Finite Group Databasebeta

← #35407959

Subgroups — #35407959

Order 21248.

882 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127748C211yesmaximal, characteristic
2#24916050C211yesmaximal, characteristic
3C2×#19127740C211yesmaximal, characteristic
4#428×D83C211yesmaximal, characteristic
5#24915879C211yesmaximal, characteristic
6#24908249C211yesmaximal, characteristic, complemented
7#24908248C211yesmaximal, characteristic, complemented
8#24908247C211yesmaximal, characteristic, complemented
9#24908245C211yesmaximal, characteristic, complemented
10#24908242C211yesmaximal, characteristic, complemented
11#24908241C211yesmaximal, characteristic, complemented
12#24908240C211yesmaximal, characteristic, complemented
13#24907200C211yesmaximal, characteristic
14#343×D83C211yesmaximal, characteristic
15C83×#1951C211yesmaximal, characteristic, complemented
16C22×#13050573C2211yescharacteristic
17C2×M16×D83C2211yescharacteristic
18C2×M16×D83C2211yescharacteristic
19C2×M16×D83C2211yescharacteristic
20C2×#14581750C2211yescharacteristic
21#117×D83C2211yescharacteristic
22C2×C4⋊C4×D83C2211yescharacteristic
23#19127748C418yescomplemented
24C2×#14581728C2211yescharacteristic
25C2×#14581727C2211yescharacteristic
26C2×#14581727C2211yescharacteristic
27#19127744C2211yescharacteristic
28#19127744C2211yescharacteristic
29#19127743C2211yescharacteristic
30#19127743C2211yescharacteristic
31#19127742—21no—
32#19127742—21no—
33#19127741C2211yescharacteristic
34#19127741C2211yescharacteristic
35#19127740—21no—
36#19127740—21no—
37#19127740—21no—
38#19127740—21no—
39#19127740—21no—
40#19127740—21no—
41#19127738—21no—
42#19127736—21no—
43C2×#14581726C2211yescharacteristic
44C2×#14581725C2211yescharacteristic
45#107×D83—21no—
46C2×#14581723C2211yescharacteristic
47#106×D83—21no—
48#19127659C2211yescharacteristic
49#19127658—21no—
50#19127657C2211yescharacteristic
51#19127656C2211yescharacteristic
52#19127655—21no—
53#19127653—21no—
54C4×#13050573—21no—
55#19127470C2211yescharacteristic
56#19127470C2211yescharacteristic
57#19127469C2211yescharacteristic
58#19127469C2211yescharacteristic
59#19127469C2211yescharacteristic
60#19127469C2211yescharacteristic
61C2×#108×C83C2211yescharacteristic
62#428×C83C2211yescharacteristic
63#428×C83C2211yescharacteristic
64C2×#106×C83C2211yescharacteristic
65#405×C83C2211yescharacteristic
66#343×C83C2211yescharacteristic
67#343×C83C2211yescharacteristic
68C22×C4×D83—21no—
69C22×C4×D83C2311yescharacteristic
70C2×#13050574C2311yescharacteristic
71C2×#13050574C2311yescharacteristic
72C2×#13050574C2311yescharacteristic
73C2×#13050573—21no—
