Finite Group Databasebeta

← #35472521

Subgroups — #35472521

Order 21248.

764 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24910466C211yesmaximal, characteristic
2#24908785C211yesmaximal, characteristic, complemented
3#24908666C211yesmaximal, characteristic
4#24908601C211yesmaximal, characteristic, complemented
5#24908471C211yesmaximal, characteristic
6#24908412C211yesmaximal, characteristic, complemented
7#24908142C211yesmaximal, characteristic, complemented
8#24908101C211yesmaximal, characteristic, complemented
9#24908071C211yesmaximal, characteristic
10#24908055C211yesmaximal, characteristic, complemented
11#24907807C211yesmaximal, characteristic, complemented
12#24907771C211yesmaximal, characteristic
13#24907551C211yesmaximal, characteristic
14C4×#14581715C211yesmaximal, characteristic, complemented
15C83×#2492C211yesmaximal, characteristic
16C22×#13050580C2211yescharacteristic
17C2×#14581771C2211yescharacteristic
18C2×C4×Dic166C2211yescharacteristic
19C22×C4⋊Dic83C2211yescharacteristic
20C2×#14581725C2211yescharacteristic
21C2×C4⋊(C83⋊C8)C2211yescharacteristic
22C2×#14581716C2211yescharacteristic
23C2×#14581716C2211yescharacteristic
24C2×#14581716C2211yescharacteristic
25C2×#14581715C2211yescharacteristic
26C2×#14581715C2211yescharacteristic
27C2×#14581715C2211yescharacteristic
28C2×C22⋊(C83⋊C8)C2211yescharacteristic
29C2×C22⋊(C83⋊C8)C2211yescharacteristic
30C2×C4×C83⋊C8C2211yescharacteristic
31#19127607C2211yescharacteristic
32#19127599C2211yescharacteristic
33#19127598—21no—
34#19127588—21no—
35#19127573—21no—
36C4×C4⋊Dic83C2211yescharacteristic
37C22⋊C4×Dic83—21no—
38#19127463C2211yescharacteristic
39#19127463C2211yescharacteristic
40#19127463C2211yescharacteristic
41#19127463C2211yescharacteristic
42#19127463C2211yescharacteristic
43#19127463C2211yescharacteristic
44#19127463C2211yescharacteristic
45#19127463C2211yescharacteristic
46#19127456C2211yescharacteristic
47#19127456C2211yescharacteristic
48C2×C4⋊Q8×C83C2211yescharacteristic
49C2×C22⋊Q8×C83C2211yescharacteristic
50#400×C83—21no—
51#393×C83—21no—
52#391×C83—21no—
53#391×C83C2211yescharacteristic
54#391×C83C2211yescharacteristic
55#388×C83—21no—
56#380×C83C2211yescharacteristic
57C4×C4⋊C4×C83C2211yescharacteristic
58C4×C22⋊C4×C83C2211yescharacteristic
59C22×Dic166C2311yescharacteristic
60C23×Dic83—21no—
61C2×#13050580—22no—
62C2×#13050580—41no—
63C2×#13050580—41no—
64C22×C83⋊C8C2311yescharacteristic
65C22×C83⋊C8C2311yescharacteristic
66C22×C83⋊C8C2311yescharacteristic
67C22×C83⋊C8C2311yescharacteristic
68#14581771—22no—
69#14581771—22no—
70C4×Dic166—22no—
71C4×Dic166—41no—
72C2×C4⋊Dic83—22no—
73C2×C4⋊Dic83—21no—
74C2×C4⋊Dic83—41no—
75C2×C4⋊Dic83C2311yescharacteristic
