Finite Group Databasebeta

← #6571312

Subgroups — #6571312

Order 512.

1774 conjugacy classes of subgroups.

The subgroup lattice has 689 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36504C211yesmaximal, characteristic
2#34183C211yesmaximal, characteristic, complemented
3#29928C212yesmaximal, complemented
4#29923C211yesmaximal, characteristic, complemented
5#27978C212yesmaximal, complemented
6#27972C211yesmaximal, characteristic, complemented
7#27647C212yesmaximal, complemented
8#27645C211yesmaximal, characteristic, complemented
9#26242C214yesmaximal, complemented
10#26234C212yesmaximal, complemented
11#24979C214yesmaximal, complemented
12#24965C212yesmaximal, complemented
13#23762C212yesmaximal, complemented
14#23755C211yesmaximal, characteristic, complemented
15#22742C212yesmaximal, complemented
16#22544C212yesmaximal, complemented
17#22532C211yesmaximal, characteristic, complemented
18C2×#556C2211yescharacteristic
19C2×#520—21no—
20C2×C4×#41C2211yescharacteristic
21C2×#481C2212yes—
22C2×#483C2214yes—
23C2×#446C2212yes—
24#2857C2211yescharacteristic
25#2850C2212yes—
26#2542C2211yescharacteristic
27#2540C2212yes—
28#2537C2211yescharacteristic
29#2536C2212yes—
30#2479—22no—
31#2467—22no—
32#2423—21no—
33#2407—21no—
34#2363—21no—
35#2355C2211yescharacteristic
36#2354—21no—
37Q8×C4⋊C4—21no—
38Q8×C4⋊C4C2212yes—
39D4×C4⋊C4—21no—
40D4×C4⋊C4—22no—
41D4×C4⋊C4C2212yes—
42D4×C4⋊C4C2214yes—
43#2317—21no—
44#2317—22no—
45C4×C22⋊Q8C2212yes—
46C4×#121C2211yescharacteristic
47#2285C2211yescharacteristic
48#2285C2212yes—
49C2×#384C2211yescharacteristic
50C2×#384C2212yes—
51#2100C2211yescharacteristic
52#2095C2212yes—
53#2094C2211yescharacteristic
54#2084C2211yescharacteristic
55#2084C2212yes—
56#2077C2212yes—
57#2075C2214yes—
58#2072C2212yes—
59#2072C2212yes—
60#2072C2212yes—
61#2071C2211yescharacteristic
62#2071C2212yes—
63#2036C2214yes—
64#2035C2214yes—
65#2035C2214yes—
66#2034C2212yes—
67#2033C2214yes—
68#1983C2214yes—
69#1981C2212yes—
70#1959C2214yes—
71#1956C2212yes—
72#1955C2214yes—
73#1954C2212yes—
74#1949C2214yes—
75#1948C2212yes—
76#1948C2214yes—
77#1947C2212yes—
78#1874C2214yes—
79#1872C2214yes—
80#1872C2214yes—
81#1836C2214yes—
82#1834C2212yes—
83#1824C2212yes—
84#1823C2214yes—
85#1803C2214yes—
86#1802C2212yes—
87#1797C2211yescharacteristic
88#1797C2212yes—
89#1776C2214yes—
90C4×#106C2212yes—
91#1763C2214yes—
92C4×#102C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C22×#41D411yescharacteristic
96#556—28no—
97C2×C4⋊Q8C2311yescharacteristic
98C2×#126—22no—
99C2×#124C2312yes—
100C2×C22⋊Q8—21no—
101C2×C22⋊Q8C2312yes—
102C2×C22⋊Q8C2312yes—
103C2×C22⋊Q8C2312yes—
104C2×#121—21no—
105C2×#121C2311yescharacteristic
106C2×#121C2312yes—
107#520—48no—
108C4×#41—28no—
109C2×C4×Q8—22no—
110C2×C4×Q8D411yescharacteristic
111C2×C4×Q8C2312yes—
112C2×C4×Q8C2311yescharacteristic
113C2×C4×D4—24no—
114C2×C4×D4—22no—
115C2×C4×D4—21no—
116C2×C4×D4D411yescharacteristic
117C2×C4×D4D412yes—
