Finite Group Databasebeta

← #35376264

Subgroups — #35376264

Order 21248.

656 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#24918059C211yesmaximal, characteristic
2#24916468C211yesmaximal, characteristic, complemented
3#24916206C211yesmaximal, characteristic, complemented
4#24906374C211yesmaximal, characteristic, complemented
5#24906350C211yesmaximal, characteristic, complemented
6#24906338C211yesmaximal, characteristic, complemented
7#24906330C211yesmaximal, characteristic, complemented
8#24906302C211yesmaximal, characteristic, complemented
9#24906289C211yesmaximal, characteristic, complemented
10#24906278C211yesmaximal, characteristic, complemented
11#24906255C211yesmaximal, characteristic, complemented
12#24906213C211yesmaximal, characteristic, complemented
13#24905881C211yesmaximal, characteristic, complemented
14#24905766C211yesmaximal, characteristic, complemented
15C83×#1483C211yesmaximal, characteristic, complemented
16C2×#14581798—21no—
17#19128251C2211yescharacteristic
18#19128111C411yescharacteristic
19#19128091C411yescharacteristic
20#19128067C411yescharacteristic
21#19128060C411yescharacteristic
22#19127952C411yescharacteristic
23#19127952C411yescharacteristic
24#19127949C411yescharacteristic
25#19127949C411yescharacteristic
26#19127832C2211yescharacteristic
27Q8×C83⋊C8C2211yescharacteristic
28C4×#13050577C2211yescharacteristic
29C4×#13050577C2211yescharacteristic
30C4×#13050575C2211yescharacteristic
31C4×#13050575C2211yescharacteristic
32#19127779—21no—
33#19127779C2211yescharacteristic
34#19127778C2211yescharacteristic
35#19127778C2211yescharacteristic
36#19127773—21no—
37#19127771—21no—
38#19127771—21no—
39D4×C83⋊C8C2211yescharacteristic
40#19127766C2211yescharacteristic
41#19127766C2211yescharacteristic
42#19127762—21no—
43#19127754C2211yescharacteristic
44#19127667—21no—
45#19127657—21no—
46C4×#13050574C2211yescharacteristic
47C4×#13050574C2211yescharacteristic
48#19127453C2211yescharacteristic
49#19127453C2211yescharacteristic
50#19127451C2211yescharacteristic
51#19127451C2211yescharacteristic
52#19127430C2211yescharacteristic
53#19127430C2211yescharacteristic
54#19127430C2211yescharacteristic
55#19127430C2211yescharacteristic
56#19127423C2211yescharacteristic
57#19127423C2211yescharacteristic
58#19127422C2211yescharacteristic
59C8×C83⋊C8C2211yescharacteristic
60#529×C83C2211yescharacteristic
61C2×C4⋊C8×C83C2211yescharacteristic
62C4×M16×C83C2211yescharacteristic
63Q8⋊C8×C83C2211yescharacteristic
64Q8⋊C8×C83C2211yescharacteristic
65D4⋊C8×C83C2211yescharacteristic
66D4⋊C8×C83C2211yescharacteristic
67C2×#13050590—21no—
68C2×#13050577C2×C411yescharacteristic
69C2×#13050577C2×C411yescharacteristic
70C2×#13050575C2×C411yescharacteristic
71C2×#13050575C2×C411yescharacteristic
72#14581798—44no—
73C2×#13050574—21no—
74C2×#13050574C2×C411yescharacteristic
75C2×#13050574C2×C411yescharacteristic
