Finite Group Databasebeta

← #29833469

Subgroups — #29833469

Order 15616.

646 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458951C211yesmaximal, characteristic
2#21454733C211yesmaximal, characteristic, complemented
3#21454701C211yesmaximal, characteristic, complemented
4#21447720C211yesmaximal, characteristic, complemented
5#21447718C211yesmaximal, characteristic, complemented
6#21447715C211yesmaximal, characteristic, complemented
7#21447712C211yesmaximal, characteristic, complemented
8#21447710C211yesmaximal, characteristic, complemented
9#21447709C211yesmaximal, characteristic, complemented
10#21447707C211yesmaximal, characteristic, complemented
11#365×D61C211yesmaximal, characteristic, complemented
12#21447565C211yesmaximal, characteristic, complemented
13#360×D61C211yesmaximal, characteristic, complemented
14#21447422C211yesmaximal, characteristic, complemented
15C61×AGL(1,ℤ/16)C211yesmaximal, characteristic, complemented
16C2×#14519796C2211yescharacteristic
17#135×D61C2211yescharacteristic
18#16840527—21no—
19#16840241—21no—
20#16840205C2211yescharacteristic
21#16840202C2211yescharacteristic
22#16840201C412yescomplemented
23#16840198C2211yescharacteristic
24#16840196C412yescomplemented
25#16840193C412yescomplemented
26#16840191C412yescomplemented
27M32×D61C2211yescharacteristic
28M32×D61C2211yescharacteristic
29#16839885C2211yescharacteristic
30#16839884C2211yescharacteristic
31#16839883C2211yescharacteristic
32#16839877—21no—
33#16839876—21no—
34#16839874C2211yescharacteristic, complemented
35#16839870C2211yescharacteristic, complemented
36#16839868C2211yescharacteristic, complemented
37#16839853—21no—
38C4×#11758200—21no—
39#108×D61C2211yescharacteristic
40#106×D61C2211yescharacteristic
41#16839777—21no—
42#16839760—21no—
43#16839719C2211yescharacteristic
44#16839586C2211yescharacteristic, complemented
45#16839585C2211yescharacteristic, complemented
46#16839584C2211yescharacteristic, complemented
47#16839579C2211yescharacteristic, complemented
48#16839577C2211yescharacteristic, complemented
49#16839575C2211yescharacteristic, complemented
50#16839573C2211yescharacteristic
51#16839572C2211yescharacteristic
52#16839571C2211yescharacteristic
53#16839540C2211yescharacteristic, complemented
54#16839539C2211yescharacteristic, complemented
55#16839539C2211yescharacteristic, complemented
56C61×#509C2211yescharacteristic
57C61×#444C2211yescharacteristic
58C61×#442C2211yescharacteristic, complemented
59C61×#365C2211yescharacteristic, complemented
60C61×#361C2211yescharacteristic, complemented
61C61×#360C2211yescharacteristic
62C61×#347C2211yescharacteristic, complemented
63C2×#11758224—21no—
64#41×D61—21no—
65#14519800C2×C411yescharacteristic, complemented
66#14519799C2311yescharacteristic
67#14519798—21no—
68#14519797—21no—
69#14519796—21no—
70#14519796—21no—
71#14519796C2×C412yes—
72#14519796C2×C412yes—
73#14519796C2×C411yescharacteristic, complemented
74C2×Dic244C2311yescharacteristic
75Q16×D61C2×C411yescharacteristic, complemented
76C2×#11758200—21no—
77SD16×D61—21no—
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×Dic122—21no—
90#14519742—21no—
91D4×Dic61—21no—
92C4×C22⋊D61—41no—
93#14519737—41no—
94#14519734D411yescharacteristic
95#14519733—41no—
96C2×#11758196—41no—
97C4⋊C4×D61—21no—
98Dic488C2×C412yescomplemented
99#14519723C2×C412yescomplemented
100#14519722C2×C412yescomplemented
101#14519721C2×C412yescomplemented
