Finite Group Databasebeta

← #366746

Subgroups — #366746

Order 512.

1964 conjugacy classes of subgroups.

The subgroup lattice has 720 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#123C211yesmaximal, characteristic
2C2×#1869C211yesmaximal, characteristic
3C2×#1869C211yesmaximal, characteristic
4C2×#1850C211yesmaximal, characteristic, complemented
5C2×#1817C211yesmaximal, characteristic, complemented
6C2×#1752C211yesmaximal, characteristic, complemented
7C22×#336C211yesmaximal, characteristic, complemented
8#11346C212yesmaximal, complemented
9#11042C212yesmaximal, complemented
10#10922C212yesmaximal, complemented
11#10835C212yesmaximal, complemented
12C24×C8C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×#123C411yescharacteristic
15C22×C4×D4C2211yescharacteristic
16C23×C42C2211yescharacteristic
17C22×C4⋊C8C2212yes—
18C22×C22⋊C8C2211yescharacteristic
19C22×C22⋊C8C2211yescharacteristic
20C22×C22⋊C8C2212yes—
21C22×C22⋊C8C2211yescharacteristic
22C22×C22⋊C8C2211yescharacteristic
23C22×#97C2212yes—
24C4×#123—24no—
25C4×#123—24no—
26C2×#388C412yes—
27C2×#387C412yes—
28C2×#386C411yescharacteristic
29C2×#385C411yescharacteristic
30C2×#381C411yescharacteristic
31C2×C4×C4⋊C4C2212yes—
32C2×C4×C22⋊C4C2212yes—
33C2×C4×C22⋊C4C2211yescharacteristic
34#1869—24no—
35#1869—24no—
36#1850—24no—
37#1817—24no—
38#1752—24no—
39C2×#336—24no—
40C2×#336C2212yes—
41C2×#336C2212yes—
42C2×#336C2212yes—
43C2×#336C2211yescharacteristic
44C2×#336C2212yes—
45C2×#336C2212yes—
46C2×#336C2212yes—
47C2×#336C2211yescharacteristic
48C2×#336C2212yes—
49C23×D4C2×C411yescharacteristic
50C24×C4C2×C411yescharacteristic
51C24×C4C2×C411yescharacteristic
52C24×C4C2311yescharacteristic
53C24×C4C2311yescharacteristic
54C23×C8—22no—
55C23×C8—22no—
56C23×C8—22no—
57C23×C8—24no—
58C23×C8—24no—
59C23×C8—22no—
60C23×C8—22no—
61C23×C8—42no—
62C23×C8—42no—
63C23×C8D414yes—
64C23×C8C2311yescharacteristic
65C23×C8C2311yescharacteristic
66C23×C8C2312yes—
67C23×C8C2311yescharacteristic
68C23×C8C2312yes—
69C23×C8C2311yescharacteristic
70C2×#123—24no—
71C2×#123—24no—
72C2×#123—24no—
73C2×C4×D4—22no—
74C2×C4×D4—22no—
75C2×C4×D4—22no—
76C2×C4×D4—22no—
77C2×C4×D4—22no—
78C2×C4×D4—22no—
79C22×C4⋊C4C2×C411yescharacteristic
80C22×C4⋊C4C2×C412yes—
81C22×C4⋊C4C2×C412yes—
82C22×C22⋊C4C2×C411yescharacteristic
83C22×C22⋊C4C2×C411yescharacteristic
84C22×C22⋊C4C2×C412yes—
85C22×C22⋊C4C2×C412yes—
86C22×C22⋊C4C2×C411yescharacteristic
87C22×C22⋊C4C2×C411yescharacteristic
88C22×C42—22no—
89C22×C42—22no—
90C22×C42—22no—
91C22×C42—22no—
92C22×C42—22no—
93C22×C42—22no—
94C22×C42C2×C412yes—
95C22×C42C2312yes—
96C22×C42C2311yescharacteristic
97C22×C42C2312yes—
98C2×C4⋊C8—24no—
99C2×C4⋊C8—24no—
100C2×C4⋊C8—42no—
101C2×C22⋊C8—22no—
102C2×C22⋊C8—22no—
103C2×C22⋊C8—22no—
104C2×C22⋊C8—22no—
105C2×C22⋊C8—22no—
106C2×C22⋊C8—24no—
107C2×C22⋊C8—22no—
108C2×C22⋊C8—22no—
109C2×C22⋊C8—24no—
110C2×C22⋊C8—24no—
111C2×C22⋊C8—42no—
112C2×C22⋊C8—42no—
113C2×C22⋊C8D414yes—
114C2×#97—24no—
115C2×#97—24no—
116C2×#97—42no—
117#388—28no—
118#387—28no—
119#386—24no—
120#385—24no—
121#381—24no—
122C4×C4⋊C4—24no—
123C4×C4⋊C4—24no—
124C4×C22⋊C4—24no—
125C4×C22⋊C4—24no—
126C4×C22⋊C4D414yes—
127C4×C22⋊C4D414yes—
128C2×#95C2×C412yes—
129C2×#95C2×C412yes—
130C2×#95C2×C412yes—
131C2×#95C2×C412yes—
132C2×#95C2×C412yes—
133#336—24no—
134#336—24no—
135#336—24no—
136#336—22no—
137#336—22no—
138#336—24no—
139#336—24no—
140#336—24no—
141#336—24no—
142C25C22×C411yescharacteristic
143C25C22×C411yescharacteristic
144C22×D4—22no—
145C22×D4—22no—
146C22×D4—22no—
147C22×D4—22no—
148C22×D4—22no—
149C22×D4—22no—
150C22×D4M1614yes—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—24no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4—42no—
176C23×C4M1614yes—
