Finite Group Databasebeta

← #35423765

Subgroups — #35423765

Order 21248.

866 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24910420C211yesmaximal, characteristic, complemented
2#24910373C211yesmaximal, characteristic
3#24908155C211yesmaximal, characteristic, complemented
4#24908141C211yesmaximal, characteristic, complemented
5#24908057C211yesmaximal, characteristic, complemented
6#24908036C211yesmaximal, characteristic, complemented
7#24907799C211yesmaximal, characteristic
8#24907799C211yesmaximal, characteristic
9#24907597C211yesmaximal, characteristic, complemented
10#24907478C211yesmaximal, characteristic, complemented
11#24907463C211yesmaximal, characteristic
12#24907335C211yesmaximal, characteristic, complemented
13#24907290C211yesmaximal, characteristic
14#102×Dic83C211yesmaximal, characteristic
15C2×#440×C83C211yesmaximal, characteristic
16C2×D4×Dic83C2211yescharacteristic
17C2×D4×Dic83C2211yescharacteristic
18C2×C4⋊(C83⋊C8)C2211yescharacteristic
19C2×#14581716C2211yescharacteristic
20C2×#14581716C2211yescharacteristic
21C2×#14581713C2211yescharacteristic
22C2×#14581713C2211yescharacteristic
23C2×#14581702C2211yescharacteristic
24C2×C8×Dic83C2211yescharacteristic
25#19127608C2211yescharacteristic
26#19127608C2211yescharacteristic
27#19127600C2211yescharacteristic
28#19127600C2211yescharacteristic
29#19127587—21no—
30#19127587—21no—
31#19127576—21no—
32#19127576—21no—
33#19127571C2211yescharacteristic
34#19127567—21no—
35#19127567C2211yescharacteristic
36#19127561—21no—
37C4⋊C4×Dic83C2211yescharacteristic
38C4⋊C4×Dic83C2211yescharacteristic
39C4×C4⋊Dic83—21no—
40C4×C4⋊Dic83C2211yescharacteristic
41C2×C83⋊#95—21no—
42#19127464C2211yescharacteristic
43#19127464C2211yescharacteristic
44#19127464C2211yescharacteristic
45#19127464C2211yescharacteristic
46#19127463C2211yescharacteristic
47#19127463C2211yescharacteristic
48#19127463C2211yescharacteristic
49#19127463C2211yescharacteristic
50#19127455C2211yescharacteristic
51#19127455C2211yescharacteristic
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
60C23×Dic83—21no—
61C23×Dic83—21no—
62C22×C83⋊C8C2311yescharacteristic
63C22×C83⋊C8C2311yescharacteristic
64D4×Dic83—22no—
65D4×Dic83—22no—
66D4×Dic83—41no—
67D4×Dic83—41no—
68C2×C4⋊Dic83—21no—
69C2×C4⋊Dic83—21no—
70C2×C4⋊Dic83—21no—
71C2×C4⋊Dic83—21no—
72C2×C4⋊Dic83D411yescharacteristic
73C2×C4⋊Dic83D411yescharacteristic
74C2×C4⋊Dic83C2311yescharacteristic
75C2×C4⋊Dic83C2311yescharacteristic
76C2×C4⋊Dic83C2311yescharacteristic
77C2×C4⋊Dic83C2311yescharacteristic
78C2×C22⋊Dic83—21no—
79C2×C22⋊Dic83—21no—
80C2×C4×Dic83—21no—
81C2×C4×Dic83—21no—
82C2×C4×Dic83C2311yescharacteristic
83C2×C4×Dic83C2311yescharacteristic
84C4⋊(C83⋊C8)—22no—
85#14581716—41no—
