Finite Group Databasebeta

← #29676835

Subgroups — #29676835

Order 15616.

684 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458522C211yesmaximal, characteristic, complemented
2#21458447C211yesmaximal, characteristic, complemented
3#21458061C211yesmaximal, characteristic, complemented
4#21445408C211yesmaximal, characteristic, complemented
5#21445357C211yesmaximal, characteristic, complemented
6#21445350C211yesmaximal, characteristic, complemented
7#21445342C211yesmaximal, characteristic, complemented
8#21445295C211yesmaximal, characteristic, complemented
9#21444946C211yesmaximal, characteristic, complemented
10#21444935C211yesmaximal, characteristic, complemented
11#21444901C211yesmaximal, characteristic, complemented
12#21444890C211yesmaximal, characteristic, complemented
13#21444710C211yesmaximal, characteristic, complemented
14#21444702C211yesmaximal, characteristic, complemented
15C61×#3274C211yesmaximal, characteristic, complemented
16#16840477C2211yescharacteristic
17#16840472C2211yescharacteristic, complemented
18#16840167C2211yescharacteristic, complemented
19#16840134C2211yescharacteristic
20#16840134C2211yescharacteristic, complemented
21#16840126—21no—
22#16840108C2211yescharacteristic
23#16840108C2211yescharacteristic, complemented
24#16840102—21no—
25#16840083—21no—
26#16840074—21no—
27#16840062—21no—
28#16840057C2211yescharacteristic, complemented
29#16840052C2211yescharacteristic
30#16840052C2211yescharacteristic
31#16840052C2211yescharacteristic
32#16840013—21no—
33#16840005—21no—
34#16839954—21no—
35#16839954C2211yescharacteristic
36#16839954C2211yescharacteristic, complemented
37#16839896—21no—
38#16839893C2211yescharacteristic, complemented
39#16839872—21no—
40C4×#11758199C2211yescharacteristic
41#16839841C2211yescharacteristic, complemented
42#16839840C2211yescharacteristic
43#16839801—21no—
44#16839767—21no—
45#16839514C2211yescharacteristic, complemented
46#16839513C2211yescharacteristic, complemented
47#16839510C2211yescharacteristic, complemented
48#16839510C2211yescharacteristic, complemented
49#16839506C2211yescharacteristic
50#16839506C2211yescharacteristic, complemented
51Q8⋊#92418C2211yescharacteristic, complemented
52D4⋊#92418C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic
54#16839489C2211yescharacteristic, complemented
55C8×#92418C2211yescharacteristic, complemented
56C2×AGL(1,ℤ/8)×C61—21no—
57C61×#550C2211yescharacteristic
58D42×C61C2211yescharacteristic
59C61×#493C2211yescharacteristic
60C61×#469—21no—
61C61×#460C2211yescharacteristic
62C61×#459C2211yescharacteristic
63C61×#449—21no—
64C61×#447—21no—
65C4×SD16×C61C2211yescharacteristic
66C61×#436C2211yescharacteristic
67C2×#11758200—41no—
68C2×#11758200—41no—
69C2×#11758199—21no—
70C2×#11758199—21no—
71C2×#11758199D411yescharacteristic
72C2×#11758194—21no—
73C2×#11758189—21no—
74#14519777C2311yescharacteristic
75#14519777C2311yescharacteristic
76#14519769—21no—
77#14519763—41no—
78#14519755—41no—
79#14519753—21no—
80#14519751—21no—
81#14519748—21no—
82C4×D244—21no—
83C4×D244C2311yescharacteristic
84#14519736—21no—
85#14519735—21no—
86C4⋊C4×D61—21no—
87#14519711—21no—
88#14519711—21no—
89#14519710—21no—
90#14519710D411yescharacteristic
91#14519708—21no—
92#14519708—21no—
93#14519708—21no—
94C4⋊#92418—21no—
95C4⋊#92418C2311yescharacteristic
96C4⋊#92418C2311yescharacteristic
