Finite Group Databasebeta

← #770059

Subgroups — #770059

Order 512.

2826 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×#2568C211yesmaximal, characteristic, complemented
2C2×#2542C211yesmaximal, characteristic
3C2×#2398C211yesmaximal, characteristic
4C2×#2327C211yesmaximal, characteristic, complemented
5#15886C214yesmaximal, complemented
6#15855C214yesmaximal, complemented
7#15815C214yesmaximal
8#15789C214yesmaximal, complemented
9#15236C212yesmaximal, complemented
10#15182C212yesmaximal, complemented
11#13969C212yesmaximal, complemented
12#13969C212yesmaximal, complemented
13#13925C211yesmaximal, characteristic, complemented
14#13905C211yesmaximal, characteristic
15#13779C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2212yes—
18C22×C22⋊Q8C2212yes—
19C22×C4×Q8C2211yescharacteristic
20C22×C4×Q8C2212yes—
21C22×C4×D4C2211yescharacteristic
22#2568—28no—
23#2542—28no—
24#2398—28no—
25C2×#400C2214yes—
26C2×#400C2212yes—
27C2×#399C2214yes—
28C2×#397C2212yes—
29C2×#394C2212yes—
30C2×#393C2212yes—
31#2327—28no—
32C2×#391C2212yes—
33C2×#391C2212yes—
34C2×#391C2212yes—
35C2×#391C2212yes—
36C2×#390C2212yes—
37C2×#387C2214yes—
38C2×#387C2214yes—
39C2×#384C2211yescharacteristic
40C2×#384C2212yes—
41C2×#382C2212yes—
42C2×#382C2211yescharacteristic
43C2×#382C2211yescharacteristic
44C2×#380C2212yes—
45C2×#380C2212yes—
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4C2212yes—
49C2×C4×C22⋊C4C2212yes—
50C2×#376C2211yescharacteristic
51#1446C2212yes—
52#1445C2212yes—
53#1445C2212yes—
54#1442C2214yes—
55#1442C2214yes—
56#1441C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1440C2214yes—
61#1440C2214yes—
62#1439C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1438C2212yes—
67#1437C2214yes—
68#1437C2214yes—
69#1437C2214yes—
70#1436C2214yes—
71#1436C2214yes—
72#1436C2214yes—
73#1436C2214yes—
74#1436C2214yes—
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C4⋊#95C2211yescharacteristic
78C4⋊#95C2211yescharacteristic
79C23×Q8C2312yes—
80C23×Q8C2311yescharacteristic
81C23×D4C2311yescharacteristic
82C24×C4C2312yes—
83C2×C4⋊Q8—24no—
84C2×C4⋊Q8—24no—
85C2×C4⋊Q8D418yes—
86C2×C22⋊Q8—28no—
87C2×C22⋊Q8—28no—
88C2×C22⋊Q8—28no—
89C2×C22⋊Q8—28no—
90C2×C22⋊Q8—44no—
91C2×C22⋊Q8—44no—
92C2×C4×Q8—24no—
93C2×C4×Q8—28no—
94C2×C4×Q8—24no—
95C2×C4×Q8—24no—
96C2×C4×Q8—44no—
97C2×C4×Q8—44no—
98C2×C4×D4—24no—
99C2×C4×D4—24no—
100C2×C4×D4—24no—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C42C2312yes—
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2314yes—
128C22×C42C2312yes—
129C22×C42C2312yes—
130C22×C42C2311yescharacteristic
131#400—216no—
132#400—28no—
133#399—216no—
134#397—28no—
135#394—28no—
136#393—28no—
137#391—28no—
138#391—28no—
139#391—28no—
140#391—28no—
141#390—28no—
142#387—216no—
143#387—216no—
144#384—28no—
145#384—24no—
146#382—28no—
147#382—24no—
148#382—24no—
149#380—44no—
150#380—44no—
151C4×C4⋊C4—24no—
152C4×C4⋊C4—24no—
153C4×C22⋊C4—28no—
154C4×C22⋊C4—28no—
155#376D418yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C25C2412yes—
182C22×Q8—24no—
183C22×Q8—22no—
184C22×Q8—22no—
185C22×Q8—24no—
186C22×Q8—24no—
187C22×Q8—24no—
188C22×Q8—22no—
189C22×Q8—24no—
190C22×Q8—24no—
191C22×Q8—24no—
192C22×Q8—42no—
193C22×Q8C2×D418yes—
194C22×D4—24no—
195C22×D4—22no—
196C22×D4—22no—
197C22×D4—22no—
198C22×D4—42no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2414yes—
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2414yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2412yes—
216C23×C4C2412yes—
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220C4⋊Q8—232no—
221C22⋊Q8—432no—
222C22⋊Q8—432no—
223C4×Q8—432no—
224C4×Q8—416no—
225C4×D4—416no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—28no—
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⋊C4C2×D418yes—
265C2×C4⋊C4C2×D418yes—
266C2×C4⋊C4#4114yes—
267C2×C4⋊C4#4114yes—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
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—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—42no—
285C2×C22⋊C4—42no—
286C2×C22⋊C4—42no—
287C2×C22⋊C4—42no—
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×C42—22no—
296C2×C42—22no—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—24no—
300C2×C42—28no—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42—44no—
306C2×C42—44no—
307C2×C42—44no—
308C2×C42—44no—
309C2×C42C2×D414yes—
