Finite Group Databasebeta

← #35412520

Subgroups — #35412520

Order 21248.

826 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24910212C211yesmaximal, characteristic, complemented
2#24910176C211yesmaximal, characteristic
3#24908425C211yesmaximal, characteristic, complemented
4#24908275C211yesmaximal, characteristic
5#24908162C211yesmaximal, characteristic
6#24908155C211yesmaximal, characteristic, complemented
7#24908148C211yesmaximal, characteristic
8#24908141C211yesmaximal, characteristic, complemented
9#24907597C211yesmaximal, characteristic, complemented
10#24907295C211yesmaximal, characteristic, complemented
11#24907290C211yesmaximal, characteristic
12#103×Dic83C211yesmaximal, characteristic, complemented
13#102×Dic83C211yesmaximal, characteristic
14#24907151C211yesmaximal, characteristic, complemented
15C2×#489×C83C211yesmaximal, characteristic
16C2×Q8×Dic83C2211yescharacteristic
17C2×D4×Dic83C2211yescharacteristic
18C2×#14581716C2211yescharacteristic
19C2×#14581716C2211yescharacteristic
20C2×#14581713C2211yescharacteristic
21C2×#14581713C2211yescharacteristic
22C2×#14581702C2211yescharacteristic
23C2×#14581702C2211yescharacteristic
24C2×C4×C83⋊C8C2211yescharacteristic
25C2×C8×Dic83C2211yescharacteristic
26C2×C8×Dic83C2211yescharacteristic
27#19127608—21no—
28#19127608—21no—
29#19127608C2211yescharacteristic
30#19127604C2211yescharacteristic
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#19127567C2211yescharacteristic
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×C4⋊Q8×C83C2211yescharacteristic
53C2×#124×C83C2211yescharacteristic
54#489×C83C418yescomplemented
55C2×#103×C83C2211yescharacteristic
56C2×#103×C83C2211yescharacteristic
57C2×#102×C83C2211yescharacteristic
58C2×#102×C83C2211yescharacteristic
59C2×#97×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⋊Dic83C2311yescharacteristic
73C2×C4⋊Dic83C2311yescharacteristic
74C2×C4⋊Dic83C2311yescharacteristic
75C2×C4⋊Dic83C2311yescharacteristic
76C2×C22⋊Dic83—21no—
77C2×C22⋊Dic83—21no—
78C2×C22⋊Dic83—21no—
79C2×C4×Dic83—21no—
80C2×C4×Dic83—21no—
81C2×C4×Dic83—21no—
82C2×C4×Dic83C2311yescharacteristic
83C2×C4×Dic83C2311yescharacteristic
84#14581716—22no—
85#14581716—22no—
86#14581713—22no—
87#14581713—22no—
88#14581702—22no—
89#14581702—22no—
90C4×C83⋊C8—22no—
91C8×Dic83—22no—
92C8×Dic83—22no—
93C83⋊#95—21no—
94C83⋊#95—21no—
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×C83—21no—
106C22×Q8×C83C2311yescharacteristic
107C22×D4×C83C2311yescharacteristic
108C22×C8×C83C2311yescharacteristic
109C22×C8×C83C2311yescharacteristic
110C4⋊Q8×C83—22no—
111C4⋊Q8×C83C2×C414yescomplemented
112#124×C83—22no—
113#124×C83C2×C414yes—
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—
124#97×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—
155C22⋊Dic83—42no—
156C22⋊Dic83—81no—
157C22⋊Dic83—81no—
158C4×Dic83—21no—
159C4×Dic83—21no—
160C4×Dic83—42no—
161C4×Dic83—81no—
162C4×Dic83—81no—
163C4×Dic83#4112yes—
164C4×Dic83#4112yes—
165C24×C83—21no—
166C2×Q8×C83—24no—
