Finite Group Databasebeta

← #35408743

Subgroups — #35408743

Order 21248.

874 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127803C211yesmaximal, characteristic
2#24909940C211yesmaximal, characteristic, complemented
3#24909927C211yesmaximal, characteristic
4#24908344C211yesmaximal, characteristic
5#24908317C211yesmaximal, characteristic
6#24908201C211yesmaximal, characteristic, complemented
7#24908196C211yesmaximal, characteristic, complemented
8#24908160C211yesmaximal, characteristic
9#24908153C211yesmaximal, characteristic, complemented
10#24908146C211yesmaximal, characteristic
11#24908139C211yesmaximal, characteristic, complemented
12#24907360C211yesmaximal, characteristic, complemented
13#24907294C211yesmaximal, characteristic, complemented
14#24907289C211yesmaximal, characteristic
15C4×#97×C83C211yesmaximal, characteristic, complemented
16C22×#13050579C2211yescharacteristic
17C2×C4×Dic166C2211yescharacteristic
18C2×C4×D332C2211yescharacteristic
19C2×#14581749—21no—
20#19127803C418yescomplemented
21C2×#14581726C2211yescharacteristic
22C2×#14581726C2211yescharacteristic
23C2×#14581724C2211yescharacteristic
24C2×#14581717C2211yescharacteristic
25C2×#14581717C2211yescharacteristic
26C2×#14581717C2211yescharacteristic
27C2×#14581714C2211yescharacteristic
28C2×#14581714C2211yescharacteristic
29C2×#14581714C2211yescharacteristic
30#19127607C2211yescharacteristic
31#19127607C2211yescharacteristic
32#19127603C2211yescharacteristic
33#19127603C2211yescharacteristic
34#19127599C2211yescharacteristic
35#19127599C2211yescharacteristic
36#19127599C2211yescharacteristic
37#19127599C2211yescharacteristic
38#19127592—21no—
39#19127592—21no—
40#19127585—21no—
41#19127585—21no—
42C4×Dic83⋊C4—21no—
43C4×C4⋊Dic83C2211yescharacteristic
44C4×C4⋊Dic83C2211yescharacteristic
45C4×(C2×D83)⋊C4—21no—
46#19127547—21no—
47#19127464C2211yescharacteristic
48#19127464C2211yescharacteristic
49#19127464C2211yescharacteristic
50#19127464C2211yescharacteristic
51#19127464C2211yescharacteristic
52#19127464C2211yescharacteristic
53C2×#97×C83C2211yescharacteristic
54C2×#97×C83C2211yescharacteristic
55C2×#97×C83C2211yescharacteristic
56C2×#97×C83C2211yescharacteristic
57C2×C4×C8×C83C2211yescharacteristic
58C2×C4×C8×C83C2211yescharacteristic
59C43×C83C2211yescharacteristic
60C22×Dic166C2311yescharacteristic
61C22×D332C2311yescharacteristic
62C22×C4×D83—21no—
63C2×#13050579—22no—
64C2×#13050579—22no—
65C2×#13050579C2×C414yes—
66C4×Dic166—22no—
67C4×Dic166C2×C414yes—
68C4×D332—22no—
69C4×D332C2×C414yes—
70#14581749—44no—
71C2×C4⋊Dic83C2311yescharacteristic
72C2×C4⋊Dic83C2311yescharacteristic
73C2×C4⋊Dic83C2311yescharacteristic
74C2×C4⋊Dic83C2311yescharacteristic
75C2×C4⋊Dic83C2311yescharacteristic
76C2×C4⋊Dic83C2311yescharacteristic
77C2×Dic83⋊C4—21no—
78C2×Dic83⋊C4—21no—
79C2×Dic83⋊C4—21no—
80C2×Dic83⋊C4—21no—
81C2×Dic83⋊C4—21no—
82C2×(C2×D83)⋊C4—21no—
83C2×(C2×D83)⋊C4—21no—
84C2×(C2×D83)⋊C4—21no—
85C2×(C2×D83)⋊C4—21no—
