Finite Group Databasebeta

← #29665265

Subgroups — #29665265

Order 15616.

650 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456950C211yesmaximal, characteristic, complemented
2#21455909C211yesmaximal, characteristic
3#21455826C211yesmaximal, characteristic, complemented
4#21454840C211yesmaximal, characteristic, complemented
5#21445253C211yesmaximal, characteristic, complemented
6#21445227C211yesmaximal, characteristic, complemented
7#21445226C211yesmaximal, characteristic, complemented
8#21445219C211yesmaximal, characteristic, complemented
9#21445218C211yesmaximal, characteristic, complemented
10#21445193C211yesmaximal, characteristic, complemented
11#21445192C211yesmaximal, characteristic, complemented
12#21444598C211yesmaximal, characteristic, complemented
13#21444526C211yesmaximal, characteristic, complemented
14#21444478C211yesmaximal, characteristic, complemented
15C61×#1681C211yesmaximal, characteristic, complemented
16#16840317C2211yescharacteristic
17#16840310C2211yescharacteristic
18C4×D4×D61C2211yescharacteristic
19#16840167C2211yescharacteristic, complemented
20#16840158C2211yescharacteristic, complemented
21#16840108C2211yescharacteristic, complemented
22#16840091—21no—
23#16840084—21no—
24#16840084—21no—
25#16840079—21no—
26#16840060—21no—
27#16840060—21no—
28#16840053C2211yescharacteristic
29#16840053C2211yescharacteristic
30#16840052C2211yescharacteristic
31#16840052C2211yescharacteristic
32#16839964C2211yescharacteristic
33#16839954C2211yescharacteristic
34#16839911—21no—
35#16839894—21no—
36#16839874—21no—
37#16839871—21no—
38SD16×Dic61—21no—
39#16839847—21no—
40C8×D244C2211yescharacteristic
41D4×#92418C2211yescharacteristic
42#16839827C2211yescharacteristic
43C8×C22⋊D61—21no—
44#16839825—21no—
45#16839818C2211yescharacteristic
46#16839814C2211yescharacteristic
47#16839814C2211yescharacteristic
48#16839813C2211yescharacteristic
49C2×#14519689—21no—
50C22⋊C8×D61—21no—
51#16839515C2211yescharacteristic, complemented
52#16839512C2211yescharacteristic, complemented
53#16839512C2211yescharacteristic, complemented
54#16839510C2211yescharacteristic, complemented
55#16839510C2211yescharacteristic, complemented
56#16839510C2211yescharacteristic, complemented
57#16839510C2211yescharacteristic, complemented
58#16839508C2211yescharacteristic, complemented
59#16839506C2211yescharacteristic, complemented
60C61×#486C2211yescharacteristic, complemented
61C61×#461C2211yescharacteristic
62C61×#459C2211yescharacteristic
63C8×D4×C61C2211yescharacteristic
64C61×#335C2211yescharacteristic, complemented
65C61×#332C2211yescharacteristic, complemented
66C61×#331C2211yescharacteristic, complemented
67C2×#11758224—21no—
68C2×D4×D61D411yescharacteristic
69C22×C4×D61—21no—
70C2×#11758201—21no—
71C2×#11758199—21no—
72C22×#92418—21no—
73C2×C8×D61—21no—
74C2×C8×D61—21no—
75#14519777C2311yescharacteristic
76#14519777C2311yescharacteristic
77#14519768—21no—
78#14519764—21no—
79#14519757—21no—
80#14519753—21no—
81#14519748—21no—
82#14519748—21no—
83Q8×Dic61—21no—
84#14519742—21no—
85C4×D244C2311yescharacteristic
86D4×Dic61—21no—
87D4×Dic61—21no—
88D4×Dic61D411yescharacteristic
89#14519739D411yescharacteristic
90C4×C22⋊D61—21no—
91#14519737—21no—
92#14519735—21no—
93C4⋊C4×D61D411yescharacteristic
94C22⋊C4×D61—21no—
95C42×D61C2311yescharacteristic
96#14519711—21no—
