Finite Group Databasebeta

← #29662327

Subgroups — #29662327

Order 15616.

716 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456143C211yesmaximal, characteristic, complemented
2#21455829C211yesmaximal, characteristic, complemented
3#21455821C211yesmaximal, characteristic
4C8×D4×D61C211yesmaximal, characteristic, complemented
5#21445267C211yesmaximal, characteristic, complemented
6#21445257C211yesmaximal, characteristic, complemented
7#21445256C211yesmaximal, characteristic, complemented
8#21445241C211yesmaximal, characteristic, complemented
9#21445240C211yesmaximal, characteristic, complemented
10#21445231C211yesmaximal, characteristic, complemented
11#21445230C211yesmaximal, characteristic, complemented
12#21444566C211yesmaximal, characteristic, complemented
13#21444473C211yesmaximal, characteristic, complemented
14#21444467C211yesmaximal, characteristic, complemented
15C61×#1683C211yesmaximal, characteristic, complemented
16C22×C8×D61—21no—
17#16840317C2211yescharacteristic
18#16840313C2211yescharacteristic
19C4×D4×D61C2211yescharacteristic
20#16840118—21no—
21#16840118—21no—
22#16840112C2211yescharacteristic, complemented
23#16840110C2211yescharacteristic, complemented
24#16840109C2211yescharacteristic, complemented
25#16840094—21no—
26#16840077—21no—
27#16840049C2211yescharacteristic
28#16840039C2211yescharacteristic
29#16839968C2211yescharacteristic
30#16839967C2211yescharacteristic
31#16839958C2211yescharacteristic
32#16839957C2211yescharacteristic
33#16839914—21no—
34#16839897—21no—
35#16839853—21no—
36#16839849—21no—
37#16839844—21no—
38#16839842—21no—
39C8×D244C2211yescharacteristic
40D4×#92418C2211yescharacteristic
41#16839827C2211yescharacteristic
42C8×C22⋊D61—21no—
43#16839825—21no—
44#16839814C2211yescharacteristic
45#16839814C2211yescharacteristic
46C4⋊C8×D61C2211yescharacteristic
47#16839757—21no—
48#16839757—21no—
49C22⋊C8×D61—21no—
50C4×C8×D61C2211yescharacteristic
51#16839515C2211yescharacteristic, complemented
52#16839512C2211yescharacteristic, complemented
53#16839512C2211yescharacteristic, complemented
54#16839511C2211yescharacteristic, complemented
55#16839509C2211yescharacteristic, complemented
56#16839508C2211yescharacteristic, complemented
57#16839507C2211yescharacteristic, complemented
58#16839505C2211yescharacteristic, complemented
59#16839505C2211yescharacteristic, complemented
60C61×#487C2211yescharacteristic, complemented
61C61×#476C2211yescharacteristic
62C61×#474C2211yescharacteristic
63C8×D4×C61C2211yescharacteristic
64C61×#335C2211yescharacteristic, complemented
65C61×#333C2211yescharacteristic, complemented
66C61×#332C2211yescharacteristic, complemented
67C2×#11758226—21no—
68C2×D4×D61D411yescharacteristic
69C22×C4×D61—21no—
70C2×#11758190—21no—
71C2×#11758200—21no—
72C2×D488—21no—
73C2×#11758199—21no—
74C22×#92418—21no—
75C2×C8×D61—21no—
76C2×C8×D61—21no—
77C2×C8×D61—21no—
78C2×C8×D61—44no—
79C2×C8×D61D411yescharacteristic
80C2×C8×D61D411yescharacteristic
81#14519777C2311yescharacteristic
82#14519774C2311yescharacteristic
83#14519768—21no—
84#14519766—21no—
85#14519757—21no—
86#14519753—21no—
87#14519752—21no—
88#14519752—21no—
89Q8×Dic61—21no—
90#14519742—21no—
91C4×D244C2311yescharacteristic
92D4×Dic61D411yescharacteristic
93#14519739—21no—
94#14519739—21no—
95#14519739D411yescharacteristic