102#14519721C2×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#14519709C2311yescharacteristic
113#14519708—21no—
114#14519705—21no—
115#14519704—21no—
116#14519703—21no—
117#14519702—21no—
118#14519700—21no—
119#14519698—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
136#41×C61—21no—
137C2×D4×C61—21no—
138C22×C4×C61—41no—
139C2×C22⋊D61—41no—
140C22×Dic61—41no—
141C22×Dic61—41no—
142#11758226—21no—
143Q8×D61—21no—
144#11758224—21no—
145#11758224—22no—
146#11758224—22no—
147#11758224—41no—
148#11758224—41no—
149#11758224—41no—
150D4×D61—21no—
151D4×D61—41no—
152#11758222—21no—
153#11758222—41no—
154C2×C4×D61—41no—
155C2×C4×D61—41no—
156C2×C4×D61C2×D411yescharacteristic
157C2×Dic122—21no—
158Q16×C61C22×C411yescharacteristic
159SD16×C61—21no—
160D8×C61—21no—
161D8×C61C22×C411yescharacteristic
162D8×C61C22×C412yes—
163M16×C61—21no—
164M16×C61—41no—
165C2×C8×C61—41no—
166C2×C8×C61C2×D411yescharacteristic
167C2×C8×C61C2411yescharacteristic
168C4⋊C4×C61—21no—
169C22⋊C4×C61—41no—
170C42×C61C2×D411yescharacteristic
171C22⋊Dic61—41no—
172#11758202—21no—
173#11758201—21no—
174#11758200—21no—
175#11758200—22no—
176#11758200—22no—
177#11758199—21no—
178#11758198—21no—
179C4⋊Dic61—21no—
180#11758196—21no—
181#11758196—41no—
182#11758196—41no—
183#11758196—41no—
184C4×Dic61—41no—
185C4×Dic61—41no—
186C4×Dic61C2×D411yescharacteristic
187#11758194—21no—
188#11758194—41no—
189C2×#92418—41no—
190C2×#92418C2×D411yescharacteristic
191C2×#92418C2×D411yescharacteristic
192Dic244—21no—
193Dic244C22×C411yescharacteristic
194Dic244C22×C412yes—
195D488C22×C411yescharacteristic
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×D61AGL(1,ℤ/8)11yescharacteristic
216Q8×C61—21no—
217D4×C61—21no—
218D4×C61—22no—
219D4×C61—41no—
220D4×C61—41no—
221C2×C4×C61—21no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61—161no—
225C2×C4×C61C22×D411yescharacteristic
226C22⋊D61—41no—
227C22⋊D61—41no—
228C22⋊D61—42no—
229C22⋊D61—42no—
230C22⋊D61—41no—
231C2×Dic61—21no—
232C2×Dic61—41no—
233C2×Dic61—41no—
234C2×Dic61—42no—
235C2×Dic61—42no—
236C2×Dic61—41no—
237C2×Dic61—161no—
238C2×Dic61AGL(1,ℤ/8)11yescharacteristic
239D244—21no—
240D244—41no—
241C4×D61—41no—
242C4×D61—41no—
243C4×D61—41no—
244C4×D61—161no—
245C4×D61C4×D411yescharacteristic
246C4×D61C4×D411yescharacteristic
247C4×D61AGL(1,ℤ/8)11yescharacteristic
248C4×D61AGL(1,ℤ/8)11yescharacteristic
249Dic122—21no—
250Dic122—22no—
251Dic122—41no—
252Dic122—41no—
253C8×C61—21no—
254C8×C61—41no—
255C8×C61C23×C411yescharacteristic
256C8×C61C2×#4111yescharacteristic
257#92418—21no—
258#92418—41no—
259#92418C4×D411yescharacteristic
260#92418C4×D411yescharacteristic
261#33469—611nomaximal, Sylow
262C22×C61—42no—
263C22×C61—41no—
264C22×C61—81no—
265C22×C61—81no—
266C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
267C2×D61—41no—
268C2×D61—41no—
269C2×D61—161no—
270C2×D61#44211yescharacteristic
271C2×D61#44211yescharacteristic
272C4×C61—41no—
273C4×C61—41no—
274C4×C61—81no—
275C4×C61C2×C4×D411yescharacteristic
276C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
277Dic61—41no—
278Dic61—41no—
279Dic61—42no—
280Dic61—81no—
281Dic61#44211yescharacteristic
282Dic61#44211yescharacteristic
283#3414—611no—
284#2957—611no—
285#2952—611no—
286AGL(1,ℤ/16)—611no—
287AGL(1,ℤ/16)—611no—
288AGL(1,ℤ/16)—611no—
289AGL(1,ℤ/16)D6111yescharacteristic, complemented
290#2180—611no—
291#2180—611no—
292#2180—611no—
293#2180—611no—
294C2×#365—611no—
295#2148—611no—
296C2×#360—611no—
297#2110—611no—
298C2×C61—21no—
299C2×C61—81no—
