Finite Group Databasebeta

← #35375555

Subgroups — #35375555

Order 21248.

650 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#24917514C211yesmaximal, characteristic
2#24916622C211yesmaximal, characteristic, complemented
3#24916496C211yesmaximal, characteristic, complemented
4#24906350C211yesmaximal, characteristic, complemented
5#24906349C211yesmaximal, characteristic, complemented
6#24906341C211yesmaximal, characteristic, complemented
7#24906327C211yesmaximal, characteristic, complemented
8#24906326C211yesmaximal, characteristic, complemented
9#24906321C211yesmaximal, characteristic, complemented
10#24906255C211yesmaximal, characteristic, complemented
11#24906245C211yesmaximal, characteristic, complemented
12#24905922C211yesmaximal, characteristic, complemented
13D4⋊C8×D83C211yesmaximal, characteristic, complemented
14#24905745C211yesmaximal, characteristic, complemented
15C8×SD16×C83C211yesmaximal, characteristic, complemented
16#19128238C2211yescharacteristic
17C4×D4×D83C2211yescharacteristic
18#19128035C411yescharacteristic
19#19128021C411yescharacteristic
20#19128000C411yescharacteristic
21#19127999C411yescharacteristic
22#19127919C411yescharacteristic
23#19127918C411yescharacteristic
24#19127850C411yescharacteristic
25#19127836C411yescharacteristic
26#19127832C2211yescharacteristic, complemented
27C4×#13050577C2211yescharacteristic
28C4×D664C2211yescharacteristic
29C4×#13050575C2211yescharacteristic
30#19127780C2211yescharacteristic
31#19127780C2211yescharacteristic, complemented
32#19127778C2211yescharacteristic
33#19127772—21no—
34#19127771—21no—
35#19127769—21no—
36C8×D332—21no—
37D4×C83⋊C8C2211yescharacteristic
38#19127766C2211yescharacteristic, complemented
39#19127760—21no—
40#19127757C2211yescharacteristic
41#19127752C2211yescharacteristic
42#19127752C2211yescharacteristic
43#19127726—21no—
44C4⋊C8×D83C2211yescharacteristic
45#19127663—21no—
46C2×(C2×D83)⋊C8—21no—
47C4×C8×D83C2211yescharacteristic
48#19127454C2211yescharacteristic, complemented
49#19127453C2211yescharacteristic, complemented
50#19127451C2211yescharacteristic, complemented
51Q8⋊(C83⋊C8)C2211yescharacteristic, complemented
52#19127430C2211yescharacteristic
53#19127430C2211yescharacteristic, complemented
54D4⋊(C83⋊C8)C2211yescharacteristic
55#19127428C2211yescharacteristic
56#19127428C2211yescharacteristic, complemented
57#19127423C2211yescharacteristic, complemented
58#19127423C2211yescharacteristic, complemented
59C8×C83⋊C8C2211yescharacteristic, complemented
60C8×Q8×C83C2211yescharacteristic, complemented
61C4×SD16×C83C2211yescharacteristic
62C8×D4×C83C2211yescharacteristic
63#334×C83C2211yescharacteristic, complemented
64Q8⋊C8×C83C2211yescharacteristic, complemented
65D4⋊C8×C83C2211yescharacteristic
66C82×C83C2211yescharacteristic, complemented
67C2×D4×D83C2×C411yescharacteristic
68C22×C4×D83—21no—
69C2×#13050577C2×C411yescharacteristic
70C2×D664C2×C411yescharacteristic
71C2×#13050575C2×C411yescharacteristic
72C2×#13050573—21no—
73C2×#13050573—21no—
74C2×#13050573—21no—
75C22×C83⋊C8—21no—
76C2×C8×D83—21no—
77C2×C8×D83—21no—
78C2×C8×D83—21no—
79#14581787—21no—
80#14581783C2×C411yescharacteristic
81#14581781C2×C411yescharacteristic
82#14581777—21no—
83#14581768C2×C411yescharacteristic
84#14581767C2×C411yescharacteristic
85C4×D332—21no—
86C4×D332C2311yescharacteristic
87C4×D332C2311yescharacteristic
88D4×Dic83C2×C411yescharacteristic
89#14581754C2×C411yescharacteristic
90C4×C22⋊D83—21no—
91(C22⋊D83)⋊C4—21no—
92#14581749—21no—
93C4⋊C4×D83C2×C411yescharacteristic
