Finite Group Databasebeta

← #764814

Subgroups — #764814

Order 512.

2956 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2486C211yesmaximal, characteristic
2C22×#398C211yesmaximal, characteristic, complemented
3C2×#2322C211yesmaximal, characteristic
4C2×#2313C211yesmaximal, characteristic, complemented
5#15874C214yesmaximal, complemented
6#15583C214yesmaximal, complemented
7#15517C214yesmaximal, complemented
8#15512C214yesmaximal, complemented
9#15073C212yesmaximal, complemented
10#15069C212yesmaximal
11#14149C211yesmaximal, characteristic, complemented
12#13996C212yesmaximal
13#13972C212yesmaximal, complemented
14#13960C211yesmaximal, characteristic
15#13904C211yesmaximal, characteristic, complemented
16C22×#123C2211yescharacteristic
17C22×#123C2212yes—
18C22×C4×D4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C4⋊C4C2212yes—
22#2486—28no—
23C2×#398—28no—
24C2×#398C2214yes—
25C2×#398C2212yes—
26C2×#398C2214yes—
27C2×#397C2212yes—
28#2322—28no—
29#2313—28no—
30C2×#389C2212yes—
31C2×#389C2212yes—
32C2×#388C2212yes—
33C2×#387C2212yes—
34C2×#387C2214yes—
35C2×#387C2214yes—
36C2×#386C2211yescharacteristic
37C2×#386C2212yes—
38C2×#385C2211yescharacteristic
39C2×#385C2212yes—
40C2×#385C2212yes—
41C2×#381C2211yescharacteristic
42C2×#381C2212yes—
43C2×#381C2212yes—
44C2×#381C2212yes—
45C2×#381C2212yes—
46C2×#380C2211yescharacteristic
47C2×#380C2212yes—
48C2×#380C2211yescharacteristic
49C2×C4×C22⋊C4C2211yescharacteristic
50C2×C4×C22⋊C4C2212yes—
51C22×#95C2211yescharacteristic
52C22×#95C2212yes—
53#1445C2214yes—
54#1445C2214yes—
55#1444C2214yes—
56#1444C2214yes—
57#1443C2214yes—
58#1442C2214yes—
59#1442C2214yes—
60#1442C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1441C2212yes—
64#1441C2214yes—
65#1439C2212yes—
66#1439C2214yes—
67#1439C2214yes—
68#1439C2214yes—
69#1437C2214yes—
70#1436C2214yes—
71#1436C2214yes—
72C4⋊#95C2214yes—
73C4⋊#95C2214yes—
74#1434C2212yes—
75#1434C2214yes—
76C22⋊#95C2212yes—
77C22⋊#95C2212yes—
78C22⋊#95C2212yes—
79C23×D4C2311yescharacteristic
80C24×C4C2312yes—
81C24×C4C2311yescharacteristic
82C24×C4C2312yes—
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C2×#123—24no—
87C2×#123—24no—
88C2×#123—28no—
89C2×#123—28no—
90C2×#123—24no—
91C2×#123—44no—
92C2×C4×D4—24no—
93C2×C4×D4—24no—
94C2×C4×D4—24no—
95C22×C4⋊C4—24no—
96C22×C4⋊C4—24no—
97C22×C4⋊C4—28no—
98C22×C4⋊C4—24no—
99C22×C4⋊C4—24no—
100C22×C4⋊C4—24no—
101C22×C4⋊C4—44no—
102C22×C4⋊C4—44no—
103C22×C4⋊C4D418yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C22⋊C4C2311yescharacteristic
118C22×C22⋊C4C2311yescharacteristic
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2311yescharacteristic
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2311yescharacteristic
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2312yes—
127C22×C42C2311yescharacteristic
128C22×C42C2314yes—
129C22×C42C2312yes—
130#398—216no—
131#398—216no—
132#398—28no—
133#397—28no—
134#389—28no—
135#389—28no—
136#388—28no—
137#387—216no—
138#387—28no—
139#387—216no—
140#386—24no—
141#386—44no—
