Finite Group Databasebeta

← #35408324

Subgroups — #35408324

Order 21248.

1782 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14581818C211yesmaximal, characteristic
2C2×#19127741C211yesmaximal, characteristic, complemented
3C2×#19127700C211yesmaximal, characteristic, complemented
4C2×#19127706C211yesmaximal, characteristic, complemented
5C2×C4×#13050574C211yesmaximal, characteristic, complemented
6#24909953C211yesmaximal, characteristic, complemented
7#24908343C212yesmaximal, complemented
8#24908338C212yesmaximal, complemented
9#24908328C212yesmaximal, complemented
10#24908316C212yesmaximal, complemented
11C2×#390×C83C211yesmaximal, characteristic, complemented
12C22×#13050590C2211yescharacteristic
13C2×#14581818—24no—
14C2×#14581818—24no—
15C2×#14581818C418yescomplemented
16C22×#13050578C2212yes—
17C22×#13050575C2212yes—
18C22×#13050574C2211yescharacteristic
19C22×#13050574C2211yescharacteristic
20C2×#14581751C2211yescharacteristic
21#19127741—24no—
22C2×#14581725C2212yescomplemented
23C2×#14581723C2212yescomplemented
24#19127706—24no—
25#19127700—24no—
26C2×#14581715C2212yescomplemented
27C2×#14581713C2212yes—
28C2×#14581713C2212yes—
29C2×#14581712C2212yescomplemented
30C4×#13050574—24no—
31C2×#14581703C2212yescomplemented
32C2×C4×C83⋊C8C2212yescomplemented
33#19127607C2211yescharacteristic, complemented
34#19127607C2211yescharacteristic, complemented
35#19127603C2211yescharacteristic, complemented
36#19127603C2211yescharacteristic, complemented
37#19127580—22no—
38C4×(C2×D83)⋊C4—22no—
39C23×D4×C83C2211yescharacteristic
40C22×C4⋊C4×C83C2211yescharacteristic
41C22×C22⋊C4×C83—22no—
42C22×C42×C83C2211yescharacteristic
43#390×C83—22no—
44#390×C83C2212yes—
45#390×C83C2212yes—
46C2×#13050590—22no—
47C2×#13050590—22no—
48C2×#13050590—22no—
49C2×#13050590—22no—
50C2×#13050590—22no—
51C2×#13050590C2×C414yes—
52C22×Dic166C2311yescharacteristic
53C22×D332C2311yescharacteristic
54C22×C22⋊D83—21no—
55C22×C4×D83—21no—
56#14581818—232no—
57C2×#13050578—24no—
58C2×#13050578—24no—
59C2×#13050578C2×C418yescomplemented
60C2×#13050575—24no—
61C2×#13050575—24no—
62C2×#13050575C2×C418yescomplemented
63C2×#13050574—22no—
64C2×#13050574—22no—
65C2×#13050574C2×C414yes—
66C2×#13050574D414yes—
67C2×#13050574D414yes—
68C2×#13050574D414yes—
69C22×C83⋊C8C2312yes—
70C22×C83⋊C8C2312yes—
71#14581751—22no—
72#14581751—22no—
73C2×C4⋊Dic83C2311yescharacteristic
74C2×C4⋊Dic83C2311yescharacteristic
75C2×C22⋊Dic83—21no—
76C2×(C2×D83)⋊C4—22no—
77C2×(C2×D83)⋊C4—22no—
78C2×C4×Dic83—21no—
79#14581725—24no—
80#14581723—24no—
81#14581715—24no—
82#14581713—24no—
83#14581713—24no—
84#14581712—24no—
85#14581703—24no—
86C4×C83⋊C8—24no—
87C83⋊#95—22no—
88C83⋊#95—22no—
89C25×C83—21no—
90C22×D4×C83—22no—
91C22×D4×C83—22no—
92C22×D4×C83—22no—
93C22×D4×C83—21no—
94C22×D4×C83—22no—
95C22×D4×C83—22no—
96C22×D4×C83C2×C414yes—
97C22×D4×C83C2312yes—
98C23×C4×C83—21no—
99C23×C4×C83D412yes—
