Finite Group Databasebeta

← #35577872

Subgroups — #35577872

Order 21248.

1986 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24915813C212yesmaximal, complemented
2#24915773C211yesmaximal, characteristic, complemented
3#24913274C214yesmaximal
4#24913132C214yesmaximal, complemented
5#24913127C214yesmaximal, complemented
6#24911178C212yesmaximal, complemented
7#24911172C212yesmaximal, complemented
8#24911157C212yesmaximal, complemented
9#24911152C212yesmaximal, complemented
10#24910194C214yesmaximal, complemented
11#24909845C212yesmaximal
12C2×C4×(C2×D83)⋊C4C211yesmaximal, characteristic
13C83×#2851C211yesmaximal, characteristic, complemented
14C23×C4×D83C2211yescharacteristic
15C2×#14581791C2212yescomplemented
16C2×#14581784C2211yescharacteristic, complemented
17C2×#14581769C2214yes—
18C2×#14581750C2212yes—
19C2×#14581750C2212yes—
20C22×(C2×D83)⋊C4C2212yes—
21C22×C4×Dic83C2211yescharacteristic
22#19127636C2214yes—
23#19127636C2214yes—
24#19127625C2214yes—
25#19127625C2214yes—
26#19127623C2214yes—
27#19127623C2214yes—
28#19127614C2214yes—
29#19127614C2214yes—
30#19127606C2214yescomplemented
31#19127605C2214yescomplemented
32#19127605C2214yescomplemented
33#19127600C2212yescomplemented
34#19127599C2212yescomplemented
35#19127598C2214yescomplemented
36#19127597C2212yescomplemented
37#19127597C2214yescomplemented
38#19127596C2214yescomplemented
39#19127586C2214yescomplemented
40#19127585C2214yescomplemented
41#19127583C2214yescomplemented
42#19127583C2214yescomplemented
43#19127582C2214yescomplemented
44#19127574C2214yes—
45#19127570C2212yescomplemented
46#19127567C2211yescharacteristic, complemented
47#19127563C2214yes—
48#19127561C2212yes—
49#19127559C2214yes—
50#19127558C2212yes—
51#19127558C2212yes—
52#19127556C2214yes—
53C22⋊C4×Dic83C2214yes—
54C4×(C2×D83)⋊C4C2212yes—
55C4×(C2×D83)⋊C4C2212yes—
56C4×(C2×D83)⋊C4C2212yes—
57C4×(C2×D83)⋊C4C2212yes—
58#19127548C2214yes—
59C2×C83⋊#95C2212yes—
60C2×#125×C83C2211yescharacteristic, complemented
61C22×C42×C83C2211yescharacteristic
62#389×C83C2212yescomplemented
63#387×C83C2214yescomplemented
64#385×C83C2214yes—
65#382×C83C2211yescharacteristic, complemented
66#380×C83C2212yes—
67C24×D83C2311yescharacteristic
68C22×Dic166C2314yes—
69C23×Dic83C2311yescharacteristic
70C23×Dic83C2312yes—
71C22×C4×D83—42no—
72C22×C4×D83—42no—
73C22×C4×D83Q818yescomplemented
74C22×C4×D83C2312yes—
75C22×C4×D83C2312yes—
76#14581791—44no—
77#14581784—42no—
78#14581769—216no—
79#14581750—44no—
80#14581750—44no—
81C2×C4⋊Dic83C2312yes—
82C2×C4⋊Dic83C2312yes—
83C2×C4⋊Dic83C2314yes—
84C2×Dic83⋊C4C2314yes—
85C2×Dic83⋊C4C2314yes—
86C2×Dic83⋊C4C2314yes—
87C2×Dic83⋊C4C2314yes—
88C2×Dic83⋊C4C2314yes—
89C2×Dic83⋊C4C2314yes—
90C2×C22⋊Dic83C2314yes—
91C2×(C2×D83)⋊C4—28no—
92C2×(C2×D83)⋊C4C2312yes—
93C2×(C2×D83)⋊C4C2312yes—
94C2×(C2×D83)⋊C4C2312yes—
95C2×(C2×D83)⋊C4C2312yes—
96C2×(C2×D83)⋊C4C2314yes—
97C2×(C2×D83)⋊C4C2314yes—
98C2×(C2×D83)⋊C4C2314yes—
99C2×(C2×D83)⋊C4C2314yes—
100C2×C4×Dic83—42no—
101C2×C4×Dic83C2312yes—
102C2×C4×Dic83C2314yes—
103C2×C4×Dic83C2314yes—
104C2×C4×Dic83C2311yescharacteristic
105C2×C4×Dic83C2311yescharacteristic
106C83⋊#95C2314yes—
