Finite Group Databasebeta

← #29671514

Subgroups — #29671514

Order 15616.

648 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#21458464C211yesmaximal, characteristic, complemented
2#21458137C211yesmaximal, characteristic, complemented
3#21457632C211yesmaximal, characteristic
4#21445419C211yesmaximal, characteristic, complemented
5#21445394C211yesmaximal, characteristic, complemented
6#21445377C211yesmaximal, characteristic, complemented
7#21445268C211yesmaximal, characteristic, complemented
8#21445261C211yesmaximal, characteristic, complemented
9#21445102C211yesmaximal, characteristic, complemented
10#21445089C211yesmaximal, characteristic, complemented
11#21444935C211yesmaximal, characteristic, complemented
12#21444890C211yesmaximal, characteristic, complemented
13#21444702C211yesmaximal, characteristic, complemented
14SD16×#92418C211yesmaximal, characteristic, complemented
15C61×#3324C211yesmaximal, characteristic, complemented
16#16840472C2211yescharacteristic, complemented
17#16840416C2211yescharacteristic
18#16840158C2211yescharacteristic, complemented
19#16840134C2211yescharacteristic
20#16840116—21no—
21#16840113C2211yescharacteristic, complemented
22#16840112C2211yescharacteristic, complemented
23#16840108C2211yescharacteristic
24#16840094—21no—
25#16840073—21no—
26#16840069—21no—
27#16840052C2211yescharacteristic
28#16840052C2211yescharacteristic
29#16840042C2211yescharacteristic
30#16840019—21no—
31#16839996—21no—
32#16839974—21no—
33#16839972C2211yescharacteristic, complemented
34#16839964C2211yescharacteristic, complemented
35#16839954C2211yescharacteristic
36#16839921—21no—
37#16839896—21no—
38Q8×#92418C2211yescharacteristic, complemented
39C4×#11758201—21no—
40C4×#11758199C2211yescharacteristic
41#16839841C2211yescharacteristic, complemented
42D4×#92418C2211yescharacteristic
43#16839760—21no—
44C2×#14519698—21no—
45#16839514C2211yescharacteristic, complemented
46#16839513C2211yescharacteristic, complemented
47#16839510C2211yescharacteristic, complemented
48#16839509C2211yescharacteristic, complemented
49#16839508C2211yescharacteristic, complemented
50#16839506C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic, complemented
52D4⋊#92418C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic
54#16839489C2211yescharacteristic, complemented
55C8×#92418C2211yescharacteristic, 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×#11758224—21no—
68C2×#11758202—21no—
69C2×#11758200—41no—
70C2×#11758201—21no—
71C2×#11758199—21no—
72C2×#11758199D411yescharacteristic
73C2×#11758189—21no—
74C22×#92418—21no—
75#14519777C2311yescharacteristic
76#14519777C2311yescharacteristic
77#14519769—21no—
78#14519761—41no—
79#14519757—21no—
80#14519755—41no—
81#14519753—21no—
82#14519751—21no—
83C4×D244—21no—
84C4×D244C2311yescharacteristic
85C2×C4⋊Dic61—21no—
86C4⋊C4×D61—21no—
87#14519710—21no—
88#14519710D411yescharacteristic
89#14519708—21no—
90#14519708—21no—
91#14519708—41no—
92C4⋊#92418—21no—
93C4⋊#92418C2311yescharacteristic
94C4⋊#92418C2311yescharacteristic
95C4⋊#92418C2311yescharacteristic
96#14519702—21no—
97#14519702—21no—
98#14519701—21no—
99#14519701—21no—
100#14519701—21no—
101#14519700—21no—
102#14519698—21no—
103#14519698—21no—
104#14519698—21no—
105#14519698—21no—
