Finite Group Databasebeta

← #35374208

Subgroups — #35374208

Order 21248.

684 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24916579C211yesmaximal, characteristic, complemented
2#24916501C211yesmaximal, characteristic, complemented
3C4×#14581819C211yesmaximal, characteristic
4#24906376C211yesmaximal, characteristic, complemented
5#24906375C211yesmaximal, characteristic, complemented
6#24906366C211yesmaximal, characteristic, complemented
7#24906363C211yesmaximal, characteristic, complemented
8#24906350C211yesmaximal, characteristic, complemented
9#24906349C211yesmaximal, characteristic, complemented
10#24906345C211yesmaximal, characteristic, complemented
11#24906212C211yesmaximal, characteristic, complemented
12#24905844C211yesmaximal, characteristic, complemented
13D4⋊C8×D83C211yesmaximal, characteristic, complemented
14C8×#13050573C211yesmaximal, characteristic, complemented
15C83×#1590C211yesmaximal, characteristic, complemented
16C2×#14581819C411yescharacteristic
17C4×#13050591C2211yescharacteristic
18C4×D4×D83C2211yescharacteristic
19#19127832C2211yescharacteristic, complemented
20#19127812C411yescharacteristic
21#19127810C411yescharacteristic
22#19127804C411yescharacteristic
23#19127802C411yescharacteristic
24C4×#13050577C2211yescharacteristic
25C4×D664C2211yescharacteristic
26C4×#13050575C2211yescharacteristic
27#19127780C2211yescharacteristic
28#19127780C2211yescharacteristic, complemented
29#19127779C2211yescharacteristic
30#19127778C2211yescharacteristic
31#19127778C2211yescharacteristic, complemented
32#19127777—21no—
33#19127776—21no—
34#19127774—21no—
35#19127771—21no—
36#19127756—21no—
37#19127753C2211yescharacteristic
38#19127752C2211yescharacteristic
39#19127740C411yescharacteristic
40#19127723—21no—
41C4⋊C8×D83C2211yescharacteristic
42#19127704C411yescharacteristic
43#19127701C411yescharacteristic
44#19127664—21no—
45#19127663—21no—
46C4×#13050573C2211yescharacteristic
47C4×C8×D83C2211yescharacteristic
48#19127454C2211yescharacteristic, complemented
49#19127453C2211yescharacteristic, complemented
50#19127451C2211yescharacteristic, complemented
51Q8⋊(C83⋊C8)C2211yescharacteristic, complemented
52#19127430C2211yescharacteristic
53#19127430C2211yescharacteristic, complemented
54D4⋊(C83⋊C8)C2211yescharacteristic
55#19127428C2211yescharacteristic
56#19127428C2211yescharacteristic, complemented
57#19127423C2211yescharacteristic, complemented
58#19127422C2211yescharacteristic, complemented
59C8×C83⋊C8C2211yescharacteristic, complemented
60#446×C83C2211yescharacteristic, complemented
61C4×SD16×C83C2211yescharacteristic
62#436×C83C2211yescharacteristic
63#334×C83C2211yescharacteristic, complemented
