Finite Group Databasebeta

← #35376262

Subgroups — #35376262

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#24918058C211yesmaximal, characteristic
2#24916467C211yesmaximal, characteristic, complemented
3#24916209C211yesmaximal, characteristic, complemented
4#24906378C211yesmaximal, characteristic, complemented
5#24906355C211yesmaximal, characteristic, complemented
6#24906343C211yesmaximal, characteristic, complemented
7#24906329C211yesmaximal, characteristic, complemented
8#24906301C211yesmaximal, characteristic, complemented
9#24906294C211yesmaximal, characteristic, complemented
10#24906277C211yesmaximal, characteristic, complemented
11C8×D664C211yesmaximal, characteristic, complemented
12#24906215C211yesmaximal, characteristic, complemented
13#24905881C211yesmaximal, characteristic, complemented
14#24905877C211yesmaximal, characteristic, complemented
15C83×#1455C211yesmaximal, characteristic, complemented
16C2×#14581800—21no—
17#19128251C2211yescharacteristic
18#19128235C2211yescharacteristic
19#19128110C411yescharacteristic
20#19128090C411yescharacteristic
21#19128066C411yescharacteristic
22#19128061C411yescharacteristic
23#19127954C411yescharacteristic
24#19127954C411yescharacteristic
25#19127947C411yescharacteristic
26#19127947C411yescharacteristic
27#19127831C2211yescharacteristic
28C8×Dic166C2211yescharacteristic
29C4×#13050579C2211yescharacteristic
30C4×#13050579C2211yescharacteristic
31C4×D664C2211yescharacteristic
32C4×D664C2211yescharacteristic
33#19127780C2211yescharacteristic
34#19127780C2211yescharacteristic
35#19127780C2211yescharacteristic
36#19127777—21no—
37#19127774—21no—
38#19127774—21no—
39#19127770—21no—
40C8×D332C2211yescharacteristic
41C8×D332C2211yescharacteristic
42C8×D332C2211yescharacteristic
43#19127763—21no—
44#19127754C2211yescharacteristic
45C2×C4⋊(C83⋊C8)C2211yescharacteristic
46#19127669—21no—
47#19127658—21no—
48#19127454C2211yescharacteristic
49#19127454C2211yescharacteristic
50#19127452C2211yescharacteristic
51#19127452C2211yescharacteristic
52#19127433C2211yescharacteristic
53#19127433C2211yescharacteristic
54#19127430C2211yescharacteristic
55#19127430C2211yescharacteristic
56#19127430C2211yescharacteristic
57#19127430C2211yescharacteristic
58#19127430C2211yescharacteristic
59#19127430C2211yescharacteristic
60C4×M16×C83C2211yescharacteristic
61C4×M16×C83C2211yescharacteristic
62C4×M16×C83C2211yescharacteristic
63#322×C83C2211yescharacteristic
64#322×C83C2211yescharacteristic
65#322×C83C2211yescharacteristic
66C82×C83C2211yescharacteristic
67C2×#13050590—21no—
68C2×#13050579C2×C411yescharacteristic
69C2×#13050579C2×C411yescharacteristic
70C2×D664C2×C411yescharacteristic
71C2×D664C2×C411yescharacteristic
72#14581800—44no—
73C2×#13050573—21no—
74C2×#13050573—21no—
75C2×#13050573—21no—
76C22×C83⋊C8—21no—
77C2×C8×D83—21no—
78C2×C8×D83—21no—
79C2×C8×D83—21no—