97C4⋊#92418C2311yescharacteristic
98#14519701—21no—
99#14519698—21no—
100#14519698—21no—
101#14519698—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—
112#14519688—21no—
113C4×#92418C2311yescharacteristic
114C4×#92418C2311yescharacteristic
115C2×#41×C61—21no—
116C22×D4×C61—21no—
117C22×D4×C61—21no—
118AGL(1,ℤ/8)×C61—44no—
119C2×SD16×C61—21no—
120C2×SD16×C61D411yescharacteristic
121C2×D8×C61—21no—
122C2×D8×C61—21no—
123C2×D8×C61—21no—
124C2×M16×C61—21no—
125#127×C61—21no—
126#127×C61C2311yescharacteristic
127#127×C61C2311yescharacteristic
128#121×C61—21no—
129#121×C61—21no—
130#121×C61—21no—
131#121×C61—41no—
132#120×C61—41no—
133C4×Q8×C61C2311yescharacteristic
134C4×D4×C61—21no—
135C4×D4×C61C2311yescharacteristic
136C4×D4×C61C2311yescharacteristic
137#106×C61D411yescharacteristic
138C4⋊C8×C61C2311yescharacteristic
139#103×C61D411yescharacteristic
140#102×C61—21no—
141#102×C61—21no—
142#102×C61D411yescharacteristic
143C22⋊C8×C61—21no—
144C4×C8×C61C2311yescharacteristic
145C24×C61—41no—
146#41×C61—42no—
147#41×C61—81no—
148C2×Q8×C61C2×D411yescharacteristic
149C2×D4×C61—21no—
150C2×D4×C61—21no—
151C2×D4×C61—21no—
152C2×D4×C61—21no—
153C2×D4×C61—21no—
154C2×D4×C61—21no—
155C2×D4×C61—41no—
156C2×D4×C61—41no—
157C2×D4×C61—41no—
158C2×D4×C61—41no—
159C2×D4×C61—42no—
160C2×D4×C61—42no—
161C2×D4×C61—81no—
162C2×D4×C61—81no—
163C2×D4×C61C2×D411yescharacteristic
164C2×D4×C61C2×D411yescharacteristic
165C22×C4×C61—21no—
166C22×C4×C61—21no—
167C22×C4×C61—41no—
168C2×C22⋊D61—41no—
169C2×D244C2×D411yescharacteristic
170C2×C4×D61—21no—
171C2×C4×D61—41no—
172C2×Dic122—21no—
173C2×Dic122—21no—
174SD16×C61—22no—
175SD16×C61—42no—
176D8×C61—42no—
177D8×C61—42no—
178D8×C61—81no—
179M16×C61—42no—
180C2×C8×C61—21no—
181C2×C8×C61C2×D411yescharacteristic
182C2×C8×C61C2×D411yescharacteristic
183C4⋊C4×C61—21no—
184C4⋊C4×C61C2×D411yescharacteristic
185C4⋊C4×C61C2×D411yescharacteristic
186C4⋊C4×C61C2×D411yescharacteristic
187C22⋊C4×C61—21no—
188C22⋊C4×C61—21no—
189C22⋊C4×C61—41no—
190C22⋊C4×C61—41no—
191C42×C61—21no—
192C42×C61C2411yescharacteristic
193C22⋊Dic61—41no—
194#11758200—82no—
195#11758200—82no—
196#11758199—41no—
197#11758199—42no—
198#11758199—81no—
199#11758198—21no—
200#11758198—41no—
201#11758198—41no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61C2×D411yescharacteristic
207#11758196—41no—
208#11758196—41no—
209C4×Dic61—41no—
210#11758194—81no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418C2×D411yescharacteristic
216C2×#92418C2×D411yescharacteristic
217#11758189—42no—
218C23×C61—21no—
219C23×C61—21no—
220C23×C61—41no—
221C23×C61—41no—
222C23×C61—82no—
223C23×C61—82no—
224C23×C61—161no—
225C22×D61—21no—
226Q8×C61—21no—
227Q8×C61—21no—
228D4×C61—21no—
229D4×C61—21no—
230D4×C61—21no—
231D4×C61—21no—
232D4×C61—41no—
233D4×C61—41no—
234D4×C61—41no—
235D4×C61—41no—
236D4×C61—41no—
237D4×C61—41no—
238D4×C61—41no—
239D4×C61—41no—
240D4×C61—81no—
241D4×C61—81no—
242D4×C61—82no—
243D4×C61—82no—
244D4×C61—82no—
245D4×C61—161no—
246C2×C4×C61—21no—
247C2×C4×C61—21no—
248C2×C4×C61—21no—
249C2×C4×C61—21no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61—82no—
255C2×C4×C61C22×D411yescharacteristic
256C2×C4×C61C22×D411yescharacteristic
257C2×C4×C61H(5,𝔽2)11yescharacteristic
258C22⋊D61—161no—
259C2×Dic61—21no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262D244—21no—
