Finite Group Databasebeta

← #35375629

Subgroups — #35375629

Order 21248.

642 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24917508C211yesmaximal, characteristic
2#24916618C211yesmaximal, characteristic, complemented
3#24916496C211yesmaximal, characteristic, complemented
4#24906374C211yesmaximal, characteristic, complemented
5#24906367C211yesmaximal, characteristic, complemented
6#24906305C211yesmaximal, characteristic, complemented
7#24906299C211yesmaximal, characteristic, complemented
8#24906293C211yesmaximal, characteristic, complemented
9#24906292C211yesmaximal, characteristic, complemented
10#24906272C211yesmaximal, characteristic, complemented
11#24906245C211yesmaximal, characteristic, complemented
12#24905920C211yesmaximal, characteristic, complemented
13D4⋊C8×D83C211yesmaximal, characteristic, complemented
14#24905754C211yesmaximal, characteristic, complemented
15C83×#1580C211yesmaximal, characteristic, complemented
16#19128233C2211yescharacteristic
17C4×D4×D83C2211yescharacteristic
18#19128038C411yescharacteristic
19#19128018C411yescharacteristic
20#19128004C411yescharacteristic
21#19128003C411yescharacteristic
22#19127915C411yescharacteristic
23#19127914C411yescharacteristic
24#19127853C411yescharacteristic
25#19127833C411yescharacteristic
26#19127831C2211yescharacteristic, complemented
27#19127825C2211yescharacteristic, complemented
28C4×#13050579C2211yescharacteristic
29C4×#13050578C2211yescharacteristic
30C4×#13050575C2211yescharacteristic
31#19127780C2211yescharacteristic
32#19127779C2211yescharacteristic, complemented
33#19127778C2211yescharacteristic
34#19127775—21no—
35#19127772—21no—
36#19127769—21no—
37D4×C83⋊C8C2211yescharacteristic
38C8×C22⋊D83—21no—
39#19127759—21no—
40#19127757C2211yescharacteristic
41#19127752C2211yescharacteristic
42#19127752C2211yescharacteristic
43#19127727—21no—
44C4⋊C8×D83C2211yescharacteristic
45#19127663—21no—
46C2×(C2×D83)⋊C8—21no—
47C4×C8×D83C2211yescharacteristic
48#19127454C2211yescharacteristic, complemented
49#19127453C2211yescharacteristic, complemented
50#19127451C2211yescharacteristic, complemented
51#19127433C2211yescharacteristic, complemented
52#19127431C2211yescharacteristic, complemented
53#19127430C2211yescharacteristic
54D4⋊(C83⋊C8)C2211yescharacteristic
55D4⋊(C83⋊C8)C2211yescharacteristic, complemented
56#19127428C2211yescharacteristic
57#19127423C2211yescharacteristic, complemented
58#19127423C2211yescharacteristic, complemented
59#19127422C2211yescharacteristic, complemented
60C4×D8×C83C2211yescharacteristic
61C8×D4×C83C2211yescharacteristic
62C8×D4×C83C2211yescharacteristic, complemented
63#334×C83C2211yescharacteristic, complemented
64D4⋊C8×C83C2211yescharacteristic
65D4⋊C8×C83C2211yescharacteristic, complemented
66#322×C83C2211yescharacteristic, complemented
67C2×D4×D83C2×C411yescharacteristic
68C22×C4×D83—21no—
69C2×#13050579C2×C411yescharacteristic
70C2×#13050578C2×C411yescharacteristic
71C2×#13050575C2×C411yescharacteristic
72C2×#13050574—21no—
73C2×#13050573—21no—
74C2×#13050573—21no—
75C22×C83⋊C8—21no—
76C2×C8×D83—21no—
77C2×C8×D83—21no—
78#14581785—21no—
79#14581783C2×C411yescharacteristic
80#14581779C2×C411yescharacteristic
81#14581774—21no—
82#14581772C2×C411yescharacteristic
83#14581763C2×C411yescharacteristic
84C4×Dic166C2311yescharacteristic
85C4×D332C2311yescharacteristic
86D4×Dic83C2×C411yescharacteristic
87#14581754C2×C411yescharacteristic
88C4×C22⋊D83—21no—
89C4×C22⋊D83—21no—
90(C22⋊D83)⋊C4—21no—
91#14581751—21no—
92C4⋊C4×D83C2×C411yescharacteristic
93C22⋊C4×D83—21no—
94C42×D83C2311yescharacteristic
95#14581725C2×C411yescharacteristic
