Finite Group Databasebeta

← #35466560

Subgroups — #35466560

Order 21248.

742 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24912636C211yesmaximal, characteristic
2#24910392C211yesmaximal, characteristic
3#24908896C211yesmaximal, characteristic, complemented
4#24908891C211yesmaximal, characteristic, complemented
5#24908891C211yesmaximal, characteristic, complemented
6#24908886C211yesmaximal, characteristic, complemented
7#24908436C211yesmaximal, characteristic, complemented
8#24908414C211yesmaximal, characteristic, complemented
9#24908233C211yesmaximal, characteristic, complemented
10#24907664C211yesmaximal, characteristic, complemented
11#24907658C211yesmaximal, characteristic, complemented
12#24907559C211yesmaximal, characteristic, complemented
13#24907550C211yesmaximal, characteristic, complemented
14#24907360C211yesmaximal, characteristic, complemented
15C83×#1955C211yesmaximal, characteristic
16C2×#14581792C2211yescharacteristic
17C2×#14581751C2211yescharacteristic
18C22×C4⋊Dic83C2211yescharacteristic
19C2×#14581726C2211yescharacteristic
20C2×#14581726C2211yescharacteristic
21C2×#14581724C2211yescharacteristic
22C2×#14581724C2211yescharacteristic
23#19127613—21no—
24#19127613—21no—
25#19127604—21no—
26#19127594—21no—
27#19127587—21no—
28#19127581—21no—
29#19127567C2211yescharacteristic
30#19127566—21no—
31#19127561C2211yescharacteristic
32#19127561C2211yescharacteristic
33#19127561C2211yescharacteristic
34#19127561C2211yescharacteristic
35#19127561C2211yescharacteristic
36#19127561C2211yescharacteristic
37C4×C4⋊Dic83C2211yescharacteristic
38C4×C4⋊Dic83C2211yescharacteristic
39C22⋊C4×Dic83—21no—
40#19127464C2211yescharacteristic
41#19127464C2211yescharacteristic
42#19127464C2211yescharacteristic
43#19127464C2211yescharacteristic
44#19127464C2211yescharacteristic
45#19127464C2211yescharacteristic
46#19127464C2211yescharacteristic
47#19127464C2211yescharacteristic
48#19127464C2211yescharacteristic
49#19127464C2211yescharacteristic
50#19127464C2211yescharacteristic
51#19127464C2211yescharacteristic
52C2×C4×D4×C83C2211yescharacteristic
53C2×C4⋊C8×C83C2211yescharacteristic
54C2×C22⋊C8×C83C2211yescharacteristic
55C2×C22⋊C8×C83C2211yescharacteristic
56C2×C22⋊C8×C83C2211yescharacteristic
57C2×C22⋊C8×C83C2211yescharacteristic
58C2×#97×C83C2211yescharacteristic
59C22×Dic166—21no—
60C23×Dic83—21no—
61#14581792—22no—
62#14581792—22no—
63#14581751—22no—
64#14581751—41no—
65C2×C4⋊Dic83—22no—
66C2×C4⋊Dic83—21no—
67C2×C4⋊Dic83—21no—
68C2×C4⋊Dic83—41no—
69C2×C4⋊Dic83C2311yescharacteristic
70C2×C4⋊Dic83C2311yescharacteristic
71C2×C4⋊Dic83C2311yescharacteristic
72C2×C4⋊Dic83C2311yescharacteristic
73C2×C4⋊Dic83C2311yescharacteristic
74C2×C4⋊Dic83C2311yescharacteristic
75C2×C4⋊Dic83C2311yescharacteristic
76C2×C4⋊Dic83C2311yescharacteristic
