Finite Group Databasebeta

← #6355372

Subgroups — #6355372

Order 512.

1762 conjugacy classes of subgroups.

The subgroup lattice has 743 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60944C211yesmaximal, characteristic, complemented
2#60558C212yesmaximal, complemented
3C2×#2974C211yesmaximal, characteristic, complemented
4#19467C214yesmaximal, complemented
5#19466C214yesmaximal, complemented
6#19461C214yesmaximal, complemented
7#19460C214yesmaximal, complemented
8#19284C211yesmaximal, characteristic, complemented
9#19283C211yesmaximal, characteristic, complemented
10#19282C212yesmaximal, complemented
11#19259C211yesmaximal, characteristic, complemented
12#19258C211yesmaximal, characteristic, complemented
13#19257C212yesmaximal, complemented
14#17819C211yesmaximal, characteristic, complemented
15#17756C212yesmaximal, complemented
16#3444C2212yes—
17C22×C4×D4C2211yescharacteristic
18#3250—21no—
19#3250—21no—
20#3231C2212yescomplemented
21#3228C2212yescomplemented
22#3210C2212yescomplemented
23#3207C2212yescomplemented
24#3191—21no—
25#3191—21no—
26#3170C2211yescharacteristic, complemented
27#3169C2211yescharacteristic, complemented
28#3167C2212yescomplemented
29#3114—24no—
30#3110C2212yes—
31#3110C2212yes—
32#3109C2212yes—
33#3109C2212yes—
34#3105—22no—
35#3105—22no—
36#3102—22no—
37#3102—22no—
38#3101C2214yes—
39#3100C2214yes—
40#3092—22no—
41#3077C2214yes—
42#3036C2214yes—
43#3003—22no—
44#2974—22no—
45#2974—22no—
46#2974C2214yes—
47#2974C2214yes—
48C4×AGL(1,ℤ/8)—24no—
49C2×#436C2211yescharacteristic
50C2×#436C2211yescharacteristic
51C2×#435—21no—
52C2×#435—21no—
53C2×C8×D4C2211yescharacteristic
54C2×C8×D4C2211yescharacteristic
55#2920C2212yes—
56C2×#432C2211yescharacteristic
57C2×#431C2211yescharacteristic
58C2×#407—21no—
59C22×C22⋊C8—21no—
60#1703C2214yescomplemented
61#1701C2214yescomplemented
62#1699C2214yescomplemented
63#1697C2214yescomplemented
64#1695C2212yescomplemented
65#1693C2212yescomplemented
66#1691C2214yescomplemented
67#1689C2212yescomplemented
68#1687C2212yescomplemented
69#1685C2214yescomplemented
70#1683C2214yescomplemented
71#1681C2214yescomplemented
72#1679C2214yescomplemented
73#1677C2214yescomplemented
74#1675C2212yescomplemented
75#1673C2212yescomplemented
76#1671C2214yescomplemented
77#1669C2212yescomplemented
78#1667C2212yescomplemented
79#1665C2214yescomplemented
80#1574C2214yescomplemented
81#1574C2214yescomplemented
82#1568C2212yescomplemented
83#1568C2211yescharacteristic, complemented
84#1568C2211yescharacteristic, complemented
85#1559C2214yescomplemented
86#1557C2214yescomplemented
87#1554C2214yescomplemented
88#1553C2214yescomplemented
89#1552C2212yescomplemented
90#1552C2212yescomplemented
