect
51. Q8⋊#92418C22semidirect
52. Q8⋊#92418C22semidirect
55. #16839483C22semidirect
56. C61×#559C22semidirect
59. C61×#486C22semidirect
61. C61×#477C22semidirect
64. C8×Q8×C61C22semidirect
299. #3361D61semidirect
320. #559C2×D61semidirect
331. #486C2×D61semidirect
335. #477C2×D61semidirect
351. C8×Q8C2×D61semidirect
371. C61#19825semidirect

Maximal subgroups

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

subgroupindex
#214588112
#214580522
#214576372
#214454552
#214453492
#214452632
#214452372
#214451582
#214450332
#214449902
#214448782
#214448722
#214446622
SD16⋊#924182
C61×#33612
#1982561