Finite Group Databasebeta

← #366747

Subgroups — #366747

Order 512.

1680 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2395C211yesmaximal, characteristic
2C2×#1870C211yesmaximal, characteristic
3C2×#1869C211yesmaximal, characteristic
4C2×#1851C211yesmaximal, characteristic, complemented
5C2×#1818C211yesmaximal, characteristic, complemented
6C2×#1752C211yesmaximal, characteristic, complemented
7C2×#1729C211yesmaximal, characteristic, complemented
8#11346C212yesmaximal, complemented
9#11042C212yesmaximal, complemented
10#10922C212yesmaximal, complemented
11#10835C212yesmaximal, complemented
12C23×M16C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C22×C4⋊C8C2212yes—
17C22×C22⋊C8C2211yescharacteristic
18C22×C22⋊C8C2211yescharacteristic
19C22×C22⋊C8C2212yes—
20C22×C22⋊C8C2211yescharacteristic
21C22×C22⋊C8C2211yescharacteristic
22C22×#97C2212yes—
23#2395—24no—
24#2395—24no—
25C2×#400C411yescharacteristic
26C2×#399C411yescharacteristic
27C2×#398C411yescharacteristic
28C2×#397C412yes—
29C2×#397C411yescharacteristic
30C2×#394C412yes—
31C2×C4×C4⋊C4C2212yes—
32C2×C4×C22⋊C4C2212yes—
33C2×C4×C22⋊C4C2211yescharacteristic
34#1870—24no—
35#1869—24no—
36#1851—24no—
37#1818—24no—
38#1752—24no—
39#1729—24no—
40C2×#336C2212yes—
41C2×#336C2212yes—
42C2×#336C2212yes—
43C2×#336C2211yescharacteristic
44C2×#336C2212yes—
45C2×#336C2212yes—
46C2×#336C2212yes—
47C2×#336C2211yescharacteristic
48C2×#336C2212yes—
49C23×D4C2×C411yescharacteristic
50C24×C4C2311yescharacteristic
51C24×C4C2311yescharacteristic
52C22×M16—22no—
53C22×M16—22no—
54C22×M16—22no—
55C22×M16—24no—
56C22×M16—42no—
57C23×C8—22no—
58C23×C8—22no—
59C23×C8—24no—
60C23×C8—42no—
61C23×C8D414yes—
62C23×C8C2311yescharacteristic
63C23×C8C2311yescharacteristic
64C23×C8C2312yes—
65C23×C8C2311yescharacteristic
66C23×C8C2312yes—
67C23×C8C2311yescharacteristic
68C2×C4×D4—22no—
69C2×C4×D4—22no—
70C2×C4×D4—22no—
71C2×C4×D4—22no—
72C2×C4×D4—22no—
73C2×C4×D4—22no—
74C2×#117—22no—
75C2×#117—22no—
76C2×#117—22no—
77C2×#117—22no—
78C2×#117—22no—
79C2×#117—22no—
80C22×C4⋊C4C2×C411yescharacteristic
81C22×C4⋊C4C2×C411yescharacteristic
82C22×C4⋊C4C2×C411yescharacteristic
83C22×C4⋊C4C2×C412yes—
84C22×C4⋊C4C2×C412yes—
85C22×C22⋊C4C2×C412yes—
86C22×C22⋊C4C2×C411yescharacteristic
87C22×C22⋊C4C2×C411yescharacteristic
88C22×C22⋊C4C2×C412yes—
89C22×C22⋊C4C2×C412yes—
90C22×C22⋊C4C2×C412yes—
91C22×C42C2312yes—
92C22×C42C2311yescharacteristic
93C22×C42C2312yes—
94C2×C4⋊C8—24no—
95C2×C4⋊C8—24no—
96C2×C4⋊C8—42no—
97C2×C22⋊C8—22no—
98C2×C22⋊C8—22no—
99C2×C22⋊C8—22no—
100C2×C22⋊C8—22no—
101C2×C22⋊C8—22no—
102C2×C22⋊C8—22no—
103C2×C22⋊C8—22no—
104C2×C22⋊C8—24no—
105C2×C22⋊C8—24no—
106C2×C22⋊C8—24no—
107C2×C22⋊C8—42no—
108C2×C22⋊C8—42no—
109C2×C22⋊C8D414yes—
110C2×#97—24no—
111C2×#97—24no—
112C2×#97—42no—
113#400—24no—
114#399—24no—
115#398—24no—
116#397—28no—
117#397—24no—
118#394—28no—
119C4×C4⋊C4—24no—
120C4×C4⋊C4—24no—
121C4×C22⋊C4—24no—
122C4×C22⋊C4—24no—
123C4×C22⋊C4D414yes—
124C4×C22⋊C4D414yes—
125C2×#95C2×C412yes—
126C2×#95C2×C412yes—
127C2×#95C2×C412yes—
128C2×#95C2×C412yes—
129C2×#95C2×C411yescharacteristic
130C2×#95C2×C411yescharacteristic
131#336—24no—
132#336—24no—
133#336—22no—
134#336—22no—
135#336—24no—
136#336—24no—
137#336—24no—
138#336—24no—
139#336—24no—
140C25C22×C411yescharacteristic
141C25C22×C411yescharacteristic
142C22×D4—22no—
143C22×D4—22no—
144C22×D4—22no—
145C22×D4—22no—
146C22×D4—22no—
147C22×D4—41no—
