Finite Group Databasebeta

← #35374523

Subgroups — #35374523

Order 21248.

678 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24917461C211yesmaximal, characteristic
2#24916540C211yesmaximal, characteristic, complemented
3#24916484C211yesmaximal, characteristic, complemented
4#24906327C211yesmaximal, characteristic, complemented
5#24906325C211yesmaximal, characteristic, complemented
6#24906294C211yesmaximal, characteristic, complemented
7#24906284C211yesmaximal, characteristic, complemented
8#24906284C211yesmaximal, characteristic, complemented
9D8×C83⋊C8C211yesmaximal, characteristic, complemented
10#24906257C211yesmaximal, characteristic, complemented
11#24906242C211yesmaximal, characteristic, complemented
12#24905904C211yesmaximal, characteristic, complemented
13#24905878C211yesmaximal, characteristic, complemented
14#24905759C211yesmaximal, characteristic, complemented
15C83×#1581C211yesmaximal, characteristic, complemented
16#19128248C2211yescharacteristic
17C4×D4×D83C2211yescharacteristic
18#19128016C411yescharacteristic
19#19128013C411yescharacteristic
20#19127980C411yescharacteristic
21#19127979C411yescharacteristic
22#19127895C411yescharacteristic
23#19127894C411yescharacteristic
24#19127836C411yescharacteristic
25#19127833C411yescharacteristic
26C4×D664C2211yescharacteristic
27C4×#13050575C2211yescharacteristic
28C4×#13050575C2211yescharacteristic
29#19127780C2211yescharacteristic, complemented
30#19127770—21no—
31#19127769—21no—
32C8×D332C2211yescharacteristic
33D4×C83⋊C8C2211yescharacteristic
34D4×C83⋊C8C2211yescharacteristic, complemented
35#19127766C2211yescharacteristic
36#19127766C2211yescharacteristic, complemented
37C8×C22⋊D83—21no—
38(C22⋊D83)⋊C8—21no—
39#19127757C2211yescharacteristic
40#19127753C2211yescharacteristic
41#19127752C2211yescharacteristic
42#19127752C2211yescharacteristic
43#19127669—21no—
44#19127666—21no—
45C2×(C2×D83)⋊C8—21no—
46C22⋊C8×D83—21no—
47C4×#13050573C2211yescharacteristic
48#19127454C2211yescharacteristic, complemented
49#19127451C2211yescharacteristic, complemented
50#19127451C2211yescharacteristic, complemented
51#19127430C2211yescharacteristic
52#19127430C2211yescharacteristic, complemented
53D4⋊(C83⋊C8)C2211yescharacteristic
54D4⋊(C83⋊C8)C2211yescharacteristic, complemented
55#19127428C2211yescharacteristic
56#19127428C2211yescharacteristic, complemented
57#19127423C2211yescharacteristic, complemented
58#19127423C2211yescharacteristic, complemented
59C8×C83⋊C8C2211yescharacteristic, complemented
60C4×D8×C83C2211yescharacteristic
61#436×C83C2211yescharacteristic, complemented
62C8×D4×C83C2211yescharacteristic
63#335×C83C2211yescharacteristic, complemented
64D4⋊C8×C83C2211yescharacteristic
65D4⋊C8×C83C2211yescharacteristic, complemented
66#322×C83C2211yescharacteristic, complemented
67C2×#13050590—21no—
68C2×D4×D83C2×C411yescharacteristic
69C22×C4×D83—21no—
70C2×D664C2×C411yescharacteristic
71C2×#13050575C2×C411yescharacteristic
72C2×#13050575C2×C411yescharacteristic
73C2×#13050573—21no—
74C2×#13050573—21no—
75C22×C83⋊C8—21no—
76C22×C83⋊C8—21no—
77C2×C8×D83—21no—
78C2×C8×D83—21no—
79C2×C8×D83—21no—
80#14581791C2×C411yescharacteristic
81#14581788C2×C411yescharacteristic
82#14581778—21no—
83#14581767C2×C411yescharacteristic
84#14581765—21no—
85#14581763C2×C411yescharacteristic
86C4×D332C2311yescharacteristic
87C4×D332C2311yescharacteristic
88D4×Dic83C2×C411yescharacteristic
89#14581754C2×C411yescharacteristic
90C4×C22⋊D83—21no—
91C4×C22⋊D83—21no—
92(C22⋊D83)⋊C4—21no—
93C4⋊C4×D83C2×C411yescharacteristic
