Finite Group Databasebeta

← #29833476

Subgroups — #29833476

Order 15616.

632 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#510×D61C211yesmaximal, characteristic
2#21454733C211yesmaximal, characteristic
3#21454684C211yesmaximal, characteristic, complemented
4#21447713C211yesmaximal, characteristic, complemented
5#21447712C211yesmaximal, characteristic, complemented
6#21447711C211yesmaximal, characteristic, complemented
7#21447710C211yesmaximal, characteristic, complemented
8#21447709C211yesmaximal, characteristic, complemented
9#21447708C211yesmaximal, characteristic, complemented
10#21447707C211yesmaximal, characteristic, complemented
11#21447590C211yesmaximal, characteristic, complemented
12#362×D61C211yesmaximal, characteristic, complemented
13#21447553C211yesmaximal, characteristic
14#21447422C211yesmaximal, characteristic, complemented
15C61×#2180C211yesmaximal, characteristic, complemented
16#135×D61C2211yescharacteristic
17C2×Q16×D61C2211yescharacteristic
18#16840527—21no—
19#16840238—21no—
20#16840205C2211yescharacteristic
21#16840204C412yescomplemented
22#16840203C2211yescharacteristic
23#16840202C2211yescharacteristic
24#16840201C412yescomplemented
25Q32×D61C412yescomplemented
26SD32×D61C412yescomplemented
27M32×D61C2211yescharacteristic
28M32×D61C2211yescharacteristic
29#16839885C2211yescharacteristic
30#16839884C2211yescharacteristic
31#16839883C2211yescharacteristic
32#16839877—21no—
33#16839876—21no—
34#16839870C2211yescharacteristic, complemented
35#16839869C2211yescharacteristic, complemented
36#16839868C2211yescharacteristic, complemented
37C4×#11758202—21no—
38#16839854—21no—
39#108×D61C2211yescharacteristic
40#16839797C2211yescharacteristic
41#16839777—21no—
42#16839769—21no—
43#16839719C2211yescharacteristic
44#16839586C2211yescharacteristic, complemented
45#16839585C2211yescharacteristic, complemented
46#16839584C2211yescharacteristic, complemented
47#16839579C2211yescharacteristic, complemented
48#16839578C2211yescharacteristic, complemented
49#16839577C2211yescharacteristic, complemented
50#16839573C2211yescharacteristic
51#16839572C2211yescharacteristic
52#16839571C2211yescharacteristic
53#16839540C2211yescharacteristic, complemented
54#16839539C2211yescharacteristic, complemented
55#16839539C2211yescharacteristic, complemented
56C61×#510C2211yescharacteristic
57C61×#444C2211yescharacteristic
58C61×#441C2211yescharacteristic, complemented
59C61×#365C2211yescharacteristic, complemented
60C61×#362C2211yescharacteristic, complemented
61C61×#360C2211yescharacteristic
62C61×#347C2211yescharacteristic, complemented
63#41×D61—21no—
64C2×Q8×D61—21no—
65#14519800C2×C411yescharacteristic
66#14519799C2311yescharacteristic
67#14519798—21no—
68#14519797—21no—
69#14519796C2×C411yescharacteristic
70C2×Dic244C2311yescharacteristic
71C2×#11758202—21no—
72Q16×D61—21no—
73Q16×D61—21no—
74Q16×D61C2×C412yes—
75Q16×D61C2×C412yes—
76Q16×D61C2×C411yescharacteristic
77SD16×D61—21no—
78D8×D61C2×C411yescharacteristic
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
89Q8×Dic61—21no—
90C4×Dic122—21no—
91C4×Dic122—41no—
92#14519742—21no—
93#14519742—41no—
94#14519735—21no—
95#14519735—41no—
96#14519734—41no—
97#14519734D411yescharacteristic
98Dic488C2×C412yes—
99#14519723C2×C412yes—
100#14519723C2×C412yes—
101#14519722C2×C412yes—
102#14519721C2×C412yes—
103#14519720C2×C412yes—
104#14519717C2311yescharacteristic
105#14519717C2311yescharacteristic
106#14519716—21no—
107#14519716C2×C412yes—
108C16×D61—21no—
109C16×D61C2×C412yes—
110#14519713C2311yescharacteristic
111#14519712—21no—
112#14519710—21no—
113#14519709C2311yescharacteristic
114#14519705—21no—
115#14519704—21no—
116#14519703—21no—
117#14519702—21no—
118#14519701—21no—
119#14519700—21no—
120#14519687C2311yescharacteristic
121C4×#92418D411yescharacteristic
122C8×Dic61D411yescharacteristic
123#135×C61C2311yescharacteristic
124C2×Q16×C61C2311yescharacteristic
125#131×C61—21no—
126C4×Q8×C61—21no—
127Q32×C61C2×C412yes—
128SD32×C61C2×C412yes—
129M32×C61C2311yescharacteristic
130M32×C61C2311yescharacteristic
