Finite Group Databasebeta

← #4420467

Subgroups — #4420467

Order 512.

2656 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2567C211yesmaximal, characteristic, complemented
2C2×#2658C211yesmaximal, characteristic
3C2×#2632C211yesmaximal, characteristic
4C2×#2356C211yesmaximal, characteristic, complemented
5#16676C212yesmaximal, complemented
6#16564C212yesmaximal, complemented
7#16507C212yesmaximal, complemented
8#16505C212yesmaximal
9#15765C214yesmaximal, complemented
10#15499C214yesmaximal, complemented
11#15128C212yesmaximal, complemented
12#15070C212yesmaximal
13#14910C212yesmaximal, complemented
14#14870C212yesmaximal, complemented
15#14663C211yesmaximal, characteristic, complemented
16#14288C211yesmaximal, characteristic
17#13952C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4⋊Q8C2212yes—
20C22×#124C2211yescharacteristic
21C22×#124C2212yes—
22C22×C4×Q8C2211yescharacteristic
23#2658—28no—
24#2632—28no—
25#2567—28no—
26C2×#399C2212yes—
27C2×#399C2212yes—
28C2×#399C2212yes—
29C2×#399C2214yes—
30C2×#398C2214yes—
31C2×#396C2214yes—
32C2×#395C2211yescharacteristic
33C2×#395C2212yes—
34C2×#395C2211yescharacteristic
35C2×#395C2212yes—
36C2×#394C2211yescharacteristic
37C2×#394C2211yescharacteristic
38C2×#393C2212yes—
39#2356—28no—
40C2×#391C2212yes—
41C2×#391C2212yes—
42C2×#389C2212yes—
43C2×#389C2212yes—
44C2×#389C2214yes—
45C2×#388C2212yes—
46C2×#388C2212yes—
47C2×#388C2214yes—
48C2×#387C2212yes—
49C2×#387C2212yes—
50C2×#384C2212yes—
51C2×#383C2211yescharacteristic
52C2×#383C2211yescharacteristic
53C2×#383C2211yescharacteristic
54C2×#382C2211yescharacteristic
55C2×C4×C4⋊C4C2211yescharacteristic
56C2×C4×C4⋊C4C2212yes—
57C2×C4×C4⋊C4C2211yescharacteristic
58C2×#376C2211yescharacteristic
59C2×#376C2211yescharacteristic
60#1446C2212yes—
61#1446C2212yes—
62#1446C2212yes—
63#1445C2212yes—
64#1445C2212yes—
65#1445C2214yes—
66#1445C2214yes—
67#1444C2212yes—
68#1444C2214yes—
69#1444C2214yes—
70#1442C2214yes—
71#1442C2212yes—
72#1441C2214yes—
73#1440C2214yes—
74#1439C2212yes—
75#1439C2214yes—
76#1439C2214yes—
77#1439C2212yes—
78#1439C2214yes—
79#1439C2214yes—
80#1438C2212yes—
81#1437C2214yes—
82#1437C2214yes—
83#1436C2214yes—
84C4⋊#95C2214yes—
85C4⋊#95C2212yes—
86#1434C2212yes—
87C23×Q8C2311yescharacteristic
88C23×Q8C2311yescharacteristic
89C23×Q8C2312yes—
90C23×D4C2311yescharacteristic
91C2×C4⋊Q8—24no—
92C2×C4⋊Q8—28no—
93C2×C4⋊Q8—24no—
94C2×C4⋊Q8—24no—
95C2×C4⋊Q8—48no—
96C2×#124—24no—
97C2×#124—28no—
98C2×#124—24no—
99C2×#124—24no—
100C2×#124—48no—
101C2×C4×Q8—24no—
102C2×C4×Q8—24no—
103C2×C4×Q8D418yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2314yes—
129C22×C42C2312yes—
130C22×C42C2312yes—
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2312yes—
134C22×C42C2312yes—
135C22×C42C2311yescharacteristic
136#399—28no—
137#399—28no—
138#399—28no—
139#399—216no—
140#398—48no—
141#396—48no—
142#395—28no—
143#395—28no—
144#395—24no—
145#395—24no—
146#394—24no—
147#394—24no—
148#393—28no—
149#391—28no—
150#391—28no—
151#389—28no—
152#389—28no—
153#389—48no—
154#388—28no—
155#388—28no—
156#388—48no—
157#387—28no—
158#387—28no—
159#384—28no—
160#383—24no—
161#383—24no—
162#383—24no—
163#382—24no—
164C4×C4⋊C4—24no—
165C4×C4⋊C4—28no—
166C4×C4⋊C4—24no—
167#376—24no—
168#376D418yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2311yescharacteristic
173C2×#95C2311yescharacteristic
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2311yescharacteristic
184C2×#95C2312yes—
185C2×#95C2311yescharacteristic
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2312yes—
196C2×#95C2311yescharacteristic
197C2×#95C2311yescharacteristic
198C25C2412yes—
199C22×Q8—22no—
200C22×Q8—24no—
201C22×Q8—22no—
202C22×Q8—22no—
203C22×Q8—24no—
204C22×Q8—22no—
205C22×Q8—22no—
206C22×Q8—24no—
207C22×Q8—22no—
208C22×Q8—22no—
209C22×Q8—41no—
210C22×Q8—41no—
211C22×Q8—44no—
212C22×Q8—44no—
213C22×Q8—44no—
214C22×Q8C2×D414yes—
215C22×D4—22no—
216C22×D4—22no—
217C22×D4—22no—
218C22×D4—44no—
219C23×C4C2412yes—
220C23×C4C2414yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C23×C4C2412yes—
228C23×C4C2411yescharacteristic
229C23×C4C2414yes—
230C23×C4C2411yescharacteristic
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233C23×C4C2412yes—
234C23×C4C2411yescharacteristic
235C4⋊Q8—432no—
236C4⋊Q8—416no—
237#124—432no—
238#124—416no—