300C2×C61—82no—
301C2×C61C2×#44211yescharacteristic
302D61—21no—
303D61—21no—
304D61—81no—
305C2×#135—611no—
306C2×#135—611no—
307#568—1221no—
308#520—1221no—
309#510—611no—
310#510—612no—
311#510—611no—
312#509—611no—
313#509—612no—
314#509C2×D6111yescharacteristic
315#508—612no—
316#508—612no—
317C2×M32—611no—
318C2×M32—611no—
319#444—611no—
320#444—611no—
321#444—611no—
322#444C2×D6111yescharacteristic
323#443—1221no—
324#443—1221no—
325#442—611no—
326#442C2×D6111yescharacteristic, complemented
327#441—611no—
328#441—611no—
329C4×SD16—1221no—
330C4×SD16—1221no—
331C2×#108—611no—
332C2×#106—611no—
333#421—1221no—
334#417—1221no—
335#405—611no—
336#365—611no—
337#365—611no—
338#365—611no—
339#365C2×D6111yescharacteristic, complemented
340#362—611no—
341#362—611no—
342#361—611no—
343#361C2×D6111yescharacteristic, complemented
344#360—611no—
345#360—611no—
346#360—611no—
347#360C2×D6111yescharacteristic
348#347—611no—
349#347—611no—
350#347—611no—
351#347C2×D6111yescharacteristic, complemented
352C61#3346911yescharacteristic, complemented, Sylow
353C2×#41—1221no—
354C2×#41—1221no—
355#135—611no—
356#135—612no—
357#135—612no—
358#135—611no—
359#135—611no—
360#135—1221no—
361#135—1221no—
362#135—1221no—
363#135—1221no—
364#135C22×D6111yescharacteristic
365C2×Q16—611no—
366C2×Q16—611no—
367C2×SD16—1221no—
368C2×SD16—1221no—
369C2×D8—611no—
370C2×D8C22×D6111yescharacteristic
371#131—21no—
372#131—1221no—
373#131—1221no—
374#131—1221no—
375#131—2441no—
376#131—2441no—
377C2×M16—611no—
378C2×M16—1221no—
379C2×M16—2441no—
380C2×M16—2441no—
381C22×C8—611no—
382C4×Q8—1221no—
383C4×Q8—1221no—
384C4×D4—21no—
385C4×D4—1221no—
386C4×D4—2441no—
387C4×D4—2441no—
388#117—611no—
389#117—2441no—
390C2×C4⋊C4—1221no—
391C2×C4⋊C4—2441no—
392Q32—612no—
393Q32—612no—
394SD32—612no—
395SD32—612no—
396SD32—612no—
397SD32C4×D6112yescomplemented
398D16—612no—
399D16C4×D6112yescomplemented
400M32—611no—
401M32—612no—
402M32—611no—
403M32—1221no—
404M32C22×D6111yescharacteristic
405M32C22×D6111yescharacteristic
406C2×C16—612no—
407C2×C16—1221no—
408#108—611no—
409#108—1221no—
410#108C22×D6111yescharacteristic
411#106—611no—
412#106—1221no—
413#106C22×D6111yescharacteristic
414#104—21no—
415#104—1221no—
416#104—1221no—
417#104—1221no—
418#103—1221no—
419#103—1221no—
420#102—21no—
421#102—1221no—
422#97—611no—
423#97C22×D6111yescharacteristic
424C4×C8—611no—
425C4×C8—611no—
426#41—21no—
427#41—1222no—
428#41—1222no—
429#41—1221no—
430#41—1221no—
431#41—1221no—
432#41—2441no—
433#41—2441no—
434#41—2441no—
435#41—2441no—
436C2×Q8—1221no—
437C2×Q8—1221no—
438C2×D4—21no—
439C2×D4—1221no—
440C2×D4—2441no—
441C2×D4—2441no—
442C22×C4—41no—
443C22×C4—611no—
444C22×C4—2441no—
445C22×C4—2441no—
446C22×C4—2441no—
447C22×C4—2441no—
448Q16—611no—
449Q16—612no—
450Q16—1221no—
451Q16—1221no—
452Q16C2×C4×D6111yescharacteristic
453SD16—21no—
454SD16—1222no—
455SD16—1222no—
456SD16—1221no—
457SD16—1221no—
458SD16—1221no—
459D8—21no—
460D8—611no—
461D8—1221no—
462D8C2×C4×D6112yes—
463D8C2×C4×D6111yescharacteristic
464M16—21no—
465M16—41no—
466M16—1221no—
467M16—1221no—
468M16—1221no—
469M16—2441no—
470M16—2441no—
471M16—2441no—
472M16—2441no—
473C2×C8—41no—
474C2×C8—611no—
475C2×C8—611no—
476C2×C8—611no—
477C2×C8—611no—
478C2×C8—611no—
479C2×C8—611no—
480C2×C8—2441no—
481C2×C8—2441no—
482C2×C8—2441no—
483C2×C8—2441no—
484C2×C8D4×D6111yescharacteristic
485C2×C8C23×D6111yescharacteristic
486C4⋊C4—21no—
487C4⋊C4—1221no—
488C4⋊C4—1221no—