77C2×C22⋊Dic83—21no—
78C2×C22⋊Dic83—21no—
79C2×C22⋊Dic83—21no—
80C2×C22⋊Dic83—21no—
81C2×C22⋊Dic83—21no—
82C2×C22⋊Dic83—21no—
83C2×C22⋊Dic83—21no—
84C2×C4×Dic83—21no—
85C2×C4×Dic83—21no—
86C2×C4×Dic83—21no—
87#14581726—22no—
88#14581726—41no—
89#14581724—22no—
90#14581724—41no—
91C83⋊#95—21no—
92C83⋊#95—21no—
93C83⋊#95—21no—
94C83⋊#95—21no—
95C83⋊#95—21no—
96C83⋊#95—21no—
97C83⋊#95—21no—
98C83⋊#95—21no—
99C83⋊#95—21no—
100C83⋊#95—21no—
101C22×D4×C83D411yescharacteristic
102C23×C4×C83C2311yescharacteristic
103C23×C4×C83C2311yescharacteristic
104C22×C8×C83C2311yescharacteristic
105C22×C8×C83C2311yescharacteristic
106C22×C8×C83C2311yescharacteristic
107C22×C8×C83C2311yescharacteristic
108C4×D4×C83—22no—
109C4×D4×C83—22no—
110C2×C4⋊C4×C83D411yescharacteristic
111C2×C22⋊C4×C83D411yescharacteristic
112C2×C22⋊C4×C83D411yescharacteristic
113C2×C42×C83C2311yescharacteristic
114C4⋊C8×C83—22no—
115C22⋊C8×C83—22no—
116C22⋊C8×C83—22no—
117C22⋊C8×C83—41no—
118C22⋊C8×C83—41no—
119#97×C83—22no—
120C2×Dic166—42no—
121C2×Dic166—81no—
122C2×Dic166—81no—
123C22×Dic83—21no—
124C22×Dic83—21no—
125C22×Dic83—21no—
126C22×Dic83—21no—
127C22×Dic83—21no—
128C22×Dic83—21no—
129C22×Dic83—21no—
130C22×Dic83—21no—
131C22×Dic83—42no—
132C22×Dic83—42no—
133C22×Dic83—81no—
134C4⋊Dic83—21no—
135C4⋊Dic83—21no—
136C4⋊Dic83—21no—
137C4⋊Dic83—21no—
138C4⋊Dic83—21no—
139C4⋊Dic83—21no—
140C4⋊Dic83—21no—
141C4⋊Dic83—21no—
142C4⋊Dic83—21no—
143C4⋊Dic83—21no—
144C4⋊Dic83—21no—
145C4⋊Dic83—21no—
146C4⋊Dic83—21no—
147C4⋊Dic83—21no—
148C4⋊Dic83—21no—
149C4⋊Dic83—42no—
150C4⋊Dic83—42no—
151C4⋊Dic83#4112yes—
152C22⋊Dic83—42no—
153C22⋊Dic83—42no—
154C22⋊Dic83—81no—
155C22⋊Dic83—81no—
156C22⋊Dic83—81no—
157C22⋊Dic83—81no—
158C22⋊Dic83—81no—
159C4×Dic83—42no—
160C4×Dic83—42no—
161C4×Dic83—81no—
162C24×C83C2×D411yescharacteristic
163C24×C83C2×D411yescharacteristic
164C2×D4×C83—22no—
165C2×D4×C83—22no—
166C2×D4×C83—41no—
167C22×C4×C83—21no—
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×C83—21no—
179C22×C4×C83C2×D411yescharacteristic
180C22×C4×C83C2×D411yescharacteristic
181C22×C4×C83C2×D411yescharacteristic
182C22×C4×C83C2×D411yescharacteristic
183C22×C4×C83C2411yescharacteristic
184C2×C8×C83—21no—
185C2×C8×C83—21no—
186C2×C8×C83—21no—
187C2×C8×C83—21no—
188C2×C8×C83—21no—
189C2×C8×C83—21no—
190C2×C8×C83—21no—
191C2×C8×C83—21no—
192C2×C8×C83—21no—
193C2×C8×C83—21no—
194C2×C8×C83—21no—
195C2×C8×C83#4112yes—
196C4⋊C4×C83—22no—
197C22⋊C4×C83—22no—
198C22⋊C4×C83—22no—
199C42×C83#4112yes—
200C42×C83#4112yes—
201C23×C83—21no—
202C23×C83—21no—
203C23×C83—21no—
204C23×C83—21no—
205C23×C83—21no—
206C23×C83—21no—
207C23×C83—21no—
208C23×C83—21no—
209C23×C83—21no—
210C23×C83—21no—
211C23×C83—21no—
212C23×C83—21no—
213C23×C83—21no—
214C23×C83#13712yes—
