Finite Group Databasebeta

← #406282

Subgroups — #406282

Order 512.

1978 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2501C211yesmaximal, characteristic
2C2×#1893C211yesmaximal, characteristic
3C2×#1874C211yesmaximal, characteristic
4C2×#1823C211yesmaximal, characteristic, complemented
5C22×#340C211yesmaximal, characteristic, complemented
6#13907C211yesmaximal, characteristic, complemented
7#11267C212yesmaximal, complemented
8#11207C212yesmaximal, complemented
9#11189C212yesmaximal, complemented
10#11066C212yesmaximal, complemented
11#10714C211yesmaximal, characteristic, complemented
12C24×C8C2211yescharacteristic
13C22×#123—21no—
14C22×#123—21no—
15C22×#121C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C22×#102C2211yescharacteristic
19C22×#102C2212yes—
20C22×#102C2211yescharacteristic
21C22×C22⋊C8C2211yescharacteristic
22#2501—24no—
23#2501—24no—
24C2×#399—22no—
25C2×#390C2212yes—
26C2×#389C2212yes—
27C2×#384C2212yes—
28C2×#380—21no—
29C2×#380C2211yescharacteristic
30C2×C4×C22⋊C4C2211yescharacteristic
31#1893—24no—
32#1874—24no—
33#1823—24no—
34C2×#340—24no—
35C2×#340C2211yescharacteristic
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2211yescharacteristic
41C2×#340C2212yes—
42C2×#336C2212yes—
43C2×#336C2212yes—
44#1441—22no—
45C4⋊#95C2212yes—
46C22⋊#95—21no—
47C23×D4—21no—
48C23×D4C2311yescharacteristic
49C24×C4—21no—
50C24×C4—21no—
51C24×C4C2311yescharacteristic
52C23×C8—22no—
53C23×C8—24no—
54C23×C8—22no—
55C23×C8—22no—
56C23×C8—42no—
57C23×C8C2311yescharacteristic
58C23×C8C2311yescharacteristic
59C23×C8C2312yes—
60C2×#123—28no—
61C2×#123—28no—
62C2×#123—44no—
63C2×#123—44no—
64C2×#123—44no—
65C2×#123—44no—
66C2×#121—22no—
67C2×#121—22no—
68C2×#121—22no—
69C2×#121—22no—
70C2×#121—22no—
71C2×#121D414yes—
72C22×C4⋊C4—22no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—22no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4—24no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—22no—
79C22×C4⋊C4—22no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4—22no—
82C22×C4⋊C4—22no—
83C22×C4⋊C4—22no—
84C22×C4⋊C4—42no—
85C22×C4⋊C4D414yes—
86C22×C4⋊C4D414yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2311yescharacteristic
92C22×C22⋊C4—21no—
93C22×C22⋊C4—22no—
94C22×C22⋊C4—22no—
95C22×C22⋊C4—21no—
96C22×C22⋊C4D411yescharacteristic
97C22×C22⋊C4D411yescharacteristic
98C22×C42C2312yes—
99C2×#102—24no—
100C2×#102—24no—
101C2×#102—22no—
102C2×#102—22no—
103C2×#102—24no—
104C2×#102—42no—
105C2×#102—42no—
106C2×C22⋊C8—22no—
107C2×C22⋊C8—22no—
108C2×C22⋊C8—22no—
109#399—216no—
110#390—24no—
111#390—24no—
112#389—24no—
113#389—42no—
114#384—24no—
115#384—24no—
116#380—22no—
117#380—22no—
118#380—44no—
119C4×C22⋊C4—22no—
120C4×C22⋊C4—22no—
121C2×#95—22no—
122C2×#95—22no—
123C2×#95—21no—
124C2×#95—22no—
125C2×#95D412yes—
126#340—24no—
127#340—24no—
128#340—22no—
129#340—22no—
130#340—24no—
131#340—24no—
132#340—24no—
133#336—24no—
134#336—42no—
135C25—21no—
136C25C2×D411yescharacteristic
137C22×D4—24no—
138C22×D4—21no—
139C22×D4—21no—
140C22×D4—21no—
141C22×D4—21no—
142C22×D4—21no—
143C22×D4—22no—
144C22×D4—22no—
145C22×D4—22no—
146C22×D4—22no—
147C22×D4—42no—
148C22×D4—42no—
149C22×D4—42no—
150C22×D4—42no—
151C22×D4—42no—
152C22×D4—42no—
153C22×D4C2×D412yes—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—21no—
159C23×C4—24no—
160C23×C4—22no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—24no—
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—82no—
175C23×C4C2×D412yes—
176C23×C4C2×D412yes—
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D412yes—
179C23×C4C2×D412yes—
180C23×C4C2×D412yes—
181C23×C4#4112yes—
