Finite Group Databasebeta

← #35416489

Subgroups — #35416489

Order 21248.

796 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128002C211yesmaximal, characteristic
2C4×#14581784C211yesmaximal, characteristic, complemented
3#24909927C211yesmaximal, characteristic
4#24908418C211yesmaximal, characteristic, complemented
5#24908233C211yesmaximal, characteristic, complemented
6#24908196C211yesmaximal, characteristic, complemented
7#24908191C211yesmaximal, characteristic, complemented
8#24908146C211yesmaximal, characteristic
9#24908146C211yesmaximal, characteristic
10#24908091C211yesmaximal, characteristic
11#24908084C211yesmaximal, characteristic
12#24908056C211yesmaximal, characteristic, complemented
13#24908049C211yesmaximal, characteristic, complemented
14#24907474C211yesmaximal, characteristic
15C4×C4⋊C8×C83C211yesmaximal, characteristic, complemented
16C2×#14581784C2211yescharacteristic
17C2×C4×D332C2211yescharacteristic
18C2×#14581749—21no—
19#19128002C418yescomplemented
20C2×#14581726C2211yescharacteristic
21C2×#14581726C2211yescharacteristic
22C2×#14581724C2211yescharacteristic
23C2×#14581724C2211yescharacteristic
24C2×#14581714C2211yescharacteristic
25C2×#14581714C2211yescharacteristic
26C2×#14581714C2211yescharacteristic
27C2×#14581714C2211yescharacteristic
28#19127603C2211yescharacteristic
29#19127603C2211yescharacteristic
30#19127599C2211yescharacteristic
31#19127599C2211yescharacteristic
32#19127599C2211yescharacteristic
33#19127599C2211yescharacteristic
34#19127592—21no—
35#19127592—21no—
36#19127585—21no—
37#19127585—21no—
38#19127567C2211yescharacteristic
39C4×Dic83⋊C4—21no—
40C4×Dic83⋊C4—21no—
41C4×C4⋊Dic83C2211yescharacteristic
42C4×C4⋊Dic83C2211yescharacteristic
43C4×C4⋊Dic83C2211yescharacteristic
44C4×(C2×D83)⋊C4—21no—
45#19127464C2211yescharacteristic
46#19127464C2211yescharacteristic
47#19127464C2211yescharacteristic
48#19127464C2211yescharacteristic
49#19127464C2211yescharacteristic
50#19127464C2211yescharacteristic
51#19127464C2211yescharacteristic
52#19127464C2211yescharacteristic
53C2×C4⋊C8×C83C2211yescharacteristic
54C2×C4⋊C8×C83C2211yescharacteristic
55C2×C4×C8×C83C2211yescharacteristic
56C2×C4×C8×C83C2211yescharacteristic
57C43×C83C2211yescharacteristic
58#336×C83C2211yescharacteristic
59#336×C83C2211yescharacteristic
60C22×D332C2311yescharacteristic
61C22×C4×D83—21no—
62#14581784—22no—
63#14581784C2×C414yes—
64C4×D332—22no—
65C4×D332C2×C414yes—
66#14581749—44no—
67C2×C4⋊Dic83C2311yescharacteristic
68C2×C4⋊Dic83C2311yescharacteristic
69C2×C4⋊Dic83C2311yescharacteristic
70C2×C4⋊Dic83C2311yescharacteristic
71C2×C4⋊Dic83C2311yescharacteristic
72C2×C4⋊Dic83C2311yescharacteristic
73C2×C4⋊Dic83C2311yescharacteristic
74C2×Dic83⋊C4—21no—
75C2×Dic83⋊C4—21no—
76C2×Dic83⋊C4—21no—
77C2×Dic83⋊C4—21no—
78C2×Dic83⋊C4—21no—
79C2×Dic83⋊C4—21no—
