Finite Group Databasebeta

← #6486076

Subgroups — #6486076

Order 512.

1562 conjugacy classes of subgroups.

The subgroup lattice has 715 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36635C211yesmaximal, characteristic, complemented
2#36631C211yesmaximal, characteristic
3#33280C212yesmaximal, complemented
4#33259C212yesmaximal, complemented
5#30365C212yesmaximal, complemented
6#30364C212yesmaximal, complemented
7#29514C212yesmaximal, complemented
8#29513C212yesmaximal, complemented
9#29223C211yesmaximal, characteristic, complemented
10#29211C211yesmaximal, characteristic, complemented
11#29145C211yesmaximal, characteristic, complemented
12#28516C211yesmaximal, characteristic, complemented
13#27897C212yesmaximal
14#27882C212yesmaximal, complemented
15#24704C214yesmaximal, complemented
16#22120C214yesmaximal, complemented
17#20463C211yesmaximal, characteristic, complemented
18C2×#529—21no—
19C2×#478C2214yes—
20C2×#431C2212yes—
21#2864C2212yes—
22#2863—21no—
23#2863—21no—
24#2862C2211yescharacteristic
25#2861—21no—
26#2860—21no—
27#2860C2212yes—
28#2859C2211yescharacteristic
29#2859C2211yescharacteristic
30#2858C2211yescharacteristic
31#2857C2211yescharacteristic
32#2857C2211yescharacteristic
33#2857C2211yescharacteristic
34#2856C2211yescharacteristic
35#2855—21no—
36#2855—21no—
37#2854—21no—
38#2854C2212yes—
39#2851C2212yes—
40#2850—21no—
41C4×C4⋊Q8C2211yescharacteristic
42C4×C4⋊Q8C2211yescharacteristic
43C4×C4⋊Q8C2211yescharacteristic
44C4×#127C2211yescharacteristic
45C4×#125C2212yes—
46C4×#124—21no—
47C4×C22⋊Q8—21no—
48C4×C22⋊Q8—21no—
49C4×#121—21no—
50C4×#121C2212yes—
51#2295—21no—
52#2295—21no—
53#2293—21no—
54#2293C2212yes—
55#2283C2212yes—
56C2×C43C2211yescharacteristic
57#2100C2212yes—
58#2099C2212yes—
59#2094C2212yes—
60#2093C2212yes—
61#2077C2214yes—
62#2076C2214yes—
63#2058C2212yes—
64#2057C2212yes—
65#2034C2214yes—
66#2033C2214yes—
67#2023C2212yes—
68#2023C2212yes—
69#1993C2212yes—
70#1993C2212yes—
71#1989C2214yes—
72#1961C2212yes—
73#1961C2212yes—
74#1949C2214yes—
75#1944C2214yes—
76#1943C2214yes—
77#1942C2214yes—
78#1941C2214yes—
79#1936C2214yes—
80#1936C2214yes—
81#1934C2214yes—
82#1934C2214yes—
83#1926C2214yes—
84#1925C2214yes—
85#1919C2214yes—
86#1918C2214yes—
87#1847C2212yes—
88#1847C2212yes—
89#1836C2212yes—
90#1836C2212yes—
91#1804C2214yes—
92#1767C2211yescharacteristic
93#1767C2211yescharacteristic
94#1767C2212yes—
95#1749C2212yes—
96#1749C2212yes—
97#1744C2212yes—
98C4×#97C2212yes—
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2311yescharacteristic
102C2×#127C2311yescharacteristic
103#529—48no—
104C2×#126—22no—
105C2×#125—21no—
106C2×#125C2312yes—
107C2×#124—21no—
108C2×#124—21no—
109C2×#123—22no—
110C2×C22⋊Q8—21no—
111C2×C22⋊Q8—21no—
112C2×C22⋊Q8—21no—
113C2×#121—21no—
114C2×#121—21no—
115C2×#121C2312yes—
116C2×C4×Q8—21no—
117C2×C4×Q8—21no—
118C2×C4×Q8—21no—
119C2×C4×D4—22no—
120C2×C4×D4—21no—
121C2×C4×D4—21no—
122C2×C4×D4C2312yes—
123C2×#117—22no—
124C22×C42D411yescharacteristic
125C22×C42D411yescharacteristic
126C22×C42D411yescharacteristic
127C22×C42D411yescharacteristic
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131#478—216no—
132#431—28no—
133C2×#107C2314yes—
134C2×#107C2314yes—
135C2×#106C2314yes—
136C2×#106C2314yes—
137C2×C4⋊C8C2312yes—
138C2×C4⋊C8C2312yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×C22⋊C8C2314yes—
144C2×#97C2312yes—
145C2×#97C2312yes—
146C2×#97C2314yes—
147C2×C4×C8C2314yes—
148#391—21no—
149#391—21no—
150#391—21no—
151#391—21no—
152#391—21no—
153#391—21no—
154#391—21no—
155#391—21no—
156#391—21no—
157#391—21no—
158#390—21no—
159#390—21no—
160#390—21no—
161#390—21no—
162#390—21no—
163#390—21no—
164#390C2312yes—
165#390C2312yes—
166#390C2312yes—
167#390C2312yes—
168#389—21no—
169#389—21no—
