Finite Group Databasebeta

← #406318

Subgroups — #406318

Order 512.

1852 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2486C211yesmaximal, characteristic
2C2×#1869C211yesmaximal, characteristic
3C2×#1823C211yesmaximal, characteristic, complemented
4C2×#1807C211yesmaximal, characteristic, complemented
5C22×#340C211yesmaximal, characteristic, complemented
6#13872C211yesmaximal, characteristic
7#11293C212yesmaximal, complemented
8#11220C212yesmaximal, complemented
9#11200C212yesmaximal, complemented
10#11095C212yesmaximal, complemented
11#10730C211yesmaximal, characteristic, complemented
12C24×C8C2211yescharacteristic
13C22×#123—21no—
14C22×C4×D4C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C22×#106C2212yes—
18C22×C22⋊C8C2211yescharacteristic
19C22×C22⋊C8C2211yescharacteristic
20C22×C22⋊C8C2211yescharacteristic
21#2486—24no—
22#2486—24no—
23C2×#398—21no—
24C2×#397—21no—
25C2×#389C2212yes—
26C2×#386—21no—
27C2×#381—21no—
28C2×#380C2211yescharacteristic
29C2×#380C2212yes—
30C2×#380C2212yes—
31C2×#380C2211yescharacteristic
32#1869—24no—
33#1823—24no—
34#1807—24no—
35C2×#340—24no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#336C2211yescharacteristic
44C2×#336C2211yescharacteristic
45#1441—21no—
46#1439—21no—
47C4⋊#95C2212yes—
48C22⋊#95—21no—
49C23×D4D411yescharacteristic
50C24×C4—21no—
51C24×C4—21no—
52C24×C4C2311yescharacteristic
53C24×C4C2311yescharacteristic
54C23×C8—22no—
55C23×C8—22no—
56C23×C8—24no—
57C23×C8—42no—
58C23×C8D414yes—
59C23×C8C2311yescharacteristic
60C23×C8C2311yescharacteristic
61C23×C8C2311yescharacteristic
62C23×C8C2311yescharacteristic
63C2×#123—28no—
64C2×#123—44no—
65C2×#123—44no—
66C2×C4×D4—22no—
67C2×C4×D4—22no—
68C2×C4×D4—22no—
69C2×C4×D4—22no—
70C2×C4×D4—22no—
71C2×C4×D4—22no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—22no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4—22no—
77C22×C4⋊C4—24no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—22no—
80C22×C4⋊C4—22no—
81C22×C4⋊C4—22no—
82C22×C4⋊C4—21no—
83C22×C4⋊C4—42no—
84C22×C4⋊C4D411yescharacteristic
85C22×C4⋊C4D414yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C22⋊C4—22no—
91C22×C22⋊C4—21no—
92C22×C22⋊C4—21no—
93C22×C22⋊C4—22no—
94C22×C22⋊C4D411yescharacteristic
95C22×C22⋊C4D411yescharacteristic
96C22×C42C2311yescharacteristic
97C2×#106—24no—
98C2×#106—24no—
99C2×#106—42no—
100C2×C22⋊C8—22no—
101C2×C22⋊C8—22no—
102C2×C22⋊C8—22no—
103C2×C22⋊C8—22no—
104C2×C22⋊C8—22no—
105C2×C22⋊C8—24no—
106C2×C22⋊C8—42no—
107#398—28no—
108#397—28no—
109#389—24no—
110#389—24no—
111#386—44no—
112#381—28no—
113#380—22no—
114#380—24no—
115#380—24no—
116#380—22no—
117#380—22no—
118#380—24no—
119#380—42no—
120#380D414yes—
121C2×#95—21no—
122C2×#95—22no—
123C2×#95—22no—
124C2×#95—22no—
125C2×#95—22no—
126C2×#95—21no—
127C2×#95—22no—
128#340—24no—
129#340—24no—
130#340—24no—
131#340—24no—
132#340—24no—
133#340—24no—
134#340—42no—
135#336—22no—
136#336—22no—
137C25C2×D411yescharacteristic
138C25C2×D411yescharacteristic
139C22×D4—22no—
140C22×D4—22no—
141C22×D4—22no—
142C22×D4—22no—
143C22×D4—22no—
144C22×D4—22no—
145C22×D4SD1614yes—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—24no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—44no—
166C23×C4—42no—
167C23×C4—42no—
168C23×C4—42no—
169C23×C4—42no—
170C23×C4—42no—
171C23×C4—42no—
172C23×C4—42no—
173C23×C4—42no—
174C23×C4—42no—
175C23×C4—82no—
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D412yes—
181C23×C4C2×D412yes—
182C23×C4C2×D412yes—
183C23×C4#4112yes—
184C23×C4#4112yes—
185C23×C4#4112yes—
186C23×C4C2411yescharacteristic
