Finite Group Databasebeta

← #29665419

Subgroups — #29665419

Order 15616.

704 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456761C211yesmaximal, characteristic, complemented
2#21455610C211yesmaximal, characteristic, complemented
3#21455296C211yesmaximal, characteristic
4#21454929C211yesmaximal, characteristic, complemented
5#21445237C211yesmaximal, characteristic, complemented
6#21445231C211yesmaximal, characteristic, complemented
7#21445225C211yesmaximal, characteristic, complemented
8#21445167C211yesmaximal, characteristic, complemented
9#21445165C211yesmaximal, characteristic, complemented
10#21445161C211yesmaximal, characteristic, complemented
11#21445159C211yesmaximal, characteristic, complemented
12#21444588C211yesmaximal, characteristic, complemented
13#21444506C211yesmaximal, characteristic, complemented
14#331×D61C211yesmaximal, characteristic, complemented
15C61×#1678C211yesmaximal, characteristic, complemented
16C2×#14519804—21no—
17#16840317C2211yescharacteristic
18#127×D61C2211yescharacteristic
19C4×#11758226C2211yescharacteristic
20#16840135C2211yescharacteristic
21#16840134C2211yescharacteristic
22#16840110C2211yescharacteristic
23#16840071—21no—
24#16840065—21no—
25#16840049C2211yescharacteristic, complemented
26#16840047C2211yescharacteristic, complemented
27#16840043C2211yescharacteristic, complemented
28#16839962C2211yescharacteristic
29#16839961—21no—
30#16839961—21no—
31#16839960—21no—
32#16839958C2211yescharacteristic
33#16839956—21no—
34#16839954C2211yescharacteristic
35#16839914—21no—
36#16839914—21no—
37Q8×#92418C2211yescharacteristic, complemented
38C8×D244—21no—
39C8×D244C2211yescharacteristic, complemented
40#16839827—21no—
41#16839827C2211yescharacteristic, complemented
42#16839813—21no—
43#16839813C2211yescharacteristic
44#16839799—21no—
45#16839784C2211yescharacteristic
46C4⋊C8×D61—21no—
47C4⋊C8×D61C2211yescharacteristic
48C4⋊C8×D61C2211yescharacteristic
49#16839765—21no—
50#16839757—21no—
51#16839513C2211yescharacteristic, complemented
52#16839512C2211yescharacteristic, complemented
53#16839512C2211yescharacteristic, complemented
54#16839511C2211yescharacteristic, complemented
55#16839509C2211yescharacteristic, complemented
56#16839507C2211yescharacteristic, complemented
57#16839506C2211yescharacteristic
58#16839505C2211yescharacteristic
59#16839505C2211yescharacteristic
60C61×#486C2211yescharacteristic
61C61×#477C2211yescharacteristic, complemented
62C61×#460C2211yescharacteristic
63C8×Q8×C61C2211yescharacteristic, complemented
64C61×#334C2211yescharacteristic, complemented
65C61×#333C2211yescharacteristic, complemented
66C61×#331C2211yescharacteristic
67C2×#11758226D411yescharacteristic
68C2×D4×D61—21no—
69C2×D4×D61—21no—
70C2×D4×D61—21no—
71#14519804—44no—
72C2×#11758190—21no—
73C2×#11758190—21no—
74C2×#11758200—21no—
75C2×#11758200—21no—
76C2×#11758201—21no—
77C2×D488—21no—
78C2×#11758199—21no—
79C2×#11758189—21no—
80C2×C8×D61—21no—
81C2×C8×D61—21no—
82C2×C8×D61—21no—
83C2×C8×D61—21no—
84#14519777C2311yescharacteristic
85#14519774C2311yescharacteristic
86#14519773—21no—
87#14519773—21no—
88#14519773—21no—
89#14519768—21no—
90#14519757—21no—
91#14519753—21no—
92Q8×Dic61—21no—
93Q8×Dic61D411yescharacteristic
94#14519742—21no—
95C4×D244—21no—
96C4×D244C2311yescharacteristic
97#14519739—21no—
98#14519739D411yescharacteristic
99#14519735—21no—
100#14519735—21no—
101#14519735D411yescharacteristic