86C2×(C2×D83)⋊C4—21no—
87C2×C4×Dic83—21no—
88C2×C4×Dic83—21no—
89C2×C4×Dic83—21no—
90#14581726—22no—
91#14581726C2×C414yes—
92#14581724—22no—
93#14581717—22no—
94#14581717—22no—
95#14581717C2×C414yes—
96#14581714—22no—
97#14581714—22no—
98#14581714C2×C414yes—
99C83⋊#95—21no—
100C83⋊#95—21no—
101C83⋊#95—21no—
102C83⋊#95—21no—
103C83⋊#95—21no—
104C83⋊#95—21no—
105C83⋊#95—21no—
106C83⋊#95—21no—
107C83⋊#95—21no—
108C83⋊#95—21no—
109C22×C8×C83C2311yescharacteristic
110C22×C8×C83C2311yescharacteristic
111C22×C8×C83C2311yescharacteristic
112C22×C8×C83C2311yescharacteristic
113C2×C42×C83D411yescharacteristic
114C2×C42×C83D411yescharacteristic
115C2×C42×C83D411yescharacteristic
116C2×C42×C83D411yescharacteristic
117C2×C42×C83C2311yescharacteristic
118C2×C42×C83C2311yescharacteristic
119C2×C42×C83C2311yescharacteristic
120#97×C83—22no—
121#97×C83—22no—
122#97×C83—22no—
123#97×C83C2×C414yes—
124C4×C8×C83—22no—
125C4×C8×C83—22no—
126C23×D83—21no—
127C2×Dic166—21no—
128C2×Dic166—21no—
129C2×Dic166—21no—
130C2×Dic166—21no—
131C2×Dic166—21no—
132C2×Dic166C22×C412yes—
133C2×D332—21no—
134C2×D332—21no—
135C2×D332—21no—
136C2×D332—21no—
137C2×D332—21no—
138C2×D332C22×C412yes—
139C22×Dic83—21no—
140C22×Dic83—21no—
141C22×Dic83—21no—
142C22×Dic83—21no—
143C22×Dic83—21no—
144C22×Dic83—21no—
145C22×Dic83—21no—
146C2×C4×D83—24no—
147C2×C4×D83—42no—
148C2×C4×D83—42no—
149#13050579—28no—
150C4⋊Dic83—21no—
151C4⋊Dic83—21no—
152C4⋊Dic83—21no—
153C4⋊Dic83—21no—
154C4⋊Dic83—21no—
155C4⋊Dic83—21no—
156C4⋊Dic83—21no—
157C4⋊Dic83—21no—
158C4⋊Dic83—21no—
159C4⋊Dic83—21no—
160C4⋊Dic83C22×C412yes—
161C4⋊Dic83C22×C412yes—
162Dic83⋊C4—24no—
163Dic83⋊C4—42no—
164Dic83⋊C4—42no—
165Dic83⋊C4—42no—
166Dic83⋊C4—42no—
167(C2×D83)⋊C4—24no—
168(C2×D83)⋊C4—42no—
169(C2×D83)⋊C4—42no—
170(C2×D83)⋊C4—42no—
171(C2×D83)⋊C4—42no—
172C4×Dic83—24no—
173C4×Dic83—42no—
174C4×Dic83—42no—
175C22×C4×C83C2×D411yescharacteristic
176C22×C4×C83C2×D411yescharacteristic
177C22×C4×C83C2×D411yescharacteristic
178C22×C4×C83C2×D411yescharacteristic
179C22×C4×C83C2×D411yescharacteristic
180C22×C4×C83C2×D411yescharacteristic
181C22×C4×C83C2411yescharacteristic
182C2×C8×C83—21no—
183C2×C8×C83—21no—
184C2×C8×C83—21no—
185C2×C8×C83—21no—
186C2×C8×C83—21no—
187C2×C8×C83—21no—
188C2×C8×C83C22×C412yes—
189C2×C8×C83C22×C412yes—
190C2×C8×C83#4112yes—
191C2×C8×C83#4112yes—
192C2×C8×C83#4112yes—
193C2×C8×C83#4112yes—
194C42×C83—22no—
195C42×C83—22no—
196C42×C83—21no—
197C42×C83—21no—
198C42×C83—21no—
199C42×C83—21no—
200C42×C83—41no—
201C42×C83—41no—
202C42×C83C22×C412yes—
203C42×C83#4112yes—
204C23×C83C22×D411yescharacteristic
205C22×D83—22no—
206C22×D83—41no—
207C22×D83—41no—
208C22×D83—41no—
209C22×D83—41no—
210C22×D83—41no—
211C22×D83—41no—
212C2×C4×C83—21no—
213C2×C4×C83—21no—
214C2×C4×C83—21no—
215C2×C4×C83—21no—
216C2×C4×C83—21no—
217C2×C4×C83—21no—
218C2×C4×C83—21no—
219C2×C4×C83—21no—
220C2×C4×C83—21no—
221C2×C4×C83—21no—
222C2×C4×C83—21no—
223C2×C4×C83—21no—
