Finite Group Databasebeta

← #35408747

Subgroups — #35408747

Order 21248.

874 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24909941C211yesmaximal, characteristic, complemented
2#24909928C211yesmaximal, characteristic
3#24908346C211yesmaximal, characteristic
4#24908319C211yesmaximal, characteristic
5#24908202C211yesmaximal, characteristic, complemented
6#24908197C211yesmaximal, characteristic, complemented
7#24908162C211yesmaximal, characteristic
8#24908155C211yesmaximal, characteristic, complemented
9#24908148C211yesmaximal, characteristic
10#24908141C211yesmaximal, characteristic, complemented
11#24907360C211yesmaximal, characteristic, complemented
12#24907293C211yesmaximal, characteristic, complemented
13#24907288C211yesmaximal, characteristic
14#97×Dic83C211yesmaximal, characteristic, complemented
15C2×#440×C83C211yesmaximal, characteristic
16C2×#14581751—21no—
17C2×#14581726C2211yescharacteristic
18C2×#14581724C2211yescharacteristic
19C2×#14581716C2211yescharacteristic
20C2×#14581716C2211yescharacteristic
21C2×#14581713C2211yescharacteristic
22C2×#14581713C2211yescharacteristic
23C2×#14581703C2211yescharacteristic
24C2×#14581702C2211yescharacteristic
25C2×#14581702C2211yescharacteristic
26C2×C8×Dic83C2211yescharacteristic
27C2×C8×Dic83C2211yescharacteristic
28#19127608C2211yescharacteristic
29#19127608C2211yescharacteristic
30#19127604C2211yescharacteristic
31#19127604C2211yescharacteristic
32#19127600C2211yescharacteristic
33#19127600C2211yescharacteristic
34#19127600C2211yescharacteristic
35#19127600C2211yescharacteristic
36#19127594—21no—
37#19127594—21no—
38#19127587—21no—
39#19127587—21no—
40C4×C4⋊Dic83—21no—
41C4×C4⋊Dic83C2211yescharacteristic
42C4×C4⋊Dic83C2211yescharacteristic
43C4×C22⋊Dic83—21no—
44#19127547—21no—
45C42×Dic83C2211yescharacteristic
46#19127464C2211yescharacteristic
47#19127464C2211yescharacteristic
48#19127463C2211yescharacteristic
49#19127463C2211yescharacteristic
50#19127463C2211yescharacteristic
51#19127463C2211yescharacteristic
52C22×SD16×C83C2211yescharacteristic
53C2×C4×Q8×C83C2211yescharacteristic
54C2×C4×D4×C83C2211yescharacteristic
55#440×C83C418yescomplemented
56C2×#107×C83C2211yescharacteristic
57C2×#103×C83C2211yescharacteristic
58C2×#102×C83C2211yescharacteristic
59C2×#97×C83C2211yescharacteristic
60C22×C83⋊C8C2311yescharacteristic
61C22×C83⋊C8C2311yescharacteristic
62#14581751—44no—
63C2×C4⋊Dic83—21no—
64C2×C4⋊Dic83—21no—
65C2×C4⋊Dic83—21no—
66C2×C4⋊Dic83—21no—
67C2×C4⋊Dic83C2311yescharacteristic
68C2×C4⋊Dic83C2311yescharacteristic
69C2×C4⋊Dic83C2311yescharacteristic
70C2×C4⋊Dic83C2311yescharacteristic
71C2×C22⋊Dic83—21no—
72C2×C22⋊Dic83—21no—
73C2×C22⋊Dic83—21no—
74C2×C22⋊Dic83—21no—
75C2×C4×Dic83—21no—
76C2×C4×Dic83—21no—
77C2×C4×Dic83D411yescharacteristic
78C2×C4×Dic83D411yescharacteristic
79C2×C4×Dic83D411yescharacteristic
80C2×C4×Dic83D411yescharacteristic
81C2×C4×Dic83C2311yescharacteristic
