Finite Group Databasebeta

← #29833484

Subgroups — #29833484

Order 15616.

668 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#509×D61C211yesmaximal, characteristic
2#21454733C211yesmaximal, characteristic, complemented
3#21454668C211yesmaximal, characteristic, complemented
4#21447720C211yesmaximal, characteristic, complemented
5#21447719C211yesmaximal, characteristic, complemented
6#21447718C211yesmaximal, characteristic, complemented
7#21447717C211yesmaximal, characteristic, complemented
8#21447716C211yesmaximal, characteristic, complemented
9#21447715C211yesmaximal, characteristic, complemented
10#21447714C211yesmaximal, characteristic, complemented
11#21447590C211yesmaximal, characteristic, complemented
12#361×D61C211yesmaximal, characteristic, complemented
13#21447553C211yesmaximal, characteristic, complemented
14#21447422C211yesmaximal, characteristic, complemented
15C61×AGL(1,ℤ/16)C211yesmaximal, characteristic, complemented
16#135×D61C2211yescharacteristic
17C2×D8×D61C2211yescharacteristic
18#16840527—21no—
19#16840234—21no—
20#16840200C2211yescharacteristic
21#16840199C412yescomplemented
22#16840198C2211yescharacteristic
23#16840197C2211yescharacteristic
24#16840196C412yescomplemented
25SD32×D61C412yescomplemented
26D16×D61C412yescomplemented
27M32×D61C2211yescharacteristic
28M32×D61C2211yescharacteristic
29#16839885C2211yescharacteristic, complemented
30#16839884C2211yescharacteristic, complemented
31#16839883C2211yescharacteristic, complemented
32#16839877—21no—
33#16839876—21no—
34#16839875C2211yescharacteristic, complemented
35#16839874C2211yescharacteristic, complemented
36#16839873C2211yescharacteristic, complemented
37C4×#11758199—21no—
38#16839842—21no—
39#108×D61C2211yescharacteristic
40#16839797C2211yescharacteristic
41#16839777—21no—
42#16839765—21no—
43#16839719C2211yescharacteristic
44#16839586C2211yescharacteristic, complemented
45#16839585C2211yescharacteristic, complemented
46#16839584C2211yescharacteristic, complemented
47#16839576C2211yescharacteristic, complemented
48#16839575C2211yescharacteristic, complemented
49#16839574C2211yescharacteristic, complemented
50#16839573C2211yescharacteristic, complemented
51#16839572C2211yescharacteristic, complemented
52#16839571C2211yescharacteristic, complemented
53#16839540C2211yescharacteristic, complemented
54#16839539C2211yescharacteristic, complemented
55#16839539C2211yescharacteristic, complemented
56C61×#509C2211yescharacteristic
57C61×#444C2211yescharacteristic, complemented
58C61×#442C2211yescharacteristic, complemented
59C61×#365C2211yescharacteristic, complemented
60C61×#361C2211yescharacteristic, complemented
61C61×#360C2211yescharacteristic, complemented
62C61×#347C2211yescharacteristic, complemented
63#41×D61—21no—
64C2×D4×D61—21no—
65#14519800C2×C411yescharacteristic, complemented
66#14519799C2311yescharacteristic
67#14519798—21no—
68#14519797—21no—
69#14519796C2×C411yescharacteristic, complemented
70Q16×D61C2×C411yescharacteristic, complemented
71SD16×D61—21no—
72C2×D488C2311yescharacteristic
73C2×#11758199—21no—
74D8×D61—21no—
75D8×D61—21no—
76D8×D61C2×C412yes—
77D8×D61C2×C412yes—
78D8×D61C2×C411yescharacteristic, complemented
79#14519785—21no—
80#14519785—41no—
81#14519784—21no—
82#14519783—21no—
83#14519783—41no—
84C2×#11758189—41no—
85C2×#11758189D411yescharacteristic
86M16×D61—21no—
87M16×D61—41no—
88C2×C8×D61C2311yescharacteristic
89C4×D244—21no—
90D4×Dic61—21no—
91#14519739—21no—