107C83⋊#95C2314yes—
108C83⋊#95C2314yes—
109C83⋊#95C2314yes—
110C83⋊#95C2314yes—
111C83⋊#95C2314yes—
112C83⋊#95C2314yes—
113C83⋊#95C2312yes—
114C83⋊#95C2312yes—
115C83⋊#95C2312yes—
116C83⋊#95C2312yes—
117C23×C4×C83C2311yescharacteristic
118C23×C4×C83C2312yes—
119#125×C83—42no—
120C2×C4⋊C4×C83C2312yes—
121C2×C4⋊C4×C83C2312yes—
122C2×C4⋊C4×C83C2314yes—
123C2×C4⋊C4×C83C2314yes—
124C2×C22⋊C4×C83C2314yes—
125C2×C42×C83—42no—
126C2×C42×C83C2311yescharacteristic
127C2×C42×C83C2311yescharacteristic
128C2×C42×C83C2312yes—
129#95×C83C2314yes—
130#95×C83C2314yes—
131#95×C83C2314yes—
132C23×D83—24no—
133C23×D83—41no—
134C23×D83—41no—
135C23×D83—42no—
136C23×D83C2×Q814yes—
137C23×D83C2412yes—
138C2×Dic166—28no—
139C2×Dic166—44no—
140C2×Dic166—44no—
141C22×Dic83—41no—
142C22×Dic83—41no—
143C22×Dic83—42no—
144C22×Dic83—42no—
145C22×Dic83C2×Q814yes—
146C22×Dic83#4118yes—
147C22×Dic83C2414yes—
148C22×Dic83C2414yes—
149C22×Dic83C2412yes—
150C22×Dic83C2412yes—
151C22×Dic83C2412yes—
152C2×C4×D83—24no—
153C2×C4×D83—24no—
154C2×C4×D83—416no—
155C2×C4×D83—42no—
156C2×C4×D83—42no—
157C2×C4×D83C2×Q818yes—
158C2×C4×D83C2×Q818yes—
159C4⋊Dic83—24no—
160C4⋊Dic83—24no—
161C4⋊Dic83—28no—
162Dic83⋊C4—28no—
163Dic83⋊C4—28no—
164Dic83⋊C4—44no—
165Dic83⋊C4—44no—
166Dic83⋊C4—44no—
167Dic83⋊C4—44no—
168C22⋊Dic83—28no—
169(C2×D83)⋊C4—24no—
170(C2×D83)⋊C4—24no—
171(C2×D83)⋊C4—24no—
172(C2×D83)⋊C4—24no—
173(C2×D83)⋊C4—44no—
174(C2×D83)⋊C4—44no—
175(C2×D83)⋊C4—44no—
176(C2×D83)⋊C4—44no—
177C4×Dic83—24no—
178C4×Dic83—41no—
179C4×Dic83—41no—
180C4×Dic83—44no—
181C4×Dic83—44no—
182C24×C83C2411yescharacteristic
183C22×C4×C83—22no—
184C22×C4×C83—22no—
185C22×C4×C83—24no—
186C22×C4×C83—42no—
187C22×C4×C83—42no—
188C22×C4×C83C2×Q814yes—
189C22×C4×C83C2414yes—
190C22×C4×C83C2412yes—
191C22×C4×C83C2412yes—
192C22×C4×C83C2414yes—
193C22×C4×C83C2411yescharacteristic
194C22×C4×C83C2411yescharacteristic
195C4⋊C4×C83—24no—
196C4⋊C4×C83—24no—
197C4⋊C4×C83—44no—
198C4⋊C4×C83—44no—
199C22⋊C4×C83—28no—
200C42×C83—22no—
201C42×C83—22no—
202C42×C83—42no—
203C23×C83—21no—
204C23×C83—21no—
205C23×C83—22no—
206C23×C83C22×Q812yes—
207C23×C83C2×#4114yes—
208C23×C83C2511yescharacteristic
209C22×D83—28no—
210C22×D83—22no—
211C22×D83—22no—
212C22×D83—22no—
213C22×D83—22no—
214C22×D83—22no—
215C22×D83—22no—
216C22×D83—44no—
217C22×D83—44no—
218C22×D83—48no—
219C22×D83—48no—
220C22×D83C22×Q814yes—
221C2×C4×C83—21no—
222C2×C4×C83—21no—
223C2×C4×C83—28no—
224C2×C4×C83—24no—
225C2×C4×C83—24no—
226C2×C4×C83—22no—
227C2×C4×C83—22no—
228C2×C4×C83—22no—
229C2×C4×C83—22no—
230C2×C4×C83—24no—
231C2×C4×C83—24no—
232C2×C4×C83—22no—
233C2×C4×C83—24no—
234C2×C4×C83—24no—
235C2×C4×C83—22no—
236C2×C4×C83—48no—
237C2×C4×C83C22×Q812yes—
238C2×C4×C83C22×Q812yes—
239C2×C4×C83H(5,𝔽2)12yes—
240C2×C4×C83H(5,𝔽2)12yes—
241C2×Dic83—22no—
242C2×Dic83—22no—
243C2×Dic83—22no—
244C2×Dic83—22no—
245C2×Dic83—22no—
246C2×Dic83—22no—
247C2×Dic83—24no—
248C2×Dic83—24no—
249C2×Dic83—24no—
250C2×Dic83—24no—
251C2×Dic83—24no—
252C2×Dic83—24no—
253C2×Dic83—44no—
254C2×Dic83—48no—
255C2×Dic83C22×Q814yes—
256C2×Dic83C2×#4114yes—
257C2×Dic83C2×#4114yes—
258C4×D83—216no—
259C4×D83—48no—