97#14519711—21no—
98#14519709—21no—
99#14519709—21no—
100#14519708—21no—
101#14519708—21no—
102#14519708—21no—
103#14519708—21no—
104C4⋊#92418C2311yescharacteristic
105C4⋊#92418C2311yescharacteristic
106C4⋊#92418C2311yescharacteristic
107#14519706D411yescharacteristic
108#14519706D411yescharacteristic
109#14519701—21no—
110#14519699—21no—
111#14519699—21no—
112#14519698—21no—
113#14519698—21no—
114#14519698—21no—
115#14519698—21no—
116#14519698—21no—
117C22⋊#92418—21no—
118#14519689—21no—
119#14519689—21no—
120#14519689—21no—
121#14519689—21no—
122#14519689—21no—
123#14519689D411yescharacteristic
124#14519689D411yescharacteristic
125#14519687—21no—
126C4×#92418C2311yescharacteristic
127C8×Dic61—21no—
128C8×Dic61—21no—
129C2×SD16×C61—21no—
130C2×D8×C61—21no—
131C22×C8×C61—21no—
132C4⋊Q8×C61C2311yescharacteristic
133#127×C61C2311yescharacteristic
134C4×D4×C61C2311yescharacteristic
135C4⋊C8×C61C2311yescharacteristic
136C4⋊C8×C61C2311yescharacteristic
137C4⋊C8×C61C2311yescharacteristic
138#103×C61—21no—
139#102×C61—21no—
140#102×C61—21no—
141#102×C61—21no—
142C22⋊C8×C61—21no—
143C4×C8×C61C2311yescharacteristic
144C23×D61—21no—
145C2×Q8×C61—21no—
146C2×D4×C61—21no—
147C2×D4×C61—21no—
148C2×D4×C61—21no—
149C2×D4×C61C2×D411yescharacteristic
150C22×C4×C61—21no—
151C2×C22⋊D61—21no—
152C2×C22⋊D61—41no—
153C22×Dic61—21no—
154C22×Dic61—41no—
155#11758224—82no—
156D4×D61—22no—
157D4×D61—22no—
158C2×D244C2×D411yescharacteristic
159C2×C4×D61—21no—
160C2×C4×D61—21no—
161C2×C4×D61—21no—
162C2×C4×D61—81no—
163C2×C4×D61—81no—
164C2×C4×D61C2×D411yescharacteristic
165C2×C4×D61C2×D411yescharacteristic
166C2×C4×D61#4111yescharacteristic
167C2×Dic122—21no—
168C2×Dic122—21no—
169SD16×C61—42no—
170D8×C61—42no—
171C2×C8×C61—21no—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61—81no—
175C2×C8×C61C2×D411yescharacteristic
176C2×C8×C61C2×D411yescharacteristic
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61C2×D411yescharacteristic
180C22⋊C4×C61—21no—
181C42×C61C2411yescharacteristic
182C22⋊Dic61—21no—
183C22⋊Dic61—41no—
184C22⋊Dic61—41no—
185#11758201—42no—
186#11758199—42no—
187#11758198—21no—
188#11758198—21no—
189#11758198—41no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—41no—
195C4⋊Dic61C2×D411yescharacteristic
196#11758196—21no—
197#11758196—41no—
198C4×Dic61—41no—
199C4×Dic61C2×D411yescharacteristic
200C4×Dic61C2×D411yescharacteristic
201C4×Dic61#4111yescharacteristic
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—81no—
206C2×#92418C2×D411yescharacteristic
207C2×#92418C2×D411yescharacteristic
208C8×D61—81no—
209C8×D61—81no—
210C23×C61—21no—
211C23×C61—41no—
212C22×D61—21no—
213C22×D61—24no—
214C22×D61—81no—
215C22×D61—81no—
216C22×D61#12111yescharacteristic
217Q8×C61—41no—
218Q8×C61—41no—
219D4×C61—21no—
220D4×C61—41no—
221D4×C61—41no—
222D4×C61—41no—
223D4×C61—41no—
224D4×C61—81no—
225D4×C61#13512yes—
226C2×C4×C61—21no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61C22×D411yescharacteristic
231C2×C4×C61C22×D411yescharacteristic
232C2×C4×C61C2×#4111yescharacteristic
233C22⋊D61—24no—
234C22⋊D61—82no—
235C2×Dic61—21no—
236C2×Dic61—41no—
237C2×Dic61—41no—
238C2×Dic61—81no—
239C2×Dic61—82no—
240C2×Dic61#12111yescharacteristic
241C2×Dic61#12111yescharacteristic
242C2×Dic61#12111yescharacteristic
243D244—21no—
244D244#13512yes—
245C4×D61—21no—
