Finite Group Databasebeta

← C83×#19947 #35247771

Subgroups — C83×#19947 #35247771

Order 21248.

658 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
1C83×#3399C211yesmaximal, characteristic, complemented
2C83×#3394C211yesmaximal, characteristic, complemented
3C83×#3353C211yesmaximal, characteristic, complemented
4C83×#3284C211yesmaximal, characteristic, complemented
5C83×#1706C211yesmaximal, characteristic, complemented
6C83×#1694C211yesmaximal, characteristic, complemented
7C83×#1687C211yesmaximal, characteristic, complemented
8C83×#1666C211yesmaximal, characteristic, complemented
9C83×#1665C211yesmaximal, characteristic, complemented
10C83×#1641C211yesmaximal, characteristic, complemented
11C83×#1640C211yesmaximal, characteristic, complemented
12C83×#1628C211yesmaximal, characteristic, complemented
13C83×#1625C211yesmaximal, characteristic, complemented
14C83×#1590C211yesmaximal, characteristic, complemented
15C8×SD16×C83C211yesmaximal, characteristic, complemented
16C2×#135×C83—21no—
17#559×C83C2211yescharacteristic
18#556×C83C2211yescharacteristic
19#550×C83C2211yescharacteristic
20#547×C83C2211yescharacteristic
21#500×C83C2211yescharacteristic
22#496×C83—21no—
23#495×C83—21no—
24#493×C83C2211yescharacteristic
25#492×C83C2211yescharacteristic
26#492×C83C2211yescharacteristic
27#486×C83C2211yescharacteristic
28#486×C83C2211yescharacteristic
29#484×C83—21no—
30#482×C83—21no—
31#475×C83C2211yescharacteristic
32#474×C83C2211yescharacteristic
33#468×C83—21no—
34#465×C83—21no—
35#464×C83—21no—
36#463×C83—21no—
37#463×C83—21no—
38#462×C83C2211yescharacteristic
39#462×C83C2211yescharacteristic
40#460×C83C2211yescharacteristic
41#459×C83—21no—
42#459×C83C2211yescharacteristic
43#459×C83C2211yescharacteristic
44#459×C83C2211yescharacteristic
45#452×C83—21no—
46#449×C83—21no—
47#446×C83C2211yescharacteristic
48C8×Q8×C83C2211yescharacteristic
49#442×C83—21no—
50C4×SD16×C83—21no—
51C4×SD16×C83C2211yescharacteristic
52C4×SD16×C83C2211yescharacteristic
53#436×C83C2211yescharacteristic
54C8×D4×C83C2211yescharacteristic
55#428×C83—21no—
56#334×C83C2211yescharacteristic
57#334×C83C2211yescharacteristic
58#332×C83C2211yescharacteristic
59#332×C83C2211yescharacteristic
60#331×C83C2211yescharacteristic
61#331×C83C2211yescharacteristic
62Q8⋊C8×C83C2211yescharacteristic
63Q8⋊C8×C83C2211yescharacteristic
64D4⋊C8×C83C2211yescharacteristic
65D4⋊C8×C83C2211yescharacteristic
66C82×C83C2211yescharacteristic
67C2×#41×C83—21no—
68C2×#41×C83—21no—
69#135×C83—44no—
70C2×Q16×C83—21no—
71C2×Q16×C83—21no—
72C2×SD16×C83—21no—
73C2×SD16×C83—21no—
74C2×SD16×C83—21no—
75C2×SD16×C83—21no—
76C2×SD16×C83—21no—
77C2×SD16×C83D411yescharacteristic
78C2×SD16×C83D411yescharacteristic
79C2×D8×C83—21no—
80C2×D8×C83—21no—
81C2×D8×C83—21no—
82C2×D8×C83—21no—
83C2×D8×C83—21no—
84C2×M16×C83—21no—
85C22×C8×C83—21no—
86C4⋊Q8×C83C2311yescharacteristic
87C4⋊Q8×C83C2311yescharacteristic
88#127×C83—21no—
89#127×C83C2311yescharacteristic
90#127×C83C2311yescharacteristic
91#126×C83—41no—
92#124×C83—21no—
93#124×C83—21no—
94#124×C83—21no—
95#123×C83—41no—
96#123×C83—41no—
97C22⋊Q8×C83—21no—
98C22⋊Q8×C83—21no—
99#121×C83—21no—
100#121×C83—21no—
101#121×C83—41no—
102C4×Q8×C83—21no—
103C4×Q8×C83C2311yescharacteristic
104C4×Q8×C83C2311yescharacteristic
105C4×D4×C83—21no—
106C4×D4×C83—21no—
107C4×D4×C83—21no—
108C4×D4×C83C2311yescharacteristic
109C4×D4×C83C2311yescharacteristic
110#117×C83—21no—
111C2×C4⋊C4×C83—21no—
112#107×C83—21no—
113#107×C83—21no—
114#107×C83—21no—
115#106×C83—21no—
116#106×C83D411yescharacteristic