96C4×C22⋊D61—21no—
97#14519737—21no—
98#14519735—21no—
99C4⋊C4×D61D411yescharacteristic
100C22⋊C4×D61—21no—
101C42×D61C2311yescharacteristic
102#14519710—21no—
103#14519708—21no—
104C4⋊#92418C2311yescharacteristic
105C4⋊#92418C2311yescharacteristic
106C4⋊#92418C2311yescharacteristic
107#14519706—21no—
108#14519702—21no—
109#14519701—21no—
110#14519701—21no—
111#14519700—21no—
112#14519699—21no—
113#14519699—21no—
114#14519699—21no—
115#14519697—21no—
116#14519697—21no—
117#14519697—21no—
118#14519697—21no—
119#14519697—21no—
120C22⋊#92418—21no—
121#14519689—21no—
122#14519689—21no—
123#14519689—21no—
124#14519689—21no—
125C4×#92418C2311yescharacteristic
126C8×Dic61—21no—
127C8×Dic61—21no—
128C8×Dic61D411yescharacteristic
129C8×Dic61D411yescharacteristic
130C22×C8×C61—21no—
131C4⋊Q8×C61C2311yescharacteristic
132C4⋊Q8×C61C2311yescharacteristic
133C4×D4×C61C2311yescharacteristic
134#107×C61—21no—
135#106×C61—21no—
136C4⋊C8×C61C2311yescharacteristic
137C4⋊C8×C61C2311yescharacteristic
138C4⋊C8×C61C2311yescharacteristic
139#103×C61—21no—
140#103×C61—21no—
141#102×C61—21no—
142#102×C61—21no—
143C22⋊C8×C61—21no—
144C4×C8×C61C2311yescharacteristic
145C23×D61—21no—
146C2×Q8×C61—21no—
147C2×Q8×C61—21no—
148C2×D4×C61C2×D411yescharacteristic
149C22×C4×C61—21no—
150C2×C22⋊D61—21no—
151C22×Dic61—21no—
152#11758226—42no—
153#11758226—42no—
154D4×D61—22no—
155D4×D61—22no—
156C2×D244—21no—
157C2×D244—21no—
158C2×D244—21no—
159C2×D244—41no—
160C2×D244C2×D411yescharacteristic
161C2×C4×D61—21no—
162C2×C4×D61—21no—
163C2×C4×D61—21no—
164C2×C4×D61—42no—
165C2×C4×D61—42no—
166C2×C4×D61—41no—
167C2×C4×D61C2×D411yescharacteristic
168C2×C4×D61C2×D411yescharacteristic
169C2×C4×D61#4111yescharacteristic
170C2×Dic122—21no—
171C2×C8×C61—21no—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61—42no—
175C2×C8×C61C2×D411yescharacteristic
176C2×C8×C61C2×D411yescharacteristic
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61C2×D411yescharacteristic
182C22⋊C4×C61—21no—
183C42×C61C2411yescharacteristic
184C22⋊Dic61—21no—
185#11758200—42no—
186#11758199—42no—
187#11758198—21no—
188#11758198—21no—
189#11758198—41no—
190#11758198—41no—
191#11758198—41no—
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—42no—
206C2×#92418C2×D411yescharacteristic
207C2×#92418C2×D411yescharacteristic
208D488—42no—
209#11758190—42no—
210C8×D61—22no—
211C8×D61—22no—
212C8×D61—42no—
213C8×D61—42no—
214C8×D61—42no—
215C23×C61—21no—
216C22×D61—24no—
217C22×D61—21no—
218C22×D61—42no—
219C22×D61—42no—
220C22×D61—41no—
221C22×D61#12111yescharacteristic
222Q8×C61—41no—
223Q8×C61—41no—
224Q8×C61—81no—
225D4×C61—21no—
226D4×C61#13512yes—
227C2×C4×C61—21no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61C22×D411yescharacteristic
233C2×C4×C61C22×D411yescharacteristic
234C2×C4×C61C2×#4111yescharacteristic
235C22⋊D61—24no—
236C2×Dic61—21no—
237C2×Dic61—42no—
238C2×Dic61—41no—
239C2×Dic61#12111yescharacteristic
240C2×Dic61#12111yescharacteristic
241C2×Dic61#12111yescharacteristic
242D244—21no—
243D244—41no—
244D244—41no—
245D244—41no—
246D244—41no—
247D244—81no—
248D244—82no—
249D244#13512yes—
250C4×D61—21no—
251C4×D61—21no—
252C4×D61—41no—
253C4×D61—42no—
254C4×D61—41no—
255C4×D61—41no—
256C4×D61—42no—
257C4×D61—82no—
258C4×D61#13512yes—
259C4×D61AGL(1,ℤ/8)12yes—
260Dic122—41no—
261Dic122—41no—
