Finite Group Databasebeta

← #35375036

Subgroups — #35375036

Order 21248.

622 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24916612C211yesmaximal, characteristic, complemented
2#24916539C211yesmaximal, characteristic, complemented
3#24906377C211yesmaximal, characteristic, complemented
4#24906366C211yesmaximal, characteristic, complemented
5#24906354C211yesmaximal, characteristic, complemented
6#24906348C211yesmaximal, characteristic, complemented
7#24906287C211yesmaximal, characteristic, complemented
8SD16⋊(C83⋊C8)C211yesmaximal, characteristic, complemented
9SD16×C83⋊C8C211yesmaximal, characteristic, complemented
10D8×C83⋊C8C211yesmaximal, characteristic, complemented
11#24906230C211yesmaximal, characteristic, complemented
12#24905923C211yesmaximal, characteristic, complemented
13#24905914C211yesmaximal, characteristic, complemented
14M16×C83⋊C8C211yesmaximal, characteristic, complemented
15C83×#3171C211yesmaximal, characteristic
16#19127832C2211yescharacteristic
17Q8×C83⋊C8C2211yescharacteristic
18#19127779—21no—
19#19127779C2211yescharacteristic
20#19127779C2211yescharacteristic
21#19127779C2211yescharacteristic
22#19127773—21no—
23#19127773—21no—
24#19127773—21no—
25D4×C83⋊C8C2211yescharacteristic
26D4×C83⋊C8C2211yescharacteristic
27D4×C83⋊C8C2211yescharacteristic
28#19127762—21no—
29#19127758C2211yescharacteristic
30#19127754C2211yescharacteristic
31#19127728—21no—
32#19127725C2211yescharacteristic
33#19127668—21no—
34#19127668—21no—
35C2×C4×C83⋊C8C2211yescharacteristic
36#19127454C2211yescharacteristic
37#19127454C2211yescharacteristic
38#19127452C2211yescharacteristic
39#19127452C2211yescharacteristic
40Q8⋊(C83⋊C8)C2211yescharacteristic
41Q8⋊(C83⋊C8)C2211yescharacteristic
42D4⋊(C83⋊C8)C2211yescharacteristic
43D4⋊(C83⋊C8)C2211yescharacteristic
44D4⋊(C83⋊C8)C2211yescharacteristic
45D4⋊(C83⋊C8)C2211yescharacteristic
46D4⋊(C83⋊C8)C2211yescharacteristic
47D4⋊(C83⋊C8)C2211yescharacteristic
48#19127422C2211yescharacteristic
49#19127422C2211yescharacteristic
50C8×C83⋊C8C2211yescharacteristic
51C8×C83⋊C8C2211yescharacteristic
52#532×C83C2211yescharacteristic
53#529×C83C2211yescharacteristic
54#501×C83C411yescharacteristic
55#496×C83C411yescharacteristic
56#490×C83C411yescharacteristic
57#489×C83C411yescharacteristic
58#469×C83C411yescharacteristic
59#469×C83C411yescharacteristic
60#468×C83C411yescharacteristic
61#468×C83C411yescharacteristic
62C4×SD16×C83C2211yescharacteristic
63C4×SD16×C83C2211yescharacteristic
64C4×D8×C83C2211yescharacteristic
65C4×D8×C83C2211yescharacteristic
66C4×M16×C83C2211yescharacteristic
67C2×#13050574—21no—
68C2×#13050574—21no—
69C2×#13050574—21no—
70C2×#13050574—21no—
71C22×C83⋊C8—21no—
72C22×C83⋊C8—21no—
73C22×C83⋊C8—21no—
74C22×C83⋊C8—21no—
75C4⋊(C83⋊C8)—21no—
76C4⋊(C83⋊C8)—21no—
77C4⋊(C83⋊C8)—21no—
78C4⋊(C83⋊C8)D411yescharacteristic
79C4⋊(C83⋊C8)D411yescharacteristic
80C4⋊(C83⋊C8)C2311yescharacteristic
81C4⋊(C83⋊C8)C2311yescharacteristic
82C4⋊(C83⋊C8)C2311yescharacteristic
83C4⋊(C83⋊C8)C2311yescharacteristic
84C22⋊(C83⋊C8)—21no—
85C22⋊(C83⋊C8)—21no—