148C22×D4—41no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4C22×C411yescharacteristic
164C23×C4C22×C412yes—
165C23×C4C22×C412yes—
166C23×C4C22×C411yescharacteristic
167C23×C4C22×C412yes—
168C23×C4C22×C412yes—
169C23×C4C22×C411yescharacteristic
170C23×C4C22×C411yescharacteristic
171C23×C4C2×D412yes—
172C23×C4C2×D412yes—
173C23×C4C2×D412yes—
174C23×C4C2×D412yes—
175C23×C4#4112yes—
176C23×C4#4112yes—
177C23×C4C2411yescharacteristic
178C2×M16—28no—
179C2×M16—48no—
180C2×M16—48no—
181C2×M16—44no—
182C2×M16—44no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—22no—
186C22×C8—28no—
187C22×C8—21no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—21no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8—22no—
202C22×C8—22no—
203C22×C8—28no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—22no—
211C22×C8—22no—
212C22×C8—22no—
213C22×C8—22no—
214C22×C8—44no—
215C22×C8—48no—
216C22×C8—48no—
217C22×C8C2×D412yes—
218C22×C8C2×D414yes—
219C22×C8C2×D412yes—
220C22×C8#4114yes—
221C22×C8#4114yes—
222C4×D4—44no—
223C4×D4—44no—
224C4×D4—44no—
225C4×D4—44no—
226#117—44no—
227#117—44no—
228#117—44no—
229#117—44no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—42no—
258C2×C22⋊C4—42no—
259C2×C22⋊C4—42no—
260C2×C22⋊C4—42no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4C22⋊C418yes—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42C2×D414yes—
280C2×C42C2×D414yes—
281C4⋊C8—48no—
282C22⋊C8—28no—
283C22⋊C8—48no—
284C22⋊C8—44no—
285C22⋊C8—44no—
286C22⋊C8—44no—
287#97—48no—
288#95—24no—
289#95—24no—
290#95—24no—
291#95—24no—
292#95C22⋊C414yes—
293#95C22⋊C414yes—
294C24—22no—
295C24—22no—
296C24—21no—
297C24—21no—
298C24—22no—
299C24—22no—
300C24—21no—
301C24—21no—
302C24—21no—
303C24—21no—
304C24—21no—
305C24—22no—
306C24—22no—
307C24—21no—
308C24—21no—
309C24—22no—
310C24—22no—
311C24—21no—
312C24C2×C22⋊C412yes—
313C24C4×D412yes—
314C24C4×D412yes—
315C24#13112yes—
316C24C23×C411yescharacteristic
317C2×D4—28no—
318C2×D4—44no—
319C2×D4—44no—
320C2×D4—44no—
321C2×D4—44no—
322C2×D4—44no—
323C2×D4—44no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—22no—
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—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—42no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4C2×C22⋊C414yes—
401C22×C4C2×C22⋊C414yes—
402C22×C4C2×C22⋊C412yes—
403C22×C4C4×D412yes—
404C22×C4C4×D412yes—
405C22×C4#12114yes—
406C22×C4#12114yes—
407C22×C4#12414yes—
408C22×C4#12714yes—
409C22×C4#13112yes—
410C22×C4C22×D411yescharacteristic
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D411yescharacteristic
414C22×C4C2×#4111yescharacteristic
415C22×C4C2×#4111yescharacteristic
416C22×C4C2×#4112yes—
417C22×C4C2×#4112yes—
418C22×C4H(5,𝔽2)12yes—
419C22×C42−1+412yes—
420M16—416no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—24no—
424C2×C8—24no—
425C2×C8—24no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—24no—
429C2×C8—22no—
430C2×C8—24no—
431C2×C8—24no—
432C2×C8—24no—
433C2×C8—24no—
434C2×C8—416no—
435C2×C8—42no—
436C2×C8—42no—
437C2×C8—42no—
438C2×C8—42no—
439C2×C8—42no—
440C2×C8—42no—
441C2×C8—42no—
442C2×C8—42no—
443C2×C8—42no—
444C2×C8—42no—
445C2×C8—41no—
446C2×C8—41no—
447C2×C8—41no—
448C2×C8—41no—
449C2×C8—41no—
450C2×C8—41no—
451C2×C8—42no—
452C2×C8—42no—
453C2×C8—42no—
454C2×C8—42no—
455C2×C8#12314yes—
456C4⋊C4—48no—
457C4⋊C4—48no—
458C4⋊C4—44no—
459C4⋊C4—44no—
460C4⋊C4—44no—
461C22⋊C4—48no—
462C22⋊C4—48no—
