Finite Group Databasebeta

← #35375556

Subgroups — #35375556

Order 21248.

650 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24917506C211yesmaximal, characteristic
2#24916497C211yesmaximal, characteristic, complemented
3#24906353C211yesmaximal, characteristic, complemented
4#24906346C211yesmaximal, characteristic, complemented
5#24906341C211yesmaximal, characteristic, complemented
6#24906327C211yesmaximal, characteristic, complemented
7#24906325C211yesmaximal, characteristic, complemented
8#24906318C211yesmaximal, characteristic, complemented
9C8×#13050577C211yesmaximal, characteristic, complemented
10C8×#13050575C211yesmaximal, characteristic, complemented
11#24906246C211yesmaximal, characteristic, complemented
12#24905923C211yesmaximal, characteristic, complemented
13C2×#19127428C211yesmaximal, characteristic, complemented
14#24905747C211yesmaximal, characteristic, complemented
15C83×#2994C211yesmaximal, characteristic, complemented
16C2×C4×D332C2211yescharacteristic
17#19128034C411yescharacteristic
18#19128017C411yescharacteristic
19#19127998C411yescharacteristic
20#19127998C411yescharacteristic
21#19127913C411yescharacteristic
22#19127913C411yescharacteristic
23#19127851C411yescharacteristic
24#19127834C411yescharacteristic
25C4×#13050577C2211yescharacteristic
26C4×#13050577C2211yescharacteristic
27C4×#13050575C2211yescharacteristic
28C4×#13050575C2211yescharacteristic
29#19127780C2211yescharacteristic
30#19127780C2211yescharacteristic
31#19127771—21no—
32#19127771—21no—
33C8×D332C2211yescharacteristic
34C8×D332C2211yescharacteristic
35#19127758C2211yescharacteristic
36#19127754C2211yescharacteristic
37#19127663—21no—
38C22⋊C8×D83—21no—
39C2×C4×C83⋊C8C2211yescharacteristic
40#19127453C2211yescharacteristic, complemented
41#19127453C2211yescharacteristic, complemented
42#19127451C2211yescharacteristic, complemented
43#19127451C2211yescharacteristic, complemented
44Q8⋊(C83⋊C8)C2211yescharacteristic, complemented
45Q8⋊(C83⋊C8)C2211yescharacteristic, complemented
46D4⋊(C83⋊C8)C2211yescharacteristic, complemented
47D4⋊(C83⋊C8)C2211yescharacteristic, complemented
48#19127428C2211yescharacteristic
49#19127428C2211yescharacteristic
50#19127428C2211yescharacteristic
51#19127428C2211yescharacteristic
52#19127423C2211yescharacteristic, complemented
53#19127423C2211yescharacteristic, complemented
54C8×C83⋊C8C2211yescharacteristic, complemented
55C8×C83⋊C8C2211yescharacteristic, complemented
56#529×C83C2211yescharacteristic
57#446×C83C2211yescharacteristic, complemented
58C8×Q8×C83C2211yescharacteristic, complemented
59#436×C83C2211yescharacteristic, complemented
60#435×C83—21no—
61C8×D4×C83—21no—