80#14581792C2×C411yescharacteristic
81#14581789C2×C411yescharacteristic
82#14581786—21no—
83#14581784C2×C411yescharacteristic
84#14581780—21no—
85#14581780C2×C411yescharacteristic
86#14581775—21no—
87#14581771C2×C411yescharacteristic
88#14581766—21no—
89#14581766C2×C411yescharacteristic
90#14581766C2×C411yescharacteristic
91#14581766C2×C411yescharacteristic
92C4×Dic166C2311yescharacteristic
93C4×D332—21no—
94C4×D332C2311yescharacteristic
95C4×D332C2311yescharacteristic
96C4×D332C2311yescharacteristic
97C4×C22⋊D83—21no—
98#14581749—21no—
99#14581726C2×C411yescharacteristic
100#14581726C2×C411yescharacteristic
101#14581724C2×C411yescharacteristic
102#14581724C2×C411yescharacteristic
103C4⋊(C83⋊C8)—21no—
104C4⋊(C83⋊C8)C2311yescharacteristic
105C4⋊(C83⋊C8)C2311yescharacteristic
106C4⋊(C83⋊C8)C2311yescharacteristic
107C4⋊(C83⋊C8)C2311yescharacteristic
108Dic83⋊C8—21no—
109Dic83⋊C8—21no—
110#14581717C2×C411yescharacteristic
111#14581717C2×C411yescharacteristic
112#14581714C2×C411yescharacteristic
113#14581714C2×C411yescharacteristic
114#14581714C2×C411yescharacteristic
115#14581714C2×C411yescharacteristic
116#14581714C2×C411yescharacteristic
117#14581714C2×C411yescharacteristic
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)⋊C8—21no—
125#14581702—21no—
126C4×C83⋊C8—21no—
127C8×Dic83—21no—
128C2×M16×C83—21no—
129C2×M16×C83C2×C411yescharacteristic
130C2×M16×C83C2×C411yescharacteristic
131C2×M16×C83D411yescharacteristic
132C2×M16×C83D411yescharacteristic
133C2×C42×C83C2311yescharacteristic
134#97×C83—21no—
135#97×C83C2×C411yescharacteristic
136#97×C83C2×C411yescharacteristic
137#97×C83D411yescharacteristic
138#97×C83D411yescharacteristic
139C4×C8×C83C2311yescharacteristic
140C4×C8×C83C2311yescharacteristic
141C4×C8×C83C2311yescharacteristic
142C4×C8×C83C2311yescharacteristic
143C4×C8×C83C2311yescharacteristic
144C4×C8×C83C2311yescharacteristic
145#13050590—42no—
146#13050590—42no—
147C2×Dic166—21no—
148C2×Dic166C22×C411yescharacteristic
149C2×D332—21no—
150C2×D332—21no—
151C2×D332C22×C411yescharacteristic
152C2×D332C22×C411yescharacteristic
153C2×D332C22×C411yescharacteristic
154C2×C22⋊D83—21no—
155C2×C22⋊D83—21no—
156C2×C22⋊D83—21no—
157C2×C4×D83—21no—
158C2×C4×D83—21no—
159C2×C4×D83—21no—
160#13050579—22no—
161#13050579—41no—
162D664—22no—
163D664—41no—
164#13050573—42no—
165#13050573—42no—
166#13050573—42no—
167C2×C83⋊C8—21no—
168C2×C83⋊C8—21no—
169C2×C83⋊C8—21no—
170C2×C83⋊C8—21no—
171C2×C83⋊C8—42no—
172C8×D83—42no—
173C8×D83—42no—
174C8×D83—42no—
175C4⋊Dic83—21no—
176C4⋊Dic83C22×C411yescharacteristic
177C4⋊Dic83C22×C411yescharacteristic
178C4⋊Dic83C22×C411yescharacteristic
179C4⋊Dic83C22×C411yescharacteristic
180Dic83⋊C4—21no—
181Dic83⋊C4—21no—
182Dic83⋊C4—21no—
183Dic83⋊C4—21no—
184C22⋊Dic83—21no—
185(C2×D83)⋊C4—21no—
186(C2×D83)⋊C4—21no—
187(C2×D83)⋊C4—21no—
188(C2×D83)⋊C4—21no—
