Finite Group Databasebeta

← #35412403

Subgroups — #35412403

Order 21248.

818 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24910205C211yesmaximal, characteristic, complemented
2#24910176C211yesmaximal, characteristic
3#24908433C211yesmaximal, characteristic, complemented
4#24908280C211yesmaximal, characteristic
5#24908162C211yesmaximal, characteristic
6#24908155C211yesmaximal, characteristic, complemented
7#24908148C211yesmaximal, characteristic
8#24908141C211yesmaximal, characteristic, complemented
9#24907591C211yesmaximal, characteristic, complemented
10#24907295C211yesmaximal, characteristic, complemented
11#24907290C211yesmaximal, characteristic
12#103×Dic83C211yesmaximal, characteristic, complemented
13#102×Dic83C211yesmaximal, characteristic
14#24907149C211yesmaximal, characteristic, complemented
15C2×#488×C83C211yesmaximal, characteristic
16C2×Q8×Dic83C2211yescharacteristic
17C2×D4×Dic83C2211yescharacteristic
18C2×#14581716C2211yescharacteristic
19C2×#14581716C2211yescharacteristic
20C2×#14581713C2211yescharacteristic
21C2×#14581713C2211yescharacteristic
22C2×#14581703C2211yescharacteristic
23C2×#14581702C2211yescharacteristic
24C2×#14581702C2211yescharacteristic
25C2×C8×Dic83C2211yescharacteristic
26C2×C8×Dic83C2211yescharacteristic
27#19127608C2211yescharacteristic
28#19127604C2211yescharacteristic
29#19127600—21no—
30#19127600—21no—
31#19127600C2211yescharacteristic
32#19127600C2211yescharacteristic
33#19127594—21no—
34#19127594—21no—
35#19127587—21no—
36#19127587—21no—
37#19127576—21no—
38#19127576—21no—
39#19127571C2211yescharacteristic
40#19127571C2211yescharacteristic
41C4⋊C4×Dic83C2211yescharacteristic
42C4⋊C4×Dic83C2211yescharacteristic
43#19127547C2211yescharacteristic
44#19127464C2211yescharacteristic
45#19127464C2211yescharacteristic
46#19127464C2211yescharacteristic
47#19127464C2211yescharacteristic
48#19127463C2211yescharacteristic
49#19127463C2211yescharacteristic
50#19127463C2211yescharacteristic
51#19127463C2211yescharacteristic
52C2×#125×C83C2211yescharacteristic
53C2×#124×C83C2211yescharacteristic
54#488×C83C418yescomplemented
55C2×#103×C83C2211yescharacteristic
56C2×#103×C83C2211yescharacteristic
57C2×#102×C83C2211yescharacteristic
58C2×#102×C83C2211yescharacteristic
59C2×C4×C8×C83C2211yescharacteristic
60C23×Dic83—21no—
61C22×C83⋊C8C2311yescharacteristic
62C22×C83⋊C8C2311yescharacteristic
63Q8×Dic83—22no—
64Q8×Dic83—22no—
65D4×Dic83—22no—
66D4×Dic83—22no—
67C2×C4⋊Dic83—21no—
68C2×C4⋊Dic83—21no—
69C2×C4⋊Dic83—21no—
70C2×C4⋊Dic83—21no—
71C2×C4⋊Dic83—21no—
72C2×C4⋊Dic83—21no—
73C2×C4⋊Dic83—21no—
74C2×C4⋊Dic83C2311yescharacteristic
75C2×C4⋊Dic83C2311yescharacteristic
76C2×C4⋊Dic83C2311yescharacteristic
77C2×C4⋊Dic83C2311yescharacteristic
78C2×C22⋊Dic83—21no—
79C2×C22⋊Dic83—21no—
80C2×C22⋊Dic83—21no—
81C2×C4×Dic83—21no—
82C2×C4×Dic83—21no—
83C2×C4×Dic83—21no—
84C2×C4×Dic83C2311yescharacteristic
