Finite Group Databasebeta

← #706050

Subgroups — #706050

Order 512.

2438 conjugacy classes of subgroups.

The subgroup lattice has 679 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2362C211yesmaximal, characteristic
2C2×#2317C211yesmaximal, characteristic
3#17065C212yesmaximal, complemented
4#16700C212yesmaximal, complemented
5#16020C214yesmaximal
6#15972C214yesmaximal
7#15196C212yesmaximal, complemented
8#15193C212yesmaximal, complemented
9#15187C212yesmaximal, complemented
10#15086C212yesmaximal, complemented
11#14258C211yesmaximal, characteristic
12#14139C211yesmaximal, characteristic
13#14130C211yesmaximal, characteristic
14#14046C212yesmaximal, complemented
15#13951C211yesmaximal, characteristic
16#13935C212yesmaximal, complemented
17#13844C211yesmaximal, characteristic
18C22×C22⋊Q8C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C22×#117C2211yescharacteristic
21C22×#117C2211yescharacteristic
22C23×C4⋊C4C2211yescharacteristic
23#2362—28no—
24#2317—28no—
25C2×#391C2211yescharacteristic
26C2×#391C2212yes—
27C2×#391C2212yes—
28C2×#389C2212yes—
29C2×#389C2212yes—
30C2×#389C2211yescharacteristic
31C2×#389C2212yes—
32C2×#389C2212yes—
33C2×#389C2214yes—
34C2×#388C2212yes—
35C2×#388C2212yes—
36C2×#387C2212yes—
37C2×#385C2214yes—
38C2×#385C2212yes—
39C2×#384C2212yes—
40C2×#382C2212yes—
41C2×#380C2211yescharacteristic
42C2×C4×C4⋊C4C2212yes—
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C22⋊C4C2211yescharacteristic
46C2×C4×C22⋊C4C2211yescharacteristic
47#1445C2214yes—
48#1445C2214yes—
49#1445C2214yes—
50#1445C2214yes—
51#1444C2211yescharacteristic
52#1444C2211yescharacteristic
53#1444C2212yes—
54#1444C2214yes—
55#1443C2214yes—
56#1442C2211yescharacteristic
57#1442C2211yescharacteristic
58#1442C2212yes—
59#1442C2214yes—
60#1442C2214yes—
61#1441C2212yes—
62#1441C2212yes—
63#1441C2214yes—
64#1441C2214yes—
65#1441C2212yes—
66#1441C2214yes—
67#1441C2212yes—
68#1441C2214yes—
69#1440C2214yes—
70#1439C2212yes—
71#1439C2212yes—
72#1439C2214yes—
73#1436C2214yes—
74C4⋊#95C2212yes—
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2214yes—
80#1434C2214yes—
81#1434C2212yes—
82C22⋊#95C2212yes—
83C22⋊#95C2214yes—
84C4×#95C2212yes—
85C23×Q8C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2312yes—
88C2×C22⋊Q8—24no—
89C2×C22⋊Q8—24no—
90C2×C22⋊Q8Q818yes—
91C2×C4×Q8—24no—
92C2×C4×Q8—28no—
93C2×#117—24no—
94C2×#117—28no—
95C2×#117—44no—
96C2×#117D418yes—
97C22×C4⋊C4—24no—
98C22×C4⋊C4—44no—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2311yescharacteristic
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2311yescharacteristic
118C22×C22⋊C4C2311yescharacteristic
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C42C2312yes—
124C22×C42C2311yescharacteristic
125C22×C42C2312yes—
126C22×C42C2312yes—
127C22×C42C2311yescharacteristic
128C22×C42C2312yes—
129C22×C42C2312yes—
130#391—24no—
131#391—44no—
132#391—44no—
133#389—28no—
134#389—28no—
135#389—24no—
136#389—28no—
137#389—44no—
138#389—48no—
139#388—28no—
140#388—28no—
141#387—28no—
142#385—216no—
143#385—44no—
