Finite Group Databasebeta

← #35405178

Subgroups — #35405178

Order 21248.

1462 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127741C211yesmaximal, characteristic, complemented
2C2×#19127737C211yesmaximal, characteristic, complemented
3C2×#19127668C211yesmaximal, characteristic
4#24910055C211yesmaximal, characteristic
5#24907662C212yesmaximal, complemented
6#24907662C212yesmaximal, complemented
7#24907656C212yesmaximal, complemented
8#24907651C212yesmaximal, complemented
9#24907184C211yesmaximal, characteristic, complemented
10C2×#19127463C211yesmaximal, characteristic, complemented
11C2×#520×C83C211yesmaximal, characteristic
12C22×#13050574C2211yescharacteristic
13C23×C83⋊C8C2211yescharacteristic
14C2×#14581751C2211yescharacteristic
15C22×C4⋊Dic83C2211yescharacteristic
16#19127741—24no—
17#19127737—24no—
18C2×#14581725C2211yescharacteristic
19C2×#14581725C2211yescharacteristic
20C2×#14581725C2212yes—
21C2×#14581723C2211yescharacteristic
22C2×#14581723C2211yescharacteristic
23C2×#14581723C2212yes—
24#19127668C418yescomplemented
25C2×C22⋊(C83⋊C8)C2212yes—
26C2×C22⋊(C83⋊C8)C2212yes—
27#19127608—21no—
28#19127604—21no—
29#19127581—22no—
30#19127561C2212yes—
31#19127561C2212yes—
32#19127463C2212yes—
33#19127463C2212yes—
34#19127463C2212yes—
35#19127463C2212yes—
36C22×#41×C83C2211yescharacteristic
37#520×C83—24no—
38#520×C83—24no—
39C2×C4×Q8×C83—21no—
40C2×C4×D4×C83—22no—
41C2×C4×D4×C83—21no—
42C2×#117×C83C2212yes—
43C2×#117×C83C2211yescharacteristic
44C22×C4⋊C4×C83C2212yes—
45C22×C4⋊C4×C83C2211yescharacteristic
46C23×Dic83—21no—
47C2×#13050574—22no—
48C2×#13050574—22no—
49C2×#13050574C2×C414yescomplemented
50C22×C83⋊C8—22no—
51C22×C83⋊C8—22no—
52C22×C83⋊C8C2×C414yescomplemented
53C22×C83⋊C8C2311yescharacteristic
54C22×C83⋊C8C2311yescharacteristic
55C22×C83⋊C8C2312yes—
56#14581751—22no—
57#14581751—22no—
58C2×C4⋊Dic83—21no—
59C2×C4⋊Dic83—22no—
60C2×C4⋊Dic83—22no—
61C2×C4⋊Dic83C2312yes—
62C2×C4⋊Dic83C2311yescharacteristic
63C2×C4⋊Dic83C2311yescharacteristic
64C2×C22⋊Dic83—21no—
65C2×C22⋊Dic83—22no—
66C2×C22⋊Dic83—22no—
67C2×C4×Dic83—21no—
68#14581725—22no—
69#14581725—22no—
70#14581725—24no—
71#14581723—22no—
72#14581723—22no—
73#14581723—24no—
74C22⋊(C83⋊C8)C2×C418yescomplemented
75C22⋊(C83⋊C8)C2×C418yescomplemented
76C83⋊#95—22no—
77C83⋊#95—22no—
78C2×#41×C83—22no—
79C2×#41×C83—22no—
80C2×#41×C83—22no—
81C2×#41×C83—22no—
82C2×#41×C83C2×C414yes—
83C2×#41×C83Q814yes—
84C22×Q8×C83D411yescharacteristic
85C22×D4×C83D412yes—
86C22×D4×C83D411yescharacteristic
87C23×C4×C83—22no—
88C23×C4×C83—21no—
89C23×C4×C83C2312yes—
90C23×C4×C83C2311yescharacteristic
91C4×Q8×C83—28no—
92C4×D4×C83—216no—
93C4×D4×C83—28no—
94#117×C83—22no—
95#117×C83—24no—
96#117×C83—24no—
97#117×C83D414yes—