92C4×C22⋊D61—41no—
93#14519737—41no—
94#14519735—21no—
95#14519734D411yescharacteristic
96C2×#11758198—41no—
97C22⋊C4×D61—41no—
98#14519722C2×C412yescomplemented
99#14519721C2×C412yescomplemented
100#14519720C2×C412yescomplemented
101D976C2×C412yescomplemented
102#14519718C2×C412yescomplemented
103#14519718C2×C412yescomplemented
104#14519717C2311yescharacteristic
105#14519717C2311yescharacteristic
106#14519716—21no—
107#14519716C2×C412yescomplemented
108C16×D61—21no—
109C16×D61C2×C412yescomplemented
110#14519713C2311yescharacteristic
111#14519712—21no—
112#14519710—21no—
113#14519709C2311yescharacteristic
114#14519705—21no—
115#14519704—21no—
116#14519703—21no—
117#14519699—21no—
118#14519698—21no—
119#14519697—21no—
120#14519687C2311yescharacteristic
121C4×#92418D411yescharacteristic
122C8×Dic61D411yescharacteristic
123#135×C61C2311yescharacteristic
124C2×D8×C61C2311yescharacteristic
125#131×C61—21no—
126C4×D4×C61—21no—
127SD32×C61C2×C412yescomplemented
128D16×C61C2×C412yescomplemented
129M32×C61C2311yescharacteristic
130M32×C61C2311yescharacteristic
131#108×C61C2311yescharacteristic
132#106×C61C2311yescharacteristic
133#104×C61—21no—
134#102×C61—21no—
135#97×C61C2311yescharacteristic
136C23×D61—41no—
137#41×C61—21no—
138C2×D4×C61—21no—
139C22×C4×C61—41no—
140C2×C22⋊D61—41no—
141C22×Dic61—41no—
142#11758226—21no—
143Q8×D61—21no—
144#11758224—21no—
145#11758224—41no—
146D4×D61—22no—
147D4×D61—22no—
148D4×D61—21no—
149D4×D61—41no—
150D4×D61—41no—
151D4×D61—41no—
152#11758222—21no—
153#11758222—41no—
154C2×D244—21no—
155C2×C4×D61—41no—
156C2×C4×D61—41no—
157C2×C4×D61—41no—
158C2×C4×D61C2×D411yescharacteristic
159Q16×C61C22×C411yescharacteristic
160SD16×C61—21no—
161D8×C61—21no—
162D8×C61C22×C411yescharacteristic
163D8×C61C22×C412yes—
164M16×C61—21no—
165M16×C61—41no—
166C2×C8×C61—41no—
167C2×C8×C61C2×D411yescharacteristic
168C2×C8×C61C2411yescharacteristic
169C4⋊C4×C61—21no—
170C22⋊C4×C61—41no—
171C42×C61C2×D411yescharacteristic
172C22⋊Dic61—41no—
173#11758202—21no—
174#11758201—21no—
175#11758200—21no—
176#11758199—22no—
177#11758199—22no—
178#11758199—21no—
179#11758198—21no—
180#11758198—41no—
181#11758198—41no—
182#11758198—41no—
183C4⋊Dic61—21no—
184#11758196—21no—
185C4×Dic61—41no—
186C4×Dic61C2×D411yescharacteristic
187#11758194—21no—
188#11758194—41no—
189C2×#92418—41no—
190C2×#92418C2×D411yescharacteristic
191C2×#92418C2×D411yescharacteristic
192Dic244C22×C411yescharacteristic
193D488—21no—
194D488C22×C411yescharacteristic
195D488C22×C412yes—
196#11758190—21no—
197#11758189—21no—
198#11758189—21no—
199#11758189—41no—
200#11758189—41no—
201#11758189—41no—
202C8×D61—41no—
203C8×D61—41no—
204C8×D61—41no—
205C8×D61C22×C411yescharacteristic
206C8×D61C22×C411yescharacteristic
207C8×D61#4111yescharacteristic
208C8×D61#4111yescharacteristic
209C16×C61—21no—
210C16×C61C22×C412yes—
211#11758185—21no—
212#11758185C22×C412yes—
213C23×C61—41no—
214C22×D61—41no—
215C22×D61—41no—
216C22×D61—42no—
217C22×D61—42no—
218C22×D61—161no—
219C22×D61—161no—
220C22×D61AGL(1,ℤ/8)11yescharacteristic
221Q8×C61—21no—
222D4×C61—22no—
223D4×C61—21no—
224D4×C61—41no—
225D4×C61—41no—
226C2×C4×C61—21no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—161no—
230C2×C4×C61C22×D411yescharacteristic
231C22⋊D61—41no—
232C22⋊D61—41no—
233C22⋊D61—41no—
234C22⋊D61—42no—
235C22⋊D61—42no—
236C2×Dic61—21no—
237C2×Dic61—41no—
238C2×Dic61—41no—
239C2×Dic61—161no—
240C2×Dic61AGL(1,ℤ/8)11yescharacteristic
