Finite Group Databasebeta

← #4353731

Subgroups — #4353731

Order 512.

3216 conjugacy classes of subgroups.

The subgroup lattice has 716 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2544C211yesmaximal, characteristic
2C2×#2409C211yesmaximal, characteristic, complemented
3#16884C212yesmaximal, complemented
4#16559C212yesmaximal, complemented
5#16368C214yesmaximal, complemented
6#15854C212yesmaximal, complemented
7#15662C212yesmaximal, complemented
8#15405C214yesmaximal, complemented
9#15089C212yesmaximal, complemented
10#14873C212yesmaximal, complemented
11#14765C212yesmaximal, complemented
12#14759C212yesmaximal, complemented
13#14473C212yesmaximal, complemented
14#14419C212yesmaximal
15#13784C211yesmaximal, characteristic, complemented
16C24×Q8C2211yescharacteristic
17C22×#127C2211yescharacteristic
18C22×#124C2211yescharacteristic
19C22×#124C2211yescharacteristic
20C22×C22⋊Q8C2212yes—
21C22×C4×D4C2211yescharacteristic
22#2544—28no—
23#2409—28no—
24C2×#395C2212yes—
25C2×#394C2212yes—
26C2×#394C2212yes—
27C2×#393C2212yes—
28C2×#393C2214yes—
29C2×#393C2212yes—
30C2×#393C2212yes—
31C2×#391C2211yescharacteristic
32C2×#391C2212yes—
33C2×#390C2211yescharacteristic
34C2×#388C2214yes—
35C2×#386C2212yes—
36C2×#386C2212yes—
37C2×#386C2212yes—
38C2×#386C2212yes—
39C2×#386C2214yes—
40C2×#386C2214yes—
41C2×#385C2212yes—
42C2×#380C2211yescharacteristic
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C22⋊C4C2211yescharacteristic
45C22×#95C2211yescharacteristic
46#1446C2212yes—
47#1446C2212yes—
48#1446C2214yes—
49#1446C2214yes—
50#1445C2214yes—
51#1443C2214yes—
52#1442C2214yes—
53#1442C2214yes—
54#1442C2212yes—
55#1442C2212yes—
56#1440C2214yes—
57#1440C2214yes—
58#1440C2214yes—
59#1440C2214yes—
60#1440C2214yes—
61#1439C2214yes—
62#1439C2212yes—
63#1439C2212yes—
64#1438C2214yes—
65#1438C2214yes—
66#1438C2214yes—
67#1437C2212yes—
68#1436C2214yes—
69#1436C2212yes—
70#1434C2214yes—
71#1434C2214yes—
72#1434C2212yes—
73C22⋊#95C2214yes—
74C22⋊#95C2212yes—
75C22⋊#95C2214yes—
76C22⋊#95C2212yes—
77C23×Q8—24no—
78C23×Q8—42no—
79C23×Q8—42no—
80C23×Q8—48no—
81C23×Q8C2312yes—
82C23×Q8C2312yes—
83C23×Q8C2311yescharacteristic
84C23×D4C2312yes—
85C23×D4C2312yes—
86C23×D4C2311yescharacteristic
87C23×D4C2311yescharacteristic
88C24×C4C2312yes—
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C24×C4C2312yes—
92C2×#127—24no—
93C2×#127—28no—
94C2×#124—24no—
95C2×#124—28no—
96C2×#124—24no—
97C2×#124—42no—
98C2×#124—42no—
99C2×C22⋊Q8—28no—
100C2×C22⋊Q8—48no—
101C2×C4×D4—42no—
102C2×C4×D4—42no—
103C2×C4×D4D418yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2311yescharacteristic
123C22×C22⋊C4C2311yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2312yes—
128#395—28no—
129#394—28no—
130#394—28no—
131#393—28no—
132#393—44no—
133#393—44no—
134#393—48no—
135#391—24no—
136#391—28no—
137#390—24no—
138#388—48no—
139#386—28no—
140#386—28no—
141#386—28no—
142#386—28no—
143#386—48no—
144#386—48no—
145#385—28no—
146#380—24no—
147C4×C4⋊C4—24no—
148C4×C22⋊C4—42no—
149C2×#95—42no—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2314yes—
174C25C2412yes—
175C25C2411yescharacteristic
176C25C2411yescharacteristic
177C22×Q8—24no—
178C22×Q8—22no—
179C22×Q8—24no—
180C22×Q8—24no—
181C22×Q8—28no—
182C22×Q8—28no—
183C22×Q8—24no—
184C22×Q8—24no—
185C22×Q8—44no—
186C22×Q8—44no—
187C22×Q8—42no—
188C22×Q8—42no—
189C22×Q8—41no—
190C22×Q8—41no—