260Dic166—88no—
261C22×C83—22no—
262C22×C83—22no—
263C22×C83—24no—
264C22×C83—24no—
265C22×C83#53314yes—
266C22×C83C23×Q811yescharacteristic
267C22×C83C22×#4112yes—
268C22×C83C2×H(5,𝔽2)11yescharacteristic
269C22×C83C2×H(5,𝔽2)11yescharacteristic
270C22×C83#58512yes—
271C2×D83—24no—
272C2×D83—24no—
273C2×D83—24no—
274C2×D83—416no—
275C2×D83—42no—
276C2×D83—42no—
277C2×D83—42no—
278C2×D83—42no—
279C4×C83—22no—
280C4×C83—44no—
281C4×C83—44no—
282C4×C83—42no—
283C4×C83—42no—
284C4×C83#54314yes—
285Dic83—24no—
286Dic83—24no—
287Dic83—24no—
288Dic83—44no—
289Dic83—44no—
290#35033—831nomaximal, Sylow
291C2×C83—24no—
292C2×C83C2×#53311yescharacteristic
293C2×C83C2×#54311yescharacteristic
294C2×C83#345811yescharacteristic
295C2×C83#350512yes—
296C2×C83#350612yes—
297D83—44no—
298#2857—832no—
299#2851—831no—
300#2851D8311yescharacteristic, complemented
301#2626—834no—
302#2608—834no—
303#2607—834no—
304#2441—832no—
305#2441—832no—
306#2439—832no—
307#2439—832no—
308#2329—834no—
309#2280—832no—
310C2×C4×C22⋊C4—831no—
311C83#3503311yescharacteristic, complemented, Sylow
312C24×C4—831no—
313C2×C4⋊Q8—832no—
314C2×#125—831no—
315C2×#125C2×D8311yescharacteristic, complemented
316C2×C22⋊Q8—834no—
317C2×#117—832no—
318C2×#117—832no—
319C22×C22⋊C4—832no—
320C22×C42—831no—
321C22×C42C2×D8311yescharacteristic
322#399—834no—
323#399—834no—
324#396—834no—
325#396—834no—
326#396—834no—
327#396—834no—
328#393—834no—
329#393—834no—
330#391—834no—
331#391—834no—
332#391—834no—
333#389—834no—
334#389—834no—
335#389—832no—
336#389—832no—
337#389—832no—
338#389—834no—
339#389C2×D8312yescomplemented
340#387—834no—
341#387—834no—
342#387—834no—
343#387—834no—
344#387—834no—
345#387C2×D8314yescomplemented
346#385—834no—
347#385C2×D8314yes—
348#384—832no—
349#382—831no—
350#382C2×D8311yescharacteristic, complemented
351#381—834no—
352#380—832no—
353#380—832no—
354#380—832no—
355#380—834no—
356#380C2×D8312yes—
357#379—834no—
358C4×C22⋊C4—834no—
359C4×C22⋊C4—832no—
360C4×C22⋊C4—832no—
361C4×C22⋊C4—832no—
362C4×C22⋊C4—832no—
363C4×C22⋊C4—834no—
364C2×#95—832no—
365C25—831no—
366C22×Q8—834no—
367C23×C4—3322no—
368C23×C4—3322no—
369C23×C4—832no—
370C23×C4—831no—
371C23×C4—838no—
372C23×C4—832no—
373C23×C4—832no—
374C23×C4C22×D8311yescharacteristic
375C23×C4C22×D8312yes—
376C4⋊Q8—3324no—
377#125—42no—
378#125—3322no—
379C22⋊Q8—16616no—
380#117—3324no—
381#117—3324no—
382C2×C4⋊C4—834no—
383C2×C4⋊C4—834no—
384C2×C4⋊C4—834no—
385C2×C4⋊C4—834no—
386C2×C4⋊C4—832no—
387C2×C4⋊C4—832no—
388C2×C4⋊C4—834no—
389C2×C4⋊C4—834no—
390C2×C4⋊C4—834no—
391C2×C4⋊C4C22×D8312yes—
392C2×C4⋊C4C22×D8312yes—
393C2×C4⋊C4C22×D8314yes—
394C2×C4⋊C4C22×D8314yes—
395C2×C22⋊C4—834no—
396C2×C22⋊C4—834no—
397C2×C22⋊C4—834no—
398C2×C22⋊C4—834no—
399C2×C22⋊C4—834no—
400C2×C22⋊C4—832no—
401C2×C22⋊C4—832no—
402C2×C22⋊C4—832no—
403C2×C22⋊C4—832no—
404C2×C22⋊C4—1668no—
405C2×C22⋊C4C22×D8314yes—
406C2×C42—42no—
407C2×C42—3322no—
408C2×C42—834no—
409C2×C42—834no—
410C2×C42—831no—
411C2×C42—831no—
412C2×C42—832no—
413C2×C42C22×D8311yescharacteristic
414C2×C42C22×D8311yescharacteristic
415C2×C42C22×D8312yes—
416#95—832no—
417#95—832no—
418#95—832no—
419#95—832no—