102C42×D61—21no—
103C42×D61C2311yescharacteristic
104#14519711—21no—
105#14519708—21no—
106#14519708—21no—
107C4⋊#92418—21no—
108C4⋊#92418C2311yescharacteristic
109C4⋊#92418C2311yescharacteristic
110C4⋊#92418C2311yescharacteristic
111#14519706—21no—
112#14519706—21no—
113#14519706—21no—
114#14519706D411yescharacteristic
115#14519706D411yescharacteristic
116#14519701—21no—
117#14519699—21no—
118#14519699—21no—
119#14519698—21no—
120#14519697—21no—
121#14519697—21no—
122#14519697—21no—
123#14519697—21no—
124#14519689—21no—
125#14519689—21no—
126#14519689D411yescharacteristic
127#14519689D411yescharacteristic
128C4×#92418—21no—
129C4×#92418C2311yescharacteristic
130C2×SD16×C61—21no—
131C4⋊Q8×C61C2311yescharacteristic
132#127×C61C2311yescharacteristic
133C4×Q8×C61C2311yescharacteristic
134#107×C61—21no—
135C4⋊C8×C61—21no—
136C4⋊C8×C61C2311yescharacteristic
137C4⋊C8×C61C2311yescharacteristic
138C4⋊C8×C61C2311yescharacteristic
139#103×C61—21no—
140#102×C61—21no—
141#102×C61—21no—
142#102×C61—21no—
143C4×C8×C61—21no—
144C4×C8×C61C2311yescharacteristic
145C23×D61—41no—
146C2×Q8×C61—21no—
147C2×Q8×C61C2×D411yescharacteristic
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61—21no—
151C2×C22⋊D61—41no—
152C2×C22⋊D61—41no—
153C2×C22⋊D61—41no—
154#11758226—22no—
155#11758226SD1614yescomplemented
156D4×D61—42no—
157D4×D61—42no—
158D4×D61—81no—
159D4×D61—81no—
160D4×D61—82no—
161C2×D244—21no—
162C2×D244—21no—
163C2×D244—21no—
164C2×D244—21no—
165C2×D244C2×D411yescharacteristic
166C2×C4×D61—21no—
167C2×C4×D61—21no—
168C2×C4×D61—21no—
169C2×C4×D61C2×D411yescharacteristic
170C2×C4×D61C2×D411yescharacteristic
171C2×C4×D61#4111yescharacteristic
172C2×Dic122—21no—
173SD16×C61—42no—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C2×C8×C61C2×D411yescharacteristic
178C2×C8×C61C2×D411yescharacteristic
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61C2×D411yescharacteristic
183C42×C61—21no—
184C42×C61C2411yescharacteristic
185#11758201—42no—
186#11758200—81no—
187#11758200—81no—
188#11758199—42no—
189#11758198—21no—
190#11758198—21no—
191#11758198—21no—
192#11758198—41no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—41no—
197C4⋊Dic61C2×D411yescharacteristic
198#11758196—41no—
199C4×Dic61—21no—
200C4×Dic61—21no—
201C4×Dic61—41no—
202C4×Dic61C2×D411yescharacteristic
203C4×Dic61C2×D411yescharacteristic
204C4×Dic61#4111yescharacteristic
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418C2×D411yescharacteristic
209C2×#92418C2×D411yescharacteristic
210D488—42no—
211#11758190—81no—
212#11758190—81no—
213#11758189—42no—
214C8×D61—42no—
215C8×D61—81no—
216C8×D61—81no—
217C8×D61—81no—
218C23×C61—41no—
219C22×D61—21no—
220C22×D61—41no—
221C22×D61—82no—
222C22×D61—82no—
223C22×D61—82no—
224C22×D61#12111yescharacteristic
225Q8×C61—21no—
226Q8×C61—81no—
227Q8×C61C2×SD1612yes—
228D4×C61—41no—
229D4×C61—41no—
230D4×C61—41no—
231D4×C61—41no—
232D4×C61—81no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—41no—
236C2×C4×C61C22×D411yescharacteristic
237C2×C4×C61C22×D411yescharacteristic
238C2×C4×C61C2×#4111yescharacteristic
239C22⋊D61—82no—
240C22⋊D61—82no—
241C22⋊D61—82no—
242C2×Dic61—21no—
243C2×Dic61—41no—
244C2×Dic61#12111yescharacteristic
245C2×Dic61#12111yescharacteristic
246C2×Dic61#12111yescharacteristic
247D244—24no—
248D244—21no—
249D244—41no—
250D244—41no—
251D244—41no—
252D244—41no—
253D244—81no—
254D244C2×SD1612yes—
255C4×D61—21no—
256C4×D61—21no—