91#1551C2212yescomplemented
92#1551C2212yescomplemented
93C23×D4D411yescharacteristic
94C24×C4—21no—
95C2×AGL(1,ℤ/8)—24no—
96C22×M16—21no—
97C22×M16—21no—
98C23×C8—21no—
99C23×C8—21no—
100#533C2312yes—
101#533C2312yes—
102#533C2312yes—
103#532C2312yes—
104#532C2312yes—
105#532C2312yes—
106#529—22no—
107#529—22no—
108#529—22no—
109#529—22no—
110#529—22no—
111#529—22no—
112#525—22no—
113#525—22no—
114#525—22no—
115C4×#41—22no—
116C4×#41—22no—
117C4×#41—22no—
118C2×C4×D4—21no—
119C2×C4×D4—22no—
120C2×C4×D4—21no—
121C2×C4×D4—22no—
122C2×C4×D4D411yescharacteristic
123C2×C4×D4D411yescharacteristic
124C2×C4×D4D414yes—
125C2×C4×D4C2314yes—
126C2×C4×D4C2311yescharacteristic
127C2×C4×D4C2311yescharacteristic
128C22×C4⋊C4D411yescharacteristic
129C22×C22⋊C4—21no—
130C22×C42C2311yescharacteristic
131#500C2311yescharacteristic, complemented
132#500C2311yescharacteristic, complemented
133#498C2312yescomplemented
134#494C2311yescharacteristic, complemented
135#493C2311yescharacteristic, complemented
136#492C2312yescomplemented
137#487C2312yescomplemented
138#487C2312yescomplemented
139#486C2312yescomplemented
140#486C2312yescomplemented
141#485—22no—
142#485—22no—
143#484—22no—
144#484—22no—
145#483—24no—
146#482—22no—
147#482—22no—
148#481—22no—
149#481—22no—
150#480—24no—
151#479—22no—
152#479—22no—
153#478—24no—
154#478—24no—
155#478—22no—
156#478—22no—
157#476C2314yes—
158#476C2314yes—
159#474C2314yes—
160#474C2314yes—
161#467—21no—
162#467—21no—
163#466—21no—
164#466—21no—
165#465—22no—
166#465—22no—
167#464—22no—
168#464—22no—
169#463—24no—
170#463—24no—
171#463—22no—
172#463—22no—
173#461C2314yes—
174#461C2314yes—
175#459C2314yes—
176#459C2314yes—
177#452—22no—
178#452—22no—
179#450—24no—
180#450—22no—
181#450—22no—
182#449—22no—
183#449—22no—
184#447—24no—
185#447—22no—
186#447—22no—
187#442—24no—
188#442—24no—
189#440—24no—
190#440—24no—
191C4×Q16—21no—
192C4×Q16—21no—
193C4×SD16—24no—
194C4×SD16—24no—
195C4×SD16—22no—
196C4×SD16—22no—
197C4×D8—24no—
198C4×D8—24no—
199C4×D8—21no—
200C4×D8—21no—
201#436—21no—
202#436—21no—
203#436—22no—
204#436C2314yes—
205#436C2312yes—
206#436C2312yes—
207#435—24no—
208#435—24no—
209#435—24no—
210#435—24no—
211C8×D4—21no—
212C8×D4—21no—
213C8×D4—22no—
214C8×D4C2314yes—
215C8×D4C2312yes—
216C8×D4C2312yes—
217#432—22no—
218#432—22no—
219#432C2312yes—
220#432C2312yes—
221#432C2314yes—
222#431—22no—
223#431D412yes—
224#431D412yes—
225#431C2314yes—
226#431C2312yes—
227#431C2312yes—
228#428—24no—
229#423C2314yes—
230C2×C4⋊C8D411yescharacteristic
231C2×C4⋊C8D411yescharacteristic
232C2×C4⋊C8C2311yescharacteristic
233C2×C4⋊C8C2311yescharacteristic
234#418—24no—
235#417—24no—
236#407—22no—
237#407—24no—
238#407—24no—
239C2×C22⋊C8—21no—
