Finite Group Databasebeta

← #6489337

Subgroups — #6489337

Order 512.

1788 conjugacy classes of subgroups.

The subgroup lattice has 723 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36640C211yesmaximal, characteristic, complemented
2C4×#532C211yesmaximal, characteristic
3#32961C214yesmaximal, complemented
4#30079C211yesmaximal, characteristic, complemented
5#30078C211yesmaximal, characteristic, complemented
6#30076C211yesmaximal, characteristic, complemented
7#30075C211yesmaximal, characteristic, complemented
8#28720C212yesmaximal, complemented
9#28513C212yesmaximal, complemented
10#28209C212yesmaximal, complemented
11#28207C212yesmaximal, complemented
12#27951C212yesmaximal, complemented
13#27587C212yesmaximal, complemented
14#27584C212yesmaximal, complemented
15#25858C212yesmaximal
16#22222C214yesmaximal, complemented
17#20777C211yesmaximal, characteristic, complemented
18C2×#532C2211yescharacteristic
19C2×C4×#41C2212yes—
20C2×#488C2214yes—
21C2×#479C2214yes—
22C2×#478C2214yes—
23#2864—21no—
24#2864—21no—
25#2859C2211yescharacteristic
26#2859C2211yescharacteristic
27#2859C2211yescharacteristic
28#2859C2211yescharacteristic
29#2851—21no—
30#2851—21no—
31#2851C2212yes—
32#2851C2212yes—
33C4×C4⋊Q8C2211yescharacteristic
34C4×C4⋊Q8C2211yescharacteristic
35C4×#127C2211yescharacteristic
36C4×#127C2211yescharacteristic
37C4×#125—21no—
38C4×#125—21no—
39C4×#125C2212yes—
40C4×#125C2212yes—
41C4×#124C2212yes—
42C4×#124C2212yes—
43C4×#121—21no—
44C4×#121—21no—
45C4×#121—21no—
46C4×#121—21no—
47#2297C2212yes—
48#2296—21no—
49#2296—21no—
50#2293—21no—
51#2293—21no—
52#2285C2211yescharacteristic
53#2285C2211yescharacteristic
54#2285C2211yescharacteristic
55C42×D4—21no—
56C42×D4—21no—
57C2×C43C2211yescharacteristic
58#2096C2212yes—
59#2095C2212yes—
60#2094C2212yes—
61#2093C2212yes—
62#2084C2212yes—
63#2084C2212yes—
64#2083C2212yes—
65#2083C2212yes—
66#2072C2212yes—
67#2072C2212yes—
68#2071C2212yes—
69#2071C2212yes—
70#2043C2212yes—
71#2043C2212yes—
72#2041C2212yes—
73#2041C2212yes—
74#1991C2214yes—
75#1958C2214yes—
76#1928C2214yes—
77#1927C2214yes—
78#1926C2214yes—
79#1925C2214yes—
80#1865C2214yes—
81#1864C2214yes—
82#1863C2214yes—
83#1862C2214yes—
84#1854C2214yes—
85#1854C2214yes—
86#1846C2214yes—
87#1839C2211yescharacteristic
88#1839C2211yescharacteristic
89#1831C2212yes—
90#1817C2212yes—
91#1805C2214yes—
92#1804C2214yes—
93#1797C2212yes—
94#1788C2214yes—
95#1788C2214yes—
96C4×C4⋊C8C2212yes—
97C4×C22⋊C8C2212yes—
98#1744C2212yes—
99C22×#41C2312yes—
100#532—28no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×#127C2311yescharacteristic
103C2×#125—21no—
104C2×#125—21no—
105C2×#125C2312yes—
106C2×#125C2312yes—
107C2×#124C2312yes—
108C2×#121—21no—
109C2×#121—21no—
110C4×#41—48no—
111C2×C4×Q8C2312yes—
112C2×C4×Q8C2312yes—
113C2×C4×D4—22no—
114C2×C4×D4—22no—
115C2×C4×D4—21no—
116C2×C4×D4—21no—
117C2×C4×D4—21no—
118C2×C4×D4—21no—
119C2×C4×D4—21no—
120C2×C4×D4—21no—
121C2×C4×D4C2312yes—
122C2×C4×D4C2312yes—
123C2×#117—21no—
124C2×#117—21no—
125C2×#117—22no—
126C2×#117C2312yes—
127C22×C42—22no—
128C22×C42D411yescharacteristic
129C22×C42D411yescharacteristic
130C22×C42D411yescharacteristic
131C22×C42D411yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135#488—216no—
136#479—48no—
137#478—48no—
138C2×#107C2314yes—
139C2×#107C2314yes—
140C2×#106C2314yes—
141C2×#106C2314yes—
142C2×C4⋊C8C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×C22⋊C8C2314yes—
152C2×C4×C8C2314yes—
153#391C2312yes—
154#391C2312yes—
155#390—21no—
156#390—21no—
157#390—21no—
158#390—21no—
159#390C2312yes—
160#390C2312yes—
161#389—21no—
162#389—21no—
163#389—21no—
164#389—21no—
165#389—21no—
166#389—21no—