189(C2×D83)⋊C4—21no—
190(C2×D83)⋊C4—21no—
191(C2×D83)⋊C4—21no—
192(C2×D83)⋊C4—21no—
193C4×Dic83—21no—
194C22×C4×C83C22×C411yescharacteristic
195C22×C4×C83C2×D411yescharacteristic
196C22×C4×C83#4111yescharacteristic
197M16×C83—22no—
198M16×C83—22no—
199M16×C83—41no—
200M16×C83—41no—
201M16×C83—42no—
202C2×C8×C83—21no—
203C2×C8×C83—21no—
204C2×C8×C83C22×C411yescharacteristic
205C2×C8×C83C22×C411yescharacteristic
206C2×C8×C83C22×C411yescharacteristic
207C2×C8×C83C22×C411yescharacteristic
208C2×C8×C83C2×D411yescharacteristic
209C2×C8×C83C2×D411yescharacteristic
210C2×C8×C83C2×D411yescharacteristic
211C2×C8×C83C2×D411yescharacteristic
212C42×C83C22×C411yescharacteristic
213C42×C83C2×D411yescharacteristic
214C42×C83#4111yescharacteristic
215C42×C83C2411yescharacteristic
216C23×C83C4×D411yescharacteristic
217C22×D83—21no—
218C22×D83—21no—
219C22×D83—21no—
220C2×C4×C83—21no—
221C2×C4×C83—21no—
222C2×C4×C83—21no—
223C2×C4×C83—21no—
224C2×C4×C83—41no—
225C2×C4×C83C4×D411yescharacteristic
226C2×C4×C83C4×D411yescharacteristic
227C2×C4×C83C4×D411yescharacteristic
228C2×C4×C83C23×C411yescharacteristic
229C2×C4×C83C22×D411yescharacteristic
230C2×C4×C83C2×#4111yescharacteristic
231C22⋊D83—42no—
232C22⋊D83—42no—
233C22⋊D83—81no—
234C2×Dic83—21no—
235C2×Dic83—21no—
236C2×Dic83—21no—
237C2×Dic83—21no—
238C2×Dic83—21no—
239D332—21no—
240D332—21no—
241D332—21no—
242D332—21no—
243D332—21no—
244D332—21no—
245D332—42no—
246D332—81no—
247C4×D83—41no—
248C4×D83—41no—
249C4×D83—41no—
250C4×D83—41no—
251C4×D83—41no—
252C4×D83—41no—
253Dic166—21no—
254Dic166—21no—
255Dic166—42no—
256C8×C83—22no—
257C8×C83—21no—
258C8×C83—21no—
259C8×C83—21no—
260C8×C83—21no—
261C8×C83—21no—
262C8×C83—21no—
263C8×C83—41no—
264C8×C83#13112yescomplemented
265C8×C83AGL(1,ℤ/8)12yes—
266C83⋊C8—41no—
267C83⋊C8—41no—
268C83⋊C8—41no—
269C83⋊C8—41no—
270C22×C83—21no—
271C22×C83—21no—
272C22×C83—21no—
273C22×C83C2×C4×D411yescharacteristic
274C2×D83—41no—
275C2×D83—41no—
276C2×D83—41no—
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×C83C2×#13111yescharacteristic
288C4×C83#56811yescharacteristic
289C4×C83C2×AGL(1,ℤ/8)11yescharacteristic
290C4×C83#57511yescharacteristic
291Dic83—41no—
292Dic83—41no—
293Dic83—41no—
294Dic83—41no—
295Dic83—41no—
296#18076—831nomaximal, Sylow
297C2×C83—41no—
298C2×C83#293511yescharacteristic
299C2×C83C4×AGL(1,ℤ/8)11yescharacteristic
300C2×C83#295611yescharacteristic
301D83—81no—
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#1483—831no—
318#1455D8311yescharacteristic, complemented
319C83#1807611yescharacteristic, complemented, Sylow
320C2×#131—1661no—
321#532—831no—
322#529—831no—
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⋊C8—831no—
350#408—1661no—
351#405—1661no—
352C4×M16C2×D8311yescharacteristic
353C4×M16C2×D8311yescharacteristic
354C4×M16C2×D8311yescharacteristic
355#335—831no—
356#335—831no—
357#334—831no—