94C22⋊C4×D83—21no—
95C42×D83C2311yescharacteristic
96#14581726C2×C411yescharacteristic
97#14581725C2×C411yescharacteristic
98#14581725C2×C411yescharacteristic
99C4⋊(C83⋊C8)C2311yescharacteristic
100C4⋊(C83⋊C8)C2311yescharacteristic
101Dic83⋊C8C2×C411yescharacteristic
102Dic83⋊C8C2×C411yescharacteristic
103Dic83⋊C8D411yescharacteristic
104Dic83⋊C8D411yescharacteristic
105#14581714C2×C411yescharacteristic
106#14581714C2×C411yescharacteristic
107#14581713C2×C411yescharacteristic
108#14581713C2×C411yescharacteristic
109#14581712C2×C411yescharacteristic
110#14581712C2×C411yescharacteristic
111C22⋊(C83⋊C8)—21no—
112C22⋊(C83⋊C8)—21no—
113(C2×D83)⋊C8—21no—
114(C2×D83)⋊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—
124#14581702—21no—
125C4×C83⋊C8C2311yescharacteristic
126C4×C83⋊C8C2311yescharacteristic
127C4×C83⋊C8C2311yescharacteristic
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—
145#13050590—81no—
146#13050590—81no—
147C2×Dic166—21no—
148C2×D332—21no—
149C2×D332C22×C411yescharacteristic
150C2×D332C22×C411yescharacteristic
151C2×C22⋊D83—21no—
152C2×C22⋊D83—21no—
153C2×C22⋊D83—21no—
154C2×C22⋊D83—21no—
155D4×D83—22no—
156D4×D83—22no—
157C22×Dic83—21no—
158C2×C4×D83—21no—
159C2×C4×D83—21no—
160C2×C4×D83—21no—
161C2×C4×D83—21no—
162C2×C4×D83—81no—
163C2×C4×D83—81no—
164C2×C4×D83C22×C411yescharacteristic
165C2×C4×D83C2×D411yescharacteristic
166C2×C4×D83#4111yescharacteristic
167D664—22no—
168#13050575—22no—
169#13050575—22no—
170#13050573—42no—
171#13050573—81no—
172C2×C83⋊C8—21no—
173C2×C83⋊C8—21no—
174C2×C83⋊C8—21no—
175C2×C83⋊C8—81no—
176C2×C83⋊C8—81no—
177C2×C83⋊C8C22×C411yescharacteristic
178C2×C83⋊C8C22×C411yescharacteristic
179C2×C83⋊C8C2×D411yescharacteristic
180C2×C83⋊C8C2×D411yescharacteristic
181C8×D83—81no—
182C8×D83—81no—
183C8×D83—81no—
184C4⋊Dic83C22×C411yescharacteristic
185C4⋊Dic83C22×C411yescharacteristic
186Dic83⋊C4—21no—
187Dic83⋊C4—21no—
188C22⋊Dic83—21no—
189C22⋊Dic83—21no—
190(C2×D83)⋊C4—21no—
191(C2×D83)⋊C4—21no—
192(C2×D83)⋊C4—21no—
193(C2×D83)⋊C4—21no—
194C4×Dic83C22×C411yescharacteristic
195C4×Dic83C2×D411yescharacteristic
196C4×Dic83#4111yescharacteristic
197C2×D4×C83C22×C411yescharacteristic
198C2×D4×C83C22×C411yescharacteristic
199C22×C4×C83—21no—
200C22×C4×C83—21no—
201D8×C83—22no—
202M16×C83—81no—
203C2×C8×C83—21no—
204C2×C8×C83—21no—
205C2×C8×C83—21no—
206C2×C8×C83—81no—
207C2×C8×C83C22×C411yescharacteristic
208C2×C8×C83C22×C411yescharacteristic
209C2×C8×C83C2×D411yescharacteristic
210C2×C8×C83C2×D411yescharacteristic
211C4⋊C4×C83C22×C411yescharacteristic
212C4⋊C4×C83C22×C411yescharacteristic
213C22⋊C4×C83—21no—
214C22⋊C4×C83—21no—
215C42×C83C2411yescharacteristic
216C23×C83—21no—
217C23×C83—21no—
218C22×D83—21no—
219C22×D83—24no—
220C22×D83—21no—
221C22×D83—81no—
222C22×D83—81no—
223C22×D83C4×D411yescharacteristic
224D4×C83—21no—
225D4×C83—21no—
226D4×C83—21no—
227D4×C83#13112yes—
228C2×C4×C83—21no—
229C2×C4×C83—21no—
230C2×C4×C83—41no—
231C2×C4×C83—41no—
232C2×C4×C83—41no—
233C2×C4×C83—41no—
234C2×C4×C83C23×C411yescharacteristic
235C2×C4×C83C22×D411yescharacteristic
236C2×C4×C83C2×#4111yescharacteristic
237C22⋊D83—24no—
238C22⋊D83—81no—
239C22⋊D83—81no—
240C22⋊D83—81no—
241C2×Dic83—21no—
242C2×Dic83—21no—
243C2×Dic83—81no—
244C2×Dic83C4×D411yescharacteristic
245C2×Dic83C4×D411yescharacteristic
246C2×Dic83C4×D411yescharacteristic
247D332—21no—
248D332—21no—
249D332—21no—
250D332—81no—
251D332#13112yes—
252C4×D83—21no—
253C4×D83—21no—
254C4×D83—21no—
255C4×D83—21no—
256C4×D83—41no—
257C4×D83—41no—
258C4×D83—41no—