82C2×C4×Dic83C2311yescharacteristic
83#14581726—22no—
84#14581724—22no—
85#14581716—22no—
86#14581716—22no—
87#14581713—22no—
88#14581713—22no—
89#14581703—22no—
90#14581702—22no—
91#14581702—22no—
92C8×Dic83—22no—
93C8×Dic83—22no—
94C83⋊#95—21no—
95C83⋊#95—21no—
96C83⋊#95—21no—
97C83⋊#95—21no—
98C83⋊#95—21no—
99C83⋊#95—21no—
100C83⋊#95—21no—
101C83⋊#95—21no—
102C83⋊#95—21no—
103C83⋊#95—21no—
104C22×Q8×C83C2311yescharacteristic
105C22×D4×C83C2311yescharacteristic
106C23×C4×C83—21no—
107C2×SD16×C83—22no—
108C2×SD16×C83—22no—
109C2×SD16×C83C2×C414yes—
110C22×C8×C83C2311yescharacteristic
111C22×C8×C83C2311yescharacteristic
112C4×Q8×C83—22no—
113C4×Q8×C83C2×C414yes—
114C4×D4×C83—22no—
115C4×D4×C83C2×C414yes—
116C2×C4⋊C4×C83—21no—
117C2×C4⋊C4×C83C2311yescharacteristic
118C2×C4⋊C4×C83C2311yescharacteristic
119C2×C22⋊C4×C83—21no—
120C2×C42×C83—21no—
121C2×C42×C83C2311yescharacteristic
122#107×C83C2×C414yes—
123#103×C83C2×C414yes—
124#102×C83C2×C414yes—
125#97×C83C2×C414yes—
126C22×Dic83—21no—
127C22×Dic83—21no—
128C22×Dic83—21no—
129C22×Dic83—21no—
130C22×Dic83C2×D411yescharacteristic
131C22×Dic83C2×D411yescharacteristic
132C22×Dic83C2×D411yescharacteristic
133C22×Dic83C2×D411yescharacteristic
134C2×C83⋊C8—21no—
135C2×C83⋊C8—21no—
136C2×C83⋊C8—21no—
137C2×C83⋊C8—21no—
138C2×C83⋊C8#4112yes—
139C2×C83⋊C8#4112yes—
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⋊Dic83—42no—
149C4⋊Dic83—42no—
150C4⋊Dic83—42no—
151C4⋊Dic83—42no—
152C22⋊Dic83—42no—
153C22⋊Dic83—42no—
154C22⋊Dic83—42no—
155C22⋊Dic83—42no—
156C4×Dic83—21no—
157C4×Dic83—21no—
158C4×Dic83—21no—
159C4×Dic83—21no—
160C4×Dic83—22no—
161C4×Dic83—22no—
162C4×Dic83—41no—
163C4×Dic83—41no—
164C4×Dic83—42no—
165C4×Dic83—42no—
166C24×C83—21no—
167C2×Q8×C83—21no—
168C2×Q8×C83—21no—
169C2×Q8×C83—21no—
170C2×Q8×C83—21no—
171C2×Q8×C83—21no—
172C2×Q8×C83C22×C412yes—
173C2×D4×C83—21no—
174C2×D4×C83—21no—
175C2×D4×C83—21no—
176C2×D4×C83—21no—
177C2×D4×C83—21no—
178C2×D4×C83C22×C412yes—
179C22×C4×C83—21no—
180C22×C4×C83—21no—
181C22×C4×C83—21no—
182C22×C4×C83—24no—
183C22×C4×C83—42no—
184C22×C4×C83—42no—
185C22×C4×C83C2×D411yescharacteristic
186C22×C4×C83C2×D411yescharacteristic
187C22×C4×C83C2411yescharacteristic
188SD16×C83—28no—
189C2×C8×C83—21no—
190C2×C8×C83—21no—
191C2×C8×C83C22×C412yes—
192C2×C8×C83C22×C412yes—
193C2×C8×C83#4112yes—
194C2×C8×C83#4112yes—
195C4⋊C4×C83—24no—
196C4⋊C4×C83—21no—
197C4⋊C4×C83—21no—
198C4⋊C4×C83C22×C412yes—
199C4⋊C4×C83C22×C412yes—
200C22⋊C4×C83—24no—
201C42×C83—24no—
202C42×C83C22×C412yes—
203C42×C83#4112yes—
204C23×C83—22no—
205C23×C83—41no—
206C23×C83—41no—
207C23×C83—41no—
208C23×C83—41no—
209C23×C83—41no—
210C23×C83—41no—
211C23×C83C22×D411yescharacteristic
212Q8×C83—22no—
213Q8×C83—22no—
214Q8×C83—41no—
215Q8×C83—41no—
216D4×C83—22no—
217D4×C83—22no—
218D4×C83—41no—
219D4×C83—41no—
220C2×C4×C83—21no—
221C2×C4×C83—21no—
222C2×C4×C83—21no—
223C2×C4×C83—21no—
224C2×C4×C83—22no—
225C2×C4×C83—22no—
