Finite Group Databasebeta

← #626261

Subgroups — #626261

Order 512.

3454 conjugacy classes of subgroups.

The subgroup lattice has 717 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2484C211yesmaximal, characteristic, complemented
2C2×#2406C211yesmaximal, characteristic
3C2×#2311C211yesmaximal, characteristic
4C22×#387C211yesmaximal, characteristic, complemented
5#15902C214yesmaximal, complemented
6#15867C214yesmaximal, complemented
7#15525C214yesmaximal, complemented
8#15512C214yesmaximal, complemented
9#14927C212yesmaximal, complemented
10#14921C212yesmaximal, complemented
11#14706C212yesmaximal, complemented
12#14339C211yesmaximal, characteristic, complemented
13#14259C212yesmaximal, complemented
14#13958C211yesmaximal, characteristic, complemented
15#13956C211yesmaximal, characteristic
16C23×#41C2211yescharacteristic
17C22×#123C2212yes—
18C22×#117C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C42C2211yescharacteristic
22#2484—28no—
23#2406—28no—
24C2×#400C2212yes—
25C2×#398C2212yes—
26C2×#395C2212yes—
27C2×#394C2212yes—
28#2311—28no—
29C2×#391C2212yes—
30C2×#390C2212yes—
31C2×#387—28no—
32C2×#387C2214yes—
33C2×#387C2214yes—
34C2×#387C2214yes—
35C2×#386C2212yes—
36C2×#386C2212yes—
37C2×#386C2212yes—
38C2×#386C2212yes—
39C2×#386C2212yes—
40C2×#385C2212yes—
41C2×#385C2214yes—
42C2×#381C2211yescharacteristic
43C2×#381C2211yescharacteristic
44C2×#381C2212yes—
45C2×#380C2212yes—
46C2×#380C2212yes—
47C2×#380C2212yes—
48C22×#95C2211yescharacteristic
49C22×#95C2211yescharacteristic
50C22×#95C2211yescharacteristic
51C22×#95C2211yescharacteristic
52#1446C2212yes—
53#1445C2212yes—
54#1445C2212yes—
55#1443C2214yes—
56#1443C2214yes—
57#1442C2214yes—
58#1442C2214yes—
59#1441C2214yes—
60#1441C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1440C2214yes—
64#1440C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2214yes—
68#1439C2214yes—
69#1438C2212yes—
70#1437C2214yes—
71#1437C2214yes—
72#1436C2214yes—
73#1436C2214yes—
74#1434C2214yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C22⋊#95C2212yes—
78C22⋊#95C2214yes—
79C22×#41—28no—
80C22×#41—24no—
81C22×#41—24no—
82C22×#41—24no—
83C22×#41—44no—
84C23×Q8C2311yescharacteristic
85C23×D4C2312yes—
86C23×D4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2312yes—
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C24×C4C2311yescharacteristic
92C24×C4C2311yescharacteristic
93C24×C4C2311yescharacteristic
94C24×C4C2311yescharacteristic
95C2×#123—28no—
96C2×#123—28no—
97C2×#123—44no—
98C2×#117—24no—
99C2×#117—24no—
100C2×#117—24no—
101C22×C4⋊C4—24no—
102C22×C4⋊C4—24no—
103C22×C4⋊C4—44no—
104C22×C4⋊C4D418yes—
105C22×C4⋊C4D418yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C42—24no—
127C22×C42—44no—
128C22×C42C2312yes—
129C22×C42C2314yes—
130#400—28no—
131#398—28no—
132#395—28no—
133#394—28no—
134#391—28no—
135#390—28no—
136#387—216no—
137#387—216no—
138#387—216no—
139#386—28no—
140#386—28no—
141#386—28no—
142#386—28no—
143#386—44no—
144#385—28no—
145#385—216no—
146#381—24no—
147#381—24no—
148#381—28no—
149#380—28no—
150#380—28no—
151#380—44no—
152C2×#95—24no—
153C2×#95—24no—
154C2×#95—24no—
155C2×#95—24no—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C25C2412yes—
183C25C2411yescharacteristic
184C2×#41—232no—
185C2×#41—432no—
186C2×#41—432no—
187C2×#41—416no—
