Finite Group Databasebeta

← #848403

Subgroups — #848403

Order 512.

2712 conjugacy classes of subgroups.

The subgroup lattice has 695 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2542C211yesmaximal, characteristic
2C2×C4×#123C211yesmaximal, characteristic
3#16591C214yesmaximal, complemented
4#16074C212yesmaximal
5#15983C212yesmaximal
6#15916C212yesmaximal, complemented
7#15730C212yesmaximal, complemented
8#15180C212yesmaximal, complemented
9#15179C212yesmaximal, complemented
10#15179C212yesmaximal, complemented
11#14702C212yesmaximal, complemented
12#14414C214yesmaximal, complemented
13#14138C212yesmaximal, complemented
14#14042C212yesmaximal, complemented
15#13871C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×#123C2211yescharacteristic
18C22×C22⋊Q8C2212yes—
19C22×C22⋊Q8C2212yes—
20C22×C4×D4C2211yescharacteristic
21C22×#117C2211yescharacteristic
22C23×C42C2211yescharacteristic
23#2542—28no—
24C2×#399C2212yes—
25C2×#399C2212yes—
26C4×#123—28no—
27C2×#391C2212yes—
28C2×#391C2212yes—
29C2×#391C2212yes—
30C2×#391C2212yes—
31C2×#390C2212yes—
32C2×#389C2212yes—
33C2×#388C2212yes—
34C2×#387C2212yes—
35C2×#387C2214yes—
36C2×#386C2211yescharacteristic
37C2×#385C2211yescharacteristic
38C2×#384C2212yes—
39C2×#381C2211yescharacteristic
40C2×#380C2211yescharacteristic
41C2×#380C2212yes—
42C2×C4×C4⋊C4C2212yes—
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C22⋊C4C2212yes—
45C2×C4×C22⋊C4C2211yescharacteristic
46#1443C2214yes—
47#1442C2214yes—
48#1442C2214yes—
49#1442C2214yes—
50#1442C2214yes—
51#1441C2214yes—
52#1441C2214yes—
53#1441C2214yes—
54#1441C2214yes—
55#1441C2214yes—
56#1440C2214yes—
57#1440C2212yes—
58#1440C2214yes—
59#1440C2214yes—
60#1440C2214yes—
61#1439C2214yes—
62#1439C2214yes—
63#1438C2212yes—
64#1437C2212yes—
65#1436C2214yes—
66#1436C2212yes—
67#1436C2214yes—
68#1436C2214yes—
69#1436C2214yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74#1434C2214yes—
75C22⋊#95C2214yes—
76C22⋊#95C2214yes—
77C23×Q8C2312yes—
78C23×D4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C2×C4⋊Q8—24no—
84C2×C4⋊Q8—42no—
85C2×C4⋊Q8—42no—
86C2×#123—24no—
87C2×#123—28no—
88C2×C22⋊Q8—28no—
89C2×C22⋊Q8—28no—
90C2×C22⋊Q8—28no—
91C2×C22⋊Q8—28no—
92C2×C22⋊Q8—44no—
93C2×C22⋊Q8—44no—
94C2×C4×D4—42no—
95C2×C4×D4—42no—
96C2×C4×D4D418yes—
97C2×#117—44no—
98C2×#117—42no—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C22⋊C4C2311yescharacteristic
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2311yescharacteristic
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2311yescharacteristic
120C22×C22⋊C4C2311yescharacteristic
121C22×C42—24no—
122C22×C42—42no—
123C22×C42—42no—
124C22×C42C2312yes—
125C22×C42C2312yes—
126C22×C42C2312yes—
127C22×C42C2311yescharacteristic
128C22×C42C2312yes—
129C22×C42C2312yes—
130#399—28no—
131#399—28no—
132#391—44no—
133#391—44no—
134#391—44no—
135#391—44no—
136#390—28no—
137#389—28no—
138#388—28no—
139#387—28no—
140#387—216no—
141#386—24no—
142#385—24no—
143#384—28no—
144#381—24no—
145#380—44no—
146#380—42no—
147C4×C4⋊C4—24no—
148C4×C4⋊C4—28no—
149C4×C22⋊C4—28no—
150C4×C22⋊C4—24no—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C25C2411yescharacteristic
177C25C2411yescharacteristic
178C22×Q8—24no—
179C22×Q8—24no—
180C22×Q8—24no—
181C22×Q8—24no—
182C22×Q8—24no—
183C22×Q8—24no—
184C22×Q8—24no—
185C22×D4—24no—
186C22×D4—24no—
187C22×D4—41no—
188C22×D4—41no—
189C22×D4C2×D414yes—
190C23×C4—22no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—22no—
195C23×C4—24no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—22no—
199C23×C4—24no—
200C23×C4—22no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—42no—
204C23×C4—41no—
205C23×C4—41no—
206C23×C4—41no—
207C23×C4—41no—
208C23×C4C2×D414yes—
209C23×C4C2×D414yes—
210C23×C4C2414yes—
211C23×C4C2414yes—
212C23×C4C2414yes—
213C23×C4C2412yes—
214C23×C4C2411yescharacteristic
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217C23×C4C2412yes—
218C23×C4C2414yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C4⋊Q8—416no—
224#123—232no—
225C22⋊Q8—432no—
226C22⋊Q8—432no—
227C4×D4—416no—
228#117—416no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—41no—
252C2×C4⋊C4—41no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4C2×D414yes—
260C2×C4⋊C4#4118yes—
261C2×C4⋊C4#4118yes—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—42no—
277C2×C22⋊C4—42no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—41no—
280C2×C22⋊C4—41no—
281C2×C22⋊C4—41no—
282C2×C22⋊C4—41no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—42no—
289C2×C22⋊C4C2×D414yes—
