Finite Group Databasebeta

← #35504416

Subgroups — #35504416

Order 21248.

828 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24915813C211yesmaximal, characteristic
2C22⋊Q8×Dic83C211yesmaximal, characteristic
3#24909024C211yesmaximal, characteristic, complemented
4#24909004C211yesmaximal, characteristic, complemented
5#24908795C211yesmaximal, characteristic, complemented
6#24908602C211yesmaximal, characteristic, complemented
7#24908044C211yesmaximal, characteristic, complemented
8#24908040C211yesmaximal, characteristic, complemented
9#24907887C211yesmaximal, characteristic
10#24907733C211yesmaximal, characteristic, complemented
11#24907729C211yesmaximal, characteristic, complemented
12#24907488C211yesmaximal, characteristic
13#24907486C211yesmaximal, characteristic
14C2×#19127455C211yesmaximal, characteristic
15C83×#1892C211yesmaximal, characteristic
16C23×C83⋊C8C2211yescharacteristic
17C2×#14581791C2211yescharacteristic
18C2×#14581769—21no—
19C2×Q8×Dic83C2211yescharacteristic
20C22×C4×Dic83C2211yescharacteristic
21C2×#14581724C2211yescharacteristic
22C2×#14581724C2211yescharacteristic
23C2×#14581723C2211yescharacteristic
24C2×#14581723C2211yescharacteristic
25C2×#14581716C2211yescharacteristic
26C2×#14581716C2211yescharacteristic
27#19127608C2211yescharacteristic
28#19127605—21no—
29#19127600C2211yescharacteristic
30#19127600C2211yescharacteristic
31#19127598—21no—
32#19127597C2211yescharacteristic
33#19127587—21no—
34#19127576—21no—
35#19127570C2211yescharacteristic
36#19127561C2211yescharacteristic
37#19127561C2211yescharacteristic
38#19127559—21no—
39C4⋊C4×Dic83—21no—
40C4⋊C4×Dic83C2211yescharacteristic
41C22⋊C4×Dic83—21no—
42#19127464C2211yescharacteristic
43#19127464C2211yescharacteristic
44#19127463C2211yescharacteristic
45#19127463C2211yescharacteristic
46#19127463C2211yescharacteristic
47#19127463C2211yescharacteristic
48#19127455C2211yescharacteristic
49#19127455C2211yescharacteristic
50#19127455C2211yescharacteristic
51#19127455C2211yescharacteristic
52C23×C8×C83C2211yescharacteristic
53C2×C22⋊Q8×C83C2211yescharacteristic
54C2×#103×C83C2211yescharacteristic
55C2×#103×C83C2211yescharacteristic
56#380×C83C2211yescharacteristic
57#340×C83C2211yescharacteristic
58#340×C83C2211yescharacteristic
59C22×Dic166—21no—
60C23×Dic83D411yescharacteristic
61C23×Dic83D411yescharacteristic
62C22×C83⋊C8—22no—
63C22×C83⋊C8—22no—
64C22×C83⋊C8—41no—
65C22×C83⋊C8C2311yescharacteristic
66C22×C83⋊C8C2311yescharacteristic
67#14581791—41no—
68#14581791—41no—
69#14581769—44no—
70Q8×Dic83—41no—
71Q8×Dic83—41no—
72C2×C4⋊Dic83—21no—
73C2×C4⋊Dic83—21no—
74C2×C4⋊Dic83—21no—
75C2×C4⋊Dic83C2311yescharacteristic
76C2×C4⋊Dic83C2311yescharacteristic
77C2×C4⋊Dic83C2311yescharacteristic
78C2×C4⋊Dic83C2311yescharacteristic
79C2×Dic83⋊C4—21no—
80C2×Dic83⋊C4—21no—
81C2×Dic83⋊C4—21no—