74C2×#13050573—21no—
75C2×#13050573—21no—
76C2×#13050573C2×C414yes—
77C2×#13050573D411yescharacteristic
78C2×#13050573D411yescharacteristic
79C2×#13050573D411yescharacteristic
80C2×#13050573D411yescharacteristic
81C2×#13050573Q814yes—
82M16×D83—22no—
83M16×D83—22no—
84M16×D83—22no—
85M16×D83C2×C414yes—
86M16×D83C2×C414yes—
87M16×D83Q814yes—
88C22×C83⋊C8C2311yescharacteristic
89C2×C8×D83—21no—
90C2×C8×D83—21no—
91C2×C8×D83—21no—
92#14581751C2311yescharacteristic
93#14581750—21no—
94#14581750—21no—
95#14581750—21no—
96#14581750—21no—
97#14581750D411yescharacteristic
98#14581750D411yescharacteristic
99#14581749—21no—
100#14581748—21no—
101C2×C4⋊Dic83C2311yescharacteristic
102C2×C4⋊Dic83C2311yescharacteristic
103C2×Dic83⋊C4—21no—
104C4⋊C4×D83—21no—
105C4⋊C4×D83—21no—
106C4⋊C4×D83—21no—
107C4⋊C4×D83—21no—
108C4⋊C4×D83D411yescharacteristic
109C4⋊C4×D83D411yescharacteristic
110C2×(C2×D83)⋊C4—21no—
111C22⋊C4×D83—21no—
112C2×C4×Dic83—21no—
113C42×D83—21no—
114#14581728C2×C414yescomplemented
115#14581727C2×C414yescomplemented
116#14581727C2×C414yescomplemented
117#14581726—21no—
118#14581726—21no—
119#14581726—21no—
120#14581726—21no—
121#14581725—21no—
122#14581725—21no—
123#14581725—21no—
124#14581725—21no—
125#14581725—21no—
126#14581725—21no—
127#14581724—21no—
128#14581724—21no—
129#14581723—21no—
130#14581723—21no—
131#14581723—21no—
132#14581723—21no—
133#14581723—21no—
134#14581723—21no—
135#14581703—21no—
136#14581702—21no—
137#14581702—21no—
138C4×C83⋊C8—21no—
139C8×Dic83—21no—
140C8×Dic83—21no—
141C2×M16×C83C2311yescharacteristic
142C2×M16×C83C2311yescharacteristic
143C2×M16×C83C2311yescharacteristic
144C22×C8×C83C2311yescharacteristic
145#117×C83C2311yescharacteristic
146C2×C4⋊C4×C83C2311yescharacteristic
147C2×C4⋊C4×C83C2311yescharacteristic
148#108×C83C2×C414yescomplemented
149#107×C83—21no—
150#107×C83—21no—
151#106×C83—21no—
152#106×C83—21no—
153#106×C83—21no—
154#106×C83—21no—
155#97×C83—21no—
156C4×C8×C83—21no—
157C23×D83#4111yescharacteristic
158C22×Dic83—21no—
159C22×Dic83—21no—
160C22×Dic83#4111yescharacteristic
161C2×C4×D83—21no—
162C2×C4×D83—21no—
163C2×C4×D83—41no—
164C2×C4×D83—41no—
165C2×C4×D83—82no—
166C2×C4×D83C22×C412yes—
167C2×C4×D83C2×D411yescharacteristic
168C2×C4×D83C2×D411yescharacteristic
169C2×C4×D83C2×D411yescharacteristic
170C2×C4×D83C2×D411yescharacteristic
171C2×C4×D83C2×Q812yes—
172C2×C4×D83C2×Q812yes—
173C2×C4×D83#4112yes—
174#13050574—21no—
175#13050574—21no—
176#13050574—21no—
177#13050574C22×C412yes—
178#13050574C22×C412yes—
179#13050574C2×Q812yes—
180#13050573—24no—
181#13050573—24no—