262C8×C61—21no—
263C8×C61—21no—
264C8×C61—41no—
265C8×C61—41no—
266C8×C61—41no—
267#92418—21no—
268#92418—21no—
269#92418—41no—
270#92418—41no—
271#92418—41no—
272#19110—611nomaximal, Sylow
273C22×C61—22no—
274C22×C61—41no—
275C22×C61—41no—
276C22×C61C2×#12111yescharacteristic
277C2×D61—21no—
278C2×D61—21no—
279C2×D61—22no—
280C2×D61—48no—
281C2×D61—41no—
282C2×D61—41no—
283C2×D61—81no—
284C2×D61—81no—
285C2×D61—81no—
286C2×D61#46312yes—
287C4×C61—21no—
288C4×C61—41no—
289C4×C61—81no—
290C4×C61—81no—
291C4×C61C2×#13511yescharacteristic
292C4×C61C2×#13511yescharacteristic
293C4×C61C2×#13511yescharacteristic
294C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
295Dic61—21no—
296Dic61—21no—
297Dic61—41no—
298Dic61—81no—
299Dic61#46312yes—
300#3131—611no—
301#3101—611no—
302#3100—611no—
303C2×C8×D4—611no—
304#1683—611no—
305#1683D6111yescharacteristic, complemented
306#1681—611no—
307#1681—611no—
308#1679—611no—
309#1679—611no—
310#1677—611no—
311#1677—611no—
312#1564—611no—
313#1542—611no—
314#1540—611no—
315C2×C61—42no—
316C2×C61C2×#46311yescharacteristic
317C2×C61#305011yescharacteristic
318C2×C61#306711yescharacteristic
319D61—22no—
320D61—42no—
321D61—161no—
322C23×C8—1221no—
323#533—611no—
324#532—611no—
325C2×C4×D4—611no—
326#488—1221no—
327#488—1221no—
328#487—611no—
329#487C2×D6111yescharacteristic, complemented
330#486—611no—
331#486—611no—
332#483—1221no—
333#480—1221no—
334#476—611no—
335#476C2×D6111yescharacteristic
336#474—611no—
337#474C2×D6111yescharacteristic
338#461—611no—
339#461—611no—
340#459—611no—
341#459—611no—
342#450—1221no—
343#447—1221no—
344C4×SD16—1221no—
345C4×SD16—1221no—
346C4×D8—1221no—
347C4×D8—1221no—
348C8×D4—611no—
349C8×D4—611no—
350C8×D4—611no—
351C8×D4—1221no—
352C8×D4—1221no—
353C8×D4C2×D6111yescharacteristic
354#431—611no—
355#431—611no—
356C2×C4⋊C8—611no—
357#416—1221no—
358#416—1221no—
359C2×C22⋊C8—1221no—
360C2×C4×C8—611no—
361#335—611no—
362#335C2×D6111yescharacteristic, complemented
363#334—611no—
364#334—611no—
365#333—611no—
366#333C2×D6111yescharacteristic, complemented
367#332—611no—
368#332—611no—
369#332—611no—
370#332C2×D6111yescharacteristic, complemented
371#331—611no—
372#331—611no—
373C61#1911011yescharacteristic, complemented, Sylow
374C2×#41—1221no—
375C22×D4—611no—
376C23×C4—1221no—
377C2×SD16—1221no—
378C2×SD16—1221no—
379C2×D8—1221no—
380C2×D8—1221no—
381C22×C8—21no—
382C22×C8—611no—
383C22×C8—611no—
384C22×C8—1221no—
385C22×C8—1221no—
386C22×C8—1221no—
387C22×C8—1221no—
388C22×C8—2444no—
389C4⋊Q8—611no—
390C4⋊Q8C22×D6111yescharacteristic
391C4⋊Q8C22×D6111yescharacteristic
392#127—611no—
393#125—1221no—
394#124—1221no—
395C22⋊Q8—1221no—
396C22⋊Q8—1221no—
397#121—1221no—
398#121—1221no—
399C4×Q8—1221no—
400C4×Q8—1221no—
401C4×D4—611no—
402C4×D4—611no—
403C4×D4—611no—
404C4×D4—1221no—
405C4×D4—1221no—
406C4×D4—1221no—
407C4×D4—1221no—
408C4×D4C22×D6111yescharacteristic
409#117—1221no—
410C2×C4⋊C4—611no—
411C2×C22⋊C4—1221no—
412C2×C42—611no—
413#107—21no—
414#107—1221no—
415#106—21no—
416#106—1221no—
417C4⋊C8—611no—
418C4⋊C8—611no—
419C4⋊C8—611no—
420C4⋊C8—1221no—
421C4⋊C8C22×D6111yescharacteristic
422C4⋊C8C22×D6111yescharacteristic
423C4⋊C8C22×D6111yescharacteristic
424#103—21no—
425#103—21no—
426#103—1221no—
427#103—1221no—
428#103—1221no—
429#103—1221no—
430#102—21no—
431#102—21no—