86C22⋊(C83⋊C8)—21no—
87C22⋊(C83⋊C8)—21no—
88C22⋊(C83⋊C8)—21no—
89C22⋊(C83⋊C8)—21no—
90C22⋊(C83⋊C8)—21no—
91C22⋊(C83⋊C8)D411yescharacteristic
92C22⋊(C83⋊C8)D411yescharacteristic
93#14581703—21no—
94#14581703—21no—
95C4×C83⋊C8—21no—
96C4×C83⋊C8—21no—
97C4×C83⋊C8C2311yescharacteristic
98C4×C83⋊C8C2311yescharacteristic
99C4×C83⋊C8C2311yescharacteristic
100C4×C83⋊C8C2311yescharacteristic
101C2×#41×C83—21no—
102C2×SD16×C83C2×C411yescharacteristic
103C2×SD16×C83C2×C411yescharacteristic
104C2×D8×C83C2×C411yescharacteristic
105C2×D8×C83C2×C411yescharacteristic
106C2×M16×C83C2×C411yescharacteristic
107C2×M16×C83C2×C411yescharacteristic
108C4⋊Q8×C83C2×C411yescharacteristic
109#127×C83C2×C411yescharacteristic
110#126×C83—21no—
111#125×C83C2×C411yescharacteristic
112#124×C83—21no—
113#124×C83C2×C411yescharacteristic
114#123×C83—21no—
115C22⋊Q8×C83C2×C411yescharacteristic
116#121×C83—21no—
117#121×C83C2×C411yescharacteristic
118#121×C83C2×C411yescharacteristic
119#121×C83C2×C411yescharacteristic
120C4×Q8×C83C2311yescharacteristic
121C4×D4×C83—21no—
122C4×D4×C83—21no—
123C4×D4×C83C2311yescharacteristic
124C4×D4×C83C2311yescharacteristic
125C4×D4×C83C2311yescharacteristic
126#117×C83—21no—
127C2×C42×C83C2311yescharacteristic
128#107×C83C2×C411yescharacteristic
129#107×C83C2×C411yescharacteristic
130#106×C83C2×C411yescharacteristic
131#106×C83C2×C411yescharacteristic
132#103×C83C2×C411yescharacteristic
133#103×C83C2×C411yescharacteristic
134#102×C83C2×C411yescharacteristic
135#102×C83C2×C411yescharacteristic
136#102×C83C2×C411yescharacteristic
137#102×C83C2×C411yescharacteristic
138#102×C83C2×C411yescharacteristic
139#102×C83C2×C411yescharacteristic
140#97×C83C2×C411yescharacteristic
141#97×C83C2×C411yescharacteristic
142C4×C8×C83C2311yescharacteristic
143C4×C8×C83C2311yescharacteristic
144#13050574—81no—
145#13050574—81no—
146#13050574—81no—
147#13050574—81no—
148C2×C83⋊C8—21no—
149C2×C83⋊C8—21no—
150C2×C83⋊C8—21no—
151C2×C83⋊C8—21no—
152C2×C83⋊C8—21no—
153C2×C83⋊C8—21no—
154C2×C83⋊C8—42no—
155C2×C83⋊C8—81no—
156C2×C83⋊C8—81no—
157C2×C83⋊C8—81no—
158C2×C83⋊C8C2×D411yescharacteristic
159C2×C83⋊C8C2×D411yescharacteristic
160C2×C83⋊C8C2×D411yescharacteristic
161C2×C83⋊C8C2×D411yescharacteristic
162#41×C83—81no—
163#41×C83—81no—
164C2×Q8×C83—21no—
165C2×Q8×C83C22×C411yescharacteristic
166C2×D4×C83—21no—
167C2×D4×C83—21no—
168C2×D4×C83—21no—
169C2×D4×C83—21no—
170C2×D4×C83—21no—
171C2×D4×C83C22×C411yescharacteristic
172C2×D4×C83C22×C411yescharacteristic
173C2×D4×C83C22×C411yescharacteristic
174C22×C4×C83—21no—
175C22×C4×C83—21no—
176C22×C4×C83—21no—
177C22×C4×C83C22×C411yescharacteristic
178C22×C4×C83C2×D411yescharacteristic
179C22×C4×C83#4111yescharacteristic
180SD16×C83—22no—
181SD16×C83—22no—
182D8×C83—22no—
183D8×C83—22no—
184M16×C83—22no—
185M16×C83—22no—
186C2×C8×C83C22×C411yescharacteristic
187C2×C8×C83C22×C411yescharacteristic
188C2×C8×C83C22×C411yescharacteristic
189C2×C8×C83C22×C411yescharacteristic
190C4⋊C4×C83—21no—
191C4⋊C4×C83—21no—
192C4⋊C4×C83—21no—
193C4⋊C4×C83—21no—
194C4⋊C4×C83—21no—
195C4⋊C4×C83C22×C411yescharacteristic