94C22⋊C4×D83—21no—
95C42×D83C2311yescharacteristic
96#14581726C2×C411yescharacteristic
97#14581725C2×C411yescharacteristic
98#14581723C2×C411yescharacteristic
99C4⋊(C83⋊C8)—21no—
100C4⋊(C83⋊C8)C2311yescharacteristic
101C4⋊(C83⋊C8)C2311yescharacteristic
102Dic83⋊C8—21no—
103Dic83⋊C8C2×C411yescharacteristic
104Dic83⋊C8C2×C411yescharacteristic
105Dic83⋊C8D411yescharacteristic
106Dic83⋊C8D411yescharacteristic
107#14581716C2×C411yescharacteristic
108#14581714C2×C411yescharacteristic
109#14581714C2×C411yescharacteristic
110#14581713C2×C411yescharacteristic
111#14581712C2×C411yescharacteristic
112#14581712C2×C411yescharacteristic
113C22⋊(C83⋊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#14581703—21no—
124#14581702—21no—
125C4×C83⋊C8C2311yescharacteristic
126C4×C83⋊C8C2311yescharacteristic
127C4×C83⋊C8C2311yescharacteristic
128C8×Dic83—21no—
129C2×SD16×C83C2×C411yescharacteristic
130C22×C8×C83—21no—
131C4×Q8×C83C2311yescharacteristic
132C4×D4×C83C2311yescharacteristic
133#106×C83C2×C411yescharacteristic
134C4⋊C8×C83—21no—
135C4⋊C8×C83C2311yescharacteristic
136C4⋊C8×C83C2311yescharacteristic
137#103×C83C2×C411yescharacteristic
138#102×C83C2×C411yescharacteristic
139C22⋊C8×C83—21no—
140C4×C8×C83—21no—
141C4×C8×C83C2311yescharacteristic
142C4×C8×C83C2311yescharacteristic
143C4×C8×C83C2311yescharacteristic
144C23×D83—21no—
145C2×D332—21no—
146C2×D332C22×C411yescharacteristic
147C2×D332C22×C411yescharacteristic
148C2×C22⋊D83—21no—
149D4×D83—22no—
150D4×D83—41no—
151C22×Dic83—21no—
152C2×C4×D83—21no—
153C2×C4×D83—21no—
154C2×C4×D83—21no—
155C2×C4×D83—21no—
156C2×C4×D83—81no—
157C2×C4×D83—81no—
158C2×C4×D83C22×C411yescharacteristic
159C2×C4×D83C2×D411yescharacteristic
160C2×C4×D83#4111yescharacteristic
161#13050577—41no—
162D664—41no—
163#13050575—22no—
164#13050573—81no—
165#13050573—81no—
166#13050573—81no—
167C2×C83⋊C8—21no—
168C2×C83⋊C8—21no—
169C2×C83⋊C8—21no—
170C2×C83⋊C8—81no—
171C2×C83⋊C8C22×C411yescharacteristic
172C2×C83⋊C8C22×C411yescharacteristic
173C2×C83⋊C8C2×D411yescharacteristic
174C2×C83⋊C8C2×D411yescharacteristic
175C8×D83—24no—
176C8×D83—81no—
177C8×D83—81no—
178C4⋊Dic83—21no—
179C4⋊Dic83C22×C411yescharacteristic
180C4⋊Dic83C22×C411yescharacteristic
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—
192C4×Dic83C22×C411yescharacteristic
193C4×Dic83C2×D411yescharacteristic
194C4×Dic83#4111yescharacteristic
195C2×Q8×C83C22×C411yescharacteristic
196C2×D4×C83C22×C411yescharacteristic
197C22×C4×C83—21no—
198SD16×C83—22no—
199C2×C8×C83—21no—
200C2×C8×C83—21no—
201C2×C8×C83—21no—
202C2×C8×C83—81no—
203C2×C8×C83C22×C411yescharacteristic
204C2×C8×C83C22×C411yescharacteristic
205C2×C8×C83C2×D411yescharacteristic
206C2×C8×C83C2×D411yescharacteristic
207C4⋊C4×C83—21no—
208C4⋊C4×C83C22×C411yescharacteristic
209C4⋊C4×C83C22×C411yescharacteristic
210C22⋊C4×C83—21no—
211C42×C83—21no—
212C42×C83C2411yescharacteristic
213C23×C83—21no—
214C22×D83—21no—
215C22×D83—21no—
216C22×D83—42no—
217C22×D83—42no—
218C22×D83—81no—
219C22×D83C4×D411yescharacteristic
220Q8×C83—21no—
221Q8×C83—21no—
222D4×C83—21no—
223D4×C83#13112yes—
224C2×C4×C83—21no—
225C2×C4×C83—21no—
226C2×C4×C83—21no—
227C2×C4×C83—41no—
228C2×C4×C83—41no—
229C2×C4×C83C23×C411yescharacteristic
230C2×C4×C83C22×D411yescharacteristic
231C2×C4×C83C2×#4111yescharacteristic
232C22⋊D83—42no—
233C2×Dic83—21no—
234C2×Dic83—21no—
235C2×Dic83—81no—
236C2×Dic83C4×D411yescharacteristic
237C2×Dic83C4×D411yescharacteristic
238C2×Dic83C4×D411yescharacteristic
239D332—21no—
240D332—21no—
