Finite Group Databasebeta

← #5956441

Subgroups — #5956441

Order 512.

3822 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#541C211yesmaximal, characteristic, complemented
2C2×#2323C212yesmaximal, complemented
3C2×#2293C211yesmaximal, characteristic
4#15218C211yesmaximal, characteristic, complemented
5#15216C211yesmaximal, characteristic, complemented
6#15180C211yesmaximal, characteristic, complemented
7#15179C212yesmaximal, complemented
8#15178C211yesmaximal, characteristic, complemented
9#15177C212yesmaximal
10#14933C214yesmaximal, complemented
11#14931C214yesmaximal, complemented
12#14930C214yesmaximal
13#14929C214yesmaximal, complemented
14#13777C211yesmaximal, characteristic, complemented
15C4×#391C212yesmaximal, complemented
16C2×#541—216no—
17C2×#541—28no—
18C22×C4⋊Q8C2211yescharacteristic
19C22×#126C2214yes—
20C22×#125C2214yes—
21C22×#125C2211yescharacteristic
22C22×C22⋊Q8C2212yes—
23C22×C22⋊Q8C2212yes—
24C22×C4×Q8C2212yes—
25C22×C4×D4C2211yescharacteristic
26C22×#117C2211yescharacteristic
27C22×#117C2211yescharacteristic
28#2323—216no—
29C2×#391C2212yes—
30C2×#391C2212yes—
31C2×#391C2212yes—
32#2293—28no—
33C2×#390C2212yes—
34C2×#389C2212yes—
35C2×#389C2214yes—
36C2×#389C2212yes—
37C2×#388C2214yes—
38C2×#388C2214yes—
39C2×#388C2212yes—
40C2×#388C2212yes—
41C2×#387C2214yes—
42C2×#380C2212yes—
43C2×#380C2212yes—
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C22⋊C4C2211yescharacteristic
47C2×C4×C22⋊C4C2211yescharacteristic
48C2×C43C2211yescharacteristic
49#1444C2214yes—
50#1444C2214yes—
51#1444C2214yes—
52#1442C2214yes—
53#1442C2214yes—
54#1442C2214yes—
55#1441C2214yes—
56#1441C2214yes—
57#1441C2214yes—
58#1440C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1439C2214yes—
63#1439C2214yes—
64#1438C2214yes—
65#1437C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68#1436C2214yes—
69C4⋊#95C2212yes—
70C4⋊#95C2211yescharacteristic
71C4⋊#95C2212yes—
72C4⋊#95C2211yescharacteristic
73C4×#95C2214yes—
74C4×#95C2214yes—
75C23×Q8C2312yes—
76C23×D4C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79#541—2128no—
80C2×C4⋊Q8—28no—
81C2×C4⋊Q8—24no—
82C2×#126—216no—
83C2×#126—216no—
84C2×#126—216no—
85C2×#125—28no—
86C2×#125—216no—
87C2×#125—216no—
88C2×#125—24no—
89C2×#125—216no—
90C2×C22⋊Q8—28no—
91C2×C22⋊Q8—28no—
92C2×C22⋊Q8—28no—
93C2×C22⋊Q8—28no—
94C2×C22⋊Q8—28no—
95C2×C22⋊Q8—28no—
96C2×C4×Q8—28no—
97C2×C4×Q8—28no—
98C2×C4×Q8D4116yescomplemented
99C2×C4×D4—24no—
100C2×C4×D4—24no—
101C2×C4×D4—24no—
102C2×#117—28no—
103C2×#117—28no—
104C2×#117—24no—
105C2×#117D418yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2311yescharacteristic
121C22×C22⋊C4C2311yescharacteristic
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2314yes—
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131C22×C42C2312yes—
132C22×C42C2312yes—
133C22×C42C2311yescharacteristic
134C22×C42C2314yes—
135C22×C42C2311yescharacteristic
136C22×C42C2314yes—
137C22×C42C2311yescharacteristic
138#391—28no—
139#391—28no—
140#391—28no—
141#390—28no—
142#389—216no—
143#389—28no—
144#389—28no—
145#388—216no—
146#388—28no—
147#388—28no—
148#388—216no—
149#387—216no—
150#380—28no—
151#380—28no—
152C4×C4⋊C4—28no—
153C4×C4⋊C4—24no—
154C4×C22⋊C4—24no—
