Finite Group Databasebeta

← #6561795

Subgroups — #6561795

Order 512.

1831 conjugacy classes of subgroups.

The subgroup lattice has 693 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36404C211yesmaximal, characteristic
2#34183C211yesmaximal, characteristic, complemented
3#29923C211yesmaximal, characteristic, complemented
4#29908C212yesmaximal, complemented
5#27972C211yesmaximal, characteristic, complemented
6#27864C212yesmaximal, complemented
7#27645C211yesmaximal, characteristic, complemented
8#27584C212yesmaximal, complemented
9#26320C212yesmaximal, complemented
10#26220C214yesmaximal, complemented
11#24990C212yesmaximal, complemented
12#24881C214yesmaximal, complemented
13#23755C211yesmaximal, characteristic, complemented
14#23630C212yesmaximal, complemented
15#22715C212yesmaximal, complemented
16#22473C211yesmaximal, characteristic, complemented
17#22431C212yesmaximal, complemented
18C2×#547C2211yescharacteristic
19C2×#520—21no—
20C2×C4×#41C2211yescharacteristic
21C2×#481C2212yes—
22C2×#436C2212yes—
23C2×#417C2214yes—
24#2857C2211yescharacteristic
25#2855C2212yes—
26#2568C2211yescharacteristic
27#2538C2211yescharacteristic
28#2470—22no—
29#2457—22no—
30#2420—21no—
31#2415C2212yes—
32#2407—21no—
33#2407C2212yes—
34#2360—21no—
35#2354C2211yescharacteristic
36#2352—21no—
37Q8×C4⋊C4—21no—
38Q8×C4⋊C4C2212yes—
39D4×C4⋊C4—21no—
40D4×C4⋊C4—22no—
41D4×C4⋊C4C2214yes—
42D4×C4⋊C4C2212yes—
43#2317—21no—
44#2317—22no—
45C4×#121C2211yescharacteristic
46#2297C2212yes—
47#2285C2212yes—
48#2285C2211yescharacteristic
49C2×#384C2211yescharacteristic
50C2×#384C2212yes—
51#2100C2211yescharacteristic
52#2099C2214yes—
53#2096C2214yes—
54#2095C2212yes—
55#2094C2211yescharacteristic
56#2084C2211yescharacteristic
57#2084C2212yes—
58#2077C2212yes—
59#2072C2212yes—
60#2072C2212yes—
61#2072C2214yes—
62#2072C2212yes—
63#2071C2214yes—
64#2071C2211yescharacteristic
65#2071C2212yes—
66#2036C2214yes—
67#2034C2212yes—
68#1984C2212yes—
69#1981C2212yes—
70#1956C2212yes—
71#1948C2212yes—
72#1947C2212yes—
73#1927C2214yes—
74#1927C2214yes—
75#1925C2214yes—
76#1918C2214yes—
77#1894C2214yes—
78#1872C2214yes—
79#1872C2214yes—
80#1870C2214yes—
81#1854C2214yes—
82#1853C2214yes—
83#1834C2212yes—
84#1824C2212yes—
85#1817C2212yes—
86#1809C2214yes—
87#1802C2212yes—
88#1797C2211yescharacteristic
89#1788C2214yes—
90C4×#106C2212yes—
91C4×#102C2212yes—
92C4×C22⋊C8C2211yescharacteristic
93#1736C2214yes—
94#1728C2212yes—
95C22×#41D411yescharacteristic
96C22×#41C2312yes—
97C22×M16C2314yes—
98#547—28no—
99C2×C4⋊Q8C2311yescharacteristic
100C2×#123—22no—
101C2×C22⋊Q8—21no—
102C2×C22⋊Q8C2312yes—
103C2×C22⋊Q8C2312yes—
104C2×#121—21no—
105C2×#121C2311yescharacteristic
106#520—48no—
107C4×#41—28no—
108C2×C4×Q8—22no—
109C2×C4×Q8D411yescharacteristic
110C2×C4×Q8C2311yescharacteristic
111C2×C4×D4—24no—
112C2×C4×D4—22no—
113C2×C4×D4—21no—
114C2×C4×D4D411yescharacteristic
115C2×C4×D4D412yes—
116C2×C4×D4C2311yescharacteristic
117C2×C4×D4C2312yes—
118C2×C4×D4C2311yescharacteristic