100C23×C4×C83C2311yescharacteristic
101C2×C4⋊C4×C83—22no—
102C2×C4⋊C4×C83—22no—
103C2×C4⋊C4×C83—21no—
104C2×C4⋊C4×C83C2312yes—
105C2×C22⋊C4×C83—24no—
106C2×C22⋊C4×C83—24no—
107C2×C22⋊C4×C83—48no—
108C2×C42×C83—22no—
109C2×C42×C83—22no—
110C2×C42×C83D412yes—
111C2×C42×C83C2311yescharacteristic
112C2×C42×C83C2311yescharacteristic
113C23×D83—21no—
114#13050590—28no—
115#13050590—28no—
116#13050590—44no—
117#13050590—44no—
118C2×Dic166—21no—
119C2×Dic166—21no—
120C2×Dic166—21no—
121C2×Dic166—21no—
122C2×Dic166—21no—
123C2×Dic166C22×C412yes—
124C2×D332—21no—
125C2×D332—21no—
126C2×D332—21no—
127C2×D332—21no—
128C2×D332—21no—
129C2×D332C22×C412yes—
130C2×C22⋊D83—24no—
131C2×C22⋊D83—42no—
132C2×C22⋊D83—42no—
133C22×Dic83—21no—
134C22×Dic83—21no—
135C22×Dic83—21no—
136C2×C4×D83—24no—
137C2×C4×D83—42no—
138C2×C4×D83—42no—
139#13050578—216no—
140#13050575—216no—
141#13050574—28no—
142#13050574C22⋊C4116yescomplemented
143C2×C83⋊C8—22no—
144C2×C83⋊C8—22no—
145C2×C83⋊C8C22×C414yes—
146C2×C83⋊C8C2×D414yes—
147C2×C83⋊C8C2×D414yes—
148C2×C83⋊C8C2×D414yes—
149C4⋊Dic83—21no—
150C4⋊Dic83—21no—
151C4⋊Dic83—21no—
152C4⋊Dic83—21no—
153C22⋊Dic83—42no—
154(C2×D83)⋊C4—44no—
155(C2×D83)⋊C4—44no—
156C4×Dic83—42no—
157C24×C83—24no—
158C24×C83—22no—
159C24×C83—44no—
160C24×C83—44no—
161C24×C83—42no—
162C24×C83—42no—
163C24×C83C2×D411yescharacteristic
164C2×D4×C83—28no—
165C2×D4×C83—28no—
166C2×D4×C83—24no—
167C2×D4×C83—22no—
168C2×D4×C83—22no—
169C2×D4×C83—22no—
170C2×D4×C83—22no—
171C2×D4×C83—22no—
172C2×D4×C83—42no—
173C2×D4×C83—44no—
174C2×D4×C83—44no—
175C2×D4×C83—42no—
176C2×D4×C83C22×C414yes—
177C22×C4×C83—24no—
178C22×C4×C83—24no—
179C22×C4×C83—22no—
180C22×C4×C83—42no—
181C22×C4×C83—44no—
182C22×C4×C83—42no—
183C22×C4×C83C22×C412yes—
184C22×C4×C83C2×D412yes—
185C22×C4×C83C2×D412yes—
186C22×C4×C83C2×D412yes—
187C22×C4×C83C2×D412yes—
188C22×C4×C83C2×D412yes—
189C22×C4×C83C2×D411yescharacteristic
190C22×C4×C83#4112yes—
191C22×C4×C83#4112yes—
192C22×C4×C83C2411yescharacteristic
193C4⋊C4×C83—22no—
194C4⋊C4×C83—22no—
195C4⋊C4×C83—42no—
196C22⋊C4×C83—48no—
197C22⋊C4×C83—48no—
198C42×C83—24no—
199C42×C83—21no—
200C42×C83—21no—
201C42×C83—21no—
202C42×C83—21no—
203C23×C83—22no—
204C23×C83—22no—
205C23×C83—24no—
206C23×C83—416no—
207C23×C83—42no—
208C23×C83—42no—
209C23×C83—44no—
210C23×C83—44no—
211C23×C83—48no—
212C23×C83—84no—
213C23×C83—84no—
214C23×C83—88no—
215C23×C83C4×D412yes—
216C23×C83#12112yes—
217C23×C83#12712yes—
218C23×C83C22×D411yescharacteristic
219C22×D83—22no—
220C22×D83—41no—
221C22×D83—41no—
222C22×D83—42no—
223C22×D83—42no—
224D4×C83—24no—
225D4×C83—24no—
226D4×C83—42no—
227D4×C83—42no—
228D4×C83—48no—
229C2×C4×C83—22no—
230C2×C4×C83—22no—
231C2×C4×C83—22no—
232C2×C4×C83—22no—
233C2×C4×C83—22no—
234C2×C4×C83—48no—
235C2×C4×C83—42no—
236C2×C4×C83—44no—
237C2×C4×C83—84no—
238C2×C4×C83C2×C22⋊C414yes—
239C2×C4×C83C4×D414yes—
240C2×C4×C83C4×D412yes—
241C2×C4×C83#12112yes—
