Finite Group Databasebeta

← #4420462

Subgroups — #4420462

Order 512.

2732 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2542C211yesmaximal, characteristic
2C2×#2660C211yesmaximal, characteristic, complemented
3C2×#2633C211yesmaximal, characteristic
4C2×#2353C211yesmaximal, characteristic
5#17062C212yesmaximal, complemented
6#16701C212yesmaximal, complemented
7#16264C212yesmaximal, complemented
8#16262C212yesmaximal, complemented
9#15629C214yesmaximal, complemented
10#15479C214yesmaximal
11#15135C212yesmaximal, complemented
12#15016C212yesmaximal, complemented
13#14763C212yesmaximal, complemented
14#14758C212yesmaximal, complemented
15#14663C211yesmaximal, characteristic, complemented
16#14289C211yesmaximal, characteristic, complemented
17#13952C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C22×#125C2211yescharacteristic
20C22×C22⋊Q8C2212yes—
21C22×C22⋊Q8C2212yes—
22C22×C4×D4C2211yescharacteristic
23#2660—28no—
24#2633—28no—
25#2542—28no—
26C2×#399C2211yescharacteristic
27C2×#399C2211yescharacteristic
28C2×#399C2212yes—
29C2×#399C2212yes—
30C2×#398C2212yes—
31C2×#398C2214yes—
32C2×#398C2214yes—
33C2×#397C2211yescharacteristic
34C2×#397C2211yescharacteristic
35C2×#397C2214yes—
36C2×#395C2212yes—
37C2×#395C2212yes—
38C2×#394C2212yes—
39#2353—28no—
40C2×#390C2212yes—
41C2×#389C2214yes—
42C2×#388C2212yes—
43C2×#387C2212yes—
44C2×#386C2212yes—
45C2×#385C2212yes—
46C2×#385C2212yes—
47C2×#384C2212yes—
48C2×#384C2212yes—
49C2×#383C2211yescharacteristic
50C2×#383C2211yescharacteristic
51C2×#382C2212yes—
52C2×#381C2211yescharacteristic
53C2×#380C2211yescharacteristic
54C2×#380C2214yes—
55C2×C4×C4⋊C4C2211yescharacteristic
56C2×C4×C22⋊C4C2212yes—
57C2×C4×C22⋊C4C2211yescharacteristic
58C2×#376C2211yescharacteristic
59C22×#95C2211yescharacteristic
60#1446C2212yes—
61#1446C2212yes—
62#1445C2214yes—
63#1445C2212yes—
64#1445C2212yes—
65#1445C2212yes—
66#1445C2212yes—
67#1444C2214yes—
68#1444C2214yes—
69#1444C2212yes—
70#1442C2214yes—
71#1442C2212yes—
72#1441C2214yes—
73#1441C2214yes—
74#1441C2212yes—
75#1441C2212yes—
76#1440C2214yes—
77#1440C2214yes—
78#1439C2214yes—
79#1439C2214yes—
80#1439C2214yes—
81#1436C2214yes—
82#1436C2214yes—
83#1436C2214yes—
84C4⋊#95C2212yes—
85#1434C2212yes—
86C22⋊#95C2214yes—
87C23×Q8C2312yes—
88C23×D4C2311yescharacteristic
89C24×C4C2312yes—
90C24×C4C2311yescharacteristic
91C24×C4C2311yescharacteristic
92C2×C4⋊Q8—24no—
93C2×C4⋊Q8—24no—
94C2×C4⋊Q8—24no—
95C2×#125—24no—
96C2×#125—24no—
97C2×#125—24no—
98C2×C22⋊Q8—28no—
99C2×C22⋊Q8—28no—
100C2×C22⋊Q8—48no—
101C2×C22⋊Q8—48no—
102C2×C4×D4—24no—
103C2×C4×D4—24no—
104C2×C4×D4D418yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2311yescharacteristic
127C22×C22⋊C4C2311yescharacteristic
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2312yes—
131C22×C22⋊C4C2312yes—
132C22×C42C2312yes—
133C22×C42C2312yes—
134C22×C42C2312yes—
135C22×C42C2311yescharacteristic
136#399—24no—
137#399—24no—
138#399—28no—
139#399—28no—
140#398—28no—
141#398—216no—
142#398—48no—
143#397—24no—
144#397—24no—
145#397—48no—
146#395—28no—
147#395—28no—
148#394—28no—
149#390—28no—
150#389—48no—
151#388—28no—
152#387—28no—
153#386—28no—
154#385—28no—
155#385—28no—
156#384—28no—
157#384—28no—
158#383—24no—
159#383—24no—
160#382—28no—
161#381—24no—
162#380—24no—
163#380—48no—
164C4×C4⋊C4—24no—
165C4×C22⋊C4—28no—
166C4×C22⋊C4—24no—
167#376D418yes—
168C2×#95—24no—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2311yescharacteristic
