Finite Group Databasebeta

← #35374591

Subgroups — #35374591

Order 21248.

734 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24917461C211yesmaximal, characteristic
2#24916170C211yesmaximal, characteristic, complemented
3#436×D83C211yesmaximal, characteristic, complemented
4#24906348C211yesmaximal, characteristic, complemented
5#24906347C211yesmaximal, characteristic, complemented
6#24906345C211yesmaximal, characteristic, complemented
7#24906345C211yesmaximal, characteristic, complemented
8#24906294C211yesmaximal, characteristic, complemented
9#24906292C211yesmaximal, characteristic, complemented
10#24906290C211yesmaximal, characteristic, complemented
11#24906236C211yesmaximal, characteristic, complemented
12#24905886C211yesmaximal, characteristic, complemented
13#24905860C211yesmaximal, characteristic, complemented
14#24905748C211yesmaximal, characteristic, complemented
15C83×#1581C211yesmaximal, characteristic, complemented
16C2×#14581798—21no—
17C2×M16×D83—21no—
18#19128248C2211yescharacteristic
19C4×D4×D83C2211yescharacteristic
20#19128016C411yescharacteristic
21#19128013C411yescharacteristic
22#19127980C411yescharacteristic
23#19127979C411yescharacteristic
24#19127895C411yescharacteristic
25#19127894C411yescharacteristic
26#19127836C411yescharacteristic
27#19127833C411yescharacteristic
28C4×D664C2211yescharacteristic
29C4×#13050575C2211yescharacteristic
30C4×#13050575C2211yescharacteristic
31#19127780C2211yescharacteristic
32#19127779C2211yescharacteristic
33#19127779C2211yescharacteristic, complemented
34#19127778C2211yescharacteristic
35#19127778C2211yescharacteristic, complemented
36#19127777—21no—
37#19127776—21no—
38#19127776—21no—
39C8×D332C2211yescharacteristic, complemented
40C8×C22⋊D83—21no—
41#19127752C2211yescharacteristic
42#19127752C2211yescharacteristic
43#19127723C2211yescharacteristic
44C4⋊C8×D83C2211yescharacteristic
45#19127666—21no—
46C22⋊C8×D83—21no—
47C4×C8×D83C2211yescharacteristic
48#19127454C2211yescharacteristic, complemented
49#19127453C2211yescharacteristic, complemented
50#19127453C2211yescharacteristic, complemented
51#19127430C2211yescharacteristic
52#19127430C2211yescharacteristic, complemented
53D4⋊(C83⋊C8)C2211yescharacteristic
54D4⋊(C83⋊C8)C2211yescharacteristic, complemented
55#19127428C2211yescharacteristic
56#19127428C2211yescharacteristic, complemented
57#19127422C2211yescharacteristic, complemented
58C8×C83⋊C8C2211yescharacteristic, complemented
59C8×C83⋊C8C2211yescharacteristic, complemented
60C4×D8×C83C2211yescharacteristic
61#436×C83C2211yescharacteristic
62C8×D4×C83C2211yescharacteristic, complemented
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
73#14581798—44no—
74C2×#13050574—21no—
75C2×#13050574—21no—
76C2×#13050573—21no—
77C2×#13050573—21no—
78C2×#13050573—21no—
79C2×#13050573—21no—
80M16×D83—44no—
81C2×C8×D83—21no—
82C2×C8×D83—21no—
83C2×C8×D83D411yescharacteristic
84C2×C8×D83D411yescharacteristic
85#14581791C2×C411yescharacteristic
86#14581788C2×C411yescharacteristic
87#14581778—21no—
88#14581767C2×C411yescharacteristic
89#14581765—21no—
90#14581763C2×C411yescharacteristic
91C4×D332C2311yescharacteristic
