Finite Group Databasebeta

← #401887

Subgroups — #401887

Order 512.

1838 conjugacy classes of subgroups.

The subgroup lattice has 708 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2361C211yesmaximal, characteristic
2C2×#2100C211yesmaximal, characteristic, complemented
3C2×#1802C211yesmaximal, characteristic, complemented
4#15135C211yesmaximal, characteristic, complemented
5#11227C211yesmaximal, characteristic, complemented
6#11066C212yesmaximal, complemented
7#11042C211yesmaximal, characteristic, complemented
8#10969C212yesmaximal
9#10885C212yesmaximal, complemented
10#10788C211yesmaximal, characteristic, complemented
11#10730C212yesmaximal, complemented
12C22×#121C2211yescharacteristic
13C22×C4×D4—21no—
14C22×C4×D4C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C22×C4⋊C8C2211yescharacteristic
17C22×#102C2212yes—
18C22×#102C2212yes—
19C22×C22⋊C8C2212yes—
20C2×#399—21no—
21#2361—24no—
22#2361—24no—
23C2×#390C2211yescharacteristic
24C2×#389C2212yes—
25C2×#389C2211yescharacteristic
26C2×#388—21no—
27C2×#386—21no—
28C2×#385—21no—
29C2×#384C2211yescharacteristic
30C2×#380C2212yes—
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C22⋊C4C2211yescharacteristic
33#2100—24no—
34#1802—24no—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2211yescharacteristic
42C2×#336C2212yes—
43#1445—21no—
44#1444—21no—
45#1439—21no—
46C4⋊#95C2211yescharacteristic
47C23×D4—21no—
48C23×D4C2311yescharacteristic
49C24×C4—21no—
50C24×C4—21no—
51C24×C4C2311yescharacteristic
52C23×C8C2312yes—
53C23×C8C2312yes—
54C2×#121—22no—
55C2×#121—22no—
56C2×#121—22no—
57C2×#121—22no—
58C2×#121—22no—
59C2×#121Q814yes—
60C2×C4×D4—28no—
61C2×C4×D4—24no—
62C2×C4×D4—22no—
63C2×C4×D4—42no—
64C2×C4×D4—44no—
65C2×C4×D4—44no—
66C2×C4×D4D414yes—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4—24no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—42no—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2311yescharacteristic
81C22×C22⋊C4—21no—
82C22×C22⋊C4—21no—
83C22×C22⋊C4—21no—
84C22×C22⋊C4—21no—
85C22×C22⋊C4—22no—
86C22×C22⋊C4D411yescharacteristic
87C22×C22⋊C4D411yescharacteristic
88C22×C42—21no—
89C22×C42C2311yescharacteristic
90C22×C42C2311yescharacteristic
91C2×C4⋊C8—22no—
92C2×C4⋊C8—42no—
93C2×#102—24no—
94C2×#102—24no—
95C2×#102—24no—
96C2×#102—42no—
97C2×#102—42no—
98C2×#102—42no—
99C2×C22⋊C8—24no—
100C2×C22⋊C8—42no—
101C2×C22⋊C8—42no—
102#399—44no—
103#390—22no—
104#390—22no—
105#389—22no—
106#389—24no—
107#389—42no—
108#389D414yes—
109#388—28no—
110#386—28no—
111#385—28no—
112#384—22no—
113#384—22no—
114#380—24no—
115#380—42no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—22no—
118C4×C22⋊C4—22no—
119C4×C22⋊C4—22no—
120C2×#95—21no—
121C2×#95—21no—
122C2×#95—21no—
123C2×#95—21no—
124C2×#95—22no—
125C2×#95—22no—
126C2×#95D411yescharacteristic
127C2×#95D411yescharacteristic
128#340—24no—
129#340—24no—
130#340—24no—
131#340—24no—
132#340—24no—
133#340—42no—
134#336—22no—
135#336—42no—
136C25—21no—
137C25C2×D411yescharacteristic
138C22×D4—24no—
139C22×D4—24no—
140C22×D4—24no—
141C22×D4—21no—
142C22×D4—22no—
143C22×D4—22no—
144C22×D4—21no—
145C22×D4—21no—
146C22×D4—22no—
147C22×D4—22no—
148C22×D4—42no—
149C22×D4—42no—
150C22×D4—42no—
151C22×D4—42no—
152C22×D4C2×D412yes—
153C22×D4C2×Q812yes—
154C22×D4#4112yes—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—22no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—21no—
166C23×C4—22no—
167C23×C4—24no—
168C23×C4—24no—
169C23×C4—22no—
170C23×C4—44no—
171C23×C4—44no—
172C23×C4—42no—
173C23×C4—42no—
174C23×C4—42no—