142#385—24no—
143#385—28no—
144#385—44no—
145#381—24no—
146#381—28no—
147#381—28no—
148#381—28no—
149#381—44no—
150#380—24no—
151#380—28no—
152#380D418yes—
153C4×C22⋊C4—24no—
154C4×C22⋊C4—28no—
155C2×#95—24no—
156C2×#95—28no—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2312yes—
183C25C2411yescharacteristic
184C25C2411yescharacteristic
185C22×D4—22no—
186C22×D4—22no—
187C22×D4—22no—
188C22×D4—22no—
189C22×D4—22no—
190C22×D4—22no—
191C22×D4#4114yes—
192C23×C4—24no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—28no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—44no—
215C23×C4—44no—
216C23×C4—44no—
217C23×C4—44no—
218C23×C4C2×D414yes—
219C23×C4C2×D414yes—
220C23×C4C2×D414yes—
221C23×C4C2×D414yes—
222C23×C4#4114yes—
223C23×C4C2412yes—
224C23×C4C2414yes—
225C23×C4C2411yescharacteristic
226C23×C4C2414yes—
227C23×C4C2411yescharacteristic
228C23×C4C2414yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2411yescharacteristic
233C23×C4C2414yes—
234C23×C4C2411yescharacteristic
235C23×C4C2411yescharacteristic
236#123—232no—
237#123—432no—
238C4×D4—416no—
239C2×C4⋊C4—232no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—432no—
270C2×C4⋊C4—416no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4C2×D418yes—
276C2×C4⋊C4C2×D418yes—
277C2×C4⋊C4C2×D418yes—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4C2×D418yes—
306C2×C22⋊C4#4114yes—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—28no—
311C2×C42—24no—
312C2×C42—44no—
313C2×C42—44no—
314C2×C42—44no—
315#95—24no—
316#95—24no—
317#95—28no—
318#95—28no—
319#95—24no—
320#95—28no—
321#95—28no—
322#95—24no—
323#95—24no—
324#95—28no—
325#95—28no—
326#95—24no—
327#95—28no—
328#95—24no—
329#95—24no—
330#95—28no—
331#95—28no—
332#95—28no—
333#95—28no—
334#95—44no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95C2×D418yes—
341C24—24no—
342C24—24no—
343C24—21no—
344C24—21no—
345C24—21no—
346C24—21no—
347C24—21no—
348C24—21no—
349C24—24no—
350C24—24no—
351C24C22×D412yes—
352C24C22×D412yes—
353C24C2×#4112yes—
354C24C2×#4112yes—
355C24C2×#4112yes—
356C24C2×#4112yes—
357C24H(5,𝔽2)12yes—
358C24H(5,𝔽2)12yes—
359C24C2511yescharacteristic
360C2×D4—28no—
361C2×D4—28no—
362C2×D4—28no—
363C2×D4—44no—
364C2×D4—44no—
365C2×D4—44no—
366C2×D4—44no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—216no—
370C22×C4—28no—
371C22×C4—28no—
372C22×C4—28no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—28no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—48no—
434C22×C4—48no—
435C22×C4—48no—
436C22×C4—48no—
437C22×C4—416no—
438C22×C4—416no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—48no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—48no—
447C22×C4—48no—
448C22×C4—48no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4—48no—
452C22×C4—48no—
453C22×C4—44no—
454C22×C4—84no—
455C22×C4—84no—
456C22×C4—84no—
457C22×C4—84no—
458C22×C4#121116yes—
459C22×C4C22×D414yes—
460C22×C4C22×D418yes—
461C22×C4C22×D414yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C2×#4114yes—
467C22×C4C2×#4112yes—