80C2×(C2×D83)⋊C4—21no—
81C2×(C2×D83)⋊C4—21no—
82C2×(C2×D83)⋊C4—21no—
83C2×(C2×D83)⋊C4—21no—
84C2×(C2×D83)⋊C4—21no—
85C2×C4×Dic83—21no—
86C2×C4×Dic83—21no—
87C2×C4×Dic83—21no—
88#14581726—41no—
89#14581726C2×C414yes—
90#14581724—41no—
91#14581724C2×C414yes—
92#14581714—41no—
93#14581714—41no—
94#14581714C2×C414yes—
95#14581714C2×C414yes—
96C83⋊#95—21no—
97C83⋊#95—21no—
98C83⋊#95—21no—
99C83⋊#95—21no—
100C83⋊#95—21no—
101C83⋊#95—21no—
102C83⋊#95—21no—
103C83⋊#95—21no—
104C83⋊#95—21no—
105C22×C8×C83C2311yescharacteristic
106C22×C8×C83C2311yescharacteristic
107C22×C8×C83C2311yescharacteristic
108C22×C8×C83C2311yescharacteristic
109C2×C42×C83D411yescharacteristic
110C2×C42×C83D411yescharacteristic
111C2×C42×C83D411yescharacteristic
112C2×C42×C83D411yescharacteristic
113C2×C42×C83C2311yescharacteristic
114C2×C42×C83C2311yescharacteristic
115C2×C42×C83C2311yescharacteristic
116C4⋊C8×C83—22no—
117C4⋊C8×C83C2×C414yes—
118C4×C8×C83—41no—
119C4×C8×C83—41no—
120C23×D83—21no—
121C2×D332—21no—
122C2×D332—21no—
123C2×D332—21no—
124C2×D332—21no—
125C2×D332—21no—
126C2×D332C22×C412yes—
127C22×Dic83—21no—
128C22×Dic83—21no—
129C22×Dic83—21no—
130C22×Dic83—21no—
131C22×Dic83—21no—
132C22×Dic83—21no—
133C22×Dic83—21no—
134C2×C4×D83—24no—
135C2×C4×D83—42no—
136C2×C4×D83—42no—
137C4⋊Dic83—21no—
138C4⋊Dic83—21no—
139C4⋊Dic83—21no—
140C4⋊Dic83—21no—
141C4⋊Dic83—21no—
142C4⋊Dic83—21no—
143C4⋊Dic83—21no—
144C4⋊Dic83—21no—
145C4⋊Dic83—21no—
146C4⋊Dic83—21no—
147C4⋊Dic83—21no—
148C4⋊Dic83C22×C412yes—
149C4⋊Dic83C22×C412yes—
150C4⋊Dic83C22×C412yes—
151Dic83⋊C4—24no—
152Dic83⋊C4—24no—
153Dic83⋊C4—42no—
154Dic83⋊C4—42no—
155Dic83⋊C4—42no—
156Dic83⋊C4—42no—
157(C2×D83)⋊C4—24no—
158(C2×D83)⋊C4—42no—
159(C2×D83)⋊C4—42no—
160(C2×D83)⋊C4—42no—
161(C2×D83)⋊C4—42no—
162C4×Dic83—42no—
163C4×Dic83—42no—
164C4×Dic83—42no—
165C22×C4×C83C2×D411yescharacteristic
166C22×C4×C83C2×D411yescharacteristic
167C22×C4×C83C2×D411yescharacteristic
168C22×C4×C83C2×D411yescharacteristic
169C22×C4×C83C2×D411yescharacteristic
170C22×C4×C83C2×D411yescharacteristic
171C22×C4×C83C2411yescharacteristic
172C2×C8×C83—21no—
173C2×C8×C83—21no—
174C2×C8×C83—21no—
175C2×C8×C83—21no—
176C2×C8×C83—21no—
177C2×C8×C83—21no—
178C2×C8×C83—21no—
179C2×C8×C83—21no—
180C2×C8×C83—21no—
181C2×C8×C83—21no—
182C2×C8×C83C22×C412yes—
183C2×C8×C83C22×C412yes—
184C42×C83—22no—
185C42×C83—22no—
186C42×C83—22no—
187C42×C83—21no—
188C42×C83—21no—
189C42×C83—21no—
190C42×C83—21no—
191C42×C83—41no—
192C42×C83C22×C412yes—
193C42×C83#4112yes—
194C23×C83C22×D411yescharacteristic
195C22×D83—22no—
196C22×D83—41no—
197C22×D83—41no—
198C22×D83—41no—
199C22×D83—41no—
200C22×D83—41no—
201C22×D83—41no—
202C2×C4×C83—21no—
203C2×C4×C83—21no—
204C2×C4×C83—21no—
205C2×C4×C83—21no—
206C2×C4×C83—21no—
207C2×C4×C83—21no—
208C2×C4×C83—21no—
209C2×C4×C83—21no—
210C2×C4×C83—21no—