242C2×C4×C83#12114yes—
243C2×C4×C83#12414yes—
244C2×C4×C83#12712yes—
245C2×C4×C83AGL(1,ℤ/8)14yescomplemented
246C2×C4×C83C23×C411yescharacteristic
247C2×C4×C83C22×D411yescharacteristic
248C2×C4×C83C22×D411yescharacteristic
249C2×C4×C83C22×D411yescharacteristic
250C2×C4×C83C2×#4111yescharacteristic
251C2×C4×C83C2×#4111yescharacteristic
252C22⋊D83—48no—
253C2×Dic83—22no—
254C2×Dic83—41no—
255C2×Dic83—41no—
256C2×Dic83—41no—
257C2×Dic83—41no—
258C2×Dic83—42no—
259C2×Dic83—42no—
260D332—22no—
261D332—22no—
262D332—41no—
263D332—41no—
264C4×D83—48no—
265Dic166—22no—
266Dic166—22no—
267Dic166—41no—
268Dic166—41no—
269C83⋊C8—24no—
270C83⋊C8C2×C22⋊C418yescomplemented
271C22×C83—22no—
272C22×C83—22no—
273C22×C83—24no—
274C22×C83—24no—
275C22×C83—48no—
276C22×C83—44no—
277C22×C83—816no—
278C22×C83—82no—
279C22×C83—82no—
280C22×C83—84no—
281C22×C83#39014yes—
282C22×C83C2×C4×D411yescharacteristic
283C22×C83C2×#12111yescharacteristic
284C22×C83C2×#12711yescharacteristic
285C22×C83C2×AGL(1,ℤ/8)12yes—
286C22×C83#57512yes—
287C2×D83—44no—
288C2×D83—42no—
289C2×D83—81no—
290C2×D83—81no—
291C2×D83—82no—
292C4×C83—24no—
293C4×C83—42no—
294C4×C83—82no—
295C4×C83#39014yes—
296C4×C83C22×C22⋊C411yescharacteristic
297C4×C83C2×C4×D411yescharacteristic
298C4×C83C2×#12111yescharacteristic
299C4×C83C2×#12411yescharacteristic
300Dic83—42no—
301Dic83—81no—
302Dic83—81no—
303#21236—831nomaximal, Sylow
304C2×C83—24no—
305C2×C83—84no—
306C2×C83C2×#39011yescharacteristic
307C2×C83C4×AGL(1,ℤ/8)12yes—
308C2×C83#305512yes—
309C2×C83#315111yescharacteristic
310C2×C83#315311yescharacteristic
311D83—82no—
312C22×AGL(1,ℤ/8)—831no—
313C2×#428—831no—
314C2×#417—831no—
315C2×#418—831no—
316C2×C4×M16—831no—
317#2297—831no—
318C2×#390D8311yescharacteristic, complemented
319#1973—832no—
320#1972—832no—
321#1970—832no—
322#1968—832no—
323C83#2123611yescharacteristic, complemented, Sylow
324C22×#41—831no—
325C23×D4C2×D8311yescharacteristic
326C2×AGL(1,ℤ/8)—838no—
327C2×AGL(1,ℤ/8)—1664no—
328C2×AGL(1,ℤ/8)—1664no—
329C22×SD16—832no—
330C22×D8—832no—
331C22×M16—831no—
332C22×M16—831no—
333C2×#117—831no—
334C22×C4⋊C4C2×D8311yescharacteristic
335C22×C22⋊C4—22no—
336C22×C42C2×D8311yescharacteristic
337#428—1664no—
338C2×#107—832no—
339C2×#106—832no—
340#418—1664no—
341#417—1664no—
342C2×#103—832no—
343C2×#102—832no—
344C2×#102—832no—
345C2×#102—832no—
346C4×M16—1664no—
347C2×#97—832no—
348C2×C4×C8—832no—
349#391—831no—
350#391—831no—
351#390—22no—
352#390—831no—
353#390—831no—
354#390C2×D8312yes—
355#390C2×D8312yes—
356#386—1662no—
357C4×C22⋊C4—1662no—
358C25—21no—
359C2×#41—834no—
360C2×#41—1662no—
361C2×#41—1662no—
362C2×#41—1662no—
363C2×#41—1662no—
364C2×#41—1662no—
365C22×Q8—831no—
366C22×D4—22no—
367C22×D4—22no—
368C22×D4—22no—
369C22×D4—21no—
370C22×D4—22no—
371C22×D4—22no—
372C22×D4—831no—
373C22×D4—1661no—
374C22×D4C4×D8314yes—
375C22×D4C22×D8312yes—
376C23×C4—21no—
377C23×C4—1661no—
378C23×C4C22⋊D8312yes—