241D332—21no—
242D332—21no—
243D332—42no—
244C4×D83—22no—
245C4×D83—21no—
246C4×D83—21no—
247C4×D83—21no—
248C4×D83—21no—
249C4×D83—41no—
250C4×D83—41no—
251C4×D83—41no—
252C4×D83—41no—
253C4×D83—81no—
254C4×D83—81no—
255C8×C83—22no—
256C8×C83—21no—
257C8×C83—21no—
258C8×C83—21no—
259C8×C83—21no—
260C8×C83—41no—
261C8×C83—41no—
262C83⋊C8—22no—
263C83⋊C8—21no—
264C83⋊C8—21no—
265C83⋊C8—21no—
266C83⋊C8—21no—
267C83⋊C8—41no—
268C83⋊C8—41no—
269C22×C83—22no—
270C22×C83—41no—
271C22×C83—41no—
272C22×C83C2×C4×D411yescharacteristic
273C2×D83—21no—
274C2×D83—21no—
275C2×D83—41no—
276C2×D83—41no—
277C2×D83—41no—
278C2×D83—41no—
279C2×D83—41no—
280C2×D83—41no—
281C2×D83—84no—
282C2×D83#44312yes—
283C4×C83—21no—
284C4×C83—41no—
285C4×C83—41no—
286C4×C83—41no—
287C4×C83C2×#13111yescharacteristic
288C4×C83#56811yescharacteristic
289C4×C83#57611yescharacteristic
290C4×C83#57711yescharacteristic
291Dic83—21no—
292Dic83—21no—
293Dic83—21no—
294Dic83—41no—
295Dic83—41no—
296#18014—831nomaximal, Sylow
297C2×C83—42no—
298C2×C83#293311yescharacteristic
299C2×C83#294611yescharacteristic
300C2×C83C2×#44311yescharacteristic
301D83—22no—
302D83—81no—
303D83—81no—
304#3105—831no—
305#2994—831no—
306#2974—831no—
307#1590—831no—
308#1589—831no—
309#1588—831no—
310#1586—831no—
311#1586—831no—
312#1585—831no—
313C8×SD16D8311yescharacteristic, complemented
314C8×D8—831no—
315#1573—831no—
316#1494—831no—
317C2×D4⋊C8—831no—
318#1450—831no—
319C83#1801411yescharacteristic, complemented, Sylow
320#529—831no—
321C2×C4×D4—831no—
322#484—831no—
323#481—831no—
324#478—831no—
325#478—831no—
326#463—831no—
327#463—831no—
328#450—831no—
329#447—831no—
330#446—831no—
331C8×Q8C2×D8311yescharacteristic, complemented
332C4×SD16—831no—
333C4×SD16C2×D8311yescharacteristic
334C4×D8—831no—
335C4×D8—831no—
336#436—831no—
337#436—831no—
338#436—831no—
339#435—1661no—
340#435—1661no—
341#435—1661no—
342C8×D4—831no—
343C8×D4—831no—
344C8×D4—1661no—
345C8×D4C2×D8311yescharacteristic
346#433—1661no—
347#432—831no—
348#431—831no—
349#431—831no—
350#424—1661no—
351C2×C4⋊C8—831no—
352#407—1661no—
353C2×C22⋊C8—1661no—
354C2×C4×C8—831no—
355#335—831no—
356#335—831no—
357#334—831no—
358#334C2×D8311yescharacteristic, complemented
359Q8⋊C8—831no—
360Q8⋊C8C2×D8311yescharacteristic, complemented
361D4⋊C8—831no—
362D4⋊C8—831no—
363D4⋊C8—831no—
364D4⋊C8—831no—
365D4⋊C8—831no—
366D4⋊C8C2×D8311yescharacteristic
367#322—831no—
368#322—831no—
369C82—831no—
370C82C2×D8311yescharacteristic, complemented
371C22×D4—831no—
372C23×C4—1661no—
373C2×SD16—831no—
374C2×SD16C4×D8311yescharacteristic
375C2×D8—831no—
376C2×D8—831no—
377C2×M16—1661no—
378C2×M16—1661no—
379C2×M16—1661no—
380C22×C8—21no—
381C22×C8—1661no—
382C22×C8—1661no—
383C22×C8—1661no—
384C22×C8—1661no—
385#126—1661no—
386#125—831no—
387#124—831no—
388#123—1661no—
389C22⋊Q8—831no—
390#121—831no—
391C4×Q8C22×D8311yescharacteristic
392C4×D4—831no—
393C4×D4—831no—
394C4×D4—831no—
395C4×D4—831no—
396C4×D4—1661no—
397C4×D4—1661no—
398C4×D4—1661no—
399C4×D4C22×D8311yescharacteristic
400#117—1661no—
401C2×C4⋊C4—831no—
402C2×C22⋊C4—1661no—
403C2×C42—831no—
404#107—831no—
405#107—831no—
406#106—831no—
407#106C4×D8311yescharacteristic
408C4⋊C8—21no—
409C4⋊C8—831no—
410C4⋊C8—831no—
411C4⋊C8—831no—
412C4⋊C8—831no—
413C4⋊C8—831no—
414C4⋊C8—831no—
415C4⋊C8—1661no—
416C4⋊C8—1661no—
417C4⋊C8C22×D8311yescharacteristic
418C4⋊C8C22×D8311yescharacteristic
419#103—831no—
420#103C4×D8311yescharacteristic
421#102—831no—
422#102—831no—
423#102—831no—
424#102—831no—