98C2×C4⋊C4×C83—22no—
99C2×C4⋊C4×C83—21no—
100C2×C4⋊C4×C83—22no—
101C2×C4⋊C4×C83—24no—
102C2×C4⋊C4×C83—24no—
103C2×C4⋊C4×C83D414yes—
104C2×C4⋊C4×C83C2311yescharacteristic
105C2×C4⋊C4×C83C2312yes—
106C2×C4⋊C4×C83C2311yescharacteristic
107C2×C22⋊C4×C83—22no—
108C2×C22⋊C4×C83—21no—
109C2×C42×C83—22no—
110C2×C42×C83—21no—
111C22×Dic83—22no—
112C22×Dic83—21no—
113C22×Dic83—21no—
114C22×Dic83—42no—
115C22×Dic83—81no—
116C22×Dic83—81no—
117#13050574—28no—
118C2×C83⋊C8—28no—
119C2×C83⋊C8—21no—
120C2×C83⋊C8—21no—
121C2×C83⋊C8—22no—
122C2×C83⋊C8—21no—
123C2×C83⋊C8—21no—
124C2×C83⋊C8—22no—
125C2×C83⋊C8C22×C412yes—
126C2×C83⋊C8C22×C412yes—
127C2×C83⋊C8C22×C414yes—
128C4⋊Dic83—21no—
129C4⋊Dic83—21no—
130C4⋊Dic83—21no—
131C4⋊Dic83—21no—
132C4⋊Dic83—22no—
133C4⋊Dic83—22no—
134C4⋊Dic83—42no—
135C22⋊Dic83—42no—
136C22⋊Dic83—82no—
137C22⋊Dic83—82no—
138C4×Dic83—42no—
139C24×C83C2×D412yes—
140C24×C83C2×D411yescharacteristic
141#41×C83—28no—
142#41×C83—28no—
143#41×C83—28no—
144#41×C83—28no—
145C2×Q8×C83—22no—
146C2×Q8×C83—22no—
147C2×Q8×C83C4⋊C414yes—
148C2×D4×C83—24no—
149C2×D4×C83—24no—
150C2×D4×C83—22no—
151C2×D4×C83—22no—
152C2×D4×C83C4⋊C418yes—
153C2×D4×C83C4⋊C414yes—
154C22×C4×C83—22no—
155C22×C4×C83—22no—
156C22×C4×C83—22no—
157C22×C4×C83—22no—
158C22×C4×C83—21no—
159C22×C4×C83—22no—
160C22×C4×C83—21no—
161C22×C4×C83—28no—
162C22×C4×C83—24no—
163C22×C4×C83—81no—
164C22×C4×C83—82no—
165C22×C4×C83—81no—
166C22×C4×C83—82no—
167C22×C4×C83C22×C414yes—
168C22×C4×C83C22×C412yes—
169C22×C4×C83C2×D412yes—
170C22×C4×C83C2×D411yescharacteristic
171C22×C4×C83C2×D412yes—
172C22×C4×C83C2×D412yes—
173C22×C4×C83C2×Q814yes—
174C22×C4×C83C2×Q812yes—
175C22×C4×C83#4112yes—
176C22×C4×C83#4112yes—
177C22×C4×C83C2411yescharacteristic
178C4⋊C4×C83—28no—
179C4⋊C4×C83—24no—
180C4⋊C4×C83—22no—
181C4⋊C4×C83—21no—
182C4⋊C4×C83—21no—
183C4⋊C4×C83C2×D412yes—
184C4⋊C4×C83C2×D414yes—
185C4⋊C4×C83C2×D412yes—
186C22⋊C4×C83—28no—
187C22⋊C4×C83—24no—
188C42×C83—28no—
189C42×C83—24no—
190C23×C83—21no—
191C23×C83—21no—
192C23×C83—21no—
193C23×C83—21no—
194C23×C83—22no—
195C23×C83—22no—
196C23×C83—22no—
197C23×C83—22no—
198C23×C83—22no—
199C23×C83—22no—
200C23×C83C2×C4⋊C414yes—
201C23×C83C2×C4⋊C412yes—
202C23×C83#12012yes—
203C23×C83#12312yes—
204C23×C83C22×D411yescharacteristic
205Q8×C83—28no—
206D4×C83—216no—
207D4×C83—28no—
208C2×C4×C83—22no—
209C2×C4×C83—22no—
210C2×C4×C83—24no—
211C2×C4×C83—24no—
212C2×C4×C83—24no—
213C2×C4×C83—24no—
214C2×C4×C83—22no—
215C2×C4×C83—24no—
216C2×C4×C83—22no—
217C2×C4×C83—42no—
218C2×C4×C83—41no—
219C2×C4×C83—41no—
220C2×C4×C83—42no—
221C2×C4×C83—41no—
222C2×C4×C83—41no—
223C2×C4×C83—84no—
224C2×C4×C83—88no—
225C2×C4×C83C2×C4⋊C414yes—
226C2×C4×C83C2×C4⋊C412yes—
227C2×C4×C83C4×D414yes—