96#14581724C2×C411yescharacteristic
97#14581723C2×C411yescharacteristic
98C4⋊(C83⋊C8)C2311yescharacteristic
99C4⋊(C83⋊C8)C2311yescharacteristic
100Dic83⋊C8—21no—
101Dic83⋊C8—21no—
102Dic83⋊C8—21no—
103Dic83⋊C8C2×C411yescharacteristic
104Dic83⋊C8C2×C411yescharacteristic
105Dic83⋊C8D411yescharacteristic
106Dic83⋊C8D411yescharacteristic
107#14581717C2×C411yescharacteristic
108#14581715C2×C411yescharacteristic
109#14581714C2×C411yescharacteristic
110#14581713C2×C411yescharacteristic
111#14581713C2×C411yescharacteristic
112#14581712C2×C411yescharacteristic
113C22⋊(C83⋊C8)—21no—
114C22⋊(C83⋊C8)—21no—
115(C2×D83)⋊C8—21no—
116(C2×D83)⋊C8—21no—
117(C2×D83)⋊C8—21no—
118(C2×D83)⋊C8C2×C411yescharacteristic
119(C2×D83)⋊C8C2×C411yescharacteristic
120(C2×D83)⋊C8D411yescharacteristic
121(C2×D83)⋊C8D411yescharacteristic
122#14581702—21no—
123#14581702—21no—
124C4×C83⋊C8C2311yescharacteristic
125C4×C83⋊C8C2311yescharacteristic
126C4×C83⋊C8C2311yescharacteristic
127C8×Dic83—21no—
128C8×Dic83—21no—
129C2×D8×C83C2×C411yescharacteristic
130C22×C8×C83—21no—
131C22×C8×C83—21no—
132C4×D4×C83C2311yescharacteristic
133C4×D4×C83C2311yescharacteristic
134#107×C83C2×C411yescharacteristic
135C4⋊C8×C83C2311yescharacteristic
136C4⋊C8×C83C2311yescharacteristic
137#102×C83C2×C411yescharacteristic
138#102×C83C2×C411yescharacteristic
139C22⋊C8×C83—21no—
140C22⋊C8×C83—21no—
141C4×C8×C83C2311yescharacteristic
142C4×C8×C83C2311yescharacteristic
143C4×C8×C83C2311yescharacteristic
144C23×D83—21no—
145C2×Dic166C22×C411yescharacteristic
146C2×D332C22×C411yescharacteristic
147C2×C22⋊D83—21no—
148C2×C22⋊D83—21no—
149D4×D83—22no—
150D4×D83—41no—
151C22×Dic83—21no—
152C2×C4×D83—21no—
153C2×C4×D83—21no—
154C2×C4×D83—21no—
155C2×C4×D83—81no—
156C2×C4×D83—81no—
157C2×C4×D83C22×C411yescharacteristic
158C2×C4×D83C2×D411yescharacteristic
159C2×C4×D83#4111yescharacteristic
160#13050579—41no—
161#13050578—22no—
162#13050575—41no—
163#13050574—81no—
164#13050573—81no—
165#13050573—81no—
166C2×C83⋊C8—21no—
167C2×C83⋊C8—21no—
168C2×C83⋊C8—21no—
169C2×C83⋊C8—81no—
170C2×C83⋊C8C22×C411yescharacteristic
171C2×C83⋊C8C22×C411yescharacteristic
172C2×C83⋊C8C2×D411yescharacteristic
173C2×C83⋊C8C2×D411yescharacteristic
174C8×D83—42no—
175C8×D83—81no—
176C4⋊Dic83—21no—
177C4⋊Dic83—21no—
178C4⋊Dic83C22×C411yescharacteristic
179C4⋊Dic83C22×C411yescharacteristic
180Dic83⋊C4—21no—
181Dic83⋊C4—21no—
182Dic83⋊C4—21no—
183C22⋊Dic83—21no—
184C22⋊Dic83—21no—
185C22⋊Dic83—21no—
186C22⋊Dic83—21no—
187(C2×D83)⋊C4—21no—
188(C2×D83)⋊C4—21no—
189(C2×D83)⋊C4—21no—
190C4×Dic83—21no—
191C4×Dic83C22×C411yescharacteristic
192C4×Dic83C2×D411yescharacteristic
193C4×Dic83#4111yescharacteristic
194C2×D4×C83C22×C411yescharacteristic
195C2×D4×C83C22×C411yescharacteristic
196C22×C4×C83—21no—
197C22×C4×C83—21no—
198D8×C83—22no—
199C2×C8×C83—21no—
200C2×C8×C83—21no—
201C2×C8×C83—21no—
202C2×C8×C83—81no—
203C2×C8×C83—81no—
204C2×C8×C83C22×C411yescharacteristic
205C2×C8×C83C22×C411yescharacteristic
206C2×C8×C83C2×D411yescharacteristic
207C2×C8×C83C2×D411yescharacteristic
208C4⋊C4×C83C22×C411yescharacteristic
209C4⋊C4×C83C22×C411yescharacteristic
210C22⋊C4×C83—21no—
211C22⋊C4×C83—21no—
212C42×C83C2411yescharacteristic
213C23×C83—21no—
214C23×C83—21no—
215C22×D83—21no—
216C22×D83—42no—
217C22×D83—42no—
218C22×D83—81no—
219C22×D83C4×D411yescharacteristic
220D4×C83—21no—
221D4×C83—21no—
222D4×C83—21no—
223D4×C83#13112yes—
224C2×C4×C83—21no—
225C2×C4×C83—21no—
226C2×C4×C83—41no—
227C2×C4×C83—41no—
228C2×C4×C83—41no—
229C2×C4×C83—41no—
230C2×C4×C83C23×C411yescharacteristic
231C2×C4×C83C22×D411yescharacteristic
