Finite Group Databasebeta

← #35375455

Subgroups — #35375455

Order 21248.

662 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24917523C211yesmaximal, characteristic
2#24916209C211yesmaximal, characteristic, complemented
3#24906363C211yesmaximal, characteristic, complemented
4#24906353C211yesmaximal, characteristic, complemented
5#24906327C211yesmaximal, characteristic, complemented
6#24906318C211yesmaximal, characteristic, complemented
7#24906297C211yesmaximal, characteristic, complemented
8#24906292C211yesmaximal, characteristic, complemented
9#24906285C211yesmaximal, characteristic, complemented
10C8×#13050575C211yesmaximal, characteristic, complemented
11#24906249C211yesmaximal, characteristic, complemented
12#24905931C211yesmaximal, characteristic, complemented
13#24905850C211yesmaximal, characteristic, complemented
14#24905760C211yesmaximal, characteristic, complemented
15C83×#2970C211yesmaximal, characteristic, complemented
16C2×#14581800—21no—
17#19128251C2211yescharacteristic
18#19128025C411yescharacteristic
19#19128025C411yescharacteristic
20#19127996C411yescharacteristic
21#19127991C411yescharacteristic
22#19127901C411yescharacteristic
23#19127896C411yescharacteristic
24#19127846C411yescharacteristic
25#19127846C411yescharacteristic
26C4×#13050577C2211yescharacteristic
27C4×#13050577C2211yescharacteristic
28C4×#13050575C2211yescharacteristic
29C4×#13050575C2211yescharacteristic
30#19127780C2211yescharacteristic
31#19127780C2211yescharacteristic
32#19127774—21no—
33#19127774—21no—
34C8×D332C2211yescharacteristic
35C8×D332C2211yescharacteristic
36#19127754C2211yescharacteristic
37#19127754C2211yescharacteristic
38C2×C4⋊(C83⋊C8)C2211yescharacteristic
39#19127669—21no—
40#19127453C2211yescharacteristic
41#19127453C2211yescharacteristic
42#19127451C2211yescharacteristic
43#19127451C2211yescharacteristic
44Q8⋊(C83⋊C8)C2211yescharacteristic
45Q8⋊(C83⋊C8)C2211yescharacteristic
46D4⋊(C83⋊C8)C2211yescharacteristic
47D4⋊(C83⋊C8)C2211yescharacteristic
48#19127428C2211yescharacteristic
49#19127428C2211yescharacteristic
50#19127428C2211yescharacteristic
51#19127428C2211yescharacteristic
52#19127423C2211yescharacteristic
53#19127423C2211yescharacteristic
54#19127422C2211yescharacteristic
55C8×C83⋊C8C2211yescharacteristic
56C4×#41×C83C2211yescharacteristic
57#446×C83C2211yescharacteristic
58#446×C83C2211yescharacteristic
59#435×C83—21no—
60#435×C83—21no—
61C8×D4×C83C2211yescharacteristic
62C8×D4×C83C2211yescharacteristic
63#432×C83—21no—
64#432×C83C2211yescharacteristic
65C4×M16×C83C2211yescharacteristic
66C2×#97×C83—21no—
67C2×#13050590—21no—
68C2×#13050577C2×C411yescharacteristic
69C2×#13050577C2×C411yescharacteristic
70C2×#13050575C2×C411yescharacteristic
71C2×#13050575C2×C411yescharacteristic
72#14581800—44no—
73C2×#13050573—21no—
74C2×#13050573—21no—
75C22×C83⋊C8—21no—
76C2×C8×D83—21no—
77C2×C8×D83—21no—
78#14581792C2×C411yescharacteristic
79#14581789C2×C411yescharacteristic
80#14581780—21no—
81#14581766—21no—
82#14581766C2×C411yescharacteristic
83#14581766C2×C411yescharacteristic
