Finite Group Databasebeta

← #6495509

Subgroups — #6495509

Order 512.

1642 conjugacy classes of subgroups.

The subgroup lattice has 721 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36640C211yesmaximal, characteristic, complemented
2#36639C211yesmaximal, characteristic
3#33317C211yesmaximal, characteristic, complemented
4#33296C211yesmaximal, characteristic, complemented
5#33237C211yesmaximal, characteristic, complemented
6#33234C211yesmaximal, characteristic, complemented
7#29846C214yesmaximal, complemented
8#29198C212yesmaximal, complemented
9#29197C212yesmaximal, complemented
10#28661C212yesmaximal, complemented
11#28660C212yesmaximal, complemented
12#27895C212yesmaximal, complemented
13#27894C212yesmaximal, complemented
14#25677C212yesmaximal, complemented
15#25676C212yesmaximal
16#22222C214yesmaximal, complemented
17#20463C211yesmaximal, characteristic, complemented
18C2×#533C2211yescharacteristic
19C2×#532C2211yescharacteristic
20C2×#501C2212yes—
21C2×#501C2212yes—
22C2×#491C2212yes—
23C2×#490C2212yes—
24C2×#488C2214yes—
25C2×#431C2212yes—
26#2864—21no—
27#2864—21no—
28#2862C2211yescharacteristic
29#2862C2211yescharacteristic
30#2861—21no—
31#2861—21no—
32#2859C2211yescharacteristic
33#2859C2211yescharacteristic
34#2859C2211yescharacteristic
35#2859C2211yescharacteristic
36#2858C2211yescharacteristic
37#2858C2211yescharacteristic
38#2851—21no—
39#2851—21no—
40#2851C2212yes—
41#2851C2212yes—
42#2850—21no—
43#2850—21no—
44#2850C2212yes—
45#2850C2212yes—
46C4×#125—21no—
47C4×#125—21no—
48C4×#125C2212yes—
49C4×#125C2212yes—
50C4×#124—21no—
51C4×#124—21no—
52C4×#124C2212yes—
53C4×#124C2212yes—
54#2297—21no—
55#2297—21no—
56#2296—21no—
57#2296—21no—
58#2285C2211yescharacteristic
59#2285C2211yescharacteristic
60C2×C43C2211yescharacteristic
61#2068C2214yes—
62#2067C2214yes—
63#2066C2214yes—
64#2065C2214yes—
65#2036C2214yes—
66#2035C2214yes—
67#2034C2214yes—
68#2033C2214yes—
69#1993C2212yes—
70#1993C2212yes—
71#1991C2214yes—
72#1961C2212yes—
73#1960C2212yes—
74#1959C2212yes—
75#1959C2212yes—
76#1959C2212yes—
77#1959C2212yes—
78#1958C2214yes—
79#1928C2214yes—
80#1927C2214yes—
81#1926C2214yes—
82#1925C2214yes—
83#1865C2214yes—
84#1864C2214yes—
85#1863C2214yes—
86#1862C2214yes—
87#1847C2212yes—
88#1847C2212yes—
89#1846C2214yes—
90#1836C2212yes—
91#1836C2212yes—
92#1767C2212yes—
93#1767C2212yes—
94#1749C2212yes—
95#1749C2212yes—
96#1744C2212yes—
97C4×#97C2211yescharacteristic
98C4×#97C2211yescharacteristic
99C22×#41—21no—
100#533—44no—
101#532—44no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2311yescharacteristic
105C2×#127C2311yescharacteristic
106C2×#125—21no—
107C2×#125—21no—
108C2×#125—21no—
109C2×#125C2312yes—
110C2×#125C2312yes—
111C2×#124—21no—
112C2×#124—21no—
113C2×#124—21no—
114C2×#124C2312yes—
115C2×#124C2312yes—
116C2×C22⋊Q8—21no—
117C2×C22⋊Q8C2312yes—
118C2×#121—21no—