211C2×C4×C83—21no—
212C2×C4×C83—21no—
213C2×C4×C83—21no—
214C2×C4×C83—21no—
215C2×C4×C83C4×D412yes—
216C2×C4×C83C4×D412yes—
217C2×C4×C83#13512yes—
218C2×C4×C83#13512yes—
219C2×C4×C83AGL(1,ℤ/8)12yes—
220C2×C4×C83C23×C411yescharacteristic
221C2×C4×C83C2×#4111yescharacteristic
222C2×C4×C83H(5,𝔽2)11yescharacteristic
223C2×C4×C832−1+411yescharacteristic
224C2×C4×C832−1+411yescharacteristic
225C2×C4×C832−1+411yescharacteristic
226C2×Dic83—22no—
227C2×Dic83—22no—
228C2×Dic83—22no—
229C2×Dic83—41no—
230C2×Dic83—41no—
231C2×Dic83—41no—
232C2×Dic83—41no—
233C2×Dic83—41no—
234C2×Dic83—41no—
235C2×Dic83—41no—
236C2×Dic83—41no—
237C2×Dic83—41no—
238C2×Dic83—41no—
239C2×Dic83—41no—
240C2×Dic83—41no—
241C2×Dic83—41no—
242C2×Dic83—41no—
243C2×Dic83—41no—
244C2×Dic83—41no—
245C2×Dic83—41no—
246C2×Dic83—41no—
247D332—22no—
248D332—22no—
249D332—41no—
250D332—41no—
251C4×D83—48no—
252C8×C83—22no—
253C8×C83—22no—
254C8×C83—41no—
255C8×C83—41no—
256C22×C83C2×C4×D411yescharacteristic
257C22×C83#53611yescharacteristic
258C22×C83#53711yescharacteristic
259C22×C83C2×#13511yescharacteristic
260C22×C83C2×#13511yescharacteristic
261C22×C83C2×AGL(1,ℤ/8)11yescharacteristic
262C22×C83#57511yescharacteristic
263C2×D83—42no—
264C2×D83—42no—
265C2×D83—42no—
266C2×D83—81no—
267C2×D83—81no—
268C2×D83—81no—
269C2×D83—81no—
270C4×C83—22no—
271C4×C83—22no—
272C4×C83—22no—
273C4×C83—22no—
274C4×C83—22no—
275C4×C83—41no—
276C4×C83#51911yescharacteristic
277C4×C83#52011yescharacteristic
278C4×C83#54111yescharacteristic
279C4×C83#54211yescharacteristic
280Dic83—42no—
281Dic83—42no—
282Dic83—42no—
283Dic83—81no—
284Dic83—81no—
285Dic83—81no—
286Dic83—81no—
287#22114—831nomaximal, Sylow
288C2×C83#232311yescharacteristic
289C2×C83C4×#13511yescharacteristic
290C2×C83C4×AGL(1,ℤ/8)11yescharacteristic
291C2×C83#322811yescharacteristic
292C2×C83#323211yescharacteristic
293C2×C83#324711yescharacteristic
294C2×C83#325011yescharacteristic
295D83—82no—
296C2×#478—831no—
297C4×#125—831no—
298#2293—831no—
299#1989—831no—
300#1949—831no—
301#1942—831no—
302#1941—831no—
303#1934—831no—
304#1934—831no—
305#1926—831no—
306#1925—831no—
307#1921—831no—
308#1920—831no—
309#1804—831no—
310C4×C4⋊C8D8311yescharacteristic, complemented
311C83#2211411yescharacteristic, complemented, Sylow
312C2×#125—831no—
313C2×C4×D4—831no—
314C2×#117—1661no—
315#478—838no—
316C2×#107—831no—
317C2×#107—831no—
318C2×#106—831no—
319C2×#106—831no—
320C2×C4⋊C8C2×D8311yescharacteristic
321C2×C4⋊C8C2×D8311yescharacteristic
322C2×#102—831no—
323C2×#102—831no—
324C2×#102—831no—
325C2×#102—831no—
326C2×C4×C8C2×D8311yescharacteristic
327C2×C4×C8C2×D8311yescharacteristic
328#390—831no—
329#390—831no—
330#389—831no—
331#389—831no—
332#389—831no—
333#389—831no—
334#388—1661no—
335#388—1661no—
336#387—1661no—
337#387—1661no—
338#382—831no—
339C4×C4⋊C4—831no—
340C4×C4⋊C4—831no—