119C2×#117—22no—
120C2×#117—21no—
121C2×#117D411yescharacteristic
122C2×#117D412yes—
123C2×#117C2312yes—
124C22×C4⋊C4—22no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4—21no—
127C22×C4⋊C4—22no—
128C22×C4⋊C4—21no—
129C22×C4⋊C4C2312yes—
130C22×C4⋊C4C2314yes—
131C22×C42C2311yescharacteristic
132C22×C42C2312yes—
133#481—44no—
134#436—28no—
135C2×#107C2314yes—
136C2×#106C2314yes—
137C2×#106C2312yes—
138C2×#106C2312yes—
139C2×C4⋊C8C2312yes—
140#417—48no—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#102C2314yes—
146C2×#102C2312yes—
147C2×#102C2312yes—
148C2×C22⋊C8C2314yes—
149C2×C22⋊C8C2312yes—
150C2×C22⋊C8C2312yes—
151C2×C4×C8C2312yes—
152#400—22no—
153#397—22no—
154#396—22no—
155#394—22no—
156#394—22no—
157#392—22no—
158#391—21no—
159#391C2312yes—
160#391C2312yes—
161#391C2312yes—
162#390—21no—
163#390C2311yescharacteristic
164#390C2312yes—
165#389—22no—
166#389—24no—
167#389—24no—
168#389—22no—
169#389—22no—
170#389—21no—
171#389—21no—
172#389C2311yescharacteristic
173#389C2312yes—
174#388—22no—
175#388—22no—
176#388—21no—
177#387—22no—
178#387—22no—
179#386—22no—
180#386—22no—
181#386—21no—
182#386—22no—
183#385—22no—
184#385—22no—
185#385—24no—
186#385—24no—
187#385—24no—
188#384—22no—
189#384—21no—
190#384—22no—
191#384—21no—
192#384C2312yes—
193#384C2311yescharacteristic
194#384C2312yes—
195#384C2314yes—
196#384C2311yescharacteristic
197#384C2311yescharacteristic
198#382—21no—
199#382—22no—
200#382—21no—
201#381—22no—
202#380—21no—
203#380—21no—
204#380—21no—
205#380—22no—
206#380—21no—
207#380C2312yes—
208#380C2314yes—
209#380C2311yescharacteristic
210#380C2312yes—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2314yes—
216C4×C4⋊C4C2312yes—
217C4×C22⋊C4—22no—
218C4×C22⋊C4—21no—
219C4×C22⋊C4—22no—
220#340C2314yes—
221#340C2312yes—
222#340C2314yes—
223#340C2312yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2312yes—
229#336C2312yes—
230#336C2311yescharacteristic
231#336C2314yes—
232#336C2311yescharacteristic
233#336C2314yes—
234C2×#41—24no—
235C2×#41—24no—
236C2×#41—24no—
237C2×#41—24no—
238C2×#41—28no—
239C2×#41—42no—
240C2×#41—44no—
241C22×Q8C2×D411yescharacteristic
242C22×Q8C2412yes—
243C22×D4—21no—
244C22×D4—22no—
245C22×D4C2×D412yes—
246C22×D4C2×D411yescharacteristic
247C22×D4C2411yescharacteristic
248C23×C4—24no—
249C23×C4—22no—
250C23×C4—21no—
251C23×C4—22no—
252C23×C4C2×D412yes—
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D412yes—
255C23×C4C2×D411yescharacteristic
256C23×C4C2411yescharacteristic
257C23×C4C2412yes—
258C2×M16—28no—
259C2×M16—44no—
260C2×M16—44no—
261C22×C8C2412yes—
262C22×C8C2414yes—
263C22×C8C2412yes—
264C4⋊Q8—24no—
265#123—48no—
266C22⋊Q8—28no—
267C22⋊Q8—44no—
268C22⋊Q8—44no—
269#121—22no—
270#121—41no—
271#121—44no—
272C4×Q8—24no—
273C4×Q8—24no—
274C4×Q8—48no—
275C4×D4—24no—
276C4×D4—28no—
277C4×D4—28no—
278C4×D4—24no—
279C4×D4—416no—
280C4×D4—48no—
281C4×D4—44no—
282C4×D4#4118yes—
283#117—24no—
284#117—28no—
285#117—48no—
286#117—44no—