182#13050573—24no—
183#13050573C4⋊C414yes—
184#13050573C4⋊C414yes—
185#13050573C4⋊C414yes—
186#13050573C4⋊C414yes—
187C2×C83⋊C8—21no—
188C2×C83⋊C8—21no—
189C2×C83⋊C8—21no—
190C2×C83⋊C8C22×C412yes—
191C2×C83⋊C8C2×D411yescharacteristic
192C2×C83⋊C8C2×D411yescharacteristic
193C2×C83⋊C8C2×Q812yes—
194C8×D83—24no—
195C8×D83—24no—
196C8×D83—24no—
197C4⋊Dic83—21no—
198C4⋊Dic83—21no—
199C4⋊Dic83—21no—
200C4⋊Dic83—21no—
201C4⋊Dic83C2×D411yescharacteristic
202C4⋊Dic83C2×D411yescharacteristic
203Dic83⋊C4—21no—
204Dic83⋊C4—21no—
205Dic83⋊C4—41no—
206Dic83⋊C4—41no—
207C22⋊Dic83—21no—
208(C2×D83)⋊C4—21no—
209(C2×D83)⋊C4—21no—
210(C2×D83)⋊C4—41no—
211(C2×D83)⋊C4—41no—
212C4×Dic83—21no—
213C4×Dic83—21no—
214C4×Dic83—21no—
215C4×Dic83—41no—
216C4×Dic83—41no—
217C22×C4×C83—21no—
218C22×C4×C83—21no—
219C22×C4×C83C2411yescharacteristic
220M16×C83—21no—
221M16×C83—21no—
222M16×C83—21no—
223M16×C83C22×C412yes—
224M16×C83C22×C412yes—
225M16×C83C2×Q812yes—
226C2×C8×C83—21no—
227C2×C8×C83—21no—
228C2×C8×C83—21no—
229C2×C8×C83C22×C412yes—
230C2×C8×C83C2×D411yescharacteristic
231C2×C8×C83C2×D411yescharacteristic
232C2×C8×C83C2×Q812yes—
233C4⋊C4×C83—21no—
234C4⋊C4×C83—21no—
235C4⋊C4×C83—21no—
236C4⋊C4×C83—21no—
237C4⋊C4×C83C2×D411yescharacteristic
238C4⋊C4×C83C2×D411yescharacteristic
239C22⋊C4×C83—21no—
240C42×C83—21no—
241C23×C83C2×#4111yescharacteristic
242C22×D83—41no—
243C22×D83—41no—
244C22×D83C4×Q812yes—
245C22×D83#12111yescharacteristic
246C22×D83#12111yescharacteristic
247C22×D83#12512yes—
248C2×C4×C83—21no—
249C2×C4×C83—21no—
250C2×C4×C83—41no—
251C2×C4×C83—41no—
252C2×C4×C83—81no—
253C2×C4×C83—81no—
254C2×C4×C83C23×C411yescharacteristic
255C2×C4×C83C22×D411yescharacteristic
256C2×C4×C83C22×D411yescharacteristic
257C2×C4×C83C22×Q811yescharacteristic
258C2×C4×C83C22×Q811yescharacteristic
259C2×C4×C83C2×#4111yescharacteristic
260C2×Dic83—21no—
261C2×Dic83—21no—
262C2×Dic83—41no—
263C2×Dic83—41no—
264C2×Dic83—81no—
265C2×Dic83—81no—
266C2×Dic83C4×Q812yes—
267C2×Dic83#12111yescharacteristic
268C2×Dic83#12111yescharacteristic
269C2×Dic83#12512yes—
270C4×D83—81no—
271C4×D83—81no—
272C4×D83—82no—
273C4×D83—82no—
274C4×D83C2×C4⋊C412yes—
275C4×D83C2×C4⋊C412yes—
276C4×D83C4×D412yes—
277C4×D83C4×D412yes—
278C4×D83C22⋊Q812yes—
279C4×D83C22⋊Q812yes—
280C4×D83C4⋊Q812yes—
281C4×D83C4⋊Q812yes—
282C8×C83—22no—
283C8×C83—22no—
284C8×C83—22no—
285C8×C83C2×C4⋊C412yes—
286C8×C83C2×C4⋊C412yes—
287C83⋊C8—22no—
288C83⋊C8—22no—
289C83⋊C8—22no—
290C83⋊C8C2×C4⋊C412yes—
291C83⋊C8C2×C4⋊C412yes—
292C22×C83—21no—
293C22×C83—21no—
294C22×C83C2×C4×Q811yescharacteristic
295C22×C83C2×#12111yescharacteristic