215C23×C83C22×D411yescharacteristic
216D4×C83—44no—
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×C83—21no—
227C2×C4×C83—41no—
228C2×C4×C83—41no—
229C2×C4×C83—41no—
230C2×C4×C83—41no—
231C2×C4×C83—41no—
232C2×C4×C83—41no—
233C2×C4×C83—41no—
234C2×C4×C83—41no—
235C2×C4×C83#12312yes—
236C2×C4×C83#12312yes—
237C2×C4×C83C2×#4111yescharacteristic
238C2×C4×C83C2×#4111yescharacteristic
239C2×C4×C83C2×#4111yescharacteristic
240C2×C4×C83C2×#4111yescharacteristic
241C2×C4×C83H(5,𝔽2)11yescharacteristic
242C2×C4×C832−1+411yescharacteristic
243C2×Dic83—22no—
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—
257C2×Dic83—41no—
258C2×Dic83—41no—
259C2×Dic83—41no—
260C2×Dic83—41no—
261C2×Dic83—41no—
262C2×Dic83—41no—
263C2×Dic83—41no—
264C2×Dic83—41no—
265C2×Dic83—41no—
266C2×Dic83—41no—
267C2×Dic83—84no—
268Dic166—84no—
269C8×C83—22no—
270C8×C83—41no—
271C8×C83—41no—
272C8×C83—41no—
273C22×C83—22no—
274C22×C83—22no—
275C22×C83—22no—
276C22×C83—41no—
277C22×C83—41no—
278C22×C83—41no—
279C22×C83—41no—
280C22×C83—41no—
281C22×C83—41no—
282C22×C83—41no—
283C22×C83—41no—
284C22×C83—41no—
285C22×C83—41no—
286C22×C83—41no—
287C22×C83C2×#12311yescharacteristic
288C22×C83#54611yescharacteristic
289C22×C83#54711yescharacteristic
290C22×C83C2×#13711yescharacteristic
291C22×C83#57511yescharacteristic
292C22×C83#57611yescharacteristic
293C22×C83#57711yescharacteristic
294C4×C83—22no—
295C4×C83—22no—
296C4×C83—41no—
297C4×C83—41no—
298C4×C83#52911yescharacteristic
299C4×C83#54211yescharacteristic
300C4×C83#55611yescharacteristic
301C4×C83#55911yescharacteristic
302Dic83—42no—
303Dic83—81no—
304Dic83—81no—
305Dic83—81no—
306Dic83—81no—
307Dic83—81no—
308Dic83—81no—
309Dic83—81no—
310#26313—831nomaximal, Sylow
311C2×C83—42no—
312C2×C83—81no—
313C2×C83#244711yescharacteristic
314C2×C83#309511yescharacteristic
315C2×C83#309611yescharacteristic
316C2×C83#331411yescharacteristic
317C2×C83#332111yescharacteristic
318C2×C83#334411yescharacteristic
319C2×C83#334711yescharacteristic
320#2563—831no—
321#2352—831no—
322#2077—831no—
323#2076—831no—
324#2076—831no—
325#2075—831no—
326#1993—831no—
327#1988—831no—
328#1955D8311yescharacteristic
329#1949—831no—
330#1854—831no—
331#1853—831no—
332#1825—831no—
333#1823—831no—
334#1776—831no—
335C83#2631311yescharacteristic, complemented, Sylow
336C2×C4⋊Q8—831no—
337C2×C4×D4C2×D8311yescharacteristic
338C2×#117—831no—
339C22×C4⋊C4—831no—
340C2×#107—831no—
341C2×#107—831no—
342C2×#106—831no—
343C2×#106—831no—
344C2×C4⋊C8C2×D8311yescharacteristic
345C2×C22⋊C8C2×D8311yescharacteristic
346C2×C22⋊C8C2×D8311yescharacteristic
347C2×C22⋊C8C2×D8311yescharacteristic
348C2×C22⋊C8C2×D8311yescharacteristic
349C2×#97C2×D8311yescharacteristic
350#393—1661no—
351#393—1661no—
352#390—1661no—
353#388—1661no—
354#387—1661no—