241D244—21no—
242D244—22no—
243D244—41no—
244D244—41no—
245C4×D61—41no—
246C4×D61—41no—
247C4×D61—41no—
248C4×D61—161no—
249C4×D61—161no—
250C4×D61C4×D411yescharacteristic
251C4×D61C4×D411yescharacteristic
252C4×D61AGL(1,ℤ/8)11yescharacteristic
253C4×D61AGL(1,ℤ/8)11yescharacteristic
254Dic122—21no—
255Dic122—41no—
256C8×C61—21no—
257C8×C61—41no—
258C8×C61C23×C411yescharacteristic
259C8×C61C2×#4111yescharacteristic
260#92418—21no—
261#92418—41no—
262#92418C4×D411yescharacteristic
263#92418C4×D411yescharacteristic
264#33471—611nomaximal, Sylow
265C22×C61—41no—
266C22×C61—42no—
267C22×C61—81no—
268C22×C61—81no—
269C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
270C2×D61—41no—
271C2×D61—41no—
272C2×D61—42no—
273C2×D61—81no—
274C2×D61—81no—
275C2×D61—161no—
276C2×D61—162no—
277C2×D61—162no—
278C2×D61#44211yescharacteristic
279C2×D61#44211yescharacteristic
280C4×C61—41no—
281C4×C61—41no—
282C4×C61—81no—
283C4×C61C2×C4×D411yescharacteristic
284C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
285Dic61—41no—
286Dic61—41no—
287Dic61—81no—
288Dic61#44211yescharacteristic
289Dic61#44211yescharacteristic
290C2×#509—611no—
291#2957—611no—
292#2948—611no—
293AGL(1,ℤ/16)—611no—
294AGL(1,ℤ/16)—611no—
295AGL(1,ℤ/16)—611no—
296AGL(1,ℤ/16)—611no—
297AGL(1,ℤ/16)—611no—
298AGL(1,ℤ/16)—611no—
299AGL(1,ℤ/16)—611no—
300AGL(1,ℤ/16)D6111yescharacteristic, complemented
301#2155—611no—
302C2×#361—611no—
303#2145—611no—
304#2110—611no—
305C2×C61—21no—
306C2×C61—82no—
307C2×C61—81no—
308C2×C61C2×#44211yescharacteristic
309D61—21no—
310D61—21no—
311D61—82no—
312D61—81no—
313C2×#135—611no—
314C22×D8—611no—
315#568—1221no—
316#518—1221no—
317#509—611no—
318#509—611no—
319#509—612no—
320#509—612no—
321#509—611no—
322#509C2×D6111yescharacteristic
323C2×SD32—612no—
324C2×D16—612no—
325C2×M32—611no—
326C2×M32—611no—
327#444—611no—
328#444—611no—
329#444—611no—
330#444C2×D6111yescharacteristic, complemented
331#443—1221no—
332#443—1221no—
333#442—611no—
334#442—611no—
335#442—611no—
336#442C2×D6111yescharacteristic, complemented
337C4×D8—1221no—
338C4×D8—1221no—
339C2×#108—611no—
340#427—611no—
341#421—1221no—
342#418—1221no—
343#405—611no—
344#365—611no—
345#365—611no—
346#365—611no—
347#365C2×D6111yescharacteristic, complemented
348#361—611no—
349#361—611no—
350#361—611no—
351#361C2×D6111yescharacteristic, complemented
352#360—611no—
353#360—611no—
354#360—611no—
355#360C2×D6111yescharacteristic, complemented
356#347—611no—
357#347—611no—
358#347—611no—
359#347C2×D6111yescharacteristic, complemented
360C61#3347111yescharacteristic, complemented, Sylow
361C2×#41—1221no—
362C22×D4—1221no—
363#135—611no—
364#135—611no—
365#135—611no—
366#135—1221no—
367#135—1221no—
368#135C22×D6111yescharacteristic
369C2×Q16—611no—
370C2×SD16—1221no—
371C2×D8—612no—
372C2×D8—612no—
373C2×D8—611no—
374C2×D8—611no—
375C2×D8—1221no—
376C2×D8—1221no—
377C2×D8—1221no—
378C2×D8C22×D6111yescharacteristic
379#131—21no—
380#131—1221no—
381#131—1221no—
382#131—1221no—
383#131—2441no—
384#131—2441no—
385C2×M16—611no—
386C2×M16—1221no—
387C2×M16—2441no—
388C2×M16—2441no—
389C22×C8—611no—
390C4×D4—21no—
391C4×D4—1221no—
392C4×D4—1221no—
393C4×D4—1221no—
394C4×D4—2441no—
395C4×D4—2441no—
396#117—611no—
397#117—1221no—
398C2×C22⋊C4—2441no—
399C2×C22⋊C4—2441no—
400SD32—612no—
401SD32—612no—
402SD32—612no—
403SD32C4×D6112yescomplemented
404D16—612no—
405D16—612no—
406D16—612no—
407D16C4×D6112yescomplemented
408M32—611no—