257C4×D61—24no—
258C4×D61—41no—
259C4×D61—41no—
260C4×D61—41no—
261C4×D61—81no—
262C4×D61C2×SD1612yes—
263C4×D61#13512yes—
264Dic122—41no—
265Dic122—41no—
266C8×C61—21no—
267C8×C61—21no—
268C8×C61—41no—
269C8×C61—41no—
270C8×C61—41no—
271#92418—21no—
272#92418—21no—
273#92418—41no—
274#92418—41no—
275#92418—41no—
276#19409—611nomaximal, Sylow
277C22×C61—81no—
278C22×C61—81no—
279C22×C61—81no—
280C22×C61C2×#12111yescharacteristic
281C2×D61—21no—
282C2×D61—21no—
283C2×D61—22no—
284C2×D61—41no—
285C2×D61—41no—
286C2×D61—81no—
287C2×D61—81no—
288C2×D61—81no—
289C2×D61—164no—
290C2×D61#46512yes—
291C4×C61—22no—
292C4×C61—21no—
293C4×C61—41no—
294C4×C61—81no—
295C4×C61C22×SD1611yescharacteristic
296C4×C61C2×#13511yescharacteristic
297C4×C61#57611yescharacteristic
298C4×C61#57711yescharacteristic
299Dic61—21no—
300Dic61—21no—
301Dic61—41no—
302Dic61—81no—
303Dic61#46512yes—
304#3208—611no—
305#3076—611no—
306#3038—611no—
307#2988—611no—
308#1678—611no—
309#1678D6111yescharacteristic, complemented
310#1677—611no—
311#1677—611no—
312#1666—611no—
313#1666—611no—
314#1665—611no—
315#1665—611no—
316#1570—611no—
317#1550—611no—
318C2×#331—611no—
319C2×C61—161no—
320C2×C61#304511yescharacteristic
321C2×C61C2×#46511yescharacteristic
322C2×C61#306811yescharacteristic
323D61—22no—
324D61—42no—
325D61—161no—
326C2×AGL(1,ℤ/8)—1221no—
327#533—611no—
328C2×#127—611no—
329C4×#41—611no—
330#492—611no—
331#492—611no—
332#486—611no—
333#486C2×D6111yescharacteristic
334#479—1221no—
335#478—1221no—
336#477C2×D6111yescharacteristic, complemented
337#476—611no—
338#475—611no—
339#474—611no—
340#460—611no—
341#460—1221no—
342#460—1221no—
343#460—1221no—
344#460C2×D6111yescharacteristic
345#459—611no—
346#459—611no—
347#459—1221no—
348#450—1221no—
349#450—1221no—
350C8×Q8—611no—
351C8×Q8C2×D6111yescharacteristic, complemented
352C8×D4—611no—
353C8×D4—611no—
354C8×D4—1221no—
355C8×D4—1221no—
356#431—611no—
357#431—1221no—
358#427—1221no—
359#423—611no—
360C2×C4⋊C8—611no—
361C2×C4⋊C8—611no—
362C2×C4⋊C8—1221no—
363#418—1221no—
364#416—1221no—
365#334—611no—
366#334—611no—
367#334—611no—
368#334C2×D6111yescharacteristic, complemented
369#333—611no—
370#333C2×D6111yescharacteristic, complemented
371#332—611no—
372#332—611no—
373#331—611no—
374#331—611no—
375#331—611no—
376#331C2×D6111yescharacteristic
377C61#1940911yescharacteristic, complemented, Sylow
378C2×#41—611no—
379C22×D4—1221no—
380C22×D4—1221no—
381C22×D4—1221no—
382AGL(1,ℤ/8)—2444no—
383C2×SD16—21no—
384C2×SD16—1221no—
385C2×SD16—1221no—
386C2×SD16—1221no—
387C2×SD16—1221no—
388C2×SD16—1221no—
389C2×D8—1221no—
390C2×D8—1221no—
391C2×M16—1221no—
392C22×C8—1221no—
393C22×C8—1221no—
394C22×C8—1221no—
395C22×C8—1221no—
396C4⋊Q8—611no—
397C4⋊Q8C22×D6111yescharacteristic
398#127—611no—
399#127—1221no—
400#127—1221no—
401#127—1221no—
402#127C22×D6111yescharacteristic
403#125—1221no—
404C22⋊Q8—1221no—
405C22⋊Q8—1221no—
406C4×Q8—611no—
407C4×Q8—1221no—
408C4×Q8—1221no—
409C4×Q8C22×D6111yescharacteristic
410C4×D4—611no—
411C4×D4—611no—
412C4×D4—1221no—
413C4×D4—1221no—
414#117—611no—
415#117—1221no—
416#117—1221no—
417C2×C42—611no—
418C2×C42—1221no—
419#107—21no—
420#107—1221no—
421#106—1221no—
422#106—1221no—
423C4⋊C8—21no—
424C4⋊C8—611no—
425C4⋊C8—611no—
426C4⋊C8—611no—
427C4⋊C8—611no—
428C4⋊C8—611no—
429C4⋊C8—1221no—
430C4⋊C8—1221no—
431C4⋊C8—1221no—
432C4⋊C8—1221no—
433C4⋊C8C22×D6111yescharacteristic