420#95—834no—
421#95—834no—
422#95—834no—
423#95—834no—
424#95—834no—
425#95—834no—
426#95—834no—
427#95C22×D8314yes—
428#95C22×D8314yes—
429#95C22×D8314yes—
430C24—3321no—
431C24—3321no—
432C24—3322no—
433C24—834no—
434C24—832no—
435C24—1664no—
436C24C23×D8311yescharacteristic
437C2×Q8—3324no—
438C2×Q8—3324no—
439C2×Q8—1668no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—3322no—
446C22×C4—3322no—
447C22×C4—33216no—
448C22×C4—3322no—
449C22×C4—3321no—
450C22×C4—3321no—
451C22×C4—3322no—
452C22×C4—834no—
453C22×C4—838no—
454C22×C4—838no—
455C22×C4—832no—
456C22×C4—832no—
457C22×C4—832no—
458C22×C4—838no—
459C22×C4—834no—
460C22×C4—834no—
461C22×C4—1664no—
462C22×C4—1664no—
463C22×C4Q8×D8314yes—
464C22×C4C23×D8312yes—
465C22×C4C23×D8312yes—
466C22×C4C23×D8314yes—
467C22×C4C23×D8314yes—
468C22×C4C23×D8311yescharacteristic
469C22×C4C23×D8311yescharacteristic
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—44no—
473C4⋊C4—44no—
474C4⋊C4—3324no—
475C4⋊C4—3324no—
476C4⋊C4—3324no—
477C4⋊C4—3324no—
478C4⋊C4—1668no—
479C4⋊C4—1668no—
480C4⋊C4—1668no—
481C4⋊C4—1664no—
482C4⋊C4—1664no—
483C22⋊C4—28no—
484C22⋊C4—3324no—
485C22⋊C4—3324no—
486C22⋊C4—3324no—
487C22⋊C4—3324no—
488C22⋊C4—1664no—
489C22⋊C4—1664no—
490C22⋊C4—1664no—
491C22⋊C4—1664no—
492C22⋊C4—1668no—
493C42—22no—
494C42—22no—
495C42—42no—
496C42—3321no—
497C42—3321no—
498C42—3324no—
499C42—3324no—
500C42—1664no—
501C23—21no—
502C23—21no—
503C23—22no—
504C23—3328no—
505C23—3328no—
506C23—3324no—
507C23—3324no—
508C23—834no—
509C23—1668no—
510C23—1662no—
511C23—1662no—
512C23—1662no—
513C23—1662no—
514C23—1662no—
515C23—1662no—
516C23C2×Q8×D8312yes—
517C23#41×D8314yes—
518C23C24×D8311yescharacteristic
519Q8—6648no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—28no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—48no—
536C2×C4—3328no—
537C2×C4—3324no—
538C2×C4—3328no—
539C2×C4—834no—
540C2×C4—834no—
541C2×C4—834no—
542C2×C4—1664no—
543C2×C4—1664no—
544C2×C4—1664no—
545C2×C4—1664no—
546C2×C4—1664no—
547C2×C4—1664no—
548C2×C4—1662no—
549C2×C4—1662no—
550C2×C4—1662no—
551C2×C4—1662no—
552C2×C4—1662no—
553C2×C4—1662no—
554C2×C4—16616no—
555C2×C4C2×Q8×D8312yes—
556C2×C4C2×Q8×D8312yes—
557C2×C4#1458183712yes—
558C2×C4#1458183812yes—
559C22—22no—
560C22—22no—
561C22—24no—
562C22—24no—
563C22—33216no—
564C22—3322no—
565C22—3322no—
566C22—3322no—
567C22—3322no—
568C22—1664no—
569C22—1664no—
570C22—1664no—
571C22#1912825714yes—
572C22C22×Q8×D8311yescharacteristic
573C22C2×#41×D8312yes—
574C22C2×#1458183711yescharacteristic
575C22C2×#1458183811yescharacteristic
576C22#1912853712yes—
577C4—22no—
578C4—44no—
579C4—42no—
580C4—42no—
581C4—44no—
582C4—3324no—
583C4—3324no—
584C4—1664no—
585C4—1664no—
586C4—1664no—
587C4#1912832614yes—
588C2—24no—
589C2—3324no—
590C2C2×#1912825711yescharacteristic
591C2C2×#1912832611yescharacteristic
592C2#2492086611yescharacteristic
593C2#2492129812yes—
594C2#2492130612yes—
subgroupG/H
center, FrattiniC23C24×D83
commutator, socle, cosocleC23×C83C25
FittingC83×#2851C2
prime radicalC24×D83C23
hypercenter#2851D83
hypocenterC83#35033
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #2851

