Finite Group Databasebeta

← #35477145

Subgroups — #35477145

Order 21248.

708 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24911896C211yesmaximal, characteristic
2#24908871C211yesmaximal, characteristic, complemented
3#24908858C211yesmaximal, characteristic, complemented
4#24908663C211yesmaximal, characteristic
5#24908663C211yesmaximal, characteristic
6#24908221C211yesmaximal, characteristic, complemented
7#24908193C211yesmaximal, characteristic, complemented
8#24908071C211yesmaximal, characteristic
9#24908055C211yesmaximal, characteristic, complemented
10#24908052C211yesmaximal, characteristic, complemented
11#24907717C211yesmaximal, characteristic, complemented
12#24907651C211yesmaximal, characteristic
13#24907553C211yesmaximal, characteristic
14C4×#14581716C211yesmaximal, characteristic, complemented
15C83×#2360C211yesmaximal, characteristic
16C2×#14581725C2211yescharacteristic
17C2×#14581725C2211yescharacteristic
18C2×#14581725C2211yescharacteristic
19C2×#14581725C2211yescharacteristic
20C2×C4⋊(C83⋊C8)C2211yescharacteristic
21C2×#14581716C2211yescharacteristic
22C2×#14581716C2211yescharacteristic
23C2×#14581716C2211yescharacteristic
24C2×#14581716C2211yescharacteristic
25C2×C22⋊(C83⋊C8)C2211yescharacteristic
26C2×C22⋊(C83⋊C8)C2211yescharacteristic
27C2×C4×C83⋊C8C2211yescharacteristic
28#19127643—21no—
29#19127624—21no—
30#19127599C2211yescharacteristic
31#19127599C2211yescharacteristic
32#19127594—21no—
33#19127588—21no—
34#19127571C2211yescharacteristic
35#19127561C2211yescharacteristic
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
50C22×C4⋊C4×C83C2211yescharacteristic
51#391×C83C2211yescharacteristic
52#389×C83—21no—
53#389×C83C2211yescharacteristic
54#387×C83—21no—
55#385×C83—21no—
56C4×C4⋊C4×C83C2211yescharacteristic
57C4×C22⋊C4×C83—21no—
58C4×C22⋊C4×C83C2211yescharacteristic
59C22×C83⋊C8C2311yescharacteristic
60C22×C83⋊C8C2311yescharacteristic
61C22×C83⋊C8C2311yescharacteristic
62C22×C83⋊C8C2311yescharacteristic
63C2×C4⋊Dic83—21no—
64C2×C4⋊Dic83—21no—
65C2×C4⋊Dic83C2311yescharacteristic
66C2×C4⋊Dic83C2311yescharacteristic
67C2×C4⋊Dic83C2311yescharacteristic
68C2×C4⋊Dic83C2311yescharacteristic
69C2×Dic83⋊C4—21no—
70C2×Dic83⋊C4—21no—
71C2×C22⋊Dic83—21no—
72C2×C22⋊Dic83—21no—
73C2×C4×Dic83—21no—
74#14581725—22no—
75#14581725—22no—
76#14581725—22no—
77#14581725—41no—
78C4⋊(C83⋊C8)—22no—
79#14581716—22no—
80#14581716—22no—
81#14581716—22no—
82#14581716—41no—
83C22⋊(C83⋊C8)—22no—
84C22⋊(C83⋊C8)—41no—
85C4×C83⋊C8—41no—
86C83⋊#95—21no—
87C83⋊#95—21no—
88C83⋊#95—21no—
89C83⋊#95—21no—
90C83⋊#95—21no—
91C22×Q8×C83C2311yescharacteristic
92C23×C4×C83—21no—
93C23×C4×C83C2311yescharacteristic
94C22⋊Q8×C83—22no—
95C22⋊Q8×C83—22no—
96C4×Q8×C83—22no—
97C4×Q8×C83—41no—
98C2×C4⋊C4×C83—21no—
99C2×C4⋊C4×C83—22no—
100C2×C4⋊C4×C83—21no—
101C2×C4⋊C4×C83—21no—
102C2×C4⋊C4×C83—21no—
103C2×C4⋊C4×C83—41no—
104C2×C4⋊C4×C83C2311yescharacteristic
105C2×C4⋊C4×C83C2311yescharacteristic
106C2×C4⋊C4×C83C2311yescharacteristic
107C2×C22⋊C4×C83—21no—
108C2×C22⋊C4×C83D411yescharacteristic
109C2×C22⋊C4×C83D411yescharacteristic
110C2×C42×C83—21no—
111C2×C42×C83—21no—
112C2×C42×C83C2311yescharacteristic
113C2×C42×C83C2311yescharacteristic
114#95×C83—21no—