84C4×D332—21no—
85C4×D332C2311yescharacteristic
86C4×D332C2311yescharacteristic
87#14581725C2×C411yescharacteristic
88#14581725C2×C411yescharacteristic
89#14581723C2×C411yescharacteristic
90#14581723C2×C411yescharacteristic
91C4⋊(C83⋊C8)—21no—
92C4⋊(C83⋊C8)—21no—
93C4⋊(C83⋊C8)C2×C411yescharacteristic
94C4⋊(C83⋊C8)C2×C411yescharacteristic
95C4⋊(C83⋊C8)C2311yescharacteristic
96C4⋊(C83⋊C8)C2311yescharacteristic
97#14581716C2×C411yescharacteristic
98#14581716C2×C411yescharacteristic
99#14581713C2×C411yescharacteristic
100#14581713C2×C411yescharacteristic
101#14581712C2×C411yescharacteristic
102#14581712C2×C411yescharacteristic
103#14581712C2×C411yescharacteristic
104#14581712C2×C411yescharacteristic
105C22⋊(C83⋊C8)—21no—
106C22⋊(C83⋊C8)C2×C411yescharacteristic
107C22⋊(C83⋊C8)C2×C411yescharacteristic
108(C2×D83)⋊C8—21no—
109(C2×D83)⋊C8—21no—
110(C2×D83)⋊C8—21no—
111(C2×D83)⋊C8—21no—
112C4×C83⋊C8C2311yescharacteristic
113C4×C83⋊C8C2311yescharacteristic
114C4×C83⋊C8C2311yescharacteristic
115C4×C83⋊C8C2311yescharacteristic
116C2×#41×C83C2×C411yescharacteristic
117C2×M16×C83D411yescharacteristic
118C2×M16×C83D411yescharacteristic
119C22×C8×C83—21no—
120C22×C8×C83—21no—
121C4×Q8×C83C2×C411yescharacteristic
122C4×Q8×C83C2311yescharacteristic
123C4×D4×C83—21no—
124C4×D4×C83—21no—
125C4×D4×C83C2×C411yescharacteristic
126C4×D4×C83C2311yescharacteristic
127#117×C83—21no—
128#117×C83C2×C411yescharacteristic
129C2×C42×C83—21no—
130C2×C42×C83C2311yescharacteristic
131C4⋊C8×C83—21no—
132C4⋊C8×C83—21no—
133C4⋊C8×C83C2311yescharacteristic
134C4⋊C8×C83C2311yescharacteristic
135C22⋊C8×C83—21no—
136C22⋊C8×C83—21no—
137C22⋊C8×C83—21no—
138#97×C83—21no—
139#97×C83—21no—
140#97×C83—21no—
141#97×C83D411yescharacteristic
142#97×C83D411yescharacteristic
143C4×C8×C83C2311yescharacteristic
144C4×C8×C83C2311yescharacteristic
145#13050590—42no—
146#13050590—42no—
147C2×Dic166—21no—
148C2×D332—21no—
149C2×D332—21no—
150C2×D332C22×C411yescharacteristic
151C2×D332C22×C411yescharacteristic
152C2×C22⋊D83—21no—
153C2×C22⋊D83—21no—
154C2×C4×D83—21no—
155C2×C4×D83—21no—
156#13050577—41no—
157#13050577—41no—
158#13050575—22no—
159#13050575—22no—
160#13050573—42no—
161#13050573—42no—
162C2×C83⋊C8—21no—
163C2×C83⋊C8—21no—
164C2×C83⋊C8—21no—
165C2×C83⋊C8—21no—
166C2×C83⋊C8—42no—
167C2×C83⋊C8C22×C411yescharacteristic
168C2×C83⋊C8C22×C411yescharacteristic
169C2×C83⋊C8C22×C411yescharacteristic
170C2×C83⋊C8C22×C411yescharacteristic
171C8×D83—42no—
172C8×D83—42no—
173C4⋊Dic83—21no—
174C4⋊Dic83C22×C411yescharacteristic
175C4⋊Dic83C22×C411yescharacteristic
176(C2×D83)⋊C4—21no—
177(C2×D83)⋊C4—21no—
178(C2×D83)⋊C4—21no—
179(C2×D83)⋊C4—21no—
180#41×C83—22no—
181#41×C83—41no—
182C2×Q8×C83C22×C411yescharacteristic
183C2×D4×C83—21no—
184C2×D4×C83C22×C411yescharacteristic
185C22×C4×C83—21no—
186C22×C4×C83—21no—
187C22×C4×C83—21no—
188C22×C4×C83C22×C411yescharacteristic
189C22×C4×C83C2×D411yescharacteristic
190C22×C4×C83#4111yescharacteristic
191M16×C83—22no—
192M16×C83—41no—
193C2×C8×C83—21no—
194C2×C8×C83—21no—
195C2×C8×C83—42no—
196C2×C8×C83—42no—