296C22×C83C2×#12511yescharacteristic
297C2×D83—21no—
298C2×D83—21no—
299C2×D83—21no—
300C2×D83—21no—
301C2×D83—42no—
302C2×D83—42no—
303C2×D83#38912yes—
304C2×D83#38912yes—
305C4×C83—41no—
306C4×C83—41no—
307C4×C83—81no—
308C4×C83—81no—
309C4×C83C22×C4⋊C411yescharacteristic
310C4×C83C2×C4×D411yescharacteristic
311C4×C83C2×C22⋊Q811yescharacteristic
312C4×C83C2×C4⋊Q811yescharacteristic
313Dic83—41no—
314Dic83—41no—
315Dic83—81no—
316Dic83—81no—
317Dic83#38912yes—
318Dic83#38912yes—
319#21182—831nomaximal, Sylow
320C2×C83—21no—
321C2×C83—21no—
322C2×C83C2×#38911yescharacteristic
323C2×C83#306811yescharacteristic
324C2×C83#306911yescharacteristic
325D83—22no—
326D83—22no—
327C2×#430—831no—
328#2911—831no—
329C2×#428—831no—
330C2×#428—831no—
331#2873—831no—
332#1952—831no—
333#1952—831no—
334#1952—831no—
335#1952—831no—
336#1951—831no—
337#1951—831no—
338#1951—831no—
339#1951D8311yescharacteristic, complemented
340#1742—831no—
341C2×#343—831no—
342C83#2118211yescharacteristic, complemented, Sylow
343C22×M16—831no—
344C22×M16—831no—
345C22×M16—831no—
346C22×M16—831no—
347C2×#117—831no—
348C2×#117—831no—
349C22×C4⋊C4—831no—
350#430—838no—
351C2×#108—831no—
352C2×#108—831no—
353C2×#108—831no—
354C2×#108C2×D8311yescharacteristic
355#428—831no—
356#428—831no—
357#428—831no—
358#428—831no—
359#428—831no—
360#428—831no—
361#428—1661no—
362#428—1661no—
363#428—1661no—
364#428—1661no—
365#428—1661no—
366#428—1661no—
367#428—1661no—
368#428—1661no—
369#428C2×D8311yescharacteristic
370#428C2×D8311yescharacteristic
371#427—1661no—
372#427—1661no—
373C2×#107—831no—
374C2×#107—831no—
375C2×#107—1661no—
376C2×#106—831no—
377C2×#106—1661no—
378C2×#106C2×D8311yescharacteristic
379#405—831no—
380#405—831no—
381#405—831no—
382#405—1661no—
383#405—1661no—
384#405C2×D8311yescharacteristic
385C4×M16—1661no—
386C4×M16—1661no—
387#343—831no—
388#343—831no—
389#343—831no—
390#343—831no—
391#343—831no—
392#343—831no—
393#343C2×D8311yescharacteristic
394#343C2×D8311yescharacteristic
395C23×C4—831no—
396C23×C4—1661no—
397C2×M16—834no—
398C2×M16—831no—
399C2×M16—831no—
400C2×M16—831no—
401C2×M16—831no—
402C2×M16—834no—
403C2×M16—831no—
404C2×M16—831no—
405C2×M16—831no—
406C2×M16—834no—
407C2×M16—834no—
408C2×M16—834no—
409C2×M16—1661no—
410C2×M16—1661no—
411C2×M16—1661no—
412C2×M16—1662no—
413C2×M16—1662no—
414C2×M16—1662no—
415C2×M16C22×D8311yescharacteristic
416C2×M16C22×D8311yescharacteristic
417C2×M16C22×D8311yescharacteristic
418C22×C8—831no—
419C22×C8—1661no—
420C22×C8—1661no—
421C22×C8—1661no—
422C22×C8C22×D8311yescharacteristic
423#117—831no—