86#14581716—41no—
87#14581713—22no—
88#14581713—22no—
89#14581702—41no—
90C8×Dic83—41no—
91C83⋊#95—21no—
92C83⋊#95—21no—
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
103C22×Q8×C83C2311yescharacteristic
104C22×D4×C83C2311yescharacteristic
105C23×C4×C83—21no—
106C2×SD16×C83—22no—
107C2×SD16×C83—22no—
108C2×SD16×C83C2×C414yes—
109C22×C8×C83C2311yescharacteristic
110C22×C8×C83C2311yescharacteristic
111C4×Q8×C83—22no—
112C4×Q8×C83C2×C414yescomplemented
113C4×D4×C83—22no—
114C4×D4×C83C2×C414yes—
115C2×C4⋊C4×C83—21no—
116C2×C4⋊C4×C83C2311yescharacteristic
117C2×C4⋊C4×C83C2311yescharacteristic
118C2×C22⋊C4×C83—21no—
119C2×C42×C83—21no—
120C2×C42×C83C2311yescharacteristic
121#107×C83C2×C414yescomplemented
122#103×C83C2×C414yescomplemented
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—42no—
131C22×Dic83—42no—
132C22×Dic83—42no—
133C22×Dic83—81no—
134C22×Dic83—81no—
135C22×Dic83C2×D411yescharacteristic
136C22×Dic83C2×D411yescharacteristic
137C22×Dic83C2×D411yescharacteristic
138C22×Dic83C2×D411yescharacteristic
139C2×C83⋊C8—21no—
140C2×C83⋊C8—21no—
141C2×C83⋊C8—21no—
142C2×C83⋊C8—21no—
143C2×C83⋊C8—21no—
144C2×C83⋊C8—21no—
145C4⋊Dic83—22no—
146C4⋊Dic83—21no—
147C4⋊Dic83—21no—
148C4⋊Dic83—21no—
149C4⋊Dic83—21no—
150C4⋊Dic83—21no—
151C4⋊Dic83—21no—
152C4⋊Dic83—21no—
153C4⋊Dic83—21no—
154C4⋊Dic83—41no—
155C4⋊Dic83—42no—
156C4⋊Dic83—81no—
157C4⋊Dic83—81no—
158C4⋊Dic83—81no—
159C22⋊Dic83—42no—
160C22⋊Dic83—42no—
161C4×Dic83—21no—
162C4×Dic83—21no—
163C4×Dic83—21no—
164C4×Dic83—21no—
165C4×Dic83—81no—
166C4×Dic83—81no—
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×C83C22×C412yes—
178C2×D4×C83#4112yes—
179C2×D4×C83#4112yes—
180C22×C4×C83—21no—
181C22×C4×C83—21no—
182C22×C4×C83—21no—
183C22×C4×C83—24no—
184C22×C4×C83—81no—
185C22×C4×C83—81no—
186C22×C4×C83C2×D411yescharacteristic
187C22×C4×C83C2×D411yescharacteristic
188C22×C4×C83C2411yescharacteristic
189SD16×C83—28no—
190C2×C8×C83—21no—
191C2×C8×C83—21no—
192C2×C8×C83—21no—
193C2×C8×C83—21no—
194C2×C8×C83C22×C412yes—
195C2×C8×C83C22×C412yes—
196C4⋊C4×C83—24no—
197C4⋊C4×C83—21no—
198C4⋊C4×C83—21no—
199C4⋊C4×C83C22×C412yes—
200C4⋊C4×C83C22×C412yes—
201C22⋊C4×C83—24no—
202C42×C83—24no—
203C42×C83C22×C412yes—
204C42×C83#4112yes—
205C23×C83—22no—
206C23×C83—22no—
207C23×C83—22no—
208C23×C83—41no—
209C23×C83—41no—
210C23×C83—41no—
211C23×C83—41no—
212C23×C83C22×D411yescharacteristic
213Q8×C83—22no—
214Q8×C83—22no—
215Q8×C83—41no—
216Q8×C83—41no—
217D4×C83—22no—
218D4×C83—22no—
219D4×C83—22no—
220D4×C83#11714yes—
221C2×C4×C83—21no—
222C2×C4×C83—21no—
223C2×C4×C83—21no—
224C2×C4×C83—21no—
225C2×C4×C83—22no—
226C2×C4×C83—22no—
227C2×C4×C83—22no—
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×C83—84no—
235C2×C4×C83C4×D412yes—