463C22⋊C4—48no—
464C22⋊C4—48no—
465C22⋊C4—44no—
466C22⋊C4—44no—
467C42—24no—
468C42—24no—
469C42—24no—
470C42—24no—
471C42—42no—
472C42—42no—
473C42—41no—
474C42—41no—
475C42—42no—
476C42—41no—
477C42—42no—
478C42—41no—
479C23—22no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—24no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—24no—
497C23—24no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—41no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—42no—
515C23—41no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—41no—
523C23—41no—
524C23#39014yes—
525C23C22×C22⋊C411yescharacteristic
526C23C2×C4×D411yescharacteristic
527C23C2×C4×D411yescharacteristic
528C23C4×#4112yes—
529C23#52012yes—
530C23C2×#13111yescharacteristic
531C23C2×#13111yescharacteristic
532C23#56812yes—
533C23#56812yes—
534C23#56811yescharacteristic
535C23#56811yescharacteristic
536D4—416no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—28no—
544C2×C4—28no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—81no—
593C2×C4—81no—
594C2×C4—82no—
595C2×C4—82no—
596C2×C4—81no—
597C2×C4—81no—
598C2×C4#39014yes—
599C2×C4C2×#12111yescharacteristic
600C2×C4C2×#12111yescharacteristic
601C2×C4C2×#12311yescharacteristic
602C2×C4#52511yescharacteristic
603C2×C4C2×#12411yescharacteristic
604C2×C4#52912yes—
605C2×C4C2×#12711yescharacteristic
606C2×C4#53411yescharacteristic
607C2×C4#53611yescharacteristic
608C2×C4#53912yes—
609C2×C4#54612yes—
610C2×C4#54612yes—
611C2×C4#54612yes—
612C2×C4#54712yes—
613C2×C4#54712yes—
614C2×C4#54712yes—
615C2×C4#55112yes—
616C2×C4#55212yes—
617C8—24no—
618C8—42no—
619C8—44no—
620C8—44no—
621C8—81no—
622C8—81no—
623C22—24no—
624C22—24no—
625C22—24no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—44no—
631C22—44no—
632C22—44no—
633C22—44no—
634C22—42no—
635C22—42no—
636C22—44no—
637C22—42no—
638C22—44no—
639C22—81no—
640C22—81no—
641C22—82no—
642C22—81no—
643C22—81no—
644C22—82no—
645C22C2×#39011yescharacteristic
646C22C4×#12311yescharacteristic
647C22#232211yescharacteristic
648C22#233912yes—
649C22#235912yes—
650C22#287912yes—
651C22#288011yescharacteristic
652C22#288011yescharacteristic
653C22#292911yescharacteristic
654C22#293011yescharacteristic
655C22#293211yescharacteristic
656C22#293211yescharacteristic
657C22#293212yes—
658C22#293311yescharacteristic
659C22#293311yescharacteristic
660C22#296712yes—
661C22#296712yes—
662C22#296812yes—
663C22#296912yes—
664C22#299212yes—
665C22#299312yes—
666C22#299512yes—
667C22#299512yes—
668C4—24no—
669C4—42no—
670C4—44no—
671C4—44no—
672C4—81no—
673C4—82no—
674C4—81no—
675C4—82no—
676C4#241211yescharacteristic
677C4#241311yescharacteristic
678C4#242512yes—
679C4#248812yes—
680C4#250111yescharacteristic
681C4#253911yescharacteristic
682C2—44no—
683C2—82no—
684C2#1399611yescharacteristic
685C2#2118811yescharacteristic
686C2#2119111yescharacteristic
687C2#2156711yescharacteristic
688C2#2156911yescharacteristic
689C2#2206611yescharacteristic
690C2#2207011yescharacteristic
691C2#2247212yes—
692C2#2247512yes—
693C2#2290712yes—
694C2#2291312yes—
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×#1851C2semidirect
5. C2×#1818C2semidirect
6. C2×#1752C2semidirect
7. C2×#1729C2semidirect
8. #11346C2semidirect
9. #11042C2semidirect
10. #10922C2semidirect
11. #10835C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23952
C2×#18702
C2×#18692
C2×#18512
C2×#18182
C2×#17522
C2×#17292
#113462
#110422
#109222
#108352