Finite Group Databasebeta

← #29669488

Subgroups — #29669488

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#21458739C211yesmaximal, characteristic, complemented
2#21458461C211yesmaximal, characteristic, complemented
3#21457518C211yesmaximal, characteristic
4D4×#11758201C211yesmaximal, characteristic
5#21445444C211yesmaximal, characteristic, complemented
6#21445294C211yesmaximal, characteristic, complemented
7#21445245C211yesmaximal, characteristic, complemented
8#21445206C211yesmaximal, characteristic, complemented
9#21445169C211yesmaximal, characteristic, complemented
10#21445079C211yesmaximal, characteristic, complemented
11#21445000C211yesmaximal, characteristic
12#21444918C211yesmaximal, characteristic, complemented
13#21444652C211yesmaximal, characteristic, complemented
14#21444634C211yesmaximal, characteristic, complemented
15C61×#1651C211yesmaximal, characteristic, complemented
16C22×#11758201—21no—
17C2×SD16×D61—21no—
18#16840491C2211yescharacteristic, complemented
19#16840427C2211yescharacteristic
20D4×D244C2211yescharacteristic
21#16840169—21no—
22#16840165C2211yescharacteristic, complemented
23#16840136C2211yescharacteristic
24#16840136C2211yescharacteristic
25#16840114—21no—
26#16840114C2211yescharacteristic, complemented
27#16840109C2211yescharacteristic
28#16840061—21no—
29#16840055—21no—
30#16840044—21no—
31#16840044C2211yescharacteristic, complemented
32#16840043C2211yescharacteristic
33#16839994—21no—
34#16839993—21no—
35#16839967—21no—
36#16839967C2211yescharacteristic
37#16839964C2211yescharacteristic
38#16839963C2211yescharacteristic
39#16839962C2211yescharacteristic
40#16839912—21no—
41#16839911—21no—
42#16839902—21no—
43#16839901—21no—
44#16839889C2211yescharacteristic, complemented
45Q8×#92418C2211yescharacteristic, complemented
46#16839865—21no—
47C4×#11758201C2211yescharacteristic
48C4×#11758200—21no—
49C4×#11758200C2211yescharacteristic
50D4×#92418C2211yescharacteristic
51#16839827C2211yescharacteristic
52#16839514C2211yescharacteristic, complemented
53#16839512C2211yescharacteristic, complemented
54#16839509C2211yescharacteristic, complemented
55#16839508C2211yescharacteristic, complemented
56#16839505C2211yescharacteristic
57#16839494C2211yescharacteristic, complemented
58#16839491C2211yescharacteristic
59#16839484C2211yescharacteristic, complemented
60D4×Q8×C61C2211yescharacteristic
61C61×#498C2211yescharacteristic, complemented
62C61×#475C2211yescharacteristic, complemented
63C61×#461C2211yescharacteristic
64C61×#332C2211yescharacteristic, complemented
65C61×Q8⋊C8C2211yescharacteristic, complemented
66C61×D4⋊C8C2211yescharacteristic
67C2×Q8×D61—21no—
68C22×D244—21no—
69C2×D4×D61—21no—
70C2×#11758190—21no—
71C2×#11758200—21no—
72C2×#11758200D411yescharacteristic
73C2×#11758201—21no—
74C2×#11758201—21no—
75C2×#11758201—21no—
76C2×#11758201—21no—
77C2×#11758201—21no—
78C2×#11758201—21no—
79C2×#11758201—44no—
80C2×#11758201D411yescharacteristic
81SD16×D61—44no—
82C22×#92418—21no—
83C2×C8×D61—21no—
84#14519777—21no—
85#14519777C2311yescharacteristic
86#14519776—21no—
87#14519774C2311yescharacteristic
88#14519768—21no—
89#14519768—41no—
90#14519755—41no—
91#14519753—21no—
92#14519752—21no—
93#14519751—21no—
94#14519747—41no—
95Q8×Dic61—21no—
96C4×Dic122—21no—
97C4×Dic122C2311yescharacteristic