64Q8⋊C8×C83C2211yescharacteristic, complemented
65D4⋊C8×C83C2211yescharacteristic
66C82×C83C2211yescharacteristic, complemented
67C2×#13050591C2×C411yescharacteristic
68C2×D4×D83C2×C411yescharacteristic
69C22×C4×D83—21no—
70#14581819—24no—
71C2×#13050577C2×C411yescharacteristic
72C2×D664C2×C411yescharacteristic
73C2×#13050575C2×C411yescharacteristic
74C2×#13050574—21no—
75C2×#13050573—21no—
76C2×#13050573—21no—
77C2×#13050573—21no—
78C2×#13050573—21no—
79C2×#13050573C2×C411yescharacteristic
80C2×#13050573C2×C411yescharacteristic
81C2×C8×D83—21no—
82C2×C8×D83—21no—
83Q8×Dic83C2×C411yescharacteristic
84C4×D332—21no—
85C4×D332C2311yescharacteristic
86C4×D332C2311yescharacteristic
87D4×Dic83C2×C411yescharacteristic
88#14581754—21no—
89#14581754C2×C411yescharacteristic
90#14581754C2×C411yescharacteristic
91C4×C22⋊D83—21no—
92(C22⋊D83)⋊C4—21no—
93#14581750—21no—
94#14581750C2×C411yescharacteristic
95C4⋊C4×D83C2×C411yescharacteristic
96C22⋊C4×D83—21no—
97C42×D83—21no—
98C42×D83C2311yescharacteristic
99#14581726C2×C411yescharacteristic
100#14581725C2×C411yescharacteristic
101#14581723C2×C411yescharacteristic
102C4⋊(C83⋊C8)—21no—
103C4⋊(C83⋊C8)C2311yescharacteristic
104C4⋊(C83⋊C8)C2311yescharacteristic
105Dic83⋊C8—21no—
106Dic83⋊C8D411yescharacteristic
107Dic83⋊C8D411yescharacteristic
108#14581716C2×C411yescharacteristic
109#14581714C2×C411yescharacteristic
110#14581714C2×C411yescharacteristic
111#14581713C2×C411yescharacteristic
112#14581712C2×C411yescharacteristic
113#14581712C2×C411yescharacteristic
114C22⋊(C83⋊C8)—21no—
115(C2×D83)⋊C8—21no—
116(C2×D83)⋊C8—21no—
117(C2×D83)⋊C8—21no—
118(C2×D83)⋊C8—21no—
119(C2×D83)⋊C8—21no—
120(C2×D83)⋊C8D411yescharacteristic
121(C2×D83)⋊C8D411yescharacteristic
122#14581703—21no—
123#14581702C2×C411yescharacteristic
124#14581702C2×C411yescharacteristic
125C4×C83⋊C8C2311yescharacteristic
126C4×C83⋊C8C2311yescharacteristic
127C4×C83⋊C8C2311yescharacteristic
128C8×Dic83—21no—
129C8×Dic83—21no—
130C2×SD16×C83C2×C411yescharacteristic
131C2×M16×C83—21no—
132C4×Q8×C83C2311yescharacteristic
133C4×D4×C83C2311yescharacteristic
134#106×C83C2×C411yescharacteristic
135C4⋊C8×C83—21no—
136C4⋊C8×C83C2311yescharacteristic
137C4⋊C8×C83C2311yescharacteristic
138#103×C83C2×C411yescharacteristic
139#102×C83C2×C411yescharacteristic
140C22⋊C8×C83—21no—
141#97×C83—21no—
142C4×C8×C83C2311yescharacteristic
143C4×C8×C83C2311yescharacteristic
144C4×C8×C83C2311yescharacteristic
145C23×D83—21no—
146#13050591—22no—
147#13050591—22no—
148C2×D332—21no—
149C2×D332C22×C411yescharacteristic
150C2×D332C22×C411yescharacteristic
151C2×C22⋊D83—21no—
152D4×D83—22no—
153D4×D83—22no—
154C22×Dic83—21no—
155C2×C4×D83—21no—
