Finite Group Databasebeta

← #4453216

Subgroups — #4453216

Order 512.

2712 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2579C211yesmaximal, characteristic
2C2×#2487C211yesmaximal, characteristic
3C22×#398C211yesmaximal, characteristic, complemented
4C2×#2318C211yesmaximal, characteristic
5#16677C212yesmaximal, complemented
6#16517C212yesmaximal, complemented
7#16469C214yesmaximal, complemented
8#16464C214yesmaximal, complemented
9#16399C212yesmaximal, complemented
10#16393C212yesmaximal, complemented
11#14955C212yesmaximal, complemented
12#14945C212yesmaximal
13#14786C212yesmaximal, complemented
14#14776C212yesmaximal
15#14347C211yesmaximal, characteristic, complemented
16#13924C211yesmaximal, characteristic, complemented
17#13923C211yesmaximal, characteristic, complemented
18C22×C4×D4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C4⋊C4C2212yes—
22#2579—28no—
23#2487—28no—
24C2×#400C2212yes—
25C2×#399C2212yes—
26C2×#398—28no—
27C2×#398C2212yes—
28C2×#398C2212yes—
29C2×#398C2212yes—
30C2×#398C2214yes—
31C2×#398C2214yes—
32C2×#397C2214yes—
33C2×#396C2212yes—
34#2318—28no—
35C2×#389C2212yes—
36C2×#388C2212yes—
37C2×#387C2212yes—
38C2×#387C2212yes—
39C2×#386C2212yes—
40C2×#386C2211yescharacteristic
41C2×#386C2211yescharacteristic
42C2×#386C2212yes—
43C2×#385C2212yes—
44C2×#385C2212yes—
45C2×#385C2211yescharacteristic
46C2×#385C2211yescharacteristic
47C2×#385C2212yes—
48C2×#383C2211yescharacteristic
49C2×#382C2211yescharacteristic
50C2×#381C2211yescharacteristic
51C2×#380C2211yescharacteristic
52C2×#380C2212yes—
53C2×#380C2211yescharacteristic
54C2×#380C2211yescharacteristic
55C2×C4×C22⋊C4C2211yescharacteristic
56C22×#95C2212yes—
57C22×#95C2211yescharacteristic
58#1445C2212yes—
59#1445C2214yes—
60#1445C2212yes—
61#1445C2214yes—
62#1444C2214yes—
63#1443C2214yes—
64#1443C2214yes—
65#1442C2212yes—
66#1442C2214yes—
67#1442C2212yes—
68#1442C2214yes—
69#1442C2214yes—
70#1442C2214yes—
71#1441C2212yes—
72#1441C2214yes—
73#1441C2214yes—
74#1441C2212yes—
75#1441C2214yes—
76#1440C2214yes—
77#1440C2214yes—
78#1439C2214yes—
79#1436C2214yes—
80C4⋊#95C2214yes—
81C22⋊#95C2212yes—
82C22⋊#95C2212yes—
83C22⋊#95C2212yes—
84C22⋊#95C2212yes—
85C22⋊#95C2212yes—
86C22⋊#95C2212yes—
87C23×D4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2312yes—
90C24×C4C2311yescharacteristic
91C24×C4C2312yes—
92C24×C4C2311yescharacteristic
93C24×C4C2311yescharacteristic
94C2×C4×D4—24no—
95C2×C4×D4—24no—
96C2×C4×D4D418yes—
97C22×C4⋊C4—24no—
98C22×C4⋊C4—28no—
99C22×C4⋊C4—24no—
100C22×C4⋊C4—24no—
101C22×C4⋊C4—24no—
102C22×C4⋊C4—24no—
103C22×C4⋊C4—24no—
104C22×C4⋊C4—48no—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2311yescharacteristic
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2311yescharacteristic
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2311yescharacteristic
125C22×C22⋊C4C2311yescharacteristic
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2312yes—
130C22×C42C2311yescharacteristic
131C22×C42C2312yes—
132#400—28no—
133#399—28no—
134#398—28no—
135#398—28no—
136#398—28no—
137#398—216no—
138#398—48no—
139#397—48no—
140#396—28no—
141#389—28no—
142#388—28no—
143#387—28no—
144#387—28no—
145#386—24no—
146#386—24no—
147#386—28no—
148#386—28no—
149#385—28no—
150#385—28no—
151#385—24no—
152#385—24no—
153#385—28no—
154#383—24no—
155#382—24no—
156#381—24no—
157#380—24no—
158#380—28no—
159#380—24no—
160#380—24no—
161C4×C22⋊C4—24no—
162C2×#95—28no—
163C2×#95D418yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2311yescharacteristic
175C2×#95C2311yescharacteristic
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2311yescharacteristic
193C2×#95C2311yescharacteristic
194C2×#95C2312yes—
195C25C2411yescharacteristic
196C25C2411yescharacteristic
197C22×D4—22no—
198C22×D4—22no—
199C22×D4—24no—
200C22×D4—24no—
201C22×D4C2×D414yes—
202C23×C4—22no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—24no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—22no—
216C23×C4—44no—
217C23×C4—44no—
218C23×C4—42no—
219C23×C4—44no—
220C23×C4—44no—
221C23×C4—44no—
222C23×C4—44no—
223C23×C4—44no—
224C23×C4C2×D418yes—
225C23×C4C2×D414yes—
226C23×C4C2×D414yes—
227C23×C4C2×D414yes—
228C23×C4#4114yes—
229C23×C4C2414yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2414yes—
235C23×C4C2412yes—
236C23×C4C2411yescharacteristic
237C23×C4C2411yescharacteristic
238C23×C4C2412yes—
239C23×C4C2412yes—
240C23×C4C2411yescharacteristic
241C23×C4C2412yes—
242C23×C4C2411yescharacteristic
243C23×C4C2411yescharacteristic
244C4×D4—232no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—42no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—432no—
276C2×C4⋊C4—416no—