119C2×#121C2312yes—
120C2×C4×Q8—21no—
121C2×C4×Q8C2312yes—
122C2×C4×D4—21no—
123C2×C4×D4C2312yes—
124C2×#117—21no—
125C2×#117—21no—
126C2×#117—21no—
127C22×C42D411yescharacteristic
128C22×C42D411yescharacteristic
129C22×C42D411yescharacteristic
130C22×C42D411yescharacteristic
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134#501—44no—
135#501—44no—
136#491—44no—
137#490—44no—
138#488—216no—
139#431—28no—
140C2×#107C2314yes—
141C2×#107C2314yes—
142C2×#106C2314yes—
143C2×#106C2314yes—
144C2×C4⋊C8C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×C22⋊C8C2314yes—
154C2×#97C2312yes—
155C2×#97C2312yes—
156C2×#97C2312yes—
157C2×#97C2312yes—
158C2×C4×C8C2312yes—
159C2×C4×C8C2312yes—
160#391—21no—
161#391—21no—
162#391—21no—
163#391—21no—
164#391—21no—
165#391—21no—
166#391C2312yes—
167#391C2312yes—
168#390—21no—
169#390—21no—
170#390—21no—
171#390—21no—
172#390—21no—
173#390—21no—
174#390C2312yes—
175#390C2312yes—
176#389—21no—
177#389—21no—
178#389—21no—
179#389—21no—
180#389—21no—
181#389—21no—
182#389—21no—
183#389—21no—
184#389—21no—
185#389—21no—
186#389—21no—
187#389—21no—
188#389C2312yes—
189#389C2312yes—
190#389C2312yes—
191#389C2312yes—
192#388—22no—
193#388—22no—
194#388—22no—
195#388—22no—
196#388—22no—
197#388—22no—
198#388—22no—
199#388—22no—
200#387—22no—
201#387—22no—
202#387—22no—
203#387—22no—
204#387—22no—
205#387—22no—
206#387—22no—
207#387—22no—
208#386—22no—
209#386—22no—
210#386—22no—
211#386—22no—
212#385—22no—
213#385—22no—
214#385—22no—
215#385—22no—
216#384C2311yescharacteristic
217#384C2311yescharacteristic
218#384C2311yescharacteristic
219#384C2311yescharacteristic
220#382—21no—
221#382—21no—
222#382—21no—
223#382—21no—
224#382—21no—
225#382—21no—
226#382C2312yes—
227#382C2312yes—
228#382C2312yes—
229#382C2312yes—
230#380—21no—
231#380—21no—
232#380C2312yes—
233#380C2312yes—
234C4×C4⋊C4—21no—
235C4×C4⋊C4—21no—
236C4×C4⋊C4—22no—
237C4×C4⋊C4—22no—
238C4×C4⋊C4—22no—
239C4×C4⋊C4—22no—
240C4×C4⋊C4C2312yes—
241C4×C4⋊C4C2312yes—
242C4×C22⋊C4—22no—
243C4×C22⋊C4—22no—
244C4×C22⋊C4—22no—
245C4×C22⋊C4—22no—
246C43D411yescharacteristic
247C43D411yescharacteristic
248C43D411yescharacteristic
249C43D411yescharacteristic
250C43C2311yescharacteristic
251C43C2311yescharacteristic
252C43C2311yescharacteristic
253C43C2311yescharacteristic
254#340C2314yes—
255#340C2314yes—
256#340C2314yes—
257#340C2314yes—
258#336C2314yes—
259#336C2314yes—
260C2×#41—44no—
261C2×#41—44no—
262C2×#41—44no—
263C22×Q8—21no—
264C22×Q8—21no—
265C22×Q8C2412yes—
266C22×D4—22no—
267C22×D4—21no—
268C22×D4—21no—
269C22×D4C2412yes—
270C23×C4—22no—
271C23×C4C2×D411yescharacteristic
272C23×C4C2×D411yescharacteristic
273C23×C4C2×D411yescharacteristic
274C23×C4C2×D411yescharacteristic
275C23×C4C2×D411yescharacteristic
276C23×C4C2×D411yescharacteristic
277C23×C4C2411yescharacteristic
278C22×C8C2414yes—