240C2×C22⋊C8—21no—
241C2×C22⋊C8—21no—
242C2×C22⋊C8—21no—
243C2×C22⋊C8—21no—
244C2×C22⋊C8—24no—
245C2×C22⋊C8—24no—
246C2×C22⋊C8D411yescharacteristic
247C2×C22⋊C8D411yescharacteristic
248C4×M16—22no—
249C2×#97—21no—
250C2×C4×C8C2311yescharacteristic
251C2×C4×C8C2311yescharacteristic
252#335C2312yescomplemented
253#335C2312yescomplemented
254#335C2314yescomplemented
255#334C2314yescomplemented
256#334C2312yescomplemented
257#334C2312yescomplemented
258#333C2314yescomplemented
259#333C2314yescomplemented
260#332C2314yescomplemented
261#332C2314yescomplemented
262#332C2314yescomplemented
263#332C2314yescomplemented
264#331C2314yescomplemented
265#331C2314yescomplemented
266C25—21no—
267C2×#41—22no—
268C2×#41—22no—
269C2×#41—22no—
270C22×D4—21no—
271C22×D4—22no—
272C22×D4—44no—
273C22×D4C2×D411yescharacteristic
274C22×D4C2×D414yes—
275C22×D4C2×D411yescharacteristic
276C23×C4—21no—
277C23×C4—21no—
278C23×C4—21no—
279C23×C4—21no—
280C23×C4—21no—
281C23×C4—24no—
282C23×C4—24no—
283C23×C4—82no—
284C23×C4—82no—
285C23×C4C2×D411yescharacteristic
286C23×C4C2×D411yescharacteristic
287C23×C4#4111yescharacteristic
288AGL(1,ℤ/8)—416no—
289C2×Q16—21no—
290C2×Q16—21no—
291C2×SD16—24no—
292C2×SD16—24no—
293C2×SD16—22no—
294C2×SD16—22no—
295C2×D8—24no—
296C2×D8—24no—
297C2×D8—21no—
298C2×D8—21no—
299C2×M16—24no—
300C2×M16—22no—
301C2×M16—24no—
302C2×M16—22no—
303C2×M16—82no—
304C2×M16—82no—
305C22×C8—21no—
306C22×C8—21no—
307C22×C8—22no—
308C22×C8—24no—
309C22×C8—22no—
310C22×C8—82no—
311C22×C8—82no—
312C22×C8C2×D411yescharacteristic
313C22×C8C2×D411yescharacteristic
314C22×C8C2×D411yescharacteristic
315C22×C8C2×D411yescharacteristic
316C4⋊Q8C2412yes—
317C4⋊Q8C2412yes—
318C4⋊Q8C2412yes—
319#127C2412yes—
320#126—42no—
321#126—42no—
322#125—22no—
323#125—22no—
324#125—22no—
325#124—22no—
326#124—22no—
327#124—22no—
328#123—42no—
329#123—42no—
330#123—42no—
331C22⋊Q8—22no—
332C22⋊Q8—22no—
333C22⋊Q8—22no—
334C22⋊Q8—22no—
335C22⋊Q8—22no—
336C22⋊Q8—22no—
337#121—22no—
338#121—22no—
339#121—22no—
340#121—22no—
341#121—22no—
342#121—22no—
343#120—42no—
344C4×Q8—22no—
345C4×Q8—22no—
346C4×Q8—22no—
347C4×Q8—22no—
348C4×Q8—22no—
349C4×Q8—22no—
350C4×D4—24no—
351C4×D4—24no—
352C4×D4—22no—
353C4×D4—22no—
354C4×D4—21no—
355C4×D4—24no—
356C4×D4—21no—
357C4×D4—22no—
358C4×D4—22no—
359C4×D4—22no—
360C4×D4—22no—
361C4×D4—24no—
362C4×D4—21no—
363C4×D4—21no—
364C4×D4—22no—
365C4×D4—22no—
366C4×D4—42no—
367C4×D4—42no—
368C4×D4—42no—
369C4×D4—42no—
370C4×D4—42no—
371C4×D4—42no—
372C4×D4C2×D414yes—
373C4×D4C2×D414yes—
374C4×D4C2×D412yes—
375C4×D4C2×D412yes—
376C4×D4C2412yes—
377C4×D4C2412yes—
378#117—22no—
379#117—22no—
380#117—22no—
381#117—42no—
382#117—42no—
383#117—42no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—22no—