246C4×D61—21no—
247C4×D61—21no—
248C4×D61—21no—
249C4×D61—41no—
250C4×D61—41no—
251C4×D61—41no—
252C4×D61—81no—
253C4×D61—81no—
254Dic122—81no—
255Dic122—81no—
256C8×C61—21no—
257C8×C61—21no—
258C8×C61—41no—
259C8×C61—41no—
260C8×C61—41no—
261#92418—21no—
262#92418—21no—
263#92418—41no—
264#92418—41no—
265#92418—41no—
266#19394—611nomaximal, Sylow
267C22×C61—22no—
268C22×C61—41no—
269C22×C61—41no—
270C22×C61—81no—
271C22×C61—81no—
272C22×C61—81no—
273C22×C61C2×#12111yescharacteristic
274C2×D61—21no—
275C2×D61—21no—
276C2×D61—22no—
277C2×D61—21no—
278C2×D61—41no—
279C2×D61—41no—
280C2×D61—84no—
281C4×C61—21no—
282C4×C61—41no—
283C4×C61—81no—
284C4×C61C2×#13511yescharacteristic
285C4×C61C2×#13511yescharacteristic
286C4×C61#57711yescharacteristic
287C4×C61#57711yescharacteristic
288Dic61—21no—
289Dic61—21no—
290Dic61—21no—
291Dic61—41no—
292Dic61—81no—
293Dic61—81no—
294#3228—611no—
295#3109—611no—
296#3101—611no—
297#2972—611no—
298#1681—611no—
299#1681D6111yescharacteristic, complemented
300#1677—611no—
301#1677—611no—
302#1675—611no—
303#1675—611no—
304#1671—611no—
305#1671—611no—
306#1572—611no—
307#1554—611no—
308#1543—611no—
309C2×C61—42no—
310C2×C61—161no—
311C2×C61#304211yescharacteristic
312C2×C61#305011yescharacteristic
313C2×C61#306811yescharacteristic
314D61—41no—
315D61—42no—
316#533—611no—
317#532—611no—
318C2×C4×D4—611no—
319#500—611no—
320#498—611no—
321#486—611no—
322#486C2×D6111yescharacteristic, complemented
323#483—1221no—
324#482—1221no—
325#482—1221no—
326#481—1221no—
327#478—1221no—
328#478—1221no—
329#476—611no—
330#476—611no—
331#476—611no—
332#476—611no—
333#461—611no—
334#461C2×D6111yescharacteristic
335#459—611no—
336#459C2×D6111yescharacteristic
337#450—1221no—
338#447—1221no—
339#442—1221no—
340#442—1221no—
341C4×SD16—1221no—
342C4×SD16—1221no—
343C8×D4—611no—
344C8×D4—611no—
345C8×D4—611no—
346C8×D4—1221no—
347C8×D4—1221no—
348C8×D4C2×D6111yescharacteristic
349#432—611no—
350#431—611no—
351#431—611no—
352#431—611no—
353C2×C22⋊C8—1221no—
354C2×C22⋊C8—1221no—
355#335—611no—
356#335C2×D6111yescharacteristic, complemented
357#334—611no—
358#334—611no—
359#333—611no—
360#333—611no—
361#333—611no—
362#333—611no—
363#332—611no—
364#332C2×D6111yescharacteristic, complemented
365#331—611no—
366#331C2×D6111yescharacteristic, complemented
367C61#1939411yescharacteristic, complemented, Sylow
368C2×#41—1221no—
369C22×D4—611no—
370C23×C4—1221no—
371C2×SD16—21no—
372C2×SD16—1221no—
373C2×D8—21no—
374C2×D8—1221no—
375C22×C8—21no—
376C22×C8—1221no—
377C22×C8—1221no—
378C22×C8—1221no—
379C4⋊Q8—611no—
380C4⋊Q8—611no—
381C4⋊Q8C22×D6111yescharacteristic
382#127C22×D6111yescharacteristic
383#125—1221no—
384#124—1221no—
385C22⋊Q8—1221no—
386C22⋊Q8—1221no—
387#121—1221no—
388#121—1221no—
389C4×Q8—1221no—
390C4×Q8—1221no—
391C4×D4—611no—
392C4×D4—611no—
393C4×D4—611no—
394C4×D4—1221no—
395C4×D4—1221no—
396C4×D4—1221no—
397C4×D4—1221no—
398C4×D4C22×D6111yescharacteristic
399#117—1221no—
400C2×C4⋊C4—611no—
401C2×C22⋊C4—1221no—
402C2×C42—611no—
403#107—1221no—
404#107—1221no—
405#106—1221no—
406#106—1221no—
407#106—1221no—
408#106—1221no—
409#106—1221no—
410#106—1221no—
411C4⋊C8—611no—
412C4⋊C8—611no—
413C4⋊C8—611no—
414C4⋊C8—611no—
415C4⋊C8—611no—
416C4⋊C8C22×D6111yescharacteristic
417C4⋊C8C22×D6111yescharacteristic
418C4⋊C8C22×D6111yescharacteristic
