Finite Group Databasebeta

← #35376135

Subgroups — #35376135

Order 21248.

648 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24916594C211yesmaximal, characteristic, complemented
2#24916591C211yesmaximal, characteristic, complemented
3C4×#14581813C211yesmaximal, characteristic
4#24906375C211yesmaximal, characteristic, complemented
5#24906374C211yesmaximal, characteristic, complemented
6#24906308C211yesmaximal, characteristic, complemented
7#24906307C211yesmaximal, characteristic, complemented
8#24906305C211yesmaximal, characteristic, complemented
9#24906280C211yesmaximal, characteristic, complemented
10#24906279C211yesmaximal, characteristic, complemented
11#24906217C211yesmaximal, characteristic, complemented
12#24905901C211yesmaximal, characteristic, complemented
13#24905899C211yesmaximal, characteristic, complemented
14#24905751C211yesmaximal, characteristic, complemented
15C83×#1583C211yesmaximal, characteristic, complemented
16C2×#14581813C411yescharacteristic
17C4×#13050591C2211yescharacteristic
18C4×#13050589C2211yescharacteristic
19Q8×C83⋊C8C2211yescharacteristic
20C8×Dic166C2211yescharacteristic
21#19127825C2211yescharacteristic
22C4×#13050580C2211yescharacteristic
23#19127793C411yescharacteristic
24C4×#13050579C2211yescharacteristic
25SD16×Dic83C411yescharacteristic
26#19127788C411yescharacteristic
27C4×#13050575C2211yescharacteristic
28#19127781C411yescharacteristic
29#19127780—21no—
30#19127780C2211yescharacteristic
31#19127779C2211yescharacteristic
32#19127778—21no—
33#19127778C2211yescharacteristic
34#19127777—21no—
35#19127776—21no—
36#19127752—21no—
37#19127752C2211yescharacteristic
38#19127751—21no—
39#19127736C411yescharacteristic
40#19127724—21no—
41#19127723—21no—
42#19127723C2211yescharacteristic
43#19127723C2211yescharacteristic
44#19127696C411yescharacteristic
45#19127694C411yescharacteristic
46C4×#13050573C2211yescharacteristic
47C4×C8×D83C2211yescharacteristic
48#19127452C2211yescharacteristic
49#19127451C2211yescharacteristic
50#19127451C2211yescharacteristic
51#19127433C2211yescharacteristic
52Q8⋊(C83⋊C8)C2211yescharacteristic
53#19127431C2211yescharacteristic
54#19127430C2211yescharacteristic
55D4⋊(C83⋊C8)C2211yescharacteristic
56#19127428C2211yescharacteristic
57#19127423C2211yescharacteristic
58#19127423C2211yescharacteristic
59#19127423C2211yescharacteristic
60C8×Q8×C83C2211yescharacteristic
61C4×SD16×C83C2211yescharacteristic
62#436×C83C2211yescharacteristic
63#334×C83C2211yescharacteristic
64Q8⋊C8×C83C2211yescharacteristic
65D4⋊C8×C83C2211yescharacteristic
66#322×C83C2211yescharacteristic
67C2×#13050591C2×C411yescharacteristic
68C2×#13050589C2×C411yescharacteristic
69#14581813—24no—
70C2×#13050580C2×C411yescharacteristic
71C2×#13050579C2×C411yescharacteristic
72C2×#13050575C2×C411yescharacteristic
73C2×#13050574—21no—
74C2×#13050573—21no—
75C2×#13050573—21no—
76C2×#13050573—21no—
77C2×#13050573—21no—
78C2×#13050573—21no—
79C2×C8×D83C2×C411yescharacteristic
