Finite Group Databasebeta

← #764906

Subgroups — #764906

Order 512.

3484 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2409C211yesmaximal, characteristic
2C22×#395C211yesmaximal, characteristic, complemented
3C2×#2322C211yesmaximal, characteristic
4C2×#2313C211yesmaximal, characteristic, complemented
5#15867C214yesmaximal, complemented
6#15427C214yesmaximal, complemented
7#15360C214yesmaximal, complemented
8#15354C214yesmaximal, complemented
9#14919C212yesmaximal, complemented
10#14912C212yesmaximal
11#14149C211yesmaximal, characteristic, complemented
12#14043C212yesmaximal, complemented
13#14019C212yesmaximal
14#13960C211yesmaximal, characteristic
15#13922C211yesmaximal, characteristic, complemented
16C24×Q8C2211yescharacteristic
17C22×#123C2211yescharacteristic
18C22×C22⋊Q8C2212yes—
19C22×C4×D4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C4⋊C4C2212yes—
22#2409—28no—
23C2×#395—28no—
24C2×#395C2212yes—
25C2×#395C2214yes—
26C2×#395C2214yes—
27C2×#393C2212yes—
28#2322—28no—
29#2313—28no—
30C2×#391C2212yes—
31C2×#391C2214yes—
32C2×#389C2212yes—
33C2×#389C2214yes—
34C2×#388C2212yes—
35C2×#388C2214yes—
36C2×#387C2214yes—
37C2×#387C2212yes—
38C2×#387C2214yes—
39C2×#386C2211yescharacteristic
40C2×#386C2212yes—
41C2×#386C2212yes—
42C2×#385C2211yescharacteristic
43C2×#385C2212yes—
44C2×#385C2212yes—
45C2×#385C2212yes—
46C2×#381C2211yescharacteristic
47C2×#381C2212yes—
48C2×#380C2211yescharacteristic
49C2×#380C2211yescharacteristic
50C2×C4×C22⋊C4C2211yescharacteristic
51C2×C4×C22⋊C4C2212yes—
52C22×#95C2212yes—
53C22×#95C2211yescharacteristic
54#1446C2214yes—
55#1445C2214yes—
56#1444C2214yes—
57#1443C2214yes—
58#1443C2214yes—
59#1442C2214yes—
60#1442C2214yes—
61#1442C2214yes—
62#1441C2212yes—
63#1441C2214yes—
64#1441C2214yes—
65#1441C2214yes—
66#1440C2214yes—
67#1439C2214yes—
68#1439C2212yes—
69#1439C2214yes—
70#1436C2214yes—
71#1436C2214yes—
72#1436C2214yes—
73#1434C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C22⋊#95C2212yes—
78C22⋊#95C2212yes—
79C23×Q8—24no—
80C23×Q8—24no—
81C23×Q8—24no—
82C23×Q8—48no—
83C23×Q8C2312yes—
84C23×Q8C2312yes—
85C23×D4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2312yes—
88C24×C4C2312yes—
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C24×C4C2311yescharacteristic
92C2×#123—24no—
93C2×#123—24no—
94C2×#123—24no—
95C2×C22⋊Q8—28no—
96C2×C22⋊Q8—48no—
97C2×C4×D4—24no—
98C2×C4×D4—24no—
99C2×C4×D4—24no—
100C22×C4⋊C4—24no—
101C22×C4⋊C4—24no—
102C22×C4⋊C4—28no—
103C22×C4⋊C4—48no—
104C22×C4⋊C4D418yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C22⋊C4C2311yescharacteristic
117C22×C22⋊C4C2311yescharacteristic
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2311yescharacteristic
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2311yescharacteristic
125C22×C22⋊C4C2312yes—
126C22×C42C2311yescharacteristic
127C22×C42C2314yes—
128C22×C42C2312yes—
129#395—216no—
130#395—216no—
131#395—28no—
132#393—28no—
133#391—28no—
134#391—48no—
135#389—28no—
136#389—48no—
137#388—28no—
138#388—48no—
139#387—216no—
140#387—28no—
141#387—216no—
142#386—28no—
143#386—24no—
144#386—28no—
145#385—24no—
146#385—28no—
147#385—28no—
148#385—28no—
149#381—24no—
150#381—28no—
151#380—24no—
152#380D418yes—
153C4×C22⋊C4—24no—
154C4×C22⋊C4—28no—
155C2×#95—28no—