76C2×#13050574D411yescharacteristic
77C2×#13050574D411yescharacteristic
78C2×#13050573—21no—
79C2×#13050573—21no—
80C22×C83⋊C8—21no—
81C2×C8×D83—21no—
82C2×C8×D83—21no—
83#14581792C2×C411yescharacteristic
84#14581789C2×C411yescharacteristic
85#14581780—21no—
86#14581766—21no—
87#14581766C2×C411yescharacteristic
88#14581766C2×C411yescharacteristic
89C4×D332—21no—
90C4×D332C2311yescharacteristic
91C4×D332C2311yescharacteristic
92#14581725C2×C411yescharacteristic
93#14581725C2×C411yescharacteristic
94#14581723C2×C411yescharacteristic
95#14581723C2×C411yescharacteristic
96C4⋊(C83⋊C8)—21no—
97C4⋊(C83⋊C8)—21no—
98C4⋊(C83⋊C8)C2311yescharacteristic
99C4⋊(C83⋊C8)C2311yescharacteristic
100#14581716C2×C411yescharacteristic
101#14581716C2×C411yescharacteristic
102#14581713C2×C411yescharacteristic
103#14581713C2×C411yescharacteristic
104#14581712C2×C411yescharacteristic
105#14581712C2×C411yescharacteristic
106#14581712C2×C411yescharacteristic
107#14581712C2×C411yescharacteristic
108C22⋊(C83⋊C8)—21no—
109C22⋊(C83⋊C8)—21no—
110C22⋊(C83⋊C8)—21no—
111(C2×D83)⋊C8—21no—
112(C2×D83)⋊C8—21no—
113(C2×D83)⋊C8—21no—
114(C2×D83)⋊C8—21no—
115#14581703—21no—
116#14581703C2×C411yescharacteristic
117#14581703C2×C411yescharacteristic
118#14581703D411yescharacteristic
119#14581703D411yescharacteristic
120C4×C83⋊C8—21no—
121C4×C83⋊C8—21no—
122C4×C83⋊C8C2311yescharacteristic
123C4×C83⋊C8C2311yescharacteristic
124C4×C83⋊C8C2311yescharacteristic
125C4×C83⋊C8C2311yescharacteristic
126C2×M16×C83—21no—
127C22×C8×C83—21no—
128#126×C83—21no—
129#125×C83C2×C411yescharacteristic
130#124×C83C2×C411yescharacteristic
131#123×C83—21no—
132C22⋊Q8×C83C2×C411yescharacteristic
133#121×C83C2×C411yescharacteristic
134C4×Q8×C83C2311yescharacteristic
135C4×D4×C83—21no—
136C4×D4×C83C2311yescharacteristic
137#117×C83—21no—
138C2×C42×C83C2311yescharacteristic
139C4⋊C8×C83—21no—
140C4⋊C8×C83C2311yescharacteristic
141C4⋊C8×C83C2311yescharacteristic
142#97×C83—21no—
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—22no—
157#13050577—41no—
158#13050575—22no—
159#13050575—41no—
160#13050574—22no—
161#13050574—22no—
162#13050574—41no—
163#13050574—41no—
164#13050574—42no—
165#13050573—42no—
166#13050573—42no—
167C2×C83⋊C8—21no—
168C2×C83⋊C8—21no—
169C2×C83⋊C8—42no—
170C2×C83⋊C8C22×C411yescharacteristic
171C2×C83⋊C8C22×C411yescharacteristic
172C2×C83⋊C8C22×C411yescharacteristic
173C2×C83⋊C8C22×C411yescharacteristic
174C2×C83⋊C8C2×D411yescharacteristic
175C2×C83⋊C8C2×D411yescharacteristic
176C2×C83⋊C8C2×D411yescharacteristic
177C2×C83⋊C8C2×D411yescharacteristic
178C8×D83—42no—
179C8×D83—42no—
180C4⋊Dic83—21no—
181C4⋊Dic83C22×C411yescharacteristic
182C4⋊Dic83C22×C411yescharacteristic
183(C2×D83)⋊C4—21no—
184(C2×D83)⋊C4—21no—
185(C2×D83)⋊C4—21no—
186(C2×D83)⋊C4—21no—
187C2×Q8×C83C22×C411yescharacteristic
188C2×D4×C83—21no—
189C2×D4×C83C22×C411yescharacteristic
190C22×C4×C83—21no—