98C4×D244C2311yescharacteristic
99#14519739—21no—
100#14519711—21no—
101#14519710—21no—
102#14519710—21no—
103#14519708—21no—
104#14519708D411yescharacteristic
105#14519708D411yescharacteristic
106C4⋊#92418—21no—
107C4⋊#92418C2311yescharacteristic
108C4⋊#92418C2311yescharacteristic
109#14519706—21no—
110#14519701—21no—
111#14519701D411yescharacteristic
112#14519700—21no—
113#14519700—21no—
114#14519700—21no—
115#14519700—21no—
116#14519700D411yescharacteristic
117#14519699—21no—
118#14519699—21no—
119#14519699—41no—
120#14519698D411yescharacteristic
121#14519697—21no—
122#14519697D411yescharacteristic
123C22⋊#92418—21no—
124#14519689—21no—
125#14519687—21no—
126C4×#92418—21no—
127C4×#92418C2311yescharacteristic
128C4×#92418C2311yescharacteristic
129C22×Q8×C61—21no—
130C2×SD16×C61—21no—
131C4⋊Q8×C61—21no—
132C4⋊Q8×C61C2311yescharacteristic
133C4⋊Q8×C61C2311yescharacteristic
134C22⋊Q8×C61—21no—
135C22⋊Q8×C61—41no—
136C4×Q8×C61C2311yescharacteristic
137C4×D4×C61—21no—
138C4×D4×C61C2311yescharacteristic
139#106×C61—21no—
140#106×C61—41no—
141C4⋊C8×C61C2311yescharacteristic
142C4⋊C8×C61C2311yescharacteristic
143#103×C61—21no—
144#103×C61—21no—
145C4×C8×C61C2311yescharacteristic
146C23×D61—41no—
147C2×Q8×C61—21no—
148C2×Q8×C61—21no—
149C2×Q8×C61—41no—
150C2×Q8×C61—42no—
151C2×Q8×C61—81no—
152C2×Q8×C61C2×D411yescharacteristic
153C2×D4×C61C2×D411yescharacteristic
154C22×C4×C61—21no—
155C22×C4×C61—41no—
156C2×C22⋊D61—41no—
157Q8×D61—42no—
158Q8×D61—81no—
159D4×D61—42no—
160D4×D61—81no—
161C2×D244—21no—
162C2×D244—21no—
163C2×D244—21no—
164C2×D244—42no—
165C2×D244—41no—
166C2×D244—81no—
167C2×D244C2×D411yescharacteristic
168C2×C4×D61—21no—
169C2×C4×D61—41no—
170C2×Dic122—21no—
171C2×Dic122—41no—
172C2×Dic122C2×D411yescharacteristic
173SD16×C61—42no—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—41no—
183C4⋊C4×C61C2×D411yescharacteristic
184C4⋊C4×C61C2×D411yescharacteristic
185C22⋊C4×C61—21no—
186C22⋊C4×C61—41no—
187C42×C61—21no—
188C42×C61C2411yescharacteristic
189#11758201—42no—
190#11758201—42no—
191#11758201—42no—
192#11758201—42no—
193#11758201—81no—
194#11758201—81no—
195#11758201SD1614yescomplemented
196#11758200—22no—
197#11758200—42no—
198#11758198—21no—
199#11758198—41no—
200#11758198—41no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—41no—
204C4⋊Dic61—41no—
205C4⋊Dic61C2×D411yescharacteristic
206C4⋊Dic61C2×D411yescharacteristic
207#11758196—21no—
208#11758196—41no—
209#11758196—41no—
210C4×Dic61—21no—
211C4×Dic61—41no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—42no—
215C2×#92418C2×D411yescharacteristic
216C2×#92418C2×D411yescharacteristic
217C2×#92418C2×D411yescharacteristic
218C2×#92418C2×D411yescharacteristic
219#11758190—42no—
220C8×D61—42no—
221C23×C61—21no—
222C22×D61—21no—
223C22×D61—41no—
224C22×D61—82no—
225C22×D61—82no—
226C22×D61—161no—
227Q8×C61—21no—
228Q8×C61—41no—
229Q8×C61—41no—
230Q8×C61—41no—
231Q8×C61—41no—
232Q8×C61—82no—
233Q8×C61C2×SD1612yescomplemented
234D4×C61—21no—
235D4×C61AGL(1,ℤ/8)12yes—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—82no—
244C2×C4×C61C22×D411yescharacteristic
245C2×C4×C61C22×D411yescharacteristic
246C2×C4×C61H(5,𝔽2)11yescharacteristic
247C22⋊D61—82no—
248C2×Dic61—21no—
249C2×Dic61—21no—
250C2×Dic61—21no—
251C2×Dic61—41no—
252D244—21no—
253D244—41no—
254D244—41no—