80C2×C8×D83C2×C411yescharacteristic
81Q8×Dic83C2×C411yescharacteristic
82C4×Dic166C2311yescharacteristic
83#14581757C2×C411yescharacteristic
84C4×D332—21no—
85C4×D332C2311yescharacteristic
86D4×Dic83C2×C411yescharacteristic
87#14581754—21no—
88#14581754C2×C411yescharacteristic
89C4×C22⋊D83—21no—
90(C22⋊D83)⋊C4—21no—
91#14581750—21no—
92#14581750C2×C411yescharacteristic
93#14581750C2×C411yescharacteristic
94#14581748—21no—
95C2×C4×Dic83—21no—
96C42×D83—21no—
97C42×D83C2311yescharacteristic
98#14581725C2×C411yescharacteristic
99#14581725C2×C411yescharacteristic
100#14581724C2×C411yescharacteristic
101C4⋊(C83⋊C8)—21no—
102C4⋊(C83⋊C8)C2311yescharacteristic
103C4⋊(C83⋊C8)C2311yescharacteristic
104Dic83⋊C8—21no—
105Dic83⋊C8—21no—
106Dic83⋊C8—21no—
107Dic83⋊C8—21no—
108Dic83⋊C8D411yescharacteristic
109Dic83⋊C8D411yescharacteristic
110#14581717C2×C411yescharacteristic
111#14581716C2×C411yescharacteristic
112#14581715C2×C411yescharacteristic
113#14581714C2×C411yescharacteristic
114#14581713C2×C411yescharacteristic
115#14581712C2×C411yescharacteristic
116C22⋊(C83⋊C8)—21no—
117(C2×D83)⋊C8—21no—
118(C2×D83)⋊C8—21no—
119(C2×D83)⋊C8D411yescharacteristic
120(C2×D83)⋊C8D411yescharacteristic
121#14581702—21no—
122C4×C83⋊C8—21no—
123C4×C83⋊C8C2311yescharacteristic
124C4×C83⋊C8C2311yescharacteristic
125C4×C83⋊C8C2311yescharacteristic
126C8×Dic83—21no—
127C8×Dic83—21no—
128C8×Dic83C2×C411yescharacteristic
129C8×Dic83C2×C411yescharacteristic
130C2×SD16×C83C2×C411yescharacteristic
131C2×M16×C83—21no—
132C4×Q8×C83C2311yescharacteristic
133C4×D4×C83C2311yescharacteristic
134#106×C83C2×C411yescharacteristic
135C4⋊C8×C83—21no—
136C4⋊C8×C83C2311yescharacteristic
137C4⋊C8×C83C2311yescharacteristic
138#103×C83C2×C411yescharacteristic
139#102×C83C2×C411yescharacteristic
140C22⋊C8×C83—21no—
141C4×C8×C83—21no—
142C4×C8×C83C2311yescharacteristic
143C4×C8×C83C2311yescharacteristic
144C4×C8×C83C2311yescharacteristic
145#13050591—22no—
146#13050591—41no—
147#13050589—22no—
148#13050589—41no—
149C2×Dic166C22×C411yescharacteristic
150C2×D332—21no—
151C2×D332C22×C411yescharacteristic
152C2×C22⋊D83—21no—
153C22×Dic83—21no—
154C22×Dic83—21no—
155C2×C4×D83—21no—
156C2×C4×D83—21no—
157C2×C4×D83—21no—
158C2×C4×D83C22×C411yescharacteristic
159C2×C4×D83C2×D411yescharacteristic
160C2×C4×D83#4111yescharacteristic
161#13050580M1614yescomplemented
162#13050579—22no—
163#13050575—22no—
164#13050574—81no—
165#13050573—42no—
166#13050573—81no—
167#13050573—81no—
168#13050573—81no—
169#13050573—81no—
170C2×C83⋊C8—21no—
171C2×C83⋊C8—21no—
172C2×C83⋊C8—21no—
173C2×C83⋊C8C22×C411yescharacteristic
174C2×C83⋊C8C22×C411yescharacteristic
175C2×C83⋊C8C2×D411yescharacteristic
176C2×C83⋊C8C2×D411yescharacteristic
177C8×D83—22no—
178C8×D83—41no—
179C4⋊Dic83—21no—
180C4⋊Dic83C22×C411yescharacteristic
181C4⋊Dic83C22×C411yescharacteristic
182Dic83⋊C4—21no—
183Dic83⋊C4—21no—
184C22⋊Dic83—21no—
185(C2×D83)⋊C4—21no—
186(C2×D83)⋊C4—21no—
187(C2×D83)⋊C4—21no—