263D244—21no—
264C4×D61—41no—
265C4×D61—41no—
266C4×D61—161no—
267Dic122—81no—
268Dic122—81no—
269C8×C61—21no—
270C8×C61—41no—
271C8×C61AGL(1,ℤ/8)12yes—
272#92418—22no—
273#92418—21no—
274#92418—21no—
275#92418—41no—
276#92418—41no—
277#92418—41no—
278#20286—611nomaximal, Sylow
279C22×C61—41no—
280C22×C61—41no—
281C22×C61—41no—
282C22×C61—41no—
283C22×C61—41no—
284C22×C61—41no—
285C22×C61—81no—
286C22×C61—81no—
287C22×C61—81no—
288C22×C61—81no—
289C22×C61—81no—
290C22×C61—81no—
291C22×C61—164no—
292C22×C61D4211yescharacteristic
293C2×D61—41no—
294C2×D61—41no—
295C2×D61—41no—
296C4×C61—21no—
297C4×C61—41no—
298C4×C61—41no—
299C4×C61—41no—
300C4×C61—41no—
301C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
302C4×C61#57511yescharacteristic
303C4×C61#57711yescharacteristic
304C4×C61#57711yescharacteristic
305Dic61—41no—
306Dic61—81no—
307Dic61—81no—
308#3366—611no—
309#3357—611no—
310#3325—611no—
311#3274D6111yescharacteristic, complemented
312#1706—611no—
313#1698—611no—
314#1697—611no—
315#1695—611no—
316#1687—611no—
317#1631—611no—
318#1629—611no—
319#1624—611no—
320#1622—611no—
321#1590—611no—
322#1589—611no—
323C2×C61—81no—
324C2×C61—81no—
325C2×C61—161no—
326C2×C61—161no—
327C2×C61#327211yescharacteristic
328C2×C61#327511yescharacteristic
329C2×C61#329211yescharacteristic
330D61—81no—
331C2×AGL(1,ℤ/8)—21no—
332#555—611no—
333#554—611no—
334#550C2×D6111yescharacteristic
335D42C2×D6111yescharacteristic
336#500—611no—
337#493C2×D6111yescharacteristic
338#492—611no—
339#492—611no—
340#490—1221no—
341#486—611no—
342#486—611no—
343#485—1221no—
344#482—1221no—
345#480—1221no—
346#478—1221no—
347#477—611no—
348#476—611no—
349#476—611no—
350#476—611no—
351#469—21no—
352#469—1221no—
353#468—1221no—
354#460C2×D6111yescharacteristic
355#459—611no—
356#459—611no—
357#459—1221no—
358#459C2×D6111yescharacteristic
359#449—21no—
360#447—21no—
361#447—1221no—
362#446—611no—
363#442—1221no—
364C4×SD16C2×D6111yescharacteristic
365C4×D8—611no—
366#436—611no—
367#436—611no—
368#436C2×D6111yescharacteristic
369#428—1221no—
370#418—1221no—
371#335—611no—
372#334—611no—
373#333—611no—
374#333—611no—
375#331—611no—
376#331—611no—
377Q8⋊C8—611no—
378D4⋊C8—611no—
379D4⋊C8—611no—
380D4⋊C8—611no—
381C82—611no—
382C61#2028611yescharacteristic, complemented, Sylow
383C2×#41—21no—
384C22×D4—21no—
385C22×D4—21no—
386AGL(1,ℤ/8)—44no—
387C2×SD16—21no—
388C2×SD16—2441no—
389C2×SD16—2441no—
390C2×SD16C22⋊D6111yescharacteristic
391C2×D8—21no—
392C2×D8—21no—
393C2×D8—21no—
394C2×D8—611no—
395C2×D8—1221no—
396C2×D8—1221no—
397C2×M16—21no—
398C2×M16—1221no—
399C2×M16—1221no—
400C4⋊Q8—611no—
401C4⋊Q8—611no—
402#127—21no—
403#127C22×D6111yescharacteristic
404#127C22×D6111yescharacteristic
405#125—1221no—
406#124—2441no—
407C22⋊Q8—1221no—
408C22⋊Q8—2441no—
409#121—21no—
410#121—21no—
411#121—21no—
412#121—41no—
413#121—1221no—
414#121—1221no—
415#120—41no—
416C4×Q8C22×D6111yescharacteristic
417C4×D4—21no—
418C4×D4—611no—
419C4×D4—1221no—
420C4×D4C22×D6111yescharacteristic
421C4×D4C22×D6111yescharacteristic
422#117—1221no—
423#117—1221no—
424C2×C4⋊C4—1221no—
425#107—611no—
426#107—1221no—
427#107—1221no—
428#107—1221no—
429#106—1221no—
430#106—1221no—
431#106—1221no—
432#106C22⋊D6111yescharacteristic
433C4⋊C8—611no—
434C4⋊C8—611no—
435C4⋊C8—611no—
436C4⋊C8—1221no—
437C4⋊C8C22×D6111yescharacteristic
438#103—1221no—