155C4×C22⋊C4—24no—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C43—24no—
176C25C2411yescharacteristic
177C25C2411yescharacteristic
178C22×Q8—24no—
179C22×Q8—24no—
180C22×Q8—24no—
181C22×Q8—24no—
182C22×Q8—24no—
183C22×Q8—24no—
184C22×Q8C2×D418yes—
185C22×D4—22no—
186C22×D4—22no—
187C22×D4—22no—
188C22×D4—24no—
189C22×D4—42no—
190C23×C4—22no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4C2×D414yes—
199C23×C4#4114yes—
200C23×C4C2412yes—
201C23×C4C2414yes—
202C23×C4C2414yes—
203C23×C4C2412yes—
204C23×C4C2414yes—
205C23×C4C2411yescharacteristic
206C23×C4C2414yes—
207C23×C4C2411yescharacteristic
208C23×C4C2411yescharacteristic
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2411yescharacteristic
212C23×C4C2411yescharacteristic
213C4⋊Q8—232no—
214#126—2128no—
215#125—232no—
216#125—2128no—
217C22⋊Q8—264no—
218C22⋊Q8—264no—
219C4×Q8—264no—
220C4×D4—232no—
221#117—232no—
222#117—416no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4C2×D418yes—
259C2×C4⋊C4C2×D418yes—
260C2×C4⋊C4C2×D4116yes—
261C2×C4⋊C4#4118yes—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—42no—
286C2×C22⋊C4—42no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4C2×D418yes—
292C2×C42—24no—
293C2×C42—22no—
294C2×C42—28no—
295C2×C42—28no—
296C2×C42—22no—
297C2×C42—22no—
298C2×C42—24no—
299C2×C42—22no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42—28no—
306C2×C42—42no—
307C2×C42—42no—
308C2×C42—44no—
309C2×C42—44no—
310C2×C42C2×D414yes—
311C2×C42C2×D4116yes—
312C2×C42C2×D418yes—
313C2×C42C2×D414yes—
314C2×C42#4114yes—
315#95—24no—
316#95—24no—
317#95—28no—
318#95—24no—
319#95—28no—
320#95—28no—
321#95—28no—
322#95—28no—
323#95—28no—
324#95—28no—
325#95—28no—
326#95—28no—
327#95—28no—
328#95—28no—
329#95—28no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—44no—
334C24—22no—
335C24—22no—
336C24—21no—
337C24—22no—
338C24—22no—
339C24—21no—
340C24—21no—
341C24—21no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24C22×D412yes—
351C24C2×#4112yes—
352C24C2511yescharacteristic
353C2×Q8—216no—
354C2×Q8—216no—
355C2×Q8—216no—
356C2×Q8—48no—
357C2×Q8—48no—
358C2×Q8—48no—
359C2×Q8—48no—
360C2×D4—28no—
361C2×D4—48no—
362C2×D4—48no—
363C2×D4—44no—
364C2×D4—44no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—21no—
390C22×C4—24no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—28no—
415C22×C4—28no—
416C22×C4—28no—
417C22×C4—216no—
418C22×C4—28no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—44no—
434C22×C4—48no—
435C22×C4—44no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4C22×D418yes—
439C22×C4C22×D418yes—
440C22×C4C22×D414yes—
441C22×C4C22×D412yes—
442C22×C4C22×D414yes—
443C22×C4C22×D414yes—
444C22×C4C22×D414yes—
445C22×C4C22×D412yes—
446C22×C4C22×D414yes—
447C22×C4C2×#4112yes—
448C22×C4C2×#4118yes—
449C22×C4C2×#4112yes—
450C22×C4C2×#4112yes—
451C22×C4C2×#4112yes—
452C22×C4C2×#4114yes—
453C22×C4C2×#4112yes—
454C22×C42−1+414yes—
455C22×C42−1+414yes—
456C4⋊C4—28no—
457C4⋊C4—232no—
458C4⋊C4—232no—
459C4⋊C4—232no—
460C4⋊C4—232no—
461C4⋊C4—216no—
462C4⋊C4—216no—
463C4⋊C4—232no—
464C4⋊C4—216no—
465C4⋊C4—48no—
466C4⋊C4—48no—
467C4⋊C4—48no—
468C4⋊C4—88no—
469C22⋊C4—28no—
470C22⋊C4—28no—
471C22⋊C4—232no—
472C22⋊C4—232no—
