Finite Group Databasebeta

← #35403933

Subgroups — #35403933

Order 21248.

1462 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128181C211yesmaximal, characteristic
2C2×#19127703C211yesmaximal, characteristic
3C2×#19127698C211yesmaximal, characteristic
4#24910086C211yesmaximal, characteristic, complemented
5#24907479C212yesmaximal, complemented
6#24907474C212yesmaximal, complemented
7#24907469C212yesmaximal, complemented
8#24907464C212yesmaximal, complemented
9#24907183C211yesmaximal, characteristic, complemented
10C2×#19127464C211yesmaximal, characteristic, complemented
11C2×#424×C83C211yesmaximal, characteristic, complemented
12C22×#13050590C2211yescharacteristic
13#19128181—24no—
14#19128181C418yescomplemented
15C2×C4×Dic166C2212yes—
16C2×C4×D332C2212yes—
17C2×(C22⋊D83)⋊C4—22no—
18C2×#14581751C2211yescharacteristic
19C2×#14581750—21no—
20C22×C4⋊Dic83C2211yescharacteristic
21C22×C4⋊Dic83C2211yescharacteristic
22C2×C4⋊C4×D83—21no—
23#19127703—24no—
24#19127698—24no—
25C2×#14581717C2212yes—
26C2×#14581717C2211yescharacteristic
27C2×#14581717C2211yescharacteristic
28C2×#14581714C2212yes—
29C2×#14581714C2211yescharacteristic
30C2×#14581714C2211yescharacteristic
31#19127600—21no—
32#19127600—21no—
33#19127576—22no—
34#19127571C2212yes—
35#19127567C2212yes—
36#19127464C2212yes—
37#19127464C2212yes—
38#19127464C2212yes—
39#19127464C2212yes—
40C22×M16×C83C2211yescharacteristic
41C23×C8×C83C2211yescharacteristic
42C2×#117×C83C2211yescharacteristic
43#424×C83—24no—
44C2×C4⋊C8×C83C2212yes—
45C2×C4⋊C8×C83C2212yes—
46C2×#13050590—22no—
47C2×#13050590—22no—
48C2×#13050590—22no—
49C2×#13050590C2×C414yes—
50C2×#13050590D414yes—
51C2×#13050590Q814yes—
52C22×Dic166C2311yescharacteristic
53C22×D332C2311yescharacteristic
54C22×C22⋊D83—21no—
55C23×Dic83—21no—
56C23×Dic83—21no—
57C22×C4×D83—21no—
58C22×C4×D83—22no—
59C4×Dic166—24no—
60C4×Dic166C2×C418yescomplemented
61C4×D332—24no—
62C4×D332C2×C418yescomplemented
63(C22⋊D83)⋊C4—216no—
64#14581751—22no—
65#14581751—22no—
66#14581750—28no—
67C2×C4⋊Dic83—21no—
68C2×C4⋊Dic83—22no—
69C2×C4⋊Dic83—22no—
70C2×C4⋊Dic83—21no—
71C2×C4⋊Dic83—22no—
72C2×C4⋊Dic83—22no—
73C2×C4⋊Dic83—22no—
74C2×C4⋊Dic83C2×C414yes—
75C2×C4⋊Dic83C2311yescharacteristic
76C2×C4⋊Dic83C2312yes—
77C2×C4⋊Dic83C2311yescharacteristic
78C2×C4⋊Dic83C2312yes—
79C2×Dic83⋊C4—22no—
80C4⋊C4×D83—28no—
81C2×C22⋊Dic83—21no—
82C2×(C2×D83)⋊C4—22no—
83C2×C4×Dic83—22no—
84C2×C4×Dic83—21no—
85#14581717—22no—
86#14581717—22no—
87#14581717—24no—
88#14581714—22no—
89#14581714—22no—
90#14581714—24no—
91C83⋊#95—22no—
92C83⋊#95—22no—
93C23×C4×C83C2311yescharacteristic
94C2×M16×C83—22no—
95C2×M16×C83—22no—
96C2×M16×C83—22no—
97C22×C8×C83—22no—
98C22×C8×C83—22no—
99C22×C8×C83—22no—
100C22×C8×C83C2311yescharacteristic
101C22×C8×C83C2311yescharacteristic
102C22×C8×C83C2312yes—