434C4⋊C8C22×D6111yescharacteristic
435C4⋊C8C22×D6111yescharacteristic
436#103—21no—
437#103—1221no—
438#102—21no—
439#102—21no—
440#102—21no—
441#102—1221no—
442#102—1221no—
443#102—1221no—
444#102—1221no—
445#102—1221no—
446#102—1221no—
447#102—1221no—
448C22⋊C8—611no—
449C22⋊C8—611no—
450C22⋊C8—1221no—
451C22⋊C8—1221no—
452C4×C8—21no—
453C4×C8—611no—
454C4×C8—1221no—
455C4×C8C22×D6111yescharacteristic
456C24—2441no—
457#41—614no—
458#41—1222no—
459C2×Q8—21no—
460C2×Q8—1221no—
461C2×Q8D4×D6111yescharacteristic
462C2×D4—21no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—611no—
466C2×D4—1221no—
467C2×D4—1221no—
468C2×D4—1221no—
469C2×D4—1221no—
470C2×D4—4881no—
471C2×D4—4881no—
472C2×D4—4882no—
473C2×D4—2442no—
474C2×D4—2442no—
475C2×D4—2441no—
476C2×D4—2441no—
477C2×D4—2441no—
478C22×C4—611no—
479C22×C4—611no—
480C22×C4—611no—
481C22×C4—1221no—
482C22×C4—1221no—
483C22×C4—1221no—
484SD16—42no—
485SD16—4881no—
486SD16—4881no—
487SD16—4881no—
488SD16—4881no—
489SD16—2442no—
490D8—2442no—
491D8—2442no—
492M16—2442no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—611no—
497C2×C8—611no—
498C2×C8—1221no—
499C2×C8—1221no—
500C2×C8—1221no—
501C2×C8—4881no—
502C2×C8—4881no—
503C2×C8—4881no—
504C2×C8—2442no—
505C2×C8D4×D6111yescharacteristic
506C2×C8D4×D6111yescharacteristic
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—611no—
511C4⋊C4—1221no—
512C4⋊C4—1221no—
513C4⋊C4—1221no—
514C4⋊C4—2441no—
515C4⋊C4—2441no—
516C4⋊C4D4×D6111yescharacteristic
517C22⋊C4—1221no—
518C22⋊C4—1221no—
519C22⋊C4—1221no—
520C22⋊C4—2441no—
521C42—21no—
522C42—611no—
523C42—611no—
524C42—611no—
525C42—1221no—
526C42—1221no—
527C42—2441no—
528C42C23×D6111yescharacteristic
529C23—41no—
530C23—611no—
531C23—1221no—
532C23—4882no—
533C23—4882no—
534C23—4882no—
535C23—2441no—
536Q8—21no—
537Q8—81no—
538Q8—2441no—
539Q8—2441no—
540Q8SD16×D6112yes—
541D4—41no—
542D4—41no—
543D4—41no—
544D4—41no—
545D4—81no—
546D4—612no—
547D4—1224no—
548D4—1221no—
549D4—4881no—
550D4—4882no—
551D4—4882no—
552D4—4882no—
553D4—2441no—
554D4—2441no—
555D4—2441no—
556D4—2441no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—41no—
560C2×C4—611no—
561C2×C4—612no—
562C2×C4—612no—
563C2×C4—611no—
564C2×C4—611no—
565C2×C4—1224no—
566C2×C4—1221no—
567C2×C4—1221no—
568C2×C4—1221no—
569C2×C4—4881no—
570C2×C4—2441no—
571C2×C4—2441no—
572C2×C4—2441no—
573C2×C4—2441no—
574C2×C4C2×D4×D6111yescharacteristic
575C2×C4C2×D4×D6111yescharacteristic
576C2×C4#41×D6111yescharacteristic
577C8—21no—
578C8—21no—
579C8—41no—
580C8—41no—
581C8—41no—
582C8—1221no—
583C8—1221no—
584C8—2441no—
585C8—2441no—
586C8—2441no—
587C22—81no—
588C22—81no—
589C22—81no—
590C22—612no—
591C22—1221no—
592C22—1221no—
593C22—1222no—
594C22—9764no—
595C22—4881no—
596C22—4881no—
597C22—4881no—
598C22—2441no—
599C22—2441no—
600C22#121×D6111yescharacteristic
601C4—22no—
602C4—21no—
603C4—41no—
604C4—81no—
605C4—612no—
606C4—1221no—
607C4—1221no—
608C4—4881no—
609C4—2441no—
610C4C2×SD16×D6111yescharacteristic
611C4#135×D6111yescharacteristic
612C4#1684056411yescharacteristic
613C4#1684057111yescharacteristic
614C2—161no—
615C2—1222no—
616C2—9761no—
617C2—2442no—
618C2#2145536111yescharacteristic
619C2#465×D6111yescharacteristic
620C2#2145554511yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1678C2
socleC22×C61C2×#121
prime radical#21455296C2
hypercenter#1678D61
hypocenterC61#19409
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1678

