Finite Group Databasebeta

← #679932

Subgroups — #679932

Order 512.

2718 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2581C211yesmaximal, characteristic, complemented
2C2×#2580C211yesmaximal, characteristic
3C2×#2356C211yesmaximal, characteristic, complemented
4C2×C4×#124C211yesmaximal, characteristic
5#15584C214yesmaximal, complemented
6#15427C214yesmaximal, complemented
7#15226C212yesmaximal
8#15186C212yesmaximal, complemented
9#14925C214yesmaximal, complemented
10#14707C211yesmaximal, characteristic, complemented
11#14141C212yesmaximal
12#14080C214yesmaximal, complemented
13#14006C212yesmaximal, complemented
14#13941C211yesmaximal, characteristic
15C4×#391C211yesmaximal, characteristic, complemented
16C22×#124C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20#2581—28no—
21#2580—28no—
22C2×#400C2212yes—
23C2×#399C2212yes—
24C2×#395C2212yes—
25C2×#393C2212yes—
26#2356—28no—
27C4×#124—28no—
28C2×#391C2212yes—
29C2×#391C2212yes—
30C2×#391C2211yescharacteristic
31C2×#391C2212yes—
32C2×#390C2211yescharacteristic
33C2×#390C2212yes—
34C2×#390C2212yes—
35C2×#389C2212yes—
36C2×#389C2212yes—
37C2×#389C2212yes—
38C2×#388C2214yes—
39C2×#388C2214yes—
40C2×#387C2214yes—
41C2×#387C2214yes—
42C2×#385C2214yes—
43C2×#383C2212yes—
44C2×#382C2211yescharacteristic
45C2×#382C2211yescharacteristic
46C2×#382C2211yescharacteristic
47C2×C4×C4⋊C4C2212yes—
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C22⋊C4C2214yes—
50C2×#376C2212yes—
51C2×C43C2211yescharacteristic
52#1445C2214yes—
53#1443C2214yes—
54#1443C2214yes—
55#1442C2214yes—
56#1442C2214yes—
57#1441C2214yes—
58#1440C2214yes—
59#1440C2214yes—
60#1440C2214yes—
61#1439C2214yes—
62#1439C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1437C2214yes—
66#1436C2212yes—
67#1436C2214yes—
68#1436C2214yes—
69#1436C2214yes—
70#1436C2212yes—
71C4⋊#95C2214yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C4×#95C2212yes—
78C4×#95C2212yes—
79C23×Q8C2311yescharacteristic
80C23×Q8C2311yescharacteristic
81C23×D4C2311yescharacteristic
82C24×C4C2312yes—
83C2×#124—24no—
84C2×#124—24no—
85C2×#124—24no—
86C2×C4×Q8—24no—
87C2×C4×Q8—24no—
88C2×C4×Q8—24no—
89C2×C4×Q8—24no—
90C2×C4×Q8—24no—
91C2×C4×Q8D418yes—
92C2×C4×D4—24no—
93C2×C4×D4—24no—
94C2×C4×D4—24no—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2314yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C42C2312yes—
114C22×C42C2311yescharacteristic
115C22×C42C2312yes—
116C22×C42C2312yes—
117C22×C42C2314yes—
118C22×C42C2312yes—
119C22×C42C2311yescharacteristic
120C22×C42C2312yes—
121C22×C42C2311yescharacteristic
122#400—28no—
123#399—28no—
124#395—28no—
125#393—28no—
126#391—28no—
127#391—24no—
128#391—28no—
129#391—28no—
130#390—28no—
131#390—28no—
132#390—24no—
133#389—28no—
134#389—28no—
135#389—28no—
136#388—216no—
137#388—216no—
138#387—216no—
139#387—216no—
140#385—216no—
141#383—28no—
142#382—24no—
143#382—24no—
144#382D418yes—
145C4×C4⋊C4—28no—
146C4×C4⋊C4—24no—
147C4×C22⋊C4—216no—
148#376—28no—
149C2×#95C2314yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2311yescharacteristic
158C2×#95C2311yescharacteristic
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C43—24no—
178C25C2412yes—
179C22×Q8—22no—
180C22×Q8—22no—
181C22×Q8—22no—
182C22×Q8—22no—
183C22×Q8—22no—
184C22×Q8—22no—
185C22×Q8—22no—
186C22×Q8—24no—
187C22×Q8—22no—
188C22×Q8—22no—
189C22×Q8—42no—
190C22×Q8C2×D414yes—
191C22×D4—22no—
192C22×D4—22no—
193C22×D4—22no—
194C22×D4—22no—
195C22×D4—22no—
196C22×D4—22no—
197C22×D4#4114yes—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—44no—
202C23×C4—44no—
203C23×C4C2412yes—
204C23×C4C2414yes—
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2414yes—
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2411yescharacteristic
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217#124—232no—
218C4×Q8—232no—
219C4×Q8—416no—
220C4×D4—416no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4C2×D414yes—
252C2×C4⋊C4C2×D418yes—
253C2×C4⋊C4C2×D418yes—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C42—24no—
276C2×C42—22no—
277C2×C42—24no—
278C2×C42—22no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—22no—
284C2×C42—28no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—28no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—22no—
291C2×C42—22no—
292C2×C42—24no—
293C2×C42—42no—
294C2×C42—44no—
295C2×C42—42no—
296C2×C42—42no—
297C2×C42C2×D418yes—
298C2×C42C2×D414yes—
299C2×C42C2×D414yes—
300C2×C42#4114yes—
301C2×C42#4114yes—
302#95—22no—
303#95—28no—
304#95—28no—
305#95—28no—