131#108×C61C2311yescharacteristic
132#106×C61C2311yescharacteristic
133#104×C61—21no—
134#103×C61—21no—
135#97×C61C2311yescharacteristic
136#41×C61—21no—
137C2×Q8×C61—21no—
138#11758226—21no—
139Q8×D61—22no—
140Q8×D61—22no—
141Q8×D61—21no—
142Q8×D61—41no—
143Q8×D61—41no—
144#11758224—21no—
145#11758224—41no—
146D4×D61—21no—
147D4×D61—41no—
148#11758222—21no—
149#11758222—41no—
150C2×C4×D61—41no—
151C2×C4×D61—41no—
152C2×C4×D61C2×D411yescharacteristic
153C2×Dic122—21no—
154C2×Dic122—41no—
155Q16×C61—21no—
156Q16×C61C22×C412yes—
157Q16×C61C22×C411yescharacteristic
158SD16×C61—21no—
159D8×C61C22×C411yescharacteristic
160M16×C61—21no—
161M16×C61—41no—
162C2×C8×C61—41no—
163C2×C8×C61C2×D411yescharacteristic
164C2×C8×C61C2411yescharacteristic
165C4⋊C4×C61—21no—
166C4⋊C4×C61—41no—
167C42×C61—41no—
168C42×C61C2×D411yescharacteristic
169#11758202—22no—
170#11758202—22no—
171#11758202—21no—
172#11758201—21no—
173#11758200—21no—
174#11758199—21no—
175#11758198—21no—
176#11758198—41no—
177C4⋊Dic61—21no—
178C4⋊Dic61—41no—
179#11758196—21no—
180#11758196—41no—
181#11758196—41no—
182C4×Dic61—41no—
183C4×Dic61—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×C412yes—
194Dic244C22×C411yescharacteristic
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—
213C22×D61—41no—
214C22×D61AGL(1,ℤ/8)11yescharacteristic
215Q8×C61—21no—
216Q8×C61—22no—
217Q8×C61—41no—
218D4×C61—21no—
219D4×C61—41no—
220C2×C4×C61—21no—
221C2×C4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61C22×D411yescharacteristic
225C22⋊D61—41no—
226C22⋊D61—41no—
227C22⋊D61—41no—
228C2×Dic61—21no—
229C2×Dic61—41no—
230C2×Dic61—41no—
231C2×Dic61—41no—
232C2×Dic61AGL(1,ℤ/8)11yescharacteristic
233D244—21no—
234D244—41no—
235C4×D61—41no—
236C4×D61—42no—
237C4×D61—42no—
238C4×D61—41no—
239C4×D61—41no—
240C4×D61C4×D411yescharacteristic
241C4×D61C4×D411yescharacteristic
242C4×D61AGL(1,ℤ/8)11yescharacteristic
243C4×D61AGL(1,ℤ/8)11yescharacteristic
244Dic122—21no—
245Dic122—22no—
246Dic122—41no—
247Dic122—42no—
248Dic122—42no—
249Dic122—41no—
250C8×C61—21no—
251C8×C61—41no—
252C8×C61C23×C411yescharacteristic
253C8×C61C2×#4111yescharacteristic
254#92418—21no—
255#92418—41no—
256#92418C4×D411yescharacteristic
257#92418C4×D411yescharacteristic
258#33470—611nomaximal, Sylow
259C22×C61—41no—
260C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
261C2×D61—41no—
262C2×D61—41no—
263C2×D61—161no—
264C2×D61#44211yescharacteristic
265C2×D61#44211yescharacteristic
266C4×C61—41no—
267C4×C61—42no—
268C4×C61—41no—
269C4×C61—81no—
270C4×C61C2×C4×D411yescharacteristic
271C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
272Dic61—41no—
273Dic61—42no—
274Dic61—41no—
275Dic61—81no—
276Dic61#44211yescharacteristic
277Dic61#44211yescharacteristic
278C2×#510—611no—
279#2957—611no—
280#2950—611no—
281#2180—611no—
282#2180—611no—
283#2180—611no—
284#2180—611no—
285#2180—611no—
286#2180—611no—
287#2180—611no—
288#2180D6111yescharacteristic, complemented
289#2155—611no—
290C2×#362—611no—
291#2145—611no—
292#2110—611no—
293C2×C61—21no—
294C2×C61—81no—
295C2×C61C2×#44211yescharacteristic
296D61—21no—
297D61—21no—
298D61—81no—
299C2×#135—611no—
300C22×Q16—611no—
301#568—1221no—
302#519—1221no—
303#510—611no—
304#510—611no—
305#510—612no—
306#510—612no—
307#510—611no—
308#510C2×D6111yescharacteristic
309C2×Q32—612no—
310C2×SD32—612no—
311C2×M32—611no—
312C2×M32—611no—
313#444—611no—
314#444—611no—
315#444—611no—
316#444C2×D6111yescharacteristic
317#443—1221no—
318#443—1221no—
319#441—611no—
320#441—611no—
321#441—611no—
322#441C2×D6111yescharacteristic, complemented
323C4×Q16—1221no—
324C4×Q16—1221no—
325C2×#108—611no—
326#427—611no—
327#421—1221no—
328#419—1221no—
329#405—611no—
330#365—611no—
331#365—611no—
332#365—611no—
333#365C2×D6111yescharacteristic, complemented
334#362—611no—
335#362—611no—
336#362—611no—
337#362C2×D6111yescharacteristic, complemented
338#360—611no—
339#360—611no—
340#360—611no—
341#360C2×D6111yescharacteristic
342#347—611no—
343#347—611no—
344#347—611no—
345#347C2×D6111yescharacteristic, complemented