76C2×C4⋊Dic83C2311yescharacteristic
77C2×C4⋊Dic83C2311yescharacteristic
78C2×Dic83⋊C4—21no—
79C2×Dic83⋊C4—21no—
80C2×Dic83⋊C4—21no—
81C2×C22⋊Dic83—21no—
82C2×C4×Dic83—21no—
83C2×C4×Dic83—21no—
84#14581725—22no—
85C4⋊(C83⋊C8)—41no—
86#14581716—22no—
87#14581716—22no—
88#14581716—41no—
89#14581715—22no—
90#14581715—22no—
91#14581715—41no—
92C22⋊(C83⋊C8)—41no—
93C22⋊(C83⋊C8)—41no—
94C4×C83⋊C8—41no—
95C83⋊#95—21no—
96C83⋊#95—21no—
97C83⋊#95—21no—
98C83⋊#95—21no—
99C22×Q8×C83—21no—
100C22×Q8×C83C2311yescharacteristic
101C23×C4×C83C2311yescharacteristic
102C4⋊Q8×C83—22no—
103C4⋊Q8×C83—41no—
104C22⋊Q8×C83—22no—
105C22⋊Q8×C83—41no—
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×C83C2311yescharacteristic
117C2×C42×C83C2311yescharacteristic
118#95×C83—21no—
119#95×C83—21no—
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#4112yes—
132C22×Dic83—21no—
133C22×Dic83—21no—
134C22×Dic83—21no—
135C22×Dic83—21no—
136C22×Dic83—42no—
137C22×Dic83—42no—
138C22×Dic83—81no—
139#13050580—44no—
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—
149C2×C83⋊C8—21no—
150C2×C83⋊C8—21no—
151C2×C83⋊C8—21no—
152C4⋊Dic83—21no—
153C4⋊Dic83—21no—
154C4⋊Dic83—21no—
155C4⋊Dic83—21no—
156C4⋊Dic83—42no—
157C4⋊Dic83#4112yes—
158C4⋊Dic83#4112yes—
159Dic83⋊C4—24no—
160Dic83⋊C4—42no—
161Dic83⋊C4—81no—
162C22⋊Dic83—24no—
163C4×Dic83—42no—
164C4×Dic83—81no—
165C24×C83C2×D411yescharacteristic
166C2×Q8×C83—21no—
167C2×Q8×C83—21no—
168C2×Q8×C83—21no—
169C2×Q8×C83—21no—
170C2×Q8×C83—21no—
171C2×Q8×C83—21no—
172C2×Q8×C83—42no—
173C2×Q8×C83—42no—
174C2×Q8×C83—81no—
175C22×C4×C83—21no—
176C22×C4×C83—21no—
177C22×C4×C83—21no—
178C22×C4×C83—21no—
179C22×C4×C83—21no—
180C22×C4×C83—21no—
181C22×C4×C83—21no—
182C22×C4×C83—21no—
183C22×C4×C83—21no—
184C22×C4×C83—21no—
185C22×C4×C83C2×D411yescharacteristic
186C22×C4×C83C2×D411yescharacteristic
187C22×C4×C83C2×D411yescharacteristic
188C22×C4×C83C2×D411yescharacteristic
189C22×C4×C83C2×D411yescharacteristic
190C22×C4×C83C2411yescharacteristic
191C4⋊C4×C83—21no—
192C4⋊C4×C83—21no—
193C4⋊C4×C83—21no—
194C4⋊C4×C83—21no—
195C4⋊C4×C83—21no—
196C4⋊C4×C83—42no—
197C4⋊C4×C83—42no—
198C4⋊C4×C83#4112yes—
199C22⋊C4×C83—41no—
200C22⋊C4×C83—42no—
201C22⋊C4×C83—81no—
202C22⋊C4×C83Q1614yescomplemented
203C42×C83—21no—
204C42×C83—21no—
205C42×C83—21no—
206C42×C83—21no—
207C42×C83—81no—
208C23×C83—21no—
209C23×C83—21no—
210C23×C83—21no—
211C23×C83—21no—
212C23×C83—21no—
213C23×C83—21no—
214C23×C83C2×Q1612yes—
215C23×C83C22×D411yescharacteristic
216Q8×C83—41no—
217Q8×C83—41no—
218Q8×C83—41no—
219Q8×C83—41no—
220Q8×C83—84no—
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—21no—
228C2×C4×C83—22no—
229C2×C4×C83—21no—
230C2×C4×C83—21no—