176C2×#95C2311yescharacteristic
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2314yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2311yescharacteristic
184C2×#95C2311yescharacteristic
185C2×#95C2312yes—
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2311yescharacteristic
196C2×#95C2311yescharacteristic
197C2×#95C2312yes—
198C25C2411yescharacteristic
199C25C2411yescharacteristic
200C22×Q8—24no—
201C22×Q8—24no—
202C22×Q8—24no—
203C22×Q8—44no—
204C22×Q8—44no—
205C22×D4—22no—
206C22×D4—22no—
207C22×D4—22no—
208C22×D4—22no—
209C22×D4—41no—
210C22×D4—41no—
211C22×D4C2×D414yes—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—28no—
219C23×C4—44no—
220C23×C4—44no—
221C23×C4—42no—
222C23×C4—44no—
223C23×C4C2×D414yes—
224C23×C4C2×D414yes—
225C23×C4C2412yes—
226C23×C4C2414yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2411yescharacteristic
234C23×C4C2411yescharacteristic
235C23×C4C2414yes—
236C23×C4C2411yescharacteristic
237C23×C4C2411yescharacteristic
238C23×C4C2411yescharacteristic
239C23×C4C2412yes—
240C4⋊Q8—416no—
241#125—416no—
242C22⋊Q8—432no—
243C22⋊Q8—432no—
244C4×D4—232no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—28no—
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—24no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—28no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—28no—
275C2×C4⋊C4—28no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
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⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4C2×D414yes—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—28no—
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—24no—
302C2×C22⋊C4—28no—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—28no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—42no—
309C2×C22⋊C4—42no—
310C2×C22⋊C4—42no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4C2×D414yes—
317C2×C22⋊C4C2×D414yes—
318C2×C42—28no—
319C2×C42—24no—
320C2×C42—24no—
321C2×C42—42no—
322C2×C42—42no—
323C2×C42—44no—
324C2×C42C2×D418yes—
325C2×C42C2×D414yes—
326C2×C42C2×D414yes—
327C2×C42#4114yes—
328#95—22no—
329#95—24no—
330#95—22no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95—28no—
335#95—28no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—22no—
340#95—22no—
341#95—24no—
342#95—24no—
343#95—28no—
344#95—28no—
345#95—28no—
346#95—28no—
347#95—44no—
348#95—44no—
349#95—44no—
350#95—44no—
351#95—44no—
352#95—44no—
353#95—44no—
354#95C2×D414yes—
355#95C2×D414yes—
356#95C2×D418yes—
357C24—21no—
358C24—21no—
359C24—24no—
360C24—24no—
361C24—21no—
362C24—21no—
363C24—21no—
364C24—21no—
365C24—24no—
366C24—24no—
367C24—21no—
368C24—21no—
369C24—21no—
370C24—21no—
371C24C22×D412yes—
372C24C22×D412yes—
373C242−1+412yes—
374C242−1+412yes—
375C24C2511yescharacteristic
376C2×Q8—48no—
377C2×Q8—48no—
378C2×Q8—48no—
379C2×Q8—88no—
380C2×Q8—88no—
381C2×D4—28no—
382C2×D4—28no—
383C2×D4—44no—
384C2×D4—44no—
385C2×D4—44no—
386C2×D4—44no—
387C2×D4—44no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—24no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—28no—
414C22×C4—28no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
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—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4—48no—
457C22×C4—416no—
458C22×C4—48no—
459C22×C4—84no—
460C22×C4—84no—
461C22×C4—84no—
462C22×C4—84no—
