Finite Group Databasebeta

← #406319

Subgroups — #406319

Order 512.

1776 conjugacy classes of subgroups.

The subgroup lattice has 693 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2487C211yesmaximal, characteristic
2C2×#1869C211yesmaximal, characteristic
3C2×#1824C211yesmaximal, characteristic, complemented
4C2×#1807C211yesmaximal, characteristic, complemented
5C22×#340C211yesmaximal, characteristic, complemented
6#13872C211yesmaximal, characteristic
7#11292C212yesmaximal, complemented
8#11220C212yesmaximal, complemented
9#11199C212yesmaximal, complemented
10#11095C212yesmaximal, complemented
11#10730C211yesmaximal, characteristic, complemented
12C24×C8C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C22×#106C2212yes—
17C22×C22⋊C8C2211yescharacteristic
18C22×C22⋊C8C2211yescharacteristic
19C22×C22⋊C8C2211yescharacteristic
20#2487—24no—
21#2487—24no—
22C2×#400—21no—
23C2×#399—21no—
24C2×#389C2212yes—
25C2×#386—21no—
26C2×#386—21no—
27C2×#385—21no—
28C2×#380C2211yescharacteristic
29C2×#380C2212yes—
30C2×#380C2212yes—
31C2×#380C2211yescharacteristic
32#1869—24no—
33#1824—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×C4×D4—22no—
64C2×C4×D4—22no—
65C2×C4×D4—22no—
66C2×C4×D4—22no—
67C2×C4×D4—22no—
68C2×C4×D4—22no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—24no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—22no—
77C22×C4⋊C4—22no—
78C22×C4⋊C4—22no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4—42no—
81C22×C4⋊C4D411yescharacteristic
82C22×C4⋊C4D414yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C22⋊C4—21no—
88C22×C22⋊C4—22no—
89C22×C22⋊C4—21no—
90C22×C22⋊C4—22no—
91C22×C22⋊C4D411yescharacteristic
92C22×C22⋊C4D411yescharacteristic
93C22×C42C2311yescharacteristic
94C2×#106—24no—
95C2×#106—24no—
96C2×#106—42no—
97C2×C22⋊C8—22no—
98C2×C22⋊C8—22no—
99C2×C22⋊C8—22no—
100C2×C22⋊C8—22no—
101C2×C22⋊C8—22no—
102C2×C22⋊C8—24no—
103C2×C22⋊C8—42no—
104#400—28no—
105#399—28no—
106#389—24no—
107#389—24no—
108#386—28no—
109#386—44no—
110#385—28no—
111#380—22no—
112#380—24no—
113#380—24no—
114#380—22no—
115#380—22no—
116#380—24no—
117#380—42no—
118#380D414yes—
119C2×#95—21no—
120C2×#95—22no—
121C2×#95—22no—
122C2×#95—22no—
123C2×#95—22no—
124C2×#95—21no—
125C2×#95—22no—
126#340—24no—
127#340—24no—
128#340—24no—
129#340—24no—
130#340—24no—
131#340—24no—
132#340—42no—
133#336—22no—
134#336—22no—
135C25C2×D411yescharacteristic
136C25C2×D411yescharacteristic
137C22×D4—22no—
138C22×D4—22no—
139C22×D4—22no—
140C22×D4—22no—
141C22×D4—22no—
142C22×D4—22no—
143C22×D4SD1614yes—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—24no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—44no—
164C23×C4—42no—
165C23×C4—42no—
166C23×C4—42no—
167C23×C4—42no—
168C23×C4—42no—
169C23×C4—82no—
170C23×C4—82no—
171C23×C4—82no—
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D412yes—
177C23×C4C2×D412yes—
178C23×C4C2×D412yes—
179C23×C4#4112yes—
180C23×C4#4112yes—
181C23×C4#4112yes—
182C23×C4C2411yescharacteristic
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—22no—
186C22×C8—28no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—28no—
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—44no—
203C22×C8—48no—
204C22×C8—48no—
205C22×C8C2×D412yes—
206C22×C8C2×D412yes—
207C22×C8#4114yes—