425#102—831no—
426#102C4×D8311yescharacteristic
427C22⋊C8—21no—
428C22⋊C8—831no—
429C22⋊C8—831no—
430C22⋊C8—831no—
431C22⋊C8—831no—
432C22⋊C8—1661no—
433C22⋊C8—1661no—
434C22⋊C8—1661no—
435C22⋊C8—1661no—
436C22⋊C8—1661no—
437C22⋊C8—1661no—
438#97—1661no—
439#97—1661no—
440C4×C8—21no—
441C4×C8—831no—
442C4×C8—831no—
443C4×C8—831no—
444C4×C8—1661no—
445C4×C8C22×D8311yescharacteristic
446C4×C8C22×D8311yescharacteristic
447C4×C8C22×D8311yescharacteristic
448C24—1661no—
449C2×Q8C2×C4×D8311yescharacteristic
450C2×D4—3321no—
451C2×D4—831no—
452C2×D4—831no—
453C2×D4—1661no—
454C2×D4—1661no—
455C2×D4—1662no—
456C2×D4C2×C4×D8311yescharacteristic
457C22×C4—21no—
458C22×C4—831no—
459C22×C4—831no—
460C22×C4—831no—
461C22×C4—6641no—
462C22×C4—6641no—
463C22×C4—1661no—
464C22×C4—1661no—
465C22×C4—1661no—
466C22×C4—1661no—
467C22×C4—1661no—
468SD16—22no—
469SD16—3321no—
470D8—3321no—
471D8—1662no—
472M16—6641no—
473M16—6641no—
474M16—6641no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—81no—
479C2×C8—831no—
480C2×C8—831no—
481C2×C8—831no—
482C2×C8—831no—
483C2×C8—6641no—
484C2×C8—6641no—
485C2×C8—6641no—
486C2×C8—1664no—
487C2×C8—1661no—
488C2×C8—1661no—
489C2×C8—1661no—
490C2×C8C2×C4×D8311yescharacteristic
491C2×C8C2×C4×D8311yescharacteristic
492C2×C8D4×D8311yescharacteristic
493C2×C8D4×D8311yescharacteristic
494C4⋊C4—21no—
495C4⋊C4—831no—
496C4⋊C4—831no—
497C4⋊C4—1661no—
498C4⋊C4—1661no—
499C4⋊C4—1661no—
500C4⋊C4—1661no—
501C4⋊C4C2×C4×D8311yescharacteristic
502C4⋊C4C2×C4×D8311yescharacteristic
503C22⋊C4—21no—
504C22⋊C4—1661no—
505C22⋊C4—1661no—
506C22⋊C4—1661no—
507C22⋊C4—1661no—
508C22⋊C4—1661no—
509C22⋊C4—1661no—
510C22⋊C4—1661no—
511C22⋊C4—1661no—
512C42—21no—
513C42—831no—
514C42—831no—
515C42—831no—
516C42C23×D8311yescharacteristic
517C23—21no—
518C23—3322no—
519C23—3322no—
520C23—831no—
521C23—6641no—
522C23—1661no—
523C23—1661no—
524Q8—21no—
525Q8—21no—
526D4—21no—
527D4—3322no—
528D4—3322no—
529D4—1661no—
530D4—1661no—
531D4—1661no—
532D4—1661no—
533D4#1458179812yes—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—3321no—
540C2×C4—3321no—
541C2×C4—3321no—
542C2×C4—3321no—
543C2×C4—831no—
544C2×C4—831no—
545C2×C4—831no—
546C2×C4—6641no—
547C2×C4—6641no—
548C2×C4—6641no—
549C2×C4—1662no—
550C2×C4—1661no—
551C2×C4—1661no—
552C2×C4—1661no—
553C2×C4—1661no—
554C2×C4—1661no—
555C2×C4—1661no—
556C2×C4C22×C4×D8311yescharacteristic
557C2×C4C2×D4×D8311yescharacteristic
558C2×C4#41×D8311yescharacteristic
559C8—22no—
560C8—21no—
561C8—21no—
562C8—21no—
563C8—21no—
564C8—41no—
565C8—41no—
566C8—3321no—
567C8—3321no—
568C8—1662no—
569C8—1661no—
570C8—1661no—
571C8—1661no—
572C8—1661no—
573C22—22no—
574C22—41no—
575C22—41no—
576C22—3321no—
577C22—3321no—
578C22—3321no—
579C22—3321no—
580C22—3321no—
581C22—3321no—
582C22—832no—
583C22—6644no—
584C22—1661no—
585C22—1661no—
586C22C4×D4×D8311yescharacteristic
587C4—21no—
588C4—41no—
589C4—41no—
590C4—41no—
591C4—3321no—
592C4—3321no—
593C4—1661no—
594C4—1661no—
595C4—1661no—
596C4C2×#1458179811yescharacteristic
597C4#1912846611yescharacteristic
598C4#1912850311yescharacteristic
599C4#1912850711yescharacteristic
600C2—42no—
601C2—6641no—
602C2—6641no—
603C2—1662no—
604C2#2491618311yescharacteristic
605C2#2491629011yescharacteristic
606C2#443×D8311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC42C23×D83
FittingC8×SD16×C83C2
socleC22×C83C2×C4×D4
prime radical#19127836C4
cosocleC42×C83C24
hypercenterC8×SD16D83
hypocenterC83#18014
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ C8×SD16