92C4×D332C2311yescharacteristic
93D4×Dic83C2×C411yescharacteristic
94#14581754C2×C411yescharacteristic
95C4×C22⋊D83—21no—
96C4×C22⋊D83—21no—
97(C22⋊D83)⋊C4—21no—
98C4⋊C4×D83C2×C411yescharacteristic
99C22⋊C4×D83—21no—
100C42×D83C2311yescharacteristic
101#14581726C2×C411yescharacteristic
102#14581725C2×C411yescharacteristic
103#14581725C2×C411yescharacteristic
104C4⋊(C83⋊C8)C2311yescharacteristic
105C4⋊(C83⋊C8)C2311yescharacteristic
106Dic83⋊C8—21no—
107Dic83⋊C8—21no—
108Dic83⋊C8—21no—
109Dic83⋊C8C2×C411yescharacteristic
110Dic83⋊C8C2×C411yescharacteristic
111#14581714C2×C411yescharacteristic
112#14581714C2×C411yescharacteristic
113#14581713C2×C411yescharacteristic
114#14581713C2×C411yescharacteristic
115#14581712C2×C411yescharacteristic
116#14581712C2×C411yescharacteristic
117C22⋊(C83⋊C8)—21no—
118C22⋊(C83⋊C8)—21no—
119(C2×D83)⋊C8—21no—
120(C2×D83)⋊C8—21no—
121(C2×D83)⋊C8—21no—
122(C2×D83)⋊C8—21no—
123(C2×D83)⋊C8—21no—
124(C2×D83)⋊C8C2×C411yescharacteristic
125(C2×D83)⋊C8C2×C411yescharacteristic
126C4×C83⋊C8C2311yescharacteristic
127C4×C83⋊C8C2311yescharacteristic
128C4×C83⋊C8C2311yescharacteristic
129C8×Dic83D411yescharacteristic
130C8×Dic83D411yescharacteristic
131C2×D8×C83C2×C411yescharacteristic
132C2×M16×C83—21no—
133C22×C8×C83—21no—
134C4×D4×C83C2311yescharacteristic
135C4×D4×C83C2311yescharacteristic
136#107×C83C2×C411yescharacteristic
137C4⋊C8×C83C2311yescharacteristic
138C4⋊C8×C83C2311yescharacteristic
139#102×C83C2×C411yescharacteristic
140#102×C83C2×C411yescharacteristic
141C22⋊C8×C83—21no—
142C22⋊C8×C83—21no—
143C4×C8×C83C2311yescharacteristic
144C4×C8×C83C2311yescharacteristic
145C4×C8×C83C2311yescharacteristic
146C23×D83—21no—
147#13050590—42no—
148#13050590—42no—
149C2×Dic166—21no—
150C2×D332—21no—
151C2×D332C22×C411yescharacteristic
152C2×D332C22×C411yescharacteristic
153C2×C22⋊D83—21no—
154C2×C22⋊D83—21no—
155C2×C22⋊D83—21no—
156C2×C22⋊D83—21no—
157D4×D83—41no—
158D4×D83—41no—
159C22×Dic83—21no—
160C2×C4×D83—21no—
161C2×C4×D83—21no—
162C2×C4×D83—21no—
163C2×C4×D83—21no—
164C2×C4×D83—42no—
165C2×C4×D83—42no—
166C2×C4×D83C22×C411yescharacteristic
167C2×C4×D83C2×D411yescharacteristic
168C2×C4×D83#4111yescharacteristic
169D664—41no—
170#13050575—41no—
171#13050575—41no—
172#13050574—42no—
173#13050574—42no—
174#13050573—42no—
175#13050573—42no—
176#13050573—42no—
177#13050573—42no—
178C2×C83⋊C8—21no—
179C2×C83⋊C8—21no—
180C2×C83⋊C8—21no—
181C2×C83⋊C8C22×C411yescharacteristic
182C2×C83⋊C8C22×C411yescharacteristic
183C2×C83⋊C8C2×D411yescharacteristic
184C2×C83⋊C8C2×D411yescharacteristic
185C8×D83—22no—
186C8×D83—42no—
187C8×D83—42no—
188C8×D83D814yescomplemented
189C4⋊Dic83C22×C411yescharacteristic
190C4⋊Dic83C22×C411yescharacteristic
191Dic83⋊C4—21no—
192Dic83⋊C4—21no—
193C22⋊Dic83—21no—
194C22⋊Dic83—21no—
195(C2×D83)⋊C4—21no—
196(C2×D83)⋊C4—21no—
197(C2×D83)⋊C4—21no—
198(C2×D83)⋊C4—21no—
199C4×Dic83C22×C411yescharacteristic
200C4×Dic83C2×D411yescharacteristic
201C4×Dic83#4111yescharacteristic
202C2×D4×C83C22×C411yescharacteristic
203C2×D4×C83C22×C411yescharacteristic
204C22×C4×C83—21no—
205C22×C4×C83—21no—
206D8×C83—41no—
207M16×C83—42no—
208C2×C8×C83—21no—
209C2×C8×C83—21no—