255D244—41no—
256D244—41no—
257D244—81no—
258D244—82no—
259D244C2×SD1612yes—
260C4×D61—41no—
261C4×D61—41no—
262C4×D61—82no—
263Dic122—21no—
264Dic122—21no—
265Dic122—41no—
266Dic122—41no—
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—22no—
279C22×C61—41no—
280C22×C61—41no—
281C22×C61D4211yescharacteristic
282C2×D61—22no—
283C2×D61—41no—
284C2×D61—41no—
285C2×D61—81no—
286C2×D61—81no—
287C2×D61—81no—
288C2×D61—164no—
289C4×C61—22no—
290C4×C61—21no—
291C4×C61—41no—
292C4×C61—41no—
293C4×C61—81no—
294C4×C61—81no—
295C4×C61C22×SD1611yescharacteristic
296C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
297C4×C61#57711yescharacteristic
298C4×C61#57711yescharacteristic
299Dic61—41no—
300Dic61—41no—
301Dic61—41no—
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#1651D6111yescharacteristic, complemented
314#1639—611no—
315#1626—611no—
316#1582—611no—
317SD16⋊C8—611no—
318C2×C61—42no—
319C2×C61#326511yescharacteristic
320C2×C61D4×SD1611yescharacteristic
321C2×C61#328911yescharacteristic
322D61—42no—
323D61—161no—
324C22×SD16—1221no—
325C22×SD16—1221no—
326#557—611no—
327D4×Q8—611no—
328D4×Q8C2×D6111yescharacteristic
329D42—611no—
330#501—1221no—
331#500—611no—
332#498C2×D6111yescharacteristic, complemented
333#492—611no—
334#492—611no—
335#487—611no—
336#487—1221no—
337#486—611no—
338#478—1221no—
339#477—1221no—
340#475—611no—
341#475—1221no—
342#475C2×D6111yescharacteristic, complemented
343#474—611no—
344#465—1221no—
345#465—1221no—
346#461—611no—
347#461—611no—
348#461—1221no—
349#461C2×D6111yescharacteristic
350#460—611no—
351#460—611no—
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⋊C8—611no—
371Q8⋊C8C2×D6111yescharacteristic, complemented
372D4⋊C8—611no—
373D4⋊C8C2×D6111yescharacteristic
374#322—611no—
375C61#1976411yescharacteristic, complemented, Sylow
376C22×Q8—21no—
377C22×Q8—1221no—
378C22×D4—1221no—
379C22×D4—1221no—
380C2×SD16—21no—
381C2×SD16—611no—
382C2×SD16—611no—
383C2×SD16—1221no—
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—1221no—
398C4⋊Q8—1221no—
399C4⋊Q8C22×D6111yescharacteristic
400C4⋊Q8C22×D6111yescharacteristic
401#127—611no—
402#125—1221no—
403#125—2441no—
404C22⋊Q8—21no—
405C22⋊Q8—41no—
406C22⋊Q8—1221no—
407C22⋊Q8—2441no—
408#121—1221no—
409#121—1221no—
410#120—2441no—
411C4×Q8—611no—
412C4×Q8—1221no—
413C4×Q8—1221no—
414C4×Q8C22×D6111yescharacteristic
415C4×D4—21no—
416C4×D4—611no—
417C4×D4—1221no—
418C4×D4C22×D6111yescharacteristic
419#107—1221no—
420#107—1221no—
421#107—1221no—
422#106—21no—
423#106—41no—
424#106—611no—
425#106—611no—
426#106—1221no—
427C4⋊C8—611no—
428C4⋊C8—611no—
429C4⋊C8—1221no—
430C4⋊C8—1221no—
431C4⋊C8C22×D6111yescharacteristic
432C4⋊C8C22×D6111yescharacteristic
433#103—21no—
434#103—21no—
435#103—611no—
436#103—611no—
437#103—1221no—
438#103—1221no—
439#103—1221no—
440#103—1221no—
441#103—1221no—
442#102—611no—
443#102—611no—
444#102—1221no—
445#102—1221no—
446#102—1221no—
447#102—2441no—
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—21no—
458C2×Q8—41no—
459C2×Q8—42no—
460C2×Q8—81no—
461C2×Q8—611no—
462C2×Q8—1221no—
463C2×Q8—4881no—
464C2×Q8—2441no—
465C2×Q8—2442no—
466C2×Q8D4×D6111yescharacteristic
467C2×D4—611no—
468C2×D4—1221no—
469C2×D4—1221no—
470C2×D4—1221no—
471C2×D4—4881no—
472C2×D4—4881no—
473C2×D4—2441no—
474C2×D4—2442no—
475C2×D4—2441no—
