Finite Group Databasebeta

← #846394

Subgroups — #846394

Order 512.

3642 conjugacy classes of subgroups.

The subgroup lattice has 704 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2561C211yesmaximal, characteristic
2C22×#385C211yesmaximal, characteristic
3#16481C212yesmaximal, complemented
4#16470C212yesmaximal, complemented
5#16039C214yesmaximal, complemented
6#15579C212yesmaximal, complemented
7#15568C212yesmaximal, complemented
8#15073C212yesmaximal, complemented
9#14934C212yesmaximal, complemented
10#14727C212yesmaximal
11#14681C212yesmaximal
12#14525C214yesmaximal, complemented
13#13992C212yesmaximal, complemented
14#13870C212yesmaximal, complemented
15#13868C211yesmaximal, characteristic
16C25×C4C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C23×C4⋊C4C2212yes—
20C23×C22⋊C4C2211yescharacteristic
21C23×C22⋊C4C2211yescharacteristic
22#2561—28no—
23C2×#397C2211yescharacteristic
24C2×#397C2211yescharacteristic
25C2×#397C2214yes—
26C2×#393C2211yescharacteristic
27C2×#393C2211yescharacteristic
28C2×#388C2212yes—
29C2×#388C2212yes—
30C2×#385—28no—
31C2×#385C2212yes—
32C2×#385C2212yes—
33C2×#385C2212yes—
34C2×#385C2212yes—
35C2×#385C2212yes—
36C2×#381C2212yes—
37C2×#381C2212yes—
38C2×#381C2211yescharacteristic
39C2×#381C2212yes—
40C2×#381C2212yes—
41C2×#381C2212yes—
42C2×#379C2212yes—
43C2×#379C2212yes—
44C2×C4×C22⋊C4C2211yescharacteristic
45C22×#95C2212yes—
46#1446C2214yescomplemented
47#1443C2214yescomplemented
48#1442C2214yescomplemented
49#1442C2214yescomplemented
50#1442C2214yescomplemented
51#1442C2214yescomplemented
52#1442C2214yescomplemented
53#1442C2214yescomplemented
54#1441C2214yescomplemented
55#1441C2214yescomplemented
56#1441C2214yescomplemented
57#1441C2214yescomplemented
58#1440C2214yes—
59#1440C2214yes—
60#1439C2212yes—
61#1439C2212yes—
62#1439C2214yes—
63#1437C2212yes—
64#1437C2212yes—
65#1436C2214yescomplemented
66#1436C2214yescomplemented
67#1436C2214yescomplemented
68#1434C2214yescomplemented
69C22⋊#95C2212yes—
70C22⋊#95C2212yes—
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2214yes—
74C22⋊#95C2214yes—
75C22⋊#95C2214yes—
76C22⋊#95C2214yes—
77C26C2311yescharacteristic
78C23×Q8C2311yescharacteristic
79C24×C4—24no—
80C24×C4—28no—
81C24×C4—44no—
82C24×C4—42no—
83C24×C4—42no—
84C24×C4C2311yescharacteristic
85C24×C4C2312yes—
86C24×C4C2312yes—
87C24×C4C2312yes—
88C24×C4C2312yes—
89C24×C4C2311yescharacteristic
90C2×C22⋊Q8—24no—
91C2×C22⋊Q8—42no—
92C2×C22⋊Q8—42no—
93C2×#117—44no—
94C2×#117—42no—
95C22×C4⋊C4—28no—
96C22×C4⋊C4—28no—
97C22×C4⋊C4—44no—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C22⋊C4—24no—
105C22×C22⋊C4—28no—
106C22×C22⋊C4—42no—
107C22×C22⋊C4—42no—
108C22×C22⋊C4D418yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2311yescharacteristic
122C22×C22⋊C4C2311yescharacteristic
123C22×C22⋊C4C2311yescharacteristic
124C22×C22⋊C4C2311yescharacteristic
125C22×C42C2312yes—
126C22×C42C2312yes—
127#397—24no—
128#397—24no—
129#397—216no—
130#393—24no—
131#393—24no—
132#388—28no—
133#388—28no—
134#385—28no—
135#385—28no—
136#385—28no—
137#385—28no—
138#385—44no—
139#381—28no—
140#381—28no—
141#381—44no—
142#381—44no—
143#381—44no—
144#381—42no—
145#379—28no—
146#379—44no—
147C4×C22⋊C4—24no—
148C2×#95—28no—
149C2×#95C2311yescharacteristic
150C2×#95C2311yescharacteristic
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C25—24no—
177C25—22no—
178C25—22no—
179C25—24no—
180C25—24no—
181C25—24no—