191C22×C4×C83C22×C411yescharacteristic
192C22×C4×C83C2×D411yescharacteristic
193C22×C4×C83#4111yescharacteristic
194M16×C83—42no—
195C2×C8×C83—21no—
196C2×C8×C83—21no—
197C2×C8×C83—21no—
198C2×C8×C83—21no—
199C2×C8×C83—42no—
200C4⋊C4×C83—21no—
201C4⋊C4×C83—21no—
202C4⋊C4×C83—21no—
203C4⋊C4×C83—21no—
204C4⋊C4×C83C22×C411yescharacteristic
205C4⋊C4×C83C22×C411yescharacteristic
206C22⋊C4×C83—21no—
207C22⋊C4×C83—21no—
208C22⋊C4×C83—21no—
209C22⋊C4×C83—21no—
210C22⋊C4×C83—21no—
211C42×C83—21no—
212C42×C83C22×C411yescharacteristic
213C42×C83C2×D411yescharacteristic
214C42×C83#4111yescharacteristic
215C42×C83C2411yescharacteristic
216C23×C83—21no—
217C23×C83C4×D411yescharacteristic
218C22×D83—21no—
219C22×D83—21no—
220Q8×C83—21no—
221Q8×C83—21no—
222D4×C83—21no—
223D4×C83—21no—
224D4×C83—81no—
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—21no—
232C2×C4×C83—41no—
233C2×C4×C83—41no—
234C2×C4×C83—41no—
235C2×C4×C83C4×D411yescharacteristic
236C2×C4×C83C4×D411yescharacteristic
237C2×C4×C83C4×D411yescharacteristic
238C2×C4×C83C23×C411yescharacteristic
239C2×C4×C83C22×D411yescharacteristic
240C2×C4×C83C2×#4111yescharacteristic
241C22⋊D83—42no—
242C22⋊D83—42no—
243C2×Dic83—21no—
244C2×Dic83—21no—
245D332—21no—
246D332—21no—
247D332—21no—
248D332—21no—
249D332—42no—
250D332—81no—
251C4×D83—41no—
252C4×D83—41no—
253C4×D83—41no—
254C4×D83—41no—
255Dic166—42no—
256C8×C83—22no—
257C8×C83—41no—
258C8×C83—41no—
259C8×C83—41no—
260C83⋊C8—21no—
261C83⋊C8—21no—
262C83⋊C8—21no—
263C83⋊C8—21no—
264C83⋊C8—21no—
265C83⋊C8—21no—
266C83⋊C8—41no—
267C83⋊C8—41no—
268C83⋊C8#13112yescomplemented
269C83⋊C8AGL(1,ℤ/8)12yes—
270C22×C83—21no—
271C22×C83—21no—
272C22×C83—21no—
273C22×C83—41no—
274C22×C83—41no—
275C22×C83—41no—
276C22×C83C2×C4×D411yescharacteristic
277C2×D83—41no—
278C2×D83—41no—
279C2×D83—41no—
280C2×D83—41no—
281C2×D83—41no—
282C2×D83—41no—
283C4×C83—21no—
284C4×C83—21no—
285C4×C83—21no—
286C4×C83—21no—
287C4×C83—41no—
288C4×C83—41no—
289C4×C83—41no—
290C4×C83C2×#13111yescharacteristic
291C4×C83#56811yescharacteristic
292C4×C83C2×AGL(1,ℤ/8)11yescharacteristic
293C4×C83#57511yescharacteristic
294Dic83—41no—
295Dic83—41no—
296#18076—831nomaximal, Sylow
297C2×C83—41no—
298C2×C83—81no—
299C2×C83#293511yescharacteristic
300C2×C83C4×AGL(1,ℤ/8)11yescharacteristic
301C2×C83#295611yescharacteristic
302D83—81no—
303D83—81no—
304#3171—831no—
305#2969—831no—
306#2935—831no—
307#1594—831no—
308#1590—831no—
309#1588—831no—
310#1586—831no—
311#1582—831no—
312#1581—831no—
313SD16⋊C8—831no—
314C8×D8—831no—
315#1565—831no—
316#1484—831no—
317#1483D8311yescharacteristic, complemented
318#1455—831no—
319C83#1807611yescharacteristic, complemented, Sylow
320C2×#131—1661no—
321#532—831no—
322#529C2×D8311yescharacteristic
323#501—831no—
324#496—831no—
325#490—831no—
326#489—831no—
327#469—831no—
328#469—831no—
329#468—831no—
330#468—831no—
331#446—831no—
332C8×Q8—831no—