177C23×C4C22×C411yescharacteristic
178C23×C4C22×C411yescharacteristic
179C23×C4C22×C412yes—
180C23×C4C22×C412yes—
181C23×C4C22×C412yes—
182C23×C4C22×C411yescharacteristic
183C23×C4C22×C412yes—
184C23×C4C22×C411yescharacteristic
185C23×C4C2×D412yes—
186C23×C4C2×D412yes—
187C23×C4C2×D412yes—
188C23×C4C2×D412yes—
189C23×C4#4112yes—
190C23×C4#4112yes—
191C23×C4C2411yescharacteristic
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—22no—
195C22×C8—28no—
196C22×C8—28no—
197C22×C8—21no—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8—22no—
202C22×C8—21no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—22no—
207C22×C8—22no—
208C22×C8—22no—
209C22×C8—22no—
210C22×C8—22no—
211C22×C8—22no—
212C22×C8—22no—
213C22×C8—28no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—22no—
221C22×C8—22no—
222C22×C8—22no—
223C22×C8—22no—
224C22×C8—44no—
225C22×C8—48no—
226C22×C8—48no—
227C22×C8—48no—
228C22×C8—48no—
229C22×C8—44no—
230C22×C8—44no—
231C22×C8C2×D412yes—
232C22×C8C2×D414yes—
233C22×C8C2×D412yes—
234C22×C8#4114yes—
235C22×C8#4114yes—
236#123—232no—
237C4×D4—44no—
238C4×D4—44no—
239C4×D4—44no—
240C4×D4—44no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—42no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4C22⋊C414yes—
267C2×C22⋊C4C22⋊C414yes—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—22no—
272C2×C42—24no—
273C2×C42—24no—
274C2×C42—24no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—44no—
288C2×C42—44no—
289C2×C42—44no—
290C2×C42—44no—
291C2×C42C2×D414yes—
292C2×C42C2×D414yes—
293C4⋊C8—48no—
294C22⋊C8—28no—
295C22⋊C8—48no—
296C22⋊C8—44no—
297C22⋊C8—44no—
298C22⋊C8—44no—
299#97—48no—
300#95—24no—
301#95—24no—
302#95—24no—
303#95—24no—
304#95C22⋊C418yes—
305C24—22no—
306C24—22no—
307C24—22no—
308C24—22no—
309C24—21no—
310C24—21no—
311C24—21no—
312C24—22no—
313C24—22no—
314C24—21no—
315C24—22no—
316C24—21no—
317C24—21no—
318C24—22no—
319C24C2×C22⋊C412yes—
320C24C4×D412yes—
321C24C4×D412yes—
322C24C2×M1612yes—
323C24C2×M1612yes—
324C24C2×M1612yes—
325C24#13112yes—
326C24#13112yes—
327C24C23×C411yescharacteristic
328C2×D4—28no—
329C2×D4—28no—
330C2×D4—28no—
331C2×D4—44no—
332C2×D4—44no—
333C2×D4—44no—
334C2×D4—44no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—28no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—28no—
383C22×C4—28no—
384C22×C4—28no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—48no—
396C22×C4—48no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4—44no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4—44no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—42no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4C2×C22⋊C414yes—
422C22×C4C2×C22⋊C414yes—
423C22×C4C2×C22⋊C412yes—
424C22×C4C4×D412yes—
425C22×C4C4×D412yes—
426C22×C4#12114yes—
427C22×C4#12114yes—
428C22×C4#12414yes—
429C22×C4#12714yes—
430C22×C4C2×M1614yes—
431C22×C4C2×M1612yes—
432C22×C4#13114yes—
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C2×#4111yescharacteristic
438C22×C4C2×#4111yescharacteristic
439C22×C4C2×#4112yes—
440C22×C4C2×#4112yes—
441C22×C4H(5,𝔽2)12yes—
442C22×C42−1+412yes—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—24no—
446C2×C8—24no—
447C2×C8—24no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—24no—
451C2×C8—22no—
452C2×C8—24no—
453C2×C8—24no—
454C2×C8—24no—
455C2×C8—24no—
456C2×C8—416no—
457C2×C8—416no—
458C2×C8—42no—
459C2×C8—42no—
460C2×C8—42no—
461C2×C8—42no—
462C2×C8—42no—
463C2×C8—42no—
464C2×C8—42no—
465C2×C8—42no—
466C2×C8—42no—
467C2×C8—42no—
468C2×C8—41no—
469C2×C8—41no—
470C2×C8—41no—
471C2×C8—41no—
472C2×C8—41no—
473C2×C8—41no—
474C2×C8—42no—
475C2×C8—42no—
476C2×C8—42no—
477C2×C8—42no—
478C2×C8#12314yescomplemented
479C4⋊C4—216no—
480C4⋊C4—48no—
481C4⋊C4—44no—
482C22⋊C4—216no—
483C22⋊C4—28no—