489C4⋊C4—2441no—
490C4⋊C4—2441no—
491C4⋊C4—2441no—
492C22⋊C4—41no—
493C22⋊C4—1221no—
494C22⋊C4—2441no—
495C42—611no—
496C42—2441no—
497C42—2441no—
498C42D4×D6111yescharacteristic
499C16—21no—
500C16—612no—
501C16—1221no—
502C16C2×C4×D6112yes—
503C23—41no—
504C23—611no—
505C23—2441no—
506Q8—21no—
507Q8—1221no—
508Q8—1222no—
509Q8—2441no—
510Q8—2441no—
511D4—22no—
512D4—21no—
513D4—41no—
514D4—41no—
515D4—1221no—
516D4—2441no—
517D4—2441no—
518D4—2441no—
519D4—2442no—
520D4—2442no—
521D4—2441no—
522C2×C4—21no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—161no—
526C2×C4—611no—
527C2×C4—611no—
528C2×C4—611no—
529C2×C4—611no—
530C2×C4—611no—
531C2×C4—1221no—
532C2×C4—9761no—
533C2×C4—9761no—
534C2×C4—2441no—
535C2×C4—2441no—
536C2×C4—2441no—
537C2×C4—2442no—
538C2×C4—2442no—
539C2×C4—2441no—
540C2×C4—2441no—
541C2×C4—2441no—
542C2×C4C2×D4×D6111yescharacteristic
543C8—21no—
544C8—41no—
545C8—611no—
546C8—611no—
547C8—1221no—
548C8—2441no—
549C8C22×C4×D6111yescharacteristic
550C8#41×D6111yescharacteristic
551C22—41no—
552C22—42no—
553C22—81no—
554C22—81no—
555C22—611no—
556C22—611no—
557C22—9761no—
558C22—2441no—
559C22—2441no—
560C22AGL(1,ℤ/8)×D6111yescharacteristic
561C4—41no—
562C4—41no—
563C4—81no—
564C4—611no—
565C4—611no—
566C4—4881no—
567C4—2441no—
568C4—2441no—
569C4—2442no—
570C4C4×D4×D6111yescharacteristic
571C4AGL(1,ℤ/8)×D6111yescharacteristic
572C2—21no—
573C2—82no—
574C2—81no—
575C2—1221no—
576C2—1221no—
577C2—4881no—
578C2#442×D6111yescharacteristic
subgroupG/H
centerC2#442×D61
commutatorC8×C61C23×C4
FrattiniC2×C8C23×D61
FittingC61×AGL(1,ℤ/16)C2
socleC2×C61C2×#442
prime radical#21458951C2
cosocleC2×C8×C61C24
hypercenterAGL(1,ℤ/16)D61
hypocenterC61#33469
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. #21454701C2semidirect
4. #21447720C2semidirect
5. #21447718C2semidirect
6. #21447715C2semidirect
7. #21447712C2semidirect
8. #21447710C2semidirect
9. #21447709C2semidirect
10. #21447707C2semidirect
11. #365×D61C2semidirect
12. #21447565C2semidirect
13. #360×D61C2semidirect
14. #21447422C2semidirect
15. C61×AGL(1,ℤ/16)C2semidirect
22. #16840201C4semidirect
24. #16840196C4semidirect
25. #16840193C4semidirect
26. #16840191C4semidirect
34. #16839874C22semidirect
35. #16839870C22semidirect
36. #16839868C22semidirect
44. #16839586C22semidirect
45. #16839585C22semidirect
46. #16839584C22semidirect
47. #16839579C22semidirect
48. #16839577C22semidirect
49. #16839575C22semidirect
53. #16839540C22semidirect
54. #16839539C22semidirect
55. #16839539C22semidirect
58. C61×#442C22semidirect
59. C61×#365C22semidirect
60. C61×#361C22semidirect
62. C61×#347C22semidirect
65. #14519800C2×C4semidirect
73. #14519796C2×C4semidirect
75. Q16×D61C2×C4semidirect
78. D8×D61C2×C4semidirect
98. Dic488C2×C4semidirect
99. #14519723C2×C4semidirect
100. #14519722C2×C4semidirect
101. #14519721C2×C4semidirect
102. #14519721C2×C4semidirect
103. #14519718C2×C4semidirect
107. #14519716C2×C4semidirect
109. C16×D61C2×C4semidirect
127. SD32×C61C2×C4semidirect
128. D16×C61C2×C4semidirect
289. AGL(1,ℤ/16)D61semidirect
326. #442C2×D61semidirect
339. #365C2×D61semidirect
343. #361C2×D61semidirect
351. #347C2×D61semidirect
352. C61#33469semidirect
397. SD32C4×D61semidirect
399. D16C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214589512
#214547332
#214547012
#214477202
#214477182
#214477152
#214477122
#214477102
#214477092
#214477072
#365×D612
#214475652
#360×D612
#214474222
C61×AGL(1,ℤ/16)2
#3346961