Finite Group Databasebeta

← #4308787

Subgroups — #4308787

Order 512.

3582 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×#2627C211yesmaximal, characteristic, complemented
2C22×#385C211yesmaximal, characteristic
3#16414C214yesmaximal, complemented
4#16343C214yesmaximal, complemented
5#16318C214yesmaximal, complemented
6#16306C214yesmaximal, complemented
7#14919C212yesmaximal, complemented
8#14918C212yesmaximal
9#14821C211yesmaximal, characteristic, complemented
10#14820C211yesmaximal, characteristic
11#14720C211yesmaximal, characteristic, complemented
12#14717C211yesmaximal, characteristic
13#14693C212yesmaximal
14#14342C211yesmaximal, characteristic, complemented
15#14265C212yesmaximal, complemented
16C25×C4C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C22⋊C4C2211yescharacteristic
21C23×C22⋊C4C2211yescharacteristic
22#2627—28no—
23C2×#400C2212yescomplemented
24C2×#398C2212yescomplemented
25C2×#398C2212yescomplemented
26C2×#398C2212yescomplemented
27C2×#397C2212yes—
28C2×#395C2212yescomplemented
29C2×#395C2212yescomplemented
30C2×#393C2212yes—
31C2×#385—28no—
32C2×#385C2214yes—
33C2×#385C2212yes—
34C2×#385C2212yes—
35C2×#385C2214yes—
36C2×#385C2212yes—
37C2×#385C2214yes—
38C2×#385C2214yes—
39C2×#381C2212yes—
40C2×#381C2212yes—
41C2×#380C2211yescharacteristic
42C2×#379C2212yes—
43C2×#379C2212yes—
44C22×#95C2211yescharacteristic
45C22×#95C2211yescharacteristic
46#1445C2214yescomplemented
47#1445C2214yescomplemented
48#1445C2214yescomplemented
49#1445C2214yescomplemented
50#1444C2214yescomplemented
51#1443C2214yescomplemented
52#1443C2214yescomplemented
53#1443C2214yescomplemented
54#1442C2214yescomplemented
55#1442C2214yescomplemented
56#1441C2212yescomplemented
57#1441C2214yescomplemented
58#1441C2212yescomplemented
59#1441C2214yescomplemented
60#1440C2214yes—
61#1440C2214yes—
62#1440C2214yes—
63#1439C2212yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2212yes—
68C22⋊#95C2212yes—
69C22⋊#95C2211yescharacteristic
70C22⋊#95C2211yescharacteristic
71C22⋊#95C2214yes—
72C22⋊#95C2214yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C26C2311yescharacteristic
78C23×Q8C2311yescharacteristic
79C24×C4—24no—
80C24×C4—44no—
81C24×C4—44no—
82C24×C4—44no—
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2314yes—
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C2×C22⋊Q8—24no—
91C2×C22⋊Q8—44no—
92C22×C4⋊C4—24no—
93C22×C4⋊C4—44no—
94C22×C4⋊C4—44no—
95C22×C4⋊C4D418yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C22⋊C4—24no—
109C22×C22⋊C4—24no—
110C22×C22⋊C4—24no—
111C22×C22⋊C4—24no—
112C22×C22⋊C4—44no—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124#400—28no—
125#398—28no—
126#398—28no—
127#398—44no—
128#397—44no—
129#395—28no—
130#395—44no—
131#393—44no—
132#385—28no—
133#385—28no—
134#385—216no—
135#385—216no—
136#385—48no—
137#385—44no—
138#385—48no—
139#381—28no—
140#381—44no—
141#380—24no—
142#379—28no—
143#379—44no—
144C2×#95—24no—
145C2×#95—24no—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C25—22no—
174C25—22no—
175C25—24no—
176C25—24no—
177C25—24no—
178C25—24no—
179C25—42no—
180C25—42no—
181C25#4114yes—
182C25C2412yes—
183C25C2411yescharacteristic
184C22×Q8—22no—
185C22×Q8—42no—
186C22×Q8—42no—
187C22×Q8—42no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—22no—