236C2×C4×C83C4×D412yes—
237C2×C4×C83C23×C411yescharacteristic
238C2×C4×C83C2×#4111yescharacteristic
239C2×C4×C83C2×#4111yescharacteristic
240C2×C4×C83C2×#4111yescharacteristic
241C2×C4×C83H(5,𝔽2)11yescharacteristic
242C2×C4×C832−1+411yescharacteristic
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—21no—
253C2×Dic83—21no—
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×Dic83—41no—
265C2×Dic83—41no—
266C2×Dic83—84no—
267C2×Dic83—84no—
268C2×Dic83#13712yes—
269C8×C83—22no—
270C8×C83—22no—
271C83⋊C8—41no—
272C83⋊C8—41no—
273C22×C83—24no—
274C22×C83—42no—
275C22×C83—42no—
276C22×C83—42no—
277C22×C83—42no—
278C22×C83—42no—
279C22×C83—42no—
280C22×C83C2×C4×D411yescharacteristic
281C22×C83#54611yescharacteristic
282C22×C83#54811yescharacteristic
283C22×C83C2×#13711yescharacteristic
284C22×C83#57511yescharacteristic
285C22×C83#57611yescharacteristic
286C22×C83#57711yescharacteristic
287C4×C83—22no—
288C4×C83—22no—
289C4×C83—42no—
290C4×C83—42no—
291C4×C83—42no—
292C4×C83C2×#11711yescharacteristic
293C4×C83#52011yescharacteristic
294C4×C83#55211yescharacteristic
295C4×C83#55311yescharacteristic
296Dic83—22no—
297Dic83—41no—
298Dic83—41no—
299Dic83—41no—
300Dic83—81no—
301Dic83—81no—
302Dic83—81no—
303Dic83—81no—
304#22776—831nomaximal, Sylow
305C2×C83—44no—
306C2×C83#235211yescharacteristic
307C2×C83C4×#13711yescharacteristic
308C2×C83#294611yescharacteristic
309C2×C83#331111yescharacteristic
310C2×C83#331411yescharacteristic
311C2×C83#337011yescharacteristic
312C2×C83#337211yescharacteristic
313C2×#440D8311yescharacteristic
314#2355—831no—
315#2349—831no—
316#1935—831no—
317#1933—831no—
318#1921—831no—
319#1918—831no—
320#1877—831no—
321#1877—831no—
322#1836—831no—
323#1805—831no—
324#1802—831no—
325#1771—831no—
326#1762—831no—
327C4×#102—831no—
328C83#2277611yescharacteristic, complemented, Sylow
329C22×SD16C2×D8311yescharacteristic
330C2×C4×Q8C2×D8311yescharacteristic
331C2×C4×D4—831no—
332C2×C4×D4—831no—
333C2×C4×D4C2×D8311yescharacteristic
334#440Dic8318yescomplemented
335C2×#107C2×D8311yescharacteristic
336C2×C4⋊C8—831no—
337C2×#103—831no—
338C2×#103—831no—
339C2×#103C2×D8311yescharacteristic
340C2×#102—831no—
341C2×#102—831no—
342C2×#102C2×D8311yescharacteristic
343C2×#97—831no—
344C2×#97C2×D8311yescharacteristic
345C2×C4×C8—831no—
346#391—831no—
347#391—831no—
348#389—831no—
349#389—831no—
350#387—1661no—
351#387—1661no—
352#385—1661no—
353#385—1661no—
354#384—831no—
355#382—831no—
356#382—1661no—
357#380—1661no—
358C4×C4⋊C4—831no—
359C4×C4⋊C4—831no—
360C4×C4⋊C4—831no—
361C4×C4⋊C4—1661no—
362C2×#95—1661no—
363#340—831no—
364#340—831no—
365#340—831no—
366#340—831no—
367#340—831no—
368#340—831no—
369#340—831no—
370#340—831no—
371#336—831no—
372#336—831no—