115#95×C83—21no—
116#95×C83—21no—
117#95×C83—21no—
118#95×C83D411yescharacteristic
119#95×C83D411yescharacteristic
120C22×Dic83—21no—
121C22×Dic83—21no—
122C22×Dic83—21no—
123C22×Dic83—21no—
124C2×C83⋊C8—21no—
125C2×C83⋊C8—21no—
126C2×C83⋊C8—21no—
127C2×C83⋊C8—21no—
128C2×C83⋊C8—21no—
129C2×C83⋊C8—21no—
130C2×C83⋊C8—21no—
131C2×C83⋊C8—21no—
132C2×C83⋊C8—21no—
133C2×C83⋊C8—21no—
134C2×C83⋊C8—21no—
135C2×C83⋊C8#4112yes—
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—81no—
145C4⋊Dic83—81no—
146Dic83⋊C4—42no—
147Dic83⋊C4—81no—
148C22⋊Dic83—81no—
149C22⋊Dic83—81no—
150C4×Dic83—81no—
151C24×C83C2×D411yescharacteristic
152C2×Q8×C83—21no—
153C2×Q8×C83—21no—
154C2×Q8×C83—21no—
155C2×Q8×C83—21no—
156C2×Q8×C83—21no—
157C2×Q8×C83#4112yes—
158C22×C4×C83—21no—
159C22×C4×C83—21no—
160C22×C4×C83—21no—
161C22×C4×C83—21no—
162C22×C4×C83—21no—
163C22×C4×C83—21no—
164C22×C4×C83—21no—
165C22×C4×C83—21no—
166C22×C4×C83—21no—
167C22×C4×C83—21no—
168C22×C4×C83—42no—
169C22×C4×C83—42no—
170C22×C4×C83—81no—
171C22×C4×C83C2×D411yescharacteristic
172C22×C4×C83C2×D411yescharacteristic
173C22×C4×C83C2×D411yescharacteristic
174C22×C4×C83C2×D411yescharacteristic
175C22×C4×C83C2×D411yescharacteristic
176C22×C4×C83C2411yescharacteristic
177C4⋊C4×C83—24no—
178C4⋊C4×C83—21no—
179C4⋊C4×C83—21no—
180C4⋊C4×C83—21no—
181C4⋊C4×C83—21no—
182C4⋊C4×C83—42no—
183C4⋊C4×C83—42no—
184C4⋊C4×C83—81no—
185C4⋊C4×C83#4112yes—
186C4⋊C4×C83#4112yes—
187C22⋊C4×C83—24no—
188C22⋊C4×C83—41no—
189C22⋊C4×C83Q1614yes—
190C42×C83—21no—
191C42×C83—21no—
192C42×C83—21no—
193C42×C83—21no—
194C42×C83—42no—
195C42×C83—81no—
196C23×C83—21no—
197C23×C83—21no—
198C23×C83—21no—
199C23×C83—21no—
200C23×C83—21no—
201C23×C83—21no—
202C23×C83C2×Q1612yes—
203C23×C83C22×D411yescharacteristic
204Q8×C83—22no—
205Q8×C83—22no—
206Q8×C83—41no—
207Q8×C83—41no—
208C2×C4×C83—21no—
209C2×C4×C83—21no—
210C2×C4×C83—21no—
211C2×C4×C83—21no—
212C2×C4×C83—22no—
213C2×C4×C83—22no—
214C2×C4×C83—21no—
215C2×C4×C83—21no—
216C2×C4×C83—22no—
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—41no—
225C2×C4×C83—41no—
226C2×C4×C83—41no—
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—41no—
236C2×C4×C83—41no—
237C2×C4×C83—84no—
238C2×C4×C83C2×Q1612yes—
239C2×C4×C83C2×Q1612yes—
240C2×C4×C83C2×#4111yescharacteristic
241C2×C4×C83C2×#4111yescharacteristic
242C2×C4×C83C2×#4111yescharacteristic
243C2×C4×C83C2×#4111yescharacteristic
244C2×C4×C83H(5,𝔽2)11yescharacteristic
245C2×C4×C832−1+411yescharacteristic
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—
258C83⋊C8—22no—
259C83⋊C8—41no—
260C83⋊C8—41no—
261C83⋊C8—41no—
262C22×C83—22no—
263C22×C83—22no—
264C22×C83—22no—
265C22×C83—41no—
266C22×C83—41no—
267C22×C83—41no—
268C22×C83—41no—
269C22×C83#52511yescharacteristic
270C22×C83#54611yescharacteristic
271C22×C83#54711yescharacteristic
272C22×C83C22×Q1611yescharacteristic
273C22×C83#57511yescharacteristic
274C22×C83#57611yescharacteristic
275C22×C83#57711yescharacteristic
276C4×C83—22no—
277C4×C83—22no—
278C4×C83—41no—
279C4×C83—42no—
280C4×C83—41no—
281C4×C83—42no—
282C4×C83—41no—
283C4×C83—42no—
284C4×C83—81no—
285C4×C83#52911yescharacteristic
286C4×C83#53911yescharacteristic
287C4×C83#55311yescharacteristic