226C2×C4×C83—22no—
227C2×C4×C83—48no—
228C2×C4×C83—41no—
229C2×C4×C83—41no—
230C2×C4×C83—41no—
231C2×C4×C83—41no—
232C2×C4×C83—41no—
233C2×C4×C83—41no—
234C2×C4×C83C4×D412yes—
235C2×C4×C83C4×D412yes—
236C2×C4×C83C23×C411yescharacteristic
237C2×C4×C83C2×#4111yescharacteristic
238C2×C4×C83C2×#4111yescharacteristic
239C2×C4×C83C2×#4111yescharacteristic
240C2×C4×C83C2×#4111yescharacteristic
241C2×C4×C83C2×#4111yescharacteristic
242C2×Dic83—21no—
243C2×Dic83—21no—
244C2×Dic83—21no—
245C2×Dic83—21no—
246C2×Dic83—21no—
247C2×Dic83—21no—
248C2×Dic83—21no—
249C2×Dic83—21no—
250C2×Dic83—21no—
251C2×Dic83—21no—
252C2×Dic83—41no—
253C2×Dic83—41no—
254C2×Dic83—41no—
255C2×Dic83—41no—
256C2×Dic83—41no—
257C2×Dic83—41no—
258C2×Dic83—41no—
259C2×Dic83—41no—
260C2×Dic83—41no—
261C2×Dic83—41no—
262C2×Dic83—41no—
263C2×Dic83—41no—
264C2×Dic83AGL(1,ℤ/8)12yes—
265C2×Dic83#13712yes—
266C8×C83—22no—
267C8×C83#11714yescomplemented
268C83⋊C8—22no—
269C83⋊C8—22no—
270C22×C83—42no—
271C22×C83—42no—
272C22×C83—42no—
273C22×C83—81no—
274C22×C83—81no—
275C22×C83—81no—
276C22×C83—81no—
277C22×C83C2×C4×D411yescharacteristic
278C22×C83#53211yescharacteristic
279C22×C83#53211yescharacteristic
280C22×C83C2×AGL(1,ℤ/8)11yescharacteristic
281C22×C83C2×#13711yescharacteristic
282C22×C83#57511yescharacteristic
283C22×C83#57511yescharacteristic
284C4×C83—22no—
285C4×C83—22no—
286C4×C83—42no—
287C4×C83—42no—
288C4×C83—42no—
289C4×C83C2×#11711yescharacteristic
290C4×C83C4×#4111yescharacteristic
291C4×C83#52911yescharacteristic
292C4×C83#52911yescharacteristic
293Dic83—22no—
294Dic83—22no—
295Dic83—41no—
296Dic83—41no—
297Dic83—81no—
298Dic83—81no—
299Dic83—81no—
300Dic83—81no—
301#21288—831nomaximal, Sylow
302C2×C83—82no—
303C2×C83#229311yescharacteristic
304C2×C83C4×AGL(1,ℤ/8)11yescharacteristic
305C2×C83C4×#13711yescharacteristic
306C2×C83#318011yescharacteristic
307C2×C83#318111yescharacteristic
308C2×C83#318211yescharacteristic
309C2×C83#318311yescharacteristic
310C2×#440D8311yescharacteristic
311#2295—831no—
312#2293—831no—
313#1973—831no—
314#1968—831no—
315#1943—831no—
316#1942—831no—
317#1936—831no—
318#1935—831no—
319#1934—831no—
320#1933—831no—
321#1776—831no—
322#1763—831no—
323#1762—831no—
324C4×#97—831no—
325C83#2128811yescharacteristic, complemented, Sylow
326C22×SD16C2×D8311yescharacteristic
327C2×C4×Q8C2×D8311yescharacteristic
328C2×C4×D4C2×D8311yescharacteristic
329C2×#117—1661no—
330#440Dic8318yescomplemented
331C2×#107—831no—
332C2×#107C2×D8311yescharacteristic
333C2×#106—831no—
334C2×#103—831no—
335C2×#103—831no—
336C2×#103C2×D8311yescharacteristic
337C2×#102—831no—
338C2×#102—831no—
339C2×#102C2×D8311yescharacteristic
340C2×#97—831no—
341C2×#97—831no—
342C2×#97—831no—
343C2×#97C2×D8311yescharacteristic
344C2×C4×C8—831no—
345C2×C4×C8—831no—
346#391—831no—
347#391—831no—
348#390—831no—
349#390—831no—
350#389—831no—
351#389—831no—
352#389—831no—
353#389—831no—
354#388—1661no—