199C23×C4—28no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—28no—
203C23×C4—28no—
204C23×C4—42no—
205C23×C4—42no—
206C23×C4—42no—
207C23×C4—42no—
208C23×C4—44no—
209C23×C4—44no—
210C23×C4—44no—
211C23×C4—44no—
212C23×C4—42no—
213C23×C4—42no—
214C23×C4—42no—
215C23×C4—42no—
216C23×C4—42no—
217C23×C4—416no—
218C23×C4—416no—
219C23×C4—416no—
220C23×C4—464no—
221C23×C4C2×D414yes—
222C23×C4C2×D414yes—
223C23×C4C2×D414yes—
224C23×C4#4114yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2414yes—
230C23×C4C2414yes—
231C23×C4C2412yes—
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2412yes—
238C22⋊Q8—416no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—416no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—416no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4C2×D418yes—
271C2×C4⋊C4C2×D418yes—
272C2×C22⋊C4—232no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—42no—
299C2×C22⋊C4—42no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—416no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307#95—28no—
308#95—28no—
309#95—24no—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—28no—
317#95—28no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95—44no—
325#95—44no—
326#95—44no—
327#95—42no—
328#95—44no—
329#95—44no—
330#95—42no—
331#95—42no—
332#95—44no—
333#95—42no—
334C24—22no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—28no—
339C24—28no—
340C24—216no—
341C24—28no—
342C24—216no—
343C24—22no—
344C24—22no—
345C24—24no—
346C24—22no—
347C24—24no—
348C24—22no—
349C24—24no—
350C24—24no—
351C24—216no—
352C24—48no—
353C24—416no—
354C24—416no—
355C24—416no—
356C24—416no—
357C24—44no—
358C24—44no—
359C24—48no—
360C24—44no—
361C24—44no—
362C24—48no—
363C24C22×D412yes—
364C24C2×#4114yes—
365C24C2×#4112yes—
366C24C2×#4112yes—
367C24H(5,𝔽2)14yes—
368C24H(5,𝔽2)14yes—
369C242−1+414yes—
370C242−1+414yes—
371C24C2511yescharacteristic
372C2×Q8—44no—
373C2×Q8—82no—
374C2×Q8—82no—
375C2×Q8—88no—
376C22×C4—28no—
377C22×C4—28no—
378C22×C4—28no—
379C22×C4—28no—
380C22×C4—28no—
381C22×C4—28no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—28no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
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—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
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—24no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—48no—
451C22×C4—416no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—416no—
455C22×C4—416no—
456C22×C4—416no—
457C22×C4—416no—
458C22×C4—416no—
459C22×C4—44no—
460C22×C4—48no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—416no—
464C22×C4—416no—
465C22×C4—4128no—
466C22×C4—416no—
467C22×C4—44no—
468C22×C4—48no—
469C22×C4—48no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—88no—
474C22×C4—88no—
475C22×C4C22×D414yes—
476C22×C4C22×D414yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C2×#4114yes—
480C22×C4H(5,𝔽2)14yes—
481C4⋊C4—48no—
482C4⋊C4—48no—
483C4⋊C4—48no—
484C4⋊C4—48no—
485C4⋊C4—48no—
486C4⋊C4—44no—
487C4⋊C4—82no—
488C4⋊C4—82no—
489C4⋊C4—84no—
490C4⋊C4—84no—
491C4⋊C4—88no—
492C4⋊C4—88no—
493C22⋊C4—232no—
494C22⋊C4—48no—
495C22⋊C4—48no—
496C22⋊C4—416no—
497C22⋊C4—48no—
498C22⋊C4—48no—
499C22⋊C4—416no—
500C22⋊C4—48no—
501C22⋊C4—48no—
502C22⋊C4—88no—
503C22⋊C4—88no—
504C23—24no—