62C8×D4×C83C2211yescharacteristic, complemented
63#433×C83—21no—
64#431×C83C2211yescharacteristic
65#424×C83—21no—
66C2×C4⋊C8×C83C2211yescharacteristic
67C22×D332C2×C411yescharacteristic
68C22×C4×D83—21no—
69C2×#13050577C2×C411yescharacteristic
70C2×#13050577C2×C411yescharacteristic
71C2×#13050575C2×C411yescharacteristic
72C2×#13050575C2×C411yescharacteristic
73C2×#13050573—21no—
74C2×#13050573—21no—
75C22×C83⋊C8—21no—
76C2×C8×D83—21no—
77C2×C8×D83—21no—
78C4×D332—21no—
79C4×D332—21no—
80C4×D332C2×C411yescharacteristic
81C4×D332C2×C411yescharacteristic
82C4×D332C2311yescharacteristic
83C4×D332C2311yescharacteristic
84C2×C4⋊Dic83C2×C411yescharacteristic
85C2×(C2×D83)⋊C4—21no—
86#14581725C2×C411yescharacteristic
87#14581725C2×C411yescharacteristic
88#14581723C2×C411yescharacteristic
89#14581723C2×C411yescharacteristic
90C4⋊(C83⋊C8)C2×C411yescharacteristic
91C4⋊(C83⋊C8)C2×C411yescharacteristic
92C4⋊(C83⋊C8)C2311yescharacteristic
93C4⋊(C83⋊C8)C2311yescharacteristic
94#14581716C2×C411yescharacteristic
95#14581716C2×C411yescharacteristic
96#14581713C2×C411yescharacteristic
97#14581713C2×C411yescharacteristic
98#14581712C2×C411yescharacteristic
99#14581712C2×C411yescharacteristic
100#14581712C2×C411yescharacteristic
101#14581712C2×C411yescharacteristic
102C22⋊(C83⋊C8)—21no—
103C22⋊(C83⋊C8)C2×C411yescharacteristic
104C22⋊(C83⋊C8)C2×C411yescharacteristic
105(C2×D83)⋊C8—21no—
106(C2×D83)⋊C8—21no—
107(C2×D83)⋊C8—21no—
108(C2×D83)⋊C8—21no—
109#14581703—21no—
110C4×C83⋊C8—21no—
111C4×C83⋊C8C2311yescharacteristic
112C4×C83⋊C8C2311yescharacteristic
113C4×C83⋊C8C2311yescharacteristic
114C4×C83⋊C8C2311yescharacteristic
115C2×M16×C83—21no—
116C22×C8×C83—21no—
117C22×C8×C83—21no—
118#126×C83—21no—
119#125×C83C2×C411yescharacteristic
120#124×C83C2×C411yescharacteristic
121#123×C83—21no—
122C22⋊Q8×C83C2×C411yescharacteristic
123#121×C83C2×C411yescharacteristic
124C4×Q8×C83C2311yescharacteristic
125C4×D4×C83—21no—
126C4×D4×C83C2311yescharacteristic
127#117×C83—21no—
128C2×C42×C83C2311yescharacteristic
129C4⋊C8×C83—21no—
130C4⋊C8×C83—21no—
131C4⋊C8×C83—21no—
132C4⋊C8×C83D411yescharacteristic
133C4⋊C8×C83D411yescharacteristic
134C4⋊C8×C83C2311yescharacteristic
135C4⋊C8×C83C2311yescharacteristic
136C22⋊C8×C83—21no—
137C22⋊C8×C83—21no—
138C22⋊C8×C83D411yescharacteristic
139C22⋊C8×C83D411yescharacteristic
140#97×C83—21no—
141C4×C8×C83—21no—
142C4×C8×C83C2311yescharacteristic
143C4×C8×C83C2311yescharacteristic
144C23×D83—21no—
145C2×D332—22no—
146C2×D332—21no—
147C2×D332—41no—
148C2×D332C22×C411yescharacteristic
149C2×D332C22×C411yescharacteristic
150C22×Dic83—21no—
151C2×C4×D83—21no—
152C2×C4×D83—21no—
153C2×C4×D83—21no—
154C2×C4×D83—21no—
155C2×C4×D83—81no—