82C2×Dic83⋊C4—21no—
83C2×C22⋊Dic83—21no—
84C2×C22⋊Dic83—21no—
85C2×C4×Dic83—21no—
86C2×C4×Dic83—21no—
87C2×C4×Dic83—41no—
88C2×C4×Dic83—41no—
89C2×C4×Dic83D411yescharacteristic
90C2×C4×Dic83D411yescharacteristic
91C2×C4×Dic83C2311yescharacteristic
92C2×C4×Dic83C2311yescharacteristic
93#14581724—22no—
94#14581724—41no—
95#14581723—22no—
96#14581723—41no—
97#14581716—22no—
98#14581716—41no—
99C83⋊#95—21no—
100C83⋊#95—21no—
101C83⋊#95—21no—
102C83⋊#95—21no—
103C83⋊#95—21no—
104C83⋊#95—21no—
105C83⋊#95—21no—
106C83⋊#95—21no—
107C22×Q8×C83C2311yescharacteristic
108C23×C4×C83C2311yescharacteristic
109C22×C8×C83—22no—
110C22×C8×C83—22no—
111C22×C8×C83—41no—
112C22×C8×C83C2311yescharacteristic
113C22×C8×C83C2311yescharacteristic
114C22⋊Q8×C83—22no—
115C22⋊Q8×C83Q814yescomplemented
116C2×C4⋊C4×C83—21no—
117C2×C4⋊C4×C83C2311yescharacteristic
118C2×C4⋊C4×C83C2311yescharacteristic
119C2×C4⋊C4×C83C2311yescharacteristic
120C2×C22⋊C4×C83—21no—
121C2×C22⋊C4×C83—21no—
122#103×C83—22no—
123#103×C83—41no—
124#95×C83—21no—
125C2×Dic166—42no—
126C2×Dic166—42no—
127C2×Dic166—42no—
128C22×Dic83—22no—
129C22×Dic83—22no—
130C22×Dic83—21no—
131C22×Dic83—21no—
132C22×Dic83—21no—
133C22×Dic83—21no—
134C22×Dic83—41no—
135C22×Dic83—41no—
136C22×Dic83—41no—
137C22×Dic83SD1614yes—
138C22×Dic83C2×D411yescharacteristic
139C22×Dic83C2×D411yescharacteristic
140C22×Dic83C2×D411yescharacteristic
141C22×Dic83C2×D411yescharacteristic
142C2×C83⋊C8—21no—
143C2×C83⋊C8—21no—
144C2×C83⋊C8—21no—
145C2×C83⋊C8—21no—
146C2×C83⋊C8—21no—
147C2×C83⋊C8—44no—
148C2×C83⋊C8#4112yes—
149C4⋊Dic83—21no—
150C4⋊Dic83—21no—
151C4⋊Dic83—21no—
152C4⋊Dic83—21no—
153C4⋊Dic83—21no—
154C4⋊Dic83—21no—
155C4⋊Dic83—21no—
156C4⋊Dic83—21no—
157C4⋊Dic83—42no—
158C4⋊Dic83—42no—
159C4⋊Dic83—42no—
160Dic83⋊C4—42no—
161Dic83⋊C4—42no—
162Dic83⋊C4—42no—
163Dic83⋊C4—42no—
164C22⋊Dic83—42no—
165C22⋊Dic83—42no—
166C4×Dic83—21no—
167C4×Dic83—21no—
168C4×Dic83—21no—
169C4×Dic83—21no—
170C4×Dic83—41no—
171C4×Dic83—41no—
172C4×Dic83—42no—
173C4×Dic83—42no—
174C24×C83C2×D411yescharacteristic
175C2×Q8×C83—21no—
176C2×Q8×C83—21no—
177C2×Q8×C83—21no—
178C2×Q8×C83—21no—
179C2×Q8×C83—21no—
180C2×Q8×C83C2×Q812yes—
181C22×C4×C83—21no—
182C22×C4×C83—21no—
183C22×C4×C83—21no—
184C22×C4×C83—21no—
185C22×C4×C83—21no—
186C22×C4×C83—21no—
187C22×C4×C83—21no—
188C22×C4×C83—21no—
189C22×C4×C83C2×D411yescharacteristic
190C22×C4×C83C2×Q812yes—
191C22×C4×C83#4112yes—
192C22×C4×C83C2411yescharacteristic
193C2×C8×C83—21no—
194C2×C8×C83—21no—
195C2×C8×C83—21no—
196C2×C8×C83—21no—
197C2×C8×C83—21no—
198C2×C8×C83—44no—
199C2×C8×C83#4112yes—
200C4⋊C4×C83—24no—
201C4⋊C4×C83—21no—
202C4⋊C4×C83—21no—
203C4⋊C4×C83—21no—
204C4⋊C4×C83—21no—
205C4⋊C4×C83—21no—
206C4⋊C4×C83C2×Q812yes—
207C22⋊C4×C83—24no—
208C22⋊C4×C83—42no—
209C23×C83—21no—
210C23×C83—21no—
211C23×C83—21no—