103#117×C83C2×C414yes—
104#117×C83D414yes—
105C2×C4⋊C4×C83D411yescharacteristic
106C2×C4⋊C4×C83D411yescharacteristic
107C2×C22⋊C4×C83D412yes—
108C2×C42×C83C2312yes—
109C4⋊C8×C83—24no—
110C4⋊C8×C83—24no—
111C23×D83—21no—
112#13050590—28no—
113#13050590—28no—
114#13050590—28no—
115#13050590C4⋊C4116yescomplemented
116C2×Dic166—21no—
117C2×Dic166—21no—
118C2×Dic166—21no—
119C2×Dic166C22×C412yes—
120C2×Dic166C2×D412yes—
121C2×Dic166C2×Q812yes—
122C2×D332—21no—
123C2×D332—21no—
124C2×D332—21no—
125C2×D332C22×C412yes—
126C2×D332C2×D412yes—
127C2×D332C2×Q812yes—
128C2×C22⋊D83—24no—
129C2×C22⋊D83—24no—
130C2×C22⋊D83—24no—
131C22×Dic83—21no—
132C22×Dic83—22no—
133C22×Dic83—24no—
134C22×Dic83—22no—
135C22×Dic83—21no—
136C22×Dic83—21no—
137C22×Dic83—42no—
138C22×Dic83—81no—
139C22×Dic83—81no—
140C22×Dic83—81no—
141C22×Dic83—81no—
142C2×C4×D83—24no—
143C2×C4×D83—24no—
144C2×C4×D83—24no—
145C2×C4×D83—28no—
146C2×C4×D83—82no—
147C2×C4×D83—82no—
148C4⋊Dic83—24no—
149C4⋊Dic83—22no—
150C4⋊Dic83—22no—
151C4⋊Dic83—21no—
152C4⋊Dic83—21no—
153C4⋊Dic83—22no—
154C4⋊Dic83—21no—
155C4⋊Dic83—21no—
156C4⋊Dic83—42no—
157C4⋊Dic83—82no—
158C4⋊Dic83—82no—
159C4⋊Dic83C22×C414yes—
160Dic83⋊C4—28no—
161C22⋊Dic83—42no—
162(C2×D83)⋊C4—28no—
163C4×Dic83—28no—
164C4×Dic83—42no—
165C24×C83C2×D411yescharacteristic
166C22×C4×C83C22×C412yes—
167C22×C4×C83C2×D412yes—
168C22×C4×C83C2×D412yes—
169C22×C4×C83C2×D412yes—
170C22×C4×C83C2×D412yes—
171C22×C4×C83C2×D411yescharacteristic
172C22×C4×C83C2×Q812yes—
173C22×C4×C83#4112yes—
174C22×C4×C83#4112yes—
175C22×C4×C83C2411yescharacteristic
176M16×C83—44no—
177C2×C8×C83—21no—
178C2×C8×C83—21no—
179C2×C8×C83—22no—
180C2×C8×C83—21no—
181C2×C8×C83—21no—
182C2×C8×C83—22no—
183C2×C8×C83—21no—
184C2×C8×C83—21no—
185C2×C8×C83—22no—
186C2×C8×C83—44no—
187C4⋊C4×C83C22⋊C414yes—
188C4⋊C4×C83C22⋊C414yes—
189C22⋊C4×C83C22⋊C418yes—
190C42×C83C22×C414yes—
191C42×C83C2×D414yes—
192C23×C83—21no—
193C23×C83—21no—
194C23×C83—21no—
195C23×C83—21no—
196C23×C83C2×C22⋊C412yes—
197C23×C83#12112yes—
198C23×C83C22⋊Q812yes—
199C23×C83C22×D411yescharacteristic
200C22×D83—22no—
201C22×D83—22no—
202C22×D83—22no—
203C22×D83—41no—
204C22×D83—41no—
205C22×D83—41no—
206C22×D83—41no—
207C2×C4×C83—22no—
208C2×C4×C83—22no—
209C2×C4×C83—22no—
210C2×C4×C83—22no—
211C2×C4×C83C2×C22⋊C414yes—
212C2×C4×C83C2×C22⋊C414yes—
213C2×C4×C83C2×C22⋊C412yes—
214C2×C4×C83C2×C4⋊C414yes—
215C2×C4×C83#11714yes—
216C2×C4×C83#12114yes—
217C2×C4×C83#12112yes—
218C2×C4×C83C22⋊Q814yes—
219C2×C4×C83C22⋊Q812yes—
220C2×C4×C83C23×C411yescharacteristic
221C2×C4×C83C22×D411yescharacteristic
222C2×C4×C83C22×D411yescharacteristic
223C2×C4×C83C22×Q811yescharacteristic
224C2×C4×C83C2×#4111yescharacteristic
225C2×C4×C83C2×#4111yescharacteristic
226C22⋊D83—216no—
227C2×Dic83—22no—
228C2×Dic83—22no—
229C2×Dic83—22no—
