Finite Group Databasebeta

← #35374467

Subgroups — #35374467

Order 21248.

648 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24917508C211yesmaximal, characteristic
2#24916607C211yesmaximal, characteristic, complemented
3#24916484C211yesmaximal, characteristic, complemented
4#24906322C211yesmaximal, characteristic, complemented
5#24906299C211yesmaximal, characteristic, complemented
6#24906290C211yesmaximal, characteristic, complemented
7#24906284C211yesmaximal, characteristic, complemented
8#24906272C211yesmaximal, characteristic, complemented
9#24906262C211yesmaximal, characteristic, complemented
10D8×C83⋊C8C211yesmaximal, characteristic, complemented
11#24906250C211yesmaximal, characteristic, complemented
12#24905886C211yesmaximal, characteristic, complemented
13#24905868C211yesmaximal, characteristic, complemented
14#24905764C211yesmaximal, characteristic, complemented
15C83×#1586C211yesmaximal, 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#19127777—21no—
32#19127769—21no—
33C8×D332C2211yescharacteristic
34D4×C83⋊C8C2211yescharacteristic
35D4×C83⋊C8C2211yescharacteristic, complemented
36#19127766C2211yescharacteristic
37C8×C22⋊D83—21no—
38(C22⋊D83)⋊C8—21no—
39#19127759—21no—
40#19127753C2211yescharacteristic
41#19127752C2211yescharacteristic
42#19127752C2211yescharacteristic
43#19127752C2211yescharacteristic
44#19127729—21no—
45#19127723C2211yescharacteristic
46C2×(C2×D83)⋊C8—21no—
47C22⋊C8×D83—21no—
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#19127422C2211yescharacteristic, complemented
59C8×C83⋊C8C2211yescharacteristic, complemented
60C4×D8×C83C2211yescharacteristic
61#436×C83C2211yescharacteristic, complemented
62C8×D4×C83C2211yescharacteristic
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×#13050573—21no—
73C22×C83⋊C8—21no—
74C22×C83⋊C8—21no—
75C2×C8×D83—21no—
76C2×C8×D83—21no—
77#14581785—21no—
78#14581783C2×C411yescharacteristic
79#14581779C2×C411yescharacteristic
80#14581774—21no—
81#14581772C2×C411yescharacteristic
82#14581763C2×C411yescharacteristic
83C4×Dic166C2311yescharacteristic
84C4×D332C2311yescharacteristic
85D4×Dic83C2×C411yescharacteristic
86#14581754C2×C411yescharacteristic
87C4×C22⋊D83—21no—
88C4×C22⋊D83—21no—
89(C22⋊D83)⋊C4—21no—
90#14581751—21no—
91C4⋊C4×D83C2×C411yescharacteristic
92C22⋊C4×D83—21no—
93C42×D83C2311yescharacteristic
94#14581725C2×C411yescharacteristic
95#14581724C2×C411yescharacteristic
96#14581723C2×C411yescharacteristic
97C4⋊(C83⋊C8)C2311yescharacteristic
98C4⋊(C83⋊C8)C2311yescharacteristic
99Dic83⋊C8—21no—
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)⋊C8—21no—
119(C2×D83)⋊C8C2×C411yescharacteristic
120(C2×D83)⋊C8C2×C411yescharacteristic
121(C2×D83)⋊C8D411yescharacteristic
122(C2×D83)⋊C8D411yescharacteristic
123#14581702—21no—
124C4×C83⋊C8C2311yescharacteristic
125C4×C83⋊C8C2311yescharacteristic
126C4×C83⋊C8C2311yescharacteristic
127C8×Dic83—21no—
128C8×Dic83—21no—
129C2×D8×C83C2×C411yescharacteristic
130C2×M16×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—22no—
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—22no—
161#13050578—22no—
162#13050575—22no—
163#13050573—81no—
164C2×C83⋊C8—21no—
165C2×C83⋊C8—21no—
166C2×C83⋊C8—21no—
167C2×C83⋊C8—81no—
168C2×C83⋊C8—81no—
169C2×C83⋊C8C22×C411yescharacteristic
170C2×C83⋊C8C22×C411yescharacteristic
171C2×C83⋊C8C2×D411yescharacteristic
172C2×C83⋊C8C2×D411yescharacteristic
173C8×D83—81no—
174C8×D83—81no—
175C4⋊Dic83—21no—
176C4⋊Dic83—21no—
177C4⋊Dic83C22×C411yescharacteristic
178C4⋊Dic83C22×C411yescharacteristic
179Dic83⋊C4—21no—
180Dic83⋊C4—21no—
181Dic83⋊C4—21no—
182C22⋊Dic83—21no—
183C22⋊Dic83—21no—
184C22⋊Dic83—21no—
185C22⋊Dic83—21no—
186(C2×D83)⋊C4—21no—
187(C2×D83)⋊C4—21no—
188(C2×D83)⋊C4—21no—
189C4×Dic83—21no—
190C4×Dic83C22×C411yescharacteristic
191C4×Dic83C2×D411yescharacteristic
192C4×Dic83#4111yescharacteristic
193C2×D4×C83C22×C411yescharacteristic
194C2×D4×C83C22×C411yescharacteristic
195C22×C4×C83—21no—
196C22×C4×C83—21no—
197D8×C83—22no—
198M16×C83—81no—
