Finite Group Databasebeta

← #35472609

Subgroups — #35472609

Order 21248.

726 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24911804C211yesmaximal, characteristic
2#24908785C211yesmaximal, characteristic, complemented
3#24908663C211yesmaximal, characteristic
4#24908614C211yesmaximal, characteristic, complemented
5#24908471C211yesmaximal, characteristic
6#24908420C211yesmaximal, characteristic, complemented
7#24908138C211yesmaximal, characteristic, complemented
8#24908104C211yesmaximal, characteristic, complemented
9#24908071C211yesmaximal, characteristic
10#24908031C211yesmaximal, characteristic, complemented
11#24907804C211yesmaximal, characteristic, complemented
12#24907775C211yesmaximal, characteristic
13#24907558C211yesmaximal, characteristic
14C4×#14581716C211yesmaximal, characteristic, complemented
15C83×#2362C211yesmaximal, characteristic
16C22×#13050580C2211yescharacteristic
17C2×#14581784C2211yescharacteristic
18C2×#14581771C2211yescharacteristic
19C2×#14581725C2211yescharacteristic
20C2×C4⋊(C83⋊C8)C2211yescharacteristic
21C2×#14581716C2211yescharacteristic
22C2×#14581716C2211yescharacteristic
23C2×#14581716C2211yescharacteristic
24C2×#14581715C2211yescharacteristic
25C2×#14581715C2211yescharacteristic
26C2×#14581715C2211yescharacteristic
27C2×C22⋊(C83⋊C8)C2211yescharacteristic
28C2×C22⋊(C83⋊C8)C2211yescharacteristic
29C2×C4×C83⋊C8C2211yescharacteristic
30#19127642—21no—
31#19127629—21no—
32#19127607C2211yescharacteristic
33#19127607C2211yescharacteristic
34#19127589—21no—
35#19127565—21no—
36#19127561C2211yescharacteristic
37C4×C4⋊Dic83C2211yescharacteristic
38#19127463C2211yescharacteristic
39#19127463C2211yescharacteristic
40#19127463C2211yescharacteristic
41#19127463C2211yescharacteristic
42#19127463C2211yescharacteristic
43#19127463C2211yescharacteristic
44#19127463C2211yescharacteristic
45#19127463C2211yescharacteristic
46#19127456C2211yescharacteristic
47#19127456C2211yescharacteristic
48C2×C22⋊Q8×C83C2211yescharacteristic
49C2×C4×Q8×C83C2211yescharacteristic
50C2×#117×C83C2211yescharacteristic
51#391×C83C2211yescharacteristic
52#389×C83C2211yescharacteristic
53#388×C83—21no—
54#388×C83—21no—
55#387×C83—21no—
56C4×C4⋊C4×C83—21no—
57C4×C4⋊C4×C83C2211yescharacteristic
58C4×C22⋊C4×C83C2211yescharacteristic
59C22×Dic166C2311yescharacteristic
60C2×#13050580—22no—
61C2×#13050580—41no—
62C2×#13050580—41no—
63C22×C83⋊C8C2311yescharacteristic
64C22×C83⋊C8C2311yescharacteristic
65C22×C83⋊C8C2311yescharacteristic
66C22×C83⋊C8C2311yescharacteristic
67#14581784—22no—
68#14581784—41no—
69#14581771—22no—
70#14581771—41no—
71C2×C4⋊Dic83C2311yescharacteristic
72C2×C4⋊Dic83C2311yescharacteristic
73C2×C4⋊Dic83C2311yescharacteristic
74C2×Dic83⋊C4—21no—
75C2×Dic83⋊C4—21no—
76C2×Dic83⋊C4—21no—
77C2×C22⋊Dic83—21no—