232C2×C4×C83C2×#4111yescharacteristic
233C22⋊D83—42no—
234C22⋊D83—42no—
235C2×Dic83—21no—
236C2×Dic83—21no—
237C2×Dic83—21no—
238C2×Dic83—81no—
239C2×Dic83C4×D411yescharacteristic
240C2×Dic83C4×D411yescharacteristic
241C2×Dic83C4×D411yescharacteristic
242D332—21no—
243D332—21no—
244C4×D83—22no—
245C4×D83—21no—
246C4×D83—21no—
247C4×D83—21no—
248C4×D83—21no—
249C4×D83—41no—
250C4×D83—41no—
251C4×D83—81no—
252C4×D83—81no—
253Dic166—21no—
254Dic166—21no—
255C8×C83—21no—
256C8×C83—21no—
257C8×C83—21no—
258C8×C83—21no—
259C8×C83—41no—
260C8×C83—41no—
261C8×C83—41no—
262C83⋊C8—21no—
263C83⋊C8—21no—
264C83⋊C8—21no—
265C83⋊C8—21no—
266C83⋊C8—41no—
267C83⋊C8—41no—
268C83⋊C8—41no—
269C22×C83—22no—
270C22×C83—41no—
271C22×C83—41no—
272C22×C83—41no—
273C22×C83—41no—
274C22×C83—41no—
275C22×C83C2×C4×D411yescharacteristic
276C2×D83—21no—
277C2×D83—21no—
278C2×D83—41no—
279C2×D83—41no—
280C2×D83—41no—
281C2×D83—84no—
282C2×D83#44412yes—
283C4×C83—21no—
284C4×C83—41no—
285C4×C83—41no—
286C4×C83C2×#13111yescharacteristic
287C4×C83#56811yescharacteristic
288C4×C83#57611yescharacteristic
289C4×C83#57711yescharacteristic
290Dic83—21no—
291Dic83—21no—
292Dic83—21no—
293Dic83—41no—
294Dic83—41no—
295Dic83—41no—
296#18020—831nomaximal, Sylow
297C2×C83—42no—
298C2×C83—81no—
299C2×C83#293311yescharacteristic
300C2×C83#294611yescharacteristic
301C2×C83C2×#44411yescharacteristic
302D83—22no—
303D83—81no—
304#3105—831no—
305#2994—831no—
306#2974—831no—
307#1594—831no—
308#1592—831no—
309#1583—831no—
310#1582—831no—
311#1581—831no—
312#1581—831no—
313#1580D8311yescharacteristic, complemented
314SD16⋊C8—831no—
315#1573—831no—
316#1494—831no—
317C2×D4⋊C8—831no—
318#1452—831no—
319C83#1802011yescharacteristic, complemented, Sylow
320#529—831no—
321C2×C4×D4—831no—
322#484—831no—
323#481—831no—
324#478—831no—
325#478—831no—
326#463—831no—
327#463—831no—
328#450—831no—
329#447—831no—
330#446—831no—
331C8×Q8—831no—
332C4×SD16—831no—
333C4×SD16—831no—
334C4×D8—831no—
335C4×D8C2×D8311yescharacteristic
336#436—831no—
337#436—831no—
338#436—831no—
339#435—1661no—
340#435—1661no—
341#435—1661no—
342C8×D4—831no—
343C8×D4—1661no—
344C8×D4C2×D8311yescharacteristic
345C8×D4C2×D8311yescharacteristic, complemented
346#433—1661no—
347#432—831no—
348#431—831no—
349#431—831no—
350#424—1661no—
351C2×C4⋊C8—831no—
352#407—1661no—
353C2×C22⋊C8—1661no—
354C2×C4×C8—831no—
355#335—831no—
356#335—831no—
357#334—831no—
358#334C2×D8311yescharacteristic, complemented
359Q8⋊C8—831no—
360Q8⋊C8—831no—
361D4⋊C8—831no—
362D4⋊C8—831no—
363D4⋊C8—831no—
364D4⋊C8—831no—
365D4⋊C8C2×D8311yescharacteristic
366D4⋊C8C2×D8311yescharacteristic, complemented
367#322—831no—
368#322—831no—
369#322—831no—
370#322C2×D8311yescharacteristic, complemented
371C22×D4—831no—
372C23×C4—1661no—
373C2×SD16—831no—
374C2×SD16—831no—
375C2×D8—831no—
376C2×D8C4×D8311yescharacteristic
377C2×M16—1661no—
378C2×M16—1661no—
379C2×M16—1661no—
380C22×C8—21no—
381C22×C8—21no—
382C22×C8—1661no—
383C22×C8—1661no—
384C22×C8—1661no—
385#126—1661no—
386#125—831no—
387#124—831no—
388#123—1661no—
389C22⋊Q8—831no—
390#121—831no—
391C4×Q8—831no—
392C4×D4—831no—
393C4×D4—831no—
394C4×D4—831no—
395C4×D4—1661no—
396C4×D4—1661no—
397C4×D4—1661no—
398C4×D4C22×D8311yescharacteristic
399C4×D4C22×D8311yescharacteristic
400#117—1661no—
401C2×C4⋊C4—831no—
402C2×C22⋊C4—1661no—
403C2×C42—831no—
404#107—831no—
405#107C4×D8311yescharacteristic
406#106—831no—
407#106—831no—
408C4⋊C8—831no—
409C4⋊C8—831no—
410C4⋊C8—831no—
411C4⋊C8—831no—
412C4⋊C8—831no—
413C4⋊C8—831no—
414C4⋊C8—1661no—
415C4⋊C8—1661no—
416C4⋊C8—1661no—
417C4⋊C8C22×D8311yescharacteristic