386C2×C4⋊C4C2×D414yes—
387C2×C4⋊C4C2×D411yescharacteristic
388C2×C4⋊C4C2×D411yescharacteristic
389C2×C22⋊C4—21no—
390C2×C22⋊C4—21no—
391C2×C22⋊C4—24no—
392C2×C22⋊C4—21no—
393C2×C22⋊C4—21no—
394C2×C22⋊C4—24no—
395C2×C42—42no—
396C2×C42C2×D412yes—
397C2×C42C2×D411yescharacteristic
398C2×C42C2×D412yes—
399C2×C42C2×D411yescharacteristic
400C2×C42#4111yescharacteristic
401C2×C42#4112yes—
402C2×C42C2411yescharacteristic
403C2×C42C2412yes—
404#107—21no—
405#107—21no—
406#107—21no—
407#107—21no—
408#107—24no—
409#107—24no—
410#106—22no—
411#106—22no—
412#106—24no—
413#106—24no—
414C4⋊C8—21no—
415C4⋊C8—21no—
416C4⋊C8—24no—
417C4⋊C8C2×D412yes—
418C4⋊C8C2×D412yes—
419C4⋊C8C2×D412yes—
420C4⋊C8C2×D412yes—
421C4⋊C8C2412yes—
422C4⋊C8C2412yes—
423C4⋊C8C2414yes—
424C4⋊C8C2414yes—
425#103—22no—
426#103—22no—
427#103—22no—
428#103—22no—
429#103—24no—
430#103—24no—
431#102—24no—
432#102—24no—
433#102—24no—
434#102—24no—
435#102—22no—
436#102—22no—
437#102—22no—
438#102—22no—
439#102—24no—
440#102—24no—
441C22⋊C8—24no—
442C22⋊C8—22no—
443C22⋊C8—22no—
444C22⋊C8—24no—
445C22⋊C8—22no—
446C22⋊C8—24no—
447C22⋊C8—22no—
448C22⋊C8—22no—
449C22⋊C8—24no—
450C22⋊C8—22no—
451C22⋊C8C2×D412yes—
452C22⋊C8C2×D412yes—
453C22⋊C8C2×D412yes—
454C22⋊C8C2×D412yes—
455#97—24no—
456#97—22no—
457#97—22no—
458C4×C8—24no—
459C4×C8C2×D411yescharacteristic
460C4×C8C2×D411yescharacteristic
461C4×C8C2×D412yes—
462C4×C8C2411yescharacteristic
463C4×C8C2412yes—
464C4×C8C2411yescharacteristic
465C24—21no—
466C24—21no—
467C24—24no—
468C24—44no—
469C24—44no—
470C24—82no—
471C24#12111yescharacteristic
472#41—48no—
473#41—82no—
474#41—82no—
475#41—84no—
476C2×Q8—22no—
477C2×Q8—22no—
478C2×Q8—22no—
479C2×D4—24no—
480C2×D4—28no—
481C2×D4—21no—
482C2×D4—24no—
483C2×D4—21no—
484C2×D4—22no—
485C2×D4—22no—
486C2×D4—22no—
487C2×D4—44no—
488C2×D4—42no—
489C2×D4—48no—
490C2×D4—44no—
491C2×D4—42no—
492C2×D4—42no—
493C2×D4—42no—
494C2×D4C22×D412yes—
495C2×D4C22×D412yes—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—24no—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—22no—
502C22×C4—22no—
503C22×C4—24no—
504C22×C4—24no—
505C22×C4—41no—
506C22×C4—41no—
507C22×C4—42no—
508C22×C4—42no—
509C22×C4—42no—
510C22×C4—42no—
511C22×C4—42no—
512C22×C4—42no—
513C22×C4—42no—
514C22×C4—42no—
515C22×C4—42no—
516C22×C4—84no—
517C22×C4—84no—
518C22×C4—88no—
519C22×C4—82no—
520C22×C4—82no—
521C22×C4—82no—
522C22×C4#12111yescharacteristic
523C22×C4#12112yes—
524C22×C4#12111yescharacteristic
525C22×C4#12112yes—
526C22×C4#12111yescharacteristic
527C22×C4#12112yes—
528C22×C4C22×D411yescharacteristic
529C22×C4C22×D412yes—
530C22×C4C22×D412yes—
531C22×C4C22×D411yescharacteristic