476C2×D4—2442no—
477C2×D4D4×D6111yescharacteristic
478C22×C4—21no—
479C22×C4—41no—
480C22×C4—1221no—
481C22×C4—2441no—
482SD16—42no—
483SD16—614no—
484SD16—1222no—
485SD16—4881no—
486SD16—4881no—
487SD16—2442no—
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—21no—
509C4⋊C4—41no—
510C4⋊C4—611no—
511C4⋊C4—611no—
512C4⋊C4—1221no—
513C4⋊C4—1221no—
514C4⋊C4—1221no—
515C4⋊C4—2441no—
516C4⋊C4—2441no—
517C4⋊C4—2441no—
518C4⋊C4—2441no—
519C4⋊C4D4×D6111yescharacteristic
520C4⋊C4D4×D6111yescharacteristic
521C22⋊C4—21no—
522C22⋊C4—41no—
523C22⋊C4—1221no—
524C22⋊C4—2441no—
525C22⋊C4—2441no—
526C42—21no—
527C42—1221no—
528C42—2441no—
529C42C23×D6111yescharacteristic
530C23—21no—
531C23—1221no—
532C23—9761no—
533C23—4882no—
534C23—4882no—
535C23—2441no—
536Q8—21no—
537Q8—41no—
538Q8—41no—
539Q8—41no—
540Q8—41no—
541Q8—82no—
542Q8—1221no—
543Q8—1221no—
544Q8—4882no—
545Q8—2441no—
546Q8—2441no—
547Q8SD16×D6112yescomplemented
548D4—21no—
549D4—612no—
550D4—1221no—
551D4—4881no—
552D4—4882no—
553D4—4882no—
554D4—2441no—
555D4—2441no—
556D4—2441no—
557D4—2441no—
558D4#1451980412yes—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—82no—
567C2×C4—1221no—
568C2×C4—1221no—
569C2×C4—1221no—
570C2×C4—4882no—
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—22no—
587C22—41no—
588C22—41no—
589C22—1222no—
590C22—9764no—
591C22—4881no—
592C22—4881no—
593C22—4881no—
594C22—2441no—
595C22—2441no—
596C22#1684039411yescharacteristic
597C4—22no—
598C4—21no—
599C4—41no—
600C4—41no—
601C4—81no—
602C4—81no—
603C4—4881no—
604C4—2441no—
605C4—2441no—
606C4—2441no—
607C4C2×SD16×D6111yescharacteristic
608C4C2×#1451980411yescharacteristic
609C4#1684057111yescharacteristic
610C4#1684057211yescharacteristic
611C2—42no—
612C2—9761no—
613C2—2442no—
614C2#2145730711yescharacteristic
615C2#2145751311yescharacteristic
616C2#2145758911yescharacteristic
subgroupG/H
centerC22#16840394
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1651C2
socleC22×C61D42
prime radicalD4×D244C22
hypercenter#1651D61
hypocenterC61#19764
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1651

Complements

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

NH ≅ G/Ntype
1. #21458739C2semidirect
2. #21458461C2semidirect
5. #21445444C2semidirect
6. #21445294C2semidirect
7. #21445245C2semidirect
8. #21445206C2semidirect
9. #21445169C2semidirect
10. #21445079C2semidirect
12. #21444918C2semidirect
13. #21444652C2semidirect
14. #21444634C2semidirect
15. C61×#1651C2semidirect
18. #16840491C22semidirect
22. #16840165C22semidirect
26. #16840114C22semidirect
31. #16840044C22semidirect
44. #16839889C22semidirect
45. Q8×#92418C22semidirect
52. #16839514C22semidirect
53. #16839512C22semidirect
54. #16839509C22semidirect
55. #16839508C22semidirect
57. #16839494C22semidirect
59. #16839484C22semidirect
61. C61×#498C22semidirect
62. C61×#475C22semidirect
64. C61×#332C22semidirect
65. C61×Q8⋊C8C22semidirect
195. #11758201SD16semidirect
233. Q8×C61C2×SD16semidirect
276. #92418C2×SD16semidirect
313. #1651D61semidirect
332. #498C2×D61semidirect
342. #475C2×D61semidirect
368. #332C2×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
#214587392
#214584612
#214575182
D4×#117582012
#214454442
#214452942
#214452452
#214452062
#214451692
#214450792
#214450002
#214449182
#214446522
#214446342
C61×#16512
#1976461