182C23×C4#4112yes—
183C23×C4#4112yes—
184C23×C4C2411yescharacteristic
185C22×C8—28no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—22no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8—22no—
202C22×C8—48no—
203C22×C8—48no—
204C22×C8—44no—
205C22×C8—44no—
206C22×C8#4114yes—
207#123—432no—
208#123—432no—
209#121—28no—
210#121—28no—
211#121—44no—
212#121—44no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
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—44no—
242C2×C4⋊C4—48no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—82no—
255C2×C4⋊C4C2×D412yes—
256C2×C4⋊C4C2×D414yes—
257C2×C4⋊C4C2×D412yes—
258C2×C4⋊C4C2×D412yes—
259C2×C4⋊C4#4114yes—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—24no—
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⋊C4D814yes—
277C2×C22⋊C4SD1614yes—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—22no—
284#102—44no—
285#102—48no—
286#102—44no—
287C22⋊C8—44no—
288#95—24no—
289#95—44no—
290#95—42no—
291#95—44no—
292#95—82no—
293C24—22no—
294C24—22no—
295C24—22no—
296C24—22no—
297C24—22no—
298C24—41no—
299C24—41no—
300C24—42no—
301C24—42no—
302C24—41no—
303C24—41no—
304C24—41no—
305C24—41no—
306C24—42no—
307C24—42no—
308C24#12012yes—
309C24#12112yes—
310C24#12412yes—
311C24C2×D812yes—
312C24C2×SD1612yes—
313C24AGL(1,ℤ/8)12yes—
314C24AGL(1,ℤ/8)12yes—
315C24C22×D411yescharacteristic
316C2×D4—22no—
317C2×D4—22no—
318C2×D4—24no—
319C2×D4—24no—
320C2×D4—48no—
321C2×D4—48no—
322C2×D4—48no—
323C2×D4—41no—
324C2×D4—41no—
325C2×D4—41no—
326C2×D4—41no—
327C2×D4—42no—
328C2×D4—42no—
329C2×D4—41no—
330C2×D4—41no—
331C2×D4—42no—
332C2×D4—42no—
333C2×D4—42no—
334C2×D4—42no—
335C2×D4—42no—
336C2×D4—42no—
337C2×D4—84no—
338C2×D4—84no—
339C2×D4—84no—
340C2×D4—84no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—48no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—42no—
378C22×C4—41no—
379C22×C4—42no—
380C22×C4—41no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—41no—
384C22×C4—42no—
385C22×C4—41no—
386C22×C4—42no—
387C22×C4—41no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—42no—
394C22×C4—48no—
395C22×C4—48no—
396C22×C4—48no—
397C22×C4—48no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—84no—
406C22×C4—88no—
407C22×C4—88no—
408C22×C4—84no—
409C22×C4—84no—
410C22×C4—84no—
411C22×C4—84no—
412C22×C4#12012yes—
413C22×C4#12114yes—
414C22×C4#12114yes—
415C22×C4#12112yes—
416C22×C4#12314yes—
417C22×C4#12314yes—
418C22×C4#12412yes—
419C22×C4C2×D814yes—
420C22×C4C2×SD1614yes—
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C2×#4111yescharacteristic
425C22×C4C2×#4111yescharacteristic
426C22×C4C2×#4111yescharacteristic
427C22×C4C2×#4112yes—
428C22×C4C2×#4112yes—
429C22×C4H(5,𝔽2)12yes—
430C22×C42−1+412yes—
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—41no—
440C2×C8—41no—
441C2×C8—41no—
442C2×C8—41no—
443C2×C8—42no—
444C2×C8—42no—
445C2×C8—41no—
446C2×C8—41no—
447C2×C8—42no—
448C2×C8—42no—
449C4⋊C4—22no—
450C4⋊C4—24no—
451C4⋊C4—24no—
452C4⋊C4—22no—
453C4⋊C4—24no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—48no—
458C4⋊C4—416no—
459C4⋊C4—416no—
460C4⋊C4—42no—
461C4⋊C4—41no—
462C4⋊C4—42no—
463C4⋊C4—41no—
464C4⋊C4—42no—
465C4⋊C4—48no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—88no—
471C4⋊C4#12314yescomplemented
472C22⋊C4—28no—
473C22⋊C4—28no—
474C22⋊C4—416no—
475C22⋊C4—416no—
476C22⋊C4—48no—
477C22⋊C4—84no—
478C42—42no—
479C42—42no—
480C42—42no—
481C42—42no—
482C23—24no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—22no—
488C23—22no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—22no—
494C23—24no—
495C23—22no—
496C23—24no—
497C23—24no—