288C4×C83#55611yescharacteristic
289Dic83—81no—
290Dic83—81no—
291Dic83—81no—
292Dic83—81no—
293#27036—831nomaximal, Sylow
294C2×C83—42no—
295C2×C83#246611yescharacteristic
296C2×C83#310411yescharacteristic
297C2×C83#311911yescharacteristic
298C2×C83#329911yescharacteristic
299C2×C83#331911yescharacteristic
300C2×C83#333811yescharacteristic
301C2×C83#335111yescharacteristic
302#2497—831no—
303#2360D8311yescharacteristic
304#2072—831no—
305#2068—831no—
306#2036—831no—
307#2036—831no—
308#1947—831no—
309#1942—831no—
310#1923—831no—
311#1921—831no—
312#1921—831no—
313#1863—831no—
314#1852—831no—
315#1824—831no—
316C4×#103—831no—
317C83#2703611yescharacteristic, complemented, Sylow
318C2×C22⋊Q8C2×D8311yescharacteristic
319C2×C4×Q8C2×D8311yescharacteristic
320C22×C4⋊C4C2×D8311yescharacteristic
321C2×#107—831no—
322C2×#107—831no—
323C2×#107—831no—
324C2×#107—831no—
325C2×C4⋊C8—831no—
326C2×#103—831no—
327C2×#103—831no—
328C2×#103—831no—
329C2×#103—831no—
330C2×C22⋊C8—831no—
331C2×C22⋊C8—831no—
332C2×C4×C8—831no—
333#400—1661no—
334#396—1661no—
335#391C2×D8311yescharacteristic
336#389—21no—
337#389—831no—
338#389—831no—
339#389C2×D8311yescharacteristic
340#388—1661no—
341#387—21no—
342#387—1661no—
343#385—21no—
344#384—831no—
345#380—831no—
346#380—831no—
347C4×C4⋊C4—831no—
348C4×C4⋊C4C2×D8311yescharacteristic
349C4×C22⋊C4—21no—
350C4×C22⋊C4C2×D8311yescharacteristic
351#340—831no—
352#340—831no—
353#340—831no—
354#340—831no—
355#340—831no—
356#340—831no—
357#340—831no—
358#340—831no—
359#336—831no—
360#336—831no—
361C22×Q8C22×D8311yescharacteristic
362C23×C4—21no—
363C23×C4C22×D8311yescharacteristic
364C22×C8—831no—
365C22×C8—831no—
366C22×C8—831no—
367C22×C8—831no—
368C22⋊Q8—22no—
369C22⋊Q8—22no—
370C4×Q8—22no—
371C4×Q8—41no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—22no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—41no—
378C2×C4⋊C4—831no—
379C2×C4⋊C4—831no—
380C2×C4⋊C4—831no—
381C2×C4⋊C4—831no—
382C2×C4⋊C4—1661no—
383C2×C4⋊C4—1661no—
384C2×C4⋊C4—1661no—
385C2×C4⋊C4—1661no—
386C2×C4⋊C4C22×D8311yescharacteristic
387C2×C4⋊C4C22×D8311yescharacteristic
388C2×C4⋊C4C22×D8311yescharacteristic
389C2×C22⋊C4—21no—
390C2×C22⋊C4—1661no—
391C2×C22⋊C4—1661no—
392C2×C22⋊C4C22⋊D8311yescharacteristic
393C2×C22⋊C4C22⋊D8311yescharacteristic
394C2×C42—21no—
395C2×C42—21no—
396C2×C42—1661no—
397C2×C42C22×D8311yescharacteristic
398C2×C42C22×D8311yescharacteristic
399#107—3321no—
400#107—1662no—
401#107—1662no—
402#107—1662no—
403C4⋊C8—1662no—
404#103—3321no—
405#103—1662no—
406#103—1662no—
407#103—1662no—
408C22⋊C8—3321no—
409C22⋊C8—1662no—
410C4×C8—3321no—
411#95—21no—
412#95—21no—
413#95—21no—
414#95—21no—
415#95—1661no—
416#95—1661no—
417#95—1661no—
418#95—1661no—
419#95—1661no—
420#95C22⋊D8311yescharacteristic
421#95C22⋊D8311yescharacteristic
422C24C2×C22⋊D8311yescharacteristic
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8#1305059012yes—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—81no—
442C22×C4—1661no—
443C22×C4—1661no—
444C22×C4—1661no—
445C22×C4—1661no—
446C22×C4C2×C22⋊D8311yescharacteristic
447C22×C4C2×C22⋊D8311yescharacteristic
448C22×C4C2×C22⋊D8311yescharacteristic
449C22×C4C2×C22⋊D8311yescharacteristic