230C2×Dic83—24no—
231C2×Dic83—42no—
232C2×Dic83—42no—
233C2×Dic83—41no—
234C2×Dic83—41no—
235C2×Dic83—42no—
236C2×Dic83—41no—
237C2×Dic83—42no—
238C2×Dic83—41no—
239C2×Dic83—42no—
240C2×Dic83—41no—
241C2×Dic83—41no—
242C2×Dic83—84no—
243C2×Dic83—84no—
244D332—22no—
245D332—22no—
246D332—22no—
247D332C2×C4⋊C414yes—
248C4×D83—216no—
249C4×D83—88no—
250Dic166—22no—
251Dic166—22no—
252Dic166—22no—
253Dic166C2×C4⋊C414yes—
254C8×C83—41no—
255C8×C83—41no—
256C8×C83—42no—
257C22×C83—22no—
258C22×C83—22no—
259C22×C83—22no—
260C22×C83—22no—
261C22×C83—22no—
262C22×C83—22no—
263C22×C83#38014yes—
264C22×C83C22×C22⋊C411yescharacteristic
265C22×C83C2×#12111yescharacteristic
266C22×C83C2×C22⋊Q811yescharacteristic
267C22×C83#57611yescharacteristic
268C22×C83#57711yescharacteristic
269C22×C83#57711yescharacteristic
270C22×C83#57811yescharacteristic
271C2×D83—24no—
272C2×D83—42no—
273C2×D83—42no—
274C2×D83—42no—
275C2×D83—42no—
276C2×D83—42no—
277C2×D83—42no—
278C4×C83—24no—
279C4×C83—24no—
280C4×C83#38014yes—
281C4×C83C22×C4⋊C411yescharacteristic
282C4×C83C2×#11711yescharacteristic
283C4×C83C2×#12111yescharacteristic
284C4×C83C2×C22⋊Q811yescharacteristic
285Dic83—24no—
286Dic83—44no—
287Dic83—81no—
288Dic83—82no—
289Dic83—81no—
290#20652—831nomaximal, Sylow
291C2×C83—24no—
292C2×C83C2×#38011yescharacteristic
293C2×C83#289711yescharacteristic
294C2×C83#289811yescharacteristic
295C2×C83#304511yescharacteristic
296C2×C83#304611yescharacteristic
297C2×C83#308311yescharacteristic
298C2×C83#308411yescharacteristic
299D83—44no—
300C2×#520—831no—
301C2×#424D8311yescharacteristic, complemented
302C2×#417—831no—
303C2×#416—831no—
304#2317—831no—
305#1805—832no—
306#1804—832no—
307#1803—832no—
308#1802—832no—
309#1736—831no—
310C2×#340—831no—
311C83#2065211yescharacteristic, complemented, Sylow
312C22×#41—831no—
313C22×M16C2×D8311yescharacteristic
314C23×C8C2×D8311yescharacteristic
315#520—838no—
316#520—1664no—
317C2×C4×Q8—832no—
318C2×C4×D4—832no—
319C2×C4×D4—1662no—
320C2×#117—831no—
321C2×#117—1661no—
322C2×#117C2×D8311yescharacteristic
323C22×C4⋊C4—831no—
324C22×C4⋊C4—831no—
325C22×C4⋊C4—1661no—
326#424—24no—
327C2×C4⋊C8C2×D8312yes—
328C2×C4⋊C8C2×D8312yes—
329#417—1664no—
330#416—1664no—
331C2×#103—831no—
332C2×#103—832no—
333C2×#103—831no—
334C2×#102—831no—
335C2×#102—832no—
336C2×#102—831no—
337#389—1661no—
338#389—1661no—
339#385—1662no—
340#384—832no—
341#382—832no—
342#340—832no—
343#340—832no—
344#340—832no—
345#340—832no—
346C2×#41—834no—
347C2×#41—834no—
348C2×#41—834no—
349C2×#41—1662no—
350C2×#41—1662no—
351C2×#41—1662no—
352C22×Q8—831no—
353C22×D4—831no—
354C22×D4—1661no—
355C23×C4—1661no—
356C23×C4—1661no—
357C23×C4—1662no—
358C23×C4—1661no—
359C23×C4C22×D8311yescharacteristic
360C2×M16—22no—
361C2×M16—22no—
362C2×M16—22no—
363C22×C8—22no—
364C22×C8—22no—