355#386—1661no—
356#382—831no—
357#382—1661no—
358#380—831no—
359#380—831no—
360#380—831no—
361#380—831no—
362#380—831no—
363#380—831no—
364C4×C4⋊C4—831no—
365C4×C4⋊C4—831no—
366C4×C22⋊C4—1661no—
367#340—831no—
368#340—831no—
369#340—831no—
370#340—831no—
371#340—831no—
372#340—831no—
373#340—831no—
374#340—831no—
375#340—831no—
376#340—831no—
377#340—831no—
378#340—831no—
379C22×Q8—1661no—
380C22×D4D33211yescharacteristic
381C23×C4—1661no—
382C23×C4C22×D8311yescharacteristic
383C23×C4C22×D8311yescharacteristic
384C22×C8C22×D8311yescharacteristic
385C22×C8C22×D8311yescharacteristic
386C22×C8C22×D8311yescharacteristic
387C22×C8C22×D8311yescharacteristic
388C4⋊Q8—1662no—
389C4⋊Q8—1662no—
390C4×D4—22no—
391C4×D4—22no—
392#117—3321no—
393#117—1662no—
394C2×C4⋊C4—3321no—
395C2×C4⋊C4—831no—
396C2×C4⋊C4—831no—
397C2×C4⋊C4—831no—
398C2×C4⋊C4—831no—
399C2×C4⋊C4—831no—
400C2×C4⋊C4—831no—
401C2×C4⋊C4—831no—
402C2×C4⋊C4—831no—
403C2×C4⋊C4—1661no—
404C2×C4⋊C4—1661no—
405C2×C4⋊C4—1662no—
406C2×C4⋊C4D33211yescharacteristic
407C2×C22⋊C4—1661no—
408C2×C22⋊C4—1661no—
409C2×C22⋊C4—1661no—
410C2×C22⋊C4—1661no—
411C2×C22⋊C4—1661no—
412C2×C22⋊C4—1661no—
413C2×C22⋊C4—1661no—
414C2×C22⋊C4D33211yescharacteristic
415C2×C22⋊C4D33211yescharacteristic
416C2×C42—1661no—
417C2×C42—1661no—
418C2×C42—1661no—
419C2×C42C22×D8311yescharacteristic
420#107—3321no—
421#107—1662no—
422#106—3321no—
423#106—1662no—
424C4⋊C8—22no—
425C22⋊C8—22no—
426C22⋊C8—22no—
427C22⋊C8—41no—
428C22⋊C8—41no—
429#97—22no—
430#95—1661no—
431#95—1661no—
432#95—1661no—
433#95—1661no—
434#95—1661no—
435#95—1661no—
436#95—1661no—
437#95—1661no—
438#95—1661no—
439#95—1661no—
440C24C2×D33211yescharacteristic
441C24C2×D33211yescharacteristic
442C2×Q8—3322no—
443C2×Q8—6641no—
444C2×Q8—6641no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—41no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
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—3322no—
461C22×C4—3322no—
462C22×C4—6641no—
463C22×C4—1661no—
464C22×C4—1661no—
465C22×C4—1661no—
466C22×C4—1661no—
467C22×C4—1661no—
468C22×C4—1661no—
469C22×C4—1661no—
470C22×C4—1661no—
471C22×C4C2×D33211yescharacteristic
472C22×C4C2×D33211yescharacteristic
473C22×C4C2×D33211yescharacteristic
474C22×C4C2×D33211yescharacteristic
475C22×C4C23×D8311yescharacteristic
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8#1305058912yes—
488C4⋊C4—22no—
489C4⋊C4—3322no—
490C4⋊C4—3322no—
491C4⋊C4—832no—
492C4⋊C4—1661no—
493C4⋊C4—1661no—
494C4⋊C4—1661no—
495C4⋊C4—1661no—
496C4⋊C4—1661no—
497C4⋊C4—1661no—
498C4⋊C4—1661no—
499C4⋊C4—1661no—
500C4⋊C4—1661no—
501C4⋊C4—1661no—
502C4⋊C4—1661no—
503C4⋊C4—1661no—
504C4⋊C4—1661no—
505C4⋊C4—1661no—
506C4⋊C4—1661no—
507C22⋊C4—22no—
508C22⋊C4—22no—
509C22⋊C4—3322no—
510C22⋊C4—3322no—
511C22⋊C4—6641no—