156C2×C4×D83—21no—
157C2×C4×D83—21no—
158C2×C4×D83—21no—
159C2×C4×D83—21no—
160C2×C4×D83—21no—
161C2×C4×D83—81no—
162C2×C4×D83—81no—
163C2×C4×D83C22×C411yescharacteristic
164C2×C4×D83C2×D411yescharacteristic
165C2×C4×D83#4111yescharacteristic
166#13050577—22no—
167D664—22no—
168#13050575—22no—
169#13050574—81no—
170#13050573—22no—
171#13050573—81no—
172#13050573—81no—
173#13050573—81no—
174#13050573—81no—
175#13050573M1614yescomplemented
176C2×C83⋊C8—21no—
177C2×C83⋊C8—21no—
178C2×C83⋊C8—21no—
179C2×C83⋊C8C22×C411yescharacteristic
180C2×C83⋊C8C22×C411yescharacteristic
181C2×C83⋊C8C2×D411yescharacteristic
182C2×C83⋊C8C2×D411yescharacteristic
183C8×D83—42no—
184C8×D83—81no—
185C4⋊Dic83—21no—
186C4⋊Dic83C22×C411yescharacteristic
187C4⋊Dic83C22×C411yescharacteristic
188Dic83⋊C4—21no—
189C22⋊Dic83—21no—
190(C2×D83)⋊C4—21no—
191(C2×D83)⋊C4—21no—
192(C2×D83)⋊C4—21no—
193(C2×D83)⋊C4—21no—
194(C2×D83)⋊C4—21no—
195C4×Dic83—21no—
196C4×Dic83—21no—
197C4×Dic83C22×C411yescharacteristic
198C4×Dic83C2×D411yescharacteristic
199C4×Dic83#4111yescharacteristic
200C2×Q8×C83C22×C411yescharacteristic
201C2×D4×C83C22×C411yescharacteristic
202C22×C4×C83—21no—
203SD16×C83—22no—
204M16×C83—81no—
205C2×C8×C83—21no—
206C2×C8×C83—21no—
207C2×C8×C83—21no—
208C2×C8×C83C22×C411yescharacteristic
209C2×C8×C83C22×C411yescharacteristic
210C2×C8×C83C2×D411yescharacteristic
211C2×C8×C83C2×D411yescharacteristic
212C4⋊C4×C83—21no—
213C4⋊C4×C83C22×C411yescharacteristic
214C4⋊C4×C83C22×C411yescharacteristic
215C22⋊C4×C83—21no—
216C42×C83—21no—
217C42×C83C2411yescharacteristic
218C23×C83—21no—
219C22×D83—21no—
220C22×D83—21no—
221C22×D83—42no—
222C22×D83—81no—
223C22×D83—81no—
224C22×D83C4×D411yescharacteristic
225Q8×C83—21no—
226Q8×C83—21no—
227D4×C83—21no—
228D4×C83—21no—
229C2×C4×C83—21no—
230C2×C4×C83—21no—
231C2×C4×C83—21no—
232C2×C4×C83—41no—
233C2×C4×C83—41no—
234C2×C4×C83C23×C411yescharacteristic
235C2×C4×C83C22×D411yescharacteristic
236C2×C4×C83C2×#4111yescharacteristic
237C22⋊D83—42no—
238C2×Dic83—21no—
239C2×Dic83—21no—
240C2×Dic83—81no—
241C2×Dic83C4×D411yescharacteristic
242C2×Dic83C4×D411yescharacteristic
243C2×Dic83C4×D411yescharacteristic
244D332—21no—
245D332—21no—
246D332—21no—
247D332—21no—
248D332—42no—
249C4×D83—21no—
250C4×D83—21no—
251C4×D83—41no—
252C4×D83—41no—
253C4×D83—41no—
254C4×D83—41no—
255C4×D83—41no—
256C4×D83—42no—
257C4×D83—81no—
258C4×D83—81no—
259C4×D83—81no—
260C4×D83C2×M1612yes—
261C4×D83#13112yes—
262C8×C83—22no—
263C8×C83—21no—
264C8×C83—21no—
265C8×C83—21no—
266C8×C83—41no—
267C8×C83—41no—
268C8×C83C2×M1612yes—
269C83⋊C8—21no—
270C83⋊C8—21no—
271C83⋊C8—21no—
272C83⋊C8—41no—
273C83⋊C8—41no—
274C83⋊C8—41no—
275C83⋊C8C2×M1612yes—