196C4⋊C4×C83C22×C411yescharacteristic
197C4⋊C4×C83C22×C411yescharacteristic
198C4⋊C4×C83C22×C411yescharacteristic
199C22⋊C4×C83—21no—
200C22⋊C4×C83—21no—
201C22⋊C4×C83—21no—
202C22⋊C4×C83—21no—
203C22⋊C4×C83—21no—
204C22⋊C4×C83—21no—
205C22⋊C4×C83—21no—
206C22⋊C4×C83—21no—
207C22⋊C4×C83—21no—
208C42×C83—21no—
209C42×C83C22×C411yescharacteristic
210C42×C83C2×D411yescharacteristic
211C42×C83#4111yescharacteristic
212C42×C83C2411yescharacteristic
213C23×C83—21no—
214C23×C83—21no—
215C23×C83—21no—
216C23×C83C4×D411yescharacteristic
217Q8×C83—21no—
218Q8×C83—21no—
219Q8×C83—81no—
220D4×C83—21no—
221D4×C83—21no—
222D4×C83—21no—
223D4×C83—21no—
224D4×C83—21no—
225D4×C83—21no—
226D4×C83—81no—
227D4×C83—81no—
228D4×C83—81no—
229D4×C83—81no—
230D4×C83—81no—
231C2×C4×C83—21no—
232C2×C4×C83—21no—
233C2×C4×C83—21no—
234C2×C4×C83—21no—
235C2×C4×C83—21no—
236C2×C4×C83—21no—
237C2×C4×C83—21no—
238C2×C4×C83—21no—
239C2×C4×C83—21no—
240C2×C4×C83—41no—
241C2×C4×C83—41no—
242C2×C4×C83—41no—
243C2×C4×C83—41no—
244C2×C4×C83—41no—
245C2×C4×C83—41no—
246C2×C4×C83—41no—
247C2×C4×C83C4×D411yescharacteristic
248C2×C4×C83C4×D411yescharacteristic
249C2×C4×C83C4×D411yescharacteristic
250C2×C4×C83C23×C411yescharacteristic
251C2×C4×C83C22×D411yescharacteristic
252C2×C4×C83C2×#4111yescharacteristic
253C8×C83—21no—
254C8×C83—21no—
255C8×C83#13112yes—
256C8×C83#13112yes—
257C83⋊C8—22no—
258C83⋊C8—21no—
259C83⋊C8—21no—
260C83⋊C8—21no—
261C83⋊C8—21no—
262C83⋊C8—41no—
263C83⋊C8—41no—
264C83⋊C8—41no—
265C83⋊C8—41no—
266C83⋊C8—41no—
267C22×C83—21no—
268C22×C83—21no—
269C22×C83—21no—
270C22×C83—41no—
271C22×C83—41no—
272C22×C83—41no—
273C22×C83—41no—
274C22×C83—41no—
275C22×C83—41no—
276C22×C83—41no—
277C22×C83—41no—
278C22×C83—41no—
279C22×C83C2×C4×D411yescharacteristic
280C4×C83—21no—
281C4×C83—21no—
282C4×C83—21no—
283C4×C83—21no—
284C4×C83—41no—
285C4×C83—41no—
286C4×C83—41no—
287C4×C83—41no—
288C4×C83—41no—
289C4×C83C2×#13111yescharacteristic
290C4×C83C2×#13111yescharacteristic
291C4×C83#57611yescharacteristic
292C4×C83#57711yescharacteristic
293#17968—831nomaximal, Sylow
294C2×C83—41no—
295C2×C83—81no—
296C2×C83—81no—
297C2×C83—81no—
298C2×C83#293111yescharacteristic
299C2×C83#294611yescharacteristic
300C2×C83#295611yescharacteristic
301#3171D8311yescharacteristic
302#2993—831no—
303#2981—831no—
304#1594—831no—
305#1592—831no—
306#1590—831no—
307#1589—831no—
308#1580—831no—
309SD16⋊C8—831no—
310C8×SD16—831no—
311C8×D8—831no—
312#1569—831no—
313#1494—831no—
314#1492—831no—
315C8×M16—831no—
316C83#1796811yescharacteristic, complemented, Sylow
317#532C2×D8311yescharacteristic
318#529C2×D8311yescharacteristic
319#501Dic8311yescharacteristic
320#496Dic8311yescharacteristic
321#490Dic8311yescharacteristic
322#489Dic8311yescharacteristic
323#469Dic8311yescharacteristic
324#469Dic8311yescharacteristic
325#468Dic8311yescharacteristic
326#468Dic8311yescharacteristic
327#446—831no—
328C8×Q8—831no—
329C4×SD16C2×D8311yescharacteristic
330C4×SD16C2×D8311yescharacteristic
331C4×D8C2×D8311yescharacteristic
332C4×D8C2×D8311yescharacteristic