306#95—22no—
307#95—28no—
308#95—24no—
309#95—24no—
310#95—28no—
311#95—28no—
312#95—28no—
313#95—28no—
314#95—28no—
315#95—28no—
316#95—28no—
317#95—24no—
318#95—24no—
319#95—24no—
320#95—42no—
321#95—44no—
322#95—42no—
323#95—44no—
324#95—42no—
325#95—42no—
326#95—44no—
327#95C2×D418yes—
328#95C2×D418yes—
329#95#4118yes—
330C24—22no—
331C24—22no—
332C24—24no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—22no—
337C24—24no—
338C24—24no—
339C24—24no—
340C24C2×#4114yes—
341C24C2511yescharacteristic
342C2×Q8—28no—
343C2×Q8—28no—
344C2×Q8—28no—
345C2×Q8—28no—
346C2×Q8—48no—
347C2×Q8—48no—
348C2×Q8—44no—
349C2×Q8—44no—
350C2×Q8—44no—
351C2×Q8—44no—
352C2×Q8—44no—
353C2×Q8—44no—
354C2×D4—28no—
355C2×D4—28no—
356C2×D4—28no—
357C2×D4—44no—
358C2×D4—44no—
359C2×D4—44no—
360C2×D4—44no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—24no—
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—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—416no—
432C22×C4—48no—
433C22×C4—88no—
434C22×C4—84no—
435C22×C4—84no—
436C22×C4C22×D414yes—
437C22×C4C22×D418yes—
438C22×C4C22×D414yes—
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C2×#4114yes—
446C22×C4C2×#4114yes—
447C22×C4C2×#4114yes—
448C22×C4C2×#4114yes—
449C22×C4C2×#4112yes—
450C22×C4C2×#4112yes—
451C22×C4C2×#4112yes—
452C22×C4C2×#4112yes—
453C22×C4C2×#4112yes—
454C22×C42−1+412yes—
455C22×C42−1+412yes—
456C4⋊C4—28no—
457C4⋊C4—216no—
458C4⋊C4—216no—
459C4⋊C4—48no—
460C4⋊C4—416no—
461C4⋊C4—44no—
462C4⋊C4—48no—
463C4⋊C4—44no—
464C4⋊C4—48no—
465C4⋊C4—88no—
466C4⋊C4—88no—
467C22⋊C4—232no—
468C22⋊C4—416no—
469C22⋊C4—416no—
470C22⋊C4—416no—
471C22⋊C4—48no—
472C22⋊C4—88no—
473C22⋊C4—88no—
474C42—216no—
475C42—28no—
476C42—48no—
477C42—416no—
478C42—48no—
479C42—48no—
480C42—44no—
481C42—48no—
482C42#121116yes—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—28no—
487C23—28no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—44no—
494C23—44no—
495C23—42no—
496C23—44no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23—42no—
501C23—42no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23C23×D411yescharacteristic
507C23C23×D411yescharacteristic
508C23C22×#4111yescharacteristic
509C23C22×#4111yescharacteristic
510C23C22×#4111yescharacteristic
511C23C22×#4112yes—
512C23C2×2−1+411yescharacteristic
513C23C2×2−1+411yescharacteristic
514C23#58512yes—
515C23#58512yes—
516C23#58512yes—
517Q8—416no—
518Q8—416no—
519D4—416no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—28no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—28no—
538C2×C4—24no—
539C2×C4—28no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—28no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—816no—
602C2×C4C2×#12114yes—
603C2×C4C2×#12114yes—
604C2×C4C2×#12114yes—
605C2×C4#52514yes—
606C2×C4#53614yes—
607C2×C4#54714yes—
608C2×C4#54814yes—
609C2×C4#55214yes—
610C2×C4#55614yes—
611C22—44no—
612C22—44no—
613C22—48no—
614C22—44no—
615C22—48no—
616C22—82no—
617C22—82no—
618C22—84no—
619C22—84no—
620C22—82no—
621C22—82no—
622C22C22×#12111yescharacteristic
623C22C2×#52511yescharacteristic
624C22#344411yescharacteristic
625C22C2×#53611yescharacteristic
626C22#345511yescharacteristic
627C22C2×#54711yescharacteristic
628C22C2×#54811yescharacteristic
629C22D4×#4112yes—
630C22D4×#4112yes—
631C22C2×#55211yescharacteristic
632C22C2×#55611yescharacteristic
633C22#347512yes—
634C22#348012yes—
635C22#348712yes—
636C22#350012yes—
637C22#350712yes—
638C22#351011yescharacteristic
639C22#351011yescharacteristic
640C22#351012yes—
641C22#351512yes—
642C22#352112yes—
643C22#355712yes—
644C22#356512yes—
645C4—24no—
646C4—24no—
647C4—44no—
648C4—44no—
649C4—44no—
650C4—48no—
651C4—44no—
652C4—44no—
653C4—44no—
654C4—82no—
655C4—82no—
656C4—84no—
657C4—82no—
658C4#241518yes—
659C2—84no—
660C2C2×#241511yescharacteristic
661C2#3474311yescharacteristic
662C2#3567711yescharacteristic
663C2#3622912yes—
664C2#3639411yescharacteristic
665C2#3660111yescharacteristic
666C2#3897111yescharacteristic
667C2#3975811yescharacteristic
668C2#3985912yes—
669C2#5958612yes—
670C2#5982912yes—
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×#2581C2semidirect
3. C2×#2356C2semidirect
5. #15584C2semidirect
6. #15427C2semidirect
8. #15186C2semidirect
9. #14925C2semidirect
10. #14707C2semidirect
12. #14080C2semidirect
13. #14006C2semidirect
15. C4×#391C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25812
C2×#25802
C2×#23562
C2×C4×#1242
#155842
#154272
#152262
#151862
#149252
#147072
#141412
#140802
#140062
#139412
C4×#3912