259C4×D83—41no—
260C4×D83—41no—
261C4×D83—81no—
262C4×D83—81no—
263Dic166—81no—
264C8×C83—21no—
265C8×C83—21no—
266C8×C83—21no—
267C8×C83—21no—
268C8×C83—41no—
269C8×C83—41no—
270C8×C83—41no—
271C83⋊C8—22no—
272C83⋊C8—21no—
273C83⋊C8—21no—
274C83⋊C8—21no—
275C83⋊C8—21no—
276C83⋊C8—41no—
277C83⋊C8—41no—
278C22×C83—22no—
279C22×C83—41no—
280C22×C83—41no—
281C22×C83—41no—
282C22×C83—41no—
283C22×C83—41no—
284C22×C83C2×C4×D411yescharacteristic
285C2×D83—21no—
286C2×D83—22no—
287C2×D83—21no—
288C2×D83—21no—
289C2×D83—41no—
290C2×D83—41no—
291C2×D83—41no—
292C2×D83—41no—
293C2×D83—41no—
294C2×D83—84no—
295C4×C83—21no—
296C4×C83—41no—
297C4×C83—41no—
298C4×C83C2×#13111yescharacteristic
299C4×C83C2×#13111yescharacteristic
300C4×C83#57611yescharacteristic
301C4×C83#57611yescharacteristic
302Dic83—21no—
303Dic83—21no—
304Dic83—21no—
305Dic83—41no—
306Dic83—41no—
307#17922—831nomaximal, Sylow
308C2×C83—42no—
309C2×C83—81no—
310C2×C83#292911yescharacteristic
311C2×C83#294811yescharacteristic
312C2×C83#295611yescharacteristic
313D83—41no—
314D83—42no—
315D83—81no—
316#3100—831no—
317#2981—831no—
318#2972—831no—
319#1586—831no—
320#1586—831no—
321#1581—831no—
322#1581D8311yescharacteristic, complemented
323#1580—831no—
324#1580—831no—
325C8×D8—831no—
326C8×D8—831no—
327#1572—831no—
328#1490—831no—
329#1484—831no—
330#1453—831no—
331C83#1792211yescharacteristic, complemented, Sylow
332#532—831no—
333C2×C4×D4—831no—
334#480—831no—
335#480—831no—
336#474—831no—
337#474—831no—
338#459—831no—
339#459—831no—
340#447—831no—
341#447—831no—
342C4×D8—831no—
343C4×D8—831no—
344C4×D8—831no—
345C4×D8C2×D8311yescharacteristic
346#436—831no—
347#436C2×D8311yescharacteristic, complemented
348#435—1661no—
349#435—1661no—
350C8×D4—831no—
351C8×D4—831no—
352C8×D4—831no—
353C8×D4—831no—
354C8×D4—831no—
355C8×D4—1661no—
356C8×D4—1661no—
357C8×D4C2×D8311yescharacteristic
358#432—831no—
359#431—831no—
360#431—831no—
361#431—831no—
362#408—1661no—
363#408—1661no—
364C2×C22⋊C8—1661no—
365C2×C22⋊C8—1661no—
366C4×M16—831no—
367#335—831no—
368#335C2×D8311yescharacteristic, complemented
369#334—831no—
370#334—831no—
371D4⋊C8—831no—
372D4⋊C8—831no—
373D4⋊C8—831no—
374D4⋊C8—831no—
375D4⋊C8—831no—
376D4⋊C8—831no—
377D4⋊C8C2×D8311yescharacteristic
378D4⋊C8C2×D8311yescharacteristic, complemented
379#322—831no—
380#322—831no—
381#322C2×D8311yescharacteristic, complemented
382C82—831no—
383C2×#41—1661no—
384C22×D4—831no—
385C23×C4—1661no—
386C2×D8—831no—
387C2×D8—831no—
388C2×D8—831no—
389C2×D8C4×D8311yescharacteristic
390C2×M16—21no—
391C2×M16—1661no—
392C2×M16—1661no—
393C22×C8—21no—
394C22×C8—1661no—
395C22×C8—1661no—
396C22×C8—1661no—
397C22×C8—1661no—
398C22×C8—1661no—
399C4⋊Q8—831no—
400#127—831no—
401#124—1661no—
402#121—831no—
403#121—831no—
404#121—1661no—
405C4×D4—831no—
406C4×D4—831no—
407C4×D4—831no—
408C4×D4—831no—
409C4×D4—1661no—
410C4×D4—1661no—
411C4×D4—1661no—
412C4×D4C22×D8311yescharacteristic
413C4×D4C22×D8311yescharacteristic
414C2×C4⋊C4—831no—
415C2×C22⋊C4—1661no—
416C2×C42—831no—
417#107—831no—
418#107—831no—
419#107—831no—
420#107C4×D8311yescharacteristic
421C4⋊C8—831no—
422C4⋊C8—831no—
423C4⋊C8—831no—
424C4⋊C8—831no—
425C4⋊C8—831no—
426C4⋊C8—831no—
427C4⋊C8C22×D8311yescharacteristic
428C4⋊C8C22×D8311yescharacteristic
429#102—831no—
430#102—831no—
431#102—831no—
432#102—831no—
433#102—831no—
434#102—831no—
435#102C4×D8311yescharacteristic
436#102C4×D8311yescharacteristic
437C22⋊C8—21no—
438C22⋊C8—21no—
439C22⋊C8—831no—
440C22⋊C8—831no—
441C22⋊C8—831no—
442C22⋊C8—831no—
443C22⋊C8—1661no—