224C2×C4×C83—21no—
225C2×C4×C83—21no—
226C2×C4×C83C4×D412yes—
227C2×C4×C83C4×D412yes—
228C2×C4×C83AGL(1,ℤ/8)12yes—
229C2×C4×C83#13712yes—
230C2×C4×C83C23×C411yescharacteristic
231C2×C4×C83C2×#4111yescharacteristic
232C2×C4×C83C2×#4111yescharacteristic
233C2×C4×C83C2×#4111yescharacteristic
234C2×C4×C83C2×#4111yescharacteristic
235C2×C4×C83C2×#4111yescharacteristic
236C2×Dic83—22no—
237C2×Dic83—22no—
238C2×Dic83—22no—
239C2×Dic83—41no—
240C2×Dic83—41no—
241C2×Dic83—41no—
242C2×Dic83—41no—
243C2×Dic83—41no—
244C2×Dic83—41no—
245C2×Dic83—41no—
246C2×Dic83—41no—
247C2×Dic83—41no—
248C2×Dic83—41no—
249C2×Dic83—41no—
250C2×Dic83—41no—
251C2×Dic83—41no—
252C2×Dic83—41no—
253C2×Dic83—41no—
254C2×Dic83—41no—
255C2×Dic83—41no—
256C2×Dic83—41no—
257D332—22no—
258D332—22no—
259D332—41no—
260D332—41no—
261C4×D83—48no—
262Dic166—22no—
263Dic166—22no—
264Dic166—41no—
265Dic166—41no—
266C8×C83—22no—
267C8×C83—22no—
268C8×C83—22no—
269C8×C83#11714yescomplemented
270C22×C83C2×C4×D411yescharacteristic
271C22×C83#53211yescharacteristic
272C22×C83#53211yescharacteristic
273C22×C83C2×AGL(1,ℤ/8)11yescharacteristic
274C22×C83C2×#13711yescharacteristic
275C22×C83#57511yescharacteristic
276C22×C83#57511yescharacteristic
277C2×D83—42no—
278C2×D83—42no—
279C2×D83—42no—
280C2×D83—81no—
281C2×D83—81no—
282C2×D83—81no—
283C2×D83—81no—
284C4×C83—22no—
285C4×C83—22no—
286C4×C83—22no—
287C4×C83—22no—
288C4×C83—41no—
289C4×C83—41no—
290C4×C83C2×#11711yescharacteristic
291C4×C83C4×#4111yescharacteristic
292C4×C83#52911yescharacteristic
293C4×C83#52911yescharacteristic
294Dic83—42no—
295Dic83—42no—
296Dic83—42no—
297Dic83—81no—
298Dic83—81no—
299Dic83—81no—
300Dic83—81no—
301#21288—831nomaximal, Sylow
302C2×C83#229311yescharacteristic
303C2×C83C4×AGL(1,ℤ/8)11yescharacteristic
304C2×C83C4×#13711yescharacteristic
305C2×C83#318011yescharacteristic
306C2×C83#318111yescharacteristic
307C2×C83#318211yescharacteristic
308C2×C83#318311yescharacteristic
309D83—82no—
310C2×#440—831no—
311#2295—831no—
312#2293—831no—
313#1973—831no—
314#1968—831no—
315#1943—831no—
316#1942—831no—
317#1936—831no—
318#1935—831no—
319#1934—831no—
320#1933—831no—
321#1776—831no—
322#1763—831no—
323#1762—831no—
324C4×#97D8311yescharacteristic, complemented
325C83#2128811yescharacteristic, complemented, Sylow
326C22×SD16—831no—
327C2×C4×Q8—831no—
328C2×C4×D4—831no—
329C2×#117—1661no—
330#440—838no—
331C2×#107—831no—
332C2×#107—831no—
333C2×#106—831no—
334C2×#103—831no—
335C2×#103—831no—
336C2×#103—831no—
337C2×#102—831no—
338C2×#102—831no—
339C2×#102—831no—
340C2×#97C2×D8311yescharacteristic
341C2×#97C2×D8311yescharacteristic
342C2×#97C2×D8311yescharacteristic
343C2×#97C2×D8311yescharacteristic
344C2×C4×C8C2×D8311yescharacteristic
345C2×C4×C8C2×D8311yescharacteristic
346#391—831no—
347#391—831no—
348#390—831no—
349#390—831no—
350#389—831no—
351#389—831no—
352#389—831no—
353#389—831no—
354#388—1661no—
355#388—1661no—
356#387—1661no—
357#387—1661no—
358C4×C4⋊C4—831no—
