Finite Group Databasebeta

← #35412517

Subgroups — #35412517

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
1C2×#19128060C211yesmaximal, characteristic
2#24910211C211yesmaximal, characteristic, complemented
3#24908423C211yesmaximal, characteristic, complemented
4#24908275C211yesmaximal, characteristic
5#24908163C211yesmaximal, characteristic
6#24908156C211yesmaximal, characteristic, complemented
7#24908149C211yesmaximal, characteristic
8#24908142C211yesmaximal, characteristic, complemented
9#24907598C211yesmaximal, characteristic, complemented
10#24907293C211yesmaximal, characteristic, complemented
11#24907288C211yesmaximal, characteristic
12C4×#14581716C211yesmaximal, characteristic, complemented
13C4×#14581713C211yesmaximal, characteristic
14#24907150C211yesmaximal, characteristic, complemented
15C83×#2326C211yesmaximal, characteristic
16C2×#14581792C2211yescharacteristic
17#19128060C418yescomplemented
18C2×#14581716C2211yescharacteristic
19C2×#14581716C2211yescharacteristic
20C2×#14581716C2211yescharacteristic
21C2×#14581716C2211yescharacteristic
22C2×#14581713C2211yescharacteristic
23C2×#14581713C2211yescharacteristic
24C2×#14581713C2211yescharacteristic
25C2×#14581713C2211yescharacteristic
26C2×#14581703C2211yescharacteristic
27C2×#14581703C2211yescharacteristic
28C2×#14581703C2211yescharacteristic
29C2×C4×C83⋊C8C2211yescharacteristic
30C2×C4×C83⋊C8C2211yescharacteristic
31C2×C4×C83⋊C8C2211yescharacteristic
32#19127605—21no—
33#19127605—21no—
34#19127599C2211yescharacteristic
35#19127599C2211yescharacteristic
36#19127584—21no—
37#19127584—21no—
38#19127567C2211yescharacteristic
39C4×C4⋊Dic83C2211yescharacteristic
40C4×C4⋊Dic83C2211yescharacteristic
41#19127463C2211yescharacteristic
42#19127463C2211yescharacteristic
43#19127463C2211yescharacteristic
44#19127463C2211yescharacteristic
45#19127463C2211yescharacteristic
46#19127463C2211yescharacteristic
47#19127463C2211yescharacteristic
48#19127463C2211yescharacteristic
49C2×#124×C83C2211yescharacteristic
50C2×C4×Q8×C83C2211yescharacteristic
51C2×C4×D4×C83C2211yescharacteristic
52#391×C83C2211yescharacteristic
53#390×C83C2211yescharacteristic
54#388×C83—21no—
55#388×C83—21no—
56#385×C83—21no—
57#385×C83—21no—
58#384×C83C2211yescharacteristic
59#376×C83C2211yescharacteristic
60C22×Dic166—21no—
61C22×C83⋊C8C2311yescharacteristic
62C22×C83⋊C8C2311yescharacteristic
63C22×C83⋊C8C2311yescharacteristic
64C22×C83⋊C8C2311yescharacteristic
65#14581792—22no—
66#14581792C2×C414yescomplemented
67C2×C4⋊Dic83—21no—
68C2×C4⋊Dic83C2311yescharacteristic
69C2×C4⋊Dic83C2311yescharacteristic
70C2×C4⋊Dic83C2311yescharacteristic
71C2×C4⋊Dic83C2311yescharacteristic
72C2×Dic83⋊C4—21no—
73C2×Dic83⋊C4—21no—
74C2×C4×Dic83—21no—
75C2×C4×Dic83—21no—
76#14581716—22no—
77#14581716—22no—
78#14581716C2×C414yescomplemented
79#14581716C2×C414yescomplemented
80#14581713—22no—
81#14581713—22no—
82#14581713C2×C414yes—
83#14581713C2×C414yes—
84#14581703—22no—
85#14581703—22no—
86#14581703C2×C414yescomplemented
87C4×C83⋊C8—22no—
88C4×C83⋊C8—22no—
89C4×C83⋊C8—22no—
90C83⋊#95—21no—
91C83⋊#95—21no—
92C83⋊#95—21no—
93C83⋊#95—21no—
94C83⋊#95—21no—
95C83⋊#95—21no—
96C22×Q8×C83C2311yescharacteristic