144#384—28no—
145#382—28no—
146#380—24no—
147C4×C4⋊C4—24no—
148C4×C4⋊C4—28no—
149C4×C4⋊C4—44no—
150C4×C22⋊C4—24no—
151C4×C22⋊C4—24no—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2311yescharacteristic
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2311yescharacteristic
159C2×#95C2312yes—
160C2×#95C2311yescharacteristic
161C2×#95C2314yes—
162C2×#95C2311yescharacteristic
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C25C2411yescharacteristic
184C22×Q8—22no—
185C22×Q8—24no—
186C22×Q8—22no—
187C22×Q8—24no—
188C22×Q8C2×Q814yes—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—44no—
193C23×C4—42no—
194C23×C4—44no—
195C23×C4—42no—
196C23×C4C2×D414yes—
197C23×C4C2×Q814yes—
198C23×C4#4114yes—
199C23×C4C2412yes—
200C23×C4C2414yes—
201C23×C4C2411yescharacteristic
202C23×C4C2412yes—
203C23×C4C2411yescharacteristic
204C23×C4C2412yes—
205C23×C4C2414yes—
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2411yescharacteristic
210C23×C4C2412yes—
211C23×C4C2411yescharacteristic
212C23×C4C2411yescharacteristic
213C23×C4C2412yes—
214C23×C4C2411yescharacteristic
215C22⋊Q8—232no—
216C4×Q8—416no—
217#117—232no—
218#117—416no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—416no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4C2×D414yes—
261C2×C4⋊C4C2×D414yes—
262C2×C4⋊C4C2×Q814yes—
263C2×C4⋊C4C2×Q818yes—
264C2×C4⋊C4#4118yes—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—42no—
277C2×C22⋊C4—41no—
278C2×C22⋊C4—41no—
279C2×C22⋊C4—42no—
280C2×C22⋊C4—42no—
281C2×C22⋊C4—42no—
282C2×C22⋊C4—42no—
283C2×C22⋊C4C2×D418yes—
284C2×C22⋊C4C2×Q818yes—
285C2×C42—28no—
286C2×C42—24no—
287C2×C42—22no—
288C2×C42—24no—
289C2×C42—22no—
290C2×C42—24no—
291C2×C42—28no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—42no—
301C2×C42C2×D418yes—
302#95—22no—
303#95—22no—
304#95—24no—
305#95—24no—
306#95—28no—
307#95—28no—
308#95—24no—
309#95—24no—
310#95—22no—
311#95—22no—
312#95—24no—
313#95—28no—
314#95—24no—
315#95—24no—
316#95—42no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95—42no—
323#95—42no—
324#95—42no—
325#95—44no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95—44no—
333C24—22no—
334C24—21no—
335C24—22no—
336C24—21no—
337C24—22no—
338C24—21no—
339C24—21no—
340C24—22no—
341C24C22×D412yes—
342C24C22×Q812yes—
343C24C2×#4112yes—
344C24C2511yescharacteristic
345C2×Q8—28no—
346C2×Q8—216no—
347C2×Q8—44no—
348C2×Q8—44no—
349C2×Q8—48no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—28no—
356C22×C4—28no—
357C22×C4—216no—
358C22×C4—216no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—48no—
433C22×C4—84no—
434C22×C4—84no—
435C22×C4—88no—
436C22×C4—88no—
437C22×C4C22⋊Q8116yes—
438C22×C4C22×D414yes—
439C22×C4C22×D412yes—
440C22×C4C22×D414yes—
441C22×C4C22×D412yes—
442C22×C4C22×Q814yes—
443C22×C4C22×Q814yes—
444C22×C4C22×Q812yes—
445C22×C4C22×Q812yes—
446C22×C4C2×#4112yes—
447C22×C4C2×#4118yes—
448C22×C4C2×#4112yes—
449C22×C4C2×#4114yes—
450C22×C4H(5,𝔽2)12yes—
451C22×C4H(5,𝔽2)14yes—
452C22×C42−1+412yes—
453C22×C42−1+414yes—
454C4⋊C4—28no—
455C4⋊C4—216no—
456C4⋊C4—28no—
