Finite Group Databasebeta

← #4417441

Subgroups — #4417441

Order 512.

2544 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2850C211yesmaximal, characteristic
2C2×C4×C22⋊Q8C211yesmaximal, characteristic, complemented
3#17048C212yesmaximal, complemented
4#16807C212yesmaximal, complemented
5#16681C212yesmaximal, complemented
6#16632C212yesmaximal, complemented
7#16262C212yesmaximal, complemented
8#16259C212yesmaximal
9#15162C214yesmaximal, complemented
10#15112C211yesmaximal, characteristic, complemented
11#15104C211yesmaximal, characteristic, complemented
12#15095C214yesmaximal, complemented
13#14883C211yesmaximal, characteristic, complemented
14#14881C211yesmaximal, characteristic, complemented
15#14684C211yesmaximal, characteristic, complemented
16#14630C212yesmaximal, complemented
17#14625C212yesmaximal, complemented
18C22×#124C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20C22×C22⋊Q8C2211yescharacteristic
21C22×#121C2211yescharacteristic
22C22×C4×Q8C2211yescharacteristic
23C23×C42C2211yescharacteristic
24#2850—28no—
25C2×#397C2212yes—
26C2×#397C2214yes—
27C2×#395C2212yes—
28C2×#394C2212yes—
29C2×#394C2214yes—
30C2×#393C2212yes—
31C4×C22⋊Q8—28no—
32C2×#391C2211yescharacteristic
33C2×#391C2211yescharacteristic
34C2×#390C2211yescharacteristic
35C2×#389C2211yescharacteristic
36C2×#388C2212yes—
37C2×#388C2212yes—
38C2×#387C2212yes—
39C2×#386C2212yes—
40C2×#386C2212yes—
41C2×#385C2212yes—
42C2×#382C2211yescharacteristic
43C2×#381C2212yes—
44C2×#380C2211yescharacteristic
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C4⋊C4C2212yes—
47C2×C4×C22⋊C4C2212yes—
48#1446C2212yes—
49#1446C2212yes—
50#1445C2212yes—
51#1445C2214yes—
52#1444C2212yes—
53#1444C2212yes—
54#1444C2212yes—
55#1444C2214yes—
56#1443C2214yes—
57#1442C2212yes—
58#1442C2214yes—
59#1442C2214yes—
60#1442C2214yes—
61#1441C2212yes—
62#1441C2212yes—
63#1440C2214yes—
64#1440C2214yes—
65#1439C2212yes—
66#1439C2214yes—
67#1439C2212yes—
68#1439C2214yes—
69#1439C2214yes—
70#1439C2214yes—
71#1439C2214yes—
72#1438C2212yes—
73#1438C2214yes—
74#1437C2214yes—
75#1437C2214yes—
76#1436C2214yes—
77#1436C2214yes—
78#1434C2211yescharacteristic
79#1434C2211yescharacteristic
80#1434C2212yes—
81#1434C2212yes—
82#1434C2212yes—
83C22⋊#95C2212yes—
84C22⋊#95C2212yes—
85C23×Q8C2311yescharacteristic
86C23×Q8C2311yescharacteristic
87C23×D4C2311yescharacteristic
88C23×D4C2312yes—
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C24×C4C2311yescharacteristic
92C2×#124—24no—
93C2×#124—44no—
94C2×C22⋊Q8—24no—
95C2×C22⋊Q8—24no—
96C2×C22⋊Q8—24no—
97C2×C22⋊Q8—24no—
98C2×C22⋊Q8—44no—
99C2×#121—24no—
100C2×#121—44no—
101C2×C4×Q8—24no—
102C2×C4×Q8—44no—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2314yes—
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⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C42—44no—
128C22×C42D418yes—
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131C22×C42C2312yes—
132C22×C42C2312yes—
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135#397—28no—
136#397—48no—
137#395—44no—
138#394—44no—
139#394—48no—
140#393—28no—
141#391—24no—
142#391—24no—
143#390—24no—
144#389—24no—
145#388—28no—
146#388—28no—
147#387—28no—
148#386—28no—
149#386—28no—
150#385—28no—
151#382—24no—
152#381—44no—
153#380—24no—
154C4×C4⋊C4—28no—
155C4×C4⋊C4—24no—
156C4×C22⋊C4—28no—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C25C2412yes—
186C25C2411yescharacteristic
187C22×Q8—22no—
188C22×Q8—22no—
189C22×Q8—22no—
190C22×Q8—24no—
191C22×Q8—42no—
192C22×Q8—42no—
193C22×Q8—42no—
194C22×Q8—42no—
195C22×Q8#4114yes—
196C22×D4—24no—
197C22×D4—28no—
198C22×D4—22no—
199C22×D4—24no—
200C22×D4—42no—
201C22×D4—42no—
202C22×D4—42no—
203C22×D4—44no—
204C22×D4—42no—
205C23×C4—24no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—42no—
211C23×C4—42no—
212C23×C4—42no—
213C23×C4—42no—
214C23×C4—42no—
215C23×C4C2×D414yes—
216C23×C4C2×D414yes—
217C23×C4C2×D414yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2414yes—
223C23×C4C2414yes—
224C23×C4C2412yes—
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2411yescharacteristic
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233#124—416no—
234C22⋊Q8—232no—
235C22⋊Q8—416no—
236#121—416no—
237C4×Q8—416no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4#4114yes—
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—28no—
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—28no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
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×C42—24no—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—416no—
303C2×C42—42no—
304C2×C42—42no—
305C2×C42—42no—