346C61#3347011yescharacteristic, complemented, Sylow
347C2×#41—1221no—
348C22×Q8—1221no—
349#135—611no—
350#135—611no—
351#135—611no—
352#135—1221no—
353#135—1221no—
354#135C22×D6111yescharacteristic
355C2×Q16—612no—
356C2×Q16—612no—
357C2×Q16—611no—
358C2×Q16—611no—
359C2×Q16—1221no—
360C2×Q16—1221no—
361C2×Q16—1221no—
362C2×Q16C22×D6111yescharacteristic
363C2×SD16—1221no—
364C2×D8—611no—
365#131—21no—
366#131—1221no—
367#131—1221no—
368#131—1221no—
369#131—2441no—
370#131—2441no—
371C2×M16—611no—
372C2×M16—1221no—
373C2×M16—2441no—
374C2×M16—2441no—
375C22×C8—611no—
376C4×Q8—21no—
377C4×Q8—1221no—
378C4×Q8—1221no—
379C4×Q8—1221no—
380C4×Q8—2441no—
381C4×Q8—2441no—
382#117—611no—
383#117—1221no—
384#117—2441no—
385#117—2441no—
386Q32—612no—
387Q32—612no—
388Q32—612no—
389Q32C4×D6112yes—
390SD32—612no—
391SD32—612no—
392SD32—612no—
393SD32C4×D6112yes—
394M32—611no—
395M32—612no—
396M32—611no—
397M32—1221no—
398M32C22×D6111yescharacteristic
399M32C22×D6111yescharacteristic
400C2×C16—612no—
401C2×C16—1221no—
402#108—611no—
403#108—1221no—
404#108C22×D6111yescharacteristic
405#107—1221no—
406#106—611no—
407#106C22×D6111yescharacteristic
408#104—21no—
409#104—1221no—
410#104—1221no—
411#104—1221no—
412#103—21no—
413#103—1221no—
414#103—1221no—
415#103—1221no—
416#97—611no—
417#97C22×D6111yescharacteristic
418C4×C8—611no—
419C4×C8—611no—
420#41—21no—
421#41—1221no—
422#41—1221no—
423#41—1221no—
424#41—2441no—
425#41—2441no—
426C2×Q8—21no—
427C2×Q8—1221no—
428C2×Q8—1222no—
429C2×Q8—1222no—
430C2×Q8—1221no—
431C2×Q8—2441no—
432C2×Q8—2441no—
433C2×Q8—2441no—
434C2×D4—1221no—
435C2×D4—2441no—
436C22×C4—611no—
437C22×C4—2441no—
438C22×C4—2441no—
439Q16—21no—
440Q16—612no—
441Q16—611no—
442Q16—1222no—
443Q16—1222no—
444Q16—1221no—
445Q16—1221no—
446Q16C2×C4×D6112yes—
447Q16C2×C4×D6111yescharacteristic
448SD16—21no—
449SD16—1221no—
450SD16—1221no—
451SD16—1221no—
452D8—611no—
453D8—1221no—
454D8C2×C4×D6111yescharacteristic
455M16—21no—
456M16—41no—
457M16—1221no—
458M16—1221no—
459M16—1221no—
460M16—2441no—
461M16—2441no—
462M16—2441no—
463M16—2441no—
464C2×C8—41no—
465C2×C8—611no—
466C2×C8—611no—
467C2×C8—611no—
468C2×C8—611no—
469C2×C8—611no—
470C2×C8—611no—
471C2×C8—2441no—
472C2×C8—2441no—
473C2×C8—2441no—
474C2×C8—2441no—
475C2×C8D4×D6111yescharacteristic
476C2×C8C23×D6111yescharacteristic
477C4⋊C4—21no—
478C4⋊C4—41no—
479C4⋊C4—1221no—
480C4⋊C4—1221no—
481C4⋊C4—2441no—
482C4⋊C4—2441no—
483C4⋊C4—2441no—
484C22⋊C4—1221no—
485C22⋊C4—2441no—
486C42—41no—
487C42—611no—
488C42—2441no—
489C42—2441no—
490C42—2441no—
491C42—2441no—
492C42D4×D6111yescharacteristic
493C16—21no—
494C16—612no—
495C16—1221no—
496C16C2×C4×D6112yes—
497C23—611no—
498C23—2441no—
499Q8—21no—
500Q8—22no—
501Q8—41no—
502Q8—1221no—
503Q8—1222no—
504Q8—2442no—
505Q8—2442no—
506Q8—2441no—
507Q8—2441no—
508D4—21no—
509D4—41no—
510D4—1221no—
511D4—2441no—
512D4—2441no—
513D4—2441no—
514D4—2441no—
515C2×C4—21no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—611no—
520C2×C4—611no—
521C2×C4—611no—
522C2×C4—611no—
523C2×C4—611no—
524C2×C4—1221no—
525C2×C4—2441no—
526C2×C4—2441no—
527C2×C4—2442no—
528C2×C4—2442no—
529C2×C4—2441no—
530C2×C4—2441no—
531C2×C4—2441no—
532C2×C4—2441no—
533C2×C4C2×D4×D6111yescharacteristic
534C8—21no—
535C8—41no—
536C8—611no—
537C8—611no—
538C8—1221no—
539C8—2441no—
540C8C22×C4×D6111yescharacteristic
541C8#41×D6111yescharacteristic
542C22—41no—
543C22—611no—
544C22—611no—
545C22—9761no—
546C22—2441no—
547C22—2441no—
548C22AGL(1,ℤ/8)×D6111yescharacteristic
549C4—41no—
550C4—41no—
551C4—42no—
552C4—81no—
553C4—611no—
554C4—611no—
555C4—4881no—
556C4—2441no—
557C4—2441no—
558C4—2442no—
559C4C4×D4×D6111yescharacteristic
560C4AGL(1,ℤ/8)×D6111yescharacteristic
561C2—21no—
562C2—81no—
563C2—1221no—
564C2—1221no—
565C2—4881no—
566C2#442×D6111yescharacteristic
subgroupG/H
centerC2#442×D61
commutatorC8×C61C23×C4
FrattiniC2×C8C23×D61
FittingC61×#2180C2
socleC2×C61C2×#442
prime radical#135×D61C22
cosocleC2×C8×C61C24
hypercenter#2180D61
hypocenterC61#33470
LCSG ▷ C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ #97 ◁ #2180