333C4×SD16—831no—
334C4×SD16—831no—
335C4×D8—831no—
336C4×D8—831no—
337#436—831no—
338#436—831no—
339#436—831no—
340#436—1661no—
341#435—1661no—
342#435—1661no—
343#435—1661no—
344C8×D4—831no—
345C8×D4—831no—
346C8×D4—831no—
347#433—1661no—
348#431—831no—
349C2×C4⋊C8C2×D8311yescharacteristic
350#408—1661no—
351#405—1661no—
352C4×M16—831no—
353C4×M16—831no—
354C4×M16C2×D8311yescharacteristic
355#335—831no—
356#335—831no—
357#334—831no—
358#334—831no—
359Q8⋊C8C2×D8311yescharacteristic
360Q8⋊C8C2×D8311yescharacteristic
361D4⋊C8—831no—
362D4⋊C8—831no—
363D4⋊C8—831no—
364D4⋊C8—831no—
365D4⋊C8C2×D8311yescharacteristic
366D4⋊C8C2×D8311yescharacteristic
367#322—831no—
368#322—831no—
369#322—831no—
370C82—831no—
371C2×#41—1661no—
372C2×SD16—831no—
373C2×SD16—831no—
374C2×D8—831no—
375C2×D8—831no—
376#131—3324no—
377C2×M16—21no—
378C2×M16—831no—
379C2×M16—831no—
380C2×M16—831no—
381C2×M16—831no—
382C2×M16—1661no—
383C2×M16—1661no—
384C2×M16—1661no—
385C22×C8—21no—
386C22×C8—1661no—
387C22×C8—1661no—
388C22×C8—1661no—
389C4⋊Q8—831no—
390#127—831no—
391#126—21no—
392#125C4×D8311yescharacteristic
393#124—1661no—
394#124C4×D8311yescharacteristic
395#123—21no—
396C22⋊Q8C4×D8311yescharacteristic
397#121—831no—
398#121—831no—
399#121—1661no—
400#121C4×D8311yescharacteristic
401C4×Q8C22×D8311yescharacteristic
402C4×D4—21no—
403C4×D4—831no—
404C4×D4—831no—
405C4×D4—1661no—
406C4×D4C22×D8311yescharacteristic
407#117—21no—
408C2×C42C22×D8311yescharacteristic
409#107—831no—
410#107—831no—
411#106—831no—
412#106—831no—
413C4⋊C8—21no—
414C4⋊C8—831no—
415C4⋊C8—831no—
416C4⋊C8—1661no—
417C4⋊C8—1661no—
418C4⋊C8C22×D8311yescharacteristic
419C4⋊C8C22×D8311yescharacteristic
420#103—831no—
421#103—831no—
422#102—831no—
423#102—831no—
424#102—831no—
425#102—831no—
426#102—831no—
427#102—831no—
428C22⋊C8—1661no—
429C22⋊C8—1661no—
430C22⋊C8—1661no—
431C22⋊C8—1661no—
432C22⋊C8—1661no—
433C22⋊C8—1661no—
434C22⋊C8—1661no—
435#97—21no—
436#97—831no—
437#97—831no—
438#97—831no—
439#97—831no—
440#97—1661no—
441C4×C8—831no—
442C4×C8—831no—
443C4×C8—831no—
444C4×C8—831no—
445C4×C8—1661no—
446C4×C8—1661no—
447C4×C8C22×D8311yescharacteristic
448C4×C8C22×D8311yescharacteristic
449#41—3322no—
450#41—3322no—
451C2×Q8—1661no—
452C2×Q8C2×C4×D8311yescharacteristic
453C2×D4—21no—
454C2×D4—831no—
455C2×D4—831no—
456C2×D4—1661no—
457C2×D4—1661no—
458C2×D4—1661no—
459C2×D4—1661no—
460C2×D4C2×C4×D8311yescharacteristic
461C22×C4—21no—
462C22×C4—1661no—
463C22×C4—1661no—
464C22×C4C2×C4×D8311yescharacteristic
465C22×C4D4×D8311yescharacteristic
466C22×C4#1305058911yescharacteristic
467SD16—3321no—
468SD16—1662no—
469D8—3321no—
470D8—1662no—
471M16—42no—
472M16—3322no—
473M16—3322no—
474M16—3322no—
475M16—3321no—
476M16—3321no—
477M16—1662no—
478M16—1662no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
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—831no—
492C2×C8—831no—