231C2×C4×C83—21no—
232C2×C4×C83—21no—
233C2×C4×C83—21no—
234C2×C4×C83—21no—
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—41no—
247C2×C4×C83—41no—
248C2×C4×C83—41no—
249C2×C4×C83—41no—
250C2×C4×C83C2×Q1612yes—
251C2×C4×C83C2×Q1612yes—
252C2×C4×C83C2×#4111yescharacteristic
253C2×C4×C83C2×#4111yescharacteristic
254C2×C4×C83C2×#4111yescharacteristic
255C2×C4×C83C2×#4111yescharacteristic
256C2×C4×C83H(5,𝔽2)11yescharacteristic
257C2×C4×C83H(5,𝔽2)11yescharacteristic
258C2×Dic83—22no—
259C2×Dic83—22no—
260C2×Dic83—22no—
261C2×Dic83—41no—
262C2×Dic83—41no—
263C2×Dic83—41no—
264C2×Dic83—41no—
265C2×Dic83—41no—
266C2×Dic83—41no—
267C2×Dic83—41no—
268C2×Dic83—41no—
269C2×Dic83—41no—
270C2×Dic83—84no—
271Dic166—22no—
272Dic166—22no—
273Dic166—41no—
274Dic166—41no—
275C83⋊C8—41no—
276C83⋊C8—41no—
277C83⋊C8—41no—
278C83⋊C8—41no—
279C22×C83—22no—
280C22×C83—22no—
281C22×C83—22no—
282C22×C83—41no—
283C22×C83—41no—
284C22×C83—41no—
285C22×C83—41no—
286C22×C83#52511yescharacteristic
287C22×C83#54611yescharacteristic
288C22×C83#54611yescharacteristic
289C22×C83C22×Q1611yescharacteristic
290C22×C83#57511yescharacteristic
291C22×C83#57711yescharacteristic
292C22×C83#57711yescharacteristic
293C4×C83—22no—
294C4×C83—22no—
295C4×C83—41no—
296C4×C83—41no—
297C4×C83—41no—
298C4×C83—42no—
299C4×C83—81no—
300C4×C83—81no—
301C4×C83—81no—
302C4×C83#52911yescharacteristic
303C4×C83#53811yescharacteristic
304C4×C83#55511yescharacteristic
305C4×C83#55911yescharacteristic
306Dic83—42no—
307Dic83—42no—
308Dic83—42no—
309Dic83—81no—
310#26726—831nomaximal, Sylow
311C2×C83—42no—
312C2×C83#245911yescharacteristic
313C2×C83#310411yescharacteristic
314C2×C83#312511yescharacteristic
315C2×C83#329911yescharacteristic
316C2×C83#329911yescharacteristic
317C2×C83#331111yescharacteristic
318C2×C83#332311yescharacteristic
319#2492D8311yescharacteristic
320#2360—831no—
321#2056—831no—
322#2036—831no—
323#2024—831no—
324#2001—831no—
325#1987—831no—
326#1933—831no—
327#1928—831no—
328#1923—831no—
329#1921—831no—
330#1878—831no—
331#1872—831no—
332#1824—831no—
333C4×#103—831no—
334C83#2672611yescharacteristic, complemented, Sylow
335C22×Q16—831no—
336C2×C4⋊Q8C2×D8311yescharacteristic
337C2×C22⋊Q8—831no—
338C2×C22⋊Q8C2×D8311yescharacteristic
339C2×C4×Q8—831no—
340C22×C4⋊C4—831no—
341C2×#107—831no—
342C2×C4⋊C8—831no—
343C2×#103—831no—
344C2×#103—831no—
345C2×#103—831no—
346C2×#103—831no—
347C2×#103—831no—
348C2×#103—831no—
349C2×C22⋊C8—831no—
350C2×C22⋊C8—831no—
351C2×C4×C8—831no—
352#400—21no—
353#393—21no—
354#391—21no—
355#391—831no—
356#391C2×D8311yescharacteristic
357#391C2×D8311yescharacteristic
358#389—831no—
359#389—1661no—
360#388—21no—
361#387—1661no—
362#385—1661no—