498C23—42no—
499C23—44no—
500C23—42no—
501C23—42no—
502C23—44no—
503C23—82no—
504C23—82no—
505C23—82no—
506C23—82no—
507C23—81no—
508C23—81no—
509C23—82no—
510C23—81no—
511C23—81no—
512C23—82no—
513C23—81no—
514C23—81no—
515C23—82no—
516C23—81no—
517C23—82no—
518C23—81no—
519C23—82no—
520C23—82no—
521C23#39414yes—
522C23#44714yes—
523C23#45014yes—
524C23#45914yes—
525C23#46014yes—
526C23#48614yes—
527C23#49014yes—
528C23C2×#12011yescharacteristic
529C23C2×#12111yescharacteristic
530C23C2×#12411yescharacteristic
531C23#54612yes—
532C23#54712yes—
533C23C22×D811yescharacteristic
534C23C22×SD1611yescharacteristic
535C23C2×AGL(1,ℤ/8)11yescharacteristic
536C23C2×AGL(1,ℤ/8)11yescharacteristic
537C23#57712yes—
538C23#57812yes—
539D4—44no—
540D4—42no—
541D4—42no—
542D4—816no—
543D4—81no—
544D4—81no—
545D4—82no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—28no—
553C2×C4—24no—
554C2×C4—28no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—816no—
585C2×C4—82no—
586C2×C4—81no—
587C2×C4—82no—
588C2×C4—81no—
589C2×C4—82no—
590C2×C4—81no—
591C2×C4—82no—
592C2×C4—81no—
593C2×C4—82no—
594C2×C4—82no—
595C2×C4—816no—
596C2×C4—82no—
597C2×C4—82no—
598C2×C4—82no—
599C2×C4—82no—
600C2×C4—82no—
601C2×C4—82no—
602C2×C4—82no—
603C2×C4#39414yes—
604C2×C4C2×#12111yescharacteristic
605C2×C4C2×#12111yescharacteristic
606C2×C4C2×#12311yescharacteristic
607C2×C4C2×#12311yescharacteristic
608C2×C4C2×#12311yescharacteristic
609C2×C4#52511yescharacteristic
610C2×C4#52912yes—
611C2×C4#53212yes—
612C2×C4#53411yescharacteristic
613C2×C4#53611yescharacteristic
614C2×C4#53712yes—
615C2×C4#53912yes—
616C2×C4#54612yes—
617C2×C4#54712yes—
618C2×C4#55112yes—
619C2×C4#55112yes—
620C2×C4#55212yes—
621C2×C4#55212yes—
622C8—44no—
623C8—81no—
624C8—81no—
625C22—24no—
626C22—24no—
627C22—28no—
628C22—28no—
629C22—24no—
630C22—28no—
631C22—28no—
632C22—24no—
633C22—24no—
634C22—24no—
635C22—82no—
636C22—84no—
637C22—84no—
638C22—82no—
639C22—82no—
640C22—161no—
641C22—161no—
642C22—162no—
643C22—162no—
644C22—161no—
645C22—161no—
646C22#204118yes—
647C22C2×#39411yescharacteristic
648C22#241211yescharacteristic
649C22#241311yescharacteristic
650C22#243012yes—
651C22#248812yes—
652C22C2×#44711yescharacteristic
653C22C2×#45011yescharacteristic
654C22#301312yes—
655C22C2×#45911yescharacteristic
656C22C2×#46011yescharacteristic
657C22#304211yescharacteristic
658C22#304411yescharacteristic
659C22C2×#48611yescharacteristic
660C22C2×#49011yescharacteristic
661C22#313712yes—
662C22#329412yes—
663C22#329812yes—
664C22#330612yes—
665C22#330812yes—
666C22#333412yes—
667C22#333512yes—
668C22#334312yes—
669C22#334512yes—
670C4—24no—
671C4—44no—
672C4—44no—
673C4—44no—
674C4—84no—
675C4—82no—
676C4—82no—
677C4—162no—
678C4#241912yes—
679C4#248611yescharacteristic
680C4#249812yes—
681C4#253411yescharacteristic
682C4#261811yescharacteristic
683C4#285411yescharacteristic
684C2—28no—
685C2—162no—
686C2#1492711yescharacteristic
687C2C2×#204111yescharacteristic
688C2#2351211yescharacteristic
689C2#2452611yescharacteristic
690C2#2452811yescharacteristic
691C2#2457611yescharacteristic
692C2#2457811yescharacteristic
693C2#2553112yes—
694C2#2553312yes—
695C2#2769512yes—
696C2#2771712yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1823C2semidirect
5. C22×#340C2semidirect
6. #13907C2semidirect
7. #11267C2semidirect
8. #11207C2semidirect
9. #11189C2semidirect
10. #11066C2semidirect
11. #10714C2semidirect
471. C4⋊C4#123semidirect

Maximal subgroups

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

subgroupindex
C2×#25012
C2×#18932
C2×#18742
C2×#18232
C22×#3402
#139072
#112672
#112072
#111892
#110662
#107142