379C23×C4C22×D8311yescharacteristic
380AGL(1,ℤ/8)—16632no—
381C2×SD16—838no—
382C2×SD16—1664no—
383C2×SD16—1664no—
384C2×D8—838no—
385C2×D8—1664no—
386C2×D8—1664no—
387C2×M16—834no—
388C2×M16—834no—
389C2×M16—834no—
390C2×M16—834no—
391C2×M16—1662no—
392C2×M16—1662no—
393C22×C8—832no—
394C22×C8—832no—
395#117—1662no—
396#117—1662no—
397C2×C4⋊C4—22no—
398C2×C4⋊C4—22no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—831no—
401C2×C4⋊C4—831no—
402C2×C4⋊C4C22×D8312yes—
403C2×C22⋊C4—24no—
404C2×C22⋊C4—24no—
405C2×C22⋊C4—48no—
406C2×C22⋊C4—1661no—
407C2×C22⋊C4—1662no—
408C2×C22⋊C4—1662no—
409C2×C42—22no—
410C2×C42—22no—
411C2×C42—1661no—
412C2×C42C22⋊D8312yes—
413C2×C42C22×D8311yescharacteristic
414C2×C42C22×D8311yescharacteristic
415#107—1664no—
416#106—1664no—
417#103—1664no—
418#102—1664no—
419#102—1664no—
420#102—1664no—
421#97—1664no—
422C4×C8—1664no—
423#95—1662no—
424#95—1662no—
425C24—22no—
426C24—24no—
427C24—44no—
428C24—44no—
429C24—42no—
430C24—42no—
431C24—1661no—
432C24C2×C22⋊D8311yescharacteristic
433#41—3324no—
434#41—3324no—
435#41—1668no—
436#41—1668no—
437C2×Q8—832no—
438C2×Q8—1661no—
439C2×Q8—1661no—
440C2×Q8—1661no—
441C2×Q8—1661no—
442C2×Q8—1661no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—28no—
449C2×D4—28no—
450C2×D4—24no—
451C2×D4—42no—
452C2×D4—44no—
453C2×D4—44no—
454C2×D4—42no—
455C2×D4—3322no—
456C2×D4—3322no—
457C2×D4—832no—
458C2×D4—1664no—
459C2×D4—1661no—
460C2×D4—1661no—
461C2×D4—1661no—
462C2×D4—1661no—
463C2×D4—1661no—
464C2×D4C2×C4×D8314yes—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—22no—
468C22×C4—42no—
469C22×C4—44no—
470C22×C4—42no—
471C22×C4—3322no—
472C22×C4—3322no—
473C22×C4—1661no—
474C22×C4—1661no—
475C22×C4—1661no—
476C22×C4—1664no—
477C22×C4C2×C4×D8312yes—
478C22×C4D4×D8312yes—
479C22×C4D4×D8312yes—
480C22×C4D4×D8312yes—
481C22×C4C2×C22⋊D8312yes—
482C22×C4C2×C22⋊D8312yes—
483C22×C4C2×C22⋊D8311yescharacteristic
484C22×C4#1305058912yes—
485C22×C4#1305059012yes—
486C22×C4C23×D8311yescharacteristic
487SD16—16616no—
488D8—16616no—
489M16—8316no—
490M16—1668no—
491C2×C8—834no—
492C2×C8—834no—
493C2×C8—834no—
494C2×C8—834no—
495C2×C8—1662no—
496C2×C8—1662no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—42no—
500C4⋊C4—1661no—
501C4⋊C4—1661no—
502C4⋊C4—1661no—
503C4⋊C4—1661no—
504C22⋊C4—48no—
505C22⋊C4—48no—
506C22⋊C4—3324no—
507C22⋊C4—3322no—
508C22⋊C4—3324no—
509C42—24no—
510C42—21no—
511C42—21no—
512C42—21no—
513C42—21no—
514C42—3322no—
515C23—22no—
516C23—22no—
517C23—24no—
518C23—416no—
519C23—48no—
520C23—42no—
521C23—44no—
522C23—42no—
523C23—44no—
524C23—88no—
525C23—84no—
526C23—84no—
527C23—3321no—
528C23—3322no—
529C23—3321no—
530C23—3322no—
531C23—1662no—
532C23C4×C22⋊D8312yes—
533C23#1458176412yes—
534C23#1458178812yes—
535C23C22×C22⋊D8311yescharacteristic
536Q8—3321no—
537Q8—3321no—
538Q8—1662no—
539Q8—1662no—
540D4—24no—
541D4—24no—
542D4—48no—
543D4—42no—