197C2×C8×C83C2×D411yescharacteristic
198C2×C8×C83C2×D411yescharacteristic
199C2×C8×C83C2×D411yescharacteristic
200C2×C8×C83C2×D411yescharacteristic
201C4⋊C4×C83—21no—
202C4⋊C4×C83—21no—
203C4⋊C4×C83C22×C411yescharacteristic
204C4⋊C4×C83C22×C411yescharacteristic
205C22⋊C4×C83—21no—
206C22⋊C4×C83—21no—
207C22⋊C4×C83—21no—
208C42×C83—21no—
209C42×C83—21no—
210C42×C83—21no—
211C42×C83C22×C411yescharacteristic
212C42×C83C2×D411yescharacteristic
213C42×C83#4111yescharacteristic
214C42×C83C2411yescharacteristic
215C23×C83—21no—
216C23×C83C4×D411yescharacteristic
217C22×D83—21no—
218C22×D83—21no—
219Q8×C83—21no—
220Q8×C83—21no—
221D4×C83—21no—
222D4×C83—42no—
223D4×C83#13112yescomplemented
224C2×C4×C83—21no—
225C2×C4×C83—21no—
226C2×C4×C83—21no—
227C2×C4×C83—21no—
228C2×C4×C83—21no—
229C2×C4×C83—21no—
230C2×C4×C83—21no—
231C2×C4×C83—41no—
232C2×C4×C83—42no—
233C2×C4×C83—41no—
234C2×C4×C83—41no—
235C2×C4×C83—81no—
236C2×C4×C83C4×D411yescharacteristic
237C2×C4×C83C4×D411yescharacteristic
238C2×C4×C83C4×D411yescharacteristic
239C2×C4×C83C23×C411yescharacteristic
240C2×C4×C83C22×D411yescharacteristic
241C2×C4×C83C2×#4111yescharacteristic
242C22⋊D83—42no—
243C22⋊D83—42no—
244C2×Dic83—21no—
245C2×Dic83—21no—
246D332—21no—
247D332—21no—
248D332—21no—
249D332—21no—
250D332—42no—
251D332—81no—
252C4×D83—41no—
253C4×D83—41no—
254C4×D83—41no—
255C4×D83—41no—
256Dic166—42no—
257C8×C83—21no—
258C8×C83—21no—
259C8×C83—21no—
260C8×C83—41no—
261C8×C83—41no—
262C8×C83AGL(1,ℤ/8)12yes—
263C83⋊C8—21no—
264C83⋊C8—21no—
265C83⋊C8—21no—
266C83⋊C8—21no—
267C83⋊C8—41no—
268C83⋊C8—41no—
269C83⋊C8—41no—
270C83⋊C8—41no—
271C22×C83—21no—
272C22×C83—22no—
273C22×C83—21no—
274C22×C83—21no—
275C22×C83—41no—
276C22×C83—41no—
277C22×C83C2×C4×D411yescharacteristic
278C2×D83—41no—
279C2×D83—41no—
280C2×D83—41no—
281C2×D83—41no—
282C2×D83—41no—
283C2×D83—41no—
284C4×C83—21no—
285C4×C83—21no—
286C4×C83—21no—
287C4×C83—21no—
288C4×C83—41no—
289C4×C83—41no—
290C4×C83—41no—
291C4×C83C2×#13111yescharacteristic
292C4×C83#56811yescharacteristic
293C4×C83C2×AGL(1,ℤ/8)11yescharacteristic
294C4×C83#57511yescharacteristic
295Dic83—41no—
296Dic83—41no—
297#18007—831nomaximal, Sylow
298C2×C83—41no—
299C2×C83—42no—
300C2×C83#293311yescharacteristic
301C2×C83C4×AGL(1,ℤ/8)11yescharacteristic
302C2×C83#295711yescharacteristic
303D83—81no—
304D83—81no—
305#3106—831no—
306#2970D8311yescharacteristic, complemented
307#2935—831no—
308#1592—831no—
309#1590—831no—
310#1586—831no—
311#1585—831no—
312#1582—831no—
313#1581—831no—
314#1580—831no—
315C8×D8—831no—
316#1573—831no—
317#1496—831no—
318#1477—831no—
319#1453—831no—
320C83#1800711yescharacteristic, complemented, Sylow
321C2×#131—1661no—
322#532—831no—
323C4×#41C2×D8311yescharacteristic
324#482—831no—
325#482—831no—
326#477—831no—
327#476—831no—
328#460—831no—
329#459—831no—
330#449—831no—
331#449—831no—
332#446C2×D8311yescharacteristic
333#446C2×D8311yescharacteristic
334C4×SD16—831no—
335C4×SD16—831no—
336C4×D8—831no—
337C4×D8—831no—
338#436—831no—
339#436—831no—
340#435—21no—
341#435—21no—