85C2×C4×Dic83C2311yescharacteristic
86#14581716—22no—
87#14581716—22no—
88#14581713—22no—
89#14581713—22no—
90#14581703—22no—
91#14581702—22no—
92#14581702—22no—
93C8×Dic83—22no—
94C8×Dic83—22no—
95C83⋊#95—21no—
96C83⋊#95—21no—
97C83⋊#95—21no—
98C83⋊#95—21no—
99C83⋊#95—21no—
100C83⋊#95—21no—
101C83⋊#95D411yescharacteristic
102C83⋊#95D411yescharacteristic
103C83⋊#95D411yescharacteristic
104C83⋊#95D411yescharacteristic
105C22×Q8×C83C2311yescharacteristic
106C22×D4×C83C2311yescharacteristic
107C22×C8×C83C2311yescharacteristic
108C22×C8×C83C2311yescharacteristic
109#125×C83—22no—
110#125×C83C2×C414yescomplemented
111#124×C83—22no—
112#124×C83C2×C414yes—
113C2×C4⋊C4×C83—21no—
114C2×C4⋊C4×C83—21no—
115C2×C4⋊C4×C83C2311yescharacteristic
116C2×C4⋊C4×C83C2311yescharacteristic
117C2×C22⋊C4×C83—21no—
118C2×C22⋊C4×C83—21no—
119C2×C42×C83C2311yescharacteristic
120#103×C83C2×C414yescomplemented
121#103×C83C2×C414yescomplemented
122#102×C83C2×C414yes—
123#102×C83C2×C414yes—
124C4×C8×C83C2×C414yescomplemented
125C22×Dic83—21no—
126C22×Dic83—21no—
127C22×Dic83—21no—
128C22×Dic83—21no—
129C22×Dic83—42no—
130C22×Dic83—81no—
131C22×Dic83—81no—
132C22×Dic83C2×D411yescharacteristic
133C22×Dic83C2×D411yescharacteristic
134C22×Dic83C2×D411yescharacteristic
135C22×Dic83C2×D411yescharacteristic
136C2×C83⋊C8—21no—
137C2×C83⋊C8—21no—
138C2×C83⋊C8—21no—
139C2×C83⋊C8—21no—
140C2×C83⋊C8#4112yes—
141C2×C83⋊C8#4112yes—
142C4⋊Dic83—21no—
143C4⋊Dic83—21no—
144C4⋊Dic83—21no—
145C4⋊Dic83—21no—
146C4⋊Dic83—21no—
147C4⋊Dic83—21no—
148C4⋊Dic83—21no—
149C4⋊Dic83—21no—
150C4⋊Dic83—42no—
151C4⋊Dic83—81no—
152C4⋊Dic83—81no—
153C4⋊Dic83—81no—
154C4⋊Dic83—81no—
155C4⋊Dic83—81no—
156C4⋊Dic83—81no—
157C22⋊Dic83—42no—
158C22⋊Dic83—81no—
159C22⋊Dic83—81no—
160C4×Dic83—21no—
161C4×Dic83—21no—
162C4×Dic83—42no—
163C4×Dic83—81no—
164C4×Dic83—81no—
165C4×Dic83#4112yes—
166C4×Dic83#4112yes—
167C24×C83—21no—
168C2×Q8×C83—21no—
169C2×Q8×C83—21no—
170C2×Q8×C83—21no—
171C2×Q8×C83—21no—
172C2×Q8×C83—21no—
173C2×Q8×C83C22×C412yes—
174C2×D4×C83—21no—
175C2×D4×C83—21no—
176C2×D4×C83—21no—
177C2×D4×C83—21no—
178C2×D4×C83—21no—
179C2×D4×C83C22×C412yes—
180C22×C4×C83—21no—
181C22×C4×C83—21no—
182C22×C4×C83—21no—
183C22×C4×C83C2×D411yescharacteristic
184C22×C4×C83C2×D411yescharacteristic
185C22×C4×C83C2411yescharacteristic
186C2×C8×C83—21no—
187C2×C8×C83—21no—
188C2×C8×C83—21no—
189C2×C8×C83—21no—
190C2×C8×C83C22×C412yes—
191C2×C8×C83C22×C412yes—
192C4⋊C4×C83—24no—
193C4⋊C4×C83—24no—
194C4⋊C4×C83—21no—
195C4⋊C4×C83—21no—
196C4⋊C4×C83C22×C412yes—
197C4⋊C4×C83C22×C412yes—
198C22⋊C4×C83—24no—
199C22⋊C4×C83—24no—
200C42×C83C22×C412yes—
201C42×C83#4112yes—
202C23×C83—22no—
203C23×C83—41no—
204C23×C83—41no—
205C23×C83—41no—
206C23×C83—41no—
207C23×C83—41no—
208C23×C83—41no—
209C23×C83C22×D411yescharacteristic
210Q8×C83—22no—
211Q8×C83—22no—
212Q8×C83—41no—
213Q8×C83—41no—
