Finite Group Databasebeta

← #6482020

Subgroups — #6482020

Order 512.

1916 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#35172C211yesmaximal, characteristic
2C2×#2352C211yesmaximal, characteristic
3#32985C212yesmaximal, complemented
4#32983C212yesmaximal, complemented
5#29894C212yesmaximal, complemented
6#29893C212yesmaximal, complemented
7#26315C214yesmaximal, complemented
8#26249C214yesmaximal, complemented
9#25028C214yesmaximal, complemented
10#24983C214yesmaximal, complemented
11#21986C211yesmaximal, characteristic, complemented
12#21866C211yesmaximal, characteristic, complemented
13#21790C211yesmaximal, characteristic
14#21394C211yesmaximal, characteristic, complemented
15#21365C211yesmaximal, characteristic, complemented
16C2×#533C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C2×#484C2214yescomplemented
21C2×#485C2214yescomplemented
22C2×#432C2211yescharacteristic
23C2×#432C2211yescharacteristic
24#2848C2212yes—
25#2848C2212yes—
26#2626—21no—
27#2626—21no—
28#2580—22no—
29#2566—22no—
30#2563C2212yes—
31#2542C2212yes—
32#2538C2212yes—
33#2538C2212yes—
34#2352—22no—
35#2352—22no—
36#2352C2214yes—
37#2352C2214yes—
38#2317—21no—
39#2309—21no—
40C2×#388—21no—
41C2×#387—21no—
42C2×#386—21no—
43#2285C2211yescharacteristic
44C2×#382C2211yescharacteristic
45C2×#380C2211yescharacteristic
46C2×#380C2211yescharacteristic
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4—21no—
49#2077C2214yescomplemented
50#2077C2214yescomplemented
51#2077C2214yescomplemented
52#2075C2214yescomplemented
53#2036C2214yescomplemented
54#2035C2214yescomplemented
55#2035C2214yescomplemented
56#2035C2214yescomplemented
57#1993C2212yescomplemented
58#1988C2212yescomplemented
59#1981C2212yescomplemented
60#1960C2212yescomplemented
61#1955C2214yes—
62#1955C2214yes—
63#1949C2214yescomplemented
64#1948C2212yescomplemented
65#1947C2212yescomplemented
66#1873C2214yescomplemented
67#1872C2212yes—
68#1872C2212yes—
69#1872C2212yes—
70#1872C2212yes—
71#1872C2214yescomplemented
72#1854C2214yescomplemented
73#1853C2212yes—
74#1853C2212yes—
75#1852C2212yes—
76#1852C2212yes—
77#1852C2214yescomplemented
78#1847C2212yescomplemented
79#1836C2212yescomplemented
80#1825C2214yescomplemented
81#1824C2214yescomplemented
82#1805C2214yescomplemented
83#1779C2212yes—
84#1779C2212yes—
85#1776C2214yescomplemented
86#1763C2214yescomplemented
87C23×D4D411yescharacteristic
88C24×C4—21no—
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91#533—24no—
92#533—24no—
93C2×C4⋊Q8C2311yescharacteristic
94C2×C4⋊Q8C2311yescharacteristic
95C2×#125—21no—
96C2×C22⋊Q8—21no—
97C2×C22⋊Q8C2312yes—
98C2×C22⋊Q8C2312yes—
99C2×C22⋊Q8C2312yes—
100C2×C22⋊Q8C2312yes—
101C2×C22⋊Q8C2312yes—
102C2×C4×Q8—21no—
103C2×C4×Q8C2312yes—
104C2×C4×D4—24no—
105C2×C4×D4—24no—
106C2×C4×D4D412yes—
107C2×C4×D4D412yes—
108C2×C4×D4C2312yes—
109C2×C4×D4C2312yes—
110C2×#117—21no—
111C2×#117—21no—
112C2×#117—22no—
113C2×#117—44no—
114C2×#117C2314yes—
115C22×C4⋊C4—22no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—44no—