78C2×C22⋊Dic83—21no—
79C2×C4×Dic83—21no—
80#14581725—22no—
81C4⋊(C83⋊C8)—41no—
82#14581716—22no—
83#14581716—22no—
84#14581716—41no—
85#14581715—22no—
86#14581715—22no—
87#14581715—41no—
88C22⋊(C83⋊C8)—41no—
89C22⋊(C83⋊C8)—41no—
90C4×C83⋊C8—41no—
91C83⋊#95—21no—
92C83⋊#95—21no—
93C83⋊#95—21no—
94C83⋊#95—21no—
95C83⋊#95—21no—
96C83⋊#95—21no—
97C22×Q8×C83C2311yescharacteristic
98C23×C4×C83C2311yescharacteristic
99C22⋊Q8×C83—22no—
100C22⋊Q8×C83—22no—
101C4×Q8×C83—22no—
102C4×Q8×C83—41no—
103#117×C83—22no—
104#117×C83—41no—
105C2×C4⋊C4×C83—21no—
106C2×C4⋊C4×C83—21no—
107C2×C4⋊C4×C83—21no—
108C2×C4⋊C4×C83C2311yescharacteristic
109C2×C4⋊C4×C83C2311yescharacteristic
110C2×C4⋊C4×C83C2311yescharacteristic
111C2×C22⋊C4×C83—21no—
112C2×C22⋊C4×C83—21no—
113C2×C22⋊C4×C83D411yescharacteristic
114C2×C22⋊C4×C83D411yescharacteristic
115C2×C42×C83—21no—
116C2×C42×C83—21no—
117C2×C42×C83—21no—
118C2×C42×C83C2311yescharacteristic
119C2×C42×C83C2311yescharacteristic
120#95×C83—21no—
121#95×C83—21no—
122#95×C83—21no—
123#95×C83—21no—
124#95×C83D411yescharacteristic
125#95×C83D411yescharacteristic
126C2×Dic166—21no—
127C2×Dic166—21no—
128C2×Dic166—21no—
129C2×Dic166—21no—
130C2×Dic166—21no—
131C2×Dic166—21no—
132C22×Dic83—21no—
133C22×Dic83—21no—
134C22×Dic83—21no—
135C22×Dic83—21no—
136#13050580—44no—
137C2×C83⋊C8—21no—
138C2×C83⋊C8—21no—
139C2×C83⋊C8—21no—
140C2×C83⋊C8—21no—
141C2×C83⋊C8—21no—
142C2×C83⋊C8—21no—
143C2×C83⋊C8—21no—
144C2×C83⋊C8—21no—
145C2×C83⋊C8—21no—
146C2×C83⋊C8—21no—
147C2×C83⋊C8—21no—
148C2×C83⋊C8—21no—
149C4⋊Dic83—21no—
150C4⋊Dic83—21no—
151C4⋊Dic83—21no—
152C4⋊Dic83—21no—
153C4⋊Dic83—21no—
154C4⋊Dic83#4112yes—
155Dic83⋊C4—42no—
156Dic83⋊C4—42no—
157Dic83⋊C4—42no—
158C22⋊Dic83—42no—
159C22⋊Dic83—42no—
160C4×Dic83—42no—
161C24×C83C2×D411yescharacteristic
162C2×Q8×C83—21no—
163C2×Q8×C83—21no—
164C2×Q8×C83—21no—
165C2×Q8×C83—21no—
166C2×Q8×C83—21no—
167C2×Q8×C83#4112yes—
168C22×C4×C83—21no—
169C22×C4×C83—21no—
170C22×C4×C83—21no—
171C22×C4×C83—21no—
172C22×C4×C83—21no—
173C22×C4×C83—21no—
174C22×C4×C83—21no—
175C22×C4×C83—21no—
176C22×C4×C83—21no—
177C22×C4×C83—21no—
178C22×C4×C83C2×D411yescharacteristic
179C22×C4×C83C2×D411yescharacteristic
180C22×C4×C83C2×D411yescharacteristic
181C22×C4×C83C2×D411yescharacteristic
182C22×C4×C83C2×D411yescharacteristic
183C22×C4×C83C2411yescharacteristic
184C4⋊C4×C83—24no—
185C4⋊C4×C83—21no—
186C4⋊C4×C83—21no—
187C4⋊C4×C83—21no—
188C4⋊C4×C83—21no—
189C4⋊C4×C83—42no—
190C4⋊C4×C83—42no—
191C4⋊C4×C83#4112yes—
192C4⋊C4×C83#4112yes—
193C22⋊C4×C83—24no—
194C22⋊C4×C83—22no—
195C22⋊C4×C83—41no—
196C22⋊C4×C83—42no—
197C42×C83—21no—
198C42×C83—21no—
199C42×C83—21no—
200C42×C83—21no—
201C42×C83—42no—
202C42×C83—42no—
203C42×C83—42no—
204C23×C83—21no—
205C23×C83—21no—
206C23×C83—21no—