444C22⋊C8—1661no—
445C22⋊C8—1661no—
446C22⋊C8—1661no—
447C22⋊C8—1661no—
448C22⋊C8—1661no—
449C22⋊C8—1661no—
450C22⋊C8—1661no—
451#97—1661no—
452#97—1661no—
453C4×C8—831no—
454C4×C8—831no—
455C4×C8—831no—
456C4×C8—1661no—
457C4×C8C22×D8311yescharacteristic
458C4×C8C22×D8311yescharacteristic
459C4×C8C22×D8311yescharacteristic
460C24—1661no—
461#41—6641no—
462#41—6641no—
463C2×Q8—1661no—
464C2×D4—831no—
465C2×D4—831no—
466C2×D4—1661no—
467C2×D4—1661no—
468C2×D4—1661no—
469C2×D4—1661no—
470C2×D4—1662no—
471C2×D4—1661no—
472C2×D4—1662no—
473C2×D4C2×C4×D8311yescharacteristic
474C2×D4C2×C4×D8311yescharacteristic
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—831no—
478C22×C4—831no—
479C22×C4—831no—
480C22×C4—6641no—
481C22×C4—6641no—
482C22×C4—1661no—
483C22×C4—1661no—
484C22×C4—1661no—
485C22×C4—1661no—
486C22×C4—1661no—
487D8—22no—
488D8—1662no—
489D8—1662no—
490D8—1662no—
491M16—81no—
492M16—3322no—
493M16—6641no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—81no—
498C2×C8—831no—
499C2×C8—831no—
500C2×C8—831no—
501C2×C8—831no—
502C2×C8—6641no—
503C2×C8—6641no—
504C2×C8—6641no—
505C2×C8—6641no—
506C2×C8—6641no—
507C2×C8—1661no—
508C2×C8—1661no—
509C2×C8—1661no—
510C2×C8C2×C4×D8311yescharacteristic
511C2×C8C2×C4×D8311yescharacteristic
512C2×C8D4×D8311yescharacteristic
513C2×C8D4×D8311yescharacteristic
514C4⋊C4—831no—
515C4⋊C4—831no—
516C4⋊C4—1661no—
517C4⋊C4—1661no—
518C4⋊C4C2×C4×D8311yescharacteristic
519C4⋊C4C2×C4×D8311yescharacteristic
520C22⋊C4—21no—
521C22⋊C4—21no—
522C22⋊C4—1661no—
523C22⋊C4—1661no—
524C22⋊C4—1661no—
525C22⋊C4—1661no—
526C22⋊C4—1661no—
527C22⋊C4—1661no—
528C42—831no—
529C42—831no—
530C42—831no—
531C42C23×D8311yescharacteristic
532C23—21no—
533C23—21no—
534C23—831no—
535C23—6641no—
536C23—6641no—
537C23—1664no—
538C23—1661no—
539C23—1661no—
540Q8—6641no—
541D4—21no—
542D4—21no—
543D4—21no—
544D4—832no—
545D4—6641no—
546D4—6641no—
547D4—6641no—
548D4—6641no—
549D4—1664no—
550D4—1661no—
551D4—1661no—
552D4—1661no—
553D4#1458180012yes—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—3321no—
561C2×C4—3321no—
562C2×C4—3321no—
563C2×C4—3321no—
564C2×C4—3321no—
565C2×C4—831no—
566C2×C4—831no—
567C2×C4—831no—
568C2×C4—6641no—
569C2×C4—6641no—
570C2×C4—6641no—
571C2×C4—1661no—
572C2×C4—1661no—
573C2×C4—1661no—
574C2×C4—1661no—
575C2×C4—1661no—
576C2×C4—1661no—
577C2×C4C22×C4×D8311yescharacteristic
578C2×C4C2×D4×D8311yescharacteristic
579C2×C4#41×D8311yescharacteristic
580C8—21no—
581C8—21no—
582C8—21no—
583C8—21no—
584C8—41no—
585C8—41no—
586C8—41no—
587C8—3321no—
588C8—3321no—
589C8—1662no—
590C8—1661no—
591C8—1661no—
592C8—1661no—
593C8—1661no—
594C22—22no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—41no—
600C22—3321no—
601C22—3321no—
602C22—3321no—
603C22—3321no—
604C22—3321no—
605C22—6644no—
606C22—1661no—
607C22—1661no—
608C22—1661no—
609C22—1662no—
610C22C4×D4×D8311yescharacteristic
611C4—21no—
612C4—41no—
613C4—41no—
614C4—3321no—
615C4—3321no—
616C4—1661no—
617C4—1661no—
618C4—1661no—
619C4#131×D8311yescharacteristic
620C4C2×#1458180011yescharacteristic
621C4#1912850211yescharacteristic
622C4#1912850311yescharacteristic
623C2—42no—
624C2—81no—
625C2—3322no—
626C2—3321no—
627C2—6641no—
628C2#2491615411yescharacteristic
629C2#2491631011yescharacteristic
630C2#2491637111yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC42C23×D83
FittingC83×#1581C2
socleC22×C83C2×C4×D4
prime radical#24917461C2
cosocleC42×C83C24
hypercenter#1581D83
hypocenterC83#17922
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1581