117#106×C83D411yescharacteristic
118C4⋊C8×C83—21no—
119C4⋊C8×C83—21no—
120C4⋊C8×C83C2311yescharacteristic
121C4⋊C8×C83C2311yescharacteristic
122C4⋊C8×C83C2311yescharacteristic
123C4⋊C8×C83C2311yescharacteristic
124#103×C83—21no—
125#103×C83—21no—
126#103×C83—21no—
127#103×C83D411yescharacteristic
128#103×C83D411yescharacteristic
129#102×C83—21no—
130#102×C83—21no—
131#102×C83—21no—
132#102×C83—21no—
133#102×C83—21no—
134#102×C83—41no—
135#102×C83—41no—
136#102×C83D411yescharacteristic
137#102×C83D411yescharacteristic
138C22⋊C8×C83—21no—
139C22⋊C8×C83—21no—
140#97×C83—21no—
141C4×C8×C83—21no—
142C4×C8×C83C2311yescharacteristic
143C4×C8×C83C2311yescharacteristic
144C4×C8×C83C2311yescharacteristic
145#41×C83—42no—
146#41×C83—42no—
147#41×C83—81no—
148#41×C83—81no—
149C2×Q8×C83—21no—
150C2×Q8×C83—21no—
151C2×Q8×C83C2×D411yescharacteristic
152C2×Q8×C83C2×D411yescharacteristic
153C2×D4×C83—21no—
154C2×D4×C83—21no—
155C2×D4×C83—21no—
156C2×D4×C83—21no—
157C2×D4×C83—21no—
158C2×D4×C83—21no—
159C2×D4×C83—41no—
160C2×D4×C83—41no—
161C2×D4×C83—41no—
162C2×D4×C83C2×D411yescharacteristic
163C2×D4×C83C2×D411yescharacteristic
164C22×C4×C83—21no—
165C22×C4×C83—21no—
166C22×C4×C83—41no—
167C22×C4×C83—41no—
168C22×C4×C83—41no—
169Q16×C83—42no—
170Q16×C83—81no—
171SD16×C83—22no—
172SD16×C83—22no—
173SD16×C83—42no—
174SD16×C83—42no—
175SD16×C83—81no—
176SD16×C83—81no—
177SD16×C83—81no—
178D8×C83—42no—
179D8×C83—42no—
180D8×C83—81no—
181D8×C83—81no—
182D8×C83—81no—
183M16×C83—42no—
184C2×C8×C83—21no—
185C2×C8×C83—21no—
186C2×C8×C83—21no—
187C2×C8×C83—21no—
188C2×C8×C83—21no—
189C2×C8×C83—42no—
190C2×C8×C83C2×D411yescharacteristic
191C2×C8×C83C2×D411yescharacteristic
192C2×C8×C83C2×D411yescharacteristic
193C2×C8×C83C2×D411yescharacteristic
194C4⋊C4×C83—21no—
195C4⋊C4×C83—21no—
196C4⋊C4×C83—21no—
197C4⋊C4×C83—21no—
198C4⋊C4×C83—21no—
199C4⋊C4×C83—21no—
200C4⋊C4×C83—41no—
201C4⋊C4×C83—41no—
202C4⋊C4×C83C2×D411yescharacteristic
203C4⋊C4×C83C2×D411yescharacteristic
204C4⋊C4×C83C2×D411yescharacteristic
205C4⋊C4×C83C2×D411yescharacteristic
206C22⋊C4×C83—21no—
207C22⋊C4×C83—21no—
208C22⋊C4×C83—41no—
209C22⋊C4×C83—41no—
210C22⋊C4×C83—41no—
211C22⋊C4×C83—41no—
212C22⋊C4×C83—41no—
213C22⋊C4×C83—41no—
214C22⋊C4×C83—41no—
215C42×C83—21no—
216C42×C83—21no—
217C42×C83—41no—
218C42×C83C2411yescharacteristic
219C23×C83—21no—
220C23×C83—21no—
221C23×C83—41no—
222C23×C83—41no—
223Q8×C83—21no—
224Q8×C83—21no—
225Q8×C83—21no—
226Q8×C83—21no—
227Q8×C83—41no—
228Q8×C83—41no—
229Q8×C83—41no—
230Q8×C83—41no—
231D4×C83—21no—
232D4×C83—21no—
233D4×C83—21no—
234D4×C83—21no—
235D4×C83—41no—
236D4×C83—41no—
237D4×C83—41no—
238D4×C83—41no—
239D4×C83—41no—
240D4×C83—41no—
241D4×C83—41no—
242D4×C83—41no—
243D4×C83—81no—
244D4×C83—81no—
245D4×C83—82no—
246D4×C83—82no—
247D4×C83—161no—
248C2×C4×C83—21no—
249C2×C4×C83—21no—
250C2×C4×C83—21no—
251C2×C4×C83—21no—
252C2×C4×C83—21no—
253C2×C4×C83—21no—
254C2×C4×C83—41no—
255C2×C4×C83—41no—
256C2×C4×C83—41no—
257C2×C4×C83—41no—
258C2×C4×C83—41no—
259C2×C4×C83—41no—
260C2×C4×C83—82no—
261C2×C4×C83—82no—
262C2×C4×C83—161no—
263C2×C4×C83C22×D411yescharacteristic
264C2×C4×C83C22×D411yescharacteristic
265C2×C4×C83H(5,𝔽2)11yescharacteristic
266C8×C83—21no—
267C8×C83—21no—
268C8×C83—41no—
269C8×C83—41no—
270C8×C83—41no—
271C8×C83—41no—
272C8×C83—41no—
273C8×C83#13512yescomplemented
274C8×C83AGL(1,ℤ/8)12yes—
275C22×C83—41no—
276C22×C83—41no—
277C22×C83—41no—
278C22×C83—41no—
279C22×C83—41no—
280C22×C83—41no—
281C22×C83—81no—
282C22×C83—81no—
283C22×C83—81no—
284C22×C83—81no—
285C22×C83—81no—
286C22×C83—81no—