310C2×C42C2×D414yes—
311C2×C42C2×D414yes—
312C2×C42C2×D414yes—
313C2×C42#4114yes—
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—28no—
324#95—28no—
325#95—28no—
326#95—28no—
327#95—28no—
328#95—28no—
329#95—28no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95—44no—
336#95—44no—
337#95C2×D418yes—
338#95C2×D418yes—
339C24—22no—
340C24—22no—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—22no—
353C242−1+414yes—
354C24C2511yescharacteristic
355C2×Q8—28no—
356C2×Q8—216no—
357C2×Q8—216no—
358C2×Q8—216no—
359C2×Q8—48no—
360C2×Q8—48no—
361C2×Q8—44no—
362C2×Q8—44no—
363C2×Q8—48no—
364C2×Q8—48no—
365C2×Q8—48no—
366C2×Q8—48no—
367C2×D4—28no—
368C2×D4—48no—
369C2×D4—48no—
370C2×D4—44no—
371C2×D4—44no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4—48no—
446C22×C4C22×D414yes—
447C22×C4C22×D418yes—
448C22×C4C22×D414yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C2×#4114yes—
456C22×C4C2×#4114yes—
457C22×C4C2×#4112yes—
458C22×C4C2×#4112yes—
459C22×C4C2×#4112yes—
460C22×C4C2×#4112yes—
461C22×C4C2×#4112yes—
462C22×C4H(5,𝔽2)12yes—
463C22×C4H(5,𝔽2)12yes—
464C22×C42−1+414yes—
465C4⋊C4—216no—
466C4⋊C4—216no—
467C4⋊C4—28no—
468C4⋊C4—28no—
469C4⋊C4—416no—
470C4⋊C4—416no—
471C4⋊C4—416no—
472C4⋊C4—44no—
473C4⋊C4—48no—
474C4⋊C4—44no—
475C4⋊C4—48no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—88no—
479C22⋊C4—48no—
480C22⋊C4—48no—
481C22⋊C4—416no—
482C22⋊C4—416no—
483C22⋊C4—48no—
484C22⋊C4—48no—
485C22⋊C4—48no—
486C22⋊C4—88no—
487C22⋊C4—88no—
488C42—28no—
489C42—28no—
490C42—48no—
491C42—48no—
492C42—416no—
493C42—48no—
494C42#121116yes—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23C23×D411yescharacteristic
531C23C23×D411yescharacteristic
532C23C22×#4111yescharacteristic
533C23C22×#4111yescharacteristic
534C23C22×#4111yescharacteristic
535C23C2×H(5,𝔽2)11yescharacteristic
536C23C2×H(5,𝔽2)11yescharacteristic
537C23C2×2−1+412yes—
538C23C2×2−1+412yes—
539C23#58512yes—
540C23#58512yes—
541Q8—416no—
542Q8—432no—
543D4—416no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—28no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—28no—
558C2×C4—28no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—28no—
567C2×C4—28no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
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—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—42no—
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—816no—
622C2×C4C2×#12114yes—
623C2×C4C2×#12114yes—
624C2×C4C2×#12114yes—
625C2×C4#53214yes—
626C2×C4#53514yes—
627C2×C4#54614yes—
628C2×C4#54614yes—
629C2×C4#55914yes—
630C2×C4#55914yes—
631C22—44no—
632C22—44no—
633C22—44no—
634C22—44no—
635C22—44no—
636C22—44no—
637C22—44no—
638C22—82no—
639C22—82no—
640C22—82no—
641C22—82no—
642C22—82no—
643C22—82no—
644C22—82no—
645C22—82no—
646C22C22×#12111yescharacteristic
647C22C2×#53211yescharacteristic
648C22C2×#53511yescharacteristic
649C22C2×#54611yescharacteristic
650C22C2×#54611yescharacteristic
651C22C2×#55911yescharacteristic
652C22C2×#55911yescharacteristic
653C22#348611yescharacteristic
654C22#348611yescharacteristic
655C22#348711yescharacteristic
656C22#348711yescharacteristic
657C22#348912yes—
658C22#348912yes—
659C22#349512yes—
660C22#349512yes—
661C22#351512yes—
662C22#351612yes—
663C22#352112yes—
664C22#352312yes—
665C22#355212yes—
666C22#355412yes—
667C22#356112yes—
668C22#356212yes—
669C4—24no—
670C4—24no—
671C4—44no—
672C4—44no—
673C4—44no—
674C4—48no—
675C4—44no—
676C4—44no—
677C4—84no—
678C4—82no—
679C4—82no—
680C4—82no—
681C4—82no—
682C4#254418yes—
683C2—84no—
684C2C2×#254411yescharacteristic
685C2#3563512yes—
686C2#3566111yescharacteristic
687C2#3566111yescharacteristic
688C2#3737912yes—
689C2#4520112yes—
690C2#4522212yes—
691C2#4526811yescharacteristic
692C2#4526911yescharacteristic
693C2#4533511yescharacteristic
694C2#4533611yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
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×#2568C2semidirect
4. C2×#2327C2semidirect
5. #15886C2semidirect
6. #15855C2semidirect
8. #15789C2semidirect
9. #15236C2semidirect
10. #15182C2semidirect
11. #13969C2semidirect
12. #13969C2semidirect
13. #13925C2semidirect
15. #13779C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25682
C2×#25422
C2×#23982
C2×#23272
#158862
#158552
#158152
#157892
#152362
#151822
#139692
#139252
#139052
#137792