188C2×#41—416no—
189C22×Q8—22no—
190C22×Q8—22no—
191C22×Q8—24no—
192C22×Q8—22no—
193C22×Q8—42no—
194C22×D4—24no—
195C22×D4—24no—
196C22×D4—24no—
197C22×D4—24no—
198C22×D4—24no—
199C22×D4—24no—
200C22×D4—24no—
201C22×D4—22no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4—42no—
205C22×D4#4118yes—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—22no—
222C23×C4—24no—
223C23×C4—24no—
224C23×C4—24no—
225C23×C4—24no—
226C23×C4—42no—
227C23×C4—42no—
228C23×C4—42no—
229C23×C4—42no—
230C23×C4—42no—
231C23×C4—42no—
232C23×C4—42no—
233C23×C4—42no—
234C23×C4—42no—
235C23×C4—42no—
236C23×C4—42no—
237C23×C4C2×D414yes—
238C23×C4C2×D414yes—
239C23×C4C2×D414yes—
240C23×C4C2×D414yes—
241C23×C4C2×D414yes—
242C23×C4C2×D414yes—
243C23×C4#4114yes—
244C23×C4C2412yes—
245C23×C4C2414yes—
246C23×C4C2414yes—
247C23×C4C2412yes—
248C23×C4C2412yes—
249C23×C4C2412yes—
250C23×C4C2412yes—
251C23×C4C2412yes—
252C23×C4C2412yes—
253C23×C4C2412yes—
254C23×C4C2412yes—
255C23×C4C2411yescharacteristic
256C23×C4C2411yescharacteristic
257#123—432no—
258#117—416no—
259C2×C4⋊C4—232no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—416no—
274C2×C4⋊C4—42no—
275C2×C4⋊C4—42no—
276C2×C4⋊C4—42no—
277C2×C4⋊C4—42no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4C2×D418yes—
285C2×C4⋊C4C2×D418yes—
286C2×C4⋊C4C2×D418yes—
287C2×C4⋊C4C2×D418yes—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—28no—
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⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—28no—
320C2×C42—24no—
321C2×C42—44no—
322C2×C42—44no—
323C2×C42—416no—
324#95—28no—
325#95—24no—
326#95—24no—
327#95—28no—
328#95—28no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95—24no—
335#95—28no—
336#95—28no—
337#95—24no—
338#95—24no—
339#95—28no—
340#95—28no—
341#95—28no—
342#95—28no—
343#95—44no—
344#95—44no—
345#95—44no—
346#95—44no—
347#95—44no—
348#95—44no—
349#95—44no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—22no—
360C24—24no—
361C24—24no—
362C24—24no—
363C24—24no—
364C24C22×D412yes—
365C24C22×D412yes—
366C24C2×#4114yes—
367C24C2×#4112yes—
368C24C2×#4114yes—
369C24H(5,𝔽2)14yes—
370C24C2511yescharacteristic
371#41—4128no—
372C2×Q8—28no—
373C2×Q8—48no—
374C2×Q8—48no—
375C2×Q8—44no—
376C2×Q8—44no—
377C2×D4—28no—
378C2×D4—216no—
379C2×D4—216no—
380C2×D4—216no—
381C2×D4—48no—
382C2×D4—48no—
383C2×D4—44no—
384C2×D4—48no—
385C2×D4—48no—
386C2×D4—48no—
387C2×D4—48no—
388C2×D4—44no—
389C22×C4—28no—
390C22×C4—28no—
391C22×C4—216no—
392C22×C4—216no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—28no—
398C22×C4—216no—
399C22×C4—28no—
400C22×C4—28no—
401C22×C4—28no—
402C22×C4—216no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—28no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—48no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—48no—
465C22×C4—48no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—416no—
469C22×C4—48no—
470C22×C4—48no—
471C22×C4—48no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—48no—
476C22×C4—416no—
477C22×C4—44no—
478C22×C4—44no—
479C22×C4—44no—
480C22×C4—48no—
481C22×C4—48no—
482C22×C4—44no—
483C22×C4—44no—
484C22×C4—44no—
485C22×C4—88no—
486C22×C4—88no—
487C22×C4#121116yes—
488C22×C4C22×D414yes—
489C22×C4C22×D414yes—
490C22×C4C22×D414yes—
491C22×C4C22×D414yes—
492C22×C4C22×D412yes—
