Finite Group Databasebeta

← #729333

Subgroups — #729333

Order 512.

2708 conjugacy classes of subgroups.

The subgroup lattice has 707 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2487C211yesmaximal, characteristic
2C22×#400C211yesmaximal, characteristic, complemented
3C2×#2345C211yesmaximal, characteristic, complemented
4C2×#2322C211yesmaximal, characteristic
5#17039C212yesmaximal, complemented
6#17010C212yesmaximal, complemented
7#15813C214yesmaximal, complemented
8#15796C214yesmaximal, complemented
9#15175C212yesmaximal, complemented
10#15090C212yesmaximal, complemented
11#14927C212yesmaximal
12#14920C212yesmaximal, complemented
13#14339C211yesmaximal, characteristic, complemented
14#14020C212yesmaximal
15#13972C212yesmaximal, complemented
16#13966C211yesmaximal, characteristic, complemented
17#13960C211yesmaximal, characteristic
18C22×#123C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22#2487—28no—
23C2×#400—28no—
24C2×#400C2212yes—
25C2×#400C2211yescharacteristic
26C2×#400C2212yes—
27C2×#400C2214yes—
28C2×#400C2211yescharacteristic
29C2×#399C2211yescharacteristic
30C2×#399C2211yescharacteristic
31#2345—28no—
32#2322—28no—
33C2×#389C2212yes—
34C2×#389C2212yes—
35C2×#389C2212yes—
36C2×#388C2212yes—
37C2×#387C2212yes—
38C2×#387C2214yes—
39C2×#386C2211yescharacteristic
40C2×#386C2212yes—
41C2×#386C2212yes—
42C2×#385C2211yescharacteristic
43C2×#385C2212yes—
44C2×#385C2212yes—
45C2×#381C2212yes—
46C2×#381C2211yescharacteristic
47C2×#380C2211yescharacteristic
48C2×#380C2211yescharacteristic
49C2×#380C2212yes—
50C2×#380C2212yes—
51C2×C4×C4⋊C4C2212yes—
52C2×C4×C22⋊C4C2212yes—
53C2×C4×C22⋊C4C2211yescharacteristic
54C22×#95C2212yes—
55C22×#95C2212yes—
56C22×#95C2211yescharacteristic
57#1445C2214yes—
58#1443C2214yes—
59#1443C2214yes—
60#1442C2214yes—
61#1442C2214yes—
62#1442C2214yes—
63#1442C2214yes—
64#1441C2212yes—
65#1441C2212yes—
66#1441C2214yes—
67#1441C2214yes—
68#1441C2214yes—
69#1441C2214yes—
70#1441C2212yes—
71#1440C2214yes—
72#1440C2214yes—
73#1439C2212yes—
74#1439C2214yes—
75#1437C2214yes—
76#1436C2214yes—
77#1434C2212yes—
78#1434C2212yes—
79#1434C2212yes—
80#1434C2214yes—
81#1434C2214yes—
82C22⋊#95C2212yes—
83C22⋊#95C2212yes—
84C22⋊#95C2212yes—
85C22⋊#95C2212yes—
86C22⋊#95C2212yes—
87C23×D4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C24×C4C2312yes—
92C24×C4C2312yes—
93C24×C4C2311yescharacteristic
94C2×#123—24no—
95C2×#123—24no—
96C2×#123—24no—
97C2×C4×D4—24no—
98C2×C4×D4—24no—
99C2×C4×D4—24no—
100C22×C4⋊C4—24no—
101C22×C4⋊C4—24no—
102C22×C4⋊C4—24no—
103C22×C4⋊C4—24no—
104C22×C4⋊C4—24no—
105C22×C4⋊C4D418yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C22⋊C4C2311yescharacteristic
120C22×C22⋊C4C2311yescharacteristic
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2311yescharacteristic
125C22×C22⋊C4C2311yescharacteristic
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2312yes—
130C22×C42C2311yescharacteristic
131C22×C42C2314yes—
132C22×C42C2312yes—
133#400—216no—
134#400—28no—
135#400—28no—
136#400—24no—
137#400—24no—
138#399—24no—
139#399—24no—
140#389—28no—
141#389—28no—
142#389—28no—
143#388—28no—
144#387—28no—
145#387—216no—
146#386—28no—
147#386—28no—
148#386—24no—
149#385—28no—
150#385—24no—
151#385—28no—
152#381—28no—
153#381—24no—
154#380—24no—
155#380—24no—
156#380—28no—
157#380—28no—
158C4×C4⋊C4—28no—
159C4×C22⋊C4—28no—