493C2×C8—831no—
494C2×C8—831no—
495C2×C8—1661no—
496C2×C8—1661no—
497C4⋊C4—21no—
498C4⋊C4—21no—
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—21no—
510C22⋊C4—21no—
511C22⋊C4—1661no—
512C22⋊C4—1661no—
513C22⋊C4—1661no—
514C22⋊C4—1661no—
515C42—21no—
516C42C2×C4×D8311yescharacteristic
517C42D4×D8311yescharacteristic
518C42#1305058911yescharacteristic
519C42C23×D8311yescharacteristic
520C23—21no—
521C23—1661no—
522C23—1661no—
523C23(C22⋊D83)⋊C411yescharacteristic
524Q8—21no—
525Q8—21no—
526Q8—3322no—
527D4—21no—
528D4—21no—
529D4—81no—
530D4—3322no—
531D4—3322no—
532D4—3322no—
533D4—6641no—
534D4—1661no—
535D4—1661no—
536D4—1661no—
537D4—1661no—
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—41no—
547C2×C4—41no—
548C2×C4—3321no—
549C2×C4—3321no—
550C2×C4—3321no—
551C2×C4—3321no—
552C2×C4—1661no—
553C2×C4—1661no—
554C2×C4(C22⋊D83)⋊C411yescharacteristic
555C2×C4(C22⋊D83)⋊C411yescharacteristic
556C2×C4(C22⋊D83)⋊C411yescharacteristic
557C2×C4C22×C4×D8311yescharacteristic
558C2×C4C2×D4×D8311yescharacteristic
559C2×C4C2×#1305058911yescharacteristic
560C8—22no—
561C8—41no—
562C8—41no—
563C8—41no—
564C8—3321no—
565C8—3321no—
566C8—832no—
567C8—832no—
568C8—1661no—
569C8—1661no—
570C8—1661no—
571C8—1661no—
572C8—1661no—
573C8—1661no—
574C22—21no—
575C22—21no—
576C22—21no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—3321no—
581C22—3321no—
582C22—3321no—
583C22—3321no—
584C22—3321no—
585C22—3321no—
586C22C2×(C22⋊D83)⋊C411yescharacteristic
587C4—21no—
588C4—21no—
589C4—21no—
590C4—21no—
591C4—41no—
592C4—41no—
593C4—41no—
594C4—3321no—
595C4—3321no—
596C4#131×D8311yescharacteristic
597C4#1912846611yescharacteristic
598C4AGL(1,ℤ/8)×D8311yescharacteristic
599C4#1912850011yescharacteristic
600C2—41no—
601C2—81no—
602C2—6641no—
603C2—6641no—
604C2#2491620311yescharacteristic
605C2#2491627611yescharacteristic
606C2#2491636911yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC42C23×D83
FittingC83×#1483C2
socleC22×C83C2×C4×D4
prime radical#24918059C2
cosocleC42×C83C24
hypercenter#1483D83
hypocenterC83#18076
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1483

Complements

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

NH ≅ G/Ntype
2. #24916468C2semidirect
3. #24916206C2semidirect
4. #24906374C2semidirect
5. #24906350C2semidirect
6. #24906338C2semidirect
7. #24906330C2semidirect
8. #24906302C2semidirect
9. #24906289C2semidirect
10. #24906278C2semidirect
11. #24906255C2semidirect
12. #24906213C2semidirect
13. #24905881C2semidirect
14. #24905766C2semidirect
15. C83×#1483C2semidirect
268. C83⋊C8#131semidirect
317. #1483D83semidirect
319. C83#18076semidirect

Maximal subgroups

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

subgroupindex
#249180592
#249164682
#249162062
#249063742
#249063502
#249063382
#249063302
#249063022
#249062892
#249062782
#249062552
#249062132
#249058812
#249057662
C83×#14832
#1807683