167C2×Q8×C83—21no—
168C2×Q8×C83—21no—
169C2×Q8×C83—21no—
170C2×Q8×C83—21no—
171C2×Q8×C83—21no—
172C2×Q8×C83—81no—
173C2×Q8×C83—81no—
174C2×Q8×C83C22×C412yes—
175C2×D4×C83—21no—
176C2×D4×C83—21no—
177C2×D4×C83—21no—
178C2×D4×C83—21no—
179C2×D4×C83—21no—
180C2×D4×C83C22×C412yes—
181C22×C4×C83—21no—
182C22×C4×C83—21no—
183C22×C4×C83—21no—
184C22×C4×C83C2×D411yescharacteristic
185C22×C4×C83C2×D411yescharacteristic
186C22×C4×C83C2411yescharacteristic
187C2×C8×C83—21no—
188C2×C8×C83—21no—
189C2×C8×C83—21no—
190C2×C8×C83—21no—
191C2×C8×C83C22×C412yes—
192C2×C8×C83C22×C412yes—
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—
214Q8×C83—84no—
215D4×C83—22no—
216D4×C83—22no—
217D4×C83—41no—
218D4×C83—41no—
219C2×C4×C83—21no—
220C2×C4×C83—21no—
221C2×C4×C83—21no—
222C2×C4×C83—21no—
223C2×C4×C83—22no—
224C2×C4×C83—22no—
225C2×C4×C83—22no—
226C2×C4×C83—41no—
227C2×C4×C83—41no—
228C2×C4×C83—41no—
229C2×C4×C83—41no—
230C2×C4×C83—41no—
231C2×C4×C83—41no—
232C2×C4×C83C4×D412yes—
233C2×C4×C83C4×D412yes—
234C2×C4×C83C23×C411yescharacteristic
235C2×C4×C83C2×#4111yescharacteristic
236C2×C4×C83C2×#4111yescharacteristic
237C2×C4×C83C2×#4111yescharacteristic
238C2×C4×C83C2×#4111yescharacteristic
239C2×C4×C83C2×#4111yescharacteristic
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—21no—
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—41no—
262C2×Dic83—84no—
263C2×Dic83#12412yes—
264C2×Dic83#12412yes—
265C8×C83—22no—
266C8×C83—22no—
267C83⋊C8—22no—
268C83⋊C8—22no—
269C22×C83—42no—
270C22×C83—42no—
271C22×C83—42no—
272C22×C83—81no—
273C22×C83—81no—
274C22×C83—81no—
275C22×C83—81no—
276C22×C83C2×C4×D411yescharacteristic
277C22×C83C2×#12411yescharacteristic
278C22×C83#53211yescharacteristic
279C22×C83#57611yescharacteristic
280C22×C83#57711yescharacteristic
281C22×C83#57711yescharacteristic
282C22×C83#57811yescharacteristic
283C4×C83—22no—
284C4×C83—22no—
285C4×C83—42no—
286C4×C83—42no—
287C4×C83—42no—
288C4×C83C4×#4111yescharacteristic
289C4×C83C4×#4111yescharacteristic
290C4×C83#52911yescharacteristic
291C4×C83#52911yescharacteristic
292Dic83—22no—
293Dic83—22no—
294Dic83—41no—
295Dic83—41no—
296Dic83—81no—
297Dic83—81no—
298Dic83—81no—
299Dic83—81no—
300#21654—831nomaximal, Sylow
301C2×C83—82no—
302C2×C83C4×#12411yescharacteristic
303C2×C83#294611yescharacteristic
304C2×C83#294711yescharacteristic
305C2×C83#313611yescharacteristic
306C2×C83#313711yescharacteristic
307C2×C83#317311yescharacteristic
308C2×C83#317511yescharacteristic
309C2×#489D8311yescharacteristic
310#2331—831no—
311#2326—831no—
312#1990—831no—
313#1958—831no—
314#1936—831no—
315#1935—831no—
316#1934—831no—
317#1933—831no—
318#1836—831no—
319#1763—831no—
320#1762—831no—
321C4×#103—831no—
322C4×#102—831no—
323#1726—831no—
324C83#2165411yescharacteristic, complemented, Sylow
325C2×C4⋊Q8C2×D8311yescharacteristic
326C2×#124C2×D8311yescharacteristic
327C2×C4×Q8—831no—
328C2×C4×D4—831no—
329#489Dic8318yescomplemented
330C2×#103—831no—
331C2×#103—831no—
332C2×#103C2×D8311yescharacteristic