432#102—1221no—
433#102—1221no—
434#102—1221no—
435#102—1221no—
436#102—1221no—
437#102—1221no—
438#102—1221no—
439#102—1221no—
440C22⋊C8—21no—
441C22⋊C8—1221no—
442C22⋊C8—1221no—
443C22⋊C8—1221no—
444C22⋊C8—1221no—
445C22⋊C8—1221no—
446C4×C8—611no—
447C4×C8—611no—
448C4×C8—611no—
449C4×C8—1221no—
450C4×C8—1221no—
451C4×C8C22×D6111yescharacteristic
452C24—1221no—
453#41—2442no—
454#41—2442no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—1221no—
458C2×D4—611no—
459C2×D4—1221no—
460C2×D4—1222no—
461C2×D4—1222no—
462C2×D4—1221no—
463C2×D4—1221no—
464C2×D4—1221no—
465C2×D4—2441no—
466C2×D4D4×D6111yescharacteristic
467C22×C4—21no—
468C22×C4—611no—
469C22×C4—611no—
470C22×C4—611no—
471C22×C4—1221no—
472C22×C4—1221no—
473C22×C4—1221no—
474C22×C4—1221no—
475C22×C4—2441no—
476C22×C4—2442no—
477C22×C4—2442no—
478SD16—2442no—
479SD16—2442no—
480D8—2442no—
481D8—2442no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—42no—
486C2×C8—611no—
487C2×C8—611no—
488C2×C8—1222no—
489C2×C8—1222no—
490C2×C8—1221no—
491C2×C8—1221no—
492C2×C8—1221no—
493C2×C8—2442no—
494C2×C8—2442no—
495C2×C8—2442no—
496C2×C8—2442no—
497C2×C8D4×D6111yescharacteristic
498C2×C8D4×D6111yescharacteristic
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—611no—
504C4⋊C4—1221no—
505C4⋊C4—1221no—
506C4⋊C4—1221no—
507C4⋊C4—2441no—
508C4⋊C4—2441no—
509C4⋊C4D4×D6111yescharacteristic
510C22⋊C4—21no—
511C22⋊C4—1221no—
512C22⋊C4—1221no—
513C22⋊C4—1221no—
514C22⋊C4—2441no—
515C22⋊C4—2441no—
516C22⋊C4—2441no—
517C42—611no—
518C42—611no—
519C42—611no—
520C42—2441no—
521C42C23×D6111yescharacteristic
522C23—21no—
523C23—611no—
524C23—1224no—
525C23—1221no—
526C23—2442no—
527C23—2442no—
528C23—2441no—
529Q8—41no—
530Q8—41no—
531Q8—81no—
532Q8—2441no—
533Q8—2441no—
534D4—21no—
535D4—612no—
536D4—1224no—
537D4—1221no—
538D4—4881no—
539D4—4882no—
540D4—2441no—
541D4—2441no—
542D4—2441no—
543D4—2441no—
544D4#1451980012yes—
545C2×C4—21no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—611no—
551C2×C4—612no—
552C2×C4—612no—
553C2×C4—611no—
554C2×C4—611no—
555C2×C4—1221no—
556C2×C4—1221no—
557C2×C4—1221no—
558C2×C4—4882no—
559C2×C4—2441no—
560C2×C4—2442no—
561C2×C4—2442no—
562C2×C4—2442no—
563C2×C4—2441no—
564C2×C4—2441no—
565C2×C4—2441no—
566C2×C4C2×D4×D6111yescharacteristic
567C2×C4C2×D4×D6111yescharacteristic
568C2×C4#41×D6111yescharacteristic
569C8—21no—
570C8—21no—
571C8—41no—
572C8—41no—
573C8—41no—
574C8—1221no—
575C8—1221no—
576C8—2441no—
577C8—2441no—
578C8—2441no—
579C22—22no—
580C22—41no—
581C22—41no—
582C22—612no—
583C22—1221no—
584C22—1221no—
585C22—1222no—
586C22—4881no—
587C22—4881no—
588C22—4881no—
589C22—2448no—
590C22—2441no—
591C22—2441no—
592C22#121×D6111yescharacteristic
593C4—21no—
594C4—41no—
595C4—81no—
596C4—81no—
597C4—612no—
598C4—1221no—
599C4—1221no—
600C4—4881no—
601C4—2441no—
602C4#135×D6111yescharacteristic
603C4#135×D6111yescharacteristic
604C4C2×#1451980011yescharacteristic
605C4AGL(1,ℤ/8)×D6111yescharacteristic
606C2—42no—
607C2—1222no—
608C2—9761no—
609C2—2442no—
610C2#463×D6111yescharacteristic
611C2#2145540211yescharacteristic
612C2#2145553411yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1683C2
socleC22×C61C2×#121
prime radical#21455821C2
hypercenter#1683D61
hypocenterC61#19110
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1683