156C2×C4×D83—81no—
157#13050577—22no—
158#13050577—41no—
159#13050575—22no—
160#13050575—41no—
161#13050573—81no—
162#13050573—81no—
163C2×C83⋊C8—24no—
164C2×C83⋊C8—21no—
165C2×C83⋊C8—21no—
166C2×C83⋊C8—21no—
167C2×C83⋊C8—21no—
168C2×C83⋊C8C22×C411yescharacteristic
169C2×C83⋊C8C22×C411yescharacteristic
170C2×C83⋊C8C22×C411yescharacteristic
171C2×C83⋊C8C22×C411yescharacteristic
172C8×D83—81no—
173C8×D83—81no—
174C4⋊Dic83—21no—
175C4⋊Dic83C22×C411yescharacteristic
176C4⋊Dic83C22×C411yescharacteristic
177(C2×D83)⋊C4—21no—
178(C2×D83)⋊C4—21no—
179(C2×D83)⋊C4—21no—
180(C2×D83)⋊C4—21no—
181C2×Q8×C83C22×C411yescharacteristic
182C2×D4×C83—21no—
183C2×D4×C83C22×C411yescharacteristic
184C22×C4×C83—21no—
185C22×C4×C83C22×C411yescharacteristic
186C22×C4×C83C2×D411yescharacteristic
187C22×C4×C83#4111yescharacteristic
188M16×C83—81no—
189C2×C8×C83—21no—
190C2×C8×C83—21no—
191C2×C8×C83—81no—
192C2×C8×C83—81no—
193C2×C8×C83C2×D411yescharacteristic
194C2×C8×C83C2×D411yescharacteristic
195C2×C8×C83C2×D411yescharacteristic
196C2×C8×C83C2×D411yescharacteristic
197C4⋊C4×C83—21no—
198C4⋊C4×C83—21no—
199C4⋊C4×C83—21no—
200C4⋊C4×C83—21no—
201C4⋊C4×C83C22×C411yescharacteristic
202C4⋊C4×C83C22×C411yescharacteristic
203C22⋊C4×C83—21no—
204C22⋊C4×C83—21no—
205C22⋊C4×C83—21no—
206C22⋊C4×C83—21no—
207C22⋊C4×C83—21no—
208C42×C83—21no—
209C42×C83C22×C411yescharacteristic
210C42×C83C2×D411yescharacteristic
211C42×C83#4111yescharacteristic
212C42×C83C2411yescharacteristic
213C23×C83—21no—
214C23×C83C4×D411yescharacteristic
215C22×D83—21no—
216C22×D83—21no—
217C22×D83—42no—
218C22×D83—42no—
219C22×D83—81no—
220Q8×C83—21no—
221Q8×C83—21no—
222D4×C83—21no—
223D4×C83—21no—
224D4×C83—42no—
225C2×C4×C83—22no—
226C2×C4×C83—21no—
227C2×C4×C83—21no—
228C2×C4×C83—21no—
229C2×C4×C83—21no—
230C2×C4×C83—21no—
231C2×C4×C83—21no—
232C2×C4×C83—21no—
233C2×C4×C83—41no—
234C2×C4×C83—41no—
235C2×C4×C83C4×D411yescharacteristic
236C2×C4×C83C4×D411yescharacteristic
237C2×C4×C83C4×D411yescharacteristic
238C2×C4×C83C23×C411yescharacteristic
239C2×C4×C83C22×D411yescharacteristic
240C2×C4×C83C2×#4111yescharacteristic
241C2×Dic83—21no—
242C2×Dic83—21no—
243C2×Dic83—81no—
244D332—21no—
245D332—21no—
246D332—21no—
247D332—42no—
248D332#13112yes—
249C4×D83—41no—
250C4×D83—41no—
251C4×D83—41no—
252C4×D83—41no—
253C4×D83—81no—
254C4×D83—81no—
255C8×C83—21no—
256C8×C83—21no—
257C8×C83—21no—
258C8×C83—21no—
259C8×C83—41no—
260C8×C83—41no—
261C83⋊C8—22no—
262C83⋊C8—22no—
263C83⋊C8—21no—
264C83⋊C8—21no—
265C83⋊C8—21no—
266C83⋊C8—21no—
267C83⋊C8—41no—
268C83⋊C8—41no—
269C22×C83—21no—
270C22×C83—21no—
271C22×C83—41no—
272C22×C83—41no—
273C22×C83—41no—
274C22×C83#44312yes—