167#389—21no—
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#389C2312yes—
178#389C2312yes—
179#389C2312yes—
180#389C2312yes—
181#388—22no—
182#388—22no—
183#388—22no—
184#388—22no—
185#387—22no—
186#387—22no—
187#387—22no—
188#387—22no—
189#387—22no—
190#387—22no—
191#387—22no—
192#387—22no—
193#386—22no—
194#386—22no—
195#385—22no—
196#385—22no—
197#385—22no—
198#385—22no—
199#384C2311yescharacteristic
200#384C2311yescharacteristic
201#384C2311yescharacteristic
202#384C2311yescharacteristic
203#384C2311yescharacteristic
204#384C2311yescharacteristic
205#382—21no—
206#382—21no—
207#382—21no—
208#382—21no—
209#382C2312yes—
210#382C2312yes—
211#382C2312yes—
212#380—21no—
213#380—21no—
214#380—21no—
215#380—21no—
216#380C2312yes—
217#380C2312yes—
218C4×C4⋊C4—21no—
219C4×C4⋊C4—21no—
220C4×C4⋊C4—22no—
221C4×C4⋊C4—22no—
222C4×C4⋊C4—22no—
223C4×C4⋊C4—22no—
224C4×C4⋊C4—21no—
225C4×C4⋊C4—21no—
226C4×C4⋊C4—21no—
227C4×C4⋊C4—21no—
228C4×C4⋊C4—21no—
229C4×C4⋊C4—21no—
230C4×C4⋊C4C2312yes—
231C4×C4⋊C4C2312yes—
232C4×C22⋊C4—22no—
233C4×C22⋊C4—22no—
234C4×C22⋊C4—22no—
235C4×C22⋊C4—22no—
236C4×C22⋊C4—22no—
237C4×C22⋊C4—22no—
238C43D411yescharacteristic
239C43D411yescharacteristic
240C43D411yescharacteristic
241C43D411yescharacteristic
242C43C2311yescharacteristic
243C43C2311yescharacteristic
244C43C2311yescharacteristic
245C43C2311yescharacteristic
246#340C2314yes—
247#340C2314yes—
248#340C2314yes—
249#340C2314yes—
250#336C2312yes—
251#336C2312yes—
252#336C2312yes—
253#336C2312yes—
254#336C2314yes—
255#336C2312yes—
256#336C2312yes—
257C2×#41—24no—
258C2×#41—24no—
259C2×#41—28no—
260C2×#41—44no—
261C22×Q8C2412yes—
262C22×D4—22no—
263C22×D4—21no—
264C22×D4—21no—
265C22×D4C2412yes—
266C23×C4—22no—
267C23×C4—22no—
268C23×C4—22no—
269C23×C4C2×D411yescharacteristic
270C23×C4C2×D411yescharacteristic
271C23×C4C2×D411yescharacteristic
272C23×C4C2×D411yescharacteristic
273C23×C4C2×D411yescharacteristic
274C23×C4C2×D411yescharacteristic
275C23×C4C2411yescharacteristic
276C22×C8C2414yes—
277C22×C8C2414yes—
278C4⋊Q8—24no—
279#127—24no—
280#125—28no—
281#125—24no—
282#125—24no—
283#125—44no—
284#125—44no—
285#124—24no—
286#124#4118yes—
287#121—28no—
288#121—28no—
289C4×Q8—28no—
290C4×Q8—44no—
291C4×D4—28no—
292C4×D4—28no—
293C4×D4—28no—
294C4×D4—44no—
295C4×D4—44no—
296C4×D4—44no—
297C4×D4—44no—
298C4×D4—48no—
299C4×D4—48no—
300C4×D4—44no—
301#117—48no—
302#117—44no—
303#117—44no—
304#117—44no—
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—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
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⋊C4—22no—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2412yes—
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×C42—24no—
347C2×C42—24no—
348C2×C42—24no—
349C2×C42—24no—
350C2×C42—22no—
351C2×C42—22no—
352C2×C42—22no—
353C2×C42—22no—
354C2×C42—22no—
355C2×C42—42no—
356C2×C42—42no—
357C2×C42—42no—
358C2×C42—48no—
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×C42C2×D411yescharacteristic
380C2×C42C2×D411yescharacteristic
381C2×C42C2411yescharacteristic
382C2×C42C2411yescharacteristic
383C2×C42C2411yescharacteristic
384C2×C42C2411yescharacteristic
385C2×C42C2411yescharacteristic
386C2×C42C2411yescharacteristic
387#107—44no—
388#107—44no—
389#106—44no—
390#106—44no—
391C4⋊C8—28no—
392#103—28no—
393#103—28no—
394#103—44no—
395#103—44no—
396#102—28no—
397#102—28no—
398#102—44no—
399#102—44no—
400C22⋊C8—28no—
401C4×C8—28no—
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—
413#95—22no—
414#95—22no—
415#95—22no—
416#95—22no—
417#95—22no—
418#95—22no—
419#95—22no—
420C24—22no—
421C24C22×D411yescharacteristic
422#41—416no—