484C22⋊C4—28no—
485C22⋊C4—48no—
486C22⋊C4—44no—
487C22⋊C4—44no—
488C42—24no—
489C42—24no—
490C42—24no—
491C42—24no—
492C42—42no—
493C42—42no—
494C42—41no—
495C42—41no—
496C42—42no—
497C42—41no—
498C42—42no—
499C42—41no—
500C42—48no—
501C23—22no—
502C23—24no—
503C23—24no—
504C23—24no—
505C23—22no—
506C23—22no—
507C23—24no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—24no—
521C23—24no—
522C23—41no—
523C23—42no—
524C23—42no—
525C23—41no—
526C23—42no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—41no—
539C23—41no—
540C23#39014yes—
541C23#40714yes—
542C23#40814yes—
543C23#43514yes—
544C23#43514yes—
545C23#43514yes—
546C23#43514yes—
547C23C22×C22⋊C411yescharacteristic
548C23C2×C4×D411yescharacteristic
549C23C2×C4×D411yescharacteristic
550C23C4×#4112yes—
551C23#52012yes—
552C23C22×M1611yescharacteristic
553C23C22×M1611yescharacteristic
554C23C2×#13111yescharacteristic
555C23C2×#13111yescharacteristic
556C23C2×#13112yes—
557C23#56812yes—
558D4—416no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—28no—
566C2×C4—28no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—416no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—416no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—81no—
617C2×C4—81no—
618C2×C4—82no—
619C2×C4—82no—
620C2×C4—81no—
621C2×C4—81no—
622C2×C4#39014yes—
623C2×C4C2×#12111yescharacteristic
624C2×C4C2×#12111yescharacteristic
625C2×C4C2×#12311yescharacteristic
626C2×C4#52511yescharacteristic
627C2×C4C2×#12411yescharacteristic
628C2×C4#52912yes—
629C2×C4C2×#12711yescharacteristic
630C2×C4#53411yescharacteristic
631C2×C4#53611yescharacteristic
632C2×C4#53912yes—
633C2×C4#54612yes—
634C2×C4#54612yes—
635C2×C4#54612yes—
636C2×C4#54712yes—
637C2×C4#54712yes—
638C2×C4#54712yes—
639C2×C4#55112yes—
640C2×C4#55212yes—
641C8—24no—
642C8—42no—
643C8—44no—
644C8—44no—
645C8—81no—
646C8—81no—
647C22—24no—
648C22—24no—
649C22—24no—
650C22—24no—
651C22—28no—
652C22—28no—
653C22—24no—
654C22—28no—
655C22—24no—
656C22—28no—
657C22—42no—
658C22—42no—
659C22—44no—
660C22—42no—
661C22—44no—
662C22—81no—
663C22—81no—
664C22—82no—
665C22—81no—
666C22—82no—
667C22—81no—
668C22#195518yes—
669C22C2×#39011yescharacteristic
670C22C4×#12311yescharacteristic
671C22#232211yescharacteristic
672C22#233912yes—
673C22#235912yes—
674C22C2×#40711yescharacteristic
675C22C2×#40811yescharacteristic
676C22#288012yes—
677C22C2×#43511yescharacteristic
678C22C2×#43511yescharacteristic
679C22C2×#43511yescharacteristic
680C22C2×#43511yescharacteristic
681C22#292911yescharacteristic
682C22#293011yescharacteristic
683C22#293312yes—
684C22#296512yes—
685C22#296612yes—
686C22#296812yes—
687C22#297012yes—
688C22#298912yes—
689C22#299012yes—
690C22#299212yes—
691C22#299412yes—
692C4—24no—
693C4—44no—
694C4—42no—
695C4—44no—
696C4—44no—
697C4—44no—
698C4—81no—
699C4—81no—
700C4#241211yescharacteristic
701C4#241311yescharacteristic
702C4#242512yes—
703C4#248812yes—
704C4#250111yescharacteristic
705C4#253911yescharacteristic
706C2—28no—
707C2—82no—
708C2#1399611yescharacteristic
709C2C2×#195511yescharacteristic
710C2#2119111yescharacteristic
711C2#2156111yescharacteristic
712C2#2156311yescharacteristic
713C2#2206511yescharacteristic
714C2#2206911yescharacteristic
715C2#2247012yes—
716C2#2247112yes—
717C2#2290212yes—
718C2#2290812yes—
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutator, socleC24C23×C4
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1850C2semidirect
5. C2×#1817C2semidirect
6. C2×#1752C2semidirect
7. C22×#336C2semidirect
8. #11346C2semidirect
9. #11042C2semidirect
10. #10922C2semidirect
11. #10835C2semidirect
478. C2×C8#123semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1232
C2×#18692
C2×#18502
C2×#18172
C2×#17522
C22×#3362
#113462
#110422
#109222
#108352