214D4×C83—22no—
215D4×C83—22no—
216D4×C83—41no—
217D4×C83—41no—
218C2×C4×C83—21no—
219C2×C4×C83—21no—
220C2×C4×C83—21no—
221C2×C4×C83—21no—
222C2×C4×C83—22no—
223C2×C4×C83—22no—
224C2×C4×C83—22no—
225C2×C4×C83—41no—
226C2×C4×C83—41no—
227C2×C4×C83—41no—
228C2×C4×C83—41no—
229C2×C4×C83—41no—
230C2×C4×C83—41no—
231C2×C4×C83C4×D412yes—
232C2×C4×C83C4×D412yes—
233C2×C4×C83C23×C411yescharacteristic
234C2×C4×C83C2×#4111yescharacteristic
235C2×C4×C83C2×#4111yescharacteristic
236C2×C4×C83C2×#4111yescharacteristic
237C2×C4×C83C2×#4111yescharacteristic
238C2×C4×C83C2×#4111yescharacteristic
239C2×Dic83—21no—
240C2×Dic83—21no—
241C2×Dic83—21no—
242C2×Dic83—21no—
243C2×Dic83—21no—
244C2×Dic83—21no—
245C2×Dic83—21no—
246C2×Dic83—21no—
247C2×Dic83—21no—
248C2×Dic83—21no—
249C2×Dic83—41no—
250C2×Dic83—41no—
251C2×Dic83—41no—
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—84no—
262C2×Dic83#12412yes—
263C2×Dic83#12412yes—
264C8×C83—22no—
265C8×C83—22no—
266C83⋊C8—22no—
267C83⋊C8—22no—
268C22×C83—42no—
269C22×C83—42no—
270C22×C83—42no—
271C22×C83—81no—
272C22×C83—81no—
273C22×C83—81no—
274C22×C83—81no—
275C22×C83C2×C4×D411yescharacteristic
276C22×C83C2×#12411yescharacteristic
277C22×C83#53211yescharacteristic
278C22×C83#57611yescharacteristic
279C22×C83#57711yescharacteristic
280C22×C83#57711yescharacteristic
281C22×C83#57811yescharacteristic
282C4×C83—22no—
283C4×C83—22no—
284C4×C83—42no—
285C4×C83—42no—
286C4×C83—42no—
287C4×C83C4×#4111yescharacteristic
288C4×C83C4×#4111yescharacteristic
289C4×C83#52911yescharacteristic
290C4×C83#52911yescharacteristic
291Dic83—22no—
292Dic83—22no—
293Dic83—41no—
294Dic83—41no—
295Dic83—81no—
296Dic83—81no—
297Dic83—81no—
298Dic83—81no—
299#21643—831nomaximal, Sylow
300C2×C83—82no—
301C2×C83C4×#12411yescharacteristic
302C2×C83#295111yescharacteristic
303C2×C83#295211yescharacteristic
304C2×C83#313611yescharacteristic
305C2×C83#313711yescharacteristic
306C2×C83#317811yescharacteristic
307C2×C83#317911yescharacteristic
308C2×#488D8311yescharacteristic
309#2330—831no—
310#2326—831no—
311#1992—831no—
312#1959—831no—
313#1936—831no—
314#1935—831no—
315#1934—831no—
316#1933—831no—
317#1834—831no—
318#1763—831no—
319#1762—831no—
320C4×#103—831no—
321C4×#102—831no—
322#1726—831no—
323C83#2164311yescharacteristic, complemented, Sylow
324C2×#125C2×D8311yescharacteristic
325C2×#124C2×D8311yescharacteristic
326C2×C4×Q8—831no—
327C2×C4×D4—831no—
328#488Dic8318yescomplemented
329C2×#103—831no—
330C2×#103—831no—
331C2×#103C2×D8311yescharacteristic
332C2×#103C2×D8311yescharacteristic
333C2×#102—831no—
334C2×#102—831no—
335C2×#102C2×D8311yescharacteristic
336C2×#102C2×D8311yescharacteristic
337C2×#97—831no—
338C2×#97—831no—
339C2×#97—831no—
340C2×C4×C8—831no—
341C2×C4×C8—831no—
342C2×C4×C8C2×D8311yescharacteristic
343#391—831no—
344#390—831no—
345#389—831no—
346#389—831no—
347#389—1661no—
348#389—1661no—