106#14519698—21no—
107#14519698—21no—
108#14519698D411yescharacteristic
109#14519697D411yescharacteristic
110C22⋊#92418—21no—
111#14519689—21no—
112C4×#92418—21no—
113C4×#92418C2311yescharacteristic
114C4×#92418C2311yescharacteristic
115C22×D4×C61—21no—
116C22×D4×C61—21no—
117C2×SD16×C61—21no—
118C2×SD16×C61D411yescharacteristic
119C2×M16×C61—21no—
120C4⋊Q8×C61C2311yescharacteristic
121#127×C61C2311yescharacteristic
122#126×C61—41no—
123#124×C61—21no—
124C22⋊Q8×C61—21no—
125#121×C61—21no—
126#121×C61—21no—
127#120×C61—41no—
128C4×Q8×C61—21no—
129C4×Q8×C61C2311yescharacteristic
130C4×D4×C61C2311yescharacteristic
131C4×D4×C61C2311yescharacteristic
132#117×C61—21no—
133#107×C61—21no—
134#106×C61D411yescharacteristic
135C4⋊C8×C61C2311yescharacteristic
136#103×C61D411yescharacteristic
137#102×C61—21no—
138#102×C61—21no—
139#102×C61—21no—
140#102×C61—21no—
141#102×C61D411yescharacteristic
142C22⋊C8×C61—21no—
143C4×C8×C61C2311yescharacteristic
144C24×C61—41no—
145C2×Q8×C61—21no—
146C2×Q8×C61C2×D411yescharacteristic
147C2×D4×C61—21no—
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61—41no—
151C2×D4×C61—41no—
152C2×D4×C61—42no—
153C2×D4×C61—42no—
154C2×D4×C61—81no—
155C2×D4×C61—81no—
156C2×D4×C61C2×D411yescharacteristic
157C2×D4×C61C2×D411yescharacteristic
158C22×C4×C61—21no—
159C22×C4×C61—21no—
160C2×C22⋊D61—41no—
161C22×Dic61—41no—
162#11758224—42no—
163#11758224—81no—
164C2×D244C2×D411yescharacteristic
165C2×C4×D61—21no—
166C2×C4×D61—41no—
167C2×Dic122—21no—
168C2×Dic122—21no—
169SD16×C61—22no—
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
190#11758202—81no—
191#11758201—42no—
192#11758200—82no—
193#11758199—41no—
194#11758199—81no—
195#11758198—21no—
196#11758198—41no—
197#11758198—41no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—41no—
203C4⋊Dic61C2×D411yescharacteristic
204#11758196—41no—
205#11758196—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—21no—
216C23×C61—41no—
217C23×C61—82no—
218C23×C61—82no—
219C23×C61—161no—
220C22×D61—21no—
221Q8×C61—21no—
222Q8×C61—21no—
223Q8×C61—41no—
224Q8×C61—41no—
225D4×C61—21no—
226D4×C61—21no—
227D4×C61—21no—
228D4×C61—41no—
229D4×C61—41no—
230D4×C61—41no—
231D4×C61—41no—
232D4×C61—81no—
233D4×C61—82no—
234D4×C61—82no—
235D4×C61#13512yes—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61C22×D411yescharacteristic
246C2×C4×C61C22×D411yescharacteristic
247C2×C4×C61H(5,𝔽2)11yescharacteristic
248C22⋊D61—82no—
249C2×Dic61—21no—
250C2×Dic61—41no—
251C2×Dic61—41no—
252C2×Dic61—82no—
253D244—21no—
254D244—21no—
255C4×D61—41no—
256C4×D61—41no—
257C4×D61—161no—
258Dic122—41no—
259Dic122—41no—
260Dic122—81no—
261C8×C61—21no—
262C8×C61—21no—
263C8×C61—41no—
264#92418—22no—
265#92418—21no—
266#92418—21no—
267#92418—41no—
268#92418—41no—
269#92418—41no—
270#19901—611nomaximal, Sylow
271C22×C61—22no—
272C22×C61—41no—
273C22×C61—41no—
274C22×C61—41no—
275C22×C61—41no—
276C22×C61—41no—
277C22×C61—81no—
278C22×C61—81no—
279C22×C61—81no—
280C22×C61—164no—
281C22×C61D4211yescharacteristic
282C2×D61—41no—
283C2×D61—41no—
284C2×D61—41no—
285C4×C61—21no—
286C4×C61—41no—
287C4×C61—41no—
288C4×C61—41no—
289C4×C61—41no—
290C4×C61—81no—
291C4×C61C2×#13511yescharacteristic
292C4×C61#57511yescharacteristic
293C4×C61#57611yescharacteristic