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—22no—
190C22×C8—28no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—28no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—22no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—44no—
207C22×C8—48no—
208C22×C8—48no—
209C22×C8C2×D412yes—
210C22×C8C2×D412yes—
211C22×C8#4114yes—
212#123—432no—
213C4×D4—44no—
214C4×D4—44no—
215C4×D4—44no—
216C4×D4—44no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—22no—
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—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—48no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4C2×D414yes—
257C2×C4⋊C4C2×D414yes—
258C2×C4⋊C4#4114yes—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—42no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4—42no—
276C2×C22⋊C4—82no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—21no—
283#106—48no—
284C22⋊C8—44no—
285C22⋊C8—44no—
286C22⋊C8—44no—
287#95—28no—
288#95—28no—
289#95—24no—
290#95—44no—
291#95—44no—
292#95—42no—
293#95—82no—
294C24—22no—
295C24—22no—
296C24—22no—
297C24—22no—
298C24—21no—
299C24—21no—
300C24—21no—
301C24—22no—
302C24—22no—
303C24—21no—
304C24—21no—
305C24—21no—
306C24—22no—
307C24—22no—
308C24#12012yes—
309C24#12112yes—
310C24#12312yes—
311C24C2×SD1612yes—
312C24C2×SD1612yes—
313C24#13512yes—
314C24AGL(1,ℤ/8)12yes—
315C24AGL(1,ℤ/8)12yes—
316C24C22×D411yescharacteristic
317C2×D4—28no—
318C2×D4—28no—
319C2×D4—28no—
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—24no—
330C22×C4—22no—
331C22×C4—22no—
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—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—48no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—48no—
379C22×C4—48no—
380C22×C4—48no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—42no—
388C22×C4—42no—
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×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—88no—
407C22×C4—88no—
408C22×C4—84no—
409C22×C4—84no—
410C22×C4—84no—
411C22×C4—84no—
412C22×C4—84no—
413C22×C4—84no—
414C22×C4#12012yes—
415C22×C4#12114yes—
416C22×C4#12114yes—
417C22×C4#12112yes—
418C22×C4#12314yes—
419C22×C4#12312yes—
420C22×C4#12414yes—
421C22×C4C2×SD1612yes—
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C2×#4111yescharacteristic
426C22×C4C2×#4111yescharacteristic
427C22×C4C2×#4111yescharacteristic
428C22×C4C2×#4112yes—
429C22×C4C2×#4112yes—
430C22×C4H(5,𝔽2)12yes—
431C22×C42−1+412yes—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—24no—
435C2×C8—24no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—416no—
440C2×C8—42no—
441C2×C8—42no—
442C2×C8—41no—
443C2×C8—41no—
444C2×C8—41no—
445C2×C8—41no—
446C2×C8—42no—
447C2×C8—42no—
448C2×C8—41no—
449C2×C8—41no—
450C2×C8—42no—
451C2×C8—42no—
452C2×C8#12314yescomplemented
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—24no—
459C4⋊C4—48no—
460C4⋊C4—416no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—48no—
468C4⋊C4—44no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C22⋊C4—416no—
474C22⋊C4—44no—
475C22⋊C4—48no—
476C22⋊C4—44no—
477C22⋊C4—88no—
478C22⋊C4—84no—
479C42—41no—
480C42—41no—
481C42—41no—
482C42—41no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—22no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—22no—
495C23—24no—
496C23—24no—
497C23—22no—
498C23—22no—
499C23—24no—
500C23—22no—
501C23—22no—
502C23—24no—
503C23—22no—
504C23—42no—
505C23—41no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—41no—
510C23—42no—
511C23—42no—
512C23—41no—
513C23—41no—
514C23—42no—