355#388—1661no—
356#387—1661no—
357#387—1661no—
358C4×C4⋊C4—831no—
359C4×C4⋊C4—831no—
360C4×C4⋊C4—1661no—
361C4×C22⋊C4—1661no—
362#376—1661no—
363C43—831no—
364#340—831no—
365#340—831no—
366#340—831no—
367#340—831no—
368#340—831no—
369#340—831no—
370C22×Q8C22×D8311yescharacteristic
371C22×D4C22×D8311yescharacteristic
372C23×C4—21no—
373C2×SD16—22no—
374C2×SD16—22no—
375C2×SD16C2×Dic8314yes—
376C22×C8—831no—
377C22×C8—831no—
378C22×C8C22×D8311yescharacteristic
379C22×C8C22×D8311yescharacteristic
380C4×Q8—22no—
381C4×Q8C2×Dic8314yes—
382C4×D4—22no—
383C4×D4C2×Dic8314yes—
384#117—3324no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—831no—
387C2×C4⋊C4—831no—
388C2×C4⋊C4—831no—
389C2×C4⋊C4—831no—
390C2×C4⋊C4—1661no—
391C2×C4⋊C4—1661no—
392C2×C4⋊C4—1661no—
393C2×C4⋊C4—1661no—
394C2×C4⋊C4C22×D8311yescharacteristic
395C2×C4⋊C4C22×D8311yescharacteristic
396C2×C22⋊C4—21no—
397C2×C22⋊C4—1661no—
398C2×C22⋊C4—1661no—
399C2×C22⋊C4—1661no—
400C2×C22⋊C4—1661no—
401C2×C42—21no—
402C2×C42—831no—
403C2×C42—831no—
404C2×C42—831no—
405C2×C42—831no—
406C2×C42—831no—
407C2×C42—831no—
408C2×C42—1661no—
409C2×C42—1661no—
410C2×C42C22×D8311yescharacteristic
411#107—1662no—
412#107C2×Dic8314yes—
413#106—1662no—
414#103—1662no—
415#103—1662no—
416#103C2×Dic8314yes—
417#102—1662no—
418#102—1662no—
419#102C2×Dic8314yes—
420#97—1662no—
421#97—1662no—
422#97—1662no—
423#97C2×Dic8314yes—
424C4×C8—1662no—
425C4×C8—1662no—
426#95—1661no—
427#95—1661no—
428#95—1661no—
429#95—1661no—
430#95—1661no—
431#95—1661no—
432#95—1661no—
433#95—1661no—
434#95—1661no—
435#95—1661no—
436C24—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8C22×Dic8312yes—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4C22×Dic8312yes—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—24no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—831no—
456C22×C4—831no—
457C22×C4—831no—
458C22×C4—831no—
459C22×C4—1661no—
460C22×C4—1661no—
461C22×C4—1661no—
462C22×C4—1661no—
463C22×C4D4×D8311yescharacteristic
464C22×C4D4×D8311yescharacteristic
465C22×C4C23×D8311yescharacteristic
466SD16—28no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—832no—
470C2×C8—832no—
471C2×C8—1661no—
472C2×C8—1661no—
473C2×C8—1661no—
474C2×C8—1661no—
475C2×C8C22×Dic8312yes—
476C2×C8C22×Dic8312yes—
477C2×C8#1305059012yes—
478C2×C8#1305059012yes—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—24no—
482C4⋊C4—3322no—
483C4⋊C4—3322no—
484C4⋊C4—3322no—
485C4⋊C4—3322no—
486C4⋊C4—1661no—
487C4⋊C4—1661no—
488C4⋊C4—1661no—
489C4⋊C4—1661no—
490C4⋊C4—1661no—
491C4⋊C4—1661no—
492C4⋊C4—1661no—
493C4⋊C4—1661no—
494C4⋊C4C22×Dic8312yes—
495C4⋊C4C22×Dic8312yes—
496C22⋊C4—24no—
497C22⋊C4—3322no—
498C22⋊C4—3322no—
499C22⋊C4—3322no—
500C22⋊C4—3322no—
501C42—24no—
502C42—3322no—
503C42—3322no—
504C42—3321no—
505C42—3321no—
506C42—1662no—
507C42—1662no—
508C42—1661no—
509C42—1661no—
510C42—1661no—
511C42—1661no—
512C42C22×Dic8312yes—
