Finite Group Databasebeta

← #29669498

Subgroups — #29669498

Order 15616.

704 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458736C211yesmaximal, characteristic, complemented
2#21458457C211yesmaximal, characteristic, complemented
3#21457510C211yesmaximal, characteristic
4#21457510C211yesmaximal, characteristic
5#21445440C211yesmaximal, characteristic, complemented
6#21445289C211yesmaximal, characteristic, complemented
7#21445241C211yesmaximal, characteristic, complemented
8#21445207C211yesmaximal, characteristic, complemented
9#21445165C211yesmaximal, characteristic, complemented
10#21445085C211yesmaximal, characteristic, complemented
11#21445085C211yesmaximal, characteristic, complemented
12#21444995C211yesmaximal, characteristic
13#21444658C211yesmaximal, characteristic, complemented
14#21444634C211yesmaximal, characteristic, complemented
15C61×#1626C211yesmaximal, characteristic, complemented
16C2×SD16×D61—21no—
17C2×SD16×D61—21no—
18Q8×D244C2211yescharacteristic
19Q8×D244C2211yescharacteristic
20#16840397C2211yescharacteristic
21#16840172—21no—
22#16840165C2211yescharacteristic, complemented
23#16840160C2211yescharacteristic, complemented
24#16840134C2211yescharacteristic
25#16840134C2211yescharacteristic
26#16840112—21no—
27#16840112C2211yescharacteristic, complemented
28#16840059—21no—
29#16840057—21no—
30#16840047—21no—
31#16840047C2211yescharacteristic, complemented
32#16840047C2211yescharacteristic, complemented
33#16840039C2211yescharacteristic
34#16839990—21no—
35#16839990—21no—
36#16839968C2211yescharacteristic
37#16839968C2211yescharacteristic
38#16839964—21no—
39#16839964C2211yescharacteristic
40#16839914—21no—
41#16839914—21no—
42#16839899—21no—
43#16839899—21no—
44#16839893C2211yescharacteristic, complemented
45Q8×#92418C2211yescharacteristic, complemented
46#16839860—21no—
47C4×#11758201—21no—
48C4×#11758201C2211yescharacteristic
49C4×#11758201C2211yescharacteristic
50#16839827C2211yescharacteristic
51#16839827C2211yescharacteristic
52#16839514C2211yescharacteristic, complemented
53#16839512C2211yescharacteristic, complemented
54#16839507C2211yescharacteristic, complemented
55#16839507C2211yescharacteristic, complemented
56#16839506C2211yescharacteristic
57#16839491C2211yescharacteristic
58#16839491C2211yescharacteristic
59#16839484C2211yescharacteristic, complemented
60C61×#557C2211yescharacteristic, complemented
61C61×#486C2211yescharacteristic
62C61×#460C2211yescharacteristic
63C61×#460C2211yescharacteristic
64C61×#332C2211yescharacteristic, complemented
65C61×Q8⋊C8C2211yescharacteristic, complemented
66C61×Q8⋊C8C2211yescharacteristic, complemented
67C2×Q8×D61—21no—
68C2×Q8×D61—21no—
69C2×D4×D61—21no—
70C2×D4×D61—21no—
71C2×#11758190—21no—
72C2×#11758190—21no—
73C2×#11758200—21no—
74C2×#11758200—21no—
75C2×#11758200—21no—
76C2×#11758200—21no—
77C2×#11758201—21no—
78C2×#11758201D411yescharacteristic
79C2×#11758201D411yescharacteristic
80SD16×D61—44no—
81SD16×D61—44no—
82C2×C8×D61—21no—
83C2×C8×D61—21no—
84#14519777—21no—
85#14519777—21no—
86#14519777C2311yescharacteristic
87#14519777C2311yescharacteristic
88#14519757—21no—
89#14519757—21no—
90#14519757—41no—
91#14519757—41no—
92#14519748—21no—
93#14519748—21no—
94#14519745—41no—
95C4×D244—21no—
96C4×D244—21no—
97C4×D244C2311yescharacteristic
98C4×D244C2311yescharacteristic
99#14519710—21no—
100#14519710—21no—
101#14519710—21no—