532C22×C4C2×#4111yescharacteristic
533C22×C4C2×#4112yes—
534Q16—81no—
535Q16—81no—
536SD16—48no—
537SD16—48no—
538SD16—82no—
539SD16—82no—
540D8—48no—
541D8—48no—
542D8—81no—
543D8—81no—
544M16—48no—
545M16—82no—
546M16—84no—
547M16—82no—
548C2×C8—24no—
549C2×C8—24no—
550C2×C8—22no—
551C2×C8—22no—
552C2×C8—41no—
553C2×C8—41no—
554C2×C8—41no—
555C2×C8—41no—
556C2×C8—42no—
557C2×C8—42no—
558C2×C8—82no—
559C2×C8—84no—
560C2×C8—82no—
561C2×C8C22×D411yescharacteristic
562C2×C8C22×D411yescharacteristic
563C2×C8C22×D412yes—
564C2×C8C22×D412yes—
565C2×C8C22×D411yescharacteristic
566C2×C8C22×D411yescharacteristic
567C4⋊C4—24no—
568C4⋊C4—21no—
569C4⋊C4—21no—
570C4⋊C4—22no—
571C4⋊C4—22no—
572C4⋊C4—22no—
573C4⋊C4—22no—
574C4⋊C4—22no—
575C4⋊C4—22no—
576C4⋊C4—42no—
577C4⋊C4—42no—
578C4⋊C4—42no—
579C4⋊C4—42no—
580C4⋊C4—42no—
581C4⋊C4—42no—
582C4⋊C4C22×D412yes—
583C4⋊C4C22×D412yes—
584C22⋊C4—22no—
585C22⋊C4—22no—
586C22⋊C4—24no—
587C22⋊C4—24no—
588C22⋊C4—22no—
589C22⋊C4—22no—
590C22⋊C4—42no—
591C22⋊C4—42no—
592C22⋊C4—42no—
593C22⋊C4—42no—
594C22⋊C4—42no—
595C22⋊C4—42no—
596C22⋊C4—42no—
597C22⋊C4—42no—
598C22⋊C4—42no—
599C42—42no—
600C42—42no—
601C42—42no—
602C42#12112yes—
603C42#12112yes—
604C42#12112yes—
605C42C22×D412yes—
606C42C22×D411yescharacteristic
607C42C22×D411yescharacteristic
608C42C22×D412yes—
609C42C2×#4112yes—
610C42C2×#4111yescharacteristic
611C42C2511yescharacteristic
612C23—24no—
613C23—22no—
614C23—22no—
615C23—22no—
616C23—41no—
617C23—41no—
618C23—48no—
619C23—42no—
620C23—42no—
621C23—42no—
622C23—48no—
623C23—42no—
624C23—42no—
625C23—42no—
626C23—84no—
627C23—816no—
628C23C2×#12112yes—
629C23C2×#12111yescharacteristic
630Q8—42no—
631Q8—42no—
632Q8—44no—
633Q8—82no—
634D4—24no—
635D4—22no—
636D4—24no—
637D4—48no—
638D4—42no—
639D4—42no—
640D4—42no—
641D4—44no—
642D4—82no—
643D4—84no—
644D4—84no—
645D4—84no—
646D4#57514yes—
647C2×C4—22no—
648C2×C4—22no—
649C2×C4—21no—
650C2×C4—21no—
651C2×C4—22no—
652C2×C4—22no—
653C2×C4—24no—
654C2×C4—22no—
655C2×C4—22no—
656C2×C4—22no—
657C2×C4—42no—
658C2×C4—41no—
659C2×C4—42no—
660C2×C4—41no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—84no—
675C2×C4—84no—
676C2×C4—84no—
677C2×C4—84no—
678C2×C4—84no—
679C2×C4—84no—
680C2×C4C2×#12111yescharacteristic
681C2×C4C2×#12112yes—
682C2×C4C2×#12112yes—
683C2×C4C2×#12112yes—
684C2×C4C2×#12111yescharacteristic
685C2×C4C2×#12111yescharacteristic
686C2×C4C23×D411yescharacteristic
687C2×C4C23×D411yescharacteristic
688C2×C4C22×#4111yescharacteristic
689C8—21no—
690C8—22no—
691C8—21no—
692C8—21no—
693C8—22no—
694C8—21no—
695C8—42no—
696C8—42no—
697C8—44no—
698C8—44no—
699C22—21no—
700C22—21no—
701C22—22no—
702C22—22no—
703C22—24no—
704C22—22no—
705C22—48no—
706C22—44no—
707C22—42no—
