Finite Group Databasebeta

← #29671484

Subgroups — #29671484

Order 15616.

666 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458124C211yesmaximal, characteristic, complemented
2#21458061C211yesmaximal, characteristic, complemented
3#21457749C211yesmaximal, characteristic, complemented
4#21445430C211yesmaximal, characteristic, complemented
5#21445364C211yesmaximal, characteristic, complemented
6#21445350C211yesmaximal, characteristic, complemented
7#21445270C211yesmaximal, characteristic, complemented
8#21445253C211yesmaximal, characteristic, complemented
9#21445046C211yesmaximal, characteristic, complemented
10#21444998C211yesmaximal, characteristic, complemented
11#21444970C211yesmaximal, characteristic, complemented
12#21444906C211yesmaximal, characteristic, complemented
13#21444704C211yesmaximal, characteristic, complemented
14C8×#11758199C211yesmaximal, characteristic, complemented
15C61×#3294C211yesmaximal, characteristic, complemented
16#16840477C2211yescharacteristic
17#16840416C2211yescharacteristic, complemented
18#16840114C2211yescharacteristic, complemented
19#16840113C2211yescharacteristic, complemented
20#16840112C2211yescharacteristic, complemented
21#16840108C2211yescharacteristic
22#16840108C2211yescharacteristic
23#16840105—21no—
24#16840095—21no—
25#16840083—21no—
26#16840052C2211yescharacteristic
27#16840052C2211yescharacteristic, complemented
28#16840042C2211yescharacteristic
29#16840013—21no—
30#16840010—21no—
31#16839990—21no—
32#16839967C2211yescharacteristic
33#16839964C2211yescharacteristic, complemented
34#16839954C2211yescharacteristic
35#16839907—21no—
36#16839904—21no—
37#16839896—21no—
38C4×#11758199C2211yescharacteristic
39#16839840C2211yescharacteristic
40#16839840C2211yescharacteristic, complemented
41C8×D244C2211yescharacteristic, complemented
42#16839767—21no—
43#16839763—21no—
44#16839755—21no—
45#16839515C2211yescharacteristic, complemented
46#16839514C2211yescharacteristic, complemented
47#16839510C2211yescharacteristic, complemented
48#16839509C2211yescharacteristic, complemented
49#16839508C2211yescharacteristic, complemented
50#16839506C2211yescharacteristic
51D4⋊#92418C2211yescharacteristic
52D4⋊#92418C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic, complemented
54#16839489C2211yescharacteristic, complemented
55C8×#92418C2211yescharacteristic, complemented
56C61×#547C2211yescharacteristic
57D42×C61C2211yescharacteristic
58C61×#486C2211yescharacteristic
59C61×#480—21no—
60C61×#474C2211yescharacteristic
61C61×#466—21no—
62C61×#459C2211yescharacteristic
63C61×#447—21no—
64C4×D8×C61C2211yescharacteristic
65C8×D4×C61C2211yescharacteristic
66C2×#102×C61—21no—
67C2×#11758224—21no—
68C2×#11758200—21no—
69C2×#11758201—21no—
70C2×#11758199—21no—
71C2×#11758199D411yescharacteristic
72C2×#11758194—21no—
73C2×#11758194—21no—
74C2×C8×D61—21no—
75#14519777C2311yescharacteristic
76#14519777C2311yescharacteristic
77#14519763—41no—
78#14519761—41no—
79#14519757—21no—
80#14519751—21no—
81#14519751—21no—
82#14519748—21no—
83C4×D244C2311yescharacteristic
84#14519736—21no—
85#14519735—21no—
86C2×C4⋊Dic61—21no—
87#14519710—21no—
88#14519710D411yescharacteristic
89#14519708—21no—
90#14519708—21no—
91C4⋊#92418C2311yescharacteristic
92C4⋊#92418C2311yescharacteristic
93C4⋊#92418C2311yescharacteristic
