Finite Group Databasebeta

← #402991

Subgroups — #402991

Order 512.

1664 conjugacy classes of subgroups.

The subgroup lattice has 728 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#124C211yesmaximal, characteristic
2C2×#2024C211yesmaximal, characteristic, complemented
3C2×#2023C211yesmaximal, characteristic
4#14714C211yesmaximal, characteristic, complemented
5#11268C212yesmaximal, complemented
6#11050C211yesmaximal, characteristic, complemented
7#11049C211yesmaximal, characteristic
8#11009C212yesmaximal
9#10795C211yesmaximal, characteristic, complemented
10#10789C212yesmaximal, complemented
11#10788C212yesmaximal
12C22×C4⋊Q8C2211yescharacteristic
13C22×#124C2211yescharacteristic
14C22×C4×Q8C2211yescharacteristic
15C22×C4×D4C2211yescharacteristic
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×#102C2212yes—
19C22×#102C2212yes—
20C4×#124—24no—
21C4×#124—24no—
22C2×#391—21no—
23C2×#391—21no—
24C2×#391C2211yescharacteristic
25C2×#390C2211yescharacteristic
26C2×#388—21no—
27C2×#388—21no—
28C2×#384C2211yescharacteristic
29C2×#382C2211yescharacteristic
30C2×C4×C22⋊C4—21no—
31C2×C4×C22⋊C4—21no—
32C2×C43C2211yescharacteristic
33#2024—24no—
34#2023—24no—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2212yes—
40C2×#336C2212yes—
41C2×#336C2211yescharacteristic
42C2×#336C2211yescharacteristic
43C4⋊#95—21no—
44C4⋊#95—21no—
45C4⋊#95C2212yes—
46C4⋊#95C2212yes—
47C23×Q8—21no—
48C23×Q8C2311yescharacteristic
49C23×D4C2311yescharacteristic
50C24×C4—21no—
51C23×C8C2312yes—
52C23×C8C2312yes—
53C2×C4⋊Q8—24no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—42no—
57C2×#124—22no—
58C2×#124—22no—
59C2×#124—22no—
60C2×#124—22no—
61C2×#124—22no—
62C2×#124Q814yescomplemented
63C2×C4×Q8—22no—
64C2×C4×Q8—22no—
65C2×C4×Q8—24no—
66C2×C4×Q8—42no—
67C2×C4×D4—22no—
68C2×C4×D4—22no—
69C2×C4×D4—24no—
70C2×C4×D4—42no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2312yes—
79C22×C22⋊C4—21no—
80C22×C22⋊C4—21no—
81C22×C22⋊C4—21no—
82C22×C22⋊C4—21no—
83C22×C22⋊C4—21no—
84C22×C42—21no—
85C22×C42D411yescharacteristic
86C22×C42D411yescharacteristic
87C22×C42D411yescharacteristic
88C22×C42D411yescharacteristic
89C22×C42C2311yescharacteristic
90C22×C42C2311yescharacteristic
91C22×C42C2311yescharacteristic
92C2×#103—24no—
93C2×#103—24no—
94C2×#103—24no—
95C2×#103—42no—
96C2×#103—42no—
97C2×#103—42no—
98C2×#102—24no—
99C2×#102—24no—
100C2×#102—24no—
101C2×#102—42no—
102C2×#102—42no—
103C2×#102—42no—
104#391—22no—
105#391—44no—
106#391—44no—
107#391D414yes—
108#390—22no—
109#390D414yes—
110#388—28no—
111#388—28no—
112#384—42no—
113#382—22no—
114#382—22no—
115C4×C22⋊C4—28no—
116C4×C22⋊C4—28no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—21no—
120C2×#95—21no—
121C2×#95—22no—
122C2×#95—22no—
123C2×#95—22no—
124C2×#95—22no—
125C43—22no—
126C43—22no—
127#340—24no—
128#340—24no—
129#340—42no—
130#340—42no—
131#336—22no—
132#336—22no—
133#336—24no—
134#336—24no—
135C25—21no—
136C22×Q8—24no—
137C22×Q8—21no—
138C22×Q8—21no—
139C22×Q8—21no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8—42no—
145C22×Q8—42no—
146C22×Q8—42no—
147C22×Q8—42no—
148C22×Q8—42no—
149C22×Q8—42no—
150C22×Q8C2×D412yes—
151C22×Q8C2×Q812yes—
152C22×Q8#4112yes—
153C22×D4—21no—
154C22×D4—22no—
155C22×D4—22no—
156C22×D4—21no—
157C22×D4—21no—
158C22×D4—22no—
159C22×D4—22no—
160C22×D4C2×D412yes—
161C22×D4C2×Q812yes—
162C22×D4#4112yes—
163C23×C4—21no—
164C23×C4—24no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—42no—
170C23×C4—42no—
171C23×C4—42no—
172C23×C4—42no—
173C23×C4—42no—
174C23×C4—42no—
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2411yescharacteristic