290C2×C22⋊C4C2×D414yes—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—28no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—42no—
303C2×C42—44no—
304C2×C42—42no—
305C2×C42—42no—
306C2×C42—416no—
307C2×C42C2×D414yes—
308#95—24no—
309#95—24no—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—28no—
315#95—28no—
316#95—28no—
317#95—28no—
318#95—28no—
319#95—28no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95—42no—
324#95—44no—
325#95—44no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95—44no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—21no—
337C24—21no—
338C24—22no—
339C24—22no—
340C24—22no—
341C24—22no—
342C24—22no—
343C24—21no—
344C24—21no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24C22×D412yes—
349C24C22×D412yes—
350C242−1+414yes—
351C24C2511yescharacteristic
352C2×Q8—48no—
353C2×Q8—48no—
354C2×Q8—48no—
355C2×Q8—48no—
356C2×Q8—48no—
357C2×Q8—48no—
358C2×Q8—48no—
359C2×D4—28no—
360C2×D4—28no—
361C2×D4—44no—
362C2×D4—48no—
363C2×D4—48no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—28no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—28no—
405C22×C4—28no—
406C22×C4—216no—
407C22×C4—216no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—44no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—48no—
446C22×C4—48no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C2×#4114yes—
457C22×C4C2×#4114yes—
458C22×C4C2×#4118yes—
459C22×C4C2×#4114yes—
460C22×C4C2×#4114yes—
461C22×C4H(5,𝔽2)12yes—
462C22×C4H(5,𝔽2)12yes—
463C22×C4H(5,𝔽2)14yes—
464C22×C42−1+414yes—
465C4⋊C4—216no—
466C4⋊C4—216no—
467C4⋊C4—44no—
468C4⋊C4—416no—
469C4⋊C4—48no—
470C4⋊C4—416no—
471C4⋊C4—416no—
472C4⋊C4—48no—
473C4⋊C4—48no—
474C4⋊C4—48no—
475C4⋊C4—48no—
476C22⋊C4—28no—
477C22⋊C4—216no—
478C22⋊C4—44no—
479C22⋊C4—44no—
480C22⋊C4—48no—
481C22⋊C4—416no—
482C22⋊C4—416no—
483C22⋊C4—48no—
484C22⋊C4—44no—
485C22⋊C4—88no—
486C22⋊C4—84no—
487C42—28no—
488C42—48no—
489C42—48no—
490C42—48no—
491C42—48no—
492C42—48no—
493C23—22no—
494C23—22no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—24no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—24no—
507C23—24no—
508C23—42no—
509C23—42no—
510C23—41no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—41no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23#53614yes—
535C23C23×D411yescharacteristic
536C23C22×#4112yes—
537C23C22×#4112yes—
538C23C2×H(5,𝔽2)11yescharacteristic
539C23C2×H(5,𝔽2)11yescharacteristic
540C23C2×H(5,𝔽2)12yes—
541C23C2×2−1+412yes—
542C23C2×2−1+412yes—
543C23#58512yes—
544Q8—816no—
545D4—416no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—28no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—28no—
560C2×C4—28no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—416no—
569C2×C4—416no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—416no—
596C2×C4—416no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4#52514yes—
626C2×C4#53414yes—
627C2×C4#54018yes—
628C2×C4#54618yes—
629C2×C4#55914yes—
630C2×C4#56014yes—
631C22—24no—
632C22—28no—
633C22—44no—
634C22—44no—
635C22—44no—
636C22—44no—
637C22—42no—
638C22—44no—
639C22—42no—
640C22—42no—
641C22—44no—
642C22—44no—
643C22—42no—
644C22—42no—
645C22—82no—
646C22—82no—
647C22—82no—
648C22—82no—
649C22C2×#52511yescharacteristic
650C22C2×#53411yescharacteristic
651C22C2×#53611yescharacteristic
652C22C2×#54012yes—
653C22C2×#54612yes—
654C22#347512yes—
655C22C2×#55911yescharacteristic
656C22#348212yes—
657C22#348912yes—
658C22#351112yes—
659C22#351612yes—
660C22#351712yes—
661C22#352012yes—
662C22#352012yes—
663C22#352312yes—
664C22C2×#56011yescharacteristic
665C22#353811yescharacteristic
666C22#354011yescharacteristic
667C22#354112yes—
668C22#355212yes—
669C22#356112yes—
670C4—24no—
671C4—44no—
672C4—44no—
673C4—48no—
674C4—42no—
675C4—42no—
676C4—42no—
677C4—44no—
678C4—84no—
679C4—84no—
680C4—82no—
681C4—84no—
682C4#270918yes—
683C2—44no—
684C2—82no—
685C2#3508411yescharacteristic
686C2#3640412yes—
687C2#3737411yescharacteristic
688C2#3802712yes—
689C2#3902312yes—
690C2#4007912yes—
691C2#4562812yes—
692C2C2×#270911yescharacteristic
693C2#5959312yes—
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
3. #16591C2semidirect
6. #15916C2semidirect
7. #15730C2semidirect
8. #15180C2semidirect
9. #15179C2semidirect
10. #15179C2semidirect
11. #14702C2semidirect
12. #14414C2semidirect
13. #14138C2semidirect
14. #14042C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25422
C2×C4×#1232
#165912
#160742
#159832
#159162
#157302
#151802
#151792
#147022
#144142
#141382
#140422
#138712