97C22×D4×C83C2311yescharacteristic
98C23×C4×C83—21no—
99#124×C83—22no—
100#124×C83C2×C414yes—
101C4×Q8×C83—22no—
102C4×Q8×C83—22no—
103C4×D4×C83—22no—
104C4×D4×C83—22no—
105C2×C4⋊C4×C83—21no—
106C2×C4⋊C4×C83—21no—
107C2×C4⋊C4×C83—21no—
108C2×C4⋊C4×C83C2311yescharacteristic
109C2×C4⋊C4×C83C2311yescharacteristic
110C2×C22⋊C4×C83—21no—
111C2×C22⋊C4×C83—21no—
112C2×C22⋊C4×C83—21no—
113C2×C22⋊C4×C83—21no—
114C2×C22⋊C4×C83—21no—
115C2×C42×C83—21no—
116C2×C42×C83C2311yescharacteristic
117C2×C42×C83C2311yescharacteristic
118C2×C42×C83C2311yescharacteristic
119#95×C83—21no—
120#95×C83—21no—
121#95×C83D411yescharacteristic
122#95×C83D411yescharacteristic
123#95×C83D411yescharacteristic
124#95×C83D411yescharacteristic
125C2×Dic166—24no—
126C2×Dic166—81no—
127C2×Dic166—81no—
128C22×Dic83—21no—
129C22×Dic83—21no—
130C22×Dic83—21no—
131C22×Dic83—21no—
132C2×C83⋊C8—21no—
133C2×C83⋊C8—21no—
134C2×C83⋊C8—21no—
135C2×C83⋊C8—21no—
136C2×C83⋊C8—21no—
137C2×C83⋊C8—21no—
138C2×C83⋊C8—21no—
139C2×C83⋊C8—21no—
140C2×C83⋊C8C22×C412yes—
141C2×C83⋊C8C22×C412yes—
142C2×C83⋊C8#4112yes—
143C2×C83⋊C8#4112yes—
144C4⋊Dic83—24no—
145C4⋊Dic83—21no—
146C4⋊Dic83—21no—
147C4⋊Dic83—21no—
148C4⋊Dic83—21no—
149C4⋊Dic83—21no—
150C4⋊Dic83—21no—
151C4⋊Dic83C22×C412yes—
152C4⋊Dic83C22×C412yes—
153Dic83⋊C4—81no—
154Dic83⋊C4—81no—
155C4×Dic83—81no—
156C4×Dic83—81no—
157C24×C83—21no—
158C2×Q8×C83—21no—
159C2×Q8×C83—21no—
160C2×Q8×C83—21no—
161C2×Q8×C83—21no—
162C2×Q8×C83—21no—
163C2×Q8×C83C22×C412yes—
164C2×D4×C83—21no—
165C2×D4×C83—21no—
166C2×D4×C83—21no—
167C2×D4×C83—21no—
168C2×D4×C83—21no—
169C2×D4×C83C22×C412yes—
170C22×C4×C83—21no—
171C22×C4×C83—21no—
172C22×C4×C83—21no—
173C22×C4×C83—42no—
174C22×C4×C83—81no—
175C22×C4×C83—81no—
176C22×C4×C83C2×D411yescharacteristic
177C22×C4×C83C2×D411yescharacteristic
178C22×C4×C83C2×D411yescharacteristic
179C22×C4×C83C2×D411yescharacteristic
180C22×C4×C83C2×D411yescharacteristic
181C22×C4×C83C2×D411yescharacteristic
182C22×C4×C83C2411yescharacteristic
183C4⋊C4×C83—21no—
184C4⋊C4×C83—21no—
185C4⋊C4×C83—21no—
186C4⋊C4×C83—21no—
187C4⋊C4×C83—42no—
188C4⋊C4×C83—81no—
189C4⋊C4×C83—81no—
190C22⋊C4×C83—24no—
191C22⋊C4×C83—24no—
192C22⋊C4×C83—42no—
193C22⋊C4×C83—81no—
194C22⋊C4×C83—81no—
195C42×C83—21no—
196C42×C83—21no—
197C42×C83—42no—
198C42×C83C22×C412yes—
199C42×C83#4112yes—
200C42×C83#4112yes—
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—21no—
223C2×C4×C83—21no—
224C2×C4×C83—21no—
225C2×C4×C83—21no—
226C2×C4×C83—22no—
227C2×C4×C83—21no—
228C2×C4×C83—21no—
229C2×C4×C83—21no—
230C2×C4×C83—21no—
231C2×C4×C83—21no—
232C2×C4×C83—21no—
233C2×C4×C83—41no—
234C2×C4×C83—41no—
235C2×C4×C83—41no—
236C2×C4×C83—41no—
237C2×C4×C83—41no—
238C2×C4×C83—41no—
239C2×C4×C83—41no—
240C2×C4×C83—41no—
241C2×C4×C83—84no—
242C2×C4×C83C4×D412yes—
243C2×C4×C83C4×D412yes—
244C2×C4×C83#12412yes—
245C2×C4×C83#12412yes—
246C2×C4×C83C23×C411yescharacteristic
247C2×C4×C83C2×#4111yescharacteristic
248C2×C4×C83C2×#4111yescharacteristic
249C2×C4×C83C2×#4111yescharacteristic
250C2×C4×C83C2×#4111yescharacteristic
251C2×C4×C83C2×#4111yescharacteristic
252C2×Dic83—22no—
253C2×Dic83—22no—
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—
264Dic166—84no—
265C83⋊C8—22no—