424#117—831no—
425#117—831no—
426#117—1661no—
427#117—1661no—
428#117—1661no—
429#117—1661no—
430#117—1661no—
431#117—1661no—
432#117C22×D8311yescharacteristic
433C2×C4⋊C4—831no—
434C2×C4⋊C4—831no—
435C2×C4⋊C4—831no—
436C2×C4⋊C4—831no—
437C2×C4⋊C4—1661no—
438C2×C4⋊C4—1661no—
439C2×C4⋊C4—1661no—
440C2×C4⋊C4—1661no—
441C2×C4⋊C4—1661no—
442C2×C4⋊C4C22×D8311yescharacteristic
443C2×C4⋊C4C22×D8311yescharacteristic
444C2×C22⋊C4—1661no—
445C2×C22⋊C4—1661no—
446C2×C42—1661no—
447C2×C42—1661no—
448#108—834no—
449#108—834no—
450#108—834no—
451#108C4×D8314yescomplemented
452#107—21no—
453#107—21no—
454#107—1661no—
455#107—1661no—
456#107—1661no—
457#107—1661no—
458#107—1661no—
459#107—1661no—
460#107—1661no—
461#107—1661no—
462#107—1661no—
463#107—1661no—
464#106—21no—
465#106—21no—
466#106—21no—
467#106—21no—
468#106—1661no—
469#106—1661no—
470#106—1661no—
471#106—1661no—
472#106—1661no—
473#106—1661no—
474#106—1661no—
475#106—1661no—
476#97—21no—
477#97—1661no—
478#97—1661no—
479#97—1661no—
480C4×C8—21no—
481C4×C8—1661no—
482C4×C8—1661no—
483C4×C8—1661no—
484C24—831no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—3321no—
488C22×C4—3321no—
489C22×C4—831no—
490C22×C4—832no—
491C22×C4—831no—
492C22×C4—831no—
493C22×C4—831no—
494C22×C4—831no—
495C22×C4—832no—
496C22×C4—832no—
497C22×C4—832no—
498C22×C4—6642no—
499C22×C4—1661no—
500C22×C4—1661no—
501C22×C4—1661no—
502C22×C4—1661no—
503C22×C4C23×D8311yescharacteristic
504M16—21no—
505M16—21no—
506M16—21no—
507M16—834no—
508M16—834no—
509M16—834no—
510M16—834no—
511M16—832no—
512M16—832no—
513M16—832no—
514M16—1664no—
515M16—1664no—
516M16—1664no—
517M16—1661no—
518M16—1661no—
519M16—1661no—
520M16C2×C4×D8312yes—
521M16C2×C4×D8312yes—
522M16Q8×D8312yes—
523C2×C8—21no—
524C2×C8—21no—
525C2×C8—21no—
526C2×C8—832no—
527C2×C8—831no—
528C2×C8—831no—
529C2×C8—832no—
530C2×C8—1664no—
531C2×C8—1664no—
532C2×C8—1664no—
533C2×C8—1661no—
534C2×C8—1661no—
535C2×C8—1661no—
536C2×C8C2×C4×D8312yes—
537C2×C8D4×D8311yescharacteristic
538C2×C8D4×D8311yescharacteristic
539C2×C8Q8×D8312yes—
540C4⋊C4—21no—
541C4⋊C4—21no—
542C4⋊C4—21no—
543C4⋊C4—21no—
544C4⋊C4—3321no—
545C4⋊C4—3321no—
546C4⋊C4—831no—
547C4⋊C4—831no—
548C4⋊C4—1661no—
549C4⋊C4—1661no—
550C4⋊C4—1661no—
551C4⋊C4—1661no—
552C4⋊C4—1661no—
553C4⋊C4—1661no—
554C4⋊C4D4×D8311yescharacteristic
555C4⋊C4D4×D8311yescharacteristic
556C22⋊C4—21no—
557C22⋊C4—3321no—
558C22⋊C4—3321no—
559C22⋊C4—1661no—
560C22⋊C4—1661no—
561C22⋊C4—1661no—
562C42—21no—
563C42—3321no—
564C42—3321no—
565C42—1661no—
566C42—1661no—
567C42—1661no—
568C23—3321no—
569C23—3321no—
570C23—832no—