191C22×Q8—42no—
192C22×Q8—416no—
193C22×Q8—432no—
194C22×Q8—432no—
195C22×Q8—432no—
196C22×D4—24no—
197C22×D4—24no—
198C22×D4—28no—
199C22×D4—28no—
200C22×D4—22no—
201C22×D4—24no—
202C22×D4—24no—
203C22×D4—24no—
204C22×D4—41no—
205C22×D4—41no—
206C22×D4—42no—
207C22×D4—41no—
208C22×D4—41no—
209C22×D4—42no—
210C22×D4—42no—
211C22×D4—42no—
212C22×D4—44no—
213C22×D4—44no—
214C22×D4C2×D414yes—
215C22×D4#4118yes—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—22no—
220C23×C4—24no—
221C23×C4—24no—
222C23×C4—28no—
223C23×C4—44no—
224C23×C4—42no—
225C23×C4—42no—
226C23×C4—44no—
227C23×C4—44no—
228C23×C4—42no—
229C23×C4—42no—
230C23×C4—44no—
231C23×C4—44no—
232C23×C4C2×D414yes—
233C23×C4C2×D414yes—
234C23×C4C2414yes—
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2414yes—
238C23×C4C2414yes—
239C23×C4C2411yescharacteristic
240C23×C4C2411yescharacteristic
241C23×C4C2411yescharacteristic
242C23×C4C2414yes—
243C23×C4C2412yes—
244C23×C4C2411yescharacteristic
245C23×C4C2411yescharacteristic
246#127—416no—
247#124—416no—
248#124—416no—
249C22⋊Q8—432no—
250C4×D4—416no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—41no—
258C2×C4⋊C4—41no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4C2×D414yes—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—42no—
280C2×C22⋊C4—42no—
281C2×C22⋊C4—42no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—41no—
285C2×C22⋊C4—41no—
286C2×C22⋊C4—41no—
287C2×C22⋊C4—41no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—42no—
290C2×C22⋊C4—42no—
291C2×C22⋊C4—42no—
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×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4C2×D414yes—
305C2×C22⋊C4C2×D414yes—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—42no—
313C2×C42—44no—
314C2×C42C2×D414yes—
315#95—24no—
316#95—24no—
317#95—28no—
318#95—24no—
319#95—24no—
320#95—28no—
321#95—28no—
322#95—24no—
323#95—24no—
324#95—28no—
325#95—28no—
326#95—28no—
327#95—42no—
328#95—44no—
329#95—44no—
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—
339C24—24no—
340C24—21no—
341C24—21no—
342C24—21no—
343C24—21no—
344C24—24no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—24no—
353C24—22no—
354C24—22no—
355C24—24no—
356C24—24no—
357C24—22no—
358C24—22no—
359C24—24no—
360C24—22no—
361C24—22no—
362C24C22×D412yes—
363C24C22×D412yes—
364C24C2×#4114yes—
365C24C2×#4114yes—
366C24C2511yescharacteristic
367C2×Q8—216no—
368C2×Q8—416no—
369C2×Q8—416no—
370C2×Q8—416no—
371C2×Q8—48no—
372C2×Q8—48no—
373C2×Q8—48no—
374C2×Q8—416no—
375C2×Q8—44no—
376C2×Q8—44no—
377C2×Q8—4128no—
378C2×Q8—48no—
379C2×Q8—44no—
380C2×Q8—48no—
381C2×Q8—84no—
382C2×Q8—84no—
383C2×D4—44no—
384C2×D4—44no—
385C2×D4—48no—
386C2×D4—48no—
387C2×D4—48no—
388C2×D4—48no—
389C2×D4—416no—
390C2×D4—416no—
391C2×D4—44no—
392C2×D4—44no—
393C2×D4—48no—
394C2×D4—48no—
395C2×D4—44no—
396C2×D4—84no—
397C2×D4—84no—
398C2×D4—84no—
399C2×D4—84no—
400C2×D4—88no—
401C2×D4—88no—
402C2×D4#124116yes—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—21no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—24no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—28no—
428C22×C4—28no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
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—22no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—48no—
461C22×C4—48no—
462C22×C4—48no—
463C22×C4—416no—
464C22×C4—48no—
465C22×C4—48no—
466C22×C4—416no—