439#103C22⋊D6111yescharacteristic
440#102—21no—
441#102—21no—
442#102—611no—
443#102—611no—
444#102—1221no—
445#102—1221no—
446#102—1221no—
447#102—1221no—
448#102—1221no—
449#102—1221no—
450#102—1221no—
451#102—1221no—
452#102—1221no—
453#102C22⋊D6111yescharacteristic
454C22⋊C8—21no—
455C22⋊C8—1221no—
456C22⋊C8—1221no—
457#97—1221no—
458C4×C8—611no—
459C4×C8—611no—
460C4×C8C22×D6111yescharacteristic
461C24—41no—
462#41—42no—
463#41—81no—
464C2×Q8—1221no—
465C2×Q8—1221no—
466C2×Q8C2×C22⋊D6111yescharacteristic
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4—21no—
473C2×D4—41no—
474C2×D4—41no—
475C2×D4—41no—
476C2×D4—41no—
477C2×D4—42no—
478C2×D4—42no—
479C2×D4—81no—
480C2×D4—81no—
481C2×D4—611no—
482C2×D4—2441no—
483C2×D4D4×D6111yescharacteristic
484C2×D4C2×C22⋊D6111yescharacteristic
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—41no—
488C22×C4—1221no—
489C22×C4—2441no—
490SD16—22no—
491SD16—42no—
492SD16—4882no—
493SD16—4882no—
494D8—42no—
495D8—42no—
496D8—81no—
497D8—4881no—
498D8—2441no—
499D8—2442no—
500M16—42no—
501M16—4881no—
502M16—2442no—
503C2×C8—21no—
504C2×C8—611no—
505C2×C8—611no—
506C2×C8—1221no—
507C2×C8—1221no—
508C2×C8—1221no—
509C2×C8—1221no—
510C2×C8C2×C22⋊D6111yescharacteristic
511C2×C8C2×C22⋊D6111yescharacteristic
512C4⋊C4—21no—
513C4⋊C4—611no—
514C4⋊C4—1221no—
515C4⋊C4—1221no—
516C4⋊C4—1221no—
517C4⋊C4—1221no—
518C4⋊C4—2441no—
519C4⋊C4—2441no—
520C4⋊C4D4×D6111yescharacteristic
521C4⋊C4C2×C22⋊D6111yescharacteristic
522C4⋊C4C2×C22⋊D6111yescharacteristic
523C22⋊C4—21no—
524C22⋊C4—21no—
525C22⋊C4—41no—
526C22⋊C4—41no—
527C22⋊C4—1221no—
528C22⋊C4—2441no—
529C22⋊C4—2441no—
530C22⋊C4—2441no—
531C42—21no—
532C42—2441no—
533C42C23×D6111yescharacteristic
534C23—21no—
535C23—21no—
536C23—41no—
537C23—41no—
538C23—82no—
539C23—82no—
540C23—161no—
541C23—1221no—
542Q8—21no—
543Q8—21no—
544Q8—4881no—
545Q8—4881no—
546D4—21no—
547D4—21no—
548D4—21no—
549D4—21no—
550D4—41no—
551D4—41no—
552D4—41no—
553D4—41no—
554D4—41no—
555D4—41no—
556D4—41no—
557D4—41no—
558D4—81no—
559D4—81no—
560D4—82no—
561D4—82no—
562D4—82no—
563D4—161no—
564D4—1221no—
565D4—1221no—
566D4—9761no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—82no—
576C2×C4—1221no—
577C2×C4—9761no—
578C2×C4—2441no—
579C2×C4—2441no—
580C2×C4—2441no—
581C2×C4—2441no—
582C2×C4C2×D4×D6111yescharacteristic
583C2×C4C22×C22⋊D6111yescharacteristic
584C2×C4#1451982211yescharacteristic
585C8—21no—
586C8—41no—
587C8—1222no—
588C8—1221no—
589C8—1221no—
590C8—2441no—
591C8—2441no—
592C8—2441no—
593C8#1451980312yes—
594C22—41no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—41no—
600C22—81no—
601C22—81no—
602C22—81no—
603C22—81no—
604C22—81no—
605C22—81no—
606C22—164no—
607C22—2441no—
608C22—2441no—
609C22—2441no—
610C22D4×C22⋊D6111yescharacteristic
611C4—21no—
612C4—41no—
613C4—41no—
614C4—41no—
615C4—41no—
616C4—4881no—
617C4—4881no—
618C4—2441no—
619C4C2×#1451980311yescharacteristic
620C4#1684055811yescharacteristic
621C4#1684057211yescharacteristic
622C4#1684057311yescharacteristic
623C2—81no—
624C2—81no—
625C2—161no—
626C2—161no—
627C2—4881no—
628C2#2145741511yescharacteristic
629C2#2145745011yescharacteristic
630C2#2145761611yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3274C2
socleC22×C61D42
hypercenter#3274D61
hypocenterC61#20286
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3274