373C22×Q8C22×D8311yescharacteristic
374C22×D4C22×D8311yescharacteristic
375C23×C4—21no—
376C23×C4—1661no—
377C23×C4—1661no—
378C2×SD16—22no—
379C2×SD16—22no—
380C2×SD16C2×Dic8314yes—
381C22×C8—831no—
382C22×C8—831no—
383C22×C8C22×D8311yescharacteristic
384C22×C8C22×D8311yescharacteristic
385C4×Q8—22no—
386C4×Q8C2×Dic8314yescomplemented
387C4×D4—22no—
388C4×D4—3321no—
389C4×D4—3321no—
390C4×D4—1662no—
391C4×D4—1662no—
392C4×D4C2×Dic8314yes—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—831no—
395C2×C4⋊C4—831no—
396C2×C4⋊C4—831no—
397C2×C4⋊C4—831no—
398C2×C4⋊C4—831no—
399C2×C4⋊C4—831no—
400C2×C4⋊C4—1661no—
401C2×C4⋊C4—1661no—
402C2×C4⋊C4—1661no—
403C2×C4⋊C4—1661no—
404C2×C4⋊C4C22×D8311yescharacteristic
405C2×C4⋊C4C22×D8311yescharacteristic
406C2×C22⋊C4—21no—
407C2×C22⋊C4—1661no—
408C2×C22⋊C4—1661no—
409C2×C42—21no—
410C2×C42—831no—
411C2×C42—831no—
412C2×C42—1661no—
413C2×C42—1661no—
414C2×C42C22×D8311yescharacteristic
415#107C2×Dic8314yescomplemented
416C4⋊C8—1662no—
417#103—3321no—
418#103—3321no—
419#103C2×Dic8314yescomplemented
420#102—1662no—
421#102—1662no—
422#102C2×Dic8314yes—
423#97—3321no—
424#97C2×Dic8314yescomplemented
425C4×C8—3321no—
426#95—831no—
427#95—831no—
428#95—1661no—
429#95—1661no—
430#95—1661no—
431#95—1661no—
432#95—1661no—
433#95—1661no—
434#95—1661no—
435#95—1661no—
436#95—1661no—
437#95—1661no—
438C24—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8C22×Dic8312yes—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4C22×Dic8312yes—
449C2×D4#1305059012yes—
450C2×D4#1305059012yes—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—24no—
455C22×C4—81no—
456C22×C4—81no—
457C22×C4—3322no—
458C22×C4—3322no—
459C22×C4—3322no—
460C22×C4—3322no—
461C22×C4—831no—
462C22×C4—831no—
463C22×C4—831no—
464C22×C4—831no—
465C22×C4—6641no—
466C22×C4—6641no—
467C22×C4—1661no—
468C22×C4—1661no—
469C22×C4—1661no—
470C22×C4—1661no—
471C22×C4D4×D8311yescharacteristic
472C22×C4D4×D8311yescharacteristic
473C22×C4C23×D8311yescharacteristic
474SD16—28no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—1661no—
480C2×C8—1661no—
481C2×C8—1661no—
482C2×C8—1661no—
483C2×C8—1661no—
484C2×C8—1661no—
485C2×C8C22×Dic8312yes—
486C2×C8C22×Dic8312yes—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—24no—
490C4⋊C4—3322no—
491C4⋊C4—3321no—
492C4⋊C4—6641no—
493C4⋊C4—6641no—
494C4⋊C4—6641no—
495C4⋊C4—1662no—
496C4⋊C4—1661no—
497C4⋊C4—1661no—
498C4⋊C4—1661no—
499C4⋊C4—1661no—
500C4⋊C4—1661no—
501C4⋊C4—1661no—
502C4⋊C4—1661no—
503C4⋊C4—1661no—
504C4⋊C4C22×Dic8312yes—
505C4⋊C4C22×Dic8312yes—
506C22⋊C4—24no—
507C22⋊C4—3322no—
508C22⋊C4—3322no—
509C42—24no—
510C42—6641no—
511C42—6641no—
512C42—1661no—
513C42—1661no—
514C42—1661no—
515C42—1661no—
516C42C22×Dic8312yes—