170#389—21no—
171#389—21no—
172#389—21no—
173#389—21no—
174#389—21no—
175#389—21no—
176#389—21no—
177#389—21no—
178#389—21no—
179#389—21no—
180#389C2312yes—
181#389C2312yes—
182#389C2312yes—
183#389C2312yes—
184#389C2312yes—
185#389C2312yes—
186#389C2312yes—
187#389C2312yes—
188#388—22no—
189#388—22no—
190#388—22no—
191#388—22no—
192#388—22no—
193#388—22no—
194#388—22no—
195#388—22no—
196#387—22no—
197#387—22no—
198#387—22no—
199#387—22no—
200#387—22no—
201#387—22no—
202#387—22no—
203#387—22no—
204#386—22no—
205#386—22no—
206#385—22no—
207#385—22no—
208#384C2311yescharacteristic
209#384C2311yescharacteristic
210#384C2311yescharacteristic
211#384C2311yescharacteristic
212#383—22no—
213#382—21no—
214#382C2312yes—
215#382C2312yes—
216#382C2312yes—
217#381—22no—
218#380—21no—
219#380—21no—
220#380—21no—
221#380C2312yes—
222#380C2312yes—
223#380C2312yes—
224C4×C4⋊C4—22no—
225C4×C4⋊C4—22no—
226C4×C4⋊C4—21no—
227C4×C4⋊C4—21no—
228C4×C4⋊C4—21no—
229C4×C4⋊C4C2312yes—
230C4×C4⋊C4C2312yes—
231C4×C4⋊C4C2312yes—
232C4×C22⋊C4—22no—
233C4×C22⋊C4—22no—
234C4×C22⋊C4—22no—
235#376—22no—
236C43D411yescharacteristic
237C43D411yescharacteristic
238C43D411yescharacteristic
239C43D411yescharacteristic
240C43C2311yescharacteristic
241C43C2311yescharacteristic
242C43C2311yescharacteristic
243C43C2311yescharacteristic
244#340C2314yes—
245#340C2314yes—
246#340C2314yes—
247#340C2314yes—
248#340C2314yes—
249#340C2314yes—
250#340C2314yes—
251#340C2314yes—
252#336C2312yes—
253#336C2312yes—
254#336C2314yes—
255C22×Q8—21no—
256C22×Q8—21no—
257C22×Q8—21no—
258C22×D4—22no—
259C22×D4—21no—
260C22×D4—21no—
261C22×D4C2412yes—
262C23×C4—22no—
263C23×C4C2×D411yescharacteristic
264C23×C4C2×D411yescharacteristic
265C23×C4C2×D411yescharacteristic
266C23×C4C2×D411yescharacteristic
267C23×C4C2×D411yescharacteristic
268C23×C4C2×D411yescharacteristic
269C23×C4C2411yescharacteristic
270C22×C8C2414yes—
271C22×C8C2414yes—
272C4⋊Q8—22no—
273C4⋊Q8—22no—
274C4⋊Q8—42no—
275C4⋊Q8—42no—
276#127—22no—
277#127—22no—
278#126—48no—
279#125—28no—
280#125—44no—
281#124—44no—
282#124—44no—
283#123—48no—
284C22⋊Q8—44no—
285C22⋊Q8—44no—
286C22⋊Q8—44no—
287#121—44no—
288#121—44no—
289#121—44no—
290C4×Q8—44no—
291C4×Q8—44no—
292C4×Q8—44no—
293C4×D4—28no—
294C4×D4—44no—
295C4×D4—44no—
296C4×D4—48no—
297#117—48no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C42—24no—
340C2×C42—24no—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—22no—
344C2×C42—22no—
345C2×C42—22no—
346C2×C42—42no—
347C2×C42—42no—
348C2×C42—42no—
349C2×C42—42no—
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D411yescharacteristic
360C2×C42C2×D411yescharacteristic
361C2×C42C2×D411yescharacteristic
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D411yescharacteristic
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D411yescharacteristic
366C2×C42C2×D411yescharacteristic
367C2×C42C2×D411yescharacteristic
368C2×C42C2×D411yescharacteristic
369C2×C42C2×D411yescharacteristic
370C2×C42C2×D411yescharacteristic
371C2×C42C2×D411yescharacteristic
372C2×C42C2411yescharacteristic
373C2×C42C2411yescharacteristic
374C2×C42C2411yescharacteristic
375C2×C42C2411yescharacteristic
376C2×C42C2411yescharacteristic
377C2×C42C2411yescharacteristic
378#107—28no—
379#107—44no—
380#106—28no—
381#106—44no—
382C4⋊C8—24no—
383C4⋊C8#4118yes—
384#102—28no—
385#102—28no—
386#102—44no—
387#102—44no—
388C22⋊C8—28no—
389#97—24no—
390#97—24no—
391#97—44no—
392C4×C8—28no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—22no—
402#95—22no—
403#95—22no—
404#95—22no—
405#95—22no—
406#95—22no—
407#95—22no—
408#95—22no—
409#95—22no—
410#95—22no—
411#95—22no—
412#95—22no—
413C24—22no—
414C24C22×D411yescharacteristic
415C2×Q8—42no—
416C2×Q8—42no—
417C2×Q8—42no—
418C2×Q8—42no—