342#435—1661no—
343#435—1661no—
344C8×D4—831no—
345C8×D4—831no—
346C8×D4C2×D8311yescharacteristic
347C8×D4C2×D8311yescharacteristic
348#432—21no—
349#432C2×D8311yescharacteristic
350#431—831no—
351#431—831no—
352C2×C4⋊C8—831no—
353#408—1661no—
354C4×M16C2×D8311yescharacteristic
355C2×#97—21no—
356#335—831no—
357#335—831no—
358#334—831no—
359#334—831no—
360Q8⋊C8—831no—
361Q8⋊C8—831no—
362D4⋊C8—831no—
363D4⋊C8—831no—
364D4⋊C8—831no—
365D4⋊C8—831no—
366D4⋊C8—831no—
367D4⋊C8—831no—
368#322—831no—
369#322—831no—
370#322—831no—
371C82—831no—
372C2×#41—1661no—
373C2×#41C4×D8311yescharacteristic
374C2×SD16—831no—
375C2×SD16—831no—
376C2×D8—831no—
377C2×D8—831no—
378#131—3324no—
379C2×M16—1661no—
380C2×M16—1661no—
381C2×M16C22⋊D8311yescharacteristic
382C2×M16C22⋊D8311yescharacteristic
383C22×C8—21no—
384C22×C8—21no—
385C22×C8—1661no—
386C22×C8—1661no—
387C22×C8—1661no—
388C4⋊Q8—831no—
389#127—831no—
390#124—1661no—
391#121—831no—
392#121—831no—
393#121—1661no—
394C4×Q8C4×D8311yescharacteristic
395C4×Q8C22×D8311yescharacteristic
396C4×D4—21no—
397C4×D4—21no—
398C4×D4—831no—
399C4×D4—831no—
400C4×D4—1661no—
401C4×D4C4×D8311yescharacteristic
402C4×D4C22×D8311yescharacteristic
403#117—21no—
404#117C4×D8311yescharacteristic
405C2×C42—21no—
406C2×C42C22×D8311yescharacteristic
407#107—831no—
408#107—831no—
409#106—831no—
410#106—831no—
411C4⋊C8—21no—
412C4⋊C8—21no—
413C4⋊C8—831no—
414C4⋊C8—831no—
415C4⋊C8—831no—
416C4⋊C8—831no—
417C4⋊C8—1661no—
418C4⋊C8—1661no—
419C4⋊C8C22×D8311yescharacteristic
420C4⋊C8C22×D8311yescharacteristic
421#103—831no—
422#103—831no—
423#102—831no—
424#102—831no—
425#102—831no—
426#102—831no—
427#102—831no—
428#102—831no—
429C22⋊C8—21no—
430C22⋊C8—21no—
431C22⋊C8—21no—
432C22⋊C8—831no—
433C22⋊C8—831no—
434C22⋊C8—1661no—
435C22⋊C8—1661no—
436C22⋊C8—1661no—
437C22⋊C8—1661no—
438C22⋊C8—1661no—
439#97—21no—
440#97—21no—
441#97—21no—
442#97C22⋊D8311yescharacteristic
443#97C22⋊D8311yescharacteristic
444C4×C8—831no—
445C4×C8—831no—
446C4×C8—831no—
447C4×C8—831no—
448C4×C8C22×D8311yescharacteristic
449C4×C8C22×D8311yescharacteristic
450#41—22no—
451#41—41no—
452#41—3322no—
453#41—3322no—
454C2×Q8—1661no—
455C2×Q8C2×C4×D8311yescharacteristic
456C2×D4—21no—
457C2×D4—831no—
458C2×D4—831no—
459C2×D4—1661no—
460C2×D4—1661no—
461C2×D4—1661no—
462C2×D4—1661no—
463C2×D4C2×C4×D8311yescharacteristic
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—1661no—
468C22×C4—1661no—
469C22×C4C2×C4×D8311yescharacteristic
470C22×C4C2×C22⋊D8311yescharacteristic
471C22×C4#1305059011yescharacteristic
472SD16—3321no—
473SD16—3321no—
474D8—1662no—
475D8—1662no—
476M16—22no—
477M16—41no—
478M16—3322no—
479M16—3322no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—42no—
483C2×C8—42no—
484C2×C8—3322no—
485C2×C8—3322no—
486C2×C8—3322no—
487C2×C8—831no—
488C2×C8—831no—
489C2×C8—831no—
490C2×C8—831no—
491C2×C8—1661no—
492C2×C8—1661no—
493C2×C8—1661no—
494C2×C8—1661no—
495C2×C8C2×C22⋊D8311yescharacteristic
496C2×C8C2×C22⋊D8311yescharacteristic
497C2×C8C2×C22⋊D8311yescharacteristic
498C2×C8C2×C22⋊D8311yescharacteristic