457C4⋊C4—416no—
458C4⋊C4—48no—
459C4⋊C4—48no—
460C4⋊C4—48no—
461C4⋊C4—48no—
462C4⋊C4—44no—
463C4⋊C4—44no—
464C4⋊C4—48no—
465C4⋊C4—48no—
466C4⋊C4—48no—
467C4⋊C4—88no—
468C4⋊C4—88no—
469C4⋊C4—88no—
470C4⋊C4#123116yes—
471C22⋊C4—216no—
472C22⋊C4—216no—
473C22⋊C4—44no—
474C22⋊C4—44no—
475C22⋊C4—48no—
476C22⋊C4—48no—
477C22⋊C4—48no—
478C42—216no—
479C42—48no—
480C42—48no—
481C42—44no—
482C42—44no—
483C42—48no—
484C42—48no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—24no—
490C23—24no—
491C23—22no—
492C23—22no—
493C23—24no—
494C23—22no—
495C23—22no—
496C23—24no—
497C23—22no—
498C23—22no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23C2×C22⋊Q814yes—
508C23C23×D411yescharacteristic
509C23C23×Q811yescharacteristic
510C23C22×#4111yescharacteristic
511C23C22×#4112yes—
512C23C2×H(5,𝔽2)12yes—
513C23C2×H(5,𝔽2)11yescharacteristic
514C23C2×2−1+411yescharacteristic
515C23C2×2−1+412yes—
516C23#58511yescharacteristic
517C23#58511yescharacteristic
518C23#58512yes—
519Q8—416no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—28no—
527C2×C4—28no—
528C2×C4—232no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—28no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—41no—
545C2×C4—42no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—816no—
609C2×C4C2×C22⋊Q814yes—
610C2×C4C2×C22⋊Q814yes—
611C2×C4C2×#12318yes—
612C2×C4C2×#12314yes—
613C2×C4#53514yes—
614C2×C4#55318yes—
615C2×C4#55418yes—
616C22—24no—
617C22—24no—
618C22—24no—
619C22—44no—
620C22—44no—
621C22—42no—
622C22—44no—
623C22—42no—
624C22—42no—
625C22—44no—
626C22—42no—
627C22C22×C22⋊Q811yescharacteristic
628C22C22×#12311yescharacteristic
629C22C2×#53511yescharacteristic
630C22#346011yescharacteristic
631C22#346111yescharacteristic
632C22C2×#55312yes—
633C22C2×#55412yes—
634C22#348811yescharacteristic
635C22#348911yescharacteristic
636C22#349512yes—
637C22#349911yescharacteristic
638C22#350011yescharacteristic
639C22#350212yes—
640C22#350311yescharacteristic
641C22#350411yescharacteristic
642C22#351512yes—
643C22#351612yes—
644C22#352012yes—
645C22#352412yes—
646C22#354112yes—
647C22#354212yes—
648C22#355012yes—
649C22#355112yes—
650C4—28no—
651C4—24no—
652C4—44no—
653C4—42no—
654C4—44no—
655C4—44no—
656C4—44no—
657C4—42no—
658C4—82no—
659C4—84no—
660C4—81no—
661C4—82no—
662C4—81no—
663C4—82no—
664C4—84no—
665C4#248718yes—
666C2—44no—
667C2C2×#248711yescharacteristic
668C2#3525711yescharacteristic
669C2#3566811yescharacteristic
670C2#3598311yescharacteristic
671C2#3599811yescharacteristic
672C2#3821712yes—
673C2#3905712yes—
674C2#4291111yescharacteristic
675C2#4296211yescharacteristic
676C2#4499712yes—
677C2#4518012yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #17065C2semidirect
4. #16700C2semidirect
7. #15196C2semidirect
8. #15193C2semidirect
9. #15187C2semidirect
10. #15086C2semidirect
14. #14046C2semidirect
16. #13935C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23622
C2×#23172
#170652
#167002
#160202
#159722
#151962
#151932
#151872
#150862
#142582
#141392
#141302
#140462
#139512
#139352
#138442