210C2×C8×C83—21no—
211C2×C8×C83—42no—
212C2×C8×C83C22×C411yescharacteristic
213C2×C8×C83C22×C411yescharacteristic
214C2×C8×C83C2×D411yescharacteristic
215C2×C8×C83C2×D411yescharacteristic
216C4⋊C4×C83C22×C411yescharacteristic
217C4⋊C4×C83C22×C411yescharacteristic
218C22⋊C4×C83—21no—
219C22⋊C4×C83—21no—
220C42×C83C2411yescharacteristic
221C23×C83—21no—
222C23×C83—21no—
223C22×D83—21no—
224C22×D83—21no—
225C22×D83—42no—
226C22×D83—42no—
227C22×D83—42no—
228C22×D83C4×D411yescharacteristic
229D4×C83—21no—
230D4×C83—21no—
231D4×C83—21no—
232D4×C83—21no—
233C2×C4×C83—21no—
234C2×C4×C83—21no—
235C2×C4×C83—41no—
236C2×C4×C83—41no—
237C2×C4×C83—41no—
238C2×C4×C83—41no—
239C2×C4×C83C23×C411yescharacteristic
240C2×C4×C83C22×D411yescharacteristic
241C2×C4×C83C2×#4111yescharacteristic
242C22⋊D83—42no—
243C22⋊D83—42no—
244C22⋊D83—42no—
245C22⋊D83—42no—
246C2×Dic83—21no—
247C2×Dic83—21no—
248C2×Dic83—42no—
249C2×Dic83C4×D411yescharacteristic
250C2×Dic83C4×D411yescharacteristic
251C2×Dic83C4×D411yescharacteristic
252D332—21no—
253D332—21no—
254D332—21no—
255D332—21no—
256D332—42no—
257C4×D83—21no—
258C4×D83—21no—
259C4×D83—41no—
260C4×D83—42no—
261C4×D83—41no—
262C4×D83—41no—
263C4×D83—41no—
264C4×D83—41no—
265C4×D83—42no—
266C4×D83C2×D812yes—
267C4×D83#13512yes—
268Dic166—42no—
269C8×C83—21no—
270C8×C83—21no—
271C8×C83—21no—
272C8×C83—41no—
273C8×C83—41no—
274C8×C83—41no—
275C8×C83C2×D812yes—
276C83⋊C8—21no—
277C83⋊C8—21no—
278C83⋊C8—21no—
279C83⋊C8—41no—
280C83⋊C8—41no—
281C83⋊C8—41no—
282C83⋊C8C2×D812yes—
283C22×C83—41no—
284C22×C83—41no—
285C22×C83—41no—
286C22×C83—41no—
287C22×C83—41no—
288C22×C83—41no—
289C22×C83C2×C4×D411yescharacteristic
290C2×D83—21no—
291C2×D83—21no—
292C2×D83—41no—
293C2×D83—41no—
294C2×D83—41no—
295C2×D83—41no—
296C2×D83—41no—
297C2×D83—41no—
298C2×D83—84no—
299C2×D83C4×D812yes—
300C4×C83—21no—
301C4×C83—41no—
302C4×C83—41no—
303C4×C83#56811yescharacteristic
304C4×C83#56811yescharacteristic
305C4×C83C22×D811yescharacteristic
306C4×C83C2×#13511yescharacteristic
307Dic83—21no—
308Dic83—21no—
309Dic83—41no—
310Dic83—41no—
311Dic83C4×D812yes—
312#17928—831nomaximal, Sylow
313C2×C83—81no—
314C2×C83—81no—
315C2×C83#293011yescharacteristic
316C2×C83C2×C4×D811yescharacteristic
317C2×C83#295511yescharacteristic
318D83—22no—
319D83—81no—
320D83—81no—
321#3100—831no—
322#2931—831no—
323C2×#436—831no—
324#1589—831no—
325#1589—831no—
326#1589—831no—
327#1589—831no—
328#1581—831no—
329#1581—831no—
330#1581—831no—
331#1581D8311yescharacteristic, complemented
332#1571—831no—
333#1486—831no—
334#1480—831no—
335#1450—831no—
336C83#1792811yescharacteristic, complemented, Sylow
337C2×#131—1661no—
338C22×M16—1661no—
339#532—831no—
340C2×C4×D4—831no—
341#480—831no—
342#480—831no—
343#474—831no—
344#474—831no—
345#459—831no—
346#459—831no—
347#447—831no—
348#447—831no—
349C4×D8—831no—
350C4×D8—831no—
351C4×D8—831no—
352C4×D8C2×D8311yescharacteristic
353#436—831no—
354#436—831no—
355#436—831no—
356#436—831no—
357#436—831no—
358#436—1661no—
359#436—1661no—
360#436—1661no—
361#436C2×D8311yescharacteristic
362C8×D4—831no—
363C8×D4—1661no—
364C8×D4C2×D8311yescharacteristic, complemented
365#431—831no—