94#14519702—21no—
95#14519702—21no—
96#14519701—21no—
97#14519701—21no—
98#14519701—21no—
99#14519700—21no—
100#14519700—41no—
101#14519698—21no—
102#14519698—21no—
103#14519698—21no—
104#14519698—21no—
105#14519698—21no—
106#14519698—21no—
107#14519698—41no—
108#14519698D411yescharacteristic
109#14519697D411yescharacteristic
110C22⋊#92418—21no—
111C22⋊#92418—21no—
112#14519689—21no—
113C4×#92418C2311yescharacteristic
114C4×#92418C2311yescharacteristic
115C2×#41×C61—21no—
116C22×D4×C61—21no—
117C22×D4×C61—21no—
118C2×D8×C61—21no—
119C2×D8×C61D411yescharacteristic
120C22×C8×C61—21no—
121C4⋊Q8×C61C2311yescharacteristic
122#127×C61C2311yescharacteristic
123#123×C61—41no—
124C22⋊Q8×C61—21no—
125#121×C61—21no—
126#121×C61—21no—
127#121×C61—21no—
128#120×C61—41no—
129C4×D4×C61C2311yescharacteristic
130C4×D4×C61C2311yescharacteristic
131C4×D4×C61C2311yescharacteristic
132C2×C4⋊C4×C61—21no—
133#107×C61—21no—
134#107×C61D411yescharacteristic
135C4⋊C8×C61C2311yescharacteristic
136#102×C61—21no—
137#102×C61—21no—
138#102×C61—21no—
139#102×C61—21no—
140#102×C61D411yescharacteristic
141#102×C61D411yescharacteristic
142C22⋊C8×C61—21no—
143C4×C8×C61C2311yescharacteristic
144C24×C61—41no—
145#41×C61—42no—
146#41×C61—81no—
147C2×Q8×C61—21no—
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—41no—
156C2×D4×C61—81no—
157C2×D4×C61—81no—
158C2×D4×C61C2×D411yescharacteristic
159C2×D4×C61C2×D411yescharacteristic
160C2×D4×C61C2×D411yescharacteristic
161C22×C4×C61—21no—
162C22×C4×C61—21no—
163C22×C4×C61—21no—
164C22×C4×C61—41no—
165C2×C22⋊D61—41no—
166C2×C22⋊D61—41no—
167C22×Dic61—41no—
168#11758224—42no—
169#11758224—81no—
170C2×D244C2×D411yescharacteristic
171C2×C4×D61—21no—
172C2×Dic122—21no—
173C2×Dic122—21no—
174D8×C61—41no—
175D8×C61—81no—
176C2×C8×C61—21no—
177C2×C8×C61—81no—
178C2×C8×C61C2×D411yescharacteristic
179C2×C8×C61C2×D411yescharacteristic
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—41no—
183C4⋊C4×C61C2×D411yescharacteristic
184C4⋊C4×C61C2×D411yescharacteristic
185C4⋊C4×C61C2×D411yescharacteristic
186C22⋊C4×C61—21no—
187C22⋊C4×C61—21no—
188C22⋊C4×C61—21no—
189C22⋊C4×C61—41no—
190C22⋊C4×C61—41no—
191C42×C61C2411yescharacteristic
192C22⋊Dic61—41no—
193#11758201—81no—
194#11758200—81no—
195#11758199—22no—
196#11758199—81no—
197#11758198—21no—
198#11758198—41no—
199#11758198—41no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—41no—
205C4⋊Dic61C2×D411yescharacteristic
206C4×Dic61—41no—
207#11758194—81no—
208#11758194—81no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418C2×D411yescharacteristic
214C2×#92418C2×D411yescharacteristic
215C8×D61—81no—
216C23×C61—21no—
217C23×C61—21no—
218C23×C61—21no—
219C23×C61—41no—
220C23×C61—82no—
221C23×C61—82no—
222C23×C61—161no—
223C22×D61—21no—
224Q8×C61—41no—
225Q8×C61—41no—
226D4×C61—21no—
227D4×C61—21no—
228D4×C61—21no—
229D4×C61—21no—
230D4×C61—21no—
231D4×C61—41no—
232D4×C61—41no—
233D4×C61—41no—
234D4×C61—41no—
235D4×C61—81no—
236D4×C61—82no—
237D4×C61—82no—
238D4×C61—82no—
239D4×C61#13512yes—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—82no—
251C2×C4×C61C22×D411yescharacteristic
252C2×C4×C61C22×D411yescharacteristic
253C2×C4×C61H(5,𝔽2)11yescharacteristic
254C22⋊D61—82no—
255C22⋊D61—161no—
256C2×Dic61—21no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—82no—