239C4×Q8—232no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4C2×D414yes—
279C2×C4⋊C4C2×D414yes—
280C2×C4⋊C4C2×D414yes—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—42no—
301C2×C22⋊C4—42no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—28no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—28no—
318C2×C42—24no—
319C2×C42—42no—
320C2×C42—42no—
321C2×C42—42no—
322C2×C42C2×D414yes—
323C2×C42C2×D414yes—
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—24no—
335#95—24no—
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—22no—
358C24—22no—
359C24—22no—
360C24—24no—
361C24—24no—
362C24—22no—
363C24—22no—
364C24—24no—
365C24—24no—
366C24—22no—
367C24—22no—
368C24C2511yescharacteristic
369C2×Q8—28no—
370C2×Q8—28no—
371C2×Q8—44no—
372C2×Q8—44no—
373C2×Q8—44no—
374C2×Q8—44no—
375C2×Q8—44no—
376C2×Q8—48no—
377C2×Q8—48no—
378C2×Q8—48no—
379C2×Q8—44no—
380C2×Q8—48no—
381C2×Q8—84no—
382C2×Q8—88no—
383C2×Q8—84no—
384C2×Q8—88no—
385C2×D4—48no—
386C2×D4—44no—
387C2×D4—84no—
388C2×D4—84no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—24no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4C22×D412yes—
457C22×C4C22×D414yes—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D414yes—
467C22×C4C22×D412yes—
468C22×C4C2×#4112yes—
469C22×C4C2×#4112yes—
470C22×C4C2×#4112yes—
471C22×C4H(5,𝔽2)18yes—
472C22×C42−1+418yes—
473C22×C42−1+412yes—
474C22×C42−1+412yes—
475C4⋊C4—216no—
476C4⋊C4—28no—
477C4⋊C4—28no—
478C4⋊C4—28no—
479C4⋊C4—48no—
480C4⋊C4—416no—
481C4⋊C4—416no—
482C4⋊C4—48no—
483C4⋊C4—48no—
484C4⋊C4—48no—
485C4⋊C4—416no—
486C4⋊C4—88no—
487C4⋊C4—88no—
488C4⋊C4—84no—
489C4⋊C4—84no—
490C22⋊C4—48no—
491C22⋊C4—416no—
492C22⋊C4—416no—
493C22⋊C4—48no—
494C22⋊C4—48no—
495C22⋊C4—416no—
496C22⋊C4—48no—
497C22⋊C4—84no—
498C22⋊C4—84no—
499C22⋊C4—88no—
500C42—28no—
501C42—28no—
502C42—216no—
503C42—48no—
504C42—48no—
505C42—48no—
506C42#121116yes—
507C23—42no—
508C23—44no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23C23×D411yescharacteristic
531C23C23×D411yescharacteristic
532C23C22×#4111yescharacteristic
533C23C2×H(5,𝔽2)14yes—
534C23C2×2−1+411yescharacteristic
535C23C2×2−1+411yescharacteristic
536C23C2×2−1+414yes—
537C23#58511yescharacteristic
538C23#58511yescharacteristic
539Q8—416no—
540Q8—816no—
541Q8—88no—
542D4—88no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—28no—
564C2×C4—28no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—28no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—28no—
577C2×C4—28no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
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—44no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
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—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4C2×#12114yes—
625C2×C4C2×#12114yes—
626C2×C4C2×#12114yes—
627C2×C4#53518yes—
628C2×C4#53614yes—
629C22—84no—
630C22—82no—
631C22—82no—
632C22—84no—
633C22—82no—
634C22—84no—
635C22—82no—
636C22—82no—
637C22—82no—
638C22—84no—
639C22—82no—
640C22C22×#12111yescharacteristic
641C22C2×#53512yes—
642C22C2×#53611yescharacteristic
643C22#344911yescharacteristic
644C22#348612yes—
645C22#348911yescharacteristic
646C22#348911yescharacteristic
647C22#349511yescharacteristic
648C22#349511yescharacteristic
649C22#349812yes—
650C22#349912yes—
651C22#350812yes—
652C22#350912yes—
653C22#353514yes—
654C22#354114yes—
655C22#354414yes—
656C22#355014yes—
657C4—28no—
658C4—24no—
659C4—24no—
660C4—24no—
661C4—24no—
662C4—44no—
663C4—42no—
664C4—42no—
665C4—48no—
666C4—42no—
667C4—82no—
668C4—82no—
669C4—82no—
670C4—82no—
671C4—84no—
672C4—82no—
673C2—162no—
674C2#3527211yescharacteristic
675C2#3530611yescharacteristic
676C2#3566711yescharacteristic
677C2#4195712yes—
678C2#4229712yes—
679C2#4278912yes—
680C2#4291112yes—
681C2#5091712yes—
682C2#5122512yes—
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×#2567C2semidirect
4. C2×#2356C2semidirect
5. #16676C2semidirect
6. #16564C2semidirect
7. #16507C2semidirect
9. #15765C2semidirect
10. #15499C2semidirect
11. #15128C2semidirect
13. #14910C2semidirect
14. #14870C2semidirect
15. #14663C2semidirect
17. #13952C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25672
C2×#26582
C2×#26322
C2×#23562
#166762
#165642
#165072
#165052
#157652
#154992
#151282
#150702
#149102
#148702
#146632
#142882
#139522