228C2×C4×C83C4×D414yes—
229C2×C4×C83#12012yes—
230C2×C4×C83C22⋊Q814yes—
231C2×C4×C83C22⋊Q814yes—
232C2×C4×C83#12312yes—
233C2×C4×C83C23×C411yescharacteristic
234C2×C4×C83C22×D411yescharacteristic
235C2×C4×C83C22×D411yescharacteristic
236C2×C4×C83C22×Q811yescharacteristic
237C2×C4×C83C2×#4111yescharacteristic
238C2×C4×C83C2×#4111yescharacteristic
239C2×Dic83—41no—
240C2×Dic83—41no—
241C2×Dic83—42no—
242C2×Dic83—41no—
243C2×Dic83—42no—
244C2×Dic83—41no—
245C2×Dic83—42no—
246C2×Dic83—41no—
247C2×Dic83—41no—
248C2×Dic83—84no—
249C83⋊C8—22no—
250C83⋊C8—22no—
251C83⋊C8—24no—
252C22×C83—22no—
253C22×C83—22no—
254C22×C83—22no—
255C22×C83—22no—
256C22×C83—22no—
257C22×C83—22no—
258C22×C83—24no—
259C22×C83—24no—
260C22×C83—24no—
261C22×C83—42no—
262C22×C83—42no—
263C22×C83—42no—
264C22×C83—42no—
265C22×C83#38514yes—
266C22×C83C22×C4⋊C411yescharacteristic
267C22×C83C2×#12011yescharacteristic
268C22×C83C2×#12311yescharacteristic
269C22×C83#57611yescharacteristic
270C22×C83#57711yescharacteristic
271C22×C83#57711yescharacteristic
272C22×C83#57811yescharacteristic
273C4×C83—24no—
274C4×C83—44no—
275C4×C83—42no—
276C4×C83—42no—
277C4×C83#38514yes—
278C4×C83C2×C4×D411yescharacteristic
279C4×C83C2×C4×D411yescharacteristic
280C4×C83C2×C22⋊Q811yescharacteristic
281C4×C83C2×C22⋊Q811yescharacteristic
282Dic83—82no—
283Dic83—81no—
284Dic83—81no—
285#20849—831nomaximal, Sylow
286C2×C83—24no—
287C2×C83—44no—
288C2×C83C2×#38511yescharacteristic
289C2×C83#291211yescharacteristic
290C2×C83#291311yescharacteristic
291C2×C83#301211yescharacteristic
292C2×C83#301311yescharacteristic
293C2×C83#309611yescharacteristic
294C2×C83#309711yescharacteristic
295C2×#520D8311yescharacteristic
296C2×#428—831no—
297C2×#427—831no—
298C2×#408—831no—
299#2311—831no—
300#1854—832no—
301#1854—832no—
302#1853—832no—
303#1852—832no—
304#1736—831no—
305C2×#340—831no—
306C83#2084911yescharacteristic, complemented, Sylow
307C22×#41C2×D8311yescharacteristic
308C22×M16—831no—
309C23×C8—831no—
310#520—24no—
311#520—24no—
312C2×C4×Q8—21no—
313C2×C4×D4—22no—
314C2×C4×D4—21no—
315C2×#117—831no—
316C2×#117C2×D8312yes—
317C2×#117C2×D8311yescharacteristic
318C22×C4⋊C4—831no—
319C22×C4⋊C4C2×D8312yes—
320C22×C4⋊C4C2×D8311yescharacteristic
321#428—1664no—
322#427—1664no—
323C2×#107—831no—
324C2×#107—832no—
325C2×#107—831no—
326C2×#106—831no—
327C2×#106—832no—
328C2×#106—831no—
329#408—838no—
330C2×C22⋊C8—832no—
331C2×C22⋊C8—832no—
332#391—1661no—
333#390—1661no—
334#386—1662no—
335#380—832no—
336#380—832no—
337#340—832no—
338#340—832no—
339#340—832no—
340#340—832no—
341C2×#41—22no—
342C2×#41—22no—
343C2×#41—22no—
344C2×#41—22no—
345C2×#41Dic16614yes—
346C2×#41C4×D8314yes—
347C22×Q8C22⋊D8311yescharacteristic
348C22×D4C22⋊D8312yes—