Complements

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

NH ≅ G/Ntype
3. #21454684C2semidirect
4. #21447713C2semidirect
5. #21447712C2semidirect
6. #21447711C2semidirect
7. #21447710C2semidirect
8. #21447709C2semidirect
9. #21447708C2semidirect
10. #21447707C2semidirect
11. #21447590C2semidirect
12. #362×D61C2semidirect
14. #21447422C2semidirect
15. C61×#2180C2semidirect
21. #16840204C4semidirect
24. #16840201C4semidirect
25. Q32×D61C4semidirect
26. SD32×D61C4semidirect
34. #16839870C22semidirect
35. #16839869C22semidirect
36. #16839868C22semidirect
44. #16839586C22semidirect
45. #16839585C22semidirect
46. #16839584C22semidirect
47. #16839579C22semidirect
48. #16839578C22semidirect
49. #16839577C22semidirect
53. #16839540C22semidirect
54. #16839539C22semidirect
55. #16839539C22semidirect
58. C61×#441C22semidirect
59. C61×#365C22semidirect
60. C61×#362C22semidirect
62. C61×#347C22semidirect
288. #2180D61semidirect
322. #441C2×D61semidirect
333. #365C2×D61semidirect
337. #362C2×D61semidirect
345. #347C2×D61semidirect
346. C61#33470semidirect

Maximal subgroups

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

subgroupindex
#510×D612
#214547332
#214546842
#214477132
#214477122
#214477112
#214477102
#214477092
#214477082
#214477072
#214475902
#362×D612
#214475532
#214474222
C61×#21802
#3347061