419C2×Q8—42no—
420C2×Q8—42no—
421C2×Q8—42no—
422C2×Q8—42no—
423C2×Q8—42no—
424C2×D4—24no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—24no—
428C2×D4—42no—
429C2×D4—42no—
430C2×D4—42no—
431C2×D4—42no—
432C2×D4—44no—
433C2×D4—44no—
434C2×D4—44no—
435C2×D4—42no—
436C2×D4—42no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—41no—
461C22×C4—41no—
462C22×C4—41no—
463C22×C4—41no—
464C22×C4—41no—
465C22×C4—41no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C2511yescharacteristic
483C2×C8—24no—
484C2×C8—24no—
485C2×C8—24no—
486C2×C8—24no—
487C2×C8—24no—
488C2×C8C2×#4118yes—
489C4⋊C4—24no—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—44no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—42no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—42no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—42no—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C22⋊C4—44no—
521C22⋊C4—44no—
522C22⋊C4—44no—
523C22⋊C4—44no—
524C22⋊C4—44no—
525C22⋊C4—44no—
526C22⋊C4—44no—
527C22⋊C4—44no—
528C22⋊C4—44no—
529C22⋊C4—44no—
530C22⋊C4—44no—
531C42—22no—
532C42—22no—
533C42—22no—
534C42—22no—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—22no—
539C42—22no—
540C42—22no—
541C42—22no—
542C42—22no—
543C42—21no—
544C42—21no—
545C42—22no—
546C42—22no—
547C42—41no—
548C42—41no—
549C42—41no—
550C42—41no—
551C42—41no—
552C42—41no—
553C42—41no—
554C42—41no—
555C42—41no—
556C42—41no—
557C42—44no—
558C42—44no—
559C42—44no—
560C42AGL(1,ℤ/8)14yes—
561C42C2×#4114yes—
562C422−1+412yes—
563C422−1+412yes—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—21no—
569C23—21no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23#57512yes—
578C23C23×D411yescharacteristic
579Q8—84no—
580Q8—84no—
581Q8—84no—
582D4—44no—
583D4—44no—
584D4—84no—
585D4—84no—
586D4—88no—
587C2×C4—28no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—88no—
647C2×C4#53212yes—
648C2×C4#53212yes—
649C2×C4#53612yes—
650C2×C4#53612yes—
651C2×C4C2×AGL(1,ℤ/8)12yes—
652C2×C4C2×AGL(1,ℤ/8)12yes—
653C2×C4C2×AGL(1,ℤ/8)12yes—
654C2×C4#57512yes—
655C2×C4#57512yes—
656C2×C4C22×#4112yes—
657C2×C4C2×2−1+411yescharacteristic
658C2×C4C2×2−1+411yescharacteristic
659C2×C4#58512yes—
660C8—28no—
661C8—44no—
662C22—22no—
663C22—22no—
664C22—22no—
665C22—41no—
666C22—41no—
667C22—41no—
668C22—41no—
669C22—82no—
670C22—82no—
671C22—82no—
672C22—82no—
673C22—82no—
674C22—82no—
675C22—82no—
676C22C2×#53211yescharacteristic
677C22C2×#53611yescharacteristic
678C22#344911yescharacteristic
679C22C22×AGL(1,ℤ/8)11yescharacteristic
680C22C2×#57511yescharacteristic
681C22C2×#57511yescharacteristic
682C22C2×#57511yescharacteristic
683C4—22no—
684C4—22no—
685C4—22no—
686C4—22no—
687C4—22no—
688C4—22no—
689C4—22no—
690C4—22no—
691C4—22no—
692C4—41no—
693C4—41no—
694C4—41no—
695C4—41no—
696C4—82no—
697C4—82no—
698C4—82no—
699C4—82no—
700C4—82no—
701C4—82no—
702C4—82no—
703C4#351512yes—
704C4#351812yes—
705C2—42no—
706C2—162no—
707C2#3520111yescharacteristic
708C2#6068811yescharacteristic
709C2#6070011yescharacteristic
710C2#6086411yescharacteristic
711C2#6086611yescharacteristic
712C2#6094911yescharacteristic
713C2#6095411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36635C2semidirect
3. #33280C2semidirect
4. #33259C2semidirect
5. #30365C2semidirect
6. #30364C2semidirect
7. #29514C2semidirect
8. #29513C2semidirect
9. #29223C2semidirect
10. #29211C2semidirect
11. #29145C2semidirect
12. #28516C2semidirect
14. #27882C2semidirect
15. #24704C2semidirect
16. #22120C2semidirect
17. #20463C2semidirect

Maximal subgroups

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

subgroupindex
#366352
#366312
#332802
#332592
#303652
#303642
#295142
#295132
#292232
#292112
#291452
#285162
#278972
#278822
#247042
#221202
#204632