182C25—44no—
183C25C2×D414yes—
184C25C2411yescharacteristic
185C25C2411yescharacteristic
186C25C2411yescharacteristic
187C22×Q8—22no—
188C22×Q8—41no—
189C22×Q8—41no—
190C22×Q8—44no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—28no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—28no—
210C23×C4—28no—
211C23×C4—42no—
212C23×C4—42no—
213C23×C4—42no—
214C23×C4—42no—
215C23×C4—44no—
216C23×C4—44no—
217C23×C4—44no—
218C23×C4—416no—
219C23×C4—44no—
220C23×C4—432no—
221C23×C4—432no—
222C23×C4—416no—
223C23×C4—416no—
224C23×C4C2×D414yes—
225C23×C4C2×D414yes—
226C23×C4#4118yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2414yes—
231C23×C4C2414yes—
232C23×C4C2412yes—
233C23×C4C2411yescharacteristic
234C23×C4C2411yescharacteristic
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237C23×C4C2414yes—
238C23×C4C2414yes—
239C22⋊Q8—416no—
240#117—416no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—41no—
255C2×C4⋊C4—41no—
256C2×C4⋊C4—432no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C22⋊C4—232no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—416no—
285C2×C22⋊C4—41no—
286C2×C22⋊C4—41no—
287C2×C22⋊C4—41no—
288C2×C22⋊C4—41no—
289C2×C22⋊C4—41no—
290C2×C22⋊C4—41no—
291C2×C22⋊C4—41no—
292C2×C22⋊C4—41no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4C2×D414yes—
300C2×C22⋊C4C2×D414yes—
301C2×C22⋊C4C2×D414yes—
302C2×C22⋊C4C2×D414yes—
303C2×C22⋊C4#4118yes—
304C2×C42—24no—
305C2×C42—42no—
306C2×C42—44no—
307C2×C42—44no—
308#95—28no—
309#95—28no—
310#95—22no—
311#95—22no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—28no—
317#95—28no—
318#95—28no—
319#95—28no—
320#95—28no—
321#95—44no—
322#95—44no—
323#95—44no—
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—24no—
338C24—216no—
339C24—216no—
340C24—28no—
341C24—28no—
342C24—28no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—21no—
347C24—21no—
348C24—21no—
349C24—21no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—24no—
354C24—24no—
355C24—216no—
356C24—48no—
357C24—416no—
358C24—416no—
359C24—416no—
360C24—416no—
361C24—48no—
362C24—48no—
363C24—48no—
364C24—44no—
365C24—44no—
366C24C22×D412yes—
367C24C22×D412yes—
368C24C22×D412yes—
369C24C2×#4114yes—
370C24C2×#4114yes—
371C24H(5,𝔽2)12yes—
372C24H(5,𝔽2)12yes—
373C24H(5,𝔽2)14yes—
374C242−1+414yes—
375C24C2511yescharacteristic
376C2×Q8—44no—
377C2×Q8—84no—
378C2×Q8—84no—
379C2×Q8—84no—
380C22×C4—216no—
381C22×C4—28no—
382C22×C4—216no—
383C22×C4—216no—
384C22×C4—216no—
385C22×C4—216no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—28no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—48no—
448C22×C4—48no—
449C22×C4—48no—
450C22×C4—416no—
451C22×C4—416no—
452C22×C4—416no—
453C22×C4—416no—
454C22×C4—416no—
455C22×C4—416no—
456C22×C4—48no—
457C22×C4—48no—
458C22×C4—48no—
459C22×C4—4128no—
460C22×C4—48no—
461C22×C4—48no—
462C22×C4—48no—
463C22×C4—48no—
464C22×C4—48no—
465C22×C4—48no—
466C22×C4—48no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—48no—
470C22×C4—84no—
471C22×C4—84no—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C2×#4118yes—
477C22×C4C2×#4118yes—
478C4⋊C4—416no—
479C4⋊C4—416no—
480C4⋊C4—48no—
481C4⋊C4—44no—
482C4⋊C4—48no—
483C4⋊C4—88no—
484C22⋊C4—216no—
485C22⋊C4—216no—
486C22⋊C4—44no—
487C22⋊C4—44no—
488C22⋊C4—44no—
489C22⋊C4—44no—
490C22⋊C4—48no—
491C22⋊C4—48no—
492C22⋊C4—48no—
493C22⋊C4—48no—
494C22⋊C4—48no—
495C22⋊C4—48no—
496C22⋊C4—84no—
497C22⋊C4—84no—