260D244—21no—
261D244—21no—
262C4×D61—41no—
263C4×D61—41no—
264Dic122—41no—
265Dic122—41no—
266Dic122—81no—
267C8×C61—21no—
268C8×C61—21no—
269C8×C61—41no—
270#92418—21no—
271#92418—21no—
272#92418—41no—
273#92418—41no—
274#92418—41no—
275#92418—41no—
276#19899—611nomaximal, Sylow
277C22×C61—22no—
278C22×C61—41no—
279C22×C61—41no—
280C22×C61—41no—
281C22×C61—41no—
282C22×C61—41no—
283C22×C61—41no—
284C22×C61—41no—
285C22×C61—41no—
286C22×C61—81no—
287C22×C61—81no—
288C22×C61—81no—
289C22×C61—164no—
290C22×C61D4211yescharacteristic
291C2×D61—41no—
292C2×D61—41no—
293C2×D61—41no—
294C4×C61—21no—
295C4×C61—41no—
296C4×C61—41no—
297C4×C61—41no—
298C4×C61—81no—
299C4×C61C2×#13511yescharacteristic
300C4×C61#57511yescharacteristic
301C4×C61#57611yescharacteristic
302C4×C61#57611yescharacteristic
303Dic61—41no—
304Dic61—81no—
305Dic61—81no—
306#3330—611no—
307#3325—611no—
308#3303—611no—
309#3294D6111yescharacteristic, complemented
310#1711—611no—
311#1699—611no—
312#1697—611no—
313#1683—611no—
314#1681—611no—
315#1646—611no—
316#1639—611no—
317#1635—611no—
318#1625—611no—
319#1589—611no—
320C8×D8—611no—
321C2×C61—42no—
322C2×C61—81no—
323C2×C61—81no—
324C2×C61—161no—
325C2×C61#327311yescharacteristic
326C2×C61#328011yescharacteristic
327C2×C61#329011yescharacteristic
328D61—81no—
329#555—611no—
330#547—611no—
331#547C2×D6111yescharacteristic
332D42C2×D6111yescharacteristic
333#487—611no—
334#487—611no—
335#487—611no—
336#486—611no—
337#486—611no—
338#486C2×D6111yescharacteristic
339#485—1221no—
340#484—1221no—
341#482—1221no—
342#480—21no—
343#476—611no—
344#476—611no—
345#474—611no—
346#474C2×D6111yescharacteristic
347#469—1221no—
348#468—1221no—
349#466—21no—
350#465—1221no—
351#461—611no—
352#461—611no—
353#459—611no—
354#459C2×D6111yescharacteristic
355#449—1221no—
356#449—1221no—
357#447—21no—
358#447—1221no—
359C4×D8—611no—
360C4×D8C2×D6111yescharacteristic
361#436—611no—
362#436—611no—
363C8×D4—611no—
364C8×D4C2×D6111yescharacteristic
365#418—1221no—
366#417—1221no—
367#416—1221no—
368C2×#102—21no—
369#335—611no—
370#335—611no—
371#333—611no—
372#332—611no—
373#332—611no—
374#331—611no—
375D4⋊C8—611no—
376D4⋊C8—611no—
377D4⋊C8—611no—
378D4⋊C8—611no—
379C82—611no—
380C61#1989911yescharacteristic, complemented, Sylow
381C2×#41—21no—
382C2×#41—1221no—
383C22×D4—21no—
384C22×D4—21no—
385C2×SD16—1221no—
386C2×SD16—1221no—
387C2×D8—21no—
388C2×D8—611no—
389C2×D8—1221no—
390C2×D8C22⋊D6111yescharacteristic
391C2×M16—1221no—
392C2×M16—1221no—
393C22×C8—21no—
394C22×C8—1221no—
395C4⋊Q8—611no—
396C4⋊Q8—611no—
397C4⋊Q8C22×D6111yescharacteristic
398#127C22×D6111yescharacteristic
399#124—2441no—
400#123—41no—
401#123—2441no—
402C22⋊Q8—21no—
403C22⋊Q8—1221no—
404#121—21no—
405#121—21no—
406#121—21no—
407#121—1221no—
408#121—1221no—
409#121—1221no—
410#120—41no—
411C4×D4—611no—
412C4×D4C22×D6111yescharacteristic
413C4×D4C22×D6111yescharacteristic
414C4×D4C22×D6111yescharacteristic
415#117—1221no—
416#117—1221no—
417C2×C4⋊C4—21no—
418C2×C4⋊C4—1221no—
419#107—21no—
420#107—611no—
421#107—1221no—
422#107C22⋊D6111yescharacteristic
423#106—1221no—
424#106—1221no—
425C4⋊C8—611no—
426C4⋊C8—611no—
427C4⋊C8—611no—
428C4⋊C8C22×D6111yescharacteristic
429#103—1221no—
430#103—1221no—
431#103—1221no—
432#103—1221no—
433#103—1221no—
434#103—1221no—
435#103—2441no—