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
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—
196C4⋊Q8—48no—
197C4⋊Q8—48no—
198#124—28no—
199#124—28no—
200#124—28no—
201#124—28no—
202C4×Q8—48no—
203C4×Q8—48no—
204C4×D4—48no—
205C4×D4—48no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4C2×D412yes—
236C2×C4⋊C4C2×D412yes—
237C2×C4⋊C4#4114yes—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—42no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—42no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—24no—
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—22no—
272C2×C42—22no—
273C2×C42—41no—
274C2×C42—41no—
275C2×C42—42no—
276C2×C42—42no—
277C2×C42C2×D412yes—
278C2×C42C2×D412yes—
279C2×C42C2×Q812yes—
280C2×C42#4112yes—
281C2×C42#4114yes—
282#103—48no—
283#103—48no—
284#102—48no—
285#102—48no—
286#95—24no—
287#95—24no—
288#95—24no—
289#95—24no—
290#95—44no—
291#95—44no—
292#95—44no—
293#95—44no—
294C24—22no—
295C24—22no—
296C24—22no—
297C24—41no—
298C24—41no—
299C24—41no—
300C24—41no—
301C24—41no—
302C24—41no—
303C24—41no—
304C24—41no—
305C24—41no—
306C24—41no—
307C24—41no—
308C24—41no—
309C24C22×D411yescharacteristic
310C2×Q8—22no—
311C2×Q8—22no—
312C2×Q8—22no—
313C2×Q8—24no—
314C2×Q8—24no—
315C2×Q8—24no—
316C2×Q8—24no—
317C2×Q8—24no—
318C2×Q8—24no—
319C2×Q8—48no—
320C2×Q8—48no—
321C2×Q8—48no—
322C2×Q8—42no—
323C2×Q8—42no—
324C2×Q8—42no—
325C2×Q8—42no—
326C2×Q8—42no—
327C2×Q8—42no—
328C2×Q8—84no—
329C2×Q8—84no—
330C2×Q8—84no—
331C2×Q8—84no—
332C2×Q8C22⋊Q814yes—
333C2×D4—22no—
334C2×D4—22no—
335C2×D4—22no—
336C2×D4—24no—
337C2×D4—24no—
338C2×D4—24no—
339C2×D4—24no—
340C2×D4—24no—
341C2×D4—24no—
342C2×D4—42no—
343C2×D4—42no—
344C2×D4—42no—
345C2×D4—42no—
346C2×D4—42no—
347C2×D4—42no—
348C2×D4C22⋊Q814yes—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—48no—
386C22×C4—48no—
387C22×C4—48no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—42no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—42no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—84no—
418C22×C4—84no—
419C22×C4—84no—
420C22×C4—84no—
421C22×C4#12012yes—
422C22×C4#12012yes—
423C22×C4C22⋊Q812yes—
424C22×C4C22⋊Q812yes—
425C22×C4#12412yes—
426C22×C4#12412yes—
427C22×C4#13512yes—
428C22×C4#13512yes—
429C22×C4AGL(1,ℤ/8)12yes—
430C22×C4AGL(1,ℤ/8)12yes—
431C22×C4#13712yes—
432C22×C4#13712yes—
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C22×Q811yescharacteristic
436C22×C4C2×#4111yescharacteristic
437C22×C4C2×#4111yescharacteristic
438C22×C4C2×#4111yescharacteristic
439C22×C4C2×#4112yes—
440C22×C4C2×#4112yes—
441C22×C4H(5,𝔽2)12yes—
442C22×C4H(5,𝔽2)12yes—
443C2×C8—42no—
444C2×C8—42no—
445C2×C8—42no—
446C2×C8—42no—
447C2×C8—42no—
448C2×C8—42no—
449C2×C8—42no—
450C2×C8—42no—
451C2×C8—42no—
452C2×C8—42no—
453C2×C8—42no—
454C2×C8—42no—
455C2×C8—42no—
456C2×C8—42no—
457C4⋊C4—24no—
458C4⋊C4—24no—
459C4⋊C4—24no—
460C4⋊C4—24no—
461C4⋊C4—48no—
462C4⋊C4—48no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—84no—
472C4⋊C4—84no—
473C22⋊C4—216no—
474C22⋊C4—216no—
475C22⋊C4—48no—
476C22⋊C4—48no—
477C22⋊C4—48no—
478C42—22no—
479C42—22no—
480C42—22no—
481C42—22no—
482C42—24no—
483C42—24no—
484C42—24no—
485C42—24no—
486C42—44no—
487C42—44no—
488C42—48no—
489C42—44no—
490C42—44no—
491C23—24no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—81no—
511C23—81no—
512C23—81no—
513C23—81no—
514C23—81no—
515C23—81no—
516C23—81no—
517C23—81no—
518C23—81no—
519C23—81no—
520C23—81no—
521C23—81no—
522C23—81no—
523C23—81no—
524C23—81no—
525C23—81no—
526C23C2×#12011yescharacteristic
527C23C2×C22⋊Q811yescharacteristic
528C23#52511yescharacteristic