118C22×C4⋊C4D411yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2314yes—
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C22⋊C4—21no—
125C22×C22⋊C4—21no—
126C22×C22⋊C4—21no—
127C22×C22⋊C4D411yescharacteristic
128C22×C22⋊C4D411yescharacteristic
129C22×C42—21no—
130C22×C42—21no—
131C22×C42C2311yescharacteristic
132#485—48no—
133#484—48no—
134#432—24no—
135#432—24no—
136C2×#107C2314yes—
137C2×#107C2314yes—
138C2×#106C2314yes—
139C2×#106C2314yes—
140C2×C4⋊C8C2312yes—
141C2×C4⋊C8C2312yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×C22⋊C8C2312yes—
147C2×C22⋊C8C2312yes—
148C2×C22⋊C8C2314yes—
149C2×C22⋊C8C2314yes—
150C2×C22⋊C8C2314yes—
151C2×C22⋊C8C2314yes—
152C2×#97C2312yes—
153C2×#97C2312yes—
154#399—22no—
155#399—22no—
156#396—22no—
157#396—22no—
158#396—22no—
159#396—22no—
160#393—22no—
161#393—22no—
162#390—22no—
163#390—22no—
164#390—22no—
165#390—22no—
166#389—22no—
167#389—22no—
168#388—24no—
169#388—24no—
170#387—24no—
171#387—24no—
172#386—24no—
173#386—24no—
174#385—22no—
175#385—22no—
176#385—22no—
177#385—22no—
178#385—22no—
179#385—22no—
180#384—22no—
181#384C2311yescharacteristic
182#384C2311yescharacteristic
183#382—22no—
184#382—21no—
185#382—21no—
186#382—21no—
187#382C2312yes—
188#382C2312yes—
189#380—22no—
190#380—22no—
191#380—21no—
192#380C2312yes—
193#380C2312yes—
194#380C2312yes—
195#380C2312yes—
196#380C2314yes—
197#380C2314yes—
198#380C2312yes—
199#379—22no—
200#379—22no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4C2312yes—
205C4×C4⋊C4C2312yes—
206C4×C4⋊C4C2312yes—
207C4×C22⋊C4—24no—
208C4×C22⋊C4—24no—
209C2×#95—21no—
210C2×#95—21no—
211C2×#95—21no—
212C2×#95—21no—
213C2×#95—21no—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222C25C2×D411yescharacteristic
223C25C2×D411yescharacteristic
224C22×Q8—21no—
225C22×Q8C2412yes—
226C22×D4—24no—
227C22×D4—24no—
228C22×D4—42no—
229C22×D4C2×D412yes—
230C22×D4C2×D412yes—
231C23×C4—21no—
232C23×C4—22no—
233C23×C4—22no—
234C23×C4—22no—
235C23×C4—22no—
236C23×C4—21no—
237C23×C4—21no—
238C23×C4—21no—
239C23×C4—24no—
240C23×C4—42no—
241C23×C4—42no—
242C23×C4—42no—
243C23×C4—42no—
244C23×C4—44no—
245C23×C4—44no—
246C23×C4—82no—
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D412yes—
251C23×C4C2×D412yes—
252C23×C4C2×D411yescharacteristic
253C23×C4C2412yes—
254C23×C4C2412yes—
255C23×C4C2411yescharacteristic
256C22×C8C2414yes—
257C22×C8C2414yes—
258C4⋊Q8—22no—
259C4⋊Q8—22no—
260C4⋊Q8—22no—
261C4⋊Q8—22no—
262#125—44no—
263C22⋊Q8—24no—
264C22⋊Q8—24no—
265C22⋊Q8—44no—
266C22⋊Q8—44no—
267C22⋊Q8—44no—
268C22⋊Q8—44no—
269C22⋊Q8—44no—
270C4×Q8—24no—
271C4×Q8—24no—
272C4×Q8—44no—
273C4×D4—28no—
274C4×D4—28no—
275C4×D4—24no—
276C4×D4—24no—
277C4×D4—24no—
278C4×D4—24no—
279#117—28no—
280#117—44no—
281#117—48no—
282#117—44no—
283#117—44no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—28no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4—42no—
307C2×C4⋊C4—42no—
308C2×C4⋊C4—42no—
309C2×C4⋊C4—48no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4C2×D412yes—
312C2×C4⋊C4C2×D412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—24no—