Complements

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

NH ≅ G/Ntype
1. #21456143C2semidirect
2. #21455829C2semidirect
4. C8×D4×D61C2semidirect
5. #21445267C2semidirect
6. #21445257C2semidirect
7. #21445256C2semidirect
8. #21445241C2semidirect
9. #21445240C2semidirect
10. #21445231C2semidirect
11. #21445230C2semidirect
12. #21444566C2semidirect
13. #21444473C2semidirect
14. #21444467C2semidirect
15. C61×#1683C2semidirect
22. #16840112C22semidirect
23. #16840110C22semidirect
24. #16840109C22semidirect
51. #16839515C22semidirect
52. #16839512C22semidirect
53. #16839512C22semidirect
54. #16839511C22semidirect
55. #16839509C22semidirect
56. #16839508C22semidirect
57. #16839507C22semidirect
58. #16839505C22semidirect
59. #16839505C22semidirect
60. C61×#487C22semidirect
64. C61×#335C22semidirect
65. C61×#333C22semidirect
66. C61×#332C22semidirect
305. #1683D61semidirect
329. #487C2×D61semidirect
362. #335C2×D61semidirect
366. #333C2×D61semidirect
370. #332C2×D61semidirect
373. C61#19110semidirect

Maximal subgroups

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

subgroupindex
#214561432
#214558292
#214558212
C8×D4×D612
#214452672
#214452572
#214452562
#214452412
#214452402
#214452312
#214452302
#214445662
#214444732
#214444672
C61×#16832
#1911061