212C23×C83—21no—
213C23×C83C22⋊Q812yes—
214C23×C83C2×SD1612yes—
215C23×C83#13712yes—
216C23×C83C22×D411yescharacteristic
217Q8×C83—22no—
218Q8×C83—22no—
219Q8×C83—41no—
220Q8×C83—41no—
221C2×C4×C83—21no—
222C2×C4×C83—21no—
223C2×C4×C83—22no—
224C2×C4×C83—22no—
225C2×C4×C83—22no—
226C2×C4×C83—22no—
227C2×C4×C83—22no—
228C2×C4×C83—22no—
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—41no—
238C2×C4×C83—41no—
239C2×C4×C83C22⋊Q812yes—
240C2×C4×C83C22×Q811yescharacteristic
241C2×C4×C83C2×#4111yescharacteristic
242C2×C4×C83C2×#4111yescharacteristic
243C2×C4×C83C2×#4111yescharacteristic
244C2×C4×C83H(5,𝔽2)11yescharacteristic
245C2×C4×C83H(5,𝔽2)11yescharacteristic
246C2×Dic83—21no—
247C2×Dic83—21no—
248C2×Dic83—21no—
249C2×Dic83—21no—
250C2×Dic83—21no—
251C2×Dic83—21no—
252C2×Dic83—21no—
253C2×Dic83—21no—
254C2×Dic83—21no—
255C2×Dic83—44no—
256C2×Dic83—44no—
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—41no—
268C2×Dic83—41no—
269C2×Dic83C2×SD1612yes—
270C2×Dic83C2×SD1612yes—
271C2×Dic83#13512yes—
272Dic166—84no—
273C8×C83—22no—
274C8×C83—41no—
275C83⋊C8—22no—
276C83⋊C8—41no—
277C22×C83—22no—
278C22×C83—22no—
279C22×C83—22no—
280C22×C83—22no—
281C22×C83—22no—
282C22×C83—22no—
283C22×C83#47514yes—
284C22×C83C2×C22⋊Q811yescharacteristic
285C22×C83#53211yescharacteristic
286C22×C83#53411yescharacteristic
287C22×C83C22×SD1611yescharacteristic
288C22×C83C2×#13511yescharacteristic
289C22×C83C2×#13711yescharacteristic
290C22×C83#57511yescharacteristic
291C4×C83—22no—
292C4×C83—42no—
293C4×C83—42no—
294C4×C83—81no—
295C4×C83—81no—
296C4×C83#55111yescharacteristic
297C4×C83#55111yescharacteristic
298C4×C83#55411yescharacteristic
299C4×C83#55411yescharacteristic
300Dic83—22no—
301Dic83—22no—
302Dic83—22no—
303Dic83—41no—
304Dic83—81no—
305Dic83—81no—
306Dic83—81no—
307Dic83—81no—
308#29159—831nomaximal, Sylow
309C2×C83—24no—
310C2×C83#256411yescharacteristic
311C2×C83C2×#47511yescharacteristic
312C2×C83#307611yescharacteristic
313C2×C83#316711yescharacteristic
314C2×C83#318111yescharacteristic
315C2×C83#318711yescharacteristic
316C2×C83#319011yescharacteristic
317#2857—831no—
318C4×C22⋊Q8—831no—
319#2099—831no—
320#2096—831no—
321#2058—831no—
322#2024—831no—
323#1919—831no—
324#1919—831no—
325#1892—831no—
326#1892D8311yescharacteristic
327#1865—831no—
328#1865—831no—
329#1807—831no—
330#1807—831no—
331C2×#336—831no—
332C83#2915911yescharacteristic, complemented, Sylow
333C23×C8—831no—
334C23×C8C2×D8311yescharacteristic
335C2×C4⋊Q8—831no—
336C2×C22⋊Q8—1661no—
337C2×C22⋊Q8C2×D8311yescharacteristic
338C2×C4×Q8—831no—
339C22×C42—831no—
340C2×#106—831no—
341C2×#106—831no—
342C2×#106—831no—
343C2×#106—831no—
344C2×#103—831no—
345C2×#103—831no—
346C2×#103C2×D8311yescharacteristic
347C2×#103C2×D8311yescharacteristic
348#391—831no—
349#391—1661no—
350#389—831no—