188(C2×D83)⋊C4—21no—
189C4×Dic83—21no—
190C4×Dic83—21no—
191C4×Dic83—21no—
192C4×Dic83—21no—
193C4×Dic83—21no—
194C4×Dic83C22×C411yescharacteristic
195C4×Dic83C2×D411yescharacteristic
196C4×Dic83#4111yescharacteristic
197C2×Q8×C83C22×C411yescharacteristic
198C2×D4×C83C22×C411yescharacteristic
199C22×C4×C83—21no—
200SD16×C83—22no—
201M16×C83—81no—
202C2×C8×C83—21no—
203C2×C8×C83—21no—
204C2×C8×C83—21no—
205C2×C8×C83C22×C411yescharacteristic
206C2×C8×C83C22×C411yescharacteristic
207C2×C8×C83C2×D411yescharacteristic
208C2×C8×C83C2×D411yescharacteristic
209C4⋊C4×C83—21no—
210C4⋊C4×C83C22×C411yescharacteristic
211C4⋊C4×C83C22×C411yescharacteristic
212C22⋊C4×C83—21no—
213C42×C83—21no—
214C42×C83C2411yescharacteristic
215C23×C83—21no—
216C22×D83—21no—
217C22×D83C4×D411yescharacteristic
218Q8×C83—21no—
219Q8×C83C2×M1612yes—
220D4×C83—21no—
221D4×C83—21no—
222C2×C4×C83—21no—
223C2×C4×C83—21no—
224C2×C4×C83—21no—
225C2×C4×C83—41no—
226C2×C4×C83—41no—
227C2×C4×C83C23×C411yescharacteristic
228C2×C4×C83C22×D411yescharacteristic
229C2×C4×C83C2×#4111yescharacteristic
230C22⋊D83—42no—
231C2×Dic83—21no—
232C2×Dic83—21no—
233C2×Dic83—21no—
234C2×Dic83—42no—
235C2×Dic83—42no—
236C2×Dic83C4×D411yescharacteristic
237C2×Dic83C4×D411yescharacteristic
238C2×Dic83C4×D411yescharacteristic
239D332—21no—
240D332—21no—
241D332—42no—
242C4×D83—21no—
243C4×D83—21no—
244C4×D83—21no—
245C4×D83—21no—
246C4×D83—22no—
247C4×D83—42no—
248C4×D83—41no—
249C4×D83—41no—
250C4×D83—42no—
251Dic166—21no—
252Dic166C2×M1612yes—
253C8×C83—21no—
254C8×C83—21no—
255C8×C83—21no—
256C8×C83—21no—
257C8×C83—41no—
258C8×C83—41no—
259C8×C83—41no—
260C83⋊C8—21no—
261C83⋊C8—21no—
262C83⋊C8—21no—
263C83⋊C8—41no—
264C83⋊C8—41no—
265C83⋊C8—41no—
266C83⋊C8C2×M1612yes—
267C22×C83—41no—
268C22×C83—41no—
269C22×C83—41no—
270C22×C83C2×C4×D411yescharacteristic
271C2×D83—21no—
272C2×D83—21no—
273C2×D83—21no—
274C2×D83—41no—
275C2×D83—41no—
276C2×D83—41no—
277C4×C83—22no—
278C4×C83—21no—
279C4×C83—41no—
280C4×C83—41no—
281C4×C83C22×M1611yescharacteristic
282C4×C83#56811yescharacteristic
283C4×C83#57711yescharacteristic
284C4×C83#57711yescharacteristic
285Dic83—22no—
286Dic83—21no—
287Dic83—21no—
288Dic83—41no—
289Dic83—41no—
290Dic83#44412yes—
291#18063—831nomaximal, Sylow
292C2×C83—81no—
293C2×C83D4×M1611yescharacteristic
294C2×C83#295211yescharacteristic
295C2×C83C2×#44411yescharacteristic
296D83—41no—
297D83—81no—
298#2991—831no—
299#2991—831no—
300C4×#135—831no—
301#1594—831no—
302#1594—831no—
303#1583—831no—
304#1583—831no—
305#1583—831no—
306#1583D8311yescharacteristic, complemented
307#1579—831no—
308#1579—831no—
309#1566—831no—
310#1489—831no—
311#1489—831no—
312#1451—831no—
313C83#1806311yescharacteristic, complemented, Sylow
314C2×#135—831no—
315C4×#41—831no—
316C4×#41—831no—
317C8×Q8—831no—
318C8×Q8—831no—
319C8×Q8—831no—
320C8×Q8C2×D8311yescharacteristic
321C4×Q16—831no—
322C4×Q16—831no—
323C4×SD16—831no—
324C4×SD16—831no—
325C4×SD16—831no—