118C2×C4×D4C2311yescharacteristic
119C2×C4×D4C2311yescharacteristic
120C2×C4×D4C2312yes—
121C2×#117—22no—
122C2×#117—21no—
123C2×#117—21no—
124C2×#117D411yescharacteristic
125C2×#117D412yes—
126C22×C4⋊C4—22no—
127C22×C4⋊C4—21no—
128C22×C4⋊C4—22no—
129C22×C4⋊C4—21no—
130C22×C4⋊C4C2314yes—
131C22×C4⋊C4C2312yes—
132C22×C42C2312yes—
133C22×C42C2311yescharacteristic
134#483—48no—
135#481—44no—
136#446—28no—
137C2×#107C2314yes—
138C2×#106C2312yes—
139C2×#106C2314yes—
140C2×#106C2312yes—
141C2×C4⋊C8C2312yes—
142C2×C4⋊C8C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#102C2314yes—
148C2×#102C2312yes—
149C2×#102C2312yes—
150C2×C22⋊C8C2314yes—
151C2×C22⋊C8C2312yes—
152C2×C22⋊C8C2314yes—
153C2×C22⋊C8C2312yes—
154C2×#97C2314yes—
155C2×C4×C8C2312yes—
156#399—22no—
157#399—22no—
158#398—22no—
159#394—22no—
160#394—22no—
161#392—22no—
162#391—21no—
163#391C2312yes—
164#390—21no—
165#390—22no—
166#390—22no—
167#390C2311yescharacteristic
168#389—22no—
169#389—24no—
170#389—22no—
171#389—24no—
172#389—21no—
173#389—21no—
174#389C2311yescharacteristic
175#388—22no—
176#388—22no—
177#388—22no—
178#388—22no—
179#388—21no—
180#387—22no—
181#387—22no—
182#386—22no—
183#386—22no—
184#386—21no—
185#385—22no—
186#385—24no—
187#385—24no—
188#385—24no—
189#384—21no—
190#384—22no—
191#384—21no—
192#384—21no—
193#384—22no—
194#384C2312yes—
195#384C2314yes—
196#384C2311yescharacteristic
197#384C2312yes—
198#384C2311yescharacteristic
199#384C2311yescharacteristic
200#382—22no—
201#382—21no—
202#382C2312yes—
203#380—21no—
204#380—21no—
205#380—22no—
206#380—21no—
207#380—21no—
208#380C2312yes—
209#380C2314yes—
210#380C2311yescharacteristic
211#380C2312yes—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2314yes—
218C4×C4⋊C4C2312yes—
219C4×C4⋊C4C2311yescharacteristic
220C4×C22⋊C4—22no—
221C4×C22⋊C4—21no—
222#340C2314yes—
223#340C2312yes—
224#340C2312yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#340C2312yes—
231#336C2312yes—
232#336C2311yescharacteristic
233#336C2311yescharacteristic
234C2×#41—24no—
235C2×#41—24no—
236C2×#41—24no—
237C22×Q8C2×D411yescharacteristic
238C22×Q8C2412yes—
239C22×D4—21no—
240C22×D4—22no—
241C22×D4C2×D412yes—
242C22×D4C2×D411yescharacteristic
243C22×D4C2411yescharacteristic
244C23×C4—22no—
245C23×C4—24no—
246C23×C4—21no—
247C23×C4C2×D412yes—
248C23×C4C2×D412yes—
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2411yescharacteristic
252C23×C4C2412yes—
253C22×C8C2412yes—
254C22×C8C2412yes—
255C22×C8C2414yes—
256C4⋊Q8—24no—
257#126—48no—
258#124—28no—
259C22⋊Q8—24no—
260C22⋊Q8—42no—
261C22⋊Q8—44no—
262C22⋊Q8—44no—
263C22⋊Q8—44no—
264#121—22no—
265#121—41no—
266#121—44no—
267#121—44no—
268C4×Q8—24no—
269C4×Q8—28no—
270C4×Q8—48no—
271C4×Q8#4118yes—
272C4×D4—24no—
273C4×D4—28no—
274C4×D4—24no—
275C4×D4—28no—
276C4×D4—24no—
277C4×D4—416no—
278C4×D4—48no—
279C4×D4—44no—
280#117—24no—
281#117—28no—
282#117—48no—
283#117—44no—
284#117—44no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4—42no—