Complements

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

NH ≅ G/Ntype
2. #24916540C2semidirect
3. #24916484C2semidirect
4. #24906327C2semidirect
5. #24906325C2semidirect
6. #24906294C2semidirect
7. #24906284C2semidirect
8. #24906284C2semidirect
9. D8×C83⋊C8C2semidirect
10. #24906257C2semidirect
11. #24906242C2semidirect
12. #24905904C2semidirect
13. #24905878C2semidirect
14. #24905759C2semidirect
15. C83×#1581C2semidirect
29. #19127780C22semidirect
34. D4×C83⋊C8C22semidirect
36. #19127766C22semidirect
48. #19127454C22semidirect
49. #19127451C22semidirect
50. #19127451C22semidirect
52. #19127430C22semidirect
54. D4⋊(C83⋊C8)C22semidirect
56. #19127428C22semidirect
57. #19127423C22semidirect
58. #19127423C22semidirect
59. C8×C83⋊C8C22semidirect
61. #436×C83C22semidirect
63. #335×C83C22semidirect
65. D4⋊C8×C83C22semidirect
66. #322×C83C22semidirect
322. #1581D83semidirect
331. C83#17922semidirect
347. #436C2×D83semidirect
368. #335C2×D83semidirect
378. D4⋊C8C2×D83semidirect
381. #322C2×D83semidirect

Maximal subgroups

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

subgroupindex
#249174612
#249165402
#249164842
#249063272
#249063252
#249062942
#249062842
D8×C83⋊C82
#249062572
#249062422
#249059042
#249058782
#249057592
C83×#15812
#1792283