419#103—21no—
420#103—1221no—
421#102—21no—
422#102—21no—
423#102—21no—
424#102—1221no—
425#102—1221no—
426#102—1221no—
427#102—1221no—
428#102—1221no—
429#102—1221no—
430#102—1221no—
431C22⋊C8—21no—
432C22⋊C8—611no—
433C22⋊C8—611no—
434C22⋊C8—1221no—
435C22⋊C8—1221no—
436C22⋊C8—1221no—
437C22⋊C8—1221no—
438C22⋊C8—1221no—
439C22⋊C8—1221no—
440#97—1221no—
441C4×C8—611no—
442C4×C8—1221no—
443C4×C8—1221no—
444C4×C8C22×D6111yescharacteristic
445C24—1221no—
446#41—4882no—
447C2×Q8—21no—
448C2×Q8—1221no—
449C2×Q8—1221no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—611no—
454C2×D4—1221no—
455C2×D4—1222no—
456C2×D4—1222no—
457C2×D4—2441no—
458C2×D4D4×D6111yescharacteristic
459C22×C4—21no—
460C22×C4—611no—
461C22×C4—611no—
462C22×C4—611no—
463C22×C4—1221no—
464C22×C4—1221no—
465C22×C4—1221no—
466C22×C4—1221no—
467C22×C4—4881no—
468C22×C4—4881no—
469C22×C4—2441no—
470SD16—42no—
471SD16—2442no—
472D8—42no—
473D8—2442no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—81no—
478C2×C8—611no—
479C2×C8—611no—
480C2×C8—1221no—
481C2×C8—1221no—
482C2×C8—1221no—
483C2×C8—4881no—
484C2×C8—4881no—
485C2×C8—4881no—
486C2×C8D4×D6111yescharacteristic
487C2×C8D4×D6111yescharacteristic
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—611no—
491C4⋊C4—1221no—
492C4⋊C4—1221no—
493C4⋊C4—1221no—
494C4⋊C4—1221no—
495C4⋊C4—1221no—
496C4⋊C4—2441no—
497C4⋊C4—2441no—
498C4⋊C4D4×D6111yescharacteristic
499C22⋊C4—21no—
500C22⋊C4—1221no—
501C22⋊C4—1221no—
502C22⋊C4—1221no—
503C22⋊C4—2441no—
504C22⋊C4—2441no—
505C22⋊C4—2441no—
506C42—611no—
507C42—611no—
508C42—611no—
509C42—2441no—
510C42C23×D6111yescharacteristic
511C23—21no—
512C23—41no—
513C23—611no—
514C23—1224no—
515C23—1221no—
516C23—4881no—
517C23—4881no—
518Q8—41no—
519Q8—41no—
520Q8—4881no—
521Q8—4881no—
522D4—21no—
523D4—41no—
524D4—41no—
525D4—41no—
526D4—41no—
527D4—81no—
528D4—612no—
529D4—1224no—
530D4—1221no—
531D4—4882no—
532D4#1451979612yes—
533C2×C4—21no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—611no—
538C2×C4—611no—
539C2×C4—611no—
540C2×C4—1221no—
541C2×C4—1221no—
542C2×C4—1221no—
543C2×C4—1221no—
544C2×C4—1221no—
545C2×C4—4881no—
546C2×C4—4881no—
547C2×C4—4882no—
548C2×C4—4881no—
549C2×C4—2441no—
550C2×C4—2441no—
551C2×C4—2441no—
552C2×C4—2441no—
553C2×C4—2441no—
554C2×C4C2×D4×D6111yescharacteristic
555C2×C4C2×D4×D6111yescharacteristic
556C2×C4#41×D6111yescharacteristic
557C8—21no—
558C8—21no—
559C8—41no—
560C8—41no—
561C8—41no—
562C8—1221no—
563C8—1221no—
564C8—2441no—
565C8—2441no—
566C8—2441no—
567C22—22no—
568C22—41no—
569C22—41no—
570C22—81no—
571C22—81no—
572C22—81no—
573C22—1221no—
574C22—1222no—
575C22—1221no—
576C22—1221no—
577C22—4884no—
578C22—2441no—
579C22—2441no—
580C22#121×D6111yescharacteristic
581C4—21no—
582C4—41no—
583C4—81no—
584C4—1221no—
585C4—1221no—
586C4—1221no—
587C4—4881no—
588C4—4881no—
589C4—2441no—
590C4#135×D6111yescharacteristic
591C4C2×#1451979611yescharacteristic
592C4#1684056911yescharacteristic
593C4#1684057211yescharacteristic
594C2—42no—
595C2—161no—
596C2—2441no—
597C2—2442no—
598C2#2145532911yescharacteristic
599C2#2145539811yescharacteristic
600C2#2145554211yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1681C2
socleC22×C61C2×#121
prime radical#21455909C2
hypercenter#1681D61
hypocenterC61#19394
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1681