Complements

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

NH ≅ G/Ntype
1. #24915813C2semidirect
2. #24915773C2semidirect
4. #24913132C2semidirect
5. #24913127C2semidirect
6. #24911178C2semidirect
7. #24911172C2semidirect
8. #24911157C2semidirect
9. #24911152C2semidirect
10. #24910194C2semidirect
13. C83×#2851C2semidirect
15. C2×#14581791C22semidirect
16. C2×#14581784C22semidirect
30. #19127606C22semidirect
31. #19127605C22semidirect
32. #19127605C22semidirect
33. #19127600C22semidirect
34. #19127599C22semidirect
35. #19127598C22semidirect
36. #19127597C22semidirect
37. #19127597C22semidirect
38. #19127596C22semidirect
39. #19127586C22semidirect
40. #19127585C22semidirect
41. #19127583C22semidirect
42. #19127583C22semidirect
43. #19127582C22semidirect
45. #19127570C22semidirect
46. #19127567C22semidirect
60. C2×#125×C83C22semidirect
62. #389×C83C22semidirect
63. #387×C83C22semidirect
65. #382×C83C22semidirect
73. C22×C4×D83Q8semidirect
300. #2851D83semidirect
311. C83#35033semidirect
315. C2×#125C2×D83semidirect
339. #389C2×D83semidirect
345. #387C2×D83semidirect
350. #382C2×D83semidirect

Maximal subgroups

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

subgroupindex
#249158132
#249157732
#249132742
#249131322
#249131272
#249111782
#249111722
#249111572
#249111522
#249101942
#249098452
C2×C4×(C2×D83)⋊C42
C83×#28512
#3503383