287#117—44no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—42no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—42no—
312C2×C4⋊C4—48no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4—82no—
316C2×C4⋊C4—84no—
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D412yes—
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2414yes—
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2412yes—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4C2×D412yes—
342C2×C22⋊C4C2×D412yes—
343C2×C22⋊C4C2×D412yes—
344C2×C42—28no—
345C2×C42—24no—
346C2×C42—24no—
347C2×C42—22no—
348C2×C42—21no—
349C2×C42—22no—
350C2×C42C2×D412yes—
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D412yes—
353C2×C42C2×D411yescharacteristic
354C2×C42C2412yes—
355C2×C42C2411yescharacteristic
356C2×C42C2411yescharacteristic
357#107—44no—
358#106—42no—
359#106—42no—
360#106—44no—
361C4⋊C8—24no—
362#103—44no—
363#103—44no—
364#103—44no—
365#103—44no—
366#102—24no—
367#102—28no—
368#102—42no—
369C22⋊C8—28no—
370C22⋊C8—42no—
371C22⋊C8—42no—
372C4×C8—24no—
373#95—22no—
374#95—22no—
375#95—21no—
376#95—21no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—24no—
381#95—22no—
382#95—24no—
383#95—22no—
384#95—21no—
385#95—22no—
386#95—22no—
387#95—22no—
388C24—21no—
389C24C22×D412yes—
390C24C22×D411yescharacteristic
391#41—232no—
392#41—416no—
393#41—88no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—24no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—24no—
400C2×Q8AGL(1,ℤ/8)14yes—
401C2×D4—24no—
402C2×D4—24no—
403C2×D4—24no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—21no—
409C2×D4—42no—
410C2×D4—44no—
411C2×D4—44no—
412C2×D4—42no—
413C2×D4—81no—
414C2×D4—82no—
415C2×D4#12314yes—
416C2×D4H(5,𝔽2)12yes—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—41no—
442C22×C4—42no—
443C22×C4—44no—
444C22×C4—48no—
445C22×C4—44no—
446C22×C4—42no—
447C22×C4—44no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—82no—
451C22×C4—84no—
452C22×C4—82no—
453C22×C4—82no—
454C22×C4—84no—
455C22×C4#13714yes—
456C22×C4C22×D412yes—
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D412yes—
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D412yes—
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C2×#4118yes—
465C22×C4C2511yescharacteristic
466M16—88no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—24no—
470C2×C8—24no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—24no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—24no—
482C4⋊C4—21no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—24no—
486C4⋊C4—21no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—24no—
490C4⋊C4—42no—
491C4⋊C4—44no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—44no—
496C4⋊C4—42no—
497C4⋊C4—48no—
498C4⋊C4—44no—
499C4⋊C4—42no—
500C4⋊C4—44no—
501C4⋊C4—42no—
502C4⋊C4—44no—
503C4⋊C4—42no—
504C4⋊C4—81no—
505C4⋊C4—84no—
506C4⋊C4—82no—
507C4⋊C4—82no—
508C4⋊C4—84no—
509C4⋊C4—82no—
510C4⋊C4#12314yes—
511C22⋊C4—24no—
512C22⋊C4—24no—
513C22⋊C4—42no—
514C22⋊C4—48no—
515C22⋊C4—44no—
516C22⋊C4—44no—