359C4×C4⋊C4—831no—
360C4×C4⋊C4—1661no—
361C4×C22⋊C4—1661no—
362#376—1661no—
363C43C2×D8311yescharacteristic
364#340—831no—
365#340—831no—
366#340—831no—
367#340—831no—
368#340—831no—
369#340—831no—
370C22×Q8—831no—
371C22×D4—831no—
372C23×C4—1661no—
373C2×SD16—834no—
374C2×SD16—1662no—
375C2×SD16—1662no—
376C22×C8C22×D8311yescharacteristic
377C22×C8C22×D8311yescharacteristic
378C22×C8C22×D8311yescharacteristic
379C22×C8C22×D8311yescharacteristic
380C4×Q8—834no—
381C4×Q8—1662no—
382C4×D4—834no—
383C4×D4—1662no—
384#117—3324no—
385C2×C4⋊C4—831no—
386C2×C4⋊C4—831no—
387C2×C4⋊C4—831no—
388C2×C4⋊C4—831no—
389C2×C4⋊C4—831no—
390C2×C4⋊C4—831no—
391C2×C4⋊C4—1661no—
392C2×C4⋊C4—1661no—
393C2×C4⋊C4—1661no—
394C2×C4⋊C4—1661no—
395C2×C4⋊C4—1661no—
396C2×C22⋊C4—1661no—
397C2×C22⋊C4—1661no—
398C2×C22⋊C4—1661no—
399C2×C22⋊C4—1661no—
400C2×C22⋊C4—1661no—
401C2×C42—1661no—
402C2×C42—1661no—
403C2×C42—1661no—
404C2×C42D33211yescharacteristic
405C2×C42D33211yescharacteristic
406C2×C42D33211yescharacteristic
407C2×C42D33211yescharacteristic
408C2×C42C22×D8311yescharacteristic
409C2×C42C22×D8311yescharacteristic
410C2×C42C22×D8311yescharacteristic
411#107—834no—
412#107—1662no—
413#106—1662no—
414#103—834no—
415#103—1662no—
416#103—1662no—
417#102—834no—
418#102—1662no—
419#102—1662no—
420#97—22no—
421#97—22no—
422#97—22no—
423#97C4×D8314yes—
424C4×C8—22no—
425C4×C8—22no—
426#95—1661no—
427#95—1661no—
428#95—1661no—
429#95—1661no—
430#95—1661no—
431#95—1661no—
432#95—1661no—
433#95—1661no—
434#95—1661no—
435#95—1661no—
436C24—1661no—
437C2×Q8—832no—
438C2×Q8—1661no—
439C2×Q8—1661no—
440C2×Q8—1661no—
441C2×Q8—1661no—
442C2×Q8—1661no—
443C2×D4—832no—
444C2×D4—1661no—
445C2×D4—1661no—
446C2×D4—1661no—
447C2×D4—1661no—
448C2×D4—1661no—
449C22×C4—3322no—
450C22×C4—3322no—
451C22×C4—1661no—
452C22×C4—1661no—
453C22×C4—1661no—
454C22×C4—1661no—
455C22×C4—1661no—
456C22×C4—1661no—
457C22×C4—1661no—
458C22×C4—1664no—
459C22×C4C2×D33211yescharacteristic
460C22×C4C2×D33211yescharacteristic
461C22×C4C2×D33211yescharacteristic
462C22×C4C2×D33211yescharacteristic
463C22×C4C2×D33211yescharacteristic
464C22×C4C2×D33211yescharacteristic
465C22×C4C23×D8311yescharacteristic
466SD16—1668no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8C2×C4×D8312yes—
474C2×C8C2×C4×D8312yes—
475C2×C8#1305059012yes—
476C2×C8#1305059012yes—
477C2×C8#1305059012yes—
478C2×C8#1305059012yes—
479C4⋊C4—3322no—
480C4⋊C4—3322no—
481C4⋊C4—3322no—
482C4⋊C4—3322no—
483C4⋊C4—832no—
484C4⋊C4—832no—
485C4⋊C4—1661no—
486C4⋊C4—1661no—
487C4⋊C4—1661no—
488C4⋊C4—1661no—
489C4⋊C4—1661no—
490C4⋊C4—1661no—
491C4⋊C4—1661no—
492C4⋊C4—1661no—
493C4⋊C4—1661no—
494C4⋊C4—1661no—
495C4⋊C4—1664no—
496C22⋊C4—3322no—
497C22⋊C4—3322no—
498C22⋊C4—3322no—
499C22⋊C4—3322no—
500C22⋊C4—1664no—
501C42—22no—
502C42—22no—
503C42—21no—
504C42—21no—
505C42—21no—
506C42—21no—
507C42—41no—
508C42—41no—
509C42—3322no—
510C42—3322no—
511C42—1664no—
512C42C2×C4×D8312yes—
513C42#1305059012yes—