160C4×C22⋊C4D418yes—
161C2×#95—28no—
162C2×#95—28no—
163C2×#95—24no—
164C2×#95C2311yescharacteristic
165C2×#95C2311yescharacteristic
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2314yes—
181C2×#95C2312yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2312yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2312yes—
195C25C2411yescharacteristic
196C25C2411yescharacteristic
197C22×D4—22no—
198C22×D4—22no—
199C22×D4—22no—
200C22×D4—22no—
201C22×D4—22no—
202C22×D4—22no—
203C22×D4#4114yes—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—22no—
213C23×C4—24no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—22no—
219C23×C4—22no—
220C23×C4—22no—
221C23×C4—44no—
222C23×C4—44no—
223C23×C4—44no—
224C23×C4—44no—
225C23×C4—44no—
226C23×C4—44no—
227C23×C4—44no—
228C23×C4C2×D418yes—
229C23×C4C2×D414yes—
230C23×C4C2×D414yes—
231C23×C4#4114yes—
232C23×C4#4114yes—
233C23×C4C2411yescharacteristic
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2414yes—
237C23×C4C2411yescharacteristic
238C23×C4C2412yes—
239C23×C4C2414yes—
240C23×C4C2412yes—
241C23×C4C2412yes—
242C23×C4C2412yes—
243C23×C4C2412yes—
244C23×C4C2412yes—
245C23×C4C2411yescharacteristic
246C23×C4C2411yescharacteristic
247C23×C4C2411yescharacteristic
248#123—232no—
249C4×D4—416no—
250C2×C4⋊C4—232no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
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—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—28no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—416no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4C2×D418yes—
284C2×C4⋊C4C2×D414yes—
285C2×C4⋊C4C2×D414yes—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—28no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4C2×D414yes—
315C2×C22⋊C4C2×D414yes—
316C2×C42—22no—
317C2×C42—22no—
318C2×C42—22no—
319C2×C42—28no—
320C2×C42—44no—
321C2×C42—44no—
322C2×C42—44no—
323C2×C42C2×D418yes—
324#95—28no—
325#95—28no—
326#95—24no—
327#95—24no—
328#95—28no—
329#95—28no—
330#95—28no—
331#95—28no—
332#95—24no—
333#95—24no—
334#95—24no—
335#95—22no—
336#95—22no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—28no—
344#95—28no—
345#95—28no—
346#95—44no—
347#95—44no—
348#95—44no—
349#95—44no—
350#95—44no—
351#95—44no—
352#95—44no—
353#95—44no—
354#95C2×D418yes—
355C24—24no—
356C24—24no—
357C24—21no—
358C24—21no—
359C24—21no—
360C24—21no—
361C24—21no—
362C24—21no—
363C24—24no—
364C24—24no—
365C24C22×D412yes—
366C24C22×D412yes—
367C24C2×#4112yes—
368C24C2×#4112yes—
369C24C2×#4112yes—
370C24C2×#4112yes—
371C242−1+412yes—
372C242−1+412yes—
373C24C2511yescharacteristic
374C2×D4—28no—
375C2×D4—28no—
376C2×D4—28no—
377C2×D4—44no—
378C2×D4—44no—
379C2×D4—44no—
380C2×D4—44no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—28no—
384C22×C4—28no—
385C22×C4—28no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—216no—
415C22×C4—24no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—21no—
431C22×C4—21no—
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—24no—
452C22×C4—48no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4—48no—
457C22×C4—48no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—48no—
463C22×C4—48no—
464C22×C4—48no—
465C22×C4—44no—
466C22×C4—48no—
467C22×C4—44no—
468C22×C4—84no—
469C22×C4—84no—
470C22×C4—88no—
471C22×C4—88no—
472C22×C4—84no—
473C22×C4—84no—
474C22×C4—88no—
475C22×C4—88no—