308C2×C4⋊C4—42no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—48no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4—84no—
313C2×C4⋊C4—82no—
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2×D412yes—
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2×D412yes—
318C2×C4⋊C4C2414yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2411yescharacteristic
325C2×C22⋊C4—22no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4C2×D412yes—
340C2×C22⋊C4C2×D412yes—
341C2×C22⋊C4C2×D412yes—
342C2×C42—24no—
343C2×C42—28no—
344C2×C42—24no—
345C2×C42—22no—
346C2×C42—22no—
347C2×C42—21no—
348C2×C42—22no—
349C2×C42C2×D412yes—
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D412yes—
353C2×C42C2411yescharacteristic
354C2×C42C2411yescharacteristic
355C2×C42C2412yes—
356#107—44no—
357#106—28no—
358#106—24no—
359#106—42no—
360C4⋊C8—24no—
361C4⋊C8—28no—
362#103—44no—
363#103—44no—
364#103—44no—
365#103—44no—
366#102—42no—
367#102—42no—
368#102—44no—
369C22⋊C8—42no—
370C22⋊C8—44no—
371C22⋊C8—42no—
372C22⋊C8—44no—
373#97—28no—
374C4×C8—24no—
375#95—21no—
376#95—22no—
377#95—21no—
378#95—22no—
379#95—22no—
380#95—24no—
381#95—22no—
382#95—22no—
383#95—24no—
384#95—22no—
385#95—22no—
386#95—21no—
387#95—22no—
388C24—21no—
389C24C22×D412yes—
390C24C22×D411yescharacteristic
391#41—232no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—24no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—24no—
398C2×Q8#12314yes—
399C2×D4—24no—
400C2×D4—24no—
401C2×D4—24no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—22no—
408C2×D4—42no—
409C2×D4—42no—
410C2×D4—44no—
411C2×D4—81no—
412C2×D4—82no—
413C2×D4—82no—
414C2×D4AGL(1,ℤ/8)14yes—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—21no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—42no—
441C22×C4—41no—
442C22×C4—44no—
443C22×C4—48no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—84no—
448C22×C4—82no—
449C22×C4—84no—
450C22×C4—82no—
451C22×C4#13714yes—
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D412yes—
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C2511yescharacteristic
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—24no—
464C2×C8—24no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—24no—
469C2×C8—22no—
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—21no—
473C4⋊C4—22no—
474C4⋊C4—24no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—24no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—24no—
482C4⋊C4—44no—
483C4⋊C4—42no—
484C4⋊C4—44no—
485C4⋊C4—42no—
486C4⋊C4—48no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—44no—
490C4⋊C4—42no—
491C4⋊C4—44no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—44no—
496C4⋊C4—81no—
497C4⋊C4—82no—
498C4⋊C4—84no—
499C4⋊C4—84no—
500C4⋊C4—82no—
501C4⋊C4—82no—
502C4⋊C4—82no—
503C4⋊C4#12314yes—
504C4⋊C4#12318yes—
505C4⋊C4H(5,𝔽2)12yes—
506C22⋊C4—24no—
507C22⋊C4—24no—
508C22⋊C4—24no—
509C22⋊C4—42no—
510C22⋊C4—44no—
511C22⋊C4—44no—
512C22⋊C4—48no—