349#388—1661no—
350#388—1661no—
351#387—1661no—
352#387—1661no—
353#385—1661no—
354#385—1661no—
355#384—831no—
356#384—831no—
357C4×C4⋊C4—831no—
358C4×C4⋊C4—831no—
359#376—831no—
360#340—831no—
361#340—831no—
362#340—831no—
363#340—831no—
364#340—831no—
365#340—831no—
366#340—831no—
367#340—831no—
368C22×Q8C22×D8311yescharacteristic
369C22×D4C22×D8311yescharacteristic
370C23×C4—1661no—
371C22×C8—831no—
372C22×C8—831no—
373C22×C8C22×D8311yescharacteristic
374C22×C8C22×D8311yescharacteristic
375#125—22no—
376#125C2×Dic8314yescomplemented
377#124—22no—
378#124C2×Dic8314yes—
379C4×Q8—1662no—
380C4×Q8—1662no—
381C4×D4—1662no—
382C4×D4—1662no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—831no—
386C2×C4⋊C4—831no—
387C2×C4⋊C4—831no—
388C2×C4⋊C4—831no—
389C2×C4⋊C4—1661no—
390C2×C4⋊C4—1661no—
391C2×C4⋊C4—1661no—
392C2×C4⋊C4—1661no—
393C2×C4⋊C4—1661no—
394C2×C4⋊C4—1661no—
395C2×C4⋊C4—1661no—
396C2×C4⋊C4C22×D8311yescharacteristic
397C2×C4⋊C4C22×D8311yescharacteristic
398C2×C22⋊C4—21no—
399C2×C22⋊C4—21no—
400C2×C22⋊C4—1661no—
401C2×C22⋊C4—1661no—
402C2×C22⋊C4—1661no—
403C2×C42—831no—
404C2×C42—831no—
405C2×C42—1661no—
406C2×C42—1661no—
407C2×C42—1661no—
408C2×C42C22×D8311yescharacteristic
409#103—1662no—
410#103—1662no—
411#103C2×Dic8314yescomplemented
412#103C2×Dic8314yescomplemented
413#102—1662no—
414#102—1662no—
415#102C2×Dic8314yes—
416#102C2×Dic8314yes—
417#97—1662no—
418#97—1662no—
419#97—1662no—
420C4×C8—1662no—
421C4×C8—1662no—
422C4×C8C2×Dic8314yescomplemented
423#95—831no—
424#95—831no—
425#95—831no—
426#95—831no—
427#95—1661no—
428#95—1661no—
429#95—1661no—
430#95—1661no—
431#95—1661no—
432#95—1661no—
433C24—21no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8C22×Dic8312yes—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4C22×Dic8312yes—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—3322no—
450C22×C4—831no—
451C22×C4—831no—
452C22×C4—831no—
453C22×C4—831no—
454C22×C4—6641no—
455C22×C4—6641no—
456C22×C4—1661no—
457C22×C4—1661no—
458C22×C4—1661no—
459C22×C4—1661no—
460C22×C4D4×D8311yescharacteristic
461C22×C4D4×D8311yescharacteristic
462C22×C4C23×D8311yescharacteristic
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—832no—
468C2×C8—832no—
469C2×C8—1661no—
470C2×C8—1661no—
471C2×C8—1661no—
472C2×C8—1661no—
473C2×C8C22×Dic8312yes—
474C2×C8C22×Dic8312yes—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—3322no—
480C4⋊C4—6641no—
481C4⋊C4—6641no—
482C4⋊C4—6641no—
483C4⋊C4—6641no—
484C4⋊C4—6641no—
485C4⋊C4—6641no—
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—24no—
498C22⋊C4—3322no—
499C22⋊C4—6641no—
500C22⋊C4—6641no—
501C42—3322no—
502C42—832no—
503C42—832no—
504C42—6641no—
505C42—6641no—
506C42—1661no—
507C42—1661no—
508C42C22×Dic8312yes—
509C42#1305058912yes—
510C23—22no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23C2×D4×D8311yescharacteristic
518Q8—22no—
519Q8—22no—