175C23×C4—42no—
176C23×C4—82no—
177C23×C4—82no—
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×Q812yes—
184C23×C4#4112yes—
185C23×C4C2411yescharacteristic
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—22no—
200#121—28no—
201#121—28no—
202#121—28no—
203#121—28no—
204C4×D4—28no—
205C4×D4—44no—
206C4×D4—48no—
207C4×D4—432no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—21no—
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—48no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—82no—
242C2×C4⋊C4—82no—
243C2×C4⋊C4C2×D412yes—
244C2×C4⋊C4C2×D412yes—
245C2×C4⋊C4C2×D412yes—
246C2×C4⋊C4C2×Q812yes—
247C2×C4⋊C4#4114yes—
248C2×C4⋊C4#4112yes—
249C2×C4⋊C4#4114yes—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—41no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—42no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—82no—
267C2×C22⋊C4—82no—
268C2×C22⋊C4SD1614yes—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—24no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42—42no—
277C2×C42—42no—
278C2×C42C2×D412yes—
279C2×C42C2×D412yes—
280C4⋊C8—44no—
281#102—48no—
282#102—48no—
283C22⋊C8—48no—
284#95—24no—
285#95—24no—
286#95—22no—
287#95—22no—
288#95—44no—
289#95—44no—
290#95—42no—
291#95—42no—
292C24—21no—
293C24—21no—
294C24—22no—
295C24—22no—
296C24—21no—
297C24—21no—
298C24—22no—
299C24—22no—
300C24—22no—
301C24—22no—
302C24—22no—
303C24—42no—
304C24—42no—
305C24—41no—
306C24—41no—
307C24—41no—
308C24—41no—
309C24—42no—
310C24—42no—
311C24C22⋊Q812yes—
312C24C2×SD1612yes—
313C24AGL(1,ℤ/8)12yes—
314C24C22×D411yescharacteristic
315C2×D4—216no—
316C2×D4—22no—
317C2×D4—22no—
318C2×D4—22no—
319C2×D4—22no—
320C2×D4—22no—
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—24no—
324C2×D4—24no—
325C2×D4—24no—
326C2×D4—24no—
327C2×D4—48no—
328C2×D4—48no—
329C2×D4—48no—
330C2×D4—48no—
331C2×D4—48no—
332C2×D4—48no—
333C2×D4—42no—
334C2×D4—42no—
335C2×D4—42no—
336C2×D4—42no—
337C2×D4—42no—
338C2×D4—42no—
339C2×D4C22⋊Q814yes—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—48no—
375C22×C4—48no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—42no—
383C22×C4—41no—
384C22×C4—42no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—48no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—41no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—41no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—48no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—84no—
413C22×C4—88no—
414C22×C4—88no—
415C22×C4—84no—
416C22×C4—88no—
417C22×C4—88no—
418C22×C4#12012yes—
419C22×C4#12012yes—
420C22×C4C22⋊Q812yes—
421C22×C4C2×SD1612yes—
422C22×C4C2×SD1612yes—
423C22×C4AGL(1,ℤ/8)12yes—
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4C22×Q811yescharacteristic
427C22×C4C2×#4111yescharacteristic
428C22×C4C2×#4111yescharacteristic
429C22×C4C2×#4112yes—
430C22×C4C2×#4111yescharacteristic
431C22×C4C2×#4112yes—
432C22×C4H(5,𝔽2)12yes—
433C22×C4H(5,𝔽2)12yes—
434C2×C8—42no—
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—42no—
446C2×C8—42no—
447C2×C8—42no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—48no—
457C4⋊C4—41no—
458C4⋊C4—41no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—48no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—84no—
467C4⋊C4—84no—
468C4⋊C4—88no—
469C4⋊C4C22⋊Q814yes—
470C4⋊C4#12314yes—
471C22⋊C4—216no—
472C22⋊C4—28no—
473C22⋊C4—48no—
474C22⋊C4—44no—
475C22⋊C4—48no—
476C22⋊C4—84no—
477C22⋊C4—88no—
478C42—22no—
479C42—22no—
480C42—24no—
481C42—48no—
482C42—41no—
483C42—41no—
484C42—42no—
485C23—22no—