493C22×C4C22×D412yes—
494C22×C4C22×D412yes—
495C22×C4C22×D412yes—
496C22×C4C2×#4114yes—
497C22×C4C2×#4112yes—
498C22×C4C2×#4112yes—
499C22×C42−1+414yes—
500C4⋊C4—216no—
501C4⋊C4—216no—
502C4⋊C4—48no—
503C4⋊C4—48no—
504C4⋊C4—416no—
505C4⋊C4—416no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—88no—
509C22⋊C4—416no—
510C22⋊C4—416no—
511C22⋊C4—48no—
512C22⋊C4—48no—
513C22⋊C4—48no—
514C22⋊C4—84no—
515C22⋊C4—84no—
516C22⋊C4—84no—
517C22⋊C4—84no—
518C22⋊C4—88no—
519C22⋊C4—88no—
520C42—416no—
521C42—48no—
522C23—24no—
523C23—24no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—24no—
529C23—24no—
530C23—28no—
531C23—28no—
532C23—28no—
533C23—28no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—28no—
538C23—28no—
539C23—42no—
540C23—42no—
541C23—44no—
542C23—44no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23C2×#12114yes—
558C23#52514yes—
559C23#53414yes—
560C23#54618yes—
561C23#55118yes—
562C23C23×D411yescharacteristic
563C23C23×D411yescharacteristic
564C23C22×#4111yescharacteristic
565C23C22×#4112yes—
566C23C2×H(5,𝔽2)12yes—
567C23C2×2−1+412yes—
568C23#58512yes—
569C23#58512yes—
570C23#58512yes—
571Q8—416no—
572D4—416no—
573D4—432no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—232no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—28no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—28no—
599C2×C4—432no—
600C2×C4—416no—
601C2×C4—416no—
602C2×C4—416no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
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—416no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—88no—
652C2×C4—88no—
653C2×C4C2×#12114yes—
654C2×C4C2×#12114yes—
655C22—24no—
656C22—24no—
657C22—24no—
658C22—24no—
659C22—24no—
660C22—24no—
661C22—216no—
662C22—216no—
663C22—44no—
664C22—44no—
665C22—44no—
666C22—48no—
667C22—48no—
668C22—82no—
669C22—82no—
670C22—82no—
671C22—82no—
672C22—84no—
673C22—84no—
674C22#253418yes—
675C22C22×#12111yescharacteristic
676C22C2×#52511yescharacteristic
677C22C2×#53411yescharacteristic
678C22#344912yes—
679C22#346112yes—
680C22C2×#54612yes—
681C22C2×#55112yes—
682C22#348511yescharacteristic
683C22#348511yescharacteristic
684C22#348611yescharacteristic
685C22#348711yescharacteristic
686C22#349312yes—
687C22#349712yes—
688C22#351514yes—
689C22#352114yes—
690C22#353314yes—
691C22#353514yes—
692C4—24no—
693C4—24no—
694C4—44no—
695C4—44no—
696C4—44no—
697C4—44no—
698C4—44no—
699C4—44no—
700C4—82no—
701C4—82no—
702C4—82no—
703C4—84no—
704C4—84no—
705C2—28no—
706C2—84no—
707C2C2×#253411yescharacteristic
708C2#3515011yescharacteristic
709C2#3515311yescharacteristic
710C2#3545311yescharacteristic
711C2#3545411yescharacteristic
712C2#3695511yescharacteristic
713C2#3718611yescharacteristic
714C2#3808714yes—
715C2#3903914yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×#41C22
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×#2484C2semidirect
4. C22×#387C2semidirect
5. #15902C2semidirect
6. #15867C2semidirect
7. #15525C2semidirect
8. #15512C2semidirect
9. #14927C2semidirect
10. #14921C2semidirect
11. #14706C2semidirect
12. #14339C2semidirect
13. #14259C2semidirect
14. #13958C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24842
C2×#24062
C2×#23112
C22×#3872
#159022
#158672
#155252
#155122
#149272
#149212
#147062
#143392
#142592
#139582
#139562