520Q8—41no—
521Q8—41no—
522D4—22no—
523D4—22no—
524D4—41no—
525D4—41no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—3321no—
540C2×C4—3321no—
541C2×C4—3321no—
542C2×C4—3321no—
543C2×C4—3321no—
544C2×C4—3321no—
545C2×C4—3321no—
546C2×C4—3321no—
547C2×C4—3321no—
548C2×C4—3321no—
549C2×C4—3321no—
550C2×C4—3321no—
551C2×C4—832no—
552C2×C4—832no—
553C2×C4—6644no—
554C2×C4—1661no—
555C2×C4—1661no—
556C2×C4—1661no—
557C2×C4—1661no—
558C2×C4—1661no—
559C2×C4—1661no—
560C2×C4—1661no—
561C2×C4—1661no—
562C2×C4—1661no—
563C2×C4—1661no—
564C2×C4D4×Dic8312yes—
565C2×C4D4×Dic8312yes—
566C2×C4C23×Dic8311yescharacteristic
567C2×C4#41×D8311yescharacteristic
568C2×C4#41×D8311yescharacteristic
569C2×C4#41×D8311yescharacteristic
570C2×C4#41×D8311yescharacteristic
571C2×C4C2×#1305058911yescharacteristic
572C8—22no—
573C8—22no—
574C8—1662no—
575C8—1662no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22—81no—
580C22—81no—
581C22—81no—
582C22—81no—
583C22C2×D4×Dic8311yescharacteristic
584C22#124×D8311yescharacteristic
585C22#1912825011yescharacteristic
586C22#1912850311yescharacteristic
587C22#1912850811yescharacteristic
588C22#1912850811yescharacteristic
589C22#1912851511yescharacteristic
590C4—22no—
591C4—22no—
592C4—42no—
593C4—42no—
594C4—42no—
595C4—3321no—
596C4—3321no—
597C4—6641no—
598C4—6641no—
599C4—6641no—
600C4—6641no—
601C4—1662no—
602C4—1662no—
603C4#41×Dic8311yescharacteristic
604C4#41×Dic8311yescharacteristic
605C4#1912823011yescharacteristic
606C4#1912823011yescharacteristic
607C2—82no—
608C2#124×Dic8311yescharacteristic
609C2#2491633911yescharacteristic
610C2#2491634711yescharacteristic
611C2#2491781611yescharacteristic
612C2#2491782711yescharacteristic
613C2#2491813011yescharacteristic
614C2#2491814511yescharacteristic
subgroupG/H
centerC23C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC2×#488×C83C2
socleC23×C83C22×D4
prime radicalC22×D4×C83C23
cosocleC22×C4×C83C24
hypercenterC2×#488D83
hypocenterC83#21643
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#488

Complements

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

NH ≅ G/Ntype
1. #24910205C2semidirect
3. #24908433C2semidirect
6. #24908155C2semidirect
8. #24908141C2semidirect
9. #24907591C2semidirect
10. #24907295C2semidirect
12. #103×Dic83C2semidirect
14. #24907149C2semidirect
54. #488×C83C4semidirect
110. #125×C83C2×C4semidirect
120. #103×C83C2×C4semidirect
121. #103×C83C2×C4semidirect
124. C4×C8×C83C2×C4semidirect
323. C83#21643semidirect
328. #488Dic83semidirect
376. #125C2×Dic83semidirect
411. #103C2×Dic83semidirect
412. #103C2×Dic83semidirect
422. C4×C8C2×Dic83semidirect

Maximal subgroups

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

subgroupindex
#249102052
#249101762
#249084332
#249082802
#249081622
#249081552
#249081482
#249081412
#249075912
#249072952
#249072902
#103×Dic832
#102×Dic832
#249071492
C2×#488×C832
#2164383