279C22×C8C2414yes—
280C4⋊Q8—24no—
281C4⋊Q8—24no—
282C4⋊Q8—24no—
283#127—24no—
284#125—28no—
285#125—24no—
286#125—24no—
287#125—44no—
288#125—44no—
289#125—44no—
290#124—28no—
291#124—28no—
292#124—44no—
293#124—44no—
294#124—44no—
295C22⋊Q8—44no—
296C22⋊Q8—44no—
297#121—44no—
298#121—44no—
299C4×Q8—44no—
300C4×Q8—44no—
301C4×D4—44no—
302C4×D4—44no—
303#117—44no—
304#117—44no—
305#117—44no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2412yes—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C42—24no—
348C2×C42—24no—
349C2×C42—24no—
350C2×C42—22no—
351C2×C42—22no—
352C2×C42—22no—
353C2×C42—42no—
354C2×C42—42no—
355C2×C42—42no—
356C2×C42—42no—
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×C42C2×D411yescharacteristic
373C2×C42C2×D411yescharacteristic
374C2×C42C2×D411yescharacteristic
375C2×C42C2×D411yescharacteristic
376C2×C42C2×D411yescharacteristic
377C2×C42C2×D411yescharacteristic
378C2×C42C2×D411yescharacteristic
379C2×C42C2411yescharacteristic
380C2×C42C2411yescharacteristic
381C2×C42C2411yescharacteristic
382C2×C42C2411yescharacteristic
383C2×C42C2411yescharacteristic
384C2×C42C2411yescharacteristic
385#107—44no—
386#107—44no—
387#106—44no—
388#106—44no—
389C4⋊C8—28no—
390#103—28no—
391#103—28no—
392#103—44no—
393#103—44no—
394#102—28no—
395#102—28no—
396#102—44no—
397#102—44no—
398C22⋊C8—28no—
399#97—24no—
400#97—24no—
401#97—24no—
402#97—24no—
403C4×C8—24no—
404C4×C8#4118yescomplemented
405#95—22no—
406#95—22no—
407#95—22no—
408#95—22no—
409#95—22no—
410#95—22no—
411#95—22no—
412#95—22no—
413#95—22no—
414#95—22no—
415#95—22no—
416#95—22no—
417#95—22no—
418#95—22no—
419#95—22no—
420#95—22no—
421#95—22no—
422#95—22no—
423#95—22no—
424#95—22no—
425C24—22no—
426C24C22×D411yescharacteristic
427#41—816no—
428C2×Q8—24no—
429C2×Q8—24no—
430C2×Q8—24no—
431C2×Q8—42no—
432C2×Q8—42no—
433C2×Q8—42no—
434C2×Q8—42no—
435C2×Q8—42no—
436C2×Q8—42no—
437C2×D4—24no—
438C2×D4—24no—
439C2×D4—24no—
440C2×D4—42no—
441C2×D4—42no—
442C2×D4—42no—
443C2×D4—42no—
444C2×D4—44no—
445C2×D4—44no—
446C2×D4—44no—
447C2×D4—42no—
448C2×D4—42no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
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—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—41no—
472C22×C4—41no—
473C22×C4—41no—
474C22×C4—41no—
475C22×C4—41no—
476C22×C4—41no—
477C22×C4—44no—
478C22×C4—44no—
479C22×C4—44no—
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D411yescharacteristic
493C22×C4C2511yescharacteristic
494C2×C8—24no—
495C2×C8—24no—
496C2×C8—24no—
497C2×C8—24no—
498C2×C8—24no—
499C2×C8C2×#4118yes—
500C4⋊C4—24no—
501C4⋊C4—24no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—24no—
507C4⋊C4—24no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4—44no—
529C4⋊C4—44no—
530C22⋊C4—44no—
531C22⋊C4—44no—
532C22⋊C4—44no—
533C22⋊C4—44no—
534C22⋊C4—44no—
535C22⋊C4—44no—