276C22×C83—41no—
277C22×C83—41no—
278C22×C83—41no—
279C22×C83C2×C4×D411yescharacteristic
280C2×D83—21no—
281C2×D83—21no—
282C2×D83—41no—
283C2×D83—41no—
284C2×D83—41no—
285C2×D83—41no—
286C2×D83—41no—
287C2×D83—41no—
288C2×D83—84no—
289C2×D83#43612yes—
290C4×C83—21no—
291C4×C83—41no—
292C4×C83—41no—
293C4×C83—41no—
294C4×C83C22×M1611yescharacteristic
295C4×C83C2×#13111yescharacteristic
296C4×C83#57611yescharacteristic
297C4×C83#57711yescharacteristic
298Dic83—21no—
299Dic83—21no—
300Dic83—41no—
301Dic83—41no—
302Dic83#43612yes—
303#17893—831nomaximal, Sylow
304C2×C83—81no—
305C2×C83C2×#43611yescharacteristic
306C2×C83#294611yescharacteristic
307C2×C83#295611yescharacteristic
308D83—22no—
309D83—81no—
310D83—81no—
311#2989—831no—
312#2975—831no—
313C4×AGL(1,ℤ/8)—831no—
314#1594—831no—
315#1594—831no—
316#1592—831no—
317#1592—831no—
318#1590—831no—
319#1590D8311yescharacteristic, complemented
320#1589—831no—
321#1589—831no—
322#1565—831no—
323#1476—831no—
324C2×D4⋊C8—831no—
325C8×M16—831no—
326C83#1789311yescharacteristic, complemented, Sylow
327C2×AGL(1,ℤ/8)—831no—
328C4×#41—831no—
329C2×C4×D4—831no—
330#446—831no—
331#446C2×D8311yescharacteristic, complemented
332#442—831no—
333#442—831no—
334#440—831no—
335#440—831no—
336C4×SD16—831no—
337C4×SD16C2×D8311yescharacteristic
338C4×D8—831no—
339C4×D8—831no—
340#436—831no—
341#436—831no—
342#436—831no—
343#436—831no—
344#436—831no—
345#436—1661no—
346#436—1661no—
347#436C2×D8311yescharacteristic
348#435—1661no—
349#435—1661no—
350#432—1661no—
351#431—831no—
352#431—831no—
353#428—831no—
354#423—1661no—
355C2×C4⋊C8—831no—
356#418—831no—
357#417—831no—
358#407—1661no—
359#407—1661no—
360C4×M16—831no—
361C2×C4×C8—831no—
362#335—831no—
363#335—831no—
364#334—831no—
365#334C2×D8311yescharacteristic, complemented
366Q8⋊C8—831no—
367Q8⋊C8C2×D8311yescharacteristic, complemented
368D4⋊C8—831no—
369D4⋊C8—831no—
370D4⋊C8—831no—
371D4⋊C8—831no—
372D4⋊C8—831no—
373D4⋊C8C2×D8311yescharacteristic
374#322—831no—
375#322—831no—
376C82—831no—
377C82C2×D8311yescharacteristic, complemented
378C2×#41—831no—
379C22×D4—831no—
380C23×C4—1661no—
381AGL(1,ℤ/8)—1664no—
382C2×SD16—831no—
383C2×SD16C4×D8311yescharacteristic
384C2×D8—831no—
385C2×D8—831no—
386C2×M16—21no—
387C2×M16—831no—
388C2×M16—831no—
389C2×M16—1661no—
390C2×M16—1661no—
391C2×M16—1661no—
392C2×M16—1661no—
393C2×M16—1661no—
394C22×C8—1661no—
395C22×C8—1661no—
396C4×Q8—831no—
397C4×Q8C22×D8311yescharacteristic
398C4×D4—831no—
399C4×D4—831no—
400C4×D4—831no—
401C4×D4—831no—
402C4×D4—831no—
403C4×D4—1661no—
404C4×D4—1661no—
405C4×D4—1661no—
406C4×D4—1661no—
407C4×D4C22×D8311yescharacteristic
408#117—831no—
409#117—1661no—