275C22×C83C2×C4×D411yescharacteristic
276C2×D83—22no—
277C2×D83—41no—
278C2×D83—41no—
279C2×D83—41no—
280C2×D83—41no—
281C2×D83—41no—
282C2×D83—84no—
283C4×C83—21no—
284C4×C83—21no—
285C4×C83—21no—
286C4×C83—21no—
287C4×C83—41no—
288C4×C83—41no—
289C4×C83—41no—
290C4×C83C2×#13111yescharacteristic
291C4×C83#56811yescharacteristic
292C4×C83#57611yescharacteristic
293C4×C83#57711yescharacteristic
294Dic83—41no—
295Dic83—41no—
296#18014—831nomaximal, Sylow
297C2×C83—22no—
298C2×C83—81no—
299C2×C83#293311yescharacteristic
300C2×C83#294611yescharacteristic
301C2×C83C2×#44311yescharacteristic
302D83—42no—
303D83—81no—
304#3105—831no—
305#2994D8311yescharacteristic, complemented
306#2974—831no—
307#1590—831no—
308#1589—831no—
309#1588—831no—
310#1586—831no—
311#1586—831no—
312#1585—831no—
313C8×SD16—831no—
314C8×D8—831no—
315#1573—831no—
316#1494—831no—
317C2×D4⋊C8—831no—
318#1450—831no—
319C83#1801411yescharacteristic, complemented, Sylow
320#529C2×D8311yescharacteristic
321C2×C4×D4—831no—
322#484—831no—
323#481—831no—
324#478—831no—
325#478—831no—
326#463—831no—
327#463—831no—
328#450—831no—
329#447—831no—
330#446C2×D8311yescharacteristic, complemented
331C8×Q8C2×D8311yescharacteristic, complemented
332C4×SD16—831no—
333C4×SD16—831no—
334C4×D8—831no—
335C4×D8—831no—
336#436—831no—
337#436—831no—
338#436C2×D8311yescharacteristic, complemented
339#435—21no—
340#435—1661no—
341#435—1661no—
342C8×D4—21no—
343C8×D4—831no—
344C8×D4—831no—
345C8×D4C2×D8311yescharacteristic, complemented
346#433—21no—
347#432—831no—
348#431—831no—
349#431C2×D8311yescharacteristic
350#424—21no—
351C2×C4⋊C8C2×D8311yescharacteristic
352#407—1661no—
353C2×C22⋊C8—1661no—
354C2×C4×C8—831no—
355#335—831no—
356#335—831no—
357#334—831no—
358#334—831no—
359Q8⋊C8—831no—
360Q8⋊C8—831no—
361D4⋊C8—831no—
362D4⋊C8—831no—
363D4⋊C8—831no—
364D4⋊C8—831no—
365D4⋊C8—831no—
366D4⋊C8—831no—
367#322—831no—
368#322—831no—
369C82—831no—
370C82—831no—
371C22×D4—831no—
372C23×C4—1661no—
373C2×SD16—831no—
374C2×SD16—831no—
375C2×D8—831no—
376C2×D8—831no—
377C2×M16—21no—
378C2×M16—1661no—
379C2×M16—1661no—
380C22×C8—21no—
381C22×C8—21no—
382C22×C8—1661no—
383C22×C8—1661no—
384C22×C8—1661no—
385#126—21no—
386#125C4×D8311yescharacteristic
387#124C4×D8311yescharacteristic
388#123—21no—
389C22⋊Q8C4×D8311yescharacteristic
390#121C4×D8311yescharacteristic
391C4×Q8C22×D8311yescharacteristic
392C4×D4—21no—
393C4×D4—831no—
394C4×D4—831no—
395C4×D4—831no—
396C4×D4—831no—
397C4×D4—1661no—
398C4×D4—1661no—
399C4×D4C22×D8311yescharacteristic
400#117—21no—
401C2×C4⋊C4—831no—
402C2×C22⋊C4—1661no—
403C2×C42C22×D8311yescharacteristic
404#107—831no—
405#107—831no—
406#106—831no—
407#106—831no—