505C23—24no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—24no—
511C23—24no—
512C23—28no—
513C23—28no—
514C23—28no—
515C23—28no—
516C23—232no—
517C23—24no—
518C23—28no—
519C23—24no—
520C23—28no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—28no—
526C23—28no—
527C23—28no—
528C23—28no—
529C23—464no—
530C23—416no—
531C23—416no—
532C23—416no—
533C23—416no—
534C23—464no—
535C23—416no—
536C23—416no—
537C23—42no—
538C23—42no—
539C23—44no—
540C23—42no—
541C23—42no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23#52514yes—
548C23#53414yes—
549C23#53614yes—
550C23#53918yes—
551C23#53918yes—
552C23#54218yes—
553C23C23×D411yescharacteristic
554C23C22×#4111yescharacteristic
555C23C22×#4111yescharacteristic
556C23C2×H(5,𝔽2)12yes—
557C23C2×H(5,𝔽2)12yes—
558C23C2×H(5,𝔽2)12yes—
559C23C2×2−1+412yes—
560C23C2×2−1+412yes—
561C23#58512yes—
562Q8—88no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—28no—
580C2×C4—416no—
581C2×C4—416no—
582C2×C4—416no—
583C2×C4—464no—
584C2×C4—416no—
585C2×C4—42no—
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—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
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—44no—
639C2×C4—88no—
640C2×C4—88no—
641C22—24no—
642C22—24no—
643C22—24no—
644C22—24no—
645C22—216no—
646C22—216no—
647C22—24no—
648C22—24no—
649C22—216no—
650C22—28no—
651C22—464no—
652C22—44no—
653C22—44no—
654C22—44no—
655C22—44no—
656C22—44no—
657C22—48no—
658C22—48no—
659C22—48no—
660C22—44no—
661C22#263818yes—
662C22C2×#52511yescharacteristic
663C22C2×#53411yescharacteristic
664C22C2×#53611yescharacteristic
665C22C2×#53912yes—
666C22C2×#53912yes—
667C22C2×#54212yes—
668C22#345912yes—
669C22#348411yescharacteristic
670C22#348411yescharacteristic
671C22#348611yescharacteristic
672C22#348711yescharacteristic
673C22#349212yes—
674C22#350512yes—
675C22#353214yes—
676C22#353514yes—
677C22#353514yes—
678C22#353814yes—
679C4—44no—
680C4—44no—
681C4—44no—
682C4—42no—
683C4—42no—
684C4—44no—
685C4—82no—
686C4—82no—
687C4—82no—
688C4—84no—
689C4—84no—
690C4—82no—
691C4—82no—
692C2—28no—
693C2—48no—
694C2C2×#263811yescharacteristic
695C2#3498711yescharacteristic
696C2#3498911yescharacteristic
697C2#3670811yescharacteristic
698C2#3671111yescharacteristic
699C2#3671411yescharacteristic
700C2#3677814yes—
701C2#3718611yescharacteristic
702C2#3748614yes—
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
1. C2×#2627C2semidirect
3. #16414C2semidirect
4. #16343C2semidirect
5. #16318C2semidirect
6. #16306C2semidirect
7. #14919C2semidirect
9. #14821C2semidirect
11. #14720C2semidirect
14. #14342C2semidirect
15. #14265C2semidirect
23. C2×#400C22semidirect
24. C2×#398C22semidirect
25. C2×#398C22semidirect
26. C2×#398C22semidirect
28. C2×#395C22semidirect
29. C2×#395C22semidirect
46. #1445C22semidirect
47. #1445C22semidirect
48. #1445C22semidirect
49. #1445C22semidirect
50. #1444C22semidirect
51. #1443C22semidirect
52. #1443C22semidirect
53. #1443C22semidirect
54. #1442C22semidirect
55. #1442C22semidirect
56. #1441C22semidirect
57. #1441C22semidirect
58. #1441C22semidirect
59. #1441C22semidirect

Maximal subgroups

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

subgroupindex
C2×#26272
C22×#3852
#164142
#163432
#163182
#163062
#149192
#149182
#148212
#148202
#147202
#147172
#146932
#143422
#142652