410C2×C4⋊C4—831no—
411C2×C22⋊C4—1661no—
412C2×C42—831no—
413C2×C42—1661no—
414#107—831no—
415#107—831no—
416#106—831no—
417#106C4×D8311yescharacteristic
418C4⋊C8—21no—
419C4⋊C8—831no—
420C4⋊C8—831no—
421C4⋊C8—831no—
422C4⋊C8—831no—
423C4⋊C8—1661no—
424C4⋊C8—1661no—
425C4⋊C8C22×D8311yescharacteristic
426C4⋊C8C22×D8311yescharacteristic
427#103—831no—
428#103C4×D8311yescharacteristic
429#102—831no—
430#102—831no—
431#102—831no—
432#102—831no—
433#102—831no—
434#102C4×D8311yescharacteristic
435C22⋊C8—21no—
436C22⋊C8—831no—
437C22⋊C8—831no—
438C22⋊C8—1661no—
439C22⋊C8—1661no—
440C22⋊C8—1661no—
441C22⋊C8—1661no—
442C22⋊C8—1661no—
443C22⋊C8—1661no—
444#97—21no—
445#97—831no—
446#97—831no—
447#97—1661no—
448C4×C8—831no—
449C4×C8—831no—
450C4×C8—831no—
451C4×C8—1661no—
452C4×C8—1661no—
453C4×C8C22×D8311yescharacteristic
454C4×C8C22×D8311yescharacteristic
455C4×C8C22×D8311yescharacteristic
456C24—1661no—
457#41—1662no—
458#41—1662no—
459C2×Q8C2×C4×D8311yescharacteristic
460C2×D4—831no—
461C2×D4—831no—
462C2×D4—1661no—
463C2×D4—1662no—
464C2×D4—1662no—
465C2×D4—1661no—
466C2×D4C2×C4×D8311yescharacteristic
467C22×C4—21no—
468C22×C4—831no—
469C22×C4—831no—
470C22×C4—831no—
471C22×C4—6641no—
472C22×C4—6641no—
473C22×C4—1661no—
474C22×C4—1661no—
475C22×C4—1661no—
476C22×C4—1661no—
477C22×C4—1661no—
478C22×C4—1661no—
479C22×C4—1661no—
480SD16—22no—
481SD16—1662no—
482D8—1662no—
483D8—1662no—
484M16—81no—
485M16—834no—
486M16—6641no—
487M16—6641no—
488M16—6641no—
489M16—6641no—
490M16—6641no—
491M16—1662no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—3322no—
496C2×C8—831no—
497C2×C8—831no—
498C2×C8—831no—
499C2×C8—831no—
500C2×C8—6641no—
501C2×C8—1661no—
502C2×C8—1661no—
503C2×C8—1661no—
504C2×C8C2×C4×D8311yescharacteristic
505C2×C8C2×C4×D8311yescharacteristic
506C2×C8D4×D8311yescharacteristic
507C2×C8D4×D8311yescharacteristic
508C4⋊C4—21no—
509C4⋊C4—831no—
510C4⋊C4—831no—
511C4⋊C4—1661no—
512C4⋊C4—1661no—
513C4⋊C4C2×C4×D8311yescharacteristic
514C4⋊C4C2×C4×D8311yescharacteristic
515C22⋊C4—21no—
516C22⋊C4—1661no—
517C22⋊C4—1661no—
518C22⋊C4—1661no—
519C22⋊C4—1661no—
520C22⋊C4—1661no—
521C22⋊C4—1661no—
522C42—21no—
523C42—831no—
524C42—831no—
525C42—831no—
526C42—1661no—
527C42—1661no—
528C42C23×D8311yescharacteristic
529C23—21no—
530C23—3322no—
531C23—831no—
532C23—6641no—
533C23—6641no—
534C23—1661no—
535C23—1661no—
536Q8—21no—
537Q8—21no—
538D4—21no—
539D4—21no—
540D4—3322no—
541D4—3322no—
542D4—1661no—
543D4—1661no—
544D4—1661no—
545D4—1661no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—3321no—
552C2×C4—3322no—
553C2×C4—3321no—
554C2×C4—3321no—
555C2×C4—3321no—
556C2×C4—3321no—
557C2×C4—831no—
558C2×C4—832no—
559C2×C4—832no—