277C2×C4⋊C4—416no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4C2×D414yes—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—42no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4C2×D414yes—
312C2×C22⋊C4C2×D414yes—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—24no—
316C2×C42—44no—
317C2×C42C2×D414yes—
318#95—28no—
319#95—28no—
320#95—22no—
321#95—22no—
322#95—24no—
323#95—28no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—24no—
333#95—28no—
334#95—28no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95—44no—
341#95—44no—
342#95—44no—
343#95—44no—
344#95—44no—
345#95—44no—
346#95C2×D414yes—
347#95C2×D414yes—
348#95C2×D418yes—
349C24—22no—
350C24—24no—
351C24—22no—
352C24—21no—
353C24—21no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—24no—
359C24C22×D412yes—
360C24C22×D412yes—
361C24C22×D412yes—
362C24C2×#4112yes—
363C242−1+414yes—
364C242−1+414yes—
365C24C2511yescharacteristic
366C2×D4—28no—
367C2×D4—28no—
368C2×D4—28no—
369C2×D4—48no—
370C2×D4—48no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—28no—
374C22×C4—216no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
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—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—28no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—24no—
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—48no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4—48no—
446C22×C4—48no—
447C22×C4—48no—
448C22×C4—44no—
449C22×C4—416no—
450C22×C4—416no—
451C22×C4—44no—
452C22×C4—48no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4—44no—
457C22×C4—84no—
458C22×C4—84no—
459C22×C4—84no—
460C22×C4—88no—
461C22×C4—84no—
462C22×C4—88no—
463C22×C4#121116yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D414yes—
467C22×C4C22×D414yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C2×#4112yes—
474C22×C4C2×#4112yes—
475C22×C4H(5,𝔽2)14yes—
476C4⋊C4—28no—
477C4⋊C4—48no—
478C4⋊C4—416no—
479C4⋊C4—416no—
480C4⋊C4—48no—
481C4⋊C4—48no—
482C4⋊C4—48no—
483C4⋊C4—48no—
484C4⋊C4—48no—
485C4⋊C4—84no—
486C4⋊C4—84no—
487C4⋊C4—84no—
488C4⋊C4—84no—
489C4⋊C4—88no—
490C22⋊C4—28no—
491C22⋊C4—28no—
492C22⋊C4—48no—
493C22⋊C4—48no—
494C22⋊C4—44no—
495C22⋊C4—44no—
496C22⋊C4—48no—
497C22⋊C4—84no—
498C22⋊C4—84no—
499C22⋊C4—84no—
500C22⋊C4—88no—
501C42—28no—
502C42—84no—
503C23—22no—
504C23—24no—
505C23—24no—
506C23—22no—
507C23—24no—
508C23—24no—
509C23—24no—
510C23—28no—
511C23—28no—
512C23—28no—
513C23—28no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—24no—
518C23—24no—
519C23—42no—
520C23—42no—
521C23—41no—
522C23—42no—
523C23—41no—
524C23—42no—
525C23—42no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23C2×#12114yes—
531C23#53614yes—
532C23#53614yes—
533C23#53618yes—
534C23#54118yes—
535C23C23×D411yescharacteristic
536C23C23×D411yescharacteristic
537C23C22×#4111yescharacteristic
538C23C2×H(5,𝔽2)12yes—
539C23C2×2−1+412yes—
540C23C2×2−1+412yes—
541C23#58514yes—
542C23#58512yes—
543D4—416no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—416no—
567C2×C4—432no—
568C2×C4—416no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—42no—
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—88no—
624C2×C4—88no—
625C2×C4—816no—
626C2×C4C2×#12114yes—
627C2×C4C2×#12114yes—
628C22—28no—
629C22—28no—
630C22—24no—
631C22—28no—
632C22—28no—
633C22—216no—
634C22—24no—
635C22—42no—
636C22—42no—
637C22—44no—
638C22—44no—
639C22—42no—
640C22—82no—
641C22—82no—
642C22—84no—
643C22#262118yes—
644C22C22×#12111yescharacteristic
645C22C2×#53612yes—
646C22C2×#53611yescharacteristic
647C22C2×#53611yescharacteristic
648C22C2×#54112yes—
649C22#348411yescharacteristic
650C22#348512yes—
651C22#348512yes—
652C22#348711yescharacteristic
653C22#348812yes—
654C22#349112yes—
655C22#349412yes—
656C22#354114yes—
657C22#354114yes—
658C22#355314yes—
659C22#355314yes—
660C4—24no—
661C4—24no—
662C4—42no—
663C4—44no—
664C4—44no—
665C4—42no—
666C4—42no—
667C4—44no—
668C4—82no—
669C4—82no—
670C4—82no—
671C4—84no—
672C4—82no—
673C4—82no—
674C4—84no—
675C2—28no—
676C2—82no—
677C2C2×#262111yescharacteristic
678C2#3544711yescharacteristic
679C2#3556311yescharacteristic
680C2#3670512yes—
681C2#3712012yes—
682C2#4140612yes—
683C2#4140812yes—
684C2#4142814yes—
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
3. C22×#398C2semidirect
5. #16677C2semidirect
6. #16517C2semidirect
7. #16469C2semidirect
8. #16464C2semidirect
9. #16399C2semidirect
10. #16393C2semidirect
11. #14955C2semidirect
13. #14786C2semidirect
15. #14347C2semidirect
16. #13924C2semidirect
17. #13923C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25792
C2×#24872
C22×#3982
C2×#23182
#166772
#165172
#164692
#164642
#163992
#163932
#149552
#149452
#147862
#147762
#143472
#139242
#139232