450C22×C4C2×C22⋊D8311yescharacteristic
451C22×C4C23×D8311yescharacteristic
452C2×C8—832no—
453C2×C8—1661no—
454C2×C8—1661no—
455C2×C8—1661no—
456C2×C8—1661no—
457C2×C8—1661no—
458C2×C8—1661no—
459C2×C8—1661no—
460C2×C8—1661no—
461C2×C8—1661no—
462C2×C8—1661no—
463C2×C8—1661no—
464C4⋊C4—24no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—81no—
472C4⋊C4—3322no—
473C4⋊C4—6641no—
474C4⋊C4—6641no—
475C4⋊C4—6641no—
476C4⋊C4—1661no—
477C4⋊C4—1661no—
478C4⋊C4—1661no—
479C4⋊C4—1661no—
480C4⋊C4—1661no—
481C4⋊C4—1661no—
482C4⋊C4—1661no—
483C4⋊C4—1661no—
484C4⋊C4#1305058912yes—
485C4⋊C4#1305059012yes—
486C22⋊C4—24no—
487C22⋊C4—41no—
488C22⋊C4—6641no—
489C22⋊C4—6641no—
490C22⋊C4#1305058014yes—
491C42—21no—
492C42—21no—
493C42—21no—
494C42—21no—
495C42—42no—
496C42—81no—
497C42—6641no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23C2×#1305058012yes—
505C23C22×C22⋊D8311yescharacteristic
506Q8—22no—
507Q8—22no—
508Q8—41no—
509Q8—41no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—22no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—84no—
540C2×C4—3321no—
541C2×C4—3321no—
542C2×C4—3321no—
543C2×C4—3321no—
544C2×C4—3321no—
545C2×C4—3321no—
546C2×C4—3321no—
547C2×C4—3321no—
548C2×C4—3321no—
549C2×C4—3321no—
550C2×C4—3321no—
551C2×C4—3321no—
552C2×C4C2×#1305058012yes—
553C2×C4C2×#1305058012yes—
554C2×C4#41×D8311yescharacteristic
555C2×C4C2×#1305058911yescharacteristic
556C2×C4C2×#1305059011yescharacteristic
557C2×C4C2×#1305059011yescharacteristic
558C2×C4#1458183711yescharacteristic
559C2×C4#1458183911yescharacteristic
560C8—3321no—
561C8—3321no—
562C8—3321no—
563C8—1662no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22#1912821211yescharacteristic
572C22#1912834511yescharacteristic
573C22#1912835011yescharacteristic
574C22C22×#1305058011yescharacteristic
575C22#1912849911yescharacteristic
576C22#1912850411yescharacteristic
577C22#1912850911yescharacteristic
578C4—22no—
579C4—22no—
580C4—41no—
581C4—42no—
582C4—41no—
583C4—42no—
584C4—42no—
585C4—41no—
586C4—81no—
587C4—6641no—
588C4—6641no—
589C4—6641no—
590C4—6641no—
591C4#1912823111yescharacteristic
592C4#1912829311yescharacteristic
593C4#1912840111yescharacteristic
594C4#1912842111yescharacteristic
595C2—42no—
596C2#2491149311yescharacteristic
597C2#2491750211yescharacteristic
598C2#2491766211yescharacteristic
599C2#2491933911yescharacteristic
600C2#2491961911yescharacteristic
601C2#2491985111yescharacteristic
602C2#2492002311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingC83×#2360C2
socleC23×C83C22×D4
prime radicalC24×C83C2×D4
hypercenter#2360D83
hypocenterC83#27036
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2360

Complements

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

NH ≅ G/Ntype
2. #24908871C2semidirect
3. #24908858C2semidirect
6. #24908221C2semidirect
7. #24908193C2semidirect
9. #24908055C2semidirect
10. #24908052C2semidirect
11. #24907717C2semidirect
14. C4×#14581716C2semidirect
317. C83#27036semidirect

Maximal subgroups

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

subgroupindex
#249118962
#249088712
#249088582
#249086632
#249082212
#249081932
#249080712
#249080552
#249080522
#249077172
#249076512
#249075532
C4×#145817162
C83×#23602
#2703683