358#334—831no—
359Q8⋊C8—831no—
360Q8⋊C8—831no—
361D4⋊C8—831no—
362D4⋊C8—831no—
363D4⋊C8—831no—
364D4⋊C8—831no—
365D4⋊C8—831no—
366D4⋊C8—831no—
367#322C2×D8311yescharacteristic
368#322C2×D8311yescharacteristic
369#322C2×D8311yescharacteristic
370C82C2×D8311yescharacteristic
371C2×#41—1661no—
372C2×SD16—831no—
373C2×SD16—831no—
374C2×D8—831no—
375C2×D8—831no—
376#131—3324no—
377C2×M16—21no—
378C2×M16—1661no—
379C2×M16—1661no—
380C2×M16—1661no—
381C2×M16C4×D8311yescharacteristic
382C2×M16C4×D8311yescharacteristic
383C2×M16D33211yescharacteristic
384C2×M16D33211yescharacteristic
385C22×C8—1661no—
386C22×C8—1661no—
387C22×C8—1661no—
388C22×C8—1661no—
389C4⋊Q8—831no—
390#127—831no—
391#126—1661no—
392#125—831no—
393#124—831no—
394#124—1661no—
395#123—1661no—
396C22⋊Q8—831no—
397#121—831no—
398#121—831no—
399#121—831no—
400#121—1661no—
401C4×Q8—831no—
402C4×D4—831no—
403C4×D4—831no—
404C4×D4—831no—
405C4×D4—1661no—
406C4×D4—1661no—
407#117—1661no—
408C2×C42C22×D8311yescharacteristic
409#107—831no—
410#107—831no—
411#106—831no—
412#106—831no—
413C4⋊C8—831no—
414C4⋊C8—831no—
415C4⋊C8—831no—
416C4⋊C8—831no—
417C4⋊C8—1661no—
418C4⋊C8—1661no—
419C4⋊C8—1661no—
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—1661no—
437#97C4×D8311yescharacteristic
438#97C4×D8311yescharacteristic
439#97D33211yescharacteristic
440#97D33211yescharacteristic
441C4×C8—1661no—
442C4×C8—1661no—
443C4×C8C22×D8311yescharacteristic
444C4×C8C22×D8311yescharacteristic
445C4×C8C22×D8311yescharacteristic
446C4×C8C22×D8311yescharacteristic
447C4×C8C22×D8311yescharacteristic
448C4×C8C22×D8311yescharacteristic
449#41—3322no—
450#41—3322no—
451C2×Q8—831no—
452C2×Q8—1661no—
453C2×D4—831no—
454C2×D4—831no—
455C2×D4—831no—
456C2×D4—1661no—
457C2×D4—1661no—
458C2×D4—1661no—
459C2×D4—1661no—
460C2×D4—1661no—
461C22×C4—1661no—
462C22×C4—1661no—
463C22×C4—1661no—
464C22×C4C2×C4×D8311yescharacteristic
465C22×C4C2×D33211yescharacteristic
466C22×C4#1305059011yescharacteristic
467SD16—3321no—
468SD16—1662no—
469D8—3321no—
470D8—1662no—
471M16—22no—
472M16—22no—
473M16—41no—
474M16—41no—
475M16—42no—
476M16—3322no—
477M16—3322no—
478M16—3322no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—3322no—
482C2×C8—3322no—
483C2×C8—3322no—
484C2×C8—3322no—
485C2×C8—1661no—
486C2×C8—1661no—
487C2×C8—1661no—
488C2×C8—1661no—
489C2×C8C2×C4×D8311yescharacteristic
490C2×C8C2×C4×D8311yescharacteristic
491C2×C8C2×C4×D8311yescharacteristic
492C2×C8C2×C4×D8311yescharacteristic
493C2×C8C2×D33211yescharacteristic
494C2×C8C2×D33211yescharacteristic
495C2×C8C2×D33211yescharacteristic
496C2×C8C2×D33211yescharacteristic
497C4⋊C4—831no—
498C4⋊C4—831no—
499C4⋊C4—831no—
500C4⋊C4—831no—
501C4⋊C4—1661no—
502C4⋊C4—1661no—
503C4⋊C4—1661no—
504C4⋊C4—1661no—
505C4⋊C4—1661no—
506C22⋊C4—1661no—
507C22⋊C4—1661no—
508C22⋊C4—1661no—
509C22⋊C4—1661no—
510C22⋊C4—1661no—
511C22⋊C4—1661no—