351#389—831no—
352#389—831no—
353#389—1661no—
354#387—1661no—
355#385—1661no—
356#384—831no—
357#380—831no—
358#380—831no—
359#380—1661no—
360#380C2×D8311yescharacteristic
361C4×C4⋊C4—831no—
362C4×C4⋊C4—1661no—
363C4×C22⋊C4—1661no—
364#340—831no—
365#340—831no—
366#340—831no—
367#340—831no—
368#340—831no—
369#340—831no—
370#340C2×D8311yescharacteristic
371#340C2×D8311yescharacteristic
372#336—831no—
373#336—831no—
374#336—831no—
375#336—831no—
376C22×Q8—1661no—
377C22×Q8C22×D8311yescharacteristic
378C23×C4—831no—
379C23×C4—831no—
380C23×C4C22×D8311yescharacteristic
381C22×C8—22no—
382C22×C8—22no—
383C22×C8—41no—
384C22×C8—3321no—
385C22×C8—831no—
386C22×C8—831no—
387C22×C8—1662no—
388C22×C8—1662no—
389C22×C8C22×D8311yescharacteristic
390C22×C8C22×D8311yescharacteristic
391C4⋊Q8—3321no—
392C4⋊Q8—3321no—
393C22⋊Q8—22no—
394C22⋊Q8—3324no—
395C22⋊Q8Dic16614yescomplemented
396C4×Q8—3321no—
397C4×Q8—3321no—
398C2×C4⋊C4—21no—
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—1661no—
406C2×C4⋊C4—1661no—
407C2×C4⋊C4—1661no—
408C2×C4⋊C4—1661no—
409C2×C4⋊C4—1661no—
410C2×C4⋊C4C22×D8311yescharacteristic
411C2×C4⋊C4C22×D8311yescharacteristic
412C2×C4⋊C4C22×D8311yescharacteristic
413C2×C22⋊C4—21no—
414C2×C22⋊C4—21no—
415C2×C22⋊C4—1661no—
416C2×C22⋊C4—1661no—
417C2×C42—3321no—
418C2×C42—3321no—
419C2×C42—831no—
420C2×C42—831no—
421C2×C42—831no—
422C2×C42—831no—
423C2×C42—1661no—
424C2×C42—1661no—
425#106—3321no—
426#106—3321no—
427#106—1662no—
428#106—1662no—
429#103—22no—
430#103—41no—
431#103—3321no—
432#103—1662no—
433#95—21no—
434#95—1661no—
435#95—1661no—
436#95—1661no—
437#95—1661no—
438#95—1661no—
439#95—1661no—
440#95—1661no—
441#95—1661no—
442C24D4×D8311yescharacteristic
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—3322no—
449C2×Q8—3322no—
450C2×Q8—3322no—
451C2×Q8C2×Dic16612yes—
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—3321no—
461C22×C4—3321no—
462C22×C4—3321no—
463C22×C4—834no—
464C22×C4—831no—
465C22×C4—831no—
466C22×C4—831no—
467C22×C4—831no—
468C22×C4—1661no—
469C22×C4—1661no—
470C22×C4—1661no—
471C22×C4—1661no—
472C22×C4—1662no—
473C22×C4—1662no—
474C22×C4D4×D8311yescharacteristic
475C22×C4C2×Dic16612yes—
476C22×C4#1305058912yes—
477C22×C4C23×D8311yescharacteristic
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—44no—
484C2×C8—3324no—
485C2×C8—832no—
486C2×C8—1661no—
487C2×C8—1661no—
488C2×C8—1661no—
489C2×C8—1661no—
490C2×C8—1661no—
491C2×C8#1305058912yes—
492C4⋊C4—24no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—3322no—
499C4⋊C4—3322no—
500C4⋊C4—3322no—
501C4⋊C4—3322no—
502C4⋊C4—3322no—
503C4⋊C4—3322no—
504C4⋊C4—3322no—
505C4⋊C4—1661no—
506C4⋊C4—1661no—
507C4⋊C4—1661no—
508C4⋊C4—1661no—
509C4⋊C4—1661no—
510C4⋊C4—1661no—
511C4⋊C4—1661no—
512C4⋊C4—1661no—
513C4⋊C4C2×Dic16612yes—
514C22⋊C4—24no—
515C22⋊C4—42no—
516C22⋊C4—3322no—