266C83⋊C8—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—22no—
286C4×C83—22no—
287C4×C83—41no—
288C4×C83—41no—
289C4×C83—42no—
290C4×C83—81no—
291C4×C83—81no—
292C4×C83C4×#4111yescharacteristic
293C4×C83C4×#4111yescharacteristic
294C4×C83#52911yescharacteristic
295C4×C83#52911yescharacteristic
296Dic83—42no—
297Dic83—42no—
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×#489—831no—
310#2331—831no—
311#2326D8311yescharacteristic
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⋊Q8—831no—
326C2×#124C2×D8311yescharacteristic
327C2×C4×Q8C2×D8311yescharacteristic
328C2×C4×D4C2×D8311yescharacteristic
329#489—838no—
330C2×#103—831no—
331C2×#103—831no—
332C2×#103—831no—
333C2×#103—831no—
334C2×#102—831no—
335C2×#102—831no—
336C2×#102—831no—
337C2×#102—831no—
338C2×#97—831no—
339C2×#97—831no—
340C2×#97—831no—
341C2×C4×C8—831no—
342C2×C4×C8—831no—
343C2×C4×C8—831no—
344#391—1661no—
345#391—1661no—
346#391C2×D8311yescharacteristic
347#390C2×D8311yescharacteristic
348#389—831no—
349#389—831no—
350#388—21no—
351#388—21no—
352#387—1661no—
353#387—1661no—
354#385—21no—
355#385—21no—
356#384C2×D8311yescharacteristic
357#382—831no—
358C4×C4⋊C4—831no—
359C4×C4⋊C4—831no—
360#376C2×D8311yescharacteristic
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—1661no—
370C22×Q8C22×D8311yescharacteristic
371C22×D4C22×D8311yescharacteristic
372C23×C4—21no—
373C22×C8—831no—
374C22×C8—831no—
375C22×C8—831no—
376C22×C8—831no—
377C4⋊Q8—834no—
378C4⋊Q8—1662no—
379#124—22no—
380#124C4×D8314yes—
381C4×Q8—22no—
382C4×Q8—22no—
383C4×D4—22no—
384C4×D4—22no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—831no—
389C2×C4⋊C4—831no—
390C2×C4⋊C4—831no—
391C2×C4⋊C4—831no—
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—21no—
400C2×C22⋊C4—21no—
401C2×C22⋊C4—21no—
402C2×C42—21no—
403C2×C42—1661no—
404C2×C42—1661no—
405C2×C42C22×D8311yescharacteristic
406C2×C42C22×D8311yescharacteristic
407C2×C42C22×D8311yescharacteristic
408#103—834no—
409#103—834no—
410#103—1662no—
411#103—1662no—
412#102—834no—
413#102—834no—
414#102—1662no—
415#102—1662no—
416#97—834no—
417#97—1662no—
418#97—1662no—
419C4×C8—1662no—
420C4×C8—1662no—
421C4×C8—1662no—
422#95—21no—
423#95—21no—
424#95—1661no—
425#95—1661no—
426#95—1661no—
427#95—1661no—
428#95—1661no—
429#95—1661no—
430#95C22⋊D8311yescharacteristic
431#95C22⋊D8311yescharacteristic
432#95C22⋊D8311yescharacteristic
433#95C22⋊D8311yescharacteristic
434C24—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—6641no—
441C2×Q8—6641no—
442C2×Q8—1664no—
443C2×Q8C2×C4×D8312yes—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4C2×C4×D8312yes—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—42no—
454C22×C4—81no—
455C22×C4—81no—
456C22×C4—1661no—
457C22×C4—1661no—
458C22×C4—1661no—
459C22×C4—1661no—
460C22×C4C2×C22⋊D8311yescharacteristic
461C22×C4C2×C22⋊D8311yescharacteristic
462C22×C4C2×C22⋊D8311yescharacteristic
463C22×C4C2×C22⋊D8311yescharacteristic
464C22×C4C2×C22⋊D8311yescharacteristic
465C22×C4C2×C22⋊D8311yescharacteristic
466C22×C4C23×D8311yescharacteristic
467C2×C8—832no—
468C2×C8—832no—
469C2×C8—832no—
470C2×C8—832no—
471C2×C8—1661no—
472C2×C8—1661no—
473C2×C8—1661no—
474C2×C8—1661no—
475C2×C8—1661no—