Complements

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

NH ≅ G/Ntype
1. #21456950C2semidirect
3. #21455826C2semidirect
4. #21454840C2semidirect
5. #21445253C2semidirect
6. #21445227C2semidirect
7. #21445226C2semidirect
8. #21445219C2semidirect
9. #21445218C2semidirect
10. #21445193C2semidirect
11. #21445192C2semidirect
12. #21444598C2semidirect
13. #21444526C2semidirect
14. #21444478C2semidirect
15. C61×#1681C2semidirect
19. #16840167C22semidirect
20. #16840158C22semidirect
21. #16840108C22semidirect
51. #16839515C22semidirect
52. #16839512C22semidirect
53. #16839512C22semidirect
54. #16839510C22semidirect
55. #16839510C22semidirect
56. #16839510C22semidirect
57. #16839510C22semidirect
58. #16839508C22semidirect
59. #16839506C22semidirect
60. C61×#486C22semidirect
64. C61×#335C22semidirect
65. C61×#332C22semidirect
66. C61×#331C22semidirect
299. #1681D61semidirect
322. #486C2×D61semidirect
356. #335C2×D61semidirect
364. #332C2×D61semidirect
366. #331C2×D61semidirect
367. C61#19394semidirect

Maximal subgroups

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

subgroupindex
#214569502
#214559092
#214558262
#214548402
#214452532
#214452272
#214452262
#214452192
#214452182
#214451932
#214451922
#214445982
#214445262
#214444782
C61×#16812
#1939461