363#380C2×D8311yescharacteristic
364C4×C4⋊C4—831no—
365C4×C4⋊C4C2×D8311yescharacteristic
366C4×C22⋊C4—1661no—
367C4×C22⋊C4C2×D8311yescharacteristic
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#336—831no—
377#336—831no—
378C22×Q8—21no—
379C22×Q8—831no—
380C22×Q8C22×D8311yescharacteristic
381C23×C4—1661no—
382C23×C4C22×D8311yescharacteristic
383C2×Q16—3321no—
384C2×Q16—3321no—
385C2×Q16—1662no—
386C22×C8—831no—
387C22×C8—831no—
388C22×C8—831no—
389C22×C8—831no—
390C4⋊Q8—22no—
391C4⋊Q8—41no—
392C22⋊Q8—22no—
393C22⋊Q8—41no—
394C22⋊Q8—1662no—
395C22⋊Q8—1662no—
396C4×Q8—3321no—
397C4×Q8—1662no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—3321no—
401C2×C4⋊C4—831no—
402C2×C4⋊C4—831no—
403C2×C4⋊C4—831no—
404C2×C4⋊C4—1661no—
405C2×C4⋊C4—1661no—
406C2×C4⋊C4—1662no—
407C2×C4⋊C4—1661no—
408C2×C4⋊C4—1661no—
409C2×C4⋊C4C22×D8311yescharacteristic
410C2×C4⋊C4C22×D8311yescharacteristic
411C2×C4⋊C4C22×D8311yescharacteristic
412C2×C22⋊C4—21no—
413C2×C22⋊C4—21no—
414C2×C22⋊C4—1661no—
415C2×C22⋊C4C22⋊D8311yescharacteristic
416C2×C22⋊C4C22⋊D8311yescharacteristic
417C2×C42—21no—
418C2×C42—1661no—
419C2×C42—1661no—
420C2×C42C22×D8311yescharacteristic
421C2×C42C22×D8311yescharacteristic
422#107—1662no—
423C4⋊C8—3321no—
424#103—3321no—
425#103—3321no—
426#103—1662no—
427#103—1662no—
428#103—1662no—
429#103—1662no—
430C22⋊C8—3321no—
431C22⋊C8—3321no—
432C4×C8—3321no—
433#95—21no—
434#95—21no—
435#95—21no—
436#95—21no—
437#95—21no—
438#95—21no—
439#95—1661no—
440#95—1661no—
441#95—1661no—
442#95—1661no—
443#95C22⋊D8311yescharacteristic
444#95C22⋊D8311yescharacteristic
445C24C2×C22⋊D8311yescharacteristic
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8—42no—
453C2×Q8—42no—
454C2×Q8—81no—
455C2×Q8—832no—
456C2×Q8—1661no—
457C2×Q8—1661no—
458C2×Q8—1661no—
459C2×Q8—1661no—
460C2×Q8—1661no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—3322no—
472C22×C4—3322no—
473C22×C4—6641no—
474C22×C4—1661no—
475C22×C4—1661no—
476C22×C4—1661no—
477C22×C4—1661no—
478C22×C4C2×C22⋊D8311yescharacteristic
479C22×C4C2×C22⋊D8311yescharacteristic
480C22×C4C2×C22⋊D8311yescharacteristic
481C22×C4C2×C22⋊D8311yescharacteristic
482C22×C4C2×C22⋊D8311yescharacteristic
483C22×C4C23×D8311yescharacteristic
484Q16—3324no—
485C2×C8—1661no—
486C2×C8—1661no—
487C2×C8—1661no—
488C2×C8—1661no—
489C2×C8—1661no—
490C2×C8—1661no—
491C2×C8—1661no—
492C2×C8—1661no—
493C2×C8—1661no—
494C2×C8—1661no—
495C2×C8—1661no—
496C2×C8—1661no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—3322no—
505C4⋊C4—3322no—
506C4⋊C4—832no—
507C4⋊C4—832no—
508C4⋊C4—6641no—
509C4⋊C4—1664no—
510C4⋊C4—1661no—
511C4⋊C4—1661no—
512C4⋊C4—1661no—
513C4⋊C4—1661no—
514C4⋊C4#1305058912yes—
515C22⋊C4—41no—