513C22⋊C4—44no—
514C22⋊C4—42no—
515C22⋊C4—44no—
516C22⋊C4—42no—
517C22⋊C4—44no—
518C22⋊C4—42no—
519C22⋊C4—42no—
520C22⋊C4—82no—
521C22⋊C4—82no—
522C22⋊C4—82no—
523C42—24no—
524C42—22no—
525C42—24no—
526C42—22no—
527C42—22no—
528C42—48no—
529C42—44no—
530C42—44no—
531C42—42no—
532C42—82no—
533C42C2×#4114yes—
534C42C2×#4118yes—
535C23—21no—
536C23—22no—
537C23—22no—
538C23—21no—
539C23—22no—
540C23—21no—
541C23—21no—
542C23—22no—
543C23—21no—
544C23—22no—
545C23—22no—
546C23—21no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23C2×AGL(1,ℤ/8)14yes—
555C23C2×#13712yes—
556C23C23×D411yescharacteristic
557Q8—28no—
558Q8—44no—
559Q8—44no—
560D4—216no—
561D4—28no—
562D4—42no—
563D4—42no—
564D4—84no—
565D4—164no—
566C2×C4—216no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—28no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—48no—
586C2×C4—44no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—84no—
615C2×C4—168no—
616C2×C4—164no—
617C2×C4C2×#12314yes—
618C2×C4C2×#12312yes—
619C2×C4C2×#12314yes—
620C2×C4C2×#12312yes—
621C2×C4C2×AGL(1,ℤ/8)12yes—
622C2×C4C2×#13712yes—
623C2×C4C2×#13712yes—
624C2×C4#57712yes—
625C2×C4C22×#4112yes—
626C2×C4C2×H(5,𝔽2)11yescharacteristic
627C2×C4#58512yes—
628C2×C4#58511yescharacteristic
629C8—42no—
630C8—44no—
631C8—42no—
632C22—22no—
633C22—24no—
634C22—22no—
635C22—24no—
636C22—22no—
637C22—24no—
638C22—42no—
639C22—41no—
640C22—42no—
641C22—41no—
642C22—42no—
643C22—41no—
644C22—42no—
645C22—41no—
646C22—81no—
647C22—81no—
648C22—81no—
649C22—81no—
650C22—81no—
651C22—81no—
652C22—81no—
653C22C22×#12311yescharacteristic
654C22#348811yescharacteristic
655C22#350011yescharacteristic
656C22C22×AGL(1,ℤ/8)11yescharacteristic
657C22C22×#13711yescharacteristic
658C22C2×#57711yescharacteristic
659C22#358611yescharacteristic
660C4—22no—
661C4—24no—
662C4—22no—
663C4—24no—
664C4—22no—
665C4—42no—
666C4—41no—
667C4—42no—
668C4—82no—
669C4—84no—
670C4—81no—
671C4—81no—
672C4—82no—
673C4—82no—
674C4—82no—
675C4#309214yes—
676C4#351712yes—
677C4#352412yes—
678C2—42no—
679C2—44no—
680C2—161no—
681C2#3607611yescharacteristic
682C2C2×#309211yescharacteristic
683C2#6052211yescharacteristic
684C2#6196011yescharacteristic
685C2#6198311yescharacteristic
686C2#6254211yescharacteristic
687C2#6258311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36504C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34183C2semidirect
3. #29928C2semidirect
4. #29923C2semidirect
5. #27978C2semidirect
6. #27972C2semidirect
7. #27647C2semidirect
8. #27645C2semidirect
9. #26242C2semidirect
10. #26234C2semidirect
11. #24979C2semidirect
12. #24965C2semidirect
13. #23762C2semidirect
14. #23755C2semidirect
15. #22742C2semidirect
16. #22544C2semidirect
17. #22532C2semidirect

Maximal subgroups

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

subgroupindex
#365042
#341832
#299282
#299232
#279782
#279722
#276472
#276452
#262422
#262342
#249792
#249652
#237622
#237552
#227422
#225442
#225322