476C2×C8—1661no—
477C2×C8—1661no—
478C2×C8—1661no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—42no—
484C4⋊C4—81no—
485C4⋊C4—81no—
486C4⋊C4—832no—
487C4⋊C4—832no—
488C4⋊C4—6641no—
489C4⋊C4—6641no—
490C4⋊C4—1661no—
491C4⋊C4—1661no—
492C4⋊C4—1661no—
493C4⋊C4—1661no—
494C4⋊C4—1661no—
495C4⋊C4—1661no—
496C4⋊C4—1664no—
497C22⋊C4—24no—
498C22⋊C4—24no—
499C22⋊C4—42no—
500C22⋊C4—81no—
501C22⋊C4—81no—
502C42—21no—
503C42—21no—
504C42—42no—
505C42—6641no—
506C42—6641no—
507C42C2×C4×D8312yes—
508C42#1305058912yes—
509C42#1305058912yes—
510C42#1305059012yes—
511C23—22no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23C22×C22⋊D8311yescharacteristic
519Q8—22no—
520Q8—22no—
521Q8—41no—
522Q8—41no—
523Q8—6644no—
524D4—22no—
525D4—22no—
526D4—41no—
527D4—41no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—84no—
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—3321no—
561C2×C4—3321no—
562C2×C4—1662no—
563C2×C4—1662no—
564C2×C4C4×C22⋊D8312yes—
565C2×C4C4×C22⋊D8312yes—
566C2×C4#1458177912yes—
567C2×C4#1458177912yes—
568C2×C4C22×C4×D8311yescharacteristic
569C2×C4#41×D8311yescharacteristic
570C2×C4#41×D8311yescharacteristic
571C2×C4C2×#1305058911yescharacteristic
572C2×C4C2×#1305058911yescharacteristic
573C2×C4C2×#1305059011yescharacteristic
574C8—1662no—
575C8—1662no—
576C8—1662no—
577C8—1662no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—81no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22C2×C4×C22⋊D8311yescharacteristic
586C22C2×#1458177911yescharacteristic
587C22#1912824711yescharacteristic
588C22#1912850411yescharacteristic
589C22#1912850911yescharacteristic
590C22#1912851211yescharacteristic
591C22#1912851611yescharacteristic
592C4—22no—
593C4—22no—
594C4—22no—
595C4—22no—
596C4—41no—
597C4—41no—
598C4—42no—
599C4—81no—
600C4—81no—
601C4—3322no—
602C4—3322no—
603C4—6641no—
604C4—6641no—
605C4C4×#1305058911yescharacteristic
606C4C4×#1305058911yescharacteristic
607C4#1912823311yescharacteristic
608C4#1912823311yescharacteristic
609C2—82no—
610C2C4×#1458177911yescharacteristic
611C2#2491629111yescharacteristic
612C2#2491630211yescharacteristic
613C2#2491781711yescharacteristic
614C2#2491782811yescharacteristic
615C2#2491807411yescharacteristic
616C2#2491809611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC83×#2326C2
socleC23×C83C22×D4
prime radicalC22×D4×C83C23
cosocleC22×C4×C83C24
hypercenter#2326D83
hypocenterC83#21654
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ #376 ◁ #2326

Complements

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

NH ≅ G/Ntype
2. #24910211C2semidirect
3. #24908423C2semidirect
6. #24908156C2semidirect
8. #24908142C2semidirect
9. #24907598C2semidirect
10. #24907293C2semidirect
12. C4×#14581716C2semidirect
14. #24907150C2semidirect
17. #19128060C4semidirect
66. #14581792C2×C4semidirect
78. #14581716C2×C4semidirect
79. #14581716C2×C4semidirect
86. #14581703C2×C4semidirect
324. C83#21654semidirect

Maximal subgroups

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

subgroupindex
C2×#191280602
#249102112
#249084232
#249082752
#249081632
#249081562
#249081492
#249081422
#249075982
#249072932
#249072882
C4×#145817162
C4×#145817132
#249071502
C83×#23262
#2165483