365C22×C8—22no—
366C22×C8C22×D8311yescharacteristic
367C22×C8C22×D8311yescharacteristic
368C22×C8C22×D8312yes—
369C4×Q8—838no—
370C4×Q8—1664no—
371C4×D4—838no—
372C4×D4—16616no—
373C4×D4—1664no—
374#117—1662no—
375#117—1662no—
376#117—1668no—
377#117C4×D8314yes—
378#117C22⋊D8314yes—
379C2×C4⋊C4—834no—
380C2×C4⋊C4—832no—
381C2×C4⋊C4—832no—
382C2×C4⋊C4—831no—
383C2×C4⋊C4—831no—
384C2×C4⋊C4—1661no—
385C2×C4⋊C4—1662no—
386C2×C4⋊C4—1662no—
387C2×C4⋊C4—1662no—
388C2×C4⋊C4—1662no—
389C2×C4⋊C4—1662no—
390C2×C4⋊C4—1661no—
391C2×C4⋊C4—1662no—
392C2×C4⋊C4—1668no—
393C2×C4⋊C4D33211yescharacteristic
394C2×C4⋊C4D33211yescharacteristic
395C2×C22⋊C4—1661no—
396C2×C22⋊C4—1662no—
397C2×C22⋊C4D33212yes—
398C2×C42—1661no—
399C2×C42—1662no—
400C2×C42C22×D8312yes—
401C4⋊C8—24no—
402C4⋊C8—24no—
403#103—1662no—
404#103—1662no—
405#103—1664no—
406#102—1662no—
407#102—1662no—
408#102—1664no—
409#95—1662no—
410#95—1662no—
411C24—1661no—
412C24C2×D33211yescharacteristic
413#41—8316no—
414#41—1668no—
415#41—1668no—
416#41—1668no—
417C2×Q8—832no—
418C2×Q8—832no—
419C2×Q8—832no—
420C2×Q8—1661no—
421C2×Q8—1661no—
422C2×Q8—1661no—
423C2×D4—832no—
424C2×D4—832no—
425C2×D4—832no—
426C2×D4—1664no—
427C2×D4—1664no—
428C2×D4—1664no—
429C2×D4—1661no—
430C2×D4—1661no—
431C2×D4—1661no—
432C22×C4—3322no—
433C22×C4—6642no—
434C22×C4—6642no—
435C22×C4—6641no—
436C22×C4—6641no—
437C22×C4—6641no—
438C22×C4—6641no—
439C22×C4—1661no—
440C22×C4—1662no—
441C22×C4—1661no—
442C22×C4—1662no—
443C22×C4—1664no—
444C22×C4—1668no—
445C22×C4—1661no—
446C22×C4—1664no—
447C22×C4—1664no—
448C22×C4—1664no—
449C22×C4C2×C4×D8312yes—
450C22×C4D4×D8312yes—
451C22×C4C2×C22⋊D8312yes—
452C22×C4C2×D33212yes—
453C22×C4C2×D33212yes—
454C22×C4C2×D33211yescharacteristic
455C22×C4Q8×D8312yes—
456C22×C4#1305058912yes—
457C22×C4#1305059112yes—
458C22×C4C23×D8311yescharacteristic
459M16—44no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—22no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—22no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—22no—
469C2×C8—44no—
470C4⋊C4—3322no—
471C4⋊C4—834no—
472C4⋊C4—6642no—
473C4⋊C4—6642no—
474C4⋊C4—1662no—
475C4⋊C4—1661no—
476C4⋊C4—1661no—
477C4⋊C4—1662no—
478C4⋊C4—1661no—
479C4⋊C4—1661no—
480C4⋊C4—1662no—
481C4⋊C4—1664no—
482C4⋊C4—1668no—
483C4⋊C4(C2×D83)⋊C414yes—
484C4⋊C4(C2×D83)⋊C414yes—
485C22⋊C4—3322no—
486C22⋊C4—1668no—
487C22⋊C4(C2×D83)⋊C418yes—
488C42—3322no—
489C42—1668no—
490C42C2×C4×D8314yes—
491C42C2×C22⋊D8314yes—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—3321no—
497C23—3321no—
498C23—3321no—
499C23—3321no—
500C23—1662no—
501C23—1662no—
502C23—1662no—
503C23C2×(C2×D83)⋊C412yes—
504C23#1458176712yes—
505C23#1458177212yes—
506C23C22×D33211yescharacteristic
507Q8—834no—
508Q8—1662no—
509Q8—1662no—
510Q8—1662no—
511D4—834no—
512D4—16616no—
513D4—1662no—