341C4×C4⋊C4—831no—
342C4×C4⋊C4—1661no—
343C4×C4⋊C4—1661no—
344C4×C22⋊C4—1661no—
345C43C2×D8311yescharacteristic
346#340—831no—
347#340—831no—
348#340—831no—
349#340—831no—
350#340—831no—
351#340—831no—
352#340—831no—
353#340—831no—
354#336C2×D8311yescharacteristic
355#336C2×D8311yescharacteristic
356C22×D4—831no—
357C23×C4—1661no—
358C22×C8C22×D8311yescharacteristic
359C22×C8C22×D8311yescharacteristic
360C22×C8C22×D8311yescharacteristic
361C22×C8C22×D8311yescharacteristic
362#125—834no—
363#125—1662no—
364C4×D4—834no—
365C4×D4—1662no—
366#117—3324no—
367C2×C4⋊C4—831no—
368C2×C4⋊C4—831no—
369C2×C4⋊C4—831no—
370C2×C4⋊C4—831no—
371C2×C4⋊C4—831no—
372C2×C4⋊C4—831no—
373C2×C4⋊C4—831no—
374C2×C4⋊C4—1661no—
375C2×C4⋊C4—1661no—
376C2×C4⋊C4—1661no—
377C2×C4⋊C4—1661no—
378C2×C4⋊C4—1661no—
379C2×C4⋊C4—1661no—
380C2×C22⋊C4—1661no—
381C2×C22⋊C4—1661no—
382C2×C22⋊C4—1661no—
383C2×C22⋊C4—1661no—
384C2×C22⋊C4—1661no—
385C2×C42—1661no—
386C2×C42—1661no—
387C2×C42—1661no—
388C2×C42D33211yescharacteristic
389C2×C42D33211yescharacteristic
390C2×C42D33211yescharacteristic
391C2×C42D33211yescharacteristic
392C2×C42C22×D8311yescharacteristic
393C2×C42C22×D8311yescharacteristic
394C2×C42C22×D8311yescharacteristic
395#107—3321no—
396#107—834no—
397#106—3321no—
398#106—834no—
399C4⋊C8—22no—
400C4⋊C8C4×D8314yes—
401#102—3321no—
402#102—3321no—
403#102—834no—
404#102—834no—
405C4×C8—41no—
406C4×C8—41no—
407#95—1661no—
408#95—1661no—
409#95—1661no—
410#95—1661no—
411#95—1661no—
412#95—1661no—
413#95—1661no—
414#95—1661no—
415#95—1661no—
416C24—1661no—
417C2×D4—832no—
418C2×D4—1661no—
419C2×D4—1661no—
420C2×D4—1661no—
421C2×D4—1661no—
422C2×D4—1661no—
423C22×C4—3322no—
424C22×C4—3322no—
425C22×C4—1661no—
426C22×C4—1661no—
427C22×C4—1661no—
428C22×C4—1661no—
429C22×C4—1661no—
430C22×C4—1661no—
431C22×C4—1661no—
432C22×C4—1664no—
433C22×C4C2×D33211yescharacteristic
434C22×C4C2×D33211yescharacteristic
435C22×C4C2×D33211yescharacteristic
436C22×C4C2×D33211yescharacteristic
437C22×C4C2×D33211yescharacteristic
438C22×C4C2×D33211yescharacteristic
439C22×C4C23×D8311yescharacteristic
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8C2×C4×D8312yes—
451C2×C8C2×C4×D8312yes—
452C4⋊C4—3322no—
453C4⋊C4—3322no—
454C4⋊C4—3322no—
455C4⋊C4—3322no—
456C4⋊C4—832no—
457C4⋊C4—832no—
458C4⋊C4—832no—
459C4⋊C4—1661no—
460C4⋊C4—1661no—
461C4⋊C4—1661no—
462C4⋊C4—1661no—
463C4⋊C4—1661no—
464C4⋊C4—1661no—
465C4⋊C4—1661no—
466C4⋊C4—1661no—
467C4⋊C4—1661no—
468C4⋊C4—1661no—
469C4⋊C4—1661no—
470C4⋊C4—1664no—
471C4⋊C4—1664no—
472C22⋊C4—3322no—
473C22⋊C4—3322no—
474C22⋊C4—3322no—
475C22⋊C4—3322no—
476C22⋊C4—1664no—
477C42—22no—
478C42—22no—
479C42—22no—
480C42—21no—
481C42—21no—
482C42—21no—
483C42—21no—
484C42—41no—
485C42—3322no—
486C42—3322no—
487C42—3322no—
488C42C2×C4×D8312yes—
489C42#1305059012yes—
490C23—3321no—