418C4⋊C8C22×D8311yescharacteristic
419#103—831no—
420#103—831no—
421#102—831no—
422#102—831no—
423#102—831no—
424#102—831no—
425#102C4×D8311yescharacteristic
426#102C4×D8311yescharacteristic
427C22⋊C8—21no—
428C22⋊C8—21no—
429C22⋊C8—831no—
430C22⋊C8—831no—
431C22⋊C8—831no—
432C22⋊C8—831no—
433C22⋊C8—1661no—
434C22⋊C8—1661no—
435C22⋊C8—1661no—
436C22⋊C8—1661no—
437C22⋊C8—1661no—
438#97—1661no—
439#97—1661no—
440C4×C8—831no—
441C4×C8—831no—
442C4×C8—831no—
443C4×C8—1661no—
444C4×C8—1661no—
445C4×C8C22×D8311yescharacteristic
446C4×C8C22×D8311yescharacteristic
447C4×C8C22×D8311yescharacteristic
448C24—1661no—
449C2×Q8—831no—
450C2×D4—3321no—
451C2×D4—831no—
452C2×D4—1661no—
453C2×D4—1661no—
454C2×D4—1662no—
455C2×D4C2×C4×D8311yescharacteristic
456C2×D4C2×C4×D8311yescharacteristic
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—831no—
460C22×C4—831no—
461C22×C4—831no—
462C22×C4—6641no—
463C22×C4—6641no—
464C22×C4—1661no—
465C22×C4—1661no—
466C22×C4—1661no—
467C22×C4—1661no—
468SD16—3321no—
469SD16—1662no—
470D8—22no—
471D8—3321no—
472M16—6641no—
473M16—6641no—
474M16—6641no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—81no—
479C2×C8—81no—
480C2×C8—3322no—
481C2×C8—831no—
482C2×C8—831no—
483C2×C8—831no—
484C2×C8—831no—
485C2×C8—6641no—
486C2×C8—6641no—
487C2×C8—1661no—
488C2×C8—1661no—
489C2×C8—1661no—
490C2×C8C2×C4×D8311yescharacteristic
491C2×C8C2×C4×D8311yescharacteristic
492C2×C8D4×D8311yescharacteristic
493C2×C8D4×D8311yescharacteristic
494C4⋊C4—831no—
495C4⋊C4—831no—
496C4⋊C4—1661no—
497C4⋊C4—1661no—
498C4⋊C4—1661no—
499C4⋊C4—1661no—
500C4⋊C4—1661no—
501C4⋊C4C2×C4×D8311yescharacteristic
502C4⋊C4C2×C4×D8311yescharacteristic
503C22⋊C4—21no—
504C22⋊C4—21no—
505C22⋊C4—1661no—
506C22⋊C4—1661no—
507C22⋊C4—1661no—
508C22⋊C4—1661no—
509C22⋊C4—1661no—
510C22⋊C4—1661no—
511C22⋊C4—1661no—
512C42—831no—
513C42—831no—
514C42—831no—
515C42—1661no—
516C42C23×D8311yescharacteristic
517C23—21no—
518C23—21no—
519C23—3322no—
520C23—3322no—
521C23—831no—
522C23—6641no—
523C23—1661no—
524Q8—1661no—
525Q8—1661no—
526D4—21no—
527D4—21no—
528D4—21no—
529D4—3322no—
530D4—3322no—
531D4—1661no—
532D4—1661no—
533D4#1458179812yes—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—3321no—
541C2×C4—3321no—
542C2×C4—831no—
543C2×C4—831no—
544C2×C4—831no—
545C2×C4—6641no—
546C2×C4—6641no—
547C2×C4—6641no—
548C2×C4—1662no—
549C2×C4—1661no—
550C2×C4—1661no—
551C2×C4—1661no—
552C2×C4—1661no—
553C2×C4—1661no—
554C2×C4—1661no—
555C2×C4—1661no—
556C2×C4C22×C4×D8311yescharacteristic
557C2×C4C2×D4×D8311yescharacteristic
558C2×C4#41×D8311yescharacteristic
559C8—21no—
560C8—21no—
561C8—21no—
562C8—21no—
563C8—41no—
564C8—41no—
565C8—41no—
566C8—3321no—
567C8—3321no—
568C8—3321no—
569C8—1661no—
570C8—1661no—
571C8—1661no—
572C8—1661no—
573C22—22no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—3321no—
580C22—3321no—
581C22—3321no—
582C22—832no—
583C22—6644no—
584C22—1661no—
585C22—1661no—
586C22C4×D4×D8311yescharacteristic
587C4—21no—
588C4—41no—
589C4—41no—
590C4—3321no—
591C4—3321no—
592C4—3321no—
593C4—1661no—
594C4—1661no—
595C4—1661no—
596C4C2×#1458179811yescharacteristic
597C4#1912846611yescharacteristic
598C4#1912850511yescharacteristic
599C4#1912851111yescharacteristic
600C2—42no—
601C2—81no—
602C2—6641no—
603C2—1662no—
604C2#2491618311yescharacteristic
605C2#2491629211yescharacteristic
606C2#444×D8311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC42C23×D83
FittingC83×#1580C2
socleC22×C83C2×C4×D4
prime radical#19127833C4
cosocleC42×C83C24
hypercenter#1580D83
hypocenterC83#18020
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1580