512C22⋊C4—1661no—
513C22⋊C4—1661no—
514C22⋊C4—1661no—
515C42—1661no—
516C42C2×C4×D8311yescharacteristic
517C42C2×D33211yescharacteristic
518C42#1305059011yescharacteristic
519C42C23×D8311yescharacteristic
520C23—1661no—
521C23—1661no—
522C23—1661no—
523C23C4×D33211yescharacteristic
524Q8—3322no—
525Q8—1661no—
526Q8—1661no—
527D4—3322no—
528D4—3322no—
529D4—3322no—
530D4—6641no—
531D4—6641no—
532D4—1661no—
533D4—1661no—
534D4—1661no—
535D4—1661no—
536D4—1661no—
537D4—1661no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—41no—
543C2×C4—3321no—
544C2×C4—3321no—
545C2×C4—3321no—
546C2×C4—3321no—
547C2×C4—3321no—
548C2×C4—3321no—
549C2×C4—1661no—
550C2×C4—1661no—
551C2×C4—1661no—
552C2×C4—1661no—
553C2×C4—1661no—
554C2×C4C4×D33211yescharacteristic
555C2×C4C4×D33211yescharacteristic
556C2×C4C4×D33211yescharacteristic
557C2×C4C22×C4×D8311yescharacteristic
558C2×C4C22×D33211yescharacteristic
559C2×C4C2×#1305059011yescharacteristic
560C8—22no—
561C8—21no—
562C8—21no—
563C8—21no—
564C8—21no—
565C8—21no—
566C8—21no—
567C8—41no—
568C8—3321no—
569C8—3321no—
570C8—3321no—
571C8—3321no—
572C8#1458180012yescomplemented
573C8#1458182012yes—
574C22—21no—
575C22—21no—
576C22—21no—
577C22—3321no—
578C22—3321no—
579C22—3321no—
580C22—3321no—
581C22—3321no—
582C22—3321no—
583C22—3321no—
584C22—3321no—
585C22—3321no—
586C22C2×C4×D33211yescharacteristic
587C4—21no—
588C4—21no—
589C4—21no—
590C4—21no—
591C4—3321no—
592C4—3321no—
593C4—3321no—
594C4—3321no—
595C4—3321no—
596C4C2×#1458180011yescharacteristic
597C4#1912846711yescharacteristic
598C4C2×#1458182011yescharacteristic
599C4#1912850111yescharacteristic
600C2—41no—
601C2—6641no—
602C2—6641no—
603C2—6641no—
604C2#2491620911yescharacteristic
605C2C4×#1458182011yescharacteristic
606C2#2491637411yescharacteristic
subgroupG/H
centerC2×C4C22×D332
commutatorC2×C4×C83C23×C4
FrattiniC42C23×D83
FittingC83×#1455C2
socleC22×C83C2×C4×D4
prime radical#24918058C2
cosocleC42×C83C24
hypercenter#1455D83
hypocenterC83#18076
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C4×M16 ◁ #1455

Complements

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

NH ≅ G/Ntype
2. #24916467C2semidirect
3. #24916209C2semidirect
4. #24906378C2semidirect
5. #24906355C2semidirect
6. #24906343C2semidirect
7. #24906329C2semidirect
8. #24906301C2semidirect
9. #24906294C2semidirect
10. #24906277C2semidirect
11. C8×D664C2semidirect
12. #24906215C2semidirect
13. #24905881C2semidirect
14. #24905877C2semidirect
15. C83×#1455C2semidirect
264. C8×C83#131semidirect
318. #1455D83semidirect
319. C83#18076semidirect
572. C8#14581800semidirect

Maximal subgroups

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

subgroupindex
#249180582
#249164672
#249162092
#249063782
#249063552
#249063432
#249063292
#249063012
#249062942
#249062772
C8×D6642
#249062152
#249058812
#249058772
C83×#14552
#1807683