207C23×C83—21no—
208C23×C83—21no—
209C23×C83—21no—
210C23×C83—21no—
211C23×C83C22×D411yescharacteristic
212Q8×C83—22no—
213Q8×C83—22no—
214Q8×C83—41no—
215Q8×C83—41no—
216C2×C4×C83—21no—
217C2×C4×C83—21no—
218C2×C4×C83—22no—
219C2×C4×C83—22no—
220C2×C4×C83—21no—
221C2×C4×C83—21no—
222C2×C4×C83—21no—
223C2×C4×C83—21no—
224C2×C4×C83—22no—
225C2×C4×C83—21no—
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×C83—41no—
236C2×C4×C83—41no—
237C2×C4×C83—41no—
238C2×C4×C83—41no—
239C2×C4×C83—41no—
240C2×C4×C83—41no—
241C2×C4×C83—41no—
242C2×C4×C83—41no—
243C2×C4×C83—41no—
244C2×C4×C83—41no—
245C2×C4×C83—41no—
246C2×C4×C83#13512yes—
247C2×C4×C83C2×#4111yescharacteristic
248C2×C4×C83C2×#4111yescharacteristic
249C2×C4×C83C2×#4111yescharacteristic
250C2×C4×C83C2×#4111yescharacteristic
251C2×C4×C83H(5,𝔽2)11yescharacteristic
252C2×C4×C83H(5,𝔽2)11yescharacteristic
253C2×Dic83—22no—
254C2×Dic83—41no—
255C2×Dic83—41no—
256C2×Dic83—41no—
257C2×Dic83—41no—
258C2×Dic83—41no—
259C2×Dic83—41no—
260C2×Dic83—41no—
261C2×Dic83—41no—
262C2×Dic83—41no—
263C2×Dic83—41no—
264C2×Dic83—41no—
265Dic166—41no—
266Dic166—41no—
267Dic166—41no—
268Dic166—41no—
269C83⋊C8—41no—
270C83⋊C8—41no—
271C83⋊C8—41no—
272C83⋊C8—41no—
273C22×C83—41no—
274C22×C83—41no—
275C22×C83—41no—
276C22×C83—41no—
277C22×C83—41no—
278C22×C83—41no—
279C22×C83—41no—
280C22×C83#52511yescharacteristic
281C22×C83#54611yescharacteristic
282C22×C83#54611yescharacteristic
283C22×C83C2×#13511yescharacteristic
284C22×C83#57511yescharacteristic
285C22×C83#57811yescharacteristic
286C22×C83#57811yescharacteristic
287C4×C83—22no—
288C4×C83—42no—
289C4×C83—41no—
290C4×C83—42no—
291C4×C83—41no—
292C4×C83—41no—
293C4×C83—42no—
294C4×C83—41no—
295C4×C83—81no—
296C4×C83#52911yescharacteristic
297C4×C83#53811yescharacteristic
298C4×C83#55511yescharacteristic
299C4×C83#55911yescharacteristic
300Dic83—42no—
301Dic83—81no—
302Dic83—81no—
303Dic83—81no—
304#26732—831nomaximal, Sylow
305C2×C83—81no—
306C2×C83#245911yescharacteristic
307C2×C83#310211yescharacteristic
308C2×C83#312711yescharacteristic
309C2×C83#330011yescharacteristic
310C2×C83#330211yescharacteristic
311C2×C83#331511yescharacteristic
312C2×C83#332711yescharacteristic
313#2489—831no—
314#2362D8311yescharacteristic
315#2056—831no—
316#2036—831no—
317#2026—831no—
318#2001—831no—
319#1989—831no—
320#1933—831no—
321#1928—831no—
322#1923—831no—
323#1918—831no—
324#1878—831no—
325#1873—831no—
326#1825—831no—
327C4×#103—831no—
328C83#2673211yescharacteristic, complemented, Sylow
329C22×Q16—831no—
330C2×#125—831no—
331C2×C22⋊Q8—831no—
332C2×C22⋊Q8C2×D8311yescharacteristic
333C2×C4×Q8C2×D8311yescharacteristic
334C2×#117C2×D8311yescharacteristic
335C2×#107—831no—
336C2×C4⋊C8—831no—
337C2×#103—831no—
338C2×#103—831no—
339C2×#103—831no—