423#41—416no—
424C2×Q8—22no—
425C2×Q8—24no—
426C2×Q8—24no—
427C2×Q8C2×#4114yes—
428C2×D4—24no—
429C2×D4—24no—
430C2×D4—28no—
431C2×D4—22no—
432C2×D4—24no—
433C2×D4—24no—
434C2×D4—42no—
435C2×D4—42no—
436C2×D4—44no—
437C2×D4—44no—
438C2×D4—42no—
439C2×D4—42no—
440C2×D4C2×#4114yes—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—28no—
447C22×C4—22no—
448C22×C4—22no—
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—41no—
464C22×C4—41no—
465C22×C4—41no—
466C22×C4—41no—
467C22×C4—41no—
468C22×C4—41no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
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×C42−1+414yes—
491C22×C4C2511yescharacteristic
492C2×C8—24no—
493C2×C8—24no—
494C2×C8—24no—
495C2×C8—24no—
496C2×C8—24no—
497C2×C8—24no—
498C4⋊C4—24no—
499C4⋊C4—24no—
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—44no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—44no—
530C22⋊C4—28no—
531C22⋊C4—28no—
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—
541C42—22no—
542C42—22no—
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—22no—
559C42—22no—
560C42—41no—
561C42—41no—
562C42—41no—
563C42—41no—
564C42—41no—
565C42—41no—
566C42—44no—
567C42—44no—
568C42—44no—
569C42—44no—
570C42—44no—
571C42#13514yes—
572C42C2×#4114yes—
573C42H(5,𝔽2)14yes—
574C23—21no—
575C23—21no—
576C23—21no—
577C23—21no—
578C23—21no—
579C23—21no—
580C23—24no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23—42no—
586C23—42no—
587C23#53612yes—
588C23C23×D411yescharacteristic
589Q8—28no—
590Q8—44no—
591D4—28no—
592D4—48no—
593D4—48no—
594D4—416no—
595D4—44no—
596C2×C4—28no—
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—21no—
619C2×C4—28no—
620C2×C4—24no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—416no—
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—88no—
654C2×C4—88no—
655C2×C4#52512yes—
656C2×C4#52512yes—
657C2×C4#53512yes—
658C2×C4#53512yes—
659C2×C4#53612yes—
660C2×C4C2×#13512yes—
661C2×C4C2×#13512yes—
662C2×C4C2×#13512yes—
663C2×C4#57512yes—
664C2×C4#57512yes—
665C2×C4#57512yes—
666C2×C4C22×#4112yes—
667C2×C4C2×H(5,𝔽2)12yes—
668C2×C4C2×2−1+412yes—
669C8—44no—
670C8—44no—
671C22—22no—
672C22—22no—
673C22—22no—
674C22—41no—
675C22—41no—
676C22—41no—
677C22—41no—
678C22—44no—
679C22—44no—
680C22—44no—
681C22—82no—
682C22—82no—
683C22—82no—
684C22—82no—
685C22C2×#52511yescharacteristic
686C22C2×#53511yescharacteristic
687C22C2×#53611yescharacteristic
688C22C22×#13511yescharacteristic
689C22C2×#57511yescharacteristic
690C22C2×#57511yescharacteristic
691C22C2×#57511yescharacteristic
692C4—22no—
693C4—22no—
694C4—22no—
695C4—22no—
696C4—22no—
697C4—22no—
698C4—22no—
699C4—22no—
700C4—22no—
701C4—22no—
702C4—22no—
703C4—41no—
704C4—44no—
705C4—41no—
706C4—82no—
707C4—82no—
708C4—82no—
709C4—82no—
710C4—82no—
711C4—82no—
712C4#352014yes—
713C2—42no—
714C2—84no—
715C2#3524011yescharacteristic
716C2#6053611yescharacteristic
717C2#6054611yescharacteristic
718C2#6078211yescharacteristic
719C2#6080211yescharacteristic
720C2#6084611yescharacteristic
721C2#6086111yescharacteristic
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. #32961C2semidirect
4. #30079C2semidirect
5. #30078C2semidirect
6. #30076C2semidirect
7. #30075C2semidirect
8. #28720C2semidirect
9. #28513C2semidirect
10. #28209C2semidirect
11. #28207C2semidirect
12. #27951C2semidirect
13. #27587C2semidirect
14. #27584C2semidirect
16. #22222C2semidirect
17. #20777C2semidirect

Maximal subgroups

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

subgroupindex
#366402
C4×#5322
#329612
#300792
#300782
#300762
#300752
#287202
#285132
#282092
#282072
#279512
#275872
#275842
#258582
#222222
#207772