Finite Group Databasebeta

← #847900

Subgroups — #847900

Order 512.

3054 conjugacy classes of subgroups.

The subgroup lattice has 705 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2409C211yesmaximal, characteristic
2C2×C4×#123C211yesmaximal, characteristic
3#16617C214yesmaximal, complemented
4#16014C212yesmaximal
5#15930C212yesmaximal, complemented
6#15568C212yesmaximal, complemented
7#15463C212yesmaximal
8#15237C212yesmaximal, complemented
9#15179C212yesmaximal, complemented
10#15132C212yesmaximal, complemented
11#14702C212yesmaximal, complemented
12#14542C214yesmaximal, complemented
13#14021C212yesmaximal, complemented
14#13922C212yesmaximal, complemented
15#13905C211yesmaximal, characteristic
16C24×Q8C2211yescharacteristic
17C22×#123C2211yescharacteristic
18C22×C22⋊Q8C2212yes—
19C22×C4×D4C2211yescharacteristic
20C23×C42C2211yescharacteristic
21#2409—28no—
22C2×#397C2214yes—
23C2×#395C2212yes—
24C2×#393C2212yes—
25C4×#123—28no—
26C2×#391C2212yes—
27C2×#391C2212yes—
28C2×#391C2212yes—
29C2×#391C2212yes—
30C2×#391C2212yes—
31C2×#389C2212yes—
32C2×#388C2212yes—
33C2×#387C2212yes—
34C2×#386C2212yes—
35C2×#386C2211yescharacteristic
36C2×#385C2212yes—
37C2×#385C2211yescharacteristic
38C2×#382C2211yescharacteristic
39C2×#381C2211yescharacteristic
40C2×#380C2211yescharacteristic
41C2×#380C2212yes—
42C2×#380C2212yes—
43C2×C4×C4⋊C4C2212yes—
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C22⋊C4C2212yes—
46C2×C4×C22⋊C4C2211yescharacteristic
47#1446C2214yes—
48#1446C2214yes—
49#1445C2212yes—
50#1443C2214yes—
51#1443C2214yes—
52#1443C2214yes—
53#1443C2214yes—
54#1442C2214yes—
55#1442C2214yes—
56#1442C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1440C2214yes—
61#1440C2214yes—
62#1439C2214yes—
63#1438C2212yes—
64#1437C2212yes—
65#1437C2212yes—
66#1436C2214yes—
67#1436C2214yes—
68#1436C2214yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72#1434C2214yes—
73C22⋊#95C2214yes—
74C22⋊#95C2214yes—
75C22⋊#95C2214yes—
76C22⋊#95C2214yes—
77C23×Q8—28no—
78C23×Q8—24no—
79C23×Q8—42no—
80C23×Q8—42no—
81C23×Q8—44no—
82C23×Q8C2312yes—
83C23×Q8C2312yes—
84C23×D4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2312yes—
89C24×C4C2311yescharacteristic
90C2×#123—24no—
91C2×#123—28no—
92C2×C22⋊Q8—28no—
93C2×C22⋊Q8—28no—
94C2×C22⋊Q8—44no—
95C2×C4×D4—42no—
96C2×C4×D4—42no—
97C2×C4×D4D418yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2311yescharacteristic
108C22×C22⋊C4C2311yescharacteristic
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2311yescharacteristic
115C22×C22⋊C4C2311yescharacteristic
116C22×C42—24no—
117C22×C42—42no—
118C22×C42—42no—
119C22×C42C2312yes—
120C22×C42C2312yes—
121C22×C42C2311yescharacteristic
122C22×C42C2312yes—
123C22×C42C2312yes—
124#397—216no—
125#395—28no—
126#393—28no—
127#391—28no—
128#391—28no—
129#391—44no—
130#391—44no—
131#391—44no—
132#389—44no—
133#388—28no—
134#387—28no—
135#386—28no—
136#386—24no—
137#385—28no—
138#385—24no—
139#382—42no—
140#381—24no—
141#380—24no—
142#380—44no—
143#380—44no—
144C4×C4⋊C4—28no—
145C4×C4⋊C4—42no—