287C22×C83D4211yescharacteristic
288C4×C83—21no—
289C4×C83—41no—
290C4×C83—41no—
291C4×C83—41no—
292C4×C83—41no—
293C4×C83—41no—
294C4×C83—41no—
295C4×C83—81no—
296C4×C83—81no—
297C4×C83C2×#13511yescharacteristic
298C4×C83C2×AGL(1,ℤ/8)11yescharacteristic
299C4×C83#57611yescharacteristic
300C4×C83#57711yescharacteristic
301#19947C8311yesmaximal, characteristic, direct product factor, Sylow
302C2×C83—81no—
303C2×C83—81no—
304C2×C83—161no—
305C2×C83—161no—
306C2×C83#327411yescharacteristic
307C2×C83#328411yescharacteristic
308C2×C83#329011yescharacteristic
309#3399C2×C8311yescharacteristic, complemented
310#3394C2×C8311yescharacteristic, complemented
311#3353C2×C8311yescharacteristic, complemented
312#3284C2×C8311yescharacteristic, complemented
313#1706C2×C8311yescharacteristic, complemented
314#1694C2×C8311yescharacteristic, complemented
315#1687C2×C8311yescharacteristic, complemented
316#1666C2×C8311yescharacteristic, complemented
317#1665C2×C8311yescharacteristic, complemented
318#1641C2×C8311yescharacteristic, complemented
319#1640C2×C8311yescharacteristic, complemented
320#1628C2×C8311yescharacteristic, complemented
321#1625C2×C8311yescharacteristic, complemented
322#1590C2×C8311yescharacteristic, complemented
323C8×SD16C2×C8311yescharacteristic, complemented
324C83#1994711yescharacteristic, direct product factor, Sylow
325C2×#135—21no—
326#559C22×C8311yescharacteristic
327#556C22×C8311yescharacteristic
328#550C22×C8311yescharacteristic
329#547C22×C8311yescharacteristic
330#500C22×C8311yescharacteristic
331#496—21no—
332#495—21no—
333#493C22×C8311yescharacteristic
334#492C22×C8311yescharacteristic
335#492C22×C8311yescharacteristic
336#486C22×C8311yescharacteristic
337#486C22×C8311yescharacteristic
338#484—21no—
339#482—21no—
340#475C22×C8311yescharacteristic
341#474C22×C8311yescharacteristic
342#468—21no—
343#465—21no—
344#464—21no—
345#463—21no—
346#463—21no—
347#462C22×C8311yescharacteristic
348#462C22×C8311yescharacteristic
349#460C22×C8311yescharacteristic
350#459—21no—
351#459C22×C8311yescharacteristic
352#459C22×C8311yescharacteristic
353#459C22×C8311yescharacteristic
354#452—21no—
355#449—21no—
356#446C22×C8311yescharacteristic
357C8×Q8C22×C8311yescharacteristic
358#442—21no—
359C4×SD16—21no—
360C4×SD16C22×C8311yescharacteristic
361C4×SD16C22×C8311yescharacteristic
362#436C22×C8311yescharacteristic
363C8×D4C22×C8311yescharacteristic
364#428—21no—
365#334C22×C8311yescharacteristic
366#334C22×C8311yescharacteristic
367#332C22×C8311yescharacteristic
368#332C22×C8311yescharacteristic
369#331C22×C8311yescharacteristic
370#331C22×C8311yescharacteristic
371Q8⋊C8C22×C8311yescharacteristic
372Q8⋊C8C22×C8311yescharacteristic
373D4⋊C8C22×C8311yescharacteristic
374D4⋊C8C22×C8311yescharacteristic
375C82C22×C8311yescharacteristic
376C2×#41—21no—
377C2×#41—21no—
378#135—44no—
379C2×Q16—21no—
380C2×Q16—21no—
381C2×SD16—21no—
382C2×SD16—21no—
383C2×SD16—21no—
384C2×SD16—21no—
385C2×SD16—21no—
386C2×SD16D4×C8311yescharacteristic
387C2×SD16D4×C8311yescharacteristic
388C2×D8—21no—
389C2×D8—21no—
390C2×D8—21no—
391C2×D8—21no—
392C2×D8—21no—
393C2×M16—21no—
394C22×C8—21no—
395C4⋊Q8C23×C8311yescharacteristic
396C4⋊Q8C23×C8311yescharacteristic
397#127—21no—
398#127C23×C8311yescharacteristic
399#127C23×C8311yescharacteristic
400#126—41no—
401#124—21no—
402#124—21no—
403#124—21no—
404#123—41no—
405#123—41no—
406C22⋊Q8—21no—
407C22⋊Q8—21no—
408#121—21no—
409#121—21no—
410#121—41no—
411C4×Q8—21no—
412C4×Q8C23×C8311yescharacteristic
413C4×Q8C23×C8311yescharacteristic
414C4×D4—21no—