199C2×C8×C83—21no—
200C2×C8×C83—21no—
201C2×C8×C83—21no—
202C2×C8×C83—81no—
203C2×C8×C83C22×C411yescharacteristic
204C2×C8×C83C22×C411yescharacteristic
205C2×C8×C83C2×D411yescharacteristic
206C2×C8×C83C2×D411yescharacteristic
207C4⋊C4×C83C22×C411yescharacteristic
208C4⋊C4×C83C22×C411yescharacteristic
209C22⋊C4×C83—21no—
210C22⋊C4×C83—21no—
211C42×C83C2411yescharacteristic
212C23×C83—21no—
213C23×C83—21no—
214C22×D83—21no—
215C22×D83—24no—
216C22×D83—81no—
217C22×D83—81no—
218C22×D83C4×D411yescharacteristic
219D4×C83—21no—
220D4×C83—21no—
221D4×C83—21no—
222D4×C83#13112yes—
223C2×C4×C83—21no—
224C2×C4×C83—21no—
225C2×C4×C83—41no—
226C2×C4×C83—41no—
227C2×C4×C83—41no—
228C2×C4×C83—41no—
229C2×C4×C83C23×C411yescharacteristic
230C2×C4×C83C22×D411yescharacteristic
231C2×C4×C83C2×#4111yescharacteristic
232C22⋊D83—24no—
233C22⋊D83—81no—
234C2×Dic83—21no—
235C2×Dic83—21no—
236C2×Dic83—21no—
237C2×Dic83—81no—
238C2×Dic83C4×D411yescharacteristic
239C2×Dic83C4×D411yescharacteristic
240C2×Dic83C4×D411yescharacteristic
241D332—21no—
242D332#13112yes—
243C4×D83—21no—
244C4×D83—21no—
245C4×D83—21no—
246C4×D83—21no—
247C4×D83—41no—
248C4×D83—41no—
249C4×D83—41no—
250C4×D83—81no—
251C4×D83—81no—
252Dic166—21no—
253Dic166—21no—
254C8×C83—21no—
255C8×C83—21no—
256C8×C83—21no—
257C8×C83—21no—
258C8×C83—41no—
259C8×C83—41no—
260C8×C83—41no—
261C83⋊C8—21no—
262C83⋊C8—21no—
263C83⋊C8—21no—
264C83⋊C8—21no—
265C83⋊C8—41no—
266C83⋊C8—41no—
267C83⋊C8—41no—
268C22×C83—22no—
269C22×C83—41no—
270C22×C83—41no—
271C22×C83—41no—
272C22×C83—41no—
273C22×C83—41no—
274C22×C83C2×C4×D411yescharacteristic
275C2×D83—21no—
276C2×D83—22no—
277C2×D83—21no—
278C2×D83—21no—
279C2×D83—41no—
280C2×D83—41no—
281C2×D83—84no—
282C4×C83—21no—
283C4×C83—41no—
284C4×C83—41no—
285C4×C83C2×#13111yescharacteristic
286C4×C83C2×#13111yescharacteristic
287C4×C83#57611yescharacteristic
288C4×C83#57711yescharacteristic
289Dic83—21no—
290Dic83—21no—
291Dic83—21no—
292Dic83—41no—
293Dic83—41no—
294Dic83—41no—
295#17918—831nomaximal, Sylow
296C2×C83—42no—
297C2×C83—81no—
298C2×C83#292911yescharacteristic
299C2×C83#294611yescharacteristic
300C2×C83#295711yescharacteristic
301D83—41no—
302D83—42no—
303#3105—831no—
304#2993—831no—
305#2972—831no—
306#1586D8311yescharacteristic, complemented
307#1585—831no—
308#1582—831no—
309#1581—831no—
310#1580—831no—
311SD16⋊C8—831no—
312C8×SD16—831no—
313C8×D8—831no—
314#1574—831no—
315#1486—831no—
316#1482—831no—
317#1454—831no—
318C83#1791811yescharacteristic, complemented, Sylow
319#529—831no—
320C2×C4×D4—831no—
321#484—831no—
322#481—831no—
323#478—831no—
324#478—831no—
325#463—831no—
326#463—831no—
327#450—831no—
328#447—831no—
329#446—831no—
330C8×Q8—831no—
331C4×SD16—831no—
332C4×SD16—831no—
333C4×D8—831no—
334C4×D8C2×D8311yescharacteristic
335#436—1661no—
336#436C2×D8311yescharacteristic, complemented
337#435—1661no—
338C8×D4—831no—
339C8×D4—831no—
340C8×D4—831no—
341C8×D4—831no—
342C8×D4—1661no—
343C8×D4—1661no—
344C8×D4C2×D8311yescharacteristic
345#433—1661no—
346#431—831no—
347#431—831no—
348#431—831no—
349#431—831no—
350#424—1661no—
351#423—831no—
352C2×C22⋊C8—1661no—
353C2×C22⋊C8—1661no—
354#335—831no—
355#335—831no—
356#334—831no—
357#334C2×D8311yescharacteristic, complemented
358Q8⋊C8—831no—
359Q8⋊C8—831no—
360D4⋊C8—831no—
361D4⋊C8—831no—
362D4⋊C8—831no—
363D4⋊C8—831no—
364D4⋊C8C2×D8311yescharacteristic
365D4⋊C8C2×D8311yescharacteristic, complemented
366#322—831no—
367#322—831no—
368#322C2×D8311yescharacteristic, complemented
369C82—831no—
370C22×D4—831no—
371C23×C4—1661no—
372C2×SD16—831no—
373C2×SD16—831no—
374C2×D8—831no—
375C2×D8C4×D8311yescharacteristic
376C2×M16—21no—
377C2×M16—1661no—
378C22×C8—21no—
379C22×C8—1661no—
380C22×C8—1661no—
381C22×C8—1661no—
382C22×C8—1661no—
383#126—1661no—
384#125—831no—
385#124—831no—
386#123—1661no—
387C22⋊Q8—831no—
388#121—831no—
389C4×Q8—831no—
390C4×D4—831no—
391C4×D4—831no—
392C4×D4—831no—
393C4×D4—1661no—
394C4×D4—1661no—
395C4×D4—1661no—
396C4×D4C22×D8311yescharacteristic
