Finite Group Databasebeta

← #2807610

Subgroups — #2807610

Order 512.

4104 conjugacy classes of subgroups.

The subgroup lattice has 733 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2673C214yesmaximal, complemented
2C2×#2537C211yesmaximal, characteristic, complemented
3C2×#2501C211yesmaximal, characteristic, complemented
4#16551C212yesmaximal, complemented
5#16508C214yesmaximal, complemented
6#15643C212yesmaximal, complemented
7#15380C214yesmaximal, complemented
8#15368C214yesmaximal, complemented
9#15310C212yesmaximal, complemented
10#14811C212yesmaximal, complemented
11#14719C212yesmaximal, complemented
12#14424C211yesmaximal, characteristic, complemented
13#14175C212yesmaximal, complemented
14C22×#127C2212yes—
15C22×#123C2214yes—
16C22×#121C2214yes—
17C22×#121C2212yes—
18C22×#121C2211yescharacteristic
19C22×C4×D4C2212yes—
20C23×C4⋊C4C2211yescharacteristic
21#2673—232no—
22#2537—28no—
23#2501—28no—
24C2×#399C2214yes—
25C2×#394C2214yes—
26C2×#394C2212yes—
27C2×#394C2214yes—
28C2×#394C2212yes—
29C2×#394C2212yes—
30C2×#392C2214yes—
31C2×#392C2214yes—
32C2×#392C2214yes—
33C2×#392C2212yes—
34C2×#392C2214yes—
35C2×#392C2212yes—
36C2×#390C2211yescharacteristic
37C2×#390C2211yescharacteristic
38C2×#388C2214yes—
39C2×#387C2214yes—
40C2×#386C2214yes—
41C2×#386C2214yes—
42C2×#386C2214yes—
43C2×#386C2214yes—
44C2×#386C2214yes—
45C2×#384C2211yescharacteristic
46C2×#384C2211yescharacteristic
47C2×#380C2214yes—
48C2×C4×C22⋊C4C2211yescharacteristic
49#1445C2214yes—
50#1442C2214yes—
51#1442C2212yes—
52#1441C2212yes—
53#1441C2212yes—
54#1440C2214yes—
55#1439C2212yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1438C2212yes—
61#1438C2214yes—
62#1438C2214yes—
63#1438C2212yes—
64#1437C2212yes—
65#1437C2214yes—
66#1436C2212yes—
67#1434C2212yes—
68C22⋊#95C2212yes—
69C22⋊#95C2214yes—
70C23×D4C2312yes—
71C23×D4C2311yescharacteristic
72C23×D4C2312yes—
73C23×D4C2312yes—
74C23×D4C2314yes—
75C23×D4C2312yes—
76C23×D4C2314yes—
77C23×D4C2314yes—
78C24×C4C2311yescharacteristic
79C24×C4C2312yes—
80C24×C4C2314yes—
81C2×#127—216no—
82C2×#127—44no—
83C2×#123—216no—
84C2×#123—216no—
85C2×#123—48no—
86C2×#121—216no—
87C2×#121—216no—
88C2×#121—24no—
89C2×#121—216no—
90C2×#121—42no—
91C2×#121—48no—
92C2×#121—42no—
93C2×#121—44no—
94C2×C4×D4—28no—
95C2×C4×D4—48no—
96C22×C4⋊C4—24no—
97C22×C4⋊C4—42no—
98C22×C4⋊C4—42no—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C42C2312yes—
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125#399—216no—
126#394—28no—
127#394—28no—
128#394—216no—
129#394—216no—
130#394—44no—
131#392—216no—
132#392—216no—
133#392—216no—
134#392—48no—
135#392—44no—
136#392D4116yescomplemented
137#390—24no—
138#390—24no—
139#388—216no—
140#387—216no—
141#386—216no—
142#386—216no—
143#386—216no—
144#386—48no—
145#386—48no—
146#384—24no—
147#384D418yescomplemented
148#380—216no—
149C4×C22⋊C4—24no—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C25C2412yes—
167C25C2414yes—
168C25C2411yescharacteristic
169C22×D4—28no—
170C22×D4—28no—
171C22×D4—28no—
172C22×D4—28no—
173C22×D4—28no—
174C22×D4—24no—
175C22×D4—28no—
176C22×D4—28no—
177C22×D4—24no—
178C22×D4—24no—
179C22×D4—22no—
180C22×D4—24no—
181C22×D4—24no—
182C22×D4—28no—
183C22×D4—24no—
184C22×D4—24no—
185C22×D4—44no—
186C22×D4—44no—
187C22×D4—41no—
188C22×D4—41no—
189C22×D4—42no—
190C22×D4—44no—
191C22×D4—44no—
192C22×D4—42no—
193C22×D4—44no—
194C22×D4—44no—
195C22×D4—42no—
196C22×D4—42no—
197C22×D4—44no—
198C22×D4—44no—
199C22×D4—44no—
200C22×D4—44no—
201C22×D4—44no—
202C22×D4—44no—
203C22×D4—44no—
204C22×D4—44no—
205C22×D4—44no—
206C22×D4—44no—