326C4×SD16C2×D8311yescharacteristic
327C4×D8—831no—
328C4×D8—831no—
329#436—831no—
330#436—831no—
331#436—831no—
332#436—1661no—
333#436—1661no—
334#436—1661no—
335#436—1661no—
336#436C2×D8311yescharacteristic
337#431—831no—
338#431—1661no—
339#431—1661no—
340#427—831no—
341#423—831no—
342#423—831no—
343#423—1661no—
344#423—1661no—
345#416—831no—
346#416—831no—
347C4×M16—831no—
348C2×C4×C8—831no—
349#334—831no—
350#334—831no—
351#334—831no—
352#334C2×D8311yescharacteristic
353Q8⋊C8—831no—
354Q8⋊C8—831no—
355Q8⋊C8—831no—
356Q8⋊C8C2×D8311yescharacteristic
357D4⋊C8—831no—
358D4⋊C8—831no—
359D4⋊C8—831no—
360D4⋊C8C2×D8311yescharacteristic
361#322—831no—
362#322—831no—
363#322—831no—
364#322C2×D8311yescharacteristic
365C2×#41—831no—
366C2×#41—831no—
367#135—1664no—
368C2×Q16—831no—
369C2×SD16—831no—
370C2×SD16C4×D8311yescharacteristic
371C2×D8—831no—
372C2×M16—21no—
373C2×M16—1661no—
374C2×M16—1661no—
375C2×M16—1661no—
376C2×M16—1661no—
377C2×M16—1661no—
378C2×M16—1661no—
379C22×C8—831no—
380C22×C8—831no—
381C4×Q8—831no—
382C4×Q8—831no—
383C4×Q8—831no—
384C4×Q8C22×D8311yescharacteristic
385C4×D4—831no—
386C4×D4—831no—
387C4×D4—831no—
388C4×D4—1661no—
389C4×D4—1661no—
390C4×D4—1661no—
391C4×D4—1661no—
392C4×D4C22×D8311yescharacteristic
393#117—831no—
394#117—831no—
395#117—1661no—
396#117—1661no—
397C2×C42—831no—
398C2×C42—1661no—
399C2×C42—1661no—
400#107—831no—
401#107—831no—
402#106—831no—
403#106C4×D8311yescharacteristic
404C4⋊C8—21no—
405C4⋊C8—831no—
406C4⋊C8—831no—
407C4⋊C8—831no—
408C4⋊C8—831no—
409C4⋊C8—1661no—
410C4⋊C8—1661no—
411C4⋊C8—1661no—
412C4⋊C8—1661no—
413C4⋊C8—1661no—
414C4⋊C8C22×D8311yescharacteristic
415C4⋊C8C22×D8311yescharacteristic
416#103—831no—
417#103—831no—
418#103—831no—
419#103C4×D8311yescharacteristic
420#102—831no—
421#102—831no—
422#102—831no—
423#102C4×D8311yescharacteristic
424C22⋊C8—21no—
425C22⋊C8—831no—
426C22⋊C8—831no—
427C22⋊C8—1661no—
428C22⋊C8—1661no—
429C22⋊C8—1661no—
430#97—1661no—
431C4×C8—21no—
432C4×C8—831no—
433C4×C8—831no—
434C4×C8—831no—
435C4×C8—831no—
436C4×C8—831no—
437C4×C8—1661no—
438C4×C8—1661no—
439C4×C8—1661no—
440C4×C8C22×D8311yescharacteristic
441C4×C8C22×D8311yescharacteristic
442C4×C8C22×D8311yescharacteristic
443#41—3321no—
444#41—3321no—
445#41—1662no—
446#41—1662no—
447C2×Q8—831no—
448C2×Q8C2×C4×D8311yescharacteristic
449C2×D4—831no—
450C2×D4—1661no—
451C2×D4—1661no—
452C2×D4C2×C4×D8311yescharacteristic
453C22×C4—21no—
454C22×C4—831no—
455C22×C4—831no—
456C22×C4—831no—
457C22×C4—1661no—
458C22×C4—1661no—
459C22×C4—1661no—
460C22×C4—1661no—
461C22×C4—1661no—
462Q16—834no—
463SD16—22no—
464SD16—1662no—
465D8—1662no—
466M16—81no—
467M16—3322no—
468M16—6641no—
469M16—6641no—
470M16—6641no—
471M16—6641no—
472M16—6641no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—3321no—
477C2×C8—831no—
478C2×C8—831no—
479C2×C8—831no—
480C2×C8—831no—
481C2×C8—1662no—
482C2×C8—1661no—
483C2×C8—1661no—
484C2×C8—1661no—
485C2×C8C2×C4×D8311yescharacteristic