397C4×D4C22×D8311yescharacteristic
398#117—1661no—
399C2×C4⋊C4—831no—
400C2×C22⋊C4—1661no—
401C2×C42—831no—
402#107—831no—
403#107C4×D8311yescharacteristic
404#106—831no—
405#106—831no—
406C4⋊C8—831no—
407C4⋊C8—831no—
408C4⋊C8—831no—
409C4⋊C8—831no—
410C4⋊C8—831no—
411C4⋊C8—831no—
412C4⋊C8—1661no—
413C4⋊C8—1661no—
414C4⋊C8—1661no—
415C4⋊C8—1661no—
416C4⋊C8C22×D8311yescharacteristic
417C4⋊C8C22×D8311yescharacteristic
418#103—831no—
419#103—831no—
420#102—831no—
421#102—831no—
422#102—831no—
423#102—831no—
424#102C4×D8311yescharacteristic
425#102C4×D8311yescharacteristic
426C22⋊C8—21no—
427C22⋊C8—21no—
428C22⋊C8—831no—
429C22⋊C8—831no—
430C22⋊C8—831no—
431C22⋊C8—831no—
432C22⋊C8—1661no—
433C22⋊C8—1661no—
434C22⋊C8—1661no—
435C22⋊C8—1661no—
436C22⋊C8—1661no—
437C22⋊C8—1661no—
438#97—1661no—
439C4×C8—831no—
440C4×C8—831no—
441C4×C8—831no—
442C4×C8—1661no—
443C4×C8—1661no—
444C4×C8C22×D8311yescharacteristic
445C4×C8C22×D8311yescharacteristic
446C4×C8C22×D8311yescharacteristic
447C24—1661no—
448C2×Q8—831no—
449C2×D4—831no—
450C2×D4—1661no—
451C2×D4—1662no—
452C2×D4—1661no—
453C2×D4—1662no—
454C2×D4C2×C4×D8311yescharacteristic
455C2×D4C2×C4×D8311yescharacteristic
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—831no—
459C22×C4—831no—
460C22×C4—831no—
461C22×C4—6641no—
462C22×C4—6641no—
463C22×C4—1661no—
464C22×C4—1661no—
465C22×C4—1661no—
466C22×C4—1661no—
467SD16—1662no—
468SD16—1662no—
469D8—22no—
470D8—1662no—
471M16—81no—
472M16—6641no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—81no—
477C2×C8—831no—
478C2×C8—831no—
479C2×C8—831no—
480C2×C8—831no—
481C2×C8—6641no—
482C2×C8—6641no—
483C2×C8—6641no—
484C2×C8—6641no—
485C2×C8—1661no—
486C2×C8—1661no—
487C2×C8—1661no—
488C2×C8C2×C4×D8311yescharacteristic
489C2×C8C2×C4×D8311yescharacteristic
490C2×C8D4×D8311yescharacteristic
491C2×C8D4×D8311yescharacteristic
492C4⋊C4—831no—
493C4⋊C4—831no—
494C4⋊C4—1661no—
495C4⋊C4—1661no—
496C4⋊C4—1661no—
497C4⋊C4—1661no—
498C4⋊C4—1661no—
499C4⋊C4C2×C4×D8311yescharacteristic
500C4⋊C4C2×C4×D8311yescharacteristic
501C22⋊C4—21no—
502C22⋊C4—21no—
503C22⋊C4—1661no—
504C22⋊C4—1661no—
505C22⋊C4—1661no—
506C22⋊C4—1661no—
507C22⋊C4—1661no—
508C22⋊C4—1661no—
509C22⋊C4—1661no—
510C42—831no—
511C42—831no—
512C42—831no—
513C42—1661no—
514C42C23×D8311yescharacteristic
515C23—21no—
516C23—21no—
517C23—831no—
518C23—6641no—
519C23—6641no—
520C23—1664no—
521C23—1661no—
522Q8—1661no—
523Q8—1661no—
524D4—21no—
525D4—21no—
526D4—21no—
527D4—832no—
528D4—6641no—
529D4—1664no—
530D4—1661no—
531D4#1458180012yes—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—3321no—
539C2×C4—3321no—
540C2×C4—3321no—
541C2×C4—831no—
542C2×C4—831no—
543C2×C4—831no—
544C2×C4—6641no—
545C2×C4—6641no—
546C2×C4—6641no—
547C2×C4—1661no—
548C2×C4—1661no—
549C2×C4—1661no—
550C2×C4—1661no—
551C2×C4—1661no—
552C2×C4—1661no—
553C2×C4—1661no—
554C2×C4C22×C4×D8311yescharacteristic
555C2×C4C2×D4×D8311yescharacteristic
556C2×C4#41×D8311yescharacteristic
557C8—21no—
558C8—21no—
559C8—21no—
560C8—21no—
561C8—41no—
562C8—41no—
563C8—41no—
564C8—3321no—
565C8—3321no—
566C8—3321no—
567C8—1661no—
568C8—1661no—
569C8—1661no—
570C8—1661no—
571C22—22no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—3321no—
578C22—3321no—
579C22—6644no—
580C22—1661no—
581C22—1661no—
582C22—1661no—
583C22—1662no—
584C22C4×D4×D8311yescharacteristic
585C4—21no—
586C4—41no—
587C4—41no—
588C4—3321no—
589C4—3321no—
590C4—3321no—
591C4—1661no—
592C4—1661no—
593C4—1661no—
594C4#131×D8311yescharacteristic
595C4C2×#1458180011yescharacteristic
596C4#1912850511yescharacteristic
597C4#1912851111yescharacteristic
598C2—42no—
599C2—81no—
600C2—3322no—
601C2—3321no—
602C2#2491615411yescharacteristic
603C2#2491629211yescharacteristic
604C2#2491637511yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC42C23×D83
FittingC83×#1586C2
socleC22×C83C2×C4×D4
prime radical#19127833C4
cosocleC42×C83C24
hypercenter#1586D83
hypocenterC83#17918
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1586

