Finite Group Databasebeta

← #729121

Subgroups — #729121

Order 512.

2708 conjugacy classes of subgroups.

The subgroup lattice has 705 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2487C211yesmaximal, characteristic
2C22×#400C211yesmaximal, characteristic, complemented
3C2×#2353C211yesmaximal, characteristic
4C2×#2322C211yesmaximal, characteristic
5#16678C212yesmaximal, complemented
6#16394C212yesmaximal, complemented
7#15758C214yesmaximal, complemented
8#15530C214yesmaximal, complemented
9#14945C212yesmaximal
10#14927C212yesmaximal
11#14908C212yesmaximal, complemented
12#14841C212yesmaximal, complemented
13#14339C211yesmaximal, characteristic, complemented
14#14149C212yesmaximal, complemented
15#13992C212yesmaximal, complemented
16#13924C211yesmaximal, characteristic, complemented
17#13908C211yesmaximal, characteristic, complemented
18C22×#123C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22#2487—28no—
23C2×#400—28no—
24C2×#400C2212yes—
25C2×#400C2212yes—
26C2×#400C2214yes—
27C2×#400C2212yes—
28C2×#399C2212yes—
29#2353—28no—
30#2322—28no—
31C2×#389C2212yes—
32C2×#389C2212yes—
33C2×#388C2212yes—
34C2×#388C2212yes—
35C2×#387C2212yes—
36C2×#387C2212yes—
37C2×#386C2211yescharacteristic
38C2×#386C2212yes—
39C2×#386C2212yes—
40C2×#385C2212yes—
41C2×#385C2211yescharacteristic
42C2×#385C2212yes—
43C2×#383C2211yescharacteristic
44C2×#381C2212yes—
45C2×#381C2211yescharacteristic
46C2×#381C2211yescharacteristic
47C2×#380C2211yescharacteristic
48C2×#380C2211yescharacteristic
49C2×#380C2212yes—
50C2×#380C2212yes—
51C2×C4×C22⋊C4C2211yescharacteristic
52C2×C4×C22⋊C4C2212yes—
53C2×C4×C22⋊C4C2211yescharacteristic
54C2×#376C2211yescharacteristic
55C22×#95C2212yes—
56C22×#95C2212yes—
57C22×#95C2211yescharacteristic
58#1445C2212yes—
59#1445C2212yes—
60#1444C2212yes—
61#1443C2214yes—
62#1443C2214yes—
63#1443C2214yes—
64#1442C2214yes—
65#1442C2214yes—
66#1442C2212yes—
67#1441C2212yes—
68#1441C2212yes—
69#1441C2214yes—
70#1441C2214yes—
71#1441C2214yes—
72#1441C2214yes—
73#1440C2214yes—
74#1440C2214yes—
75#1440C2214yes—
76#1439C2214yes—
77#1436C2214yes—
78#1436C2214yes—
79#1436C2214yes—
80#1436C2214yes—
81C4⋊#95C2214yes—
82#1434C2214yes—
83C22⋊#95C2212yes—
84C22⋊#95C2212yes—
85C22⋊#95C2212yes—
86C22⋊#95C2212yes—
87C23×D4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2312yes—
91C24×C4C2311yescharacteristic
92C24×C4C2312yes—
93C24×C4C2311yescharacteristic
94C2×#123—24no—
95C2×#123—24no—
96C2×#123—24no—
97C2×C4×D4—24no—
98C2×C4×D4—24no—
99C2×C4×D4D418yes—
100C22×C4⋊C4—24no—
101C22×C4⋊C4—24no—
102C22×C4⋊C4—24no—
103C22×C4⋊C4—24no—
104C22×C4⋊C4—24no—
105C22×C4⋊C4—24no—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2311yescharacteristic
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2311yescharacteristic
122C22×C22⋊C4C2311yescharacteristic
123C22×C22⋊C4C2311yescharacteristic
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2312yes—
128C22×C42C2311yescharacteristic
129C22×C42C2312yes—
130C22×C42C2312yes—
131C22×C42C2312yes—
132#400—28no—
133#400—28no—
134#400—28no—
135#400—216no—
136#399—28no—
137#389—28no—
138#389—28no—
139#388—28no—
140#388—28no—
141#387—28no—
142#387—28no—
143#386—24no—
144#386—28no—
145#386—28no—
146#385—28no—
147#385—24no—
148#385—28no—
149#383—24no—
150#381—28no—
151#381—24no—
152#381—24no—