409M32—612no—
410M32—611no—
411M32—1221no—
412M32C22×D6111yescharacteristic
413M32C22×D6111yescharacteristic
414C2×C16—612no—
415C2×C16—1221no—
416#108—611no—
417#108—1221no—
418#108C22×D6111yescharacteristic
419#107—1221no—
420#106—611no—
421#106C22×D6111yescharacteristic
422#104—21no—
423#104—1221no—
424#104—1221no—
425#104—1221no—
426#102—21no—
427#102—1221no—
428#102—1221no—
429#102—1221no—
430#97—611no—
431#97C22×D6111yescharacteristic
432C4×C8—611no—
433C4×C8—611no—
434C24—2441no—
435#41—21no—
436#41—1221no—
437#41—1221no—
438#41—1221no—
439#41—2441no—
440#41—2441no—
441C2×Q8—1221no—
442C2×D4—21no—
443C2×D4—1222no—
444C2×D4—1222no—
445C2×D4—1221no—
446C2×D4—1221no—
447C2×D4—2441no—
448C2×D4—2441no—
449C2×D4—2441no—
450C2×D4—2441no—
451C22×C4—41no—
452C22×C4—611no—
453C22×C4—2441no—
454C22×C4—2441no—
455C22×C4—2441no—
456C22×C4—2441no—
457Q16—611no—
458Q16—1221no—
459Q16C2×C4×D6111yescharacteristic
460SD16—21no—
461SD16—1221no—
462SD16—1221no—
463SD16—1221no—
464D8—21no—
465D8—611no—
466D8—612no—
467D8—1222no—
468D8—1222no—
469D8—1221no—
470D8—1221no—
471D8C2×C4×D6111yescharacteristic
472D8C2×C4×D6112yes—
473M16—21no—
474M16—41no—
475M16—1221no—
476M16—1221no—
477M16—1221no—
478M16—2441no—
479M16—2441no—
480M16—2441no—
481M16—2441no—
482C2×C8—41no—
483C2×C8—611no—
484C2×C8—611no—
485C2×C8—611no—
486C2×C8—611no—
487C2×C8—611no—
488C2×C8—611no—
489C2×C8—2441no—
490C2×C8—2441no—
491C2×C8—2441no—
492C2×C8—2441no—
493C2×C8D4×D6111yescharacteristic
494C2×C8C23×D6111yescharacteristic
495C4⋊C4—21no—
496C4⋊C4—1221no—
497C4⋊C4—1221no—
498C22⋊C4—41no—
499C22⋊C4—1221no—
500C22⋊C4—2441no—
501C22⋊C4—2441no—
502C22⋊C4—2441no—
503C22⋊C4—2441no—
504C42—611no—
505C42—2441no—
506C42D4×D6111yescharacteristic
507C16—21no—
508C16—612no—
509C16—1221no—
510C16C2×C4×D6112yes—
511C23—41no—
512C23—611no—
513C23—9761no—
514C23—9761no—
515C23—2441no—
516C23—2442no—
517C23—2442no—
518C23—2441no—
519Q8—21no—
520Q8—1221no—
521Q8—2441no—
522D4—22no—
523D4—21no—
524D4—41no—
525D4—41no—
526D4—1222no—
527D4—1221no—
528D4—2441no—
529D4—2442no—
530D4—2442no—
531D4—2441no—
532D4—2441no—
533D4—2441no—
534D4—2441no—
535C2×C4—21no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—161no—
539C2×C4—611no—
540C2×C4—611no—
541C2×C4—611no—
542C2×C4—611no—
543C2×C4—611no—
544C2×C4—1221no—
545C2×C4—9761no—
546C2×C4—9761no—
547C2×C4—9761no—
548C2×C4—2441no—
549C2×C4—2441no—
550C2×C4—2441no—
551C2×C4—2441no—
552C2×C4—2441no—
553C2×C4C2×D4×D6111yescharacteristic
554C8—21no—
555C8—41no—
556C8—611no—
557C8—611no—
558C8—1221no—
559C8—2441no—
560C8C22×C4×D6111yescharacteristic
561C8#41×D6111yescharacteristic
562C22—41no—
563C22—42no—
564C22—81no—
565C22—81no—
566C22—611no—
567C22—611no—
568C22—9762no—
569C22—9762no—
570C22—9761no—
571C22—4881no—
572C22—4881no—
573C22—2441no—
574C22—2441no—
575C22—2442no—
576C22AGL(1,ℤ/8)×D6111yescharacteristic
577C4—41no—
578C4—41no—
579C4—81no—
580C4—611no—
581C4—611no—
582C4—4881no—
583C4—2441no—
584C4—2441no—
585C4C4×D4×D6111yescharacteristic
586C4AGL(1,ℤ/8)×D6111yescharacteristic
587C2—21no—
588C2—82no—
589C2—81no—
590C2—1221no—
591C2—1221no—
592C2—4882no—
593C2—4881no—
594C2#442×D6111yescharacteristic
subgroupG/H
centerC2#442×D61
commutatorC8×C61C23×C4
FrattiniC2×C8C23×D61
FittingC61×AGL(1,ℤ/16)C2
socleC2×C61C2×#442
prime radical#509×D61C2
cosocleC2×C8×C61C24
hypercenterAGL(1,ℤ/16)D61
hypocenterC61#33471
LCSG ▷ C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ #97 ◁ AGL(1,ℤ/16)