294C4×C61#57711yescharacteristic
295Dic61—41no—
296Dic61—81no—
297Dic61—81no—
298#3360—611no—
299#3331—611no—
300#3324D6111yescharacteristic, complemented
301#3294—611no—
302#1708—611no—
303#1703—611no—
304#1701—611no—
305#1683—611no—
306#1682—611no—
307#1654—611no—
308#1652—611no—
309#1629—611no—
310#1622—611no—
311#1589—611no—
312C8×SD16—611no—
313C2×C61—42no—
314C2×C61—81no—
315C2×C61—161no—
316C2×C61#327511yescharacteristic
317C2×C61#328011yescharacteristic
318C2×C61#329111yescharacteristic
319D61—81no—
320#556C2×D6111yescharacteristic
321#554—611no—
322#547—611no—
323D42C2×D6111yescharacteristic
324#498—611no—
325#492—611no—
326#488—1221no—
327#487—611no—
328#487—611no—
329#486—611no—
330#486C2×D6111yescharacteristic
331#483—1221no—
332#482—21no—
333#480—1221no—
334#479—1221no—
335#476—611no—
336#476—611no—
337#474—611no—
338#474C2×D6111yescharacteristic
339#470—1221no—
340#468—21no—
341#466—1221no—
342#463—1221no—
343#462—611no—
344#461—611no—
345#460C2×D6111yescharacteristic
346#459—611no—
347#452—1221no—
348#450—21no—
349#447—1221no—
350C8×Q8—611no—
351C4×SD16—1221no—
352C4×SD16C2×D6111yescharacteristic
353C4×D8—611no—
354#436—611no—
355#436C2×D6111yescharacteristic
356C8×D4—611no—
357#418—21no—
358#417—1221no—
359C2×#102—1221no—
360#335—611no—
361#334—611no—
362#333—611no—
363#332—611no—
364#332—611no—
365#331—611no—
366Q8⋊C8—611no—
367D4⋊C8—611no—
368D4⋊C8—611no—
369D4⋊C8—611no—
370C82—611no—
371C61#1990111yescharacteristic, complemented, Sylow
372C2×#41—1221no—
373C22×D4—21no—
374C22×D4—21no—
375C2×Q16—1221no—
376C2×SD16—21no—
377C2×SD16—1221no—
378C2×SD16—2441no—
379C2×SD16C22⋊D6111yescharacteristic
380C2×D8—611no—
381C2×D8—1221no—
382C2×M16—21no—
383C2×M16—1221no—
384C22×C8—1221no—
385C4⋊Q8—611no—
386C4⋊Q8—611no—
387C4⋊Q8C22×D6111yescharacteristic
388#127C22×D6111yescharacteristic
389#126—41no—
390#125—1221no—
391#124—21no—
392#123—2441no—
393C22⋊Q8—21no—
394C22⋊Q8—1221no—
395C22⋊Q8—1221no—
396C22⋊Q8—2441no—
397#121—21no—
398#121—21no—
399#121—1221no—
400#120—41no—
401C4×Q8—21no—
402C4×Q8C22×D6111yescharacteristic
403C4×D4—611no—
404C4×D4—1221no—
405C4×D4C22×D6111yescharacteristic
406C4×D4C22×D6111yescharacteristic
407#117—21no—
408C2×C4⋊C4—1221no—
409C2×C4⋊C4—1221no—
410#107—21no—
411#107—611no—
412#107—1221no—
413#106—1221no—
414#106—1221no—
415#106—2441no—
416#106C22⋊D6111yescharacteristic
417C4⋊C8—611no—
418C4⋊C8—611no—
419C4⋊C8—611no—
420C4⋊C8—1221no—
421C4⋊C8C22×D6111yescharacteristic
422#103—1221no—
423#103—1221no—
424#103—1221no—
425#103—1221no—
426#103—1221no—
427#103—1221no—
428#103C22⋊D6111yescharacteristic
429#102—21no—
430#102—21no—
431#102—21no—
432#102—21no—
433#102—611no—
434#102—611no—
435#102—1221no—
436#102—1221no—
437#102—1221no—
438#102—1221no—
439#102—1221no—
440#102—1221no—
441#102C22⋊D6111yescharacteristic
442C22⋊C8—21no—
443C22⋊C8—1221no—
444C22⋊C8—1221no—
445C4×C8—611no—
446C4×C8—611no—
447C4×C8—1221no—
448C4×C8C22×D6111yescharacteristic
449C24—41no—
450#41—4881no—
451#41—2442no—
452C2×Q8—21no—
453C2×Q8—1221no—
454C2×Q8—1221no—
455C2×Q8C2×C22⋊D6111yescharacteristic
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—41no—
460C2×D4—41no—
461C2×D4—42no—
462C2×D4—42no—
463C2×D4—81no—
464C2×D4—81no—
465C2×D4—611no—
466C2×D4—2441no—
467C2×D4D4×D6111yescharacteristic