349C22×D4C22⋊D8311yescharacteristic
350C23×C4—22no—
351C23×C4—21no—
352C23×C4—1661no—
353C23×C4C22×D8312yes—
354C23×C4C22×D8311yescharacteristic
355C2×M16—834no—
356C2×M16—1662no—
357C2×M16—1662no—
358C22×C8—834no—
359C22×C8—831no—
360C22×C8—831no—
361C22×C8—832no—
362C22×C8—1662no—
363C22×C8—1662no—
364C4×Q8—28no—
365C4×D4—216no—
366C4×D4—28no—
367#117—22no—
368#117—24no—
369#117—24no—
370#117—1662no—
371#117—1662no—
372#117C22⋊D8314yes—
373C2×C4⋊C4—22no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—22no—
376C2×C4⋊C4—24no—
377C2×C4⋊C4—24no—
378C2×C4⋊C4—832no—
379C2×C4⋊C4—831no—
380C2×C4⋊C4—831no—
381C2×C4⋊C4—1661no—
382C2×C4⋊C4—1662no—
383C2×C4⋊C4—1662no—
384C2×C4⋊C4C22⋊D8314yes—
385C2×C4⋊C4C22×D8311yescharacteristic
386C2×C4⋊C4C22×D8312yes—
387C2×C4⋊C4C22×D8311yescharacteristic
388C2×C22⋊C4—22no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—1661no—
391C2×C22⋊C4—1662no—
392C2×C22⋊C4—1662no—
393C2×C42—22no—
394C2×C42—21no—
395C2×C42—1661no—
396#107—1662no—
397#107—1662no—
398#107—1664no—
399#106—1662no—
400#106—1662no—
401#106—1664no—
402C22⋊C8—838no—
403C22⋊C8—838no—
404#95—1662no—
405#95—1662no—
406C24C2×C22⋊D8312yes—
407C24C2×C22⋊D8311yescharacteristic
408#41—28no—
409#41—28no—
410#41—28no—
411#41—28no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8Dic83⋊C414yes—
415C2×D4—24no—
416C2×D4—24no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4Dic83⋊C418yes—
420C2×D4Dic83⋊C414yes—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—28no—
429C22×C4—24no—
430C22×C4—81no—
431C22×C4—82no—
432C22×C4—81no—
433C22×C4—82no—
434C22×C4—3322no—
435C22×C4—6641no—
436C22×C4—6641no—
437C22×C4—1661no—
438C22×C4—1661no—
439C22×C4—1662no—
440C22×C4C2×C4×D8314yes—
441C22×C4C2×C4×D8312yes—
442C22×C4C2×C22⋊D8312yes—
443C22×C4C2×C22⋊D8311yescharacteristic
444C22×C4C2×C22⋊D8312yes—
445C22×C4C2×C22⋊D8312yes—
446C22×C4C2×Dic16614yes—
447C22×C4C2×Dic16612yes—
448C22×C4#1305059012yes—
449C22×C4#1305059012yes—
450C22×C4C23×D8311yescharacteristic
451M16—1668no—
452C2×C8—832no—
453C2×C8—832no—
454C2×C8—834no—
455C2×C8—1661no—
456C2×C8—1662no—
457C2×C8—1661no—
458C2×C8—1661no—
459C2×C8—1661no—
460C2×C8—1662no—
461C2×C8—1668no—
462C4⋊C4—28no—
463C4⋊C4—24no—
464C4⋊C4—22no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—3322no—
468C4⋊C4—1661no—
469C4⋊C4—1661no—
470C4⋊C4—1661no—
471C4⋊C4—1661no—
472C4⋊C4—1662no—
473C4⋊C4—1662no—
474C4⋊C4C2×C22⋊D8312yes—
475C4⋊C4C2×C22⋊D8314yes—
476C4⋊C4C2×C22⋊D8312yes—
477C22⋊C4—28no—
478C22⋊C4—24no—
479C22⋊C4—3322no—
480C22⋊C4—6642no—
481C22⋊C4—6642no—
482C42—28no—
483C42—24no—
484C42—3322no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23C2×Dic83⋊C414yes—
496C23C2×Dic83⋊C412yes—
497C23#1458176112yes—
498C23#1458177512yes—
499C23C22×C22⋊D8311yescharacteristic