544D4—42no—
545D4—3328no—
546D4—3321no—
547D4—3321no—
548D4—1662no—
549D4—1662no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—48no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—84no—
559C2×C4—3321no—
560C2×C4—3321no—
561C2×C4—3328no—
562C2×C4—3322no—
563C2×C4—3321no—
564C2×C4—3321no—
565C2×C4—3322no—
566C2×C4—1662no—
567C2×C4C22⋊C4×D8314yes—
568C2×C4(C22⋊D83)⋊C414yes—
569C2×C4C4×C22⋊D8312yes—
570C2×C4#1458176412yes—
571C2×C4#1458176514yes—
572C2×C4#1458177814yes—
573C2×C4#1458178812yes—
574C2×C4#1458181814yescomplemented
575C2×C4C22×C4×D8311yescharacteristic
576C2×C4C2×D4×D8311yescharacteristic
577C2×C4C2×D4×D8311yescharacteristic
578C2×C4C2×D4×D8311yescharacteristic
579C2×C4C2×#1305058911yescharacteristic
580C2×C4C2×#1305059011yescharacteristic
581C8—838no—
582C8—1664no—
583C22—24no—
584C22—24no—
585C22—22no—
586C22—22no—
587C22—48no—
588C22—44no—
589C22—816no—
590C22—84no—
591C22—82no—
592C22—82no—
593C22—3324no—
594C22—3322no—
595C22—6642no—
596C22—6641no—
597C22—6641no—
598C22#1912760114yes—
599C22C2×C4×C22⋊D8311yescharacteristic
600C22C2×#1458176411yescharacteristic
601C22C2×#1458178811yescharacteristic
602C22C2×#1458181812yes—
603C22#1912849912yes—
604C4—24no—
605C4—42no—
606C4—82no—
607C4—3322no—
608C4—6641no—
609C4—6641no—
610C4#1912760114yes—
611C4C2×C22⋊C4×D8311yescharacteristic
612C4C2×(C22⋊D83)⋊C411yescharacteristic
613C4C2×#1458176511yescharacteristic
614C4C2×#1458177811yescharacteristic
615C2—24no—
616C2—84no—
617C2—6642no—
618C2C2×#1912760111yescharacteristic
619C2C4×#1458181812yes—
620C2#2491707812yes—
621C2#2491791311yescharacteristic
622C2#2491792711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC2×#390×C83C2
socleC23×C83C22×D4
prime radicalC22×#14581818C2
cosocleC22×C4×C83C24
hypercenterC2×#390D83
hypocenterC83#21236
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
2. C2×#19127741C2semidirect
3. C2×#19127700C2semidirect
4. C2×#19127706C2semidirect
5. C2×C4×#13050574C2semidirect
6. #24909953C2semidirect
7. #24908343C2semidirect
8. #24908338C2semidirect
9. #24908328C2semidirect
10. #24908316C2semidirect
11. C2×#390×C83C2semidirect
15. C2×#14581818C4semidirect
22. C2×#14581725C22semidirect
23. C2×#14581723C22semidirect
26. C2×#14581715C22semidirect
29. C2×#14581712C22semidirect
31. C2×#14581703C22semidirect
32. C2×C4×C83⋊C8C22semidirect
33. #19127607C22semidirect
34. #19127607C22semidirect
35. #19127603C22semidirect
36. #19127603C22semidirect
59. C2×#13050578C2×C4semidirect
62. C2×#13050575C2×C4semidirect
142. #13050574C22⋊C4semidirect
245. C2×C4×C83AGL(1,ℤ/8)semidirect
270. C83⋊C8C2×C22⋊C4semidirect
318. C2×#390D83semidirect
323. C83#21236semidirect
574. C2×C4#14581818semidirect

Maximal subgroups

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

subgroupindex
C22×#145818182
C2×#191277412
C2×#191277002
C2×#191277062
C2×C4×#130505742
#249099532
#249083432
#249083382
#249083282
#249083162
C2×#390×C832
#2123683