436#102—21no—
437#102—21no—
438#102—21no—
439#102—21no—
440#102—611no—
441#102—611no—
442#102—1221no—
443#102—1221no—
444#102—1221no—
445#102—1221no—
446#102—1221no—
447#102—1221no—
448#102—2441no—
449#102C22⋊D6111yescharacteristic
450#102C22⋊D6111yescharacteristic
451C22⋊C8—21no—
452C22⋊C8—1221no—
453C22⋊C8—1221no—
454C22⋊C8—1221no—
455C4×C8—611no—
456C4×C8—611no—
457C4×C8C22×D6111yescharacteristic
458C24—41no—
459#41—42no—
460#41—81no—
461#41—4881no—
462#41—2442no—
463C2×Q8—21no—
464C2×Q8—1221no—
465C2×Q8—1221no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—41no—
470C2×D4—41no—
471C2×D4—42no—
472C2×D4—42no—
473C2×D4—41no—
474C2×D4—81no—
475C2×D4—81no—
476C2×D4—611no—
477C2×D4—2441no—
478C2×D4—2441no—
479C2×D4D4×D6111yescharacteristic
480C2×D4C2×C22⋊D6111yescharacteristic
481C2×D4C2×C22⋊D6111yescharacteristic
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—41no—
486C22×C4—1221no—
487C22×C4—2441no—
488SD16—4881no—
489SD16—4881no—
490D8—41no—
491D8—81no—
492D8—1222no—
493D8—4881no—
494M16—4881no—
495M16—4881no—
496C2×C8—21no—
497C2×C8—81no—
498C2×C8—611no—
499C2×C8—611no—
500C2×C8—1221no—
501C2×C8—1221no—
502C2×C8—1221no—
503C2×C8—1221no—
504C2×C8—4881no—
505C2×C8C2×C22⋊D6111yescharacteristic
506C2×C8C2×C22⋊D6111yescharacteristic
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—41no—
510C4⋊C4—611no—
511C4⋊C4—1221no—
512C4⋊C4—1221no—
513C4⋊C4—1221no—
514C4⋊C4—1221no—
515C4⋊C4—2441no—
516C4⋊C4D4×D6111yescharacteristic
517C4⋊C4C2×C22⋊D6111yescharacteristic
518C4⋊C4C2×C22⋊D6111yescharacteristic
519C22⋊C4—21no—
520C22⋊C4—21no—
521C22⋊C4—21no—
522C22⋊C4—41no—
523C22⋊C4—41no—
524C22⋊C4—1221no—
525C22⋊C4—2441no—
526C22⋊C4—2441no—
527C22⋊C4—2441no—
528C42—2441no—
529C42C23×D6111yescharacteristic
530C23—21no—
531C23—21no—
532C23—21no—
533C23—41no—
534C23—82no—
535C23—82no—
536C23—161no—
537C23—1221no—
538Q8—41no—
539Q8—41no—
540Q8—4881no—
541Q8—2441no—
542Q8—2441no—
543D4—21no—
544D4—21no—
545D4—21no—
546D4—21no—
547D4—21no—
548D4—41no—
549D4—41no—
550D4—41no—
551D4—41no—
552D4—81no—
553D4—82no—
554D4—82no—
555D4—82no—
556D4—1221no—
557D4—1221no—
558D4—9761no—
559D4—4882no—
560D4#1451979612yes—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—82no—
572C2×C4—1221no—
573C2×C4—4882no—
574C2×C4—2441no—
575C2×C4—2441no—
576C2×C4—2441no—
577C2×C4—2441no—
578C2×C4C2×D4×D6111yescharacteristic
579C2×C4C22×C22⋊D6111yescharacteristic
580C2×C4#1451982211yescharacteristic
581C8—21no—
582C8—21no—
583C8—41no—
584C8—1221no—
585C8—1221no—
586C8—2441no—
587C8—2441no—
588C8—2441no—
589C8—2441no—
590C22—22no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—81no—
600C22—81no—
601C22—81no—
602C22—164no—
603C22—2441no—
604C22—2441no—
605C22—2441no—
606C22D4×C22⋊D6111yescharacteristic
607C4—21no—
608C4—41no—
609C4—41no—
610C4—41no—
611C4—81no—
612C4—4881no—
613C4—4881no—
614C4—2441no—
615C4C2×#1451979611yescharacteristic
616C4#1684056011yescharacteristic
617C4#1684056311yescharacteristic
618C4#1684056511yescharacteristic
619C2—42no—
620C2—81no—
621C2—81no—
622C2—161no—
623C2—4881no—
624C2#2145742611yescharacteristic
625C2#2145750411yescharacteristic
626C2#2145759911yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3294C2
socleC22×C61D42
hypercenter#3294D61
hypocenterC61#19899
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3294