156C2×#95—24no—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2312yes—
183C25C2411yescharacteristic
184C25C2411yescharacteristic
185C22×Q8—24no—
186C22×Q8—24no—
187C22×Q8—24no—
188C22×Q8—24no—
189C22×Q8—24no—
190C22×Q8—24no—
191C22×Q8—28no—
192C22×Q8—28no—
193C22×Q8—44no—
194C22×Q8—44no—
195C22×Q8—432no—
196C22×Q8—432no—
197C22×Q8—432no—
198C22×Q8—416no—
199C22×D4—22no—
200C22×D4—22no—
201C22×D4—22no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4—22no—
205C22×D4#4114yes—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—28no—
217C23×C4—28no—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—24no—
222C23×C4—22no—
223C23×C4—22no—
224C23×C4—22no—
225C23×C4—44no—
226C23×C4—44no—
227C23×C4—44no—
228C23×C4—44no—
229C23×C4—44no—
230C23×C4C2×D414yes—
231C23×C4C2×D414yes—
232C23×C4C2×D414yes—
233C23×C4C2×D414yes—
234C23×C4#4114yes—
235C23×C4C2414yes—
236C23×C4C2411yescharacteristic
237C23×C4C2412yes—
238C23×C4C2411yescharacteristic
239C23×C4C2414yes—
240C23×C4C2412yes—
241C23×C4C2412yes—
242C23×C4C2412yes—
243C23×C4C2411yescharacteristic
244C23×C4C2414yes—
245C23×C4C2414yes—
246C23×C4C2411yescharacteristic
247C23×C4C2411yescharacteristic
248#123—232no—
249C22⋊Q8—432no—
250C4×D4—416no—
251C2×C4⋊C4—232no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—432no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4C2×D418yes—
281C2×C4⋊C4C2×D418yes—
282C2×C4⋊C4C2×D418yes—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—28no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—28no—
305C2×C22⋊C4—28no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4C2×D418yes—
311C2×C22⋊C4#4114yes—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—28no—
316C2×C42—24no—
317C2×C42—44no—
318C2×C42—44no—
319C2×C42—44no—
320#95—24no—
321#95—24no—
322#95—28no—
323#95—28no—
324#95—24no—
325#95—28no—
326#95—24no—
327#95—24no—
328#95—28no—
329#95—28no—
330#95—24no—
331#95—24no—
332#95—28no—
333#95—24no—
334#95—28no—
335#95—28no—
336#95—28no—
337#95—28no—
338#95—28no—
339#95—28no—
340#95—28no—
341#95—44no—
342#95—44no—
343#95—44no—
344#95—44no—
345#95C2×D418yes—
346C24—24no—
347C24—24no—
348C24—21no—
349C24—21no—
350C24—21no—
351C24—21no—
352C24—24no—
353C24—24no—
354C24—21no—
355C24—21no—
356C24C22×D412yes—
357C24C22×D412yes—
358C24C2×#4112yes—
359C24C2×#4112yes—
360C24C2×#4112yes—
361C24C2×#4112yes—
362C242−1+412yes—
363C242−1+412yes—
364C24C2511yescharacteristic
365C2×Q8—216no—
366C2×Q8—216no—
367C2×Q8—416no—
368C2×Q8—416no—
369C2×Q8—416no—
370C2×Q8—416no—
371C2×Q8—48no—
372C2×Q8—48no—
373C2×Q8—4128no—
374C2×Q8—48no—
375C2×Q8—48no—
376C2×D4—28no—
377C2×D4—28no—
378C2×D4—28no—
379C2×D4—44no—
380C2×D4—44no—
381C2×D4—44no—
382C2×D4—44no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—216no—
386C22×C4—216no—
387C22×C4—28no—
388C22×C4—28no—
389C22×C4—28no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—28no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—24no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4—48no—
452C22×C4—48no—
453C22×C4—416no—
454C22×C4—416no—
455C22×C4—416no—
456C22×C4—416no—
457C22×C4—48no—
458C22×C4—48no—
459C22×C4—48no—
460C22×C4—48no—
461C22×C4—48no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—84no—
468C22×C4—84no—
469C22×C4—84no—
470C22×C4—84no—