468C2×D4C2×C22⋊D6111yescharacteristic
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—1221no—
472C22×C4—2441no—
473C22×C4—2441no—
474Q16—4881no—
475SD16—22no—
476SD16—81no—
477SD16—4882no—
478SD16—2442no—
479D8—4881no—
480D8—2441no—
481M16—81no—
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—41no—
497C4⋊C4—611no—
498C4⋊C4—1221no—
499C4⋊C4—1221no—
500C4⋊C4—1221no—
501C4⋊C4—1221no—
502C4⋊C4—2441no—
503C4⋊C4—2441no—
504C4⋊C4—2441no—
505C4⋊C4D4×D6111yescharacteristic
506C4⋊C4C2×C22⋊D6111yescharacteristic
507C4⋊C4C2×C22⋊D6111yescharacteristic
508C22⋊C4—21no—
509C22⋊C4—21no—
510C22⋊C4—41no—
511C22⋊C4—41no—
512C22⋊C4—41no—
513C22⋊C4—1221no—
514C22⋊C4—2441no—
515C22⋊C4—2441no—
516C42—21no—
517C42—41no—
518C42C23×D6111yescharacteristic
519C23—21no—
520C23—21no—
521C23—41no—
522C23—82no—
523C23—82no—
524C23—161no—
525C23—1221no—
526Q8—21no—
527Q8—21no—
528Q8—41no—
529Q8—41no—
530Q8—4881no—
531Q8—2441no—
532Q8—2441no—
533D4—21no—
534D4—21no—
535D4—21no—
536D4—41no—
537D4—41no—
538D4—41no—
539D4—41no—
540D4—81no—
541D4—82no—
542D4—82no—
543D4—1221no—
544D4—1221no—
545D4—4882no—
546D4#1451979712yes—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—1221no—
557C2×C4—9761no—
558C2×C4—4882no—
559C2×C4—2441no—
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—1222no—
570C8—1221no—
571C8—1221no—
572C8—2441no—
573C8—2441no—
574C8—2441no—
575C22—22no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22—81no—
582C22—81no—
583C22—81no—
584C22—164no—
585C22—2441no—
586C22—2441no—
587C22—2441no—
588C22D4×C22⋊D6111yescharacteristic
589C4—21no—
590C4—41no—
591C4—41no—
592C4—41no—
593C4—41no—
594C4—81no—
595C4—4881no—
596C4—4881no—
597C4—2441no—
598C4C2×#1451979711yescharacteristic
599C4#1684055811yescharacteristic
600C4#1684056511yescharacteristic
601C4#1684057211yescharacteristic
602C2—42no—
603C2—81no—
604C2—161no—
605C2—4881no—
606C2#2145745011yescharacteristic
607C2#2145750011yescharacteristic
608C2#2145761011yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3324C2
socleC22×C61D42
prime radical#21457632C2
hypercenter#3324D61
hypocenterC61#19901
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. #21458464C2semidirect
2. #21458137C2semidirect
4. #21445419C2semidirect
5. #21445394C2semidirect
6. #21445377C2semidirect
7. #21445268C2semidirect
8. #21445261C2semidirect
9. #21445102C2semidirect
10. #21445089C2semidirect
11. #21444935C2semidirect
12. #21444890C2semidirect
13. #21444702C2semidirect
14. SD16×#92418C2semidirect
15. C61×#3324C2semidirect
16. #16840472C22semidirect
18. #16840158C22semidirect
21. #16840113C22semidirect
22. #16840112C22semidirect
33. #16839972C22semidirect
34. #16839964C22semidirect
38. Q8×#92418C22semidirect
41. #16839841C22semidirect
45. #16839514C22semidirect
46. #16839513C22semidirect
47. #16839510C22semidirect
48. #16839509C22semidirect
49. #16839508C22semidirect
51. Q8⋊#92418C22semidirect
54. #16839489C22semidirect
55. C8×#92418C22semidirect
300. #3324D61semidirect
371. C61#19901semidirect

Maximal subgroups

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

subgroupindex
#214584642
#214581372
#214576322
#214454192
#214453942
#214453772
#214452682
#214452612
#214451022
#214450892
#214449352
#214448902
#214447022
SD16×#924182
C61×#33242
#1990161