571C23—832no—
572C23—831no—
573C23—831no—
574C23#41×D8311yescharacteristic
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—81no—
580C2×C4—81no—
581C2×C4—3321no—
582C2×C4—3321no—
583C2×C4—832no—
584C2×C4—831no—
585C2×C4—831no—
586C2×C4—832no—
587C2×C4—832no—
588C2×C4—832no—
589C2×C4—832no—
590C2×C4—832no—
591C2×C4—832no—
592C2×C4—832no—
593C2×C4—832no—
594C2×C4—832no—
595C2×C4—6641no—
596C2×C4—6641no—
597C2×C4—6642no—
598C2×C4—6642no—
599C2×C4—6641no—
600C2×C4—6641no—
601C2×C4—1661no—
602C2×C4—1661no—
603C2×C4C22×C4×D8311yescharacteristic
604C2×C4C2×D4×D8311yescharacteristic
605C2×C4C2×D4×D8311yescharacteristic
606C2×C4C2×Q8×D8311yescharacteristic
607C2×C4C2×Q8×D8311yescharacteristic
608C2×C4#41×D8311yescharacteristic
609C8—22no—
610C8—22no—
611C8—22no—
612C8—832no—
613C8—832no—
614C8—1662no—
615C8—1662no—
616C8—1662no—
617C8C4⋊C4×D8312yes—
618C8C4⋊C4×D8312yes—
619C22—21no—
620C22—21no—
621C22—3322no—
622C22—3322no—
623C22—832no—
624C22—832no—
625C22—1661no—
626C22—1661no—
627C22—1661no—
628C22—1661no—
629C22C4×Q8×D8311yescharacteristic
630C22#121×D8311yescharacteristic
631C22#125×D8311yescharacteristic
632C4—41no—
633C4—41no—
634C4—81no—
635C4—81no—
636C4—3321no—
637C4—3321no—
638C4—832no—
639C4—832no—
640C4—6641no—
641C4—6641no—
642C4C2×C4⋊C4×D8311yescharacteristic
643C4C4×D4×D8311yescharacteristic
644C4C22⋊Q8×D8311yescharacteristic
645C4C4⋊Q8×D8311yescharacteristic
646C2—21no—
647C2—21no—
648C2—1662no—
649C2—1662no—
650C2#389×D8311yescharacteristic
651C2#2491718611yescharacteristic
652C2#2491719511yescharacteristic
subgroupG/H
centerC22#121×D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC83×#1951C2
socleC22×C83C2×#121
prime radicalC23×D83#41
cosocleC22×C4×C83C24
hypercenter#1951D83
hypocenterC83#21182
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1951

Complements

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

NH ≅ G/Ntype
6. #24908249C2semidirect
7. #24908248C2semidirect
8. #24908247C2semidirect
9. #24908245C2semidirect
10. #24908242C2semidirect
11. #24908241C2semidirect
12. #24908240C2semidirect
15. C83×#1951C2semidirect
23. #19127748C4semidirect
114. #14581728C2×C4semidirect
115. #14581727C2×C4semidirect
116. #14581727C2×C4semidirect
148. #108×C83C2×C4semidirect
339. #1951D83semidirect
342. C83#21182semidirect
451. #108C4×D83semidirect

Maximal subgroups

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

subgroupindex
C2×#191277482
#249160502
C2×#191277402
#428×D832
#249158792
#249082492
#249082482
#249082472
#249082452
#249082422
#249082412
#249082402
#249072002
#343×D832
C83×#19512
#2118283