408C4⋊C8—21no—
409C4⋊C8—21no—
410C4⋊C8—21no—
411C4⋊C8—831no—
412C4⋊C8—831no—
413C4⋊C8—831no—
414C4⋊C8—831no—
415C4⋊C8C22⋊D8311yescharacteristic
416C4⋊C8C22⋊D8311yescharacteristic
417C4⋊C8C22×D8311yescharacteristic
418C4⋊C8C22×D8311yescharacteristic
419#103—831no—
420#103—831no—
421#102—831no—
422#102—831no—
423#102—831no—
424#102—831no—
425#102—831no—
426#102—831no—
427C22⋊C8—21no—
428C22⋊C8—21no—
429C22⋊C8—831no—
430C22⋊C8—831no—
431C22⋊C8—1661no—
432C22⋊C8—1661no—
433C22⋊C8—1661no—
434C22⋊C8—1661no—
435C22⋊C8—1661no—
436C22⋊C8C22⋊D8311yescharacteristic
437C22⋊C8C22⋊D8311yescharacteristic
438#97—21no—
439#97—1661no—
440C4×C8—21no—
441C4×C8—831no—
442C4×C8—831no—
443C4×C8—831no—
444C4×C8—831no—
445C4×C8—1661no—
446C4×C8C22×D8311yescharacteristic
447C4×C8C22×D8311yescharacteristic
448C24—1661no—
449C2×Q8C2×C4×D8311yescharacteristic
450C2×D4—21no—
451C2×D4—3321no—
452C2×D4—831no—
453C2×D4—831no—
454C2×D4—1662no—
455C2×D4—1661no—
456C2×D4C2×C4×D8311yescharacteristic
457C22×C4—21no—
458C22×C4—6641no—
459C22×C4—6641no—
460C22×C4—1661no—
461C22×C4—1661no—
462C22×C4—1661no—
463C22×C4—1661no—
464C22×C4—1661no—
465C22×C4C2×C4×D8311yescharacteristic
466C22×C4C2×C22⋊D8311yescharacteristic
467C22×C4#1305059011yescharacteristic
468SD16—3321no—
469SD16—1662no—
470D8—3321no—
471D8—1662no—
472M16—81no—
473M16—6641no—
474M16—6641no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—81no—
478C2×C8—81no—
479C2×C8—831no—
480C2×C8—831no—
481C2×C8—831no—
482C2×C8—831no—
483C2×C8—6641no—
484C2×C8—6641no—
485C2×C8—1661no—
486C2×C8—1661no—
487C2×C8—1664no—
488C2×C8—1661no—
489C2×C8—1661no—
490C2×C8C2×C22⋊D8311yescharacteristic
491C2×C8C2×C22⋊D8311yescharacteristic
492C2×C8C2×C22⋊D8311yescharacteristic
493C2×C8C2×C22⋊D8311yescharacteristic
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—831no—
499C4⋊C4—831no—
500C4⋊C4—1661no—
501C4⋊C4C2×C4×D8311yescharacteristic
502C4⋊C4C2×C4×D8311yescharacteristic
503C22⋊C4—21no—
504C22⋊C4—21no—
505C22⋊C4—21no—
506C22⋊C4—21no—
507C22⋊C4—21no—
508C22⋊C4—1661no—
509C22⋊C4—1661no—
510C22⋊C4—1661no—
511C22⋊C4—1661no—
512C42—21no—
513C42C2×C4×D8311yescharacteristic
514C42C2×C22⋊D8311yescharacteristic
515C42#1305059011yescharacteristic
516C42C23×D8311yescharacteristic
517C23—21no—
518C23—3322no—
519C23—3322no—
520C23—6641no—
521C23—1661no—
522C23—1661no—
523C23C4×C22⋊D8311yescharacteristic
524Q8—21no—
525Q8—21no—
526D4—21no—
527D4—21no—
528D4—42no—
529D4—3322no—
530D4—832no—
531D4—1661no—
532D4—1661no—
533D4—1661no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—3321no—
545C2×C4—3321no—
546C2×C4—3321no—
547C2×C4—3321no—
548C2×C4—6641no—
549C2×C4—6641no—
550C2×C4—6641no—
551C2×C4—1661no—