333C2×#103C2×D8311yescharacteristic
334C2×#102—831no—
335C2×#102—831no—
336C2×#102C2×D8311yescharacteristic
337C2×#102C2×D8311yescharacteristic
338C2×#97—831no—
339C2×#97—831no—
340C2×#97C2×D8311yescharacteristic
341C2×C4×C8—831no—
342C2×C4×C8—831no—
343C2×C4×C8—831no—
344#391—831no—
345#391—1661no—
346#391—1661no—
347#390—831no—
348#389—831no—
349#389—831no—
350#388—1661no—
351#388—1661no—
352#387—1661no—
353#387—1661no—
354#385—1661no—
355#385—1661no—
356#384—831no—
357#382—831no—
358C4×C4⋊C4—831no—
359C4×C4⋊C4—831no—
360#376—831no—
361#340—831no—
362#340—831no—
363#340—831no—
364#340—831no—
365#340—831no—
366#340—831no—
367#340—831no—
368#340—831no—
369C22×Q8—21no—
370C22×Q8C22×D8311yescharacteristic
371C22×D4C22×D8311yescharacteristic
372C23×C4—1661no—
373C22×C8—831no—
374C22×C8—831no—
375C22×C8C22×D8311yescharacteristic
376C22×C8C22×D8311yescharacteristic
377C4⋊Q8—22no—
378C4⋊Q8C2×Dic8314yescomplemented
379#124—22no—
380#124C2×Dic8314yes—
381C4×Q8—1662no—
382C4×Q8—1662no—
383C4×D4—1662no—
384C4×D4—1662no—
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⋊C4—1661no—
395C2×C4⋊C4C22×D8311yescharacteristic
396C2×C4⋊C4C22×D8311yescharacteristic
397C2×C22⋊C4—21no—
398C2×C22⋊C4—21no—
399C2×C22⋊C4—1661no—
400C2×C22⋊C4—1661no—
401C2×C22⋊C4—1661no—
402C2×C42—831no—
403C2×C42—831no—
404C2×C42—1661no—
405C2×C42—1661no—
406C2×C42—1661no—
407C2×C42C22×D8311yescharacteristic
408#103—1662no—
409#103—1662no—
410#103C2×Dic8314yescomplemented
411#103C2×Dic8314yescomplemented
412#102—1662no—
413#102—1662no—
414#102C2×Dic8314yes—
415#102C2×Dic8314yes—
416#97—1662no—
417#97—1662no—
418#97C2×Dic8314yescomplemented
419C4×C8—1662no—
420C4×C8—1662no—
421C4×C8—1662no—
422#95—831no—
423#95—831no—
424#95—831no—
425#95—831no—
426#95—1661no—
427#95—1661no—
428#95—1661no—
429#95—1661no—
430#95—1661no—
431#95—1661no—
432#95—1661no—
433#95—1661no—
434C24—21no—
435C2×Q8—21no—
436C2×Q8—24no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—81no—
442C2×Q8—81no—
443C2×Q8C22×Dic8312yes—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4C22×Dic8312yes—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—3322no—
454C22×C4—831no—
455C22×C4—831no—
456C22×C4—831no—
457C22×C4—831no—
458C22×C4—6641no—
459C22×C4—6641no—
460C22×C4—1661no—
461C22×C4—1661no—
462C22×C4—1661no—
463C22×C4—1661no—
464C22×C4D4×D8311yescharacteristic
465C22×C4D4×D8311yescharacteristic
466C22×C4C23×D8311yescharacteristic
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—832no—
472C2×C8—832no—
473C2×C8—1661no—
474C2×C8—1661no—
475C2×C8—1661no—
476C2×C8—1661no—
477C2×C8C22×Dic8312yes—
478C2×C8C22×Dic8312yes—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—24no—
482C4⋊C4—3322no—
483C4⋊C4—6641no—
484C4⋊C4—6641no—
485C4⋊C4—6641no—
486C4⋊C4—6641no—
487C4⋊C4—1661no—
488C4⋊C4—1661no—
489C4⋊C4—1661no—
490C4⋊C4—1661no—
491C4⋊C4—1661no—
492C4⋊C4—1661no—
493C4⋊C4—1661no—
494C4⋊C4—1661no—
495C4⋊C4C22×Dic8312yes—
496C4⋊C4C22×Dic8312yes—
497C22⋊C4—24no—
498C22⋊C4—24no—
499C22⋊C4—3322no—