517C22⋊C4—3322no—
518C42—3321no—
519C42—3321no—
520C42—3322no—
521C42—3322no—
522C42—1661no—
523C42—1661no—
524C42—1661no—
525C42—1661no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23#1458176912yes—
531C23SD16×D8312yes—
532C23#1458182412yes—
533C23C2×D4×D8311yescharacteristic
534Q8—22no—
535Q8—22no—
536Q8—41no—
537Q8—41no—
538Q8—6644no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—41no—
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—41no—
557C2×C4—3324no—
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—3324no—
571C2×C4—832no—
572C2×C4—832no—
573C2×C4—832no—
574C2×C4—1661no—
575C2×C4—1661no—
576C2×C4—1661no—
577C2×C4—1661no—
578C2×C4—1661no—
579C2×C4—1661no—
580C2×C4—1661no—
581C2×C4—1661no—
582C2×C4—1661no—
583C2×C4#1458176912yes—
584C2×C4C22×Dic16611yescharacteristic
585C2×C4#41×D8311yescharacteristic
586C2×C4C2×#1305058911yescharacteristic
587C2×C4C2×#1305058911yescharacteristic
588C2×C4#1458183711yescharacteristic
589C2×C4#1458183811yescharacteristic
590C8—22no—
591C8—41no—
592C8—3321no—
593C8—1662no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22#1912798414yes—
601C22C2×#1458176911yescharacteristic
602C22#1912824611yescharacteristic
603C22#1912825911yescharacteristic
604C22C2×SD16×D8311yescharacteristic
605C22#135×D8311yescharacteristic
606C22C2×#1458182411yescharacteristic
607C22#1912849811yescharacteristic
608C4—22no—
609C4—42no—
610C4—42no—
611C4—81no—
612C4—81no—
613C4—3321no—
614C4—6641no—
615C4—6641no—
616C4—6641no—
617C4—6641no—
618C4—1662no—
619C4—1662no—
620C4—1662no—
621C4#1912837411yescharacteristic
622C4#1912838211yescharacteristic
623C4#1912840811yescharacteristic
624C4#1912841011yescharacteristic
625C2—24no—
626C2#2491264711yescharacteristic
627C2C2×#1912798411yescharacteristic
628C2#2491724711yescharacteristic
629C2#2491801711yescharacteristic
630C2#2491816711yescharacteristic
631C2#2491821511yescharacteristic
632C2#2491824511yescharacteristic
subgroupG/H
centerC23C2×D4×D83
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingC83×#1892C2
socleC23×C83C22×D4
prime radicalC24×C83C2×D4
hypercenter#1892D83
hypocenterC83#29159
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1892

Complements

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

NH ≅ G/Ntype
3. #24909024C2semidirect
4. #24909004C2semidirect
5. #24908795C2semidirect
6. #24908602C2semidirect
7. #24908044C2semidirect
8. #24908040C2semidirect
10. #24907733C2semidirect
11. #24907729C2semidirect
115. C22⋊Q8×C83Q8semidirect
332. C83#29159semidirect
395. C22⋊Q8Dic166semidirect

Maximal subgroups

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

subgroupindex
#249158132
C22⋊Q8×Dic832
#249090242
#249090042
#249087952
#249086022
#249080442
#249080402
#249078872
#249077332
#249077292
#249074882
#249074862
C2×#191274552
C83×#18922
#2915983