517C42#1305058912yes—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23C2×D4×D8311yescharacteristic
526Q8—22no—
527Q8—22no—
528Q8—41no—
529Q8—41no—
530D4—22no—
531D4—22no—
532D4—22no—
533D4#1458175114yes—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—84no—
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—3321no—
557C2×C4—3321no—
558C2×C4—3321no—
559C2×C4—3321no—
560C2×C4—832no—
561C2×C4—6644no—
562C2×C4—6644no—
563C2×C4—1661no—
564C2×C4—1661no—
565C2×C4—1661no—
566C2×C4—1661no—
567C2×C4—1661no—
568C2×C4—1661no—
569C2×C4—1661no—
570C2×C4—1661no—
571C2×C4—1661no—
572C2×C4—1661no—
573C2×C4—1661no—
574C2×C4D4×Dic8312yes—
575C2×C4D4×Dic8312yes—
576C2×C4C23×Dic8311yescharacteristic
577C2×C4C2×#1305058911yescharacteristic
578C2×C4C2×#1305059011yescharacteristic
579C2×C4C2×#1305059011yescharacteristic
580C2×C4#1458183811yescharacteristic
581C2×C4#1458183911yescharacteristic
582C8—22no—
583C8—22no—
584C8—3321no—
585C8—3321no—
586C22—24no—
587C22—42no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—42no—
592C22—42no—
593C22C2×D4×Dic8311yescharacteristic
594C22#1912834711yescharacteristic
595C22#1912836011yescharacteristic
596C22#137×D8311yescharacteristic
597C22#1912850011yescharacteristic
598C22#1912850511yescharacteristic
599C22#1912851111yescharacteristic
600C4—22no—
601C4—22no—
602C4—42no—
603C4—42no—
604C4—42no—
605C4—3321no—
606C4—3321no—
607C4—3321no—
608C4—6641no—
609C4—6641no—
610C4—6641no—
611C4—6641no—
612C4—1662no—
613C4C2×#1458175111yescharacteristic
614C4#1912818411yescharacteristic
615C4#1912839111yescharacteristic
616C4#1912839711yescharacteristic
617C2—44no—
618C2#2491039211yescharacteristic
619C2#137×Dic8311yescharacteristic
620C2#2491629611yescharacteristic
621C2#2491950411yescharacteristic
622C2#2491954911yescharacteristic
623C2#2492021711yescharacteristic
624C2#2492024511yescharacteristic
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#22776
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. #24910420C2semidirect
3. #24908155C2semidirect
4. #24908141C2semidirect
5. #24908057C2semidirect
6. #24908036C2semidirect
9. #24907597C2semidirect
10. #24907478C2semidirect
12. #24907335C2semidirect
55. #440×C83C4semidirect
112. C4×Q8×C83C2×C4semidirect
121. #107×C83C2×C4semidirect
122. #103×C83C2×C4semidirect
124. #97×C83C2×C4semidirect
328. C83#22776semidirect
334. #440Dic83semidirect
386. C4×Q8C2×Dic83semidirect
415. #107C2×Dic83semidirect
419. #103C2×Dic83semidirect
424. #97C2×Dic83semidirect

Maximal subgroups

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

subgroupindex
#249104202
#249103732
#249081552
#249081412
#249080572
#249080362
#249077992
#249075972
#249074782
#249074632
#249073352
#249072902
#102×Dic832
C2×#440×C832
#2277683