Finite Group Databasebeta

← #647560

Subgroups — #647560

Order 512.

2730 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×#2851C211yesmaximal, characteristic, complemented
2C2×C4×#121C211yesmaximal, characteristic
3#17046C212yesmaximal, complemented
4#16815C212yesmaximal, complemented
5#16682C212yesmaximal, complemented
6#16501C212yesmaximal
7#15111C211yesmaximal, characteristic, complemented
8#15106C211yesmaximal, characteristic, complemented
9#14868C211yesmaximal, characteristic, complemented
10#14842C212yesmaximal, complemented
11#14839C212yesmaximal, complemented
12#14807C211yesmaximal, characteristic, complemented
13#14731C214yesmaximal, complemented
14#14328C212yesmaximal, complemented
15#14325C212yesmaximal, complemented
16#14075C214yesmaximal, complemented
17C4×#381C211yesmaximal, characteristic, complemented
18C22×#125C2211yescharacteristic
19C22×#121C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C22×C4×D4C2212yes—
22C23×C42C2211yescharacteristic
23#2851—28no—
24C2×#399C2212yes—
25C2×#398C2212yes—
26C2×#397C2212yes—
27C2×#397C2214yes—
28C2×#394C2212yes—
29C2×#394C2214yes—
30C4×#121—28no—
31C2×#390C2211yescharacteristic
32C2×#389C2211yescharacteristic
33C2×#389C2211yescharacteristic
34C2×#389C2211yescharacteristic
35C2×#388C2212yes—
36C2×#387C2212yes—
37C2×#387C2212yes—
38C2×#386C2212yes—
39C2×#385C2212yes—
40C2×#385C2212yes—
41C2×#385C2214yes—
42C2×#382C2211yescharacteristic
43C2×#381C2212yes—
44C2×#380C2211yescharacteristic
45C2×#380C2211yescharacteristic
46C2×#380C2211yescharacteristic
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4C2212yes—
49C2×C4×C22⋊C4C2212yes—
50#1445C2212yes—
51#1445C2212yes—
52#1445C2212yes—
53#1444C2212yes—
54#1444C2214yes—
55#1444C2212yes—
56#1443C2214yes—
57#1442C2212yes—
58#1442C2212yes—
59#1441C2212yes—
60#1441C2212yes—
61#1441C2214yes—
62#1441C2214yes—
63#1440C2214yes—
64#1440C2214yes—
65#1439C2212yes—
66#1439C2214yes—
67#1439C2214yes—
68#1439C2214yes—
69#1439C2212yes—
70#1438C2214yes—
71#1438C2212yes—
72#1437C2214yes—
73#1437C2214yes—
74#1436C2214yes—
75#1436C2214yes—
76C4⋊#95C2214yes—
77#1434C2211yescharacteristic
78#1434C2211yescharacteristic
79#1434C2212yes—
80#1434C2214yes—
81C22⋊#95C2214yes—
82C22⋊#95C2212yes—
83C4×#95C2212yes—
84C4×#95C2212yes—
85C23×D4C2312yes—
86C23×D4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2312yes—
90C24×C4C2311yescharacteristic
91C2×#125—24no—
92C2×#125—44no—
93C2×#121—24no—
94C2×#121—24no—
95C2×#121—24no—
96C2×C4×D4—28no—
97C2×C4×D4—24no—
98C2×C4×D4—44no—
99C2×C4×D4—48no—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2312yes—
126C22×C42—44no—
127C22×C42D418yes—
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2312yes—
131C22×C42C2314yes—
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134#399—44no—
135#398—28no—
136#397—44no—
137#397—48no—
138#394—28no—
139#394—48no—
140#390—24no—
141#389—24no—
142#389—24no—
143#389—24no—
144#388—28no—
145#387—28no—
146#387—28no—
147#386—28no—
148#385—28no—
149#385—28no—
150#385—216no—
151#382—24no—
152#381—28no—
153#380—24no—
154#380—24no—
155#380—24no—
156C4×C4⋊C4—24no—
157C4×C22⋊C4—28no—
158C4×C22⋊C4—44no—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2312yes—
186C25C2412yes—
187C25C2411yescharacteristic
188C22×D4—24no—
189C22×D4—24no—
190C22×D4—28no—
191C22×D4—22no—
192C22×D4—24no—
193C22×D4—22no—
194C22×D4—42no—
195C22×D4—42no—
196C22×D4—44no—
197C22×D4#4114yes—
198C23×C4—24no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—28no—
206C23×C4—42no—
207C23×C4—42no—
208C23×C4—42no—
209C23×C4—42no—
210C23×C4—42no—
211C23×C4—42no—
212C23×C4—44no—
213C23×C4C2×D414yes—
214C23×C4C2×D414yes—
215C23×C4C2×D414yes—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2414yes—
222C23×C4C2412yes—
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C23×C4C2414yes—
226C23×C4C2412yes—
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C23×C4C2411yescharacteristic
231#125—416no—
232#121—232no—
233C4×D4—416no—
234C4×D4—432no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4#4114yes—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
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—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
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×C42—24no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—28no—
303C2×C42—416no—
304C2×C42—44no—
305C2×C42—42no—
306C2×C42—42no—
307C2×C42—44no—
308C2×C42—44no—
309C2×C42C2×D414yes—
310C2×C42C2×D414yes—
311C2×C42C2×D414yes—
312C2×C42C2×D414yes—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—24no—