146C4×C22⋊C4—28no—
147C4×C22⋊C4—24no—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C25C2411yescharacteristic
177C25C2411yescharacteristic
178C22×Q8—24no—
179C22×Q8—24no—
180C22×Q8—24no—
181C22×Q8—28no—
182C22×Q8—24no—
183C22×Q8—24no—
184C22×Q8—24no—
185C22×Q8—24no—
186C22×Q8—44no—
187C22×Q8—42no—
188C22×Q8—42no—
189C22×Q8—432no—
190C22×Q8—416no—
191C22×Q8—416no—
192C22×Q8—432no—
193C22×Q8—416no—
194C22×Q8#4118yes—
195C22×D4—24no—
196C22×D4—24no—
197C22×D4—41no—
198C22×D4—41no—
199C22×D4C2×D414yes—
200C23×C4—22no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—22no—
212C23×C4—24no—
213C23×C4—28no—
214C23×C4—24no—
215C23×C4—42no—
216C23×C4—42no—
217C23×C4—42no—
218C23×C4—44no—
219C23×C4—41no—
220C23×C4—41no—
221C23×C4—41no—
222C23×C4—41no—
223C23×C4C2×D414yes—
224C23×C4C2×D414yes—
225C23×C4C2414yes—
226C23×C4C2414yes—
227C23×C4C2414yes—
228C23×C4C2412yes—
229C23×C4C2411yescharacteristic
230C23×C4C2411yescharacteristic
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233C23×C4C2412yes—
234C23×C4C2414yes—
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2411yescharacteristic
238#123—232no—
239C22⋊Q8—432no—
240C4×D4—416no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—41no—
260C2×C4⋊C4—41no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4C2×D414yes—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—42no—
280C2×C22⋊C4—41no—
281C2×C22⋊C4—41no—
282C2×C22⋊C4—41no—
283C2×C22⋊C4—41no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4C2×D414yes—
290C2×C22⋊C4C2×D414yes—
291C2×C22⋊C4#4118yes—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—42no—
302C2×C42—42no—
303C2×C42—416no—
304C2×C42—42no—
305C2×C42—44no—
306C2×C42C2×D414yes—
307#95—24no—
308#95—24no—
309#95—24no—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—28no—
314#95—28no—
315#95—28no—
316#95—28no—
317#95—28no—
318#95—28no—
319#95—28no—
320#95—28no—
321#95—44no—
322#95—44no—
323#95—42no—
324#95—44no—
325#95—44no—
326#95—42no—
327#95—42no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—44no—
334#95—44no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—21no—
341C24—21no—
342C24—21no—
343C24—21no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24C22×D412yes—
350C24C22×D412yes—
351C24C2×#4114yes—
352C242−1+414yes—
353C24C2511yescharacteristic
354C2×Q8—216no—
355C2×Q8—216no—
356C2×Q8—216no—
357C2×Q8—416no—
358C2×Q8—48no—
359C2×Q8—48no—
360C2×Q8—416no—
361C2×Q8—4128no—
362C2×Q8—48no—
363C2×Q8—48no—
364C2×Q8—48no—
365C2×Q8—48no—
366C2×Q8—48no—
367C2×D4—28no—
368C2×D4—28no—
369C2×D4—44no—
370C2×D4—48no—
371C2×D4—48no—
372C22×C4—28no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—28no—
414C22×C4—28no—
415C22×C4—216no—
416C22×C4—216no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—44no—
446C22×C4—48no—
447C22×C4—48no—
448C22×C4—48no—
449C22×C4—48no—
450C22×C4—416no—
451C22×C4—48no—
452C22×C4—416no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—48no—
456C22×C4—44no—
457C22×C4—48no—
458C22×C4—44no—
459C22×C4—48no—
460C22×C4—48no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C2×#4114yes—