Complements

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

NH ≅ G/Ntype
1. #21458124C2semidirect
2. #21458061C2semidirect
3. #21457749C2semidirect
4. #21445430C2semidirect
5. #21445364C2semidirect
6. #21445350C2semidirect
7. #21445270C2semidirect
8. #21445253C2semidirect
9. #21445046C2semidirect
10. #21444998C2semidirect
11. #21444970C2semidirect
12. #21444906C2semidirect
13. #21444704C2semidirect
14. C8×#11758199C2semidirect
15. C61×#3294C2semidirect
17. #16840416C22semidirect
18. #16840114C22semidirect
19. #16840113C22semidirect
20. #16840112C22semidirect
27. #16840052C22semidirect
33. #16839964C22semidirect
40. #16839840C22semidirect
41. C8×D244C22semidirect
45. #16839515C22semidirect
46. #16839514C22semidirect
47. #16839510C22semidirect
48. #16839509C22semidirect
49. #16839508C22semidirect
53. D4⋊#92418C22semidirect
54. #16839489C22semidirect
55. C8×#92418C22semidirect
309. #3294D61semidirect
380. C61#19899semidirect

Maximal subgroups

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

subgroupindex
#214581242
#214580612
#214577492
#214454302
#214453642
#214453502
#214452702
#214452532
#214450462
#214449982
#214449702
#214449062
#214447042
C8×#117581992
C61×#32942
#1989961