500Q8—28no—
501D4—216no—
502D4—28no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—24no—
511C2×C4—22no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—84no—
519C2×C4—88no—
520C2×C4—3321no—
521C2×C4—3321no—
522C2×C4—3322no—
523C2×C4—3321no—
524C2×C4—3322no—
525C2×C4—3321no—
526C2×C4—3322no—
527C2×C4—3321no—
528C2×C4—3321no—
529C2×C4—6644no—
530C2×C4C2×Dic83⋊C414yes—
531C2×C4C2×Dic83⋊C412yes—
532C2×C4C4×C22⋊D8314yes—
533C2×C4C4×C22⋊D8314yes—
534C2×C4#1458176112yes—
535C2×C4#1458177114yes—
536C2×C4#1458177114yes—
537C2×C4#1458177512yes—
538C2×C4C22×C4×D8311yescharacteristic
539C2×C4C22×C22⋊D8311yescharacteristic
540C2×C4C22×C22⋊D8311yescharacteristic
541C2×C4C22×Dic16611yescharacteristic
542C2×C4C2×#1305059011yescharacteristic
543C2×C4C2×#1305059011yescharacteristic
544C8—1662no—
545C8—1662no—
546C8—1664no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—22no—
552C22—22no—
553C22—24no—
554C22—24no—
555C22—24no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22#1912757414yes—
561C22C22×Dic83⋊C411yescharacteristic
562C22C2×#1458176111yescharacteristic
563C22C2×#1458177511yescharacteristic
564C22#1912850411yescharacteristic
565C22#1912850911yescharacteristic
566C22#1912851211yescharacteristic
567C22#1912851611yescharacteristic
568C4—24no—
569C4—44no—
570C4—42no—
571C4—42no—
572C4—6641no—
573C4—6641no—
574C4—6642no—
575C4#1912757414yes—
576C4C2×C4×C22⋊D8311yescharacteristic
577C4C2×C4×C22⋊D8311yescharacteristic
578C4C2×#1458177111yescharacteristic
579C4C2×#1458177111yescharacteristic
580C2—24no—
581C2—44no—
582C2C2×#1912757411yescharacteristic
583C2#2491605611yescharacteristic
584C2#2491606111yescharacteristic
585C2#2491676711yescharacteristic
586C2#2491677811yescharacteristic
587C2#2491742111yescharacteristic
588C2#2491743211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC2×#520×C83C2
socleC23×C83C22×D4
prime radicalC22×#41×C83C22
cosocleC22×C4×C83C24
hypercenterC2×#520D83
hypocenterC83#20849
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C22×#41 ◁ C2×#520

Complements

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

NH ≅ G/Ntype
1. C2×#19127741C2semidirect
2. C2×#19127737C2semidirect
5. #24907662C2semidirect
6. #24907662C2semidirect
7. #24907656C2semidirect
8. #24907651C2semidirect
9. #24907184C2semidirect
10. C2×#19127463C2semidirect
24. #19127668C4semidirect
49. C2×#13050574C2×C4semidirect
52. C22×C83⋊C8C2×C4semidirect
74. C22⋊(C83⋊C8)C2×C4semidirect
75. C22⋊(C83⋊C8)C2×C4semidirect
306. C83#20849semidirect

Maximal subgroups

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

subgroupindex
C2×#191277412
C2×#191277372
C2×#191276682
#249100552
#249076622
#249076562
#249076512
#249071842
C2×#191274632
C2×#520×C832
#2084983