102#14519709—41no—
103#14519708—21no—
104#14519708—21no—
105#14519708D411yescharacteristic
106#14519708D411yescharacteristic
107C4⋊#92418—21no—
108C4⋊#92418—21no—
109C4⋊#92418C2311yescharacteristic
110C4⋊#92418C2311yescharacteristic
111#14519702—21no—
112#14519702—21no—
113#14519701—21no—
114#14519701—21no—
115#14519701—21no—
116#14519701—21no—
117#14519701—21no—
118#14519701D411yescharacteristic
119#14519701D411yescharacteristic
120#14519697—21no—
121#14519697D411yescharacteristic
122#14519697D411yescharacteristic
123#14519689—21no—
124#14519689—21no—
125#14519688—21no—
126C4×#92418—21no—
127C4×#92418C2311yescharacteristic
128C4×#92418C2311yescharacteristic
129C2×SD16×C61—21no—
130C2×SD16×C61—21no—
131C4⋊Q8×C61—21no—
132C4⋊Q8×C61C2311yescharacteristic
133#127×C61C2311yescharacteristic
134#125×C61—21no—
135#125×C61—41no—
136C4×Q8×C61—21no—
137C4×Q8×C61—21no—
138C4×Q8×C61C2311yescharacteristic
139C4×Q8×C61C2311yescharacteristic
140C4⋊C8×C61C2311yescharacteristic
141C4⋊C8×C61C2311yescharacteristic
142#102×C61—21no—
143#102×C61—21no—
144#102×C61—41no—
145C4×C8×C61C2311yescharacteristic
146C23×D61—41no—
147C2×Q8×C61—21no—
148C2×Q8×C61C2×D411yescharacteristic
149C2×Q8×C61C2×D411yescharacteristic
150C2×D4×C61—21no—
151C2×D4×C61—21no—
152C2×D4×C61—21no—
153C2×C22⋊D61—41no—
154C2×C22⋊D61—41no—
155Q8×D61—42no—
156Q8×D61—42no—
157Q8×D61—81no—
158Q8×D61—81no—
159D4×D61—42no—
160D4×D61—42no—
161D4×D61—81no—
162D4×D61—81no—
163C2×D244C2×D411yescharacteristic
164C2×D244C2×D411yescharacteristic
165C2×C4×D61—21no—
166C2×C4×D61—21no—
167C2×C4×D61—41no—
168C2×C4×D61—41no—
169C2×Dic122—21no—
170C2×Dic122—21no—
171C2×Dic122—41no—
172C2×Dic122—41no—
173SD16×C61—42no—
174SD16×C61—42no—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C2×C8×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—41no—
183C4⋊C4×C61—41no—
184C4⋊C4×C61—41no—
185C4⋊C4×C61C2×D411yescharacteristic
186C4⋊C4×C61C2×D411yescharacteristic
187C42×C61—21no—
188C42×C61—21no—
189C42×C61—41no—
190C42×C61C2411yescharacteristic
191C22⋊Dic61—41no—
192#11758201—22no—
193#11758201—42no—
194#11758201SD1614yescomplemented
195#11758200—42no—
196#11758200—42no—
197#11758200—81no—
198#11758200—81no—
199#11758198—21no—
200#11758198—21no—
201#11758198—41no—
202#11758198—41no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61—41no—
208C4⋊Dic61—41no—
209C4⋊Dic61C2×D411yescharacteristic
210C4⋊Dic61C2×D411yescharacteristic
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418C2×D411yescharacteristic
214C2×#92418C2×D411yescharacteristic
215C2×#92418C2×D411yescharacteristic
216C2×#92418C2×D411yescharacteristic
217#11758190—42no—
218#11758190—42no—
219C8×D61—42no—
220C8×D61—42no—
221C23×C61—41no—
222C22×D61—21no—
223C22×D61—21no—
224C22×D61—82no—
225C22×D61—82no—
226C22×D61—161no—
227Q8×C61—21no—
228Q8×C61—21no—
229Q8×C61—21no—
230Q8×C61—41no—
231Q8×C61—41no—
232Q8×C61C2×SD1612yescomplemented
233D4×C61—41no—
234D4×C61—41no—
235D4×C61—41no—
236D4×C61—41no—
237D4×C61—81no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—41no—
243C2×C4×C61C22×D411yescharacteristic
244C2×C4×C61C22×D411yescharacteristic
245C2×C4×C61H(5,𝔽2)11yescharacteristic
246C22⋊D61—82no—
247C22⋊D61—82no—
248C2×Dic61—21no—
249C2×Dic61—21no—
250C2×Dic61—41no—
251C2×Dic61—41no—
252D244—21no—
253D244—21no—
254D244C2×SD1612yes—
255D244AGL(1,ℤ/8)12yes—
256C4×D61—41no—
257C4×D61—41no—
258C4×D61—41no—