467C22×C4—48no—
468C22×C4—48no—
469C22×C4—48no—
470C22×C4—84no—
471C22×C4—84no—
472C22×C4—88no—
473C22×C4—84no—
474C22×C4—88no—
475C22×C4—84no—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C2×#4114yes—
482C22×C4H(5,𝔽2)18yes—
483C22×C4H(5,𝔽2)12yes—
484C22×C4H(5,𝔽2)12yes—
485C22×C42−1+414yes—
486C4⋊C4—44no—
487C4⋊C4—48no—
488C4⋊C4—416no—
489C4⋊C4—48no—
490C22⋊C4—44no—
491C22⋊C4—44no—
492C22⋊C4—416no—
493C22⋊C4—48no—
494C22⋊C4—48no—
495C22⋊C4—48no—
496C22⋊C4—416no—
497C22⋊C4—48no—
498C22⋊C4—48no—
499C22⋊C4—48no—
500C22⋊C4—84no—
501C22⋊C4—84no—
502C22⋊C4—88no—
503C22⋊C4—88no—
504C22⋊C4—88no—
505C22⋊C4—88no—
506C42—28no—
507C42—44no—
508C42—44no—
509C42—84no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—28no—
523C23—28no—
524C23—28no—
525C23—28no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—44no—
531C23—44no—
532C23—42no—
533C23—42no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—44no—
539C23—44no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—44no—
562C23—44no—
563C23C2×#12414yes—
564C23C2×#12414yes—
565C23C23×D411yescharacteristic
566C23C22×#4112yes—
567C23C2×H(5,𝔽2)11yescharacteristic
568C23C2×H(5,𝔽2)11yescharacteristic
569C23C2×H(5,𝔽2)14yes—
570C23C2×2−1+412yes—
571C23#58512yes—
572C23#58512yes—
573Q8—432no—
574Q8—432no—
575Q8—88no—
576D4—416no—
577D4—88no—
578D4—816no—
579D4—816no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—28no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—28no—
596C2×C4—28no—
597C2×C4—28no—
598C2×C4—416no—
599C2×C4—432no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—42no—
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—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—88no—
646C2×C4—816no—
647C2×C4C2×#12414yes—
648C2×C4#53414yes—
649C22—28no—
650C22—28no—
651C22—24no—
652C22—24no—
653C22—42no—
654C22—42no—
655C22—48no—
656C22—48no—
657C22—42no—
658C22—42no—
659C22—44no—
660C22—44no—
661C22—44no—
662C22—44no—
663C22—44no—
664C22—44no—
665C22—82no—
666C22—84no—
667C22—84no—
668C22—82no—
669C22—84no—
670C22—82no—
671C22—82no—
672C22—84no—
673C22—82no—
674C22—82no—
675C22—82no—
676C22C22×#12411yescharacteristic
677C22C2×#53411yescharacteristic
678C22#344911yescharacteristic
679C22#348412yes—
680C22#348912yes—
681C22#350512yes—
682C22#350512yes—
683C22#351414yes—
684C22#352014yes—
685C22#353212yes—
686C22#353212yes—
687C22#354014yes—
688C22#354712yes—
689C22#355011yescharacteristic
690C22#355111yescharacteristic
691C22#355414yes—
692C4—24no—
693C4—24no—
694C4—42no—
695C4—42no—
696C4—48no—
697C4—42no—
698C4—44no—
699C4—84no—
700C4—84no—
701C4—82no—
702C4—84no—
703C4—82no—
704C2—44no—
705C2—44no—
706C2—162no—
707C2#3501911yescharacteristic
708C2#3593912yes—
709C2#3670211yescharacteristic
710C2#3737611yescharacteristic
711C2#4060712yes—
712C2#4711514yes—
713C2#4826512yes—
714C2#5027412yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#127C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2409C2semidirect
3. #16884C2semidirect
4. #16559C2semidirect
5. #16368C2semidirect
6. #15854C2semidirect
7. #15662C2semidirect
8. #15405C2semidirect
9. #15089C2semidirect
10. #14873C2semidirect
11. #14765C2semidirect
12. #14759C2semidirect
13. #14473C2semidirect
15. #13784C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25442
C2×#24092
#168842
#165592
#163682
#158542
#156622
#154052
#150892
#148732
#147652
#147592
#144732
#144192
#137842