471C22×C4#121116yes—
472C22×C4C22×D414yes—
473C22×C4C22×D418yes—
474C22×C4C22×D414yes—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C2×#4112yes—
480C22×C4C2×#4114yes—
481C22×C4C2×#4112yes—
482C22×C4C2×#4112yes—
483C22×C4H(5,𝔽2)18yes—
484C22×C42−1+418yes—
485C4⋊C4—216no—
486C4⋊C4—216no—
487C4⋊C4—216no—
488C4⋊C4—48no—
489C4⋊C4—416no—
490C4⋊C4—416no—
491C4⋊C4—416no—
492C4⋊C4—416no—
493C4⋊C4—44no—
494C4⋊C4—84no—
495C4⋊C4—84no—
496C22⋊C4—216no—
497C22⋊C4—28no—
498C22⋊C4—216no—
499C22⋊C4—416no—
500C22⋊C4—416no—
501C22⋊C4—44no—
502C22⋊C4—44no—
503C22⋊C4—48no—
504C22⋊C4—44no—
505C22⋊C4—84no—
506C42—44no—
507C42—84no—
508C42—88no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—28no—
516C23—28no—
517C23—28no—
518C23—28no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—28no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23C2×#12114yes—
537C23#52514yes—
538C23#53614yes—
539C23#54714yes—
540C23#54814yes—
541C23#55214yes—
542C23#55614yes—
543C23C23×D411yescharacteristic
544C23C23×D411yescharacteristic
545C23C22×#4111yescharacteristic
546C23C22×#4111yescharacteristic
547C23C22×#4111yescharacteristic
548C23C2×H(5,𝔽2)14yes—
549C23C2×2−1+411yescharacteristic
550C23C2×2−1+411yescharacteristic
551C23C2×2−1+414yes—
552Q8—432no—
553Q8—432no—
554D4—416no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—28no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—28no—
571C2×C4—28no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—28no—
575C2×C4—28no—
576C2×C4—28no—
577C2×C4—28no—
578C2×C4—28no—
579C2×C4—28no—
580C2×C4—28no—
581C2×C4—416no—
582C2×C4—432no—
583C2×C4—432no—
584C2×C4—41no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—416no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—88no—
631C2×C4—88no—
632C2×C4C2×#12114yes—
633C2×C4C2×#12114yes—
634C22—24no—
635C22—28no—
636C22—28no—
637C22—28no—
638C22—24no—
639C22—28no—
640C22—28no—
641C22—28no—
642C22—44no—
643C22—44no—
644C22—42no—
645C22—44no—
646C22—81no—
647C22—82no—
648C22—81no—
649C22—82no—
650C22—82no—
651C22#241518yes—
652C22C22×#12111yescharacteristic
653C22C2×#52511yescharacteristic
654C22C2×#53611yescharacteristic
655C22C2×#54711yescharacteristic
656C22C2×#54811yescharacteristic
657C22C2×#55211yescharacteristic
658C22C2×#55611yescharacteristic
659C22#348412yes—
660C22#348412yes—
661C22#348612yes—
662C22#349012yes—
663C22#349912yes—
664C22#350912yes—
665C22#352014yes—
666C22#352014yes—
667C22#355214yes—
668C22#355214yes—
669C4—24no—
670C4—24no—
671C4—48no—
672C4—44no—
673C4—44no—
674C4—48no—
675C4—44no—
676C4—42no—
677C4—48no—
678C4—81no—
679C4—81no—
680C4—82no—
681C4—84no—
682C2—28no—
683C2—82no—
684C2C2×#241511yescharacteristic
685C2#3534811yescharacteristic
686C2#3535712yes—
687C2#3566711yescharacteristic
688C2#3723312yes—
689C2#4300412yes—
690C2#4300712yes—
691C2#4308212yes—
692C2#4308512yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#395C2semidirect
4. C2×#2313C2semidirect
5. #15867C2semidirect
6. #15427C2semidirect
7. #15360C2semidirect
8. #15354C2semidirect
9. #14919C2semidirect
11. #14149C2semidirect
12. #14043C2semidirect
15. #13922C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24092
C22×#3952
C2×#23222
C2×#23132
#158672
#154272
#153602
#153542
#149192
#149122
#141492
#140432
#140192
#139602
#139222