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—44no—
336C2×C22⋊C4—44no—
337C2×C22⋊C4—82no—
338C2×C22⋊C4C2×D412yes—
339C2×C22⋊C4C2×D412yes—
340C2×C22⋊C4C2×D412yes—
341C2×C22⋊C4C2×D412yes—
342C2×C42—24no—
343C2×C42—22no—
344C2×C42—22no—
345C2×C42—22no—
346C2×C42—22no—
347C2×C42—44no—
348C2×C42—82no—
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42#4114yes—
352C2×C42#4114yes—
353C2×C42C2411yescharacteristic
354C2×C42C2411yescharacteristic
355#107—28no—
356#107—44no—
357#106—44no—
358#106—44no—
359C4⋊C8—24no—
360C4⋊C8—24no—
361#103—28no—
362#103—44no—
363#103—44no—
364#103—44no—
365C22⋊C8—24no—
366C22⋊C8—24no—
367C22⋊C8—44no—
368C22⋊C8—44no—
369C22⋊C8—44no—
370C22⋊C8—44no—
371#97—24no—
372#97—24no—
373#95—24no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—24no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387C24—22no—
388C24—22no—
389C24—22no—
390C24—22no—
391C24—22no—
392C24—41no—
393C24—41no—
394C24—41no—
395C24—41no—
396C24—41no—
397C24—41no—
398C24—41no—
399C24—41no—
400C24#13714yes—
401C24C22×D412yes—
402C24C22×D412yes—
403C24C22×D411yescharacteristic
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—24no—
407C2×Q8—24no—
408C2×Q8—42no—
409C2×Q8—81no—
410C2×Q8—81no—
411C2×D4—24no—
412C2×D4—24no—
413C2×D4—24no—
414C2×D4—24no—
415C2×D4—42no—
416C2×D4—42no—
417C2×D4—416no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—42no—
446C22×C4—41no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4—48no—
457C22×C4—48no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—48no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—84no—
471C22×C4—816no—
472C22×C4#12314yes—
473C22×C4#12314yes—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C2×#4112yes—
482C22×C4C2×#4112yes—
483C22×C4C2511yescharacteristic
484C2×C8—24no—
485C2×C8—24no—
486C2×C8—24no—
487C2×C8—24no—
488C2×C8—24no—
489C2×C8—24no—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—24no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—24no—
502C4⋊C4—48no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C4⋊C4—42no—
513C4⋊C4—81no—
514C4⋊C4—81no—
515C4⋊C4—82no—
516C4⋊C4—82no—
517C4⋊C4—82no—
518C4⋊C4—82no—
519C4⋊C4H(5,𝔽2)14yes—
520C22⋊C4—24no—
521C22⋊C4—24no—
522C22⋊C4—24no—
523C22⋊C4—24no—
524C22⋊C4—48no—
525C22⋊C4—44no—
526C22⋊C4—48no—
527C22⋊C4—44no—
528C22⋊C4—82no—
529C22⋊C4—82no—
530C22⋊C4—82no—
531C22⋊C4—82no—
532C22⋊C4—84no—
533C22⋊C4—82no—
534C22⋊C4—82no—
535C22⋊C4—82no—
536C22⋊C4—82no—
537C42—48no—
538C42—44no—
539C42—44no—
540C42—82no—
541C42—82no—
542C42#12314yes—
543C42#12314yes—
544C42C2×#4112yes—
545C42C2×#4112yes—
546C42C2×#4112yes—
547C42C2×#4112yes—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—22no—
553C23—22no—
554C23—28no—
555C23—22no—
556C23—22no—
557C23—22no—
558C23—22no—
559C23—22no—
560C23—22no—
561C23—22no—
562C23—22no—
563C23—22no—
564C23—22no—
565C23—22no—
566C23—44no—
567C23—44no—
568C23—44no—
569C23—44no—
570C23—44no—
571C23—44no—
572C23—44no—
573C23—44no—
574C23—44no—
575C23—82no—
576C23—82no—