Complements

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

NH ≅ G/Ntype
1. #21456761C2semidirect
2. #21455610C2semidirect
4. #21454929C2semidirect
5. #21445237C2semidirect
6. #21445231C2semidirect
7. #21445225C2semidirect
8. #21445167C2semidirect
9. #21445165C2semidirect
10. #21445161C2semidirect
11. #21445159C2semidirect
12. #21444588C2semidirect
13. #21444506C2semidirect
14. #331×D61C2semidirect
15. C61×#1678C2semidirect
25. #16840049C22semidirect
26. #16840047C22semidirect
27. #16840043C22semidirect
37. Q8×#92418C22semidirect
39. C8×D244C22semidirect
41. #16839827C22semidirect
51. #16839513C22semidirect
52. #16839512C22semidirect
53. #16839512C22semidirect
54. #16839511C22semidirect
55. #16839509C22semidirect
56. #16839507C22semidirect
61. C61×#477C22semidirect
63. C8×Q8×C61C22semidirect
64. C61×#334C22semidirect
65. C61×#333C22semidirect
155. #11758226SD16semidirect
309. #1678D61semidirect
336. #477C2×D61semidirect
351. C8×Q8C2×D61semidirect
368. #334C2×D61semidirect
370. #333C2×D61semidirect
377. C61#19409semidirect

Maximal subgroups

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

subgroupindex
#214567612
#214556102
#214552962
#214549292
#214452372
#214452312
#214452252
#214451672
#214451652
#214451612
#214451592
#214445882
#214445062
#331×D612
C61×#16782
#1940961