512C22⋊C4—6641no—
513C22⋊C4—6641no—
514C22⋊C4—6641no—
515C22⋊C4—6641no—
516C42—3322no—
517C42—3322no—
518C42—6641no—
519C42#1305058912yes—
520C42#1305058912yes—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23#1458182512yes—
535C23C22×D33211yescharacteristic
536Q8—6644no—
537D4—44no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—3321no—
557C2×C4—3321no—
558C2×C4—3321no—
559C2×C4—3321no—
560C2×C4—3321no—
561C2×C4—3321no—
562C2×C4—3321no—
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—3321no—
575C2×C4—3321no—
576C2×C4—3321no—
577C2×C4—3321no—
578C2×C4—3321no—
579C2×C4—6644no—
580C2×C4—1662no—
581C2×C4#1458177612yes—
582C2×C4#1458177612yes—
583C2×C4#41×D8311yescharacteristic
584C2×C4C2×#1305058911yescharacteristic
585C2×C4C2×#1305058911yescharacteristic
586C2×C4C2×#1305058911yescharacteristic
587C2×C4#1458183711yescharacteristic
588C2×C4#1458183911yescharacteristic
589C8—22no—
590C8—41no—
591C8—41no—
592C8—41no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—41no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—41no—
607C22C2×#1458177611yescharacteristic
608C22#1912834611yescharacteristic
609C22#1912835511yescharacteristic
610C22C2×#1458182511yescharacteristic
611C22#1912850111yescharacteristic
612C22#1912850611yescharacteristic
613C22#1912851311yescharacteristic
614C4—22no—
615C4—22no—
616C4—41no—
617C4—41no—
618C4—3322no—
619C4—6641no—
620C4—6641no—
621C4—6641no—
622C4—6641no—
623C4—6641no—
624C4—6641no—
625C4—6641no—
626C4#1912823211yescharacteristic
627C4#1912831811yescharacteristic
628C4#1912842011yescharacteristic
629C4#1912843711yescharacteristic
630C2—42no—
631C2—81no—
632C2#2491124811yescharacteristic
633C2#2491741211yescharacteristic
634C2#2491742211yescharacteristic
635C2#2491954611yescharacteristic
636C2#2491965111yescharacteristic
637C2#2491992011yescharacteristic
638C2#2491996511yescharacteristic
subgroupG/H
centerC23C22×D332
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingC83×#1955C2
socleC23×C83C22×D4
prime radicalC22×D4×C83D4
hypercenter#1955D83
hypocenterC83#26313
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #1955

Complements

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

NH ≅ G/Ntype
3. #24908896C2semidirect
4. #24908891C2semidirect
5. #24908891C2semidirect
6. #24908886C2semidirect
7. #24908436C2semidirect
8. #24908414C2semidirect
9. #24908233C2semidirect
10. #24907664C2semidirect
11. #24907658C2semidirect
12. #24907559C2semidirect
13. #24907550C2semidirect
14. #24907360C2semidirect
335. C83#26313semidirect

Maximal subgroups

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

subgroupindex
#249126362
#249103922
#249088962
#249088912
#249088862
#249084362
#249084142
#249082332
#249076642
#249076582
#249075592
#249075502
#249073602
C83×#19552
#2631383