208C4×D4—44no—
209C4×D4—44no—
210C4×D4—44no—
211C4×D4—44no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
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—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
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—28no—
241C2×C4⋊C4—48no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—82no—
250C2×C4⋊C4—82no—
251C2×C4⋊C4C2×D414yes—
252C2×C4⋊C4C2×D414yes—
253C2×C4⋊C4#4114yes—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—42no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—82no—
270C2×C22⋊C4—82no—
271C2×C22⋊C4—82no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—21no—
278#106—48no—
279C22⋊C8—44no—
280C22⋊C8—44no—
281C22⋊C8—44no—
282#95—28no—
283#95—28no—
284#95—24no—
285#95—44no—
286#95—42no—
287#95—44no—
288#95—82no—
289C24—22no—
290C24—22no—
291C24—22no—
292C24—22no—
293C24—21no—
294C24—21no—
295C24—21no—
296C24—22no—
297C24—22no—
298C24—21no—
299C24—21no—
300C24—21no—
301C24—22no—
302C24—22no—
303C24#12012yes—
304C24#12112yes—
305C24#12312yes—
306C24C2×SD1612yes—
307C24C2×SD1612yes—
308C24#13512yes—
309C24#13712yes—
310C24#13712yes—
311C24C22×D411yescharacteristic
312C2×D4—28no—
313C2×D4—28no—
314C2×D4—28no—
315C2×D4—44no—
316C2×D4—44no—
317C2×D4—44no—
318C2×D4—44no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—48no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
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—48no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
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—88no—
400C22×C4—88no—
401C22×C4—88no—
402C22×C4—88no—
403C22×C4—84no—
404C22×C4—84no—
405C22×C4—84no—
406C22×C4—84no—
407C22×C4#12012yes—
408C22×C4#12114yes—
409C22×C4#12114yes—
410C22×C4#12112yes—
411C22×C4#12314yes—
412C22×C4#12312yes—
413C22×C4#12414yes—
414C22×C4C2×SD1612yes—
415C22×C4C22×D411yescharacteristic
416C22×C4C22×D411yescharacteristic
417C22×C4C22×D411yescharacteristic
418C22×C4C2×#4111yescharacteristic
419C22×C4C2×#4111yescharacteristic
420C22×C4C2×#4111yescharacteristic
421C22×C4C2×#4112yes—
422C22×C4C2×#4112yes—
423C22×C4H(5,𝔽2)12yes—
424C22×C42−1+412yes—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—24no—
428C2×C8—24no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—416no—
433C2×C8—42no—
434C2×C8—42no—
435C2×C8—41no—
436C2×C8—41no—
437C2×C8—41no—
438C2×C8—41no—
439C2×C8—42no—
440C2×C8—42no—
441C2×C8—41no—
442C2×C8—41no—
443C2×C8—42no—
444C2×C8—42no—
445C2×C8#12314yes—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—24no—
451C4⋊C4—24no—
452C4⋊C4—48no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—48no—
460C4⋊C4—44no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—88no—
466C22⋊C4—44no—
467C22⋊C4—48no—
468C22⋊C4—44no—
469C22⋊C4—88no—
470C22⋊C4—88no—
471C22⋊C4—84no—
472C42—41no—
473C42—41no—
474C42—41no—
475C42—41no—
476C23—24no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—22no—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—22no—
494C23—22no—
495C23—24no—
496C23—22no—
497C23—42no—
498C23—41no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—41no—
503C23—42no—
504C23—42no—
505C23—41no—
506C23—41no—
507C23—42no—
508C23—42no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—42no—
514C23—42no—
515C23#39414yes—