552C2×C4—1661no—
553C2×C4C4×C22⋊D8311yescharacteristic
554C2×C4C4×C22⋊D8311yescharacteristic
555C2×C4C4×C22⋊D8311yescharacteristic
556C2×C4C22×C4×D8311yescharacteristic
557C2×C4C22×C22⋊D8311yescharacteristic
558C2×C4C2×#1305059011yescharacteristic
559C8—21no—
560C8—21no—
561C8—21no—
562C8—21no—
563C8—41no—
564C8—41no—
565C8—3321no—
566C8—3321no—
567C8—1662no—
568C8—1662no—
569C8—1661no—
570C8—1661no—
571C8—1661no—
572C8—1661no—
573C22—21no—
574C22—21no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—3321no—
579C22—3321no—
580C22—3321no—
581C22—3321no—
582C22—3321no—
583C22—6644no—
584C22—1662no—
585C22#1912781612yes—
586C22C2×C4×C22⋊D8311yescharacteristic
587C4—21no—
588C4—21no—
589C4—21no—
590C4—21no—
591C4—41no—
592C4—41no—
593C4—41no—
594C4—3321no—
595C4—3321no—
596C4#131×D8311yescharacteristic
597C4#1912846611yescharacteristic
598C4#1912850411yescharacteristic
599C4#1912850911yescharacteristic
600C2—22no—
601C2—81no—
602C2—3322no—
603C2—6641no—
604C2#2491618511yescharacteristic
605C2#2491629111yescharacteristic
606C2C2×#1912781611yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D83
commutatorC2×C4×C83C23×C4
FrattiniC42C23×D83
FittingC83×#2994C2
socleC22×C83C2×C4×D4
prime radical#19127834C4
cosocleC42×C83C24
hypercenter#2994D83
hypocenterC83#18014
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ #431 ◁ #2994

Complements

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

NH ≅ G/Ntype
2. #24916497C2semidirect
3. #24906353C2semidirect
4. #24906346C2semidirect
5. #24906341C2semidirect
6. #24906327C2semidirect
7. #24906325C2semidirect
8. #24906318C2semidirect
9. C8×#13050577C2semidirect
10. C8×#13050575C2semidirect
11. #24906246C2semidirect
12. #24905923C2semidirect
13. C2×#19127428C2semidirect
14. #24905747C2semidirect
15. C83×#2994C2semidirect
40. #19127453C22semidirect
41. #19127453C22semidirect
42. #19127451C22semidirect
43. #19127451C22semidirect
44. Q8⋊(C83⋊C8)C22semidirect
45. Q8⋊(C83⋊C8)C22semidirect
46. D4⋊(C83⋊C8)C22semidirect
47. D4⋊(C83⋊C8)C22semidirect
52. #19127423C22semidirect
53. #19127423C22semidirect
54. C8×C83⋊C8C22semidirect
55. C8×C83⋊C8C22semidirect
57. #446×C83C22semidirect
58. C8×Q8×C83C22semidirect
59. #436×C83C22semidirect
62. C8×D4×C83C22semidirect
305. #2994D83semidirect
319. C83#18014semidirect
330. #446C2×D83semidirect
331. C8×Q8C2×D83semidirect
338. #436C2×D83semidirect
345. C8×D4C2×D83semidirect

Maximal subgroups

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

subgroupindex
#249175062
#249164972
#249063532
#249063462
#249063412
#249063272
#249063252
#249063182
C8×#130505772
C8×#130505752
#249062462
#249059232
C2×#191274282
#249057472
C83×#29942
#1801483