498C22⋊C4—88no—
499C22⋊C4—88no—
500C42—48no—
501C42—84no—
502C23—22no—
503C23—22no—
504C23—28no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—28no—
509C23—28no—
510C23—24no—
511C23—24no—
512C23—28no—
513C23—232no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—24no—
519C23—24no—
520C23—28no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—24no—
527C23—28no—
528C23—28no—
529C23—28no—
530C23—432no—
531C23—432no—
532C23—432no—
533C23—416no—
534C23—432no—
535C23—464no—
536C23—42no—
537C23—42no—
538C23—416no—
539C23—41no—
540C23—41no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—41no—
545C23—41no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—44no—
551C23#52514yes—
552C23#53414yes—
553C23#53614yes—
554C23#53818yes—
555C23#54618yes—
556C23#55114yes—
557C23H(3,𝔽4)14yes—
558C23C23×D411yescharacteristic
559C23C22×#4112yes—
560C23C22×#4112yes—
561C23C2×H(5,𝔽2)11yescharacteristic
562C23C2×H(5,𝔽2)11yescharacteristic
563C23C2×H(5,𝔽2)12yes—
564C23C2×2−1+412yes—
565C23#58514yes—
566Q8—88no—
567C2×C4—28no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—28no—
573C2×C4—28no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—28no—
577C2×C4—28no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—28no—
581C2×C4—432no—
582C2×C4—432no—
583C2×C4—432no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—416no—
589C2×C4—432no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—88no—
639C22—28no—
640C22—28no—
641C22—24no—
642C22—28no—
643C22—28no—
644C22—24no—
645C22—216no—
646C22—28no—
647C22—28no—
648C22—24no—
649C22—24no—
650C22—464no—
651C22—44no—
652C22—44no—
653C22—44no—
654C22—42no—
655C22—42no—
656C22—42no—
657C22—42no—
658C22—44no—
659C22—44no—
660C22—44no—
661C22—48no—
662C22—48no—
663C22#268418yes—
664C22C2×#52511yescharacteristic
665C22C2×#53411yescharacteristic
666C22C2×#53611yescharacteristic
667C22C2×#53812yes—
668C22C2×#54612yes—
669C22C2×#55111yescharacteristic
670C22#347514yes—
671C22#348512yes—
672C22#349212yes—
673C22#349212yes—
674C22#349514yes—
675C22#351314yes—
676C22C2×H(3,𝔽4)11yescharacteristic
677C22#353312yes—
678C22#353312yes—
679C22#353514yes—
680C4—48no—
681C4—42no—
682C4—42no—
683C4—42no—
684C4—42no—
685C4—48no—
686C4—82no—
687C4—82no—
688C4—82no—
689C4—84no—
690C4—82no—
691C4—84no—
692C2—28no—
693C2—44no—
694C2—44no—
695C2#3505711yescharacteristic
696C2#3615912yes—
697C2#3696511yescharacteristic
698C2#3783512yes—
699C2#3839914yes—
700C2#4553912yes—
701C2C2×#268411yescharacteristic
702C2#5919412yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC26C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #16481C2semidirect
4. #16470C2semidirect
5. #16039C2semidirect
6. #15579C2semidirect
7. #15568C2semidirect
8. #15073C2semidirect
9. #14934C2semidirect
12. #14525C2semidirect
13. #13992C2semidirect
14. #13870C2semidirect
46. #1446C22semidirect
47. #1443C22semidirect
48. #1442C22semidirect
49. #1442C22semidirect
50. #1442C22semidirect
51. #1442C22semidirect
52. #1442C22semidirect
53. #1442C22semidirect
54. #1441C22semidirect
55. #1441C22semidirect
56. #1441C22semidirect
57. #1441C22semidirect
65. #1436C22semidirect
66. #1436C22semidirect
67. #1436C22semidirect
68. #1434C22semidirect

Maximal subgroups

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

subgroupindex
C2×#25612
C22×#3852
#164812
#164702
#160392
#155792
#155682
#150732
#149342
#147272
#146812
#145252
#139922
#138702
#138682