Complements

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

NH ≅ G/Ntype
2. #21454733C2semidirect
3. #21454668C2semidirect
4. #21447720C2semidirect
5. #21447719C2semidirect
6. #21447718C2semidirect
7. #21447717C2semidirect
8. #21447716C2semidirect
9. #21447715C2semidirect
10. #21447714C2semidirect
11. #21447590C2semidirect
12. #361×D61C2semidirect
13. #21447553C2semidirect
14. #21447422C2semidirect
15. C61×AGL(1,ℤ/16)C2semidirect
21. #16840199C4semidirect
24. #16840196C4semidirect
25. SD32×D61C4semidirect
26. D16×D61C4semidirect
29. #16839885C22semidirect
30. #16839884C22semidirect
31. #16839883C22semidirect
34. #16839875C22semidirect
35. #16839874C22semidirect
36. #16839873C22semidirect
44. #16839586C22semidirect
45. #16839585C22semidirect
46. #16839584C22semidirect
47. #16839576C22semidirect
48. #16839575C22semidirect
49. #16839574C22semidirect
50. #16839573C22semidirect
51. #16839572C22semidirect
52. #16839571C22semidirect
53. #16839540C22semidirect
54. #16839539C22semidirect
55. #16839539C22semidirect
57. C61×#444C22semidirect
58. C61×#442C22semidirect
59. C61×#365C22semidirect
60. C61×#361C22semidirect
61. C61×#360C22semidirect
62. C61×#347C22semidirect
65. #14519800C2×C4semidirect
69. #14519796C2×C4semidirect
70. Q16×D61C2×C4semidirect
78. D8×D61C2×C4semidirect
98. #14519722C2×C4semidirect
99. #14519721C2×C4semidirect
100. #14519720C2×C4semidirect
101. D976C2×C4semidirect
102. #14519718C2×C4semidirect
103. #14519718C2×C4semidirect
107. #14519716C2×C4semidirect
109. C16×D61C2×C4semidirect
127. SD32×C61C2×C4semidirect
128. D16×C61C2×C4semidirect
300. AGL(1,ℤ/16)D61semidirect
330. #444C2×D61semidirect
336. #442C2×D61semidirect
347. #365C2×D61semidirect
351. #361C2×D61semidirect
355. #360C2×D61semidirect
359. #347C2×D61semidirect
360. C61#33471semidirect
403. SD32C4×D61semidirect
407. D16C4×D61semidirect

Maximal subgroups

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

subgroupindex
#509×D612
#214547332
#214546682
#214477202
#214477192
#214477182
#214477172
#214477162
#214477152
#214477142
#214475902
#361×D612
#214475532
#214474222
C61×AGL(1,ℤ/16)2
#3347161