560C2×C4—831no—
561C2×C4—831no—
562C2×C4—6641no—
563C2×C4—6641no—
564C2×C4—6641no—
565C2×C4—6641no—
566C2×C4—1661no—
567C2×C4—1661no—
568C2×C4—1661no—
569C2×C4—1661no—
570C2×C4C22×C4×D8311yescharacteristic
571C2×C4C2×D4×D8311yescharacteristic
572C2×C4#41×D8311yescharacteristic
573C8—22no—
574C8—21no—
575C8—21no—
576C8—21no—
577C8—41no—
578C8—41no—
579C8—3321no—
580C8—3321no—
581C8—3321no—
582C8—832no—
583C8—1661no—
584C8—1661no—
585C8—1661no—
586C8M16×D8312yes—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—3321no—
591C22—3321no—
592C22—3321no—
593C22—3321no—
594C22—3321no—
595C22—3321no—
596C22—832no—
597C22—6644no—
598C22—1661no—
599C22—1661no—
600C22C4×D4×D8311yescharacteristic
601C4—21no—
602C4—41no—
603C4—41no—
604C4—41no—
605C4—3321no—
606C4—3321no—
607C4—832no—
608C4—1661no—
609C4—1661no—
610C4C2×M16×D8311yescharacteristic
611C4#131×D8311yescharacteristic
612C4#1912850311yescharacteristic
613C4#1912850711yescharacteristic
614C2—81no—
615C2—6641no—
616C2—6641no—
617C2—1662no—
618C2#436×D8311yescharacteristic
619C2#2491629011yescharacteristic
620C2#2491636811yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC42C23×D83
FittingC83×#1590C2
socleC22×C83C2×C4×D4
prime radicalC2×#14581819C4
cosocleC42×C83C24
hypercenter#1590D83
hypocenterC83#17893
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1590

Complements

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

NH ≅ G/Ntype
1. #24916579C2semidirect
2. #24916501C2semidirect
4. #24906376C2semidirect
5. #24906375C2semidirect
6. #24906366C2semidirect
7. #24906363C2semidirect
8. #24906350C2semidirect
9. #24906349C2semidirect
10. #24906345C2semidirect
11. #24906212C2semidirect
12. #24905844C2semidirect
13. D4⋊C8×D83C2semidirect
14. C8×#13050573C2semidirect
15. C83×#1590C2semidirect
19. #19127832C22semidirect
28. #19127780C22semidirect
31. #19127778C22semidirect
48. #19127454C22semidirect
49. #19127453C22semidirect
50. #19127451C22semidirect
51. Q8⋊(C83⋊C8)C22semidirect
53. #19127430C22semidirect
56. #19127428C22semidirect
57. #19127423C22semidirect
58. #19127422C22semidirect
59. C8×C83⋊C8C22semidirect
60. #446×C83C22semidirect
63. #334×C83C22semidirect
64. Q8⋊C8×C83C22semidirect
66. C82×C83C22semidirect
175. #13050573M16semidirect
319. #1590D83semidirect
326. C83#17893semidirect
331. #446C2×D83semidirect
365. #334C2×D83semidirect
367. Q8⋊C8C2×D83semidirect
377. C82C2×D83semidirect

Maximal subgroups

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

subgroupindex
#249165792
#249165012
C4×#145818192
#249063762
#249063752
#249063662
#249063632
#249063502
#249063492
#249063452
#249062122
#249058442
D4⋊C8×D832
C8×#130505732
C83×#15902
#1789383