536C22⋊C4—44no—
537C22⋊C4—44no—
538C22⋊C4—44no—
539C22⋊C4—44no—
540C22⋊C4—44no—
541C22⋊C4—44no—
542C22⋊C4—44no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—22no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—41no—
559C42—41no—
560C42—41no—
561C42—41no—
562C42—41no—
563C42—41no—
564C42—41no—
565C42—41no—
566C42—41no—
567C42—41no—
568C42—44no—
569C42—44no—
570C42—44no—
571C42C2×#4114yes—
572C42H(5,𝔽2)14yes—
573C422−1+414yes—
574C23—21no—
575C23—21no—
576C23—21no—
577C23—21no—
578C23—21no—
579C23—21no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23—42no—
586C23—42no—
587C23#57512yes—
588C23C23×D411yescharacteristic
589Q8—48no—
590Q8—84no—
591Q8—84no—
592D4—48no—
593D4—84no—
594D4—84no—
595D4—88no—
596C2×C4—28no—
597C2×C4—28no—
598C2×C4—28no—
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—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
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—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—88no—
655C2×C4#53212yes—
656C2×C4#53212yes—
657C2×C4#53512yes—
658C2×C4#53512yes—
659C2×C4#53612yes—
660C2×C4#53612yes—
661C2×C4#57512yes—
662C2×C4#57512yes—
663C2×C4#57512yes—
664C2×C4C22×#4112yes—
665C2×C4C2×H(5,𝔽2)12yes—
666C2×C4C2×2−1+412yes—
667C8—28no—
668C8—44no—
669C22—22no—
670C22—22no—
671C22—22no—
672C22—41no—
673C22—41no—
674C22—41no—
675C22—41no—
676C22—82no—
677C22—82no—
678C22—82no—
679C22—82no—
680C22—82no—
681C22—82no—
682C22—82no—
683C22C2×#53211yescharacteristic
684C22C2×#53511yescharacteristic
685C22C2×#53611yescharacteristic
686C22C2×#57511yescharacteristic
687C22C2×#57511yescharacteristic
688C22C2×#57511yescharacteristic
689C22C2×#57511yescharacteristic
690C4—22no—
691C4—22no—
692C4—22no—
693C4—22no—
694C4—22no—
695C4—22no—
696C4—22no—
697C4—22no—
698C4—22no—
699C4—41no—
700C4—41no—
701C4—41no—
702C4—41no—
703C4—82no—
704C4—82no—
705C4—82no—
706C4—82no—
707C4—82no—
708C4—82no—
709C4—82no—
710C4#351114yes—
711C2—42no—
712C2—162no—
713C2#3529811yescharacteristic
714C2#6067611yescharacteristic
715C2#6068811yescharacteristic
716C2#6079911yescharacteristic
717C2#6080211yescharacteristic
718C2#6086111yescharacteristic
719C2#6086311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
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. #36640C2semidirect
3. #33317C2semidirect
4. #33296C2semidirect
5. #33237C2semidirect
6. #33234C2semidirect
7. #29846C2semidirect
8. #29198C2semidirect
9. #29197C2semidirect
10. #28661C2semidirect
11. #28660C2semidirect
12. #27895C2semidirect
13. #27894C2semidirect
14. #25677C2semidirect
16. #22222C2semidirect
17. #20463C2semidirect
404. C4×C8#41semidirect

Maximal subgroups

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

subgroupindex
#366402
#366392
#333172
#332962
#332372
#332342
#298462
#291982
#291972
#286612
#286602
#278952
#278942
#256772
#256762
#222222
#204632