207C22×D4—44no—
208C22×D4—44no—
209C22×D4—42no—
210C22×D4—44no—
211C22×D4—44no—
212C22×D4—42no—
213C22×D4C2×D418yes—
214C22×D4C2×D4116yes—
215C23×C4—22no—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—28no—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—28no—
222C23×C4—24no—
223C23×C4—28no—
224C23×C4—28no—
225C23×C4—42no—
226C23×C4—42no—
227C23×C4—44no—
228C23×C4—44no—
229C23×C4—44no—
230C23×C4—44no—
231C23×C4—44no—
232C23×C4C2412yes—
233C23×C4C2414yes—
234C23×C4C2414yes—
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2412yes—
238C23×C4C2414yes—
239C23×C4C2411yescharacteristic
240C23×C4C2412yes—
241C23×C4C2411yescharacteristic
242#127—432no—
243#123—464no—
244#121—464no—
245#121—416no—
246#121—432no—
247C4×D4—432no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—41no—
258C2×C4⋊C4—41no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—41no—
261C2×C4⋊C4—41no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—41no—
264C2×C4⋊C4—41no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—416no—
268C2×C4⋊C4C2×D4116yes—
269C2×C4⋊C4C2×D414yes—
270C2×C4⋊C4C2×D414yes—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—42no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4C2×D418yes—
315C2×C22⋊C4C2×D4116yes—
316C2×C42—24no—
317C2×C42—28no—
318C2×C42—22no—
319C2×C42—42no—
320C2×C42—42no—
321C2×C42C2×D414yes—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—28no—
326#95—28no—
327#95—28no—
328#95—28no—
329#95—28no—
330#95—24no—
331#95—28no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95—42no—
336#95—42no—
337#95C2×D418yes—
338C24—21no—
339C24—22no—
340C24—22no—
341C24—21no—
342C24—24no—
343C24—22no—
344C24—24no—
345C24—24no—
346C24—24no—
347C24—24no—
348C24—24no—
349C24—24no—
350C24—24no—
351C24—24no—
352C24—24no—
353C24—24no—
354C24—24no—
355C24—24no—
356C24—24no—
357C24—24no—
358C24—22no—
359C24—24no—
360C24—24no—
361C24—22no—
362C24—24no—
363C24—22no—
364C24—24no—
365C24—24no—
366C24—24no—
367C24C22×D418yes—
368C24C22×D414yes—
369C24H(5,𝔽2)12yes—
370C24C2511yescharacteristic
371C2×D4—232no—
372C2×D4—216no—
373C2×D4—416no—
374C2×D4—48no—
375C2×D4—48no—
376C2×D4—416no—
377C2×D4—416no—
378C2×D4—416no—
379C2×D4—416no—
380C2×D4—416no—
381C2×D4—48no—
382C2×D4—416no—
383C2×D4—416no—
384C2×D4—44no—
385C2×D4—44no—
386C2×D4—416no—
387C2×D4—48no—
388C2×D4—48no—
389C2×D4—416no—
390C2×D4—48no—
391C2×D4—44no—
392C2×D4—84no—
393C2×D4—84no—
394C2×D4—84no—
395C2×D4—84no—
396C2×D4—88no—
397C2×D4—88no—
398C2×D4—88no—
399C2×D4—88no—
400C2×D4—88no—
401C2×D4—88no—
402C2×D4—88no—
403C2×D4—88no—
404C2×D4—88no—
405C2×D4—84no—
406C2×D4—88no—
407C2×D4—88no—
408C2×D4—88no—
409C2×D4—88no—
410C2×D4—88no—
411C2×D4—84no—
412C2×D4—84no—
413C2×D4—88no—
414C2×D4—88no—
415C2×D4—88no—
416C2×D4—88no—
417C22×C4—24no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—28no—
423C22×C4—24no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—21no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—48no—
465C22×C4—48no—
466C22×C4—44no—
467C22×C4—48no—
468C22×C4—416no—
469C22×C4—416no—
470C22×C4—416no—
471C22×C4—44no—
472C22×C4—48no—
473C22×C4—416no—
474C22×C4—88no—
475C22×C4—88no—
476C22×C4—88no—
477C22×C4—88no—
478C22×C4—88no—
479C22×C4—84no—
480C22×C4C22×D418yes—
481C22×C4C22×D414yes—
482C22×C4C22×D414yes—
483C22×C4C22×D414yes—
484C22×C4C22×D412yes—
485C22×C4C22×D414yes—
486C22×C4C22×D414yes—
487C22×C4H(5,𝔽2)12yes—
488C22×C4H(5,𝔽2)14yes—
489C22×C4H(5,𝔽2)14yes—
490C4⋊C4—232no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—48no—
494C4⋊C4—48no—
495C4⋊C4—416no—
496C4⋊C4—44no—
497C4⋊C4—48no—