516C22⋊C4—42no—
517C22⋊C4—81no—
518C22⋊C4—1664no—
519C22⋊C4#1305058014yescomplemented
520C42—21no—
521C42—21no—
522C42—21no—
523C42—21no—
524C42—81no—
525C42—3322no—
526C42—6641no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23C2×#1305058012yes—
534C23C22×C22⋊D8311yescharacteristic
535Q8—41no—
536Q8—41no—
537Q8—41no—
538Q8—41no—
539Q8—84no—
540Q8—3321no—
541Q8—3321no—
542Q8—1662no—
543Q8—1662no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—3321no—
574C2×C4—3321no—
575C2×C4—3321no—
576C2×C4—3321no—
577C2×C4—3321no—
578C2×C4—3321no—
579C2×C4—3321no—
580C2×C4—3321no—
581C2×C4—3321no—
582C2×C4—6644no—
583C2×C4—1662no—
584C2×C4—1662no—
585C2×C4—1662no—
586C2×C4C2×#1305058012yes—
587C2×C4C2×#1305058012yes—
588C2×C4#41×D8311yescharacteristic
589C2×C4#41×D8311yescharacteristic
590C2×C4#41×D8311yescharacteristic
591C2×C4C2×#1305058911yescharacteristic
592C2×C4#1458183711yescharacteristic
593C2×C4#1458183811yescharacteristic
594C8—3321no—
595C8—3321no—
596C8—3321no—
597C8—3321no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22#1912820811yescharacteristic
606C22#1912833911yescharacteristic
607C22#1912834511yescharacteristic
608C22C22×#1305058011yescharacteristic
609C22#1912849911yescharacteristic
610C22#1912850911yescharacteristic
611C22#1912851211yescharacteristic
612C4—22no—
613C4—22no—
614C4—41no—
615C4—41no—
616C4—42no—
617C4—41no—
618C4—81no—
619C4—81no—
620C4—81no—
621C4—3322no—
622C4—3322no—
623C4—3322no—
624C4—6641no—
625C4#1912823011yescharacteristic
626C4#1912828511yescharacteristic
627C4#1912841311yescharacteristic
628C4#1912843311yescharacteristic
629C2—42no—
630C2#2491139611yescharacteristic
631C2#2491749811yescharacteristic
632C2#2491773411yescharacteristic
633C2#2491933311yescharacteristic
634C2#2491933911yescharacteristic
635C2#2491949311yescharacteristic
636C2#2491967911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingC83×#2492C2
socleC23×C83C22×D4
prime radicalC24×C83C2×D4
hypercenter#2492D83
hypocenterC83#26726
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2492

Complements

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

NH ≅ G/Ntype
2. #24908785C2semidirect
4. #24908601C2semidirect
6. #24908412C2semidirect
7. #24908142C2semidirect
8. #24908101C2semidirect
10. #24908055C2semidirect
11. #24907807C2semidirect
14. C4×#14581715C2semidirect
202. C22⋊C4×C83Q16semidirect
334. C83#26726semidirect
519. C22⋊C4#13050580semidirect

Maximal subgroups

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

subgroupindex
#249104662
#249087852
#249086662
#249086012
#249084712
#249084122
#249081422
#249081012
#249080712
#249080552
#249078072
#249077712
#249075512
C4×#145817152
C83×#24922
#2672683