415C4×D4—21no—
416C4×D4—21no—
417C4×D4C23×C8311yescharacteristic
418C4×D4C23×C8311yescharacteristic
419#117—21no—
420C2×C4⋊C4—21no—
421#107—21no—
422#107—21no—
423#107—21no—
424#106—21no—
425#106D4×C8311yescharacteristic
426#106D4×C8311yescharacteristic
427C4⋊C8—21no—
428C4⋊C8—21no—
429C4⋊C8C23×C8311yescharacteristic
430C4⋊C8C23×C8311yescharacteristic
431C4⋊C8C23×C8311yescharacteristic
432C4⋊C8C23×C8311yescharacteristic
433#103—21no—
434#103—21no—
435#103—21no—
436#103D4×C8311yescharacteristic
437#103D4×C8311yescharacteristic
438#102—21no—
439#102—21no—
440#102—21no—
441#102—21no—
442#102—21no—
443#102—41no—
444#102—41no—
445#102D4×C8311yescharacteristic
446#102D4×C8311yescharacteristic
447C22⋊C8—21no—
448C22⋊C8—21no—
449#97—21no—
450C4×C8—21no—
451C4×C8C23×C8311yescharacteristic
452C4×C8C23×C8311yescharacteristic
453C4×C8C23×C8311yescharacteristic
454#41—42no—
455#41—42no—
456#41—81no—
457#41—81no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8C2×D4×C8311yescharacteristic
461C2×Q8C2×D4×C8311yescharacteristic
462C2×D4—21no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—41no—
469C2×D4—41no—
470C2×D4—41no—
471C2×D4C2×D4×C8311yescharacteristic
472C2×D4C2×D4×C8311yescharacteristic
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—41no—
476C22×C4—41no—
477C22×C4—41no—
478Q16—42no—
479Q16—81no—
480SD16—22no—
481SD16—22no—
482SD16—42no—
483SD16—42no—
484SD16—81no—
485SD16—81no—
486SD16—81no—
487D8—42no—
488D8—42no—
489D8—81no—
490D8—81no—
491D8—81no—
492M16—42no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—42no—
499C2×C8C2×D4×C8311yescharacteristic
500C2×C8C2×D4×C8311yescharacteristic
501C2×C8C2×D4×C8311yescharacteristic
502C2×C8C2×D4×C8311yescharacteristic
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—41no—
510C4⋊C4—41no—
511C4⋊C4C2×D4×C8311yescharacteristic
512C4⋊C4C2×D4×C8311yescharacteristic
513C4⋊C4C2×D4×C8311yescharacteristic
514C4⋊C4C2×D4×C8311yescharacteristic
515C22⋊C4—21no—
516C22⋊C4—21no—
517C22⋊C4—41no—
518C22⋊C4—41no—
519C22⋊C4—41no—
520C22⋊C4—41no—
521C22⋊C4—41no—
522C22⋊C4—41no—
523C22⋊C4—41no—
524C42—21no—
525C42—21no—
526C42—41no—
527C42C24×C8311yescharacteristic
528C23—21no—
529C23—21no—
530C23—41no—
531C23—41no—
532Q8—21no—
533Q8—21no—
534Q8—21no—
535Q8—21no—
536Q8—41no—
537Q8—41no—
538Q8—41no—
539Q8—41no—
540D4—21no—
541D4—21no—
542D4—21no—
543D4—21no—
544D4—41no—
545D4—41no—
546D4—41no—
547D4—41no—
548D4—41no—
549D4—41no—
550D4—41no—
551D4—41no—
552D4—81no—
553D4—81no—
554D4—82no—
555D4—82no—
556D4—161no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4—161no—
572C2×C4C22×D4×C8311yescharacteristic
573C2×C4C22×D4×C8311yescharacteristic
574C2×C4H(5,𝔽2)×C8311yescharacteristic
575C8—21no—
576C8—21no—
577C8—41no—
578C8—41no—
579C8—41no—
580C8—41no—
581C8—41no—
582C8#135×C8312yescomplemented
583C8AGL(1,ℤ/8)×C8312yes—
584C22—41no—
585C22—41no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—81no—
591C22—81no—
592C22—81no—
593C22—81no—
594C22—81no—
595C22—81no—
596C22D42×C8311yescharacteristic
597C4—21no—
598C4—41no—
599C4—41no—
600C4—41no—
601C4—41no—
602C4—41no—
603C4—41no—
604C4—81no—
605C4—81no—
606C4C2×#135×C8311yescharacteristic
607C4C2×AGL(1,ℤ/8)×C8311yescharacteristic
608C4#576×C8311yescharacteristic
609C4#577×C8311yescharacteristic
610C2—81no—
611C2—81no—
612C2—161no—
613C2—161no—
614C2C83×#327411yescharacteristic
615C2C83×#328411yescharacteristic
616C2C83×#329011yescharacteristic
subgroupG/H
center, socleC22×C83D42
commutator, Frattini, cosocleC42C24×C83
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C83 ◁ C42×C83 ◁ G