577C23—82no—
578C23—82no—
579C23C2×#12312yes—
580C23C2×#13714yes—
581C23C2×#13712yes—
582C23#57512yes—
583C23C23×D411yescharacteristic
584Q8—44no—
585Q8—44no—
586Q8—84no—
587D4—48no—
588D4—48no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—28no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—24no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—44no—
610C2×C4—48no—
611C2×C4—48no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—48no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—42no—
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—816no—
643C2×C4—84no—
644C2×C4—84no—
645C2×C4—84no—
646C2×C4C2×#12312yes—
647C2×C4C2×#12312yes—
648C2×C4C2×#12312yes—
649C2×C4C2×#12312yes—
650C2×C4C2×#12312yes—
651C2×C4C22×#4111yescharacteristic
652C2×C4C22×#4111yescharacteristic
653C2×C4C2×H(5,𝔽2)12yes—
654C2×C4#58512yes—
655C8—44no—
656C8—44no—
657C22—22no—
658C22—22no—
659C22—22no—
660C22—22no—
661C22—22no—
662C22—22no—
663C22—24no—
664C22—24no—
665C22—24no—
666C22—416no—
667C22—42no—
668C22—42no—
669C22—42no—
670C22—42no—
671C22—42no—
672C22—42no—
673C22—42no—
674C22—42no—
675C22—42no—
676C22—42no—
677C22—42no—
678C22—88no—
679C22#309514yes—
680C22C22×#12311yescharacteristic
681C22#348411yescharacteristic
682C22#348511yescharacteristic
683C22C22×#13711yescharacteristic
684C22C2×#57511yescharacteristic
685C22#358511yescharacteristic
686C22#358611yescharacteristic
687C4—22no—
688C4—22no—
689C4—22no—
690C4—22no—
691C4—22no—
692C4—42no—
693C4—42no—
694C4—44no—
695C4—82no—
696C4—82no—
697C4—82no—
698C4—82no—
699C4—82no—
700C4—82no—
701C4—82no—
702C4#346012yes—
703C4#350612yes—
704C2—24no—
705C2—44no—
706C2—82no—
707C2#3515211yescharacteristic
708C2C2×#309511yescharacteristic
709C2#6051711yescharacteristic
710C2#6157311yescharacteristic
711C2#6157611yescharacteristic
712C2#6170811yescharacteristic
713C2#6171111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #32985C2semidirect
4. #32983C2semidirect
5. #29894C2semidirect
6. #29893C2semidirect
7. #26315C2semidirect
8. #26249C2semidirect
9. #25028C2semidirect
10. #24983C2semidirect
11. #21986C2semidirect
12. #21866C2semidirect
14. #21394C2semidirect
15. #21365C2semidirect
20. C2×#484C22semidirect
21. C2×#485C22semidirect
49. #2077C22semidirect
50. #2077C22semidirect
51. #2077C22semidirect
52. #2075C22semidirect
53. #2036C22semidirect
54. #2035C22semidirect
55. #2035C22semidirect
56. #2035C22semidirect
57. #1993C22semidirect
58. #1988C22semidirect
59. #1981C22semidirect
60. #1960C22semidirect
63. #1949C22semidirect
64. #1948C22semidirect
65. #1947C22semidirect
66. #1873C22semidirect
71. #1872C22semidirect
72. #1854C22semidirect
77. #1852C22semidirect
78. #1847C22semidirect
79. #1836C22semidirect
80. #1825C22semidirect
81. #1824C22semidirect
82. #1805C22semidirect
85. #1776C22semidirect
86. #1763C22semidirect

Maximal subgroups

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

subgroupindex
#351722
C2×#23522
#329852
#329832
#298942
#298932
#263152
#262492
#250282
#249832
#219862
#218662
#217902
#213942
#213652