306C2×C42—42no—
307C2×C42—44no—
308C2×C42C2×D414yes—
309C2×C42C2×D414yes—
310C2×C42C2×D414yes—
311C2×C42C2×D414yes—
312#95—24no—
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—28no—
322#95—24no—
323#95—28no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—24no—
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—
339#95—44no—
340C24—21no—
341C24—21no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—24no—
351C24—24no—
352C24—24no—
353C24—24no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24C22×D412yes—
359C24H(5,𝔽2)14yes—
360C24C2511yescharacteristic
361C2×Q8—28no—
362C2×Q8—48no—
363C2×Q8—44no—
364C2×Q8—44no—
365C2×Q8—48no—
366C2×Q8—44no—
367C2×Q8—44no—
368C2×Q8—44no—
369C2×Q8—88no—
370C2×D4—44no—
371C2×D4—44no—
372C2×D4—48no—
373C2×D4—48no—
374C2×D4—48no—
375C2×D4—44no—
376C2×D4—88no—
377C2×D4—88no—
378C2×D4—88no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—28no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—28no—
407C22×C4—28no—
408C22×C4—28no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—416no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4—416no—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C2×#4114yes—
464C22×C4C2×#4114yes—
465C22×C4C2×#4112yes—
466C22×C4C2×#4112yes—
467C22×C4H(5,𝔽2)14yes—
468C22×C42−1+414yes—
469C4⋊C4—216no—
470C4⋊C4—28no—
471C4⋊C4—44no—
472C4⋊C4—48no—
473C4⋊C4—44no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4—48no—
477C4⋊C4—48no—
478C4⋊C4—88no—
479C4⋊C4—88no—
480C22⋊C4—216no—
481C22⋊C4—48no—
482C22⋊C4—48no—
483C22⋊C4—48no—
484C22⋊C4—48no—
485C22⋊C4—48no—
486C22⋊C4—48no—
487C22⋊C4—48no—
488C22⋊C4—88no—
489C22⋊C4—88no—
490C22⋊C4—88no—
491C22⋊C4—88no—
492C22⋊C4—88no—
493C42—28no—
494C42—28no—
495C42—44no—
496C42—44no—
497C42—48no—
498C42—48no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—24no—
506C23—24no—
507C23—28no—
508C23—28no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—44no—
514C23—44no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—44no—
523C23—44no—
524C23—42no—
525C23—42no—
526C23—44no—
527C23—44no—
528C23—42no—
529C23—42no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23#53514yes—
539C23C23×D411yescharacteristic
540C23C22×#4111yescharacteristic
541C23C22×#4111yescharacteristic
542C23C2×H(5,𝔽2)12yes—
543C23C2×H(5,𝔽2)12yes—
544C23C2×2−1+412yes—
545C23#58512yes—
546C23#58512yes—
547C23#58512yes—
548Q8—416no—
549Q8—88no—
550D4—88no—
551D4—816no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—28no—
561C2×C4—28no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—28no—
576C2×C4—416no—
577C2×C4—416no—
578C2×C4—416no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
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—42no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
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—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4#52514yes—
630C2×C4#53414yes—
631C2×C4#53614yes—
632C22—24no—
633C22—24no—
634C22—24no—
635C22—42no—
636C22—42no—
637C22—42no—
638C22—42no—
639C22—48no—
640C22—48no—
641C22—82no—
642C22—82no—
643C22—82no—
644C22—82no—
645C22—82no—
646C22—82no—
647C22—84no—
648C22—84no—
649C22—84no—
650C22—84no—
651C22—82no—
652C22C2×#52511yescharacteristic
653C22C2×#53411yescharacteristic
654C22C2×#53511yescharacteristic
655C22C2×#53611yescharacteristic
656C22#344911yescharacteristic
657C22#344911yescharacteristic
658C22#344911yescharacteristic
659C22#350512yes—
660C22#350612yes—
661C22#352112yes—
662C22#352212yes—
663C22#352212yes—
664C22#352412yes—
665C22#353114yes—
666C22#353414yes—
667C22#354114yes—
668C22#354514yes—
669C4—24no—
670C4—24no—
671C4—24no—
672C4—42no—
673C4—42no—
674C4—44no—
675C4—42no—
676C4—44no—
677C4—82no—
678C4—82no—
679C4—82no—
680C4—84no—
681C4—84no—
682C4—82no—
683C4—82no—
684C2—44no—
685C2—162no—
686C2#3503111yescharacteristic
687C2#3521011yescharacteristic
688C2#3524911yescharacteristic
689C2#3669911yescharacteristic
690C2#3744611yescharacteristic
691C2#3744911yescharacteristic
692C2#3760611yescharacteristic
693C2#4630014yes—
694C2#4630614yes—
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
2. C2×C4×C22⋊Q8C2semidirect
3. #17048C2semidirect
4. #16807C2semidirect
5. #16681C2semidirect
6. #16632C2semidirect
7. #16262C2semidirect
9. #15162C2semidirect
10. #15112C2semidirect
11. #15104C2semidirect
12. #15095C2semidirect
13. #14883C2semidirect
14. #14881C2semidirect
15. #14684C2semidirect
16. #14630C2semidirect
17. #14625C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28502
C2×C4×C22⋊Q82
#170482
#168072
#166812
#166322
#162622
#162592
#151622
#151122
#151042
#150952
#148832
#148812
#146842
#146302
#146252