516C23#44814yes—
517C23#45214yes—
518C23#46514yes—
519C23#47014yes—
520C23#48314yes—
521C23#48514yes—
522C23C2×#12011yescharacteristic
523C23C2×#12111yescharacteristic
524C23C2×#12311yescharacteristic
525C23#54612yes—
526C23#54712yes—
527C23C22×SD1611yescharacteristic
528C23C2×#13511yescharacteristic
529C23C2×#13711yescharacteristic
530C23C2×#13711yescharacteristic
531C23#57612yes—
532C23#57712yes—
533D4—416no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—28no—
543C2×C4—28no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—816no—
571C2×C4—816no—
572C2×C4—82no—
573C2×C4—82no—
574C2×C4—82no—
575C2×C4—82no—
576C2×C4—82no—
577C2×C4—82no—
578C2×C4—82no—
579C2×C4—82no—
580C2×C4—82no—
581C2×C4—82no—
582C2×C4—82no—
583C2×C4—82no—
584C2×C4—81no—
585C2×C4—81no—
586C2×C4—81no—
587C2×C4—81no—
588C2×C4—82no—
589C2×C4—82no—
590C2×C4—82no—
591C2×C4—82no—
592C2×C4—82no—
593C2×C4—82no—
594C2×C4—82no—
595C2×C4—82no—
596C2×C4—82no—
597C2×C4#39414yes—
598C2×C4C2×#12111yescharacteristic
599C2×C4C2×#12111yescharacteristic
600C2×C4C2×#12311yescharacteristic
601C2×C4C2×#12311yescharacteristic
602C2×C4#52511yescharacteristic
603C2×C4C2×#12411yescharacteristic
604C2×C4#52912yes—
605C2×C4#53212yes—
606C2×C4#53411yescharacteristic
607C2×C4#53611yescharacteristic
608C2×C4#53712yes—
609C2×C4#53912yes—
610C2×C4#54612yes—
611C2×C4#54712yes—
612C2×C4#55112yes—
613C2×C4#55112yes—
614C2×C4#55212yes—
615C2×C4#55212yes—
616C8—24no—
617C8—42no—
618C8—81no—
619C8—81no—
620C22—24no—
621C22—24no—
622C22—24no—
623C22—28no—
624C22—28no—
625C22—24no—
626C22—28no—
627C22—24no—
628C22—28no—
629C22—24no—
630C22—42no—
631C22—44no—
632C22—42no—
633C22—42no—
634C22—44no—
635C22—81no—
636C22—81no—
637C22—82no—
638C22—81no—
639C22—82no—
640C22—81no—
641C22#203518yes—
642C22C2×#39411yescharacteristic
643C22#241912yes—
644C22#243012yes—
645C22#248611yescharacteristic
646C22#253411yescharacteristic
647C22C2×#44811yescharacteristic
648C22C2×#45211yescharacteristic
649C22#301212yes—
650C22C2×#46511yescharacteristic
651C22C2×#47011yescharacteristic
652C22#305711yescharacteristic
653C22#305811yescharacteristic
654C22C2×#48311yescharacteristic
655C22C2×#48511yescharacteristic
656C22#309612yes—
657C22#329512yes—
658C22#329712yes—
659C22#331612yes—
660C22#332012yes—
661C22#333512yes—
662C22#333712yes—
663C22#334612yes—
664C22#334612yes—
665C4—24no—
666C4—42no—
667C4—81no—
668C4—81no—
669C4—84no—
670C4—84no—
671C4—84no—
672C4—162no—
673C4#241211yescharacteristic
674C4#241311yescharacteristic
675C4#248812yes—
676C4#249812yes—
677C4#261811yescharacteristic
678C4#285411yescharacteristic
679C2—28no—
680C2—82no—
681C2#1492711yescharacteristic
682C2C2×#203511yescharacteristic
683C2#2348611yescharacteristic
684C2#2493512yes—
685C2#2494312yes—
686C2#2553512yes—
687C2#2554312yes—
688C2#2765811yescharacteristic
689C2#2766611yescharacteristic
690C2#2809711yescharacteristic
691C2#2810811yescharacteristic
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×#1824C2semidirect
4. C2×#1807C2semidirect
5. C22×#340C2semidirect
7. #11292C2semidirect
8. #11220C2semidirect
9. #11199C2semidirect
10. #11095C2semidirect
11. #10730C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24872
C2×#18692
C2×#18242
C2×#18072
C22×#3402
#138722
#112922
#112202
#111992
#110952
#107302