333#436—831no—
334#436—831no—
335#436—831no—
336#436—1661no—
337#435—1661no—
338#435—1661no—
339#435—1661no—
340C8×D4—831no—
341C8×D4—831no—
342C8×D4—831no—
343#433—1661no—
344#432—831no—
345#431—831no—
346#424—1661no—
347#423—831no—
348#408—1661no—
349#408—1661no—
350C4×M16C2×D8311yescharacteristic
351C2×C4×C8—831no—
352#335—831no—
353#335—831no—
354#334—831no—
355#334—831no—
356Q8⋊C8—831no—
357Q8⋊C8—831no—
358D4⋊C8—831no—
359D4⋊C8—831no—
360D4⋊C8—831no—
361D4⋊C8—831no—
362D4⋊C8—831no—
363D4⋊C8—831no—
364#322—831no—
365#322—831no—
366C82—831no—
367C82—831no—
368C2×#41—21no—
369C2×SD16C2×Dic8311yescharacteristic
370C2×SD16C2×Dic8311yescharacteristic
371C2×D8C2×Dic8311yescharacteristic
372C2×D8C2×Dic8311yescharacteristic
373C2×M16—1661no—
374C2×M16—1661no—
375C2×M16—1661no—
376C2×M16—1661no—
377C2×M16C2×Dic8311yescharacteristic
378C2×M16C2×Dic8311yescharacteristic
379C22×C8—1661no—
380C22×C8—1661no—
381C22×C8—1661no—
382C22×C8—1661no—
383C4⋊Q8C2×Dic8311yescharacteristic
384#127C2×Dic8311yescharacteristic
385#126—21no—
386#125C2×Dic8311yescharacteristic
387#124—21no—
388#124C2×Dic8311yescharacteristic
389#123—21no—
390C22⋊Q8C2×Dic8311yescharacteristic
391#121—21no—
392#121C2×Dic8311yescharacteristic
393#121C2×Dic8311yescharacteristic
394#121C2×Dic8311yescharacteristic
395C4×Q8C22×D8311yescharacteristic
396C4×D4—21no—
397C4×D4—21no—
398C4×D4C22×D8311yescharacteristic
399C4×D4C22×D8311yescharacteristic
400C4×D4C22×D8311yescharacteristic
401#117—21no—
402C2×C42C22×D8311yescharacteristic
403#107C2×Dic8311yescharacteristic
404#107C2×Dic8311yescharacteristic
405#106C2×Dic8311yescharacteristic
406#106C2×Dic8311yescharacteristic
407C4⋊C8—831no—
408C4⋊C8—831no—
409C4⋊C8—831no—
410C4⋊C8—831no—
411C4⋊C8—831no—
412C4⋊C8—831no—
413C4⋊C8—1661no—
414C4⋊C8—1661no—
415C4⋊C8—1661no—
416#103C2×Dic8311yescharacteristic
417#103C2×Dic8311yescharacteristic
418#102C2×Dic8311yescharacteristic
419#102C2×Dic8311yescharacteristic
420#102C2×Dic8311yescharacteristic
421#102C2×Dic8311yescharacteristic
422#102C2×Dic8311yescharacteristic
423#102C2×Dic8311yescharacteristic
424C22⋊C8—831no—
425C22⋊C8—831no—
426C22⋊C8—1661no—
427C22⋊C8—1661no—
428C22⋊C8—1661no—
429C22⋊C8—1661no—
430C22⋊C8—1661no—
431C22⋊C8—1661no—
432C22⋊C8—1661no—
433#97—1661no—
434#97—1661no—
435#97C2×Dic8311yescharacteristic
436#97C2×Dic8311yescharacteristic
437C4×C8—831no—
438C4×C8—831no—
439C4×C8—831no—
440C4×C8—831no—
441C4×C8—1661no—
442C4×C8—1661no—
443C4×C8C22×D8311yescharacteristic
444C4×C8C22×D8311yescharacteristic
445#41—81no—
446#41—81no—
447C2×Q8—21no—
448C2×Q8C22×Dic8311yescharacteristic
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4C22×Dic8311yescharacteristic
455C2×D4C22×Dic8311yescharacteristic
456C2×D4C22×Dic8311yescharacteristic
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4C22×Dic8311yescharacteristic
461C22×C4D4×D8311yescharacteristic
462C22×C4#1305058911yescharacteristic
463SD16—22no—
464SD16—22no—
465D8—22no—
466D8—22no—
467M16—22no—
468M16—22no—
469M16—6641no—
470M16—6641no—
471M16—6641no—
472M16—6641no—
473C2×C8—3322no—