514D4—1662no—
515D4—1662no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—3322no—
521C2×C4—3322no—
522C2×C4—3321no—
523C2×C4—3321no—
524C2×C4—3322no—
525C2×C4—3321no—
526C2×C4—3322no—
527C2×C4—3321no—
528C2×C4—3322no—
529C2×C4—3321no—
530C2×C4—3321no—
531C2×C4—6648no—
532C2×C4—6644no—
533C2×C4—6644no—
534C2×C4—1664no—
535C2×C4—1662no—
536C2×C4—1662no—
537C2×C4—1662no—
538C2×C4—16616no—
539C2×C4C2×(C2×D83)⋊C414yes—
540C2×C4C2×(C2×D83)⋊C414yes—
541C2×C4C2×(C2×D83)⋊C412yes—
542C2×C4C4⋊C4×D8314yes—
543C2×C4#1458175014yes—
544C2×C4#1458176314yes—
545C2×C4#1458176712yes—
546C2×C4#1458176814yes—
547C2×C4#1458177212yes—
548C2×C4C22×C4×D8311yescharacteristic
549C2×C4C2×D4×D8311yescharacteristic
550C2×C4C22×C22⋊D8311yescharacteristic
551C2×C4C2×Q8×D8311yescharacteristic
552C2×C4C2×#1305058911yescharacteristic
553C2×C4C2×#1305059111yescharacteristic
554C8—41no—
555C8—41no—
556C8—42no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—3322no—
564C22—3322no—
565C22—3322no—
566C22—3322no—
567C22—3322no—
568C22—3322no—
569C22—1664no—
570C22#1912755814yes—
571C22C22×(C2×D83)⋊C411yescharacteristic
572C22C2×#1458176711yescharacteristic
573C22C2×#1458177211yescharacteristic
574C22#1912850611yescharacteristic
575C22#1912851311yescharacteristic
576C22#1912851311yescharacteristic
577C22#1912851811yescharacteristic
578C4—24no—
579C4—24no—
580C4—3324no—
581C4—6642no—
582C4—6641no—
583C4—6641no—
584C4—1664no—
585C4#1912755814yes—
586C4C2×C4⋊C4×D8311yescharacteristic
587C4C2×#1458175011yescharacteristic
588C4C2×#1458176311yescharacteristic
589C4C2×#1458176811yescharacteristic
590C2—24no—
591C2—3324no—
592C2C2×#1912755811yescharacteristic
593C2#2491599011yescharacteristic
594C2#2491599711yescharacteristic
595C2#2491700411yescharacteristic
596C2#2491701511yescharacteristic
597C2#2491731611yescharacteristic
598C2#2491732711yescharacteristic
subgroupG/H
centerC23C22×D332
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC2×#424×C83C2
socleC23×C83C22×D4
prime radicalC22×#13050590C22
cosocleC22×C4×C83C24
hypercenterC2×#424D83
hypocenterC83#20652
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#424

Complements

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

NH ≅ G/Ntype
4. #24910086C2semidirect
5. #24907479C2semidirect
6. #24907474C2semidirect
7. #24907469C2semidirect
8. #24907464C2semidirect
9. #24907183C2semidirect
10. C2×#19127464C2semidirect
11. C2×#424×C83C2semidirect
14. #19128181C4semidirect
60. C4×Dic166C2×C4semidirect
62. C4×D332C2×C4semidirect
115. #13050590C4⋊C4semidirect
301. C2×#424D83semidirect
311. C83#20652semidirect

Maximal subgroups

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

subgroupindex
C2×#191281812
C2×#191277032
C2×#191276982
#249100862
#249074792
#249074742
#249074692
#249074642
#249071832
C2×#191274642
C2×#424×C832
#2065283