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—831no—
502C4⋊C4—831no—
503C4⋊C4—1661no—
504C4⋊C4C2×C4×D8311yescharacteristic
505C4⋊C4C2×C4×D8311yescharacteristic
506C22⋊C4—21no—
507C22⋊C4—21no—
508C22⋊C4—21no—
509C22⋊C4—1661no—
510C22⋊C4—1661no—
511C22⋊C4—1661no—
512C22⋊C4—1661no—
513C42—21no—
514C42—21no—
515C42—21no—
516C42C2×C4×D8311yescharacteristic
517C42C2×C22⋊D8311yescharacteristic
518C42#1305059011yescharacteristic
519C42C23×D8311yescharacteristic
520C23—21no—
521C23—1661no—
522C23—1661no—
523C23C4×C22⋊D8311yescharacteristic
524Q8—21no—
525Q8—21no—
526Q8—3322no—
527D4—21no—
528D4—42no—
529D4—3322no—
530D4—3322no—
531D4—3322no—
532D4—6641no—
533D4—1661no—
534D4—1661no—
535D4—1661no—
536D4—1661no—
537D4#1458180012yescomplemented
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—81no—
550C2×C4—3321no—
551C2×C4—3321no—
552C2×C4—3321no—
553C2×C4—3321no—
554C2×C4—1661no—
555C2×C4—1661no—
556C2×C4C4×C22⋊D8311yescharacteristic
557C2×C4C4×C22⋊D8311yescharacteristic
558C2×C4C4×C22⋊D8311yescharacteristic
559C2×C4C22×C4×D8311yescharacteristic
560C2×C4C22×C22⋊D8311yescharacteristic
561C2×C4C2×#1305059011yescharacteristic
562C8—21no—
563C8—21no—
564C8—21no—
565C8—41no—
566C8—41no—
567C8—3321no—
568C8—3321no—
569C8—3321no—
570C8—3321no—
571C8—1661no—
572C8—1661no—
573C8—1661no—
574C8—1661no—
575C8#1458181712yes—
576C22—21no—
577C22—21no—
578C22—21no—
579C22—22no—
580C22—41no—
581C22—41no—
582C22—3321no—
583C22—3321no—
584C22—3321no—
585C22—3321no—
586C22—3321no—
587C22—3321no—
588C22C2×C4×C22⋊D8311yescharacteristic
589C4—21no—
590C4—21no—
591C4—21no—
592C4—21no—
593C4—41no—
594C4—41no—
595C4—41no—
596C4—3321no—
597C4—3321no—
598C4C2×#1458180011yescharacteristic
599C4#1912846711yescharacteristic
600C4C2×#1458181711yescharacteristic
601C4#1912849911yescharacteristic
602C2—41no—
603C2—42no—
604C2—6641no—
605C2—6641no—
606C2#2491618811yescharacteristic
607C2C4×#1458181711yescharacteristic
608C2#2491637811yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D83
commutatorC2×C4×C83C23×C4
FrattiniC42C23×D83
FittingC83×#2970C2
socleC22×C83C2×C4×D4
prime radical#24917523C2
cosocleC42×C83C24
hypercenter#2970D83
hypocenterC83#18007
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C4×M16 ◁ #2970

Complements

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

NH ≅ G/Ntype
2. #24916209C2semidirect
3. #24906363C2semidirect
4. #24906353C2semidirect
5. #24906327C2semidirect
6. #24906318C2semidirect
7. #24906297C2semidirect
8. #24906292C2semidirect
9. #24906285C2semidirect
10. C8×#13050575C2semidirect
11. #24906249C2semidirect
12. #24905931C2semidirect
13. #24905850C2semidirect
14. #24905760C2semidirect
15. C83×#2970C2semidirect
223. D4×C83#131semidirect
306. #2970D83semidirect
320. C83#18007semidirect
537. D4#14581800semidirect

Maximal subgroups

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

subgroupindex
#249175232
#249162092
#249063632
#249063532
#249063272
#249063182
#249062972
#249062922
#249062852
C8×#130505752
#249062492
#249059312
#249058502
#249057602
C83×#29702
#1800783