259C4×D61—41no—
260C4×D61—82no—
261C4×D61—82no—
262Dic122—41no—
263Dic122—41no—
264Dic122—41no—
265Dic122—41no—
266Dic122—82no—
267Dic122—82no—
268C8×C61—41no—
269C8×C61—41no—
270C8×C61—41no—
271#92418—21no—
272#92418—21no—
273#92418—21no—
274#92418—41no—
275#92418—41no—
276#92418C2×SD1612yescomplemented
277#19764—611nomaximal, Sylow
278C22×C61—81no—
279C22×C61—81no—
280C22×C61—81no—
281C22×C61D4211yescharacteristic
282C2×D61—22no—
283C2×D61—22no—
284C2×D61—41no—
285C2×D61—41no—
286C2×D61—41no—
287C2×D61—41no—
288C2×D61—164no—
289C4×C61—22no—
290C4×C61—21no—
291C4×C61—41no—
292C4×C61—41no—
293C4×C61—41no—
294C4×C61—81no—
295C4×C61C22×SD1611yescharacteristic
296C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
297C4×C61#57711yescharacteristic
298C4×C61#57711yescharacteristic
299Dic61—41no—
300Dic61—41no—
301Dic61—81no—
302Dic61—81no—
303#3386—611no—
304#3359—611no—
305D4×SD16—611no—
306D4×SD16—611no—
307#1714—611no—
308#1686—611no—
309#1679—611no—
310#1673—611no—
311#1666—611no—
312#1651—611no—
313#1651—611no—
314#1639—611no—
315#1626D6111yescharacteristic, complemented
316#1582—611no—
317SD16⋊C8—611no—
318C2×C61—161no—
319C2×C61#326511yescharacteristic
320C2×C61D4×SD1611yescharacteristic
321C2×C61#328911yescharacteristic
322D61—42no—
323D61—42no—
324C22×SD16—1221no—
325C22×SD16—1221no—
326#557C2×D6111yescharacteristic, complemented
327D4×Q8—611no—
328D4×Q8—611no—
329D42—611no—
330#501—1221no—
331#500—611no—
332#498—611no—
333#492—611no—
334#492—611no—
335#487—611no—
336#487—1221no—
337#486C2×D6111yescharacteristic
338#478—1221no—
339#477—1221no—
340#475—611no—
341#475—611no—
342#475—1221no—
343#474—611no—
344#465—1221no—
345#465—1221no—
346#461—611no—
347#461—611no—
348#461—611no—
349#461—1221no—
350#460C2×D6111yescharacteristic
351#460C2×D6111yescharacteristic
352#450—1221no—
353#450—1221no—
354#448—1221no—
355#448—1221no—
356#446—611no—
357C8×Q8—611no—
358#440—1221no—
359C4×SD16—611no—
360C4×SD16—611no—
361C4×SD16—1221no—
362C8×D4—611no—
363C8×D4—611no—
364#335—611no—
365#334—611no—
366#332—611no—
367#332—611no—
368#332C2×D6111yescharacteristic, complemented
369#331—611no—
370Q8⋊C8C2×D6111yescharacteristic, complemented
371Q8⋊C8C2×D6111yescharacteristic, complemented
372D4⋊C8—611no—
373D4⋊C8—611no—
374#322—611no—
375C61#1976411yescharacteristic, complemented, Sylow
376C22×Q8—1221no—
377C22×Q8—1221no—
378C22×D4—1221no—
379C22×D4—1221no—
380C2×SD16—21no—
381C2×SD16—21no—
382C2×SD16—611no—
383C2×SD16—611no—
384C2×SD16—1221no—
385C2×SD16—1221no—
386C2×SD16—1221no—
387C2×SD16—1221no—
388C2×SD16—1221no—
389C2×SD16—1221no—
390C2×SD16—1221no—
391C2×SD16—2444no—
392C2×SD16—2444no—
393C22×C8—1221no—
394C22×C8—1221no—
395C4⋊Q8—21no—
396C4⋊Q8—611no—
397C4⋊Q8—611no—
398C4⋊Q8—1221no—
399C4⋊Q8—1221no—
400C4⋊Q8C22×D6111yescharacteristic
401#127C22×D6111yescharacteristic
402#125—21no—
403#125—41no—
404C22⋊Q8—1221no—
405C22⋊Q8—1221no—
406C22⋊Q8—2441no—
407C22⋊Q8—2441no—
408#121—1221no—
409#121—1221no—
410#120—2441no—
411C4×Q8—21no—
412C4×Q8—21no—
413C4×Q8C22×D6111yescharacteristic
414C4×Q8C22×D6111yescharacteristic
415C4×D4—611no—
416C4×D4—611no—
417C4×D4—1221no—
418C4×D4—1221no—
419#107—1221no—
420#107—1221no—
421#107—1221no—
422#106—611no—
423#106—611no—
424#106—1221no—
425#106—1221no—
426#106—2441no—
427C4⋊C8—611no—
428C4⋊C8—611no—
429C4⋊C8—1221no—
430C4⋊C8—1221no—
431C4⋊C8C22×D6111yescharacteristic