Complements

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

NH ≅ G/Ntype
1. C83×#3399C2semidirect
2. C83×#3394C2semidirect
3. C83×#3353C2semidirect
4. C83×#3284C2semidirect
5. C83×#1706C2semidirect
6. C83×#1694C2semidirect
7. C83×#1687C2semidirect
8. C83×#1666C2semidirect
9. C83×#1665C2semidirect
10. C83×#1641C2semidirect
11. C83×#1640C2semidirect
12. C83×#1628C2semidirect
13. C83×#1625C2semidirect
14. C83×#1590C2semidirect
15. C8×SD16×C83C2semidirect
273. C8×C83#135semidirect
301. #19947C83direct (N × H)
309. #3399C2×C83semidirect
310. #3394C2×C83semidirect
311. #3353C2×C83semidirect
312. #3284C2×C83semidirect
313. #1706C2×C83semidirect
314. #1694C2×C83semidirect
315. #1687C2×C83semidirect
316. #1666C2×C83semidirect
317. #1665C2×C83semidirect
318. #1641C2×C83semidirect
319. #1640C2×C83semidirect
320. #1628C2×C83semidirect
321. #1625C2×C83semidirect
322. #1590C2×C83semidirect
323. C8×SD16C2×C83semidirect
324. C83#19947direct (N × H)
582. C8#135×C83semidirect

Maximal subgroups

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

subgroupindex
C83×#33992
C83×#33942
C83×#33532
C83×#32842
C83×#17062
C83×#16942
C83×#16872
C83×#16662
C83×#16652
C83×#16412
C83×#16402
C83×#16282
C83×#16252
C83×#15902
C8×SD16×C832
#1994783