476C22×C4#121116yes—
477C22×C4C22×D414yes—
478C22×C4C22×D414yes—
479C22×C4C22×D414yes—
480C22×C4C22×D414yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C2×#4114yes—
486C22×C4C2×#4112yes—
487C22×C4C2×#4112yes—
488C22×C4C2×#4112yes—
489C4⋊C4—216no—
490C4⋊C4—216no—
491C4⋊C4—28no—
492C4⋊C4—28no—
493C4⋊C4—48no—
494C4⋊C4—48no—
495C4⋊C4—48no—
496C4⋊C4—48no—
497C4⋊C4—44no—
498C4⋊C4—84no—
499C4⋊C4—84no—
500C4⋊C4—84no—
501C4⋊C4—88no—
502C22⋊C4—216no—
503C22⋊C4—28no—
504C22⋊C4—28no—
505C22⋊C4—44no—
506C22⋊C4—48no—
507C22⋊C4—48no—
508C22⋊C4—44no—
509C22⋊C4—84no—
510C22⋊C4—84no—
511C22⋊C4—84no—
512C22⋊C4—88no—
513C42—48no—
514C42—44no—
515C42—88no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—28no—
523C23—28no—
524C23—28no—
525C23—28no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—28no—
530C23—41no—
531C23—41no—
532C23—42no—
533C23—41no—
534C23—41no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23C2×#12114yes—
544C23#52514yes—
545C23#53614yes—
546C23#54814yes—
547C23#54814yes—
548C23#55214yes—
549C23#55214yes—
550C23C23×D411yescharacteristic
551C23C23×D411yescharacteristic
552C23C22×#4111yescharacteristic
553C23C22×#4111yescharacteristic
554C23C22×#4111yescharacteristic
555C23C2×2−1+411yescharacteristic
556C23C2×2−1+411yescharacteristic
557C23#58514yes—
558C23#58514yes—
559D4—416no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—416no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—416no—
601C2×C4—44no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—42no—
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—44no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—42no—
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—44no—
638C2×C4—44no—
639C2×C4—88no—
640C2×C4—816no—
641C2×C4—88no—
642C2×C4—816no—
643C2×C4C2×#12114yes—
644C2×C4C2×#12114yes—
645C22—24no—
646C22—24no—
647C22—28no—
648C22—28no—
649C22—28no—
650C22—28no—
651C22—28no—
652C22—28no—
653C22—42no—
654C22—44no—
655C22—44no—
656C22—44no—
657C22—81no—
658C22—81no—
659C22—82no—
660C22—82no—
661C22—82no—
662C22#253618yes—
663C22C22×#12111yescharacteristic
664C22C2×#52511yescharacteristic
665C22C2×#53611yescharacteristic
666C22#344912yes—
667C22C2×#54811yescharacteristic
668C22C2×#54811yescharacteristic
669C22C2×#55211yescharacteristic
670C22C2×#55211yescharacteristic
671C22#348512yes—
672C22#348512yes—
673C22#349112yes—
674C22#350012yes—
675C22#351012yes—
676C22#351214yes—
677C22#352114yes—
678C22#355314yes—
679C22#356414yes—
680C4—24no—
681C4—24no—
682C4—44no—
683C4—44no—
684C4—44no—
685C4—42no—
686C4—44no—
687C4—44no—
688C4—81no—
689C4—82no—
690C4—81no—
691C4—82no—
692C4—82no—
693C4—84no—
694C4—84no—
695C2—28no—
696C2—82no—
697C2C2×#253611yescharacteristic
698C2#3508911yescharacteristic
699C2#3561711yescharacteristic
700C2#3603212yes—
701C2#3911712yes—
702C2#3971712yes—
703C2#4137412yes—
704C2#4351312yes—
705C2#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
3. C2×#2345C2semidirect
5. #17039C2semidirect
6. #17010C2semidirect
7. #15813C2semidirect
8. #15796C2semidirect
9. #15175C2semidirect
10. #15090C2semidirect
12. #14920C2semidirect
13. #14339C2semidirect
15. #13972C2semidirect
16. #13966C2semidirect

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×#23452
C2×#23222
#170392
#170102
#158132
#157962
#151752
#150902
#149272
#149202
#143392
#140202
#139722
#139662
#139602