529C23#52511yescharacteristic
530C23C2×#12411yescharacteristic
531C23#53511yescharacteristic
532C23#53511yescharacteristic
533C23C2×#13511yescharacteristic
534C23C2×#13511yescharacteristic
535C23C2×AGL(1,ℤ/8)11yescharacteristic
536C23C2×AGL(1,ℤ/8)11yescharacteristic
537C23C2×#13711yescharacteristic
538C23C2×#13711yescharacteristic
539C23#57511yescharacteristic
540C23#57511yescharacteristic
541Q8—28no—
542Q8—44no—
543Q8—44no—
544Q8—44no—
545Q8—816no—
546D4—28no—
547D4—44no—
548D4—44no—
549D4—44no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—816no—
606C2×C4—81no—
607C2×C4—81no—
608C2×C4—81no—
609C2×C4—81no—
610C2×C4—81no—
611C2×C4—81no—
612C2×C4—82no—
613C2×C4—81no—
614C2×C4—81no—
615C2×C4—82no—
616C2×C4—82no—
617C2×C4—82no—
618C2×C4—82no—
619C2×C4—82no—
620C2×C4—82no—
621C2×C4—82no—
622C2×C4—82no—
623C2×C4—82no—
624C2×C4—82no—
625C2×C4#44914yes—
626C2×C4#45214yes—
627C2×C4#48914yes—
628C2×C4C2×C22⋊Q811yescharacteristic
629C2×C4C2×C22⋊Q811yescharacteristic
630C2×C4#52511yescharacteristic
631C2×C4#52511yescharacteristic
632C2×C4#52912yes—
633C2×C4#52912yes—
634C2×C4#53812yes—
635C2×C4#53812yes—
636C2×C4#54612yes—
637C2×C4#54612yes—
638C2×C4#54612yes—
639C2×C4#54612yes—
640C2×C4#55412yes—
641C2×C4#55412yes—
642C2×C4#55512yes—
643C2×C4#55912yes—
644C8—82no—
645C8—82no—
646C22—44no—
647C22—44no—
648C22—44no—
649C22—82no—
650C22—82no—
651C22—82no—
652C22—82no—
653C22—82no—
654C22—82no—
655C22—82no—
656C22—82no—
657C22—82no—
658C22—82no—
659C22—82no—
660C22—82no—
661C22#239811yescharacteristic
662C22#240411yescharacteristic
663C22#243511yescharacteristic
664C22#256311yescharacteristic
665C22#258011yescharacteristic
666C22#263111yescharacteristic
667C22#284811yescharacteristic
668C22C2×#44911yescharacteristic
669C22C2×#45211yescharacteristic
670C22#300611yescharacteristic
671C22#300711yescharacteristic
672C22#307611yescharacteristic
673C22#307611yescharacteristic
674C22#307911yescharacteristic
675C22#307911yescharacteristic
676C22#310511yescharacteristic
677C22#310611yescharacteristic
678C22#310711yescharacteristic
679C22#310811yescharacteristic
680C22#311411yescharacteristic
681C22#311511yescharacteristic
682C22#311611yescharacteristic
683C22#311711yescharacteristic
684C22#313111yescharacteristic
685C22C2×#48911yescharacteristic
686C22#314011yescharacteristic
687C22#314111yescharacteristic
688C22#320711yescharacteristic
689C22#321011yescharacteristic
690C22#321111yescharacteristic
691C22#321111yescharacteristic
692C22#321311yescharacteristic
693C22#321511yescharacteristic
694C22#321711yescharacteristic
695C22#321811yescharacteristic
696C4—24no—
697C4—24no—
698C4—24no—
699C4—42no—
700C4—44no—
701C4—42no—
702C4—42no—
703C4—82no—
704C4—82no—
705C4—84no—
706C4—162no—
707C4#245012yes—
708C4#245012yes—
709C4#245912yes—
710C4#245912yes—
711C2—84no—
712C2#1486511yescharacteristic
713C2#2381711yescharacteristic
714C2#2382111yescharacteristic
715C2#2393611yescharacteristic
716C2#2393711yescharacteristic
717C2#2585711yescharacteristic
718C2#2586411yescharacteristic
719C2#2914211yescharacteristic
720C2#2914811yescharacteristic
721C2#2926011yescharacteristic
722C2#2926211yescharacteristic
723C2#3004511yescharacteristic
724C2#3004611yescharacteristic
725C2#3295511yescharacteristic
726C2#3295811yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2024C2semidirect
4. #14714C2semidirect
5. #11268C2semidirect
6. #11050C2semidirect
9. #10795C2semidirect
10. #10789C2semidirect
62. C2×#124Q8semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1242
C2×#20242
C2×#20232
#147142
#112682
#110502
#110492
#110092
#107952
#107892
#107882