500C22⋊C4—6641no—
501C22⋊C4—6641no—
502C42—3322no—
503C42—832no—
504C42—832no—
505C42—6641no—
506C42—6641no—
507C42—1661no—
508C42—1661no—
509C42C22×Dic8312yes—
510C42#1305058912yes—
511C23—22no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23C2×D4×D8311yescharacteristic
519Q8—22no—
520Q8—22no—
521Q8—41no—
522Q8—41no—
523Q8—84no—
524D4—22no—
525D4—22no—
526D4—41no—
527D4—41no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
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—3321no—
552C2×C4—3321no—
553C2×C4—832no—
554C2×C4—832no—
555C2×C4—6644no—
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×C4—1661no—
565C2×C4—1661no—
566C2×C4D4×Dic8312yes—
567C2×C4D4×Dic8312yes—
568C2×C4C23×Dic8311yescharacteristic
569C2×C4#41×D8311yescharacteristic
570C2×C4#41×D8311yescharacteristic
571C2×C4#41×D8311yescharacteristic
572C2×C4#41×D8311yescharacteristic
573C2×C4C2×#1305058911yescharacteristic
574C8—22no—
575C8—22no—
576C8—1662no—
577C8—1662no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—81no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22C2×D4×Dic8311yescharacteristic
586C22#124×D8311yescharacteristic
587C22#1912825011yescharacteristic
588C22#1912850511yescharacteristic
589C22#1912851011yescharacteristic
590C22#1912851111yescharacteristic
591C22#1912851711yescharacteristic
592C4—22no—
593C4—22no—
594C4—42no—
595C4—42no—
596C4—42no—
597C4—3321no—
598C4—3321no—
599C4—6641no—
600C4—6641no—
601C4—6641no—
602C4—6641no—
603C4—1662no—
604C4—1662no—
605C4#41×Dic8311yescharacteristic
606C4#41×Dic8311yescharacteristic
607C4#1912823011yescharacteristic
608C4#1912823011yescharacteristic
609C2—82no—
610C2#124×Dic8311yescharacteristic
611C2#2491629611yescharacteristic
612C2#2491630711yescharacteristic
613C2#2491781411yescharacteristic
614C2#2491782511yescharacteristic
615C2#2491807911yescharacteristic
616C2#2491810111yescharacteristic
subgroupG/H
centerC23C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC2×#489×C83C2
socleC23×C83C22×D4
prime radicalC22×D4×C83C23
cosocleC22×C4×C83C24
hypercenterC2×#489D83
hypocenterC83#21654
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#489

Complements

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

NH ≅ G/Ntype
1. #24910212C2semidirect
3. #24908425C2semidirect
6. #24908155C2semidirect
8. #24908141C2semidirect
9. #24907597C2semidirect
10. #24907295C2semidirect
12. #103×Dic83C2semidirect
14. #24907151C2semidirect
54. #489×C83C4semidirect
111. C4⋊Q8×C83C2×C4semidirect
120. #103×C83C2×C4semidirect
121. #103×C83C2×C4semidirect
124. #97×C83C2×C4semidirect
324. C83#21654semidirect
329. #489Dic83semidirect
378. C4⋊Q8C2×Dic83semidirect
410. #103C2×Dic83semidirect
411. #103C2×Dic83semidirect
418. #97C2×Dic83semidirect

Maximal subgroups

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

subgroupindex
#249102122
#249101762
#249084252
#249082752
#249081622
#249081552
#249081482
#249081412
#249075972
#249072952
#249072902
#103×Dic832
#102×Dic832
#249071512
C2×#489×C832
#2165483