486C2×C8C2×C4×D8311yescharacteristic
487C2×C8D4×D8311yescharacteristic
488C2×C8D4×D8311yescharacteristic
489C4⋊C4—21no—
490C4⋊C4—831no—
491C4⋊C4—831no—
492C4⋊C4—1661no—
493C4⋊C4—1661no—
494C4⋊C4—1661no—
495C4⋊C4C2×C4×D8311yescharacteristic
496C4⋊C4C2×C4×D8311yescharacteristic
497C22⋊C4—21no—
498C22⋊C4—1661no—
499C22⋊C4—1661no—
500C22⋊C4—1661no—
501C22⋊C4—1661no—
502C22⋊C4—1661no—
503C42—21no—
504C42—831no—
505C42—831no—
506C42—831no—
507C42—1661no—
508C42—1661no—
509C42—1661no—
510C42—1661no—
511C42—1661no—
512C42C23×D8311yescharacteristic
513C23—21no—
514C23—831no—
515C23—1661no—
516Q8—21no—
517Q8—832no—
518Q8—1661no—
519Q8M16×D8312yes—
520D4—21no—
521D4—21no—
522D4—3322no—
523D4—3322no—
524D4—1661no—
525D4—1661no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—3322no—
532C2×C4—3322no—
533C2×C4—3322no—
534C2×C4—3321no—
535C2×C4—3321no—
536C2×C4—3322no—
537C2×C4—831no—
538C2×C4—831no—
539C2×C4—831no—
540C2×C4—1662no—
541C2×C4—1661no—
542C2×C4—1661no—
543C2×C4—1661no—
544C2×C4—1661no—
545C2×C4—1661no—
546C2×C4—1661no—
547C2×C4—1661no—
548C2×C4C22×C4×D8311yescharacteristic
549C2×C4C2×D4×D8311yescharacteristic
550C2×C4#41×D8311yescharacteristic
551C8—21no—
552C8—21no—
553C8—21no—
554C8—21no—
555C8—41no—
556C8—41no—
557C8—41no—
558C8—3321no—
559C8—3321no—
560C8—3321no—
561C8—832no—
562C8—1661no—
563C8—1661no—
564C8—1661no—
565C22—41no—
566C22—41no—
567C22—41no—
568C22—3321no—
569C22—3321no—
570C22—3321no—
571C22—1661no—
572C22—1661no—
573C22—1661no—
574C22C4×D4×D8311yescharacteristic
575C4—21no—
576C4—22no—
577C4—41no—
578C4—41no—
579C4—3321no—
580C4—3321no—
581C4—832no—
582C4—1661no—
583C4—1661no—
584C4—1662no—
585C4C2×M16×D8311yescharacteristic
586C4#1912846611yescharacteristic
587C4#1912850711yescharacteristic
588C4#1912850811yescharacteristic
589C2—81no—
590C2—3321no—
591C2—6641no—
592C2#2491619611yescharacteristic
593C2#2491634111yescharacteristic
594C2#444×D8311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC42C23×D83
FittingC83×#1583C2
socleC22×C83C2×C4×D4
prime radicalC2×#14581813C4
cosocleC42×C83C24
hypercenter#1583D83
hypocenterC83#18063
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1583

Complements

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

NH ≅ G/Ntype
1. #24916594C2semidirect
2. #24916591C2semidirect
4. #24906375C2semidirect
5. #24906374C2semidirect
6. #24906308C2semidirect
7. #24906307C2semidirect
8. #24906305C2semidirect
9. #24906280C2semidirect
10. #24906279C2semidirect
11. #24906217C2semidirect
12. #24905901C2semidirect
13. #24905899C2semidirect
14. #24905751C2semidirect
15. C83×#1583C2semidirect
161. #13050580M16semidirect
306. #1583D83semidirect
313. C83#18063semidirect

Maximal subgroups

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

subgroupindex
#249165942
#249165912
C4×#145818132
#249063752
#249063742
#249063082
#249063072
#249063052
#249062802
#249062792
#249062172
#249059012
#249058992
#249057512
C83×#15832
#1806383