153#380—24no—
154#380—28no—
155#380—24no—
156#380—28no—
157C4×C22⋊C4—24no—
158C4×C22⋊C4—28no—
159C4×C22⋊C4—24no—
160#376—24no—
161C2×#95—28no—
162C2×#95—28no—
163C2×#95D418yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2311yescharacteristic
181C2×#95C2311yescharacteristic
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2312yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2312yes—
194C2×#95C2311yescharacteristic
195C2×#95C2311yescharacteristic
196C25C2411yescharacteristic
197C25C2411yescharacteristic
198C22×D4—22no—
199C22×D4—22no—
200C22×D4—22no—
201C22×D4—22no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4C2×D414yes—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—24no—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—24no—
219C23×C4—22no—
220C23×C4—24no—
221C23×C4—44no—
222C23×C4—44no—
223C23×C4—44no—
224C23×C4—44no—
225C23×C4—44no—
226C23×C4—44no—
227C23×C4—44no—
228C23×C4C2×D414yes—
229C23×C4C2×D418yes—
230C23×C4C2×D414yes—
231C23×C4C2×D414yes—
232C23×C4#4114yes—
233C23×C4#4114yes—
234C23×C4C2414yes—
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2412yes—
238C23×C4C2414yes—
239C23×C4C2412yes—
240C23×C4C2411yescharacteristic
241C23×C4C2411yescharacteristic
242C23×C4C2412yes—
243C23×C4C2412yes—
244C23×C4C2411yescharacteristic
245C23×C4C2412yes—
246C23×C4C2412yes—
247C23×C4C2411yescharacteristic
248C23×C4C2411yescharacteristic
249#123—232no—
250C4×D4—232no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
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—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—28no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—416no—
275C2×C4⋊C4—416no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4C2×D414yes—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—28no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4C2×D414yes—
311C2×C22⋊C4C2×D414yes—
312C2×C22⋊C4#4114yes—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—44no—
319C2×C42—44no—
320C2×C42—44no—
321C2×C42C2×D414yes—
322#95—28no—
323#95—28no—
324#95—28no—
325#95—28no—
326#95—22no—
327#95—24no—
328#95—22no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—28no—
333#95—24no—
334#95—24no—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—28no—
340#95—28no—
341#95—28no—
342#95—28no—
343#95—44no—
344#95—44no—
345#95—44no—
346#95—44no—
347#95—44no—
348#95—44no—
349#95—44no—
350#95—44no—
351#95C2×D414yes—
352#95C2×D414yes—
353#95C2×D418yes—
354C24—24no—
355C24—24no—
356C24—21no—
357C24—21no—
358C24—21no—
359C24—21no—
360C24—21no—
361C24—21no—
362C24—24no—
363C24—24no—
364C24C22×D412yes—
365C24C22×D412yes—
366C24C22×D412yes—
367C24C2×#4112yes—
368C24C2×#4112yes—
369C24C2×#4112yes—
370C242−1+412yes—
371C242−1+412yes—
372C24C2511yescharacteristic
373C2×D4—28no—
374C2×D4—28no—
375C2×D4—28no—
376C2×D4—44no—
377C2×D4—44no—
378C2×D4—44no—
379C2×D4—44no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—28no—
383C22×C4—28no—
384C22×C4—28no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—216no—
416C22×C4—28no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
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—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—48no—
452C22×C4—48no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4—48no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—48no—