340C2×#103—831no—
341C2×#103—831no—
342C2×#103—831no—
343C2×C22⋊C8—831no—
344C2×C22⋊C8—831no—
345C2×C4×C8—831no—
346#400—1661no—
347#397—1661no—
348#391—831no—
349#391—831no—
350#391C2×D8311yescharacteristic
351#389C2×D8311yescharacteristic
352#388—21no—
353#388—21no—
354#388—1661no—
355#387—21no—
356#382—1661no—
357#380—831no—
358C4×C4⋊C4—21no—
359C4×C4⋊C4—831no—
360C4×C4⋊C4C2×D8311yescharacteristic
361C4×C22⋊C4C2×D8311yescharacteristic
362#340—831no—
363#340—831no—
364#340—831no—
365#340—831no—
366#340—831no—
367#340—831no—
368#340—831no—
369#340—831no—
370#336—831no—
371#336—831no—
372C22×Q8—831no—
373C22×Q8C22×D8311yescharacteristic
374C23×C4C22×D8311yescharacteristic
375C2×Q16—3321no—
376C2×Q16—3321no—
377C2×Q16—1662no—
378C22×C8—831no—
379C22×C8—831no—
380C22×C8—831no—
381C22×C8—831no—
382#125—3321no—
383#125—1662no—
384C22⋊Q8—22no—
385C22⋊Q8—22no—
386C22⋊Q8—3321no—
387C22⋊Q8—1662no—
388C4×Q8—22no—
389C4×Q8—41no—
390#117—22no—
391#117—41no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—831no—
396C2×C4⋊C4—831no—
397C2×C4⋊C4—831no—
398C2×C4⋊C4—1661no—
399C2×C4⋊C4—1661no—
400C2×C4⋊C4—1661no—
401C2×C4⋊C4C22×D8311yescharacteristic
402C2×C4⋊C4C22×D8311yescharacteristic
403C2×C4⋊C4C22×D8311yescharacteristic
404C2×C22⋊C4—21no—
405C2×C22⋊C4—21no—
406C2×C22⋊C4—1661no—
407C2×C22⋊C4—1661no—
408C2×C22⋊C4C22⋊D8311yescharacteristic
409C2×C22⋊C4C22⋊D8311yescharacteristic
410C2×C42—21no—
411C2×C42—21no—
412C2×C42—21no—
413C2×C42—1661no—
414C2×C42C22×D8311yescharacteristic
415C2×C42C22×D8311yescharacteristic
416#107—1662no—
417C4⋊C8—3321no—
418#103—3321no—
419#103—3321no—
420#103—1662no—
421#103—1662no—
422#103—1662no—
423#103—1662no—
424C22⋊C8—3321no—
425C22⋊C8—3321no—
426C4×C8—3321no—
427#95—21no—
428#95—21no—
429#95—21no—
430#95—21no—
431#95—1661no—
432#95—1661no—
433#95—1661no—
434#95—1661no—
435#95—1661no—
436#95—1661no—
437#95C22⋊D8311yescharacteristic
438#95C22⋊D8311yescharacteristic
439C24C2×C22⋊D8311yescharacteristic
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—1661no—
446C2×Q8—1661no—
447C2×Q8—1661no—
448C2×Q8—1661no—
449C2×Q8—1661no—
450C2×Q8—1661no—
451C2×Q8#1305059012yes—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—1661no—
463C22×C4—1661no—
464C22×C4—1661no—
465C22×C4—1661no—
466C22×C4C2×C22⋊D8311yescharacteristic
467C22×C4C2×C22⋊D8311yescharacteristic
468C22×C4C2×C22⋊D8311yescharacteristic
469C22×C4C2×C22⋊D8311yescharacteristic
470C22×C4C2×C22⋊D8311yescharacteristic
471C22×C4C23×D8311yescharacteristic
472Q16—3324no—
473C2×C8—1661no—
474C2×C8—1661no—
475C2×C8—1661no—
476C2×C8—1661no—
477C2×C8—1661no—
478C2×C8—1661no—
479C2×C8—1661no—
480C2×C8—1661no—
481C2×C8—1661no—
482C2×C8—1661no—
483C2×C8—1661no—
484C2×C8—1661no—
485C4⋊C4—24no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—3322no—