473C22⋊C4—216no—
474C22⋊C4—48no—
475C22⋊C4—48no—
476C22⋊C4—48no—
477C22⋊C4—88no—
478C22⋊C4—88no—
479C42—28no—
480C42—232no—
481C42—28no—
482C42—48no—
483C42—48no—
484C42—44no—
485C42—44no—
486C42—88no—
487C42#120116yes—
488C23—22no—
489C23—24no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—24no—
494C23—24no—
495C23—24no—
496C23—22no—
497C23—22no—
498C23—41no—
499C23—42no—
500C23—41no—
501C23—42no—
502C23—42no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—41no—
521C23—41no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23C23×D411yescharacteristic
531C23C23×D412yes—
532C23C22×#4111yescharacteristic
533C23C22×#4111yescharacteristic
534C23C22×#4112yes—
535C23C2×2−1+412yes—
536C23C2×2−1+412yes—
537C23#58512yes—
538C23#58512yes—
539Q8—432no—
540D4—416no—
541C2×C4—22no—
542C2×C4—22no—
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—24no—
552C2×C4—28no—
553C2×C4—28no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—28no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—28no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—28no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—28no—
568C2×C4—28no—
569C2×C4—28no—
570C2×C4—24no—
571C2×C4—28no—
572C2×C4—24no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—44no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—416no—
602C2×C4—416no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4C2×#12018yes—
618C2×C4C2×#12014yes—
619C2×C4#52514yes—
620C2×C4#52518yes—
621C2×C4#53214yes—
622C2×C4#53218yes—
623C2×C4#54114yescomplemented
624C22—44no—
625C22—44no—
626C22—44no—
627C22—44no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—82no—
635C22—82no—
636C22—82no—
637C22—82no—
638C22—82no—
639C22—82no—
640C22—82no—
641C22—82no—
642C22C22×#12011yescharacteristic
643C22C2×#52512yes—
644C22C2×#52511yescharacteristic
645C22C2×#53211yescharacteristic
646C22C2×#53212yes—
647C22C2×#54111yescharacteristic
648C22#345511yescharacteristic
649C22#346112yes—
650C22#348912yes—
651C22#348912yes—
652C22#349512yes—
653C22#349712yes—
654C22#349712yes—
655C22#349912yes—
656C22#350012yes—
657C22#350911yescharacteristic
658C22#351211yescharacteristic
659C22#351512yes—
660C22#351612yes—
661C22#352112yes—
662C22#352312yes—
663C4—28no—
664C4—24no—
665C4—48no—
666C4—44no—
667C4—44no—
668C4—48no—
669C4—42no—
670C4—42no—
671C4—48no—
672C4—42no—
673C4—82no—
674C4—84no—
675C4#238818yes—
676C2—82no—
677C2—82no—
678C2C2×#238811yescharacteristic
679C2#3520912yes—
680C2#3522012yes—
681C2#3530911yescharacteristic
682C2#3531011yescharacteristic
683C2#3562512yes—
684C2#3563512yes—
685C2#3566012yes—
686C2#3567112yes—
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. C22×#541C2semidirect
2. C2×#2323C2semidirect
4. #15218C2semidirect
5. #15216C2semidirect
6. #15180C2semidirect
7. #15179C2semidirect
8. #15178C2semidirect
10. #14933C2semidirect
11. #14931C2semidirect
13. #14929C2semidirect
14. #13777C2semidirect
15. C4×#391C2semidirect
98. C2×C4×Q8D4semidirect
623. C2×C4#541semidirect

Maximal subgroups

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

subgroupindex
C22×#5412
C2×#23232
C2×#22932
#152182
#152162
#151802
#151792
#151782
#151772
#149332
#149312
#149302
#149292
#137772
C4×#3912