517C22⋊C4—42no—
518C22⋊C4—42no—
519C22⋊C4—44no—
520C22⋊C4—42no—
521C22⋊C4—42no—
522C22⋊C4—82no—
523C22⋊C4—82no—
524C22⋊C4—82no—
525C22⋊C4—82no—
526C22⋊C4#12318yes—
527C42—24no—
528C42—22no—
529C42—24no—
530C42—22no—
531C42—22no—
532C42—24no—
533C42—48no—
534C42—44no—
535C42—42no—
536C42—82no—
537C42C2×#4114yes—
538C23—22no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—22no—
543C23—21no—
544C23—22no—
545C23—22no—
546C23—21no—
547C23—22no—
548C23—21no—
549C23—22no—
550C23—22no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23C2×#12314yes—
558C23C2×#13712yes—
559C23C23×D411yescharacteristic
560Q8—28no—
561Q8—44no—
562Q8—44no—
563D4—216no—
564D4—28no—
565D4—24no—
566D4—42no—
567D4—84no—
568D4—88no—
569C2×C4—216no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—28no—
581C2×C4—28no—
582C2×C4—28no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—44no—
590C2×C4—48no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—88no—
618C2×C4—84no—
619C2×C4—168no—
620C2×C4—164no—
621C2×C4C2×#12314yes—
622C2×C4C2×#12312yes—
623C2×C4C2×#12312yes—
624C2×C4C2×AGL(1,ℤ/8)14yes—
625C2×C4C2×AGL(1,ℤ/8)12yes—
626C2×C4C2×#13712yes—
627C2×C4C2×#13712yes—
628C2×C4#57712yes—
629C2×C4C22×#4112yes—
630C2×C4C2×H(5,𝔽2)11yescharacteristic
631C2×C4#58512yes—
632C2×C4#58511yescharacteristic
633C8—44no—
634C8—42no—
635C8—42no—
636C22—22no—
637C22—22no—
638C22—24no—
639C22—22no—
640C22—24no—
641C22—24no—
642C22—42no—
643C22—41no—
644C22—42no—
645C22—42no—
646C22—41no—
647C22—42no—
648C22—41no—
649C22—41no—
650C22—42no—
651C22—42no—
652C22—42no—
653C22—81no—
654C22—81no—
655C22—81no—
656C22—81no—
657C22C22×#12311yescharacteristic
658C22#348811yescharacteristic
659C22#350011yescharacteristic
660C22C22×AGL(1,ℤ/8)11yescharacteristic
661C22C22×#13711yescharacteristic
662C22C2×#57711yescharacteristic
663C22#358611yescharacteristic
664C4—22no—
665C4—24no—
666C4—22no—
667C4—22no—
668C4—24no—
669C4—42no—
670C4—41no—
671C4—81no—
672C4—82no—
673C4—84no—
674C4—81no—
675C4—81no—
676C4—82no—
677C4—82no—
678C4—82no—
679C4#309214yes—
680C4#351312yes—
681C4#352112yes—
682C2—42no—
683C2—44no—
684C2—82no—
685C2#3597611yescharacteristic
686C2C2×#309211yescharacteristic
687C2#6052211yescharacteristic
688C2#6196011yescharacteristic
689C2#6199911yescharacteristic
690C2#6254211yescharacteristic
691C2#6259511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36404C2
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. #29923C2semidirect
4. #29908C2semidirect
5. #27972C2semidirect
6. #27864C2semidirect
7. #27645C2semidirect
8. #27584C2semidirect
9. #26320C2semidirect
10. #26220C2semidirect
11. #24990C2semidirect
12. #24881C2semidirect
13. #23755C2semidirect
14. #23630C2semidirect
15. #22715C2semidirect
16. #22473C2semidirect
17. #22431C2semidirect

Maximal subgroups

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

subgroupindex
#364042
#341832
#299232
#299082
#279722
#278642
#276452
#275842
#263202
#262202
#249902
#248812
#237552
#236302
#227152
#224732
#224312