Complements

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

NH ≅ G/Ntype
2. #24916607C2semidirect
3. #24916484C2semidirect
4. #24906322C2semidirect
5. #24906299C2semidirect
6. #24906290C2semidirect
7. #24906284C2semidirect
8. #24906272C2semidirect
9. #24906262C2semidirect
10. D8×C83⋊C8C2semidirect
11. #24906250C2semidirect
12. #24905886C2semidirect
13. #24905868C2semidirect
14. #24905764C2semidirect
15. C83×#1586C2semidirect
26. #19127831C22semidirect
27. #19127825C22semidirect
35. D4×C83⋊C8C22semidirect
48. #19127454C22semidirect
49. #19127453C22semidirect
50. #19127451C22semidirect
51. #19127433C22semidirect
52. #19127431C22semidirect
55. D4⋊(C83⋊C8)C22semidirect
57. #19127423C22semidirect
58. #19127422C22semidirect
59. C8×C83⋊C8C22semidirect
61. #436×C83C22semidirect
63. #334×C83C22semidirect
65. D4⋊C8×C83C22semidirect
66. #322×C83C22semidirect
306. #1586D83semidirect
318. C83#17918semidirect
336. #436C2×D83semidirect
357. #334C2×D83semidirect
365. D4⋊C8C2×D83semidirect
368. #322C2×D83semidirect

Maximal subgroups

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

subgroupindex
#249175082
#249166072
#249164842
#249063222
#249062992
#249062902
#249062842
#249062722
#249062622
D8×C83⋊C82
#249062502
#249058862
#249058682
#249057642
C83×#15862
#1791883