514C23—3321no—
515C23—3321no—
516C23—3321no—
517C23—3321no—
518C23—3321no—
519C23—3321no—
520C23—1662no—
521C23C22×D33211yescharacteristic
522Q8—3321no—
523Q8—3321no—
524Q8—1662no—
525Q8—1662no—
526D4—3321no—
527D4—3321no—
528D4—1662no—
529D4—1662no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—3328no—
545C2×C4—3321no—
546C2×C4—3321no—
547C2×C4—3321no—
548C2×C4—3321no—
549C2×C4—3321no—
550C2×C4—3321no—
551C2×C4—3321no—
552C2×C4—3321no—
553C2×C4—3321no—
554C2×C4—3321no—
555C2×C4—3321no—
556C2×C4—3321no—
557C2×C4—3321no—
558C2×C4—3321no—
559C2×C4—3321no—
560C2×C4—3321no—
561C2×C4—3321no—
562C2×C4—3321no—
563C2×C4—1662no—
564C2×C4—1662no—
565C2×C4—1662no—
566C2×C4C4×D33212yes—
567C2×C4C4×D33212yes—
568C2×C4#1458182012yes—
569C2×C4#1458182512yes—
570C2×C4C22×C4×D8311yescharacteristic
571C2×C4C2×#1305059011yescharacteristic
572C2×C4C2×#1305059011yescharacteristic
573C2×C4C2×#1305059011yescharacteristic
574C2×C4C2×#1305059011yescharacteristic
575C2×C4C2×#1305059011yescharacteristic
576C8—22no—
577C8—22no—
578C8—22no—
579C8#1458174914yescomplemented
580C22—3322no—
581C22—3322no—
582C22—3322no—
583C22—6641no—
584C22—6641no—
585C22—6641no—
586C22—6641no—
587C22C2×C4×D33211yescharacteristic
588C22#1912825111yescharacteristic
589C22#1912825111yescharacteristic
590C22C2×#1458182011yescharacteristic
591C22C2×#1458182511yescharacteristic
592C22#1912850111yescharacteristic
593C22#1912850111yescharacteristic
594C4—22no—
595C4—22no—
596C4—22no—
597C4—22no—
598C4—41no—
599C4—41no—
600C4—3322no—
601C4—3322no—
602C4—3322no—
603C4—6641no—
604C4—6641no—
605C4—6641no—
606C4—6641no—
607C4C2×#1458174911yescharacteristic
608C4C4×#1305059011yescharacteristic
609C4#1912823511yescharacteristic
610C4#1912823511yescharacteristic
611C2—6642no—
612C2#2490992711yescharacteristic
613C2C4×#1458182011yescharacteristic
614C2C4×#1458182511yescharacteristic
615C2#2491816111yescharacteristic
616C2#2491817211yescharacteristic
617C2#2491818111yescharacteristic
618C2#2491819211yescharacteristic
subgroupG/H
centerC23C22×D332
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC4×#97×C83C2
socleC23×C83C22×D4
prime radicalC22×D332C23
cosocleC22×C4×C83C24
hypercenterC4×#97D83
hypocenterC83#21288
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C43 ◁ C4×#97

Complements

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

NH ≅ G/Ntype
2. #24909940C2semidirect
6. #24908201C2semidirect
7. #24908196C2semidirect
9. #24908153C2semidirect
11. #24908139C2semidirect
12. #24907360C2semidirect
13. #24907294C2semidirect
15. C4×#97×C83C2semidirect
20. #19127803C4semidirect
269. C8×C83#117semidirect
324. C4×#97D83semidirect
325. C83#21288semidirect
579. C8#14581749semidirect

Maximal subgroups

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

subgroupindex
C2×#191278032
#249099402
#249099272
#249083442
#249083172
#249082012
#249081962
#249081602
#249081532
#249081462
#249081392
#249073602
#249072942
#249072892
C4×#97×C832
#2128883