491C23—3321no—
492C23—3321no—
493C23—3321no—
494C23—3321no—
495C23—3321no—
496C23—1662no—
497C23C22×D33211yescharacteristic
498D4—3321no—
499D4—3321no—
500D4—1662no—
501D4—1662no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—3328no—
516C2×C4—3321no—
517C2×C4—3321no—
518C2×C4—3321no—
519C2×C4—3321no—
520C2×C4—3321no—
521C2×C4—3321no—
522C2×C4—3321no—
523C2×C4—3321no—
524C2×C4—3321no—
525C2×C4—3321no—
526C2×C4—3321no—
527C2×C4—3321no—
528C2×C4—3321no—
529C2×C4—3321no—
530C2×C4—3321no—
531C2×C4—3321no—
532C2×C4—3321no—
533C2×C4—3321no—
534C2×C4—1662no—
535C2×C4—1662no—
536C2×C4—1662no—
537C2×C4C4×D33212yes—
538C2×C4C4×D33212yes—
539C2×C4#1458181412yes—
540C2×C4#1458181412yes—
541C2×C4#1458182012yes—
542C2×C4C22×C4×D8311yescharacteristic
543C2×C4C2×#1305059011yescharacteristic
544C2×C4#1458183711yescharacteristic
545C2×C4#1458183911yescharacteristic
546C2×C4#1458183911yescharacteristic
547C2×C4#1458184011yescharacteristic
548C8—22no—
549C8—22no—
550C8—41no—
551C8—41no—
552C22—3322no—
553C22—3322no—
554C22—3322no—
555C22—6641no—
556C22—6641no—
557C22—6641no—
558C22—6641no—
559C22C2×C4×D33211yescharacteristic
560C22#1912827711yescharacteristic
561C22#1912828211yescharacteristic
562C22C2×#1458181411yescharacteristic
563C22C2×#1458181411yescharacteristic
564C22C2×#1458182011yescharacteristic
565C22#1912850111yescharacteristic
566C4—22no—
567C4—22no—
568C4—22no—
569C4—22no—
570C4—22no—
571C4—41no—
572C4—3322no—
573C4—3322no—
574C4—3322no—
575C4—6641no—
576C4—6641no—
577C4—6641no—
578C4—6641no—
579C4#1912817711yescharacteristic
580C4#1912818011yescharacteristic
581C4#1912830911yescharacteristic
582C4#1912831611yescharacteristic
583C2—6642no—
584C2#2491014611yescharacteristic
585C2C4×#1458181411yescharacteristic
586C2C4×#1458182011yescharacteristic
587C2#2491860111yescharacteristic
588C2#2491863311yescharacteristic
589C2#2491879011yescharacteristic
590C2#2491881511yescharacteristic
subgroupG/H
centerC23C22×D332
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC4×C4⋊C8×C83C2
socleC23×C83C22×D4
prime radicalC22×D332C23
cosocleC22×C4×C83C24
hypercenterC4×C4⋊C8D83
hypocenterC83#22114
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C43 ◁ C4×C4⋊C8

Complements

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

NH ≅ G/Ntype
2. C4×#14581784C2semidirect
4. #24908418C2semidirect
5. #24908233C2semidirect
6. #24908196C2semidirect
7. #24908191C2semidirect
12. #24908056C2semidirect
13. #24908049C2semidirect
15. C4×C4⋊C8×C83C2semidirect
19. #19128002C4semidirect
310. C4×C4⋊C8D83semidirect
311. C83#22114semidirect

Maximal subgroups

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

subgroupindex
C2×#191280022
C4×#145817842
#249099272
#249084182
#249082332
#249081962
#249081912
#249081462
#249080912
#249080842
#249080562
#249080492
#249074742
C4×C4⋊C8×C832
#2211483