515C23—42no—
516C23—41no—
517C23—42no—
518C23—41no—
519C23—42no—
520C23—41no—
521C23—41no—
522C23#39414yes—
523C23#44914yes—
524C23#45014yes—
525C23#46514yes—
526C23#46914yes—
527C23#48114yes—
528C23#48214yes—
529C23C2×#12011yescharacteristic
530C23C2×#12111yescharacteristic
531C23C2×#12311yescharacteristic
532C23#54612yes—
533C23#54712yes—
534C23C22×SD1611yescharacteristic
535C23C2×#13511yescharacteristic
536C23C2×AGL(1,ℤ/8)11yescharacteristic
537C23C2×AGL(1,ℤ/8)11yescharacteristic
538C23#57712yes—
539C23#57812yes—
540D4—416no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—28no—
550C2×C4—28no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—41no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—41no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—816no—
578C2×C4—82no—
579C2×C4—82no—
580C2×C4—82no—
581C2×C4—82no—
582C2×C4—82no—
583C2×C4—82no—
584C2×C4—82no—
585C2×C4—82no—
586C2×C4—82no—
587C2×C4—82no—
588C2×C4—82no—
589C2×C4—82no—
590C2×C4—816no—
591C2×C4—81no—
592C2×C4—81no—
593C2×C4—81no—
594C2×C4—81no—
595C2×C4—82no—
596C2×C4—82no—
597C2×C4—82no—
598C2×C4—82no—
599C2×C4—82no—
600C2×C4—82no—
601C2×C4—82no—
602C2×C4—82no—
603C2×C4—82no—
604C2×C4#39414yes—
605C2×C4C2×#12111yescharacteristic
606C2×C4C2×#12111yescharacteristic
607C2×C4C2×#12311yescharacteristic
608C2×C4C2×#12311yescharacteristic
609C2×C4#52511yescharacteristic
610C2×C4C2×#12411yescharacteristic
611C2×C4#52912yes—
612C2×C4#53212yes—
613C2×C4#53411yescharacteristic
614C2×C4#53611yescharacteristic
615C2×C4#53712yes—
616C2×C4#53912yes—
617C2×C4#54612yes—
618C2×C4#54712yes—
619C2×C4#55112yes—
620C2×C4#55112yes—
621C2×C4#55212yes—
622C2×C4#55212yes—
623C8—24no—
624C8—42no—
625C8—81no—
626C8—81no—
627C22—24no—
628C22—24no—
629C22—24no—
630C22—28no—
631C22—28no—
632C22—28no—
633C22—24no—
634C22—24no—
635C22—28no—
636C22—24no—
637C22—42no—
638C22—44no—
639C22—42no—
640C22—44no—
641C22—42no—
642C22—81no—
643C22—81no—
644C22—82no—
645C22—81no—
646C22—82no—
647C22—81no—
648C22#203418yes—
649C22C2×#39411yescharacteristic
650C22#241912yes—
651C22#243012yes—
652C22#248611yescharacteristic
653C22#253411yescharacteristic
654C22C2×#45011yescharacteristic
655C22C2×#44911yescharacteristic
656C22#301312yes—
657C22C2×#46511yescharacteristic
658C22C2×#46911yescharacteristic
659C22#305711yescharacteristic
660C22#305911yescharacteristic
661C22C2×#48211yescharacteristic
662C22C2×#48111yescharacteristic
663C22#309712yes—
664C22#329812yes—
665C22#330012yes—
666C22#330612yes—
667C22#331012yes—
668C22#333512yes—
669C22#333712yes—
670C22#334312yes—
671C22#334312yes—
672C4—24no—
673C4—42no—
674C4—81no—
675C4—84no—
676C4—84no—
677C4—81no—
678C4—84no—
679C4—162no—
680C4#241211yescharacteristic
681C4#241311yescharacteristic
682C4#248812yes—
683C4#249812yes—
684C4#261811yescharacteristic
685C4#285411yescharacteristic
686C2—28no—
687C2—82no—
688C2#1492711yescharacteristic
689C2C2×#203411yescharacteristic
690C2#2348711yescharacteristic
691C2#2493712yes—
692C2#2494512yes—
693C2#2553112yes—
694C2#2553912yes—
695C2#2765811yescharacteristic
696C2#2766711yescharacteristic
697C2#2809811yescharacteristic
698C2#2810711yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1823C2semidirect
4. C2×#1807C2semidirect
5. C22×#340C2semidirect
7. #11293C2semidirect
8. #11220C2semidirect
9. #11200C2semidirect
10. #11095C2semidirect
11. #10730C2semidirect
452. C2×C8#123semidirect

Maximal subgroups

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

subgroupindex
C2×#24862
C2×#18692
C2×#18232
C2×#18072
C22×#3402
#138722
#112932
#112202
#112002
#110952
#107302