463C22×C4—84no—
464C22×C4—88no—
465C22×C4C22×D414yes—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D414yes—
475C22×C4C2×#4112yes—
476C22×C4C2×#4112yes—
477C22×C4C2×#4112yes—
478C22×C4H(5,𝔽2)18yes—
479C22×C42−1+418yes—
480C4⋊C4—216no—
481C4⋊C4—28no—
482C4⋊C4—416no—
483C4⋊C4—416no—
484C4⋊C4—416no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—48no—
488C4⋊C4—48no—
489C4⋊C4—48no—
490C4⋊C4—48no—
491C4⋊C4—88no—
492C4⋊C4—84no—
493C4⋊C4—84no—
494C4⋊C4—84no—
495C4⋊C4—88no—
496C22⋊C4—28no—
497C22⋊C4—28no—
498C22⋊C4—48no—
499C22⋊C4—416no—
500C22⋊C4—416no—
501C22⋊C4—48no—
502C22⋊C4—48no—
503C22⋊C4—48no—
504C22⋊C4—84no—
505C22⋊C4—84no—
506C22⋊C4—84no—
507C42—216no—
508C42—48no—
509C42—84no—
510C42#121116yes—
511C23—22no—
512C23—22no—
513C23—24no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—24no—
519C23—28no—
520C23—28no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—24no—
525C23—24no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—44no—
531C23—44no—
532C23—42no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23#53614yes—
545C23C23×D411yescharacteristic
546C23C23×D411yescharacteristic
547C23C22×#4111yescharacteristic
548C23C2×H(5,𝔽2)14yes—
549C23C2×2−1+411yescharacteristic
550C23C2×2−1+411yescharacteristic
551C23C2×2−1+414yes—
552C23#58511yescharacteristic
553C23#58511yescharacteristic
554Q8—816no—
555D4—416no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—28no—
571C2×C4—28no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—28no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—28no—
581C2×C4—28no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—42no—
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—44no—
630C2×C4—44no—
631C2×C4—88no—
632C2×C4—88no—
633C2×C4—816no—
634C2×C4C2×#12114yes—
635C2×C4C2×#12114yes—
636C2×C4C2×#12114yes—
637C2×C4#53618yes—
638C22—28no—
639C22—24no—
640C22—44no—
641C22—44no—
642C22—44no—
643C22—42no—
644C22—44no—
645C22—44no—
646C22—42no—
647C22—44no—
648C22—44no—
649C22—42no—
650C22—42no—
651C22—42no—
652C22—82no—
653C22—82no—
654C22—82no—
655C22—82no—
656C22C22×#12111yescharacteristic
657C22C2×#53611yescharacteristic
658C22C2×#53612yes—
659C22#344911yescharacteristic
660C22#348412yes—
661C22#348911yescharacteristic
662C22#348911yescharacteristic
663C22#349511yescharacteristic
664C22#349511yescharacteristic
665C22#349812yes—
666C22#349912yes—
667C22#350812yes—
668C22#350912yes—
669C22#353514yes—
670C22#353814yes—
671C22#355214yes—
672C22#355914yes—
673C4—24no—
674C4—28no—
675C4—24no—
676C4—24no—
677C4—48no—
678C4—44no—
679C4—42no—
680C4—42no—
681C4—82no—
682C4—82no—
683C4—82no—
684C4—82no—
685C4—84no—
686C4—82no—
687C4—82no—
688C2—44no—
689C2—82no—
690C2#3530611yescharacteristic
691C2#3531611yescharacteristic
692C2#3552811yescharacteristic
693C2#4049512yes—
694C2#4050412yes—
695C2#4195312yes—
696C2#4291412yes—
697C2#5091412yes—
698C2#5122612yes—
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. C2×#2660C2semidirect
5. #17062C2semidirect
6. #16701C2semidirect
7. #16264C2semidirect
8. #16262C2semidirect
9. #15629C2semidirect
11. #15135C2semidirect
12. #15016C2semidirect
13. #14763C2semidirect
14. #14758C2semidirect
15. #14663C2semidirect
16. #14289C2semidirect
17. #13952C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25422
C2×#26602
C2×#26332
C2×#23532
#170622
#167012
#162642
#162622
#156292
#154792
#151352
#150162
#147632
#147582
#146632
#142892
#139522