366#431—831no—
367#423—831no—
368C2×C4⋊C8—831no—
369#408—1661no—
370C2×C22⋊C8—1661no—
371C2×C4×C8—831no—
372#335—831no—
373#335—831no—
374#335—831no—
375#335C2×D8311yescharacteristic, complemented
376D4⋊C8—831no—
377D4⋊C8—831no—
378D4⋊C8—831no—
379D4⋊C8—831no—
380D4⋊C8—831no—
381D4⋊C8—831no—
382D4⋊C8C2×D8311yescharacteristic
383D4⋊C8C2×D8311yescharacteristic, complemented
384#322—831no—
385#322C2×D8311yescharacteristic, complemented
386C82—831no—
387C82—831no—
388C2×#41—1661no—
389C22×D4—831no—
390C23×C4—1661no—
391C2×D8—831no—
392C2×D8—831no—
393C2×D8—831no—
394C2×D8C4×D8311yescharacteristic
395#131—3324no—
396C2×M16—21no—
397C2×M16—3324no—
398C2×M16—1661no—
399C2×M16—1661no—
400C2×M16—1661no—
401C2×M16—1661no—
402C2×M16—1661no—
403C2×M16—1661no—
404C22×C8—21no—
405C22×C8—831no—
406C22×C8—831no—
407C22×C8—1661no—
408C22×C8—1661no—
409C4⋊Q8—831no—
410#127—831no—
411#124—1661no—
412#121—831no—
413#121—831no—
414#121—1661no—
415C4×D4—831no—
416C4×D4—831no—
417C4×D4—831no—
418C4×D4—831no—
419C4×D4—1661no—
420C4×D4—1661no—
421C4×D4—1661no—
422C4×D4C22×D8311yescharacteristic
423C4×D4C22×D8311yescharacteristic
424C2×C4⋊C4—831no—
425C2×C22⋊C4—1661no—
426C2×C42—831no—
427#107—831no—
428#107—831no—
429#107—831no—
430#107C4×D8311yescharacteristic
431C4⋊C8—831no—
432C4⋊C8—831no—
433C4⋊C8—831no—
434C4⋊C8—831no—
435C4⋊C8—1661no—
436C4⋊C8—1661no—
437C4⋊C8—1661no—
438C4⋊C8C22×D8311yescharacteristic
439C4⋊C8C22×D8311yescharacteristic
440#102—831no—
441#102—831no—
442#102—831no—
443#102—831no—
444#102—831no—
445#102—831no—
446#102C4×D8311yescharacteristic
447#102C4×D8311yescharacteristic
448C22⋊C8—21no—
449C22⋊C8—21no—
450C22⋊C8—831no—
451C22⋊C8—831no—
452C22⋊C8—1661no—
453C22⋊C8—1661no—
454C22⋊C8—1661no—
455C22⋊C8—1661no—
456C22⋊C8—1661no—
457C22⋊C8—1661no—
458C22⋊C8—1661no—
459C4×C8—831no—
460C4×C8—831no—
461C4×C8—831no—
462C4×C8—831no—
463C4×C8—831no—
464C4×C8C22×D8311yescharacteristic
465C4×C8C22×D8311yescharacteristic
466C4×C8C22×D8311yescharacteristic
467C24—1661no—
468#41—3322no—
469#41—3322no—
470C2×Q8—1661no—
471C2×D4—3321no—
472C2×D4—3321no—
473C2×D4—831no—
474C2×D4—831no—
475C2×D4—1661no—
476C2×D4—1661no—
477C2×D4—1661no—
478C2×D4—1661no—
479C2×D4—1661no—
480C2×D4C2×C4×D8311yescharacteristic
481C2×D4C2×C4×D8311yescharacteristic
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—3322no—
485C22×C4—3322no—
486C22×C4—831no—
487C22×C4—831no—
488C22×C4—831no—
489C22×C4—1661no—
490C22×C4—1661no—
491C22×C4—1661no—
492C22×C4—1661no—
493C22×C4—1661no—
494D8—41no—
495D8—3321no—
496D8—3321no—
497D8—3321no—
498M16—42no—
499M16—3322no—
500M16—3322no—
501M16—3322no—
502M16—3322no—
503M16—3322no—
504M16—3322no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—42no—
509C2×C8—3322no—
510C2×C8—3322no—
511C2×C8—831no—
512C2×C8—834no—
513C2×C8—831no—
514C2×C8—831no—
515C2×C8—831no—
516C2×C8—1662no—
517C2×C8—1661no—
518C2×C8—1661no—
519C2×C8—1661no—
520C2×C8C2×C4×D8311yescharacteristic
521C2×C8C2×C4×D8311yescharacteristic
522C2×C8D4×D8311yescharacteristic
523C2×C8D4×D8311yescharacteristic
524C4⋊C4—831no—
525C4⋊C4—831no—
526C4⋊C4—1661no—
527C4⋊C4—1661no—
528C4⋊C4C2×C4×D8311yescharacteristic