317#95—24no—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—24no—
322#95—28no—
323#95—24no—
324#95—28no—
325#95—24no—
326#95—24no—
327#95—28no—
328#95—24no—
329#95—24no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95#4118yes—
340C24—22no—
341C24—21no—
342C24—21no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—24no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—24no—
353C24—24no—
354C24—22no—
355C24—22no—
356C24—24no—
357C24C22×D412yes—
358C24C2×#4114yes—
359C24H(5,𝔽2)14yes—
360C24C2511yescharacteristic
361C2×D4—216no—
362C2×D4—28no—
363C2×D4—48no—
364C2×D4—416no—
365C2×D4—44no—
366C2×D4—44no—
367C2×D4—48no—
368C2×D4—416no—
369C2×D4—48no—
370C2×D4—48no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—28no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—28no—
398C22×C4—28no—
399C22×C4—28no—
400C22×C4—28no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—416no—
445C22×C4—416no—
446C22×C4—48no—
447C22×C4—48no—
448C22×C4—48no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4—88no—
452C22×C4—88no—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C2×#4118yes—
460C22×C4C2×#4114yes—
461C22×C4C2×#4112yes—
462C22×C4C2×#4112yes—
463C22×C4C2×#4114yes—
464C22×C4H(5,𝔽2)14yes—
465C22×C42−1+414yes—
466C4⋊C4—28no—
467C4⋊C4—44no—
468C4⋊C4—44no—
469C4⋊C4—44no—
470C4⋊C4—44no—
471C4⋊C4—48no—
472C4⋊C4—48no—
473C4⋊C4—48no—
474C4⋊C4—48no—
475C4⋊C4—48no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—88no—
479C4⋊C4—88no—
480C22⋊C4—216no—
481C22⋊C4—416no—
482C22⋊C4—416no—
483C22⋊C4—48no—
484C22⋊C4—48no—
485C22⋊C4—48no—
486C22⋊C4—48no—
487C22⋊C4—48no—
488C22⋊C4—48no—
489C22⋊C4—88no—
490C22⋊C4—88no—
491C22⋊C4—88no—
492C42—28no—
493C42—28no—
494C42—44no—
495C42—44no—
496C42—48no—
497C42—88no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—24no—
505C23—24no—
506C23—28no—
507C23—28no—
508C23—24no—
509C23—28no—
510C23—24no—
511C23—24no—
512C23—28no—
513C23—44no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—44no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—42no—
529C23—42no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23#53514yes—
538C23C23×D411yescharacteristic
539C23C22×#4111yescharacteristic
540C23C22×#4111yescharacteristic
541C23C22×#4112yes—
542C23C2×H(5,𝔽2)12yes—
543C23C2×H(5,𝔽2)12yes—
544C23C2×2−1+412yes—
545C23#58512yes—
546C23#58512yes—
547D4—416no—
548D4—432no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—28no—
560C2×C4—28no—
561C2×C4—28no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—28no—
570C2×C4—28no—
571C2×C4—28no—
572C2×C4—28no—
573C2×C4—416no—
574C2×C4—416no—
575C2×C4—416no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
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—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—816no—
626C2×C4#52514yes—
627C2×C4#53214yes—
628C2×C4#53414yes—
629C2×C4#53614yes—
630C22—24no—
631C22—24no—
632C22—24no—
633C22—42no—
634C22—42no—
635C22—48no—
636C22—48no—
637C22—42no—
638C22—42no—
639C22—48no—
640C22—48no—
641C22—44no—
642C22—44no—
643C22—44no—
644C22—82no—
645C22—82no—
646C22—82no—
647C22—84no—
648C22—84no—
649C22—82no—
650C22C2×#52511yescharacteristic
651C22C2×#53211yescharacteristic
652C22C2×#53411yescharacteristic
653C22C2×#53511yescharacteristic
654C22C2×#53611yescharacteristic
655C22#344911yescharacteristic
656C22#344911yescharacteristic
657C22#345514yes—
658C22#348212yes—
659C22#348212yes—
660C22#350512yes—
661C22#350612yes—
662C22#351312yes—
663C22#351312yes—
664C22#351914yes—
665C22#353514yes—
666C22#355914yes—
667C4—24no—
668C4—24no—
669C4—24no—
670C4—24no—
671C4—42no—
672C4—48no—
673C4—44no—
674C4—42no—
675C4—82no—
676C4—82no—
677C4—82no—
678C4—84no—
679C4—82no—
680C4—82no—
681C4—82no—
682C2—44no—
683C2—84no—
684C2#3464311yescharacteristic
685C2#3497911yescharacteristic
686C2#3503111yescharacteristic
687C2#3507811yescharacteristic
688C2#3523211yescharacteristic
689C2#3744711yescharacteristic
690C2#3753911yescharacteristic
691C2#4006414yes—
692C2#4008814yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
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×#2851C2semidirect
3. #17046C2semidirect
4. #16815C2semidirect
5. #16682C2semidirect
7. #15111C2semidirect
8. #15106C2semidirect
9. #14868C2semidirect
10. #14842C2semidirect
11. #14839C2semidirect
12. #14807C2semidirect
13. #14731C2semidirect
14. #14328C2semidirect
15. #14325C2semidirect
16. #14075C2semidirect
17. C4×#381C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28512
C2×C4×#1212
#170462
#168152
#166822
#165012
#151112
#151062
#148682
#148422
#148392
#148072
#147312
#143282
#143252
#140752
C4×#3812