462C22×C4—48no—
463C22×C4—48no—
464C22×C4—44no—
465C22×C4—48no—
466C22×C4—84no—
467C22×C4—84no—
468C22×C4—88no—
469C22×C4—88no—
470C22×C4—84no—
471C22×C4—84no—
472C22×C4—88no—
473C22×C4—88no—
474C22×C4#121116yes—
475C22×C4C22×D414yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D414yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C2×#4114yes—
485C22×C4C2×#4114yes—
486C22×C4C2×#4112yes—
487C22×C4C2×#4112yes—
488C4⋊C4—216no—
489C4⋊C4—28no—
490C4⋊C4—48no—
491C4⋊C4—48no—
492C4⋊C4—48no—
493C4⋊C4—48no—
494C4⋊C4—48no—
495C4⋊C4—84no—
496C4⋊C4—84no—
497C4⋊C4—88no—
498C4⋊C4—88no—
499C22⋊C4—216no—
500C22⋊C4—28no—
501C22⋊C4—28no—
502C22⋊C4—28no—
503C22⋊C4—48no—
504C22⋊C4—48no—
505C22⋊C4—44no—
506C22⋊C4—48no—
507C22⋊C4—84no—
508C22⋊C4—84no—
509C22⋊C4—88no—
510C42—28no—
511C42—48no—
512C42—84no—
513C42—84no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—28no—
521C23—28no—
522C23—28no—
523C23—28no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—24no—
528C23—24no—
529C23—42no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23C2×#12114yes—
542C23#52514yes—
543C23#53614yes—
544C23#54714yes—
545C23#54814yes—
546C23#55214yes—
547C23#55614yes—
548C23C23×D411yescharacteristic
549C23C23×D411yescharacteristic
550C23C22×#4111yescharacteristic
551C23C22×#4111yescharacteristic
552C23C22×#4111yescharacteristic
553C23C2×2−1+411yescharacteristic
554C23C2×2−1+411yescharacteristic
555C23#58514yes—
556C23#58514yes—
557D4—416no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—28no—
582C2×C4—416no—
583C2×C4—42no—
584C2×C4—416no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—88no—
638C2×C4—816no—
639C2×C4—88no—
640C2×C4—816no—
641C2×C4C2×#12114yes—
642C2×C4C2×#12114yes—
643C22—28no—
644C22—24no—
645C22—28no—
646C22—28no—
647C22—28no—
648C22—28no—
649C22—28no—
650C22—24no—
651C22—42no—
652C22—42no—
653C22—44no—
654C22—44no—
655C22—42no—
656C22—82no—
657C22—82no—
658C22—82no—
659C22—82no—
660C22#241518yes—
661C22C22×#12111yescharacteristic
662C22C2×#52511yescharacteristic
663C22C2×#53611yescharacteristic
664C22#344912yes—
665C22C2×#54711yescharacteristic
666C22C2×#54811yescharacteristic
667C22C2×#55211yescharacteristic
668C22C2×#55611yescharacteristic
669C22#348512yes—
670C22#348512yes—
671C22#349112yes—
672C22#350012yes—
673C22#351012yes—
674C22#351214yes—
675C22#352114yes—
676C22#355314yes—
677C22#356414yes—
678C4—24no—
679C4—24no—
680C4—42no—
681C4—44no—
682C4—42no—
683C4—44no—
684C4—44no—
685C4—44no—
686C4—42no—
687C4—82no—
688C4—82no—
689C4—82no—
690C4—82no—
691C4—84no—
692C4—84no—
693C2—28no—
694C2—82no—
695C2C2×#241511yescharacteristic
696C2#3532711yescharacteristic
697C2#3544711yescharacteristic
698C2#3603212yes—
699C2#3804612yes—
700C2#3971712yes—
701C2#4137412yes—
702C2#4351612yes—
703C2#4446012yes—
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. C22×#400C2semidirect
5. #16678C2semidirect
6. #16394C2semidirect
7. #15758C2semidirect
8. #15530C2semidirect
11. #14908C2semidirect
12. #14841C2semidirect
13. #14339C2semidirect
14. #14149C2semidirect
15. #13992C2semidirect
16. #13924C2semidirect
17. #13908C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24872
C22×#4002
C2×#23532
C2×#23222
#166782
#163942
#157582
#155302
#149452
#149272
#149082
#148412
#143392
#141492
#139922
#139242
#139082