432C4⋊C8C22×D6111yescharacteristic
433#103—611no—
434#103—611no—
435#103—1221no—
436#103—1221no—
437#103—1221no—
438#103—1221no—
439#103—1221no—
440#103—1221no—
441#103—1221no—
442#102—21no—
443#102—21no—
444#102—41no—
445#102—611no—
446#102—611no—
447#102—1221no—
448C22⋊C8—1221no—
449C22⋊C8—1221no—
450#97—1221no—
451C4×C8—611no—
452C4×C8—611no—
453C4×C8—1221no—
454C4×C8C22×D6111yescharacteristic
455C24—2441no—
456C2×Q8—21no—
457C2×Q8—1221no—
458C2×Q8—1221no—
459C2×Q8—4881no—
460C2×Q8—4881no—
461C2×Q8—2441no—
462C2×Q8—2441no—
463C2×Q8—2442no—
464C2×Q8—2442no—
465C2×Q8D4×D6111yescharacteristic
466C2×Q8D4×D6111yescharacteristic
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4—611no—
471C2×D4—611no—
472C2×D4—4881no—
473C2×D4—4881no—
474C2×D4—2441no—
475C2×D4—2441no—
476C2×D4—2442no—
477C2×D4—2442no—
478C22×C4—1221no—
479C22×C4—1221no—
480C22×C4—2441no—
481C22×C4—2441no—
482SD16—42no—
483SD16—42no—
484SD16—614no—
485SD16—1222no—
486SD16—4881no—
487SD16—4881no—
488SD16—2442no—
489SD16—2442no—
490SD16—2442no—
491SD16—2442no—
492SD16—2442no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—611no—
497C2×C8—611no—
498C2×C8—611no—
499C2×C8—611no—
500C2×C8—1221no—
501C2×C8—1221no—
502C2×C8—2442no—
503C2×C8—2442no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—41no—
509C4⋊C4—41no—
510C4⋊C4—41no—
511C4⋊C4—611no—
512C4⋊C4—611no—
513C4⋊C4—1221no—
514C4⋊C4—1221no—
515C4⋊C4—1221no—
516C4⋊C4—1221no—
517C4⋊C4—2441no—
518C4⋊C4—2441no—
519C4⋊C4D4×D6111yescharacteristic
520C4⋊C4D4×D6111yescharacteristic
521C22⋊C4—1221no—
522C22⋊C4—1221no—
523C22⋊C4—2441no—
524C22⋊C4—2441no—
525C22⋊C4—2441no—
526C42—21no—
527C42—21no—
528C42—41no—
529C42C23×D6111yescharacteristic
530C23—41no—
531C23—1221no—
532C23—1221no—
533C23—9761no—
534C23—4882no—
535C23—4882no—
536Q8—21no—
537Q8—21no—
538Q8—21no—
539Q8—41no—
540Q8—41no—
541Q8—4882no—
542Q8—4882no—
543Q8—2441no—
544Q8—2441no—
545Q8—2441no—
546Q8—2441no—
547Q8SD16×D6112yescomplemented
548D4—41no—
549D4—41no—
550D4—41no—
551D4—41no—
552D4—81no—
553D4—612no—
554D4—612no—
555D4—1221no—
556D4—1221no—
557D4—4882no—
558D4—4882no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—41no—
564C2×C4—1221no—
565C2×C4—1221no—
566C2×C4—4882no—
567C2×C4—4882no—
568C2×C4—2441no—
569C2×C4—2441no—
570C2×C4—2441no—
571C2×C4—2441no—
572C2×C4—2441no—
573C2×C4—2441no—
574C2×C4C2×D4×D6111yescharacteristic
575C2×C4C2×D4×D6111yescharacteristic
576C2×C4#1451982211yescharacteristic
577C8—41no—
578C8—41no—
579C8—41no—
580C8—612no—
581C8—1221no—
582C8—1221no—
583C8—1221no—
584C8—2441no—
585C8—2441no—
586C22—81no—
587C22—81no—
588C22—81no—
589C22—1222no—
590C22—1222no—
591C22—9764no—
592C22—2441no—
593C22—2441no—
594C22—2441no—
595C22—2441no—
596C22#1684039411yescharacteristic
597C4—22no—
598C4—21no—
599C4—41no—
600C4—41no—
601C4—41no—
602C4—81no—
603C4—4881no—
604C4—4881no—
605C4—2441no—
606C4—2441no—
607C4C2×SD16×D6111yescharacteristic
608C4AGL(1,ℤ/8)×D6111yescharacteristic
609C4#1684056911yescharacteristic
610C4#1684057111yescharacteristic
611C2—161no—
612C2—2442no—
613C2—2442no—
614C2#2145731711yescharacteristic
615C2#2145750811yescharacteristic
616C2#2145759611yescharacteristic
subgroupG/H
centerC22#16840394
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1626C2
socleC22×C61D42
prime radical#16840397C22
hypercenter#1626D61
hypocenterC61#19764
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1626