471C22×C4C2×#4114yes—
472C22×C4C2×#4118yes—
473C22×C4C2×#4114yes—
474C22×C4H(5,𝔽2)12yes—
475C22×C4H(5,𝔽2)12yes—
476C22×C4H(5,𝔽2)14yes—
477C22×C42−1+414yes—
478C4⋊C4—216no—
479C4⋊C4—44no—
480C4⋊C4—48no—
481C4⋊C4—416no—
482C4⋊C4—48no—
483C4⋊C4—48no—
484C4⋊C4—48no—
485C4⋊C4—88no—
486C22⋊C4—28no—
487C22⋊C4—216no—
488C22⋊C4—44no—
489C22⋊C4—44no—
490C22⋊C4—48no—
491C22⋊C4—416no—
492C22⋊C4—48no—
493C22⋊C4—44no—
494C22⋊C4—88no—
495C22⋊C4—88no—
496C42—28no—
497C42—48no—
498C42—48no—
499C42—48no—
500C42—84no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—42no—
522C23—42no—
523C23—41no—
524C23—41no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23#52514yes—
542C23#53614yes—
543C23C23×D411yescharacteristic
544C23C22×#4112yes—
545C23C22×#4112yes—
546C23C2×H(5,𝔽2)11yescharacteristic
547C23C2×H(5,𝔽2)11yescharacteristic
548C23C2×H(5,𝔽2)12yes—
549C23C2×2−1+412yes—
550C23C2×2−1+412yes—
551C23#58512yes—
552Q8—432no—
553Q8—432no—
554D4—416no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—28no—
564C2×C4—28no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—28no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—416no—
577C2×C4—416no—
578C2×C4—432no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—44no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—416no—
607C2×C4—416no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—42no—
636C2×C4#53414yes—
637C2×C4#53818yes—
638C2×C4#54618yes—
639C2×C4#55114yes—
640C2×C4#56014yes—
641C22—24no—
642C22—24no—
643C22—28no—
644C22—28no—
645C22—44no—
646C22—44no—
647C22—44no—
648C22—44no—
649C22—44no—
650C22—44no—
651C22—42no—
652C22—42no—
653C22—44no—
654C22—44no—
655C22—44no—
656C22—44no—
657C22—42no—
658C22—42no—
659C22C2×#52511yescharacteristic
660C22C2×#53411yescharacteristic
661C22C2×#53611yescharacteristic
662C22C2×#53812yes—
663C22C2×#54612yes—
664C22C2×#55111yescharacteristic
665C22#348012yes—
666C22#348212yes—
667C22#348912yes—
668C22#351112yes—
669C22#351412yes—
670C22#351612yes—
671C22#352012yes—
672C22#352012yes—
673C22#352312yes—
674C22C2×#56011yescharacteristic
675C22#353811yescharacteristic
676C22#354011yescharacteristic
677C22#354112yes—
678C22#355212yes—
679C22#356112yes—
680C4—44no—
681C4—42no—
682C4—44no—
683C4—42no—
684C4—42no—
685C4—48no—
686C4—42no—
687C4—44no—
688C4—84no—
689C4—84no—
690C4—84no—
691C4—82no—
692C4#269718yes—
693C2—44no—
694C2—44no—
695C2#3508411yescharacteristic
696C2#3637412yes—
697C2#3737411yescharacteristic
698C2#3802712yes—
699C2#3923812yes—
700C2#4010012yes—
701C2#4566912yes—
702C2C2×#269711yescharacteristic
703C2#5938312yes—
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. #16617C2semidirect
5. #15930C2semidirect
6. #15568C2semidirect
8. #15237C2semidirect
9. #15179C2semidirect
10. #15132C2semidirect
11. #14702C2semidirect
12. #14542C2semidirect
13. #14021C2semidirect
14. #13922C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24092
C2×C4×#1232
#166172
#160142
#159302
#155682
#154632
#152372
#151792
#151322
#147022
#145422
#140212
#139222
#139052