529C4⋊C4C2×C4×D8311yescharacteristic
530C22⋊C4—21no—
531C22⋊C4—21no—
532C22⋊C4—1661no—
533C22⋊C4—1661no—
534C22⋊C4—1661no—
535C22⋊C4—1661no—
536C22⋊C4—1661no—
537C22⋊C4—1661no—
538C42—831no—
539C42—831no—
540C42—831no—
541C42C23×D8311yescharacteristic
542C23—21no—
543C23—21no—
544C23—3322no—
545C23—3322no—
546C23—3322no—
547C23—831no—
548C23—1661no—
549C23—1661no—
550Q8—3322no—
551D4—21no—
552D4—21no—
553D4—21no—
554D4—21no—
555D4—3322no—
556D4—3322no—
557D4—3322no—
558D4—3322no—
559D4—3322no—
560D4—1661no—
561D4—1661no—
562D4—1661no—
563D4—1661no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—3321no—
571C2×C4—3322no—
572C2×C4—3322no—
573C2×C4—3322no—
574C2×C4—3321no—
575C2×C4—3321no—
576C2×C4—3321no—
577C2×C4—3321no—
578C2×C4—831no—
579C2×C4—832no—
580C2×C4—832no—
581C2×C4—831no—
582C2×C4—831no—
583C2×C4—1661no—
584C2×C4—1661no—
585C2×C4—1661no—
586C2×C4—1661no—
587C2×C4C22×C4×D8311yescharacteristic
588C2×C4C2×D4×D8311yescharacteristic
589C2×C4#41×D8311yescharacteristic
590C8—21no—
591C8—21no—
592C8—21no—
593C8—41no—
594C8—41no—
595C8—41no—
596C8—3321no—
597C8—3321no—
598C8—3321no—
599C8—832no—
600C8—1661no—
601C8—1661no—
602C8—1661no—
603C8D8×D8312yes—
604C22—41no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—3321no—
611C22—3321no—
612C22—3321no—
613C22—3321no—
614C22—3321no—
615C22—3321no—
616C22—832no—
617C22—6644no—
618C22—1661no—
619C22—1661no—
620C22C4×D4×D8311yescharacteristic
621C4—21no—
622C4—41no—
623C4—41no—
624C4—3321no—
625C4—3321no—
626C4—832no—
627C4—1661no—
628C4—1661no—
629C4#1912846611yescharacteristic
630C4#1912846711yescharacteristic
631C4C2×D8×D8311yescharacteristic
632C4#135×D8311yescharacteristic
633C2—81no—
634C2—81no—
635C2—6641no—
636C2—6641no—
637C2—1662no—
638C2#2491616111yescharacteristic
639C2C4×D8×D8311yescharacteristic
640C2#2491636311yescharacteristic
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#17928
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. #24916170C2semidirect
3. #436×D83C2semidirect
4. #24906348C2semidirect
5. #24906347C2semidirect
6. #24906345C2semidirect
7. #24906345C2semidirect
8. #24906294C2semidirect
9. #24906292C2semidirect
10. #24906290C2semidirect
11. #24906236C2semidirect
12. #24905886C2semidirect
13. #24905860C2semidirect
14. #24905748C2semidirect
15. C83×#1581C2semidirect
33. #19127779C22semidirect
35. #19127778C22semidirect
39. C8×D332C22semidirect
48. #19127454C22semidirect
49. #19127453C22semidirect
50. #19127453C22semidirect
52. #19127430C22semidirect
54. D4⋊(C83⋊C8)C22semidirect
56. #19127428C22semidirect
57. #19127422C22semidirect
58. C8×C83⋊C8C22semidirect
59. C8×C83⋊C8C22semidirect
62. C8×D4×C83C22semidirect
63. #335×C83C22semidirect
65. D4⋊C8×C83C22semidirect
66. #322×C83C22semidirect
188. C8×D83D8semidirect
331. #1581D83semidirect
336. C83#17928semidirect
364. C8×D4C2×D83semidirect
375. #335C2×D83semidirect
383. D4⋊C8C2×D83semidirect
385. #322C2×D83semidirect

Maximal subgroups

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

subgroupindex
#249174612
#249161702
#436×D832
#249063482
#249063472
#249063452
#249062942
#249062922
#249062902
#249062362
#249058862
#249058602
#249057482
C83×#15812
#1792883