Complements

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

NH ≅ G/Ntype
1. #21458522C2semidirect
2. #21458447C2semidirect
3. #21458061C2semidirect
4. #21445408C2semidirect
5. #21445357C2semidirect
6. #21445350C2semidirect
7. #21445342C2semidirect
8. #21445295C2semidirect
9. #21444946C2semidirect
10. #21444935C2semidirect
11. #21444901C2semidirect
12. #21444890C2semidirect
13. #21444710C2semidirect
14. #21444702C2semidirect
15. C61×#3274C2semidirect
17. #16840472C22semidirect
18. #16840167C22semidirect
20. #16840134C22semidirect
23. #16840108C22semidirect
28. #16840057C22semidirect
36. #16839954C22semidirect
38. #16839893C22semidirect
41. #16839841C22semidirect
45. #16839514C22semidirect
46. #16839513C22semidirect
47. #16839510C22semidirect
48. #16839510C22semidirect
50. #16839506C22semidirect
51. Q8⋊#92418C22semidirect
54. #16839489C22semidirect
55. C8×#92418C22semidirect
311. #3274D61semidirect
382. C61#20286semidirect

Maximal subgroups

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

subgroupindex
#214585222
#214584472
#214580612
#214454082
#214453572
#214453502
#214453422
#214452952
#214449462
#214449352
#214449012
#214448902
#214447102
#214447022
C61×#32742
#2028661