493C4⋊C4—3322no—
494C4⋊C4—3322no—
495C4⋊C4—832no—
496C4⋊C4—1661no—
497C4⋊C4—1661no—
498C4⋊C4—1661no—
499C4⋊C4—1661no—
500C4⋊C4—1661no—
501C4⋊C4#1305058912yes—
502C4⋊C4#1305059012yes—
503C22⋊C4—24no—
504C22⋊C4—22no—
505C22⋊C4—41no—
506C22⋊C4—42no—
507C22⋊C4—3322no—
508C22⋊C4—3322no—
509C42—21no—
510C42—21no—
511C42—21no—
512C42—21no—
513C42—42no—
514C42—42no—
515C42—42no—
516C42—3322no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23C22×C22⋊D8311yescharacteristic
525Q8—22no—
526Q8—22no—
527Q8—41no—
528Q8—41no—
529Q8—3321no—
530Q8—3321no—
531Q8—3321no—
532Q8—3321no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—3321no—
564C2×C4—3321no—
565C2×C4—3321no—
566C2×C4—3321no—
567C2×C4—3321no—
568C2×C4—3321no—
569C2×C4—3321no—
570C2×C4—3321no—
571C2×C4—3321no—
572C2×C4—3321no—
573C2×C4—3321no—
574C2×C4—1662no—
575C2×C4#1458181212yes—
576C2×C4#41×D8311yescharacteristic
577C2×C4C2×#1305058911yescharacteristic
578C2×C4C2×#1305059011yescharacteristic
579C2×C4C2×#1305059011yescharacteristic
580C2×C4#1458183711yescharacteristic
581C2×C4#1458183811yescharacteristic
582C8—3321no—
583C8—3321no—
584C8—3321no—
585C8—3321no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22#1912821211yescharacteristic
594C22#1912833911yescharacteristic
595C22#1912834511yescharacteristic
596C22C2×#1458181211yescharacteristic
597C22#1912849911yescharacteristic
598C22#1912851611yescharacteristic
599C22#1912851611yescharacteristic
600C4—22no—
601C4—41no—
602C4—42no—
603C4—41no—
604C4—42no—
605C4—41no—
606C4—41no—
607C4—42no—
608C4—81no—
609C4—3322no—
610C4—6641no—
611C4—6641no—
612C4—6641no—
613C4#1912823111yescharacteristic
614C4#1912828411yescharacteristic
615C4#1912841611yescharacteristic
616C4#1912843411yescharacteristic
617C2—81no—
618C2#2491139411yescharacteristic
619C2#2491748111yescharacteristic
620C2#2491776311yescharacteristic
621C2#2491935011yescharacteristic
622C2#2491937411yescharacteristic
623C2#2491955911yescharacteristic
624C2#2491972511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingC83×#2362C2
socleC23×C83C22×D4
prime radicalC24×C83C2×D4
hypercenter#2362D83
hypocenterC83#26732
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2362

Complements

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

NH ≅ G/Ntype
2. #24908785C2semidirect
4. #24908614C2semidirect
6. #24908420C2semidirect
7. #24908138C2semidirect
8. #24908104C2semidirect
10. #24908031C2semidirect
11. #24907804C2semidirect
14. C4×#14581716C2semidirect
328. C83#26732semidirect

Maximal subgroups

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

subgroupindex
#249118042
#249087852
#249086632
#249086142
#249084712
#249084202
#249081382
#249081042
#249080712
#249080312
#249078042
#249077752
#249075582
C4×#145817162
C83×#23622
#2673283