468C22×C4C2×#4112yes—
469C22×C4C2×#4112yes—
470C22×C42−1+418yes—
471C4⋊C4—216no—
472C4⋊C4—216no—
473C4⋊C4—216no—
474C4⋊C4—48no—
475C4⋊C4—416no—
476C4⋊C4—416no—
477C4⋊C4—416no—
478C4⋊C4—48no—
479C4⋊C4—48no—
480C4⋊C4—44no—
481C4⋊C4—84no—
482C4⋊C4—84no—
483C4⋊C4—88no—
484C22⋊C4—216no—
485C22⋊C4—28no—
486C22⋊C4—216no—
487C22⋊C4—416no—
488C22⋊C4—44no—
489C22⋊C4—44no—
490C22⋊C4—48no—
491C22⋊C4—44no—
492C22⋊C4—84no—
493C22⋊C4—88no—
494C42—44no—
495C42—84no—
496C42—88no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—24no—
503C23—28no—
504C23—28no—
505C23—28no—
506C23—28no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—28no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23C2×#12114yes—
525C23#52514yes—
526C23#53514yes—
527C23#54614yes—
528C23#55014yes—
529C23#55314yes—
530C23#55514yes—
531C23C23×D411yescharacteristic
532C23C23×D411yescharacteristic
533C23C22×#4111yescharacteristic
534C23C22×#4111yescharacteristic
535C23C22×#4111yescharacteristic
536C23C2×H(5,𝔽2)11yescharacteristic
537C23C2×H(5,𝔽2)11yescharacteristic
538C23C2×2−1+414yes—
539C23#58514yes—
540D4—416no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—28no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—28no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—28no—
560C2×C4—28no—
561C2×C4—28no—
562C2×C4—28no—
563C2×C4—28no—
564C2×C4—416no—
565C2×C4—432no—
566C2×C4—41no—
567C2×C4—416no—
568C2×C4—44no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—88no—
616C2×C4—88no—
617C2×C4—816no—
618C2×C4C2×#12114yes—
619C2×C4C2×#12114yes—
620C22—24no—
621C22—24no—
622C22—28no—
623C22—28no—
624C22—28no—
625C22—28no—
626C22—28no—
627C22—28no—
628C22—44no—
629C22—42no—
630C22—44no—
631C22—44no—
632C22—81no—
633C22—81no—
634C22—82no—
635C22—82no—
636C22—82no—
637C22#241418yes—
638C22C22×#12111yescharacteristic
639C22C2×#52511yescharacteristic
640C22C2×#53511yescharacteristic
641C22C2×#54611yescharacteristic
642C22C2×#55011yescharacteristic
643C22C2×#55311yescharacteristic
644C22C2×#55511yescharacteristic
645C22#348512yes—
646C22#348512yes—
647C22#348612yes—
648C22#349112yes—
649C22#349912yes—
650C22#350912yes—
651C22#351514yes—
652C22#351514yes—
653C22#353514yes—
654C22#353514yes—
655C4—24no—
656C4—24no—
657C4—44no—
658C4—44no—
659C4—44no—
660C4—42no—
661C4—48no—
662C4—48no—
663C4—81no—
664C4—81no—
665C4—84no—
666C4—84no—
667C4—82no—
668C2—28no—
669C2—82no—
670C2C2×#241411yescharacteristic
671C2#3544711yescharacteristic
672C2#3545312yes—
673C2#3566711yescharacteristic
674C2#3711812yes—
675C2#4349612yes—
676C2#4349712yes—
677C2#4351412yes—
678C2#4351512yes—
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
2. C22×#398C2semidirect
4. C2×#2313C2semidirect
5. #15874C2semidirect
6. #15583C2semidirect
7. #15517C2semidirect
8. #15512C2semidirect
9. #15073C2semidirect
11. #14149C2semidirect
13. #13972C2semidirect
15. #13904C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24862
C22×#3982
C2×#23222
C2×#23132
#158742
#155832
#155172
#155122
#150732
#150692
#141492
#139962
#139722
#139602
#139042