498C22⋊C4—216no—
499C22⋊C4—416no—
500C22⋊C4—48no—
501C22⋊C4—416no—
502C22⋊C4—416no—
503C22⋊C4—416no—
504C22⋊C4—416no—
505C22⋊C4—416no—
506C22⋊C4—416no—
507C22⋊C4—88no—
508C22⋊C4—88no—
509C22⋊C4—88no—
510C22⋊C4—88no—
511C22⋊C4—88no—
512C22⋊C4—88no—
513C42—28no—
514C42—48no—
515C42—44no—
516C23—24no—
517C23—22no—
518C23—28no—
519C23—28no—
520C23—24no—
521C23—24no—
522C23—22no—
523C23—24no—
524C23—28no—
525C23—28no—
526C23—28no—
527C23—28no—
528C23—28no—
529C23—28no—
530C23—28no—
531C23—41no—
532C23—41no—
533C23—42no—
534C23—42no—
535C23—44no—
536C23—44no—
537C23—42no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23—44no—
562C23—44no—
563C23—44no—
564C23—44no—
565C23—44no—
566C23—42no—
567C23—44no—
568C23—42no—
569C23—42no—
570C23—44no—
571C23—44no—
572C23—42no—
573C23—44no—
574C23—44no—
575C23—44no—
576C23—44no—
577C23—44no—
578C23—44no—
579C23—44no—
580C23—44no—
581C23—44no—
582C23—44no—
583C23—42no—
584C23—44no—
585C23—42no—
586C23—42no—
587C23D4214yes—
588C23C23×D411yescharacteristic
589C23C23×D412yes—
590C23C2×H(5,𝔽2)12yes—
591C23C2×H(5,𝔽2)11yescharacteristic
592C23C2×H(5,𝔽2)12yes—
593C23C2×H(5,𝔽2)11yescharacteristic
594C23#58512yes—
595C23#58514yes—
596D4—432no—
597D4—88no—
598D4—832no—
599D4—816no—
600D4—816no—
601D4—816no—
602D4—832no—
603D4—832no—
604C2×C4—24no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—24no—
608C2×C4—28no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—28no—
612C2×C4—28no—
613C2×C4—24no—
614C2×C4—28no—
615C2×C4—28no—
616C2×C4—22no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—28no—
620C2×C4—28no—
621C2×C4—28no—
622C2×C4—24no—
623C2×C4—416no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—41no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—42no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—42no—
658C2×C4—44no—
659C2×C4—832no—
660C2×C4—816no—
661C2×C4#53514yes—
662C2×C4D4218yes—
663C22—28no—
664C22—24no—
665C22—44no—
666C22—44no—
667C22—42no—
668C22—42no—
669C22—44no—
670C22—48no—
671C22—48no—
672C22—48no—
673C22—48no—
674C22—48no—
675C22—48no—
676C22—48no—
677C22—48no—
678C22—84no—
679C22—84no—
680C22—84no—
681C22—84no—
682C22—84no—
683C22—84no—
684C22—84no—
685C22—84no—
686C22—84no—
687C22—84no—
688C22—84no—
689C22—82no—
690C22—82no—
691C22—84no—
692C22—84no—
693C22—84no—
694C22—82no—
695C22C2×#53511yescharacteristic
696C22C2×D4211yescharacteristic
697C22C2×D4212yes—
698C22#348411yescharacteristic
699C22#348514yes—
700C22#348711yescharacteristic
701C22#348712yes—
702C22#349614yes—
703C22#349812yes—
704C22#353412yes—
705C22#353412yes—
706C22#353914yes—
707C22#354214yes—
708C22#354311yescharacteristic
709C22#354512yes—
710C22#355112yes—
711C4—24no—
712C4—28no—
713C4—48no—
714C4—44no—
715C4—42no—
716C4—44no—
717C4—44no—
718C4—84no—
719C4—84no—
720C4—82no—
721C2—44no—
722C2—88no—
723C2—162no—
724C2#3671011yescharacteristic
725C2#3933912yes—
726C2#4301712yes—
727C2#4538712yes—
728C2#5190114yes—
729C2#5208311yescharacteristic
730C2#5485112yes—
731C2#5912611yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2673C2semidirect
2. C2×#2537C2semidirect
3. C2×#2501C2semidirect
4. #16551C2semidirect
5. #16508C2semidirect
6. #15643C2semidirect
7. #15380C2semidirect
8. #15368C2semidirect
9. #15310C2semidirect
10. #14811C2semidirect
11. #14719C2semidirect
12. #14424C2semidirect
13. #14175C2semidirect
136. #392D4semidirect
147. #384D4semidirect

Maximal subgroups

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

subgroupindex
C2×#26732
C2×#25372
C2×#25012
#165512
#165082
#156432
#153802
#153682
#153102
#148112
#147192
#144242
#141752