486C23—24no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—24no—
491C23—22no—
492C23—22no—
493C23—24no—
494C23—24no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—42no—
499C23—42no—
500C23—41no—
501C23—41no—
502C23—42no—
503C23—41no—
504C23—41no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—44no—
510C23—42no—
511C23—42no—
512C23—44no—
513C23—42no—
514C23—42no—
515C23—44no—
516C23—44no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—82no—
523C23—82no—
524C23—82no—
525C23—82no—
526C23—82no—
527C23—82no—
528C23—82no—
529C23—82no—
530C23#47614yes—
531C23C2×#12011yescharacteristic
532C23C2×C22⋊Q811yescharacteristic
533C23#52511yescharacteristic
534C23#54612yes—
535C23#54612yes—
536C23C22×SD1611yescharacteristic
537C23C2×AGL(1,ℤ/8)11yescharacteristic
538C23C2×AGL(1,ℤ/8)11yescharacteristic
539C23#57511yescharacteristic
540C23#57612yes—
541C23#57812yes—
542D4—24no—
543D4—24no—
544D4—432no—
545D4—42no—
546D4—42no—
547D4—44no—
548D4—44no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—816no—
598C2×C4—816no—
599C2×C4—82no—
600C2×C4—82no—
601C2×C4—82no—
602C2×C4—82no—
603C2×C4—82no—
604C2×C4—82no—
605C2×C4—82no—
606C2×C4—82no—
607C2×C4—82no—
608C2×C4—82no—
609C2×C4—82no—
610C2×C4—82no—
611C2×C4—82no—
612C2×C4#45014yes—
613C2×C4C2×C22⋊Q811yescharacteristic
614C2×C4C2×C22⋊Q811yescharacteristic
615C2×C4C2×#12311yescharacteristic
616C2×C4#52511yescharacteristic
617C2×C4#52511yescharacteristic
618C2×C4#52912yes—
619C2×C4#52912yes—
620C2×C4#53211yescharacteristic
621C2×C4#53411yescharacteristic
622C2×C4#53511yescharacteristic
623C2×C4#53812yes—
624C2×C4#54012yes—
625C2×C4#54612yes—
626C2×C4#55012yes—
627C2×C4#55112yes—
628C2×C4#55312yes—
629C2×C4#55412yes—
630C2×C4#55412yes—
631C8—82no—
632C8—82no—
633C22—28no—
634C22—24no—
635C22—44no—
636C22—44no—
637C22—44no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—44no—
645C22—44no—
646C22—44no—
647C22—84no—
648C22—84no—
649C22—84no—
650C22—84no—
651C22—82no—
652C22—82no—
653C22—82no—
654C22—82no—
655C22#240511yescharacteristic
656C22#243111yescharacteristic
657C22#245012yes—
658C22#245712yes—
659C22#284811yescharacteristic
660C22C2×#45011yescharacteristic
661C22#300611yescharacteristic
662C22#300811yescharacteristic
663C22#301111yescharacteristic
664C22C2×#47611yescharacteristic
665C22#307711yescharacteristic
666C22#308312yes—
667C22#310111yescharacteristic
668C22#310911yescharacteristic
669C22#312412yes—
670C22#329512yes—
671C22#329812yes—
672C22#330812yes—
673C22#330912yes—
674C22#331812yes—
675C22#331812yes—
676C22#332712yes—
677C22#333012yes—
678C4—24no—
679C4—44no—
680C4—44no—
681C4—42no—
682C4—42no—
683C4—42no—
684C4—84no—
685C4—82no—
686C4—81no—
687C4—162no—
688C4#239511yescharacteristic
689C4#245412yes—
690C4#246712yes—
691C4#248711yescharacteristic
692C4#256411yescharacteristic
693C4#263511yescharacteristic
694C2—44no—
695C2—84no—
696C2#1482511yescharacteristic
697C2#2403311yescharacteristic
698C2#2405111yescharacteristic
699C2#2557111yescharacteristic
700C2#2558111yescharacteristic
701C2#2641012yes—
702C2#2643812yes—
703C2#2665812yes—
704C2#2667312yes—
705C2#3295611yescharacteristic
706C2#3296311yescharacteristic
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
2. C2×#2100C2semidirect
3. C2×#1802C2semidirect
4. #15135C2semidirect
5. #11227C2semidirect
6. #11066C2semidirect
7. #11042C2semidirect
9. #10885C2semidirect
10. #10788C2semidirect
11. #10730C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23612
C2×#21002
C2×#18022
#151352
#112272
#110662
#110422
#109692
#108852
#107882
#107302