513C42#1305058912yes—
514C23—22no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23C2×D4×D8311yescharacteristic
522Q8—22no—
523Q8—22no—
524Q8—41no—
525Q8—41no—
526D4—22no—
527D4—22no—
528D4—41no—
529D4—41no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—48no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—3321no—
545C2×C4—3321no—
546C2×C4—3321no—
547C2×C4—3321no—
548C2×C4—3321no—
549C2×C4—3321no—
550C2×C4—3321no—
551C2×C4—3321no—
552C2×C4—3321no—
553C2×C4—3321no—
554C2×C4—3321no—
555C2×C4—3321no—
556C2×C4—832no—
557C2×C4—832no—
558C2×C4—1661no—
559C2×C4—1661no—
560C2×C4—1661no—
561C2×C4—1661no—
562C2×C4—1661no—
563C2×C4—1661no—
564C2×C4—1661no—
565C2×C4—1661no—
566C2×C4—1661no—
567C2×C4—1661no—
568C2×C4D4×Dic8312yes—
569C2×C4D4×Dic8312yes—
570C2×C4C23×Dic8311yescharacteristic
571C2×C4#41×D8311yescharacteristic
572C2×C4#41×D8311yescharacteristic
573C2×C4C2×#1305058911yescharacteristic
574C2×C4C2×#1305059011yescharacteristic
575C2×C4C2×#1305059011yescharacteristic
576C8—22no—
577C8—1662no—
578C8—1662no—
579C8#1458175114yescomplemented
580C22—42no—
581C22—42no—
582C22—42no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22—81no—
587C22C2×D4×Dic8311yescharacteristic
588C22#1912824811yescharacteristic
589C22#1912824811yescharacteristic
590C22AGL(1,ℤ/8)×D8311yescharacteristic
591C22#137×D8311yescharacteristic
592C22#1912849711yescharacteristic
593C22#1912850011yescharacteristic
594C4—22no—
595C4—22no—
596C4—42no—
597C4—42no—
598C4—42no—
599C4—3321no—
600C4—3321no—
601C4—6641no—
602C4—6641no—
603C4—6641no—
604C4—6641no—
605C4—1662no—
606C4—1662no—
607C4C2×#1458175111yescharacteristic
608C4#41×Dic8311yescharacteristic
609C4#1912823311yescharacteristic
610C4#1912823311yescharacteristic
611C2—82no—
612C2#2490992811yescharacteristic
613C2AGL(1,ℤ/8)×Dic8311yescharacteristic
614C2#137×Dic8311yescharacteristic
615C2#2491815711yescharacteristic
616C2#2491816811yescharacteristic
617C2#2491817811yescharacteristic
618C2#2491818811yescharacteristic
subgroupG/H
centerC23C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC2×#440×C83C2
socleC23×C83C22×D4
prime radicalC22×D4×C83C23
cosocleC22×C4×C83C24
hypercenterC2×#440D83
hypocenterC83#21288
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#440

Complements

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

NH ≅ G/Ntype
1. #24909941C2semidirect
5. #24908202C2semidirect
6. #24908197C2semidirect
8. #24908155C2semidirect
10. #24908141C2semidirect
11. #24907360C2semidirect
12. #24907293C2semidirect
14. #97×Dic83C2semidirect
55. #440×C83C4semidirect
267. C8×C83#117semidirect
325. C83#21288semidirect
330. #440Dic83semidirect
579. C8#14581751semidirect

Maximal subgroups

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

subgroupindex
#249099412
#249099282
#249083462
#249083192
#249082022
#249081972
#249081622
#249081552
#249081482
#249081412
#249073602
#249072932
#249072882
#97×Dic832
C2×#440×C832
#2128883