708C22—42no—
709C22—42no—
710C22—42no—
711C22—42no—
712C22—84no—
713C22—816no—
714C22—82no—
715C22—82no—
716C22—82no—
717C22#306712yes—
718C22C22×#12111yescharacteristic
719C4—21no—
720C4—22no—
721C4—22no—
722C4—21no—
723C4—21no—
724C4—22no—
725C4—21no—
726C4—42no—
727C4—42no—
728C4—82no—
729C4—82no—
730C4—82no—
731C4C2×#57512yes—
732C4#358511yescharacteristic
733C4#358511yescharacteristic
734C2—22no—
735C2—42no—
736C2—44no—
737C2—82no—
738C2—162no—
739C2#6042311yescharacteristic
740C2#6044411yescharacteristic
741C2C2×#306711yescharacteristic
subgroupG/H
center, socleC22C22×#121
commutator, Frattini, cosocleC42C25
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #60944C2semidirect
2. #60558C2semidirect
3. C2×#2974C2semidirect
4. #19467C2semidirect
5. #19466C2semidirect
6. #19461C2semidirect
7. #19460C2semidirect
8. #19284C2semidirect
9. #19283C2semidirect
10. #19282C2semidirect
11. #19259C2semidirect
12. #19258C2semidirect
13. #19257C2semidirect
14. #17819C2semidirect
15. #17756C2semidirect
20. #3231C22semidirect
21. #3228C22semidirect
22. #3210C22semidirect
23. #3207C22semidirect
26. #3170C22semidirect
27. #3169C22semidirect
28. #3167C22semidirect
60. #1703C22semidirect
61. #1701C22semidirect
62. #1699C22semidirect
63. #1697C22semidirect
64. #1695C22semidirect
65. #1693C22semidirect
66. #1691C22semidirect
67. #1689C22semidirect
68. #1687C22semidirect
69. #1685C22semidirect
70. #1683C22semidirect
71. #1681C22semidirect
72. #1679C22semidirect
73. #1677C22semidirect
74. #1675C22semidirect
75. #1673C22semidirect
76. #1671C22semidirect
77. #1669C22semidirect
78. #1667C22semidirect
79. #1665C22semidirect
80. #1574C22semidirect
81. #1574C22semidirect
82. #1568C22semidirect
83. #1568C22semidirect
84. #1568C22semidirect
85. #1559C22semidirect
86. #1557C22semidirect
87. #1554C22semidirect
88. #1553C22semidirect
89. #1552C22semidirect
90. #1552C22semidirect
91. #1551C22semidirect
92. #1551C22semidirect
131. #500C23semidirect
132. #500C23semidirect
133. #498C23semidirect
134. #494C23semidirect
135. #493C23semidirect
136. #492C23semidirect
137. #487C23semidirect
138. #487C23semidirect
139. #486C23semidirect
140. #486C23semidirect
252. #335C23semidirect
253. #335C23semidirect
254. #335C23semidirect
255. #334C23semidirect
256. #334C23semidirect
257. #334C23semidirect
258. #333C23semidirect
259. #333C23semidirect
260. #332C23semidirect
261. #332C23semidirect
262. #332C23semidirect
263. #332C23semidirect
264. #331C23semidirect
265. #331C23semidirect

Maximal subgroups

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

subgroupindex
#609442
#605582
C2×#29742
#194672
#194662
#194612
#194602
#192842
#192832
#192822
#192592
#192582
#192572
#178192
#177562