Complements

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

NH ≅ G/Ntype
2. #24916622C2semidirect
3. #24916496C2semidirect
4. #24906350C2semidirect
5. #24906349C2semidirect
6. #24906341C2semidirect
7. #24906327C2semidirect
8. #24906326C2semidirect
9. #24906321C2semidirect
10. #24906255C2semidirect
11. #24906245C2semidirect
12. #24905922C2semidirect
13. D4⋊C8×D83C2semidirect
14. #24905745C2semidirect
15. C8×SD16×C83C2semidirect
26. #19127832C22semidirect
31. #19127780C22semidirect
38. #19127766C22semidirect
48. #19127454C22semidirect
49. #19127453C22semidirect
50. #19127451C22semidirect
51. Q8⋊(C83⋊C8)C22semidirect
53. #19127430C22semidirect
56. #19127428C22semidirect
57. #19127423C22semidirect
58. #19127423C22semidirect
59. C8×C83⋊C8C22semidirect
60. C8×Q8×C83C22semidirect
63. #334×C83C22semidirect
64. Q8⋊C8×C83C22semidirect
66. C82×C83C22semidirect
313. C8×SD16D83semidirect
319. C83#18014semidirect
331. C8×Q8C2×D83semidirect
358. #334C2×D83semidirect
360. Q8⋊C8C2×D83semidirect
370. C82C2×D83semidirect

Maximal subgroups

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

subgroupindex
#249175142
#249166222
#249164962
#249063502
#249063492
#249063412
#249063272
#249063262
#249063212
#249062552
#249062452
#249059222
D4⋊C8×D832
#249057452
C8×SD16×C832
#1801483