Complements

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

NH ≅ G/Ntype
2. #24916618C2semidirect
3. #24916496C2semidirect
4. #24906374C2semidirect
5. #24906367C2semidirect
6. #24906305C2semidirect
7. #24906299C2semidirect
8. #24906293C2semidirect
9. #24906292C2semidirect
10. #24906272C2semidirect
11. #24906245C2semidirect
12. #24905920C2semidirect
13. D4⋊C8×D83C2semidirect
14. #24905754C2semidirect
15. C83×#1580C2semidirect
26. #19127831C22semidirect
27. #19127825C22semidirect
32. #19127779C22semidirect
48. #19127454C22semidirect
49. #19127453C22semidirect
50. #19127451C22semidirect
51. #19127433C22semidirect
52. #19127431C22semidirect
55. D4⋊(C83⋊C8)C22semidirect
57. #19127423C22semidirect
58. #19127423C22semidirect
59. #19127422C22semidirect
62. C8×D4×C83C22semidirect
63. #334×C83C22semidirect
65. D4⋊C8×C83C22semidirect
66. #322×C83C22semidirect
313. #1580D83semidirect
319. C83#18020semidirect
345. C8×D4C2×D83semidirect
358. #334C2×D83semidirect
366. D4⋊C8C2×D83semidirect
370. #322C2×D83semidirect

Maximal subgroups

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

subgroupindex
#249175082
#249166182
#249164962
#249063742
#249063672
#249063052
#249062992
#249062932
#249062922
#249062722
#249062452
#249059202
D4⋊C8×D832
#249057542
C83×#15802
#1802083