Complements

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

NH ≅ G/Ntype
1. #21458736C2semidirect
2. #21458457C2semidirect
5. #21445440C2semidirect
6. #21445289C2semidirect
7. #21445241C2semidirect
8. #21445207C2semidirect
9. #21445165C2semidirect
10. #21445085C2semidirect
11. #21445085C2semidirect
13. #21444658C2semidirect
14. #21444634C2semidirect
15. C61×#1626C2semidirect
22. #16840165C22semidirect
23. #16840160C22semidirect
27. #16840112C22semidirect
31. #16840047C22semidirect
32. #16840047C22semidirect
44. #16839893C22semidirect
45. Q8×#92418C22semidirect
52. #16839514C22semidirect
53. #16839512C22semidirect
54. #16839507C22semidirect
55. #16839507C22semidirect
59. #16839484C22semidirect
60. C61×#557C22semidirect
64. C61×#332C22semidirect
65. C61×Q8⋊C8C22semidirect
66. C61×Q8⋊C8C22semidirect
194. #11758201SD16semidirect
232. Q8×C61C2×SD16semidirect
276. #92418C2×SD16semidirect
315. #1626D61semidirect
326. #557C2×D61semidirect
368. #332C2×D61semidirect
370. Q8⋊C8C2×D61semidirect
371. Q8⋊C8C2×D61semidirect
375. C61#19764semidirect
547. Q8SD16×D61semidirect

Maximal subgroups

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

subgroupindex
#214587362
#214584572
#214575102
#214454402
#214452892
#214452412
#214452072
#214451652
#214450852
#214449952
#214446582
#214446342
C61×#16262
#1976461