474C2×C8—831no—
475C2×C8—831no—
476C2×C8—831no—
477C2×C8—831no—
478C2×C8—6641no—
479C2×C8—6641no—
480C2×C8—6641no—
481C2×C8—1661no—
482C2×C8—1661no—
483C2×C8—1661no—
484C2×C8—1661no—
485C2×C8—1661no—
486C2×C8—1661no—
487C2×C8C22×Dic8311yescharacteristic
488C2×C8C22×Dic8311yescharacteristic
489C2×C8C22×Dic8311yescharacteristic
490C2×C8C22×Dic8311yescharacteristic
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4C22×Dic8311yescharacteristic
497C4⋊C4C22×Dic8311yescharacteristic
498C4⋊C4C22×Dic8311yescharacteristic
499C4⋊C4C22×Dic8311yescharacteristic
500C22⋊C4—21no—
501C22⋊C4—21no—
502C22⋊C4—21no—
503C22⋊C4—21no—
504C22⋊C4—21no—
505C22⋊C4—21no—
506C22⋊C4—21no—
507C22⋊C4—21no—
508C22⋊C4—21no—
509C42—21no—
510C42C22×Dic8311yescharacteristic
511C42D4×D8311yescharacteristic
512C42#1305058911yescharacteristic
513C42C23×D8311yescharacteristic
514C23—21no—
515C23—21no—
516C23—21no—
517C23D4×Dic8311yescharacteristic
518Q8—21no—
519Q8—21no—
520Q8—81no—
521D4—21no—
522D4—21no—
523D4—21no—
524D4—21no—
525D4—21no—
526D4—21no—
527D4—81no—
528D4—81no—
529D4—81no—
530D4—81no—
531D4—81no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4D4×Dic8311yescharacteristic
549C2×C4D4×Dic8311yescharacteristic
550C2×C4D4×Dic8311yescharacteristic
551C2×C4C23×Dic8311yescharacteristic
552C2×C4C2×D4×D8311yescharacteristic
553C2×C4C2×#1305058911yescharacteristic
554C8—21no—
555C8—21no—
556C8—3321no—
557C8—3321no—
558C8—3321no—
559C8—3321no—
560C8—3321no—
561C8—1662no—
562C8—1661no—
563C8—1661no—
564C8—1661no—
565C8—1661no—
566C8#1458179912yes—
567C8#1458179912yes—
568C22—21no—
569C22—21no—
570C22—21no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22C2×D4×Dic8311yescharacteristic
581C4—21no—
582C4—21no—
583C4—21no—
584C4—21no—
585C4—41no—
586C4—41no—
587C4—41no—
588C4—41no—
589C4—41no—
590C4C2×#1458179911yescharacteristic
591C4C2×#1458179911yescharacteristic
592C4#1912850511yescharacteristic
593C4#1912851111yescharacteristic
594C2—41no—
595C2—81no—
596C2—81no—
597C2—81no—
598C2#2491617511yescharacteristic
599C2#2491629611yescharacteristic
600C2#2491637311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC42C23×D83
Fitting, prime radicalC83×#3171C2
socleC22×C83C2×C4×D4
cosocleC42×C83C24
hypercenter#3171D83
hypocenterC83#17968
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3171

Complements

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

NH ≅ G/Ntype
1. #24916612C2semidirect
2. #24916539C2semidirect
3. #24906377C2semidirect
4. #24906366C2semidirect
5. #24906354C2semidirect
6. #24906348C2semidirect
7. #24906287C2semidirect
8. SD16⋊(C83⋊C8)C2semidirect
9. SD16×C83⋊C8C2semidirect
10. D8×C83⋊C8C2semidirect
11. #24906230C2semidirect
12. #24905923C2semidirect
13. #24905914C2semidirect
14. M16×C83⋊C8C2semidirect
316. C83#17968semidirect

Maximal subgroups

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

subgroupindex
#249166122
#249165392
#249063772
#249063662
#249063542
#249063482
#249062872
SD16⋊(C83⋊C8)2
SD16×C83⋊C82
D8×C83⋊C82
#249062302
#249059232
#249059142
M16×C83⋊C82
C83×#31712
#1796883