Finite Group Databasebeta

← #396591

Subgroups — #396591

Order 512.

1608 conjugacy classes of subgroups.

The subgroup lattice has 699 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2072C211yesmaximal, characteristic, complemented
2C2×#1892C212yesmaximal
3C22×#336C211yesmaximal, characteristic
4#14702C211yesmaximal, characteristic
5#14686C211yesmaximal, characteristic
6#11357C211yesmaximal, characteristic, complemented
7#11316C212yesmaximal, complemented
8#11267C211yesmaximal, characteristic, complemented
9#11190C212yesmaximal, complemented
10#11050C211yesmaximal, characteristic, complemented
11#10721C212yesmaximal
12C24×C8C2212yes—
13C22×C22⋊Q8C2211yescharacteristic
14C23×C42C2211yescharacteristic
15C22×#103C2212yes—
16C22×#103C2212yes—
17C2×#391C2211yescharacteristic
18C2×#391C2211yescharacteristic
19C2×#384C2211yescharacteristic
20C2×#384C2211yescharacteristic
21C2×#380—21no—
22C2×#380—21no—
23C2×#380C2212yes—
24C2×#380C2211yescharacteristic
25#2072—24no—
26#1892—28no—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#336—24no—
34C2×#336C2211yescharacteristic
35C2×#336C2212yes—
36C2×#336C2211yescharacteristic
37#1436—21no—
38#1436—21no—
39C4⋊#95—21no—
40C4⋊#95—21no—
41C4⋊#95C2211yescharacteristic
42C4⋊#95C2211yescharacteristic
43C4⋊#95C2212yes—
44C22⋊#95—21no—
45C22⋊#95—21no—
46C23×Q8C2311yescharacteristic
47C24×C4D411yescharacteristic
48C24×C4D411yescharacteristic
49C24×C4C2311yescharacteristic
50C23×C8—24no—
51C23×C8—24no—
52C23×C8—24no—
53C23×C8—24no—
54C23×C8—24no—
55C23×C8—42no—
56C23×C8—42no—
57C23×C8C2312yes—
58C23×C8C2312yes—
59C2×C22⋊Q8—22no—
60C2×C22⋊Q8—24no—
61C2×C22⋊Q8—24no—
62C2×C22⋊Q8D414yes—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C22⋊C4—21no—
74C22×C22⋊C4—21no—
75C22×C22⋊C4—22no—
76C22×C42—24no—
77C22×C42—22no—
78C22×C42—22no—
79C22×C42—42no—
80C22×C42D411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84C2×#103—24no—
85C2×#103—24no—
86C2×#103—24no—
87C2×#103—24no—
88C2×#103—42no—
89C2×#103—42no—
90#391—22no—
91#391—22no—
92#391—42no—
93#384—22no—
94#384—22no—
95#384—42no—
96#380—24no—
97#380—24no—
98#380—24no—
99#380—44no—
100#380—44no—
101C2×#95—21no—
102C2×#95—21no—
103C2×#95—21no—
104C2×#95—21no—
105C2×#95—21no—
106C2×#95—21no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—21no—
113#340—24no—
114#340—24no—
115#340—24no—
116#340—24no—
117#340—42no—
118#340—42no—
119#336—24no—
120#336—22no—
121#336D414yes—
122C25C2×D411yescharacteristic
123C22×Q8—21no—
124C22×Q8—21no—
125C22×Q8—22no—
126C22×Q8—22no—
127C22×Q8—21no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8—21no—
131C22×Q8—21no—
132C22×Q8C2×D412yes—
133C23×C4—21no—
134C23×C4—22no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—22no—
138C23×C4—21no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—41no—
154C23×C4—41no—
155C23×C4—41no—
156C23×C4—41no—
157C23×C4SD1614yes—
158C23×C4Q1614yes—
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D412yes—
165C23×C4#4112yes—
166C23×C4#4114yes—
167C23×C4#4114yes—
168C23×C4C2411yescharacteristic
169C22×C8—216no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—48no—
183C22×C8—48no—
184C22×C8—48no—
185C22×C8—48no—
186C22×C8—48no—
187C22×C8—48no—
188C22×C8C2×D414yes—
189C22×C8#4114yes—
190C22⋊Q8—216no—
191C22⋊Q8—48no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4C2×D412yes—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—42no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—42no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—21no—
247C2×C42—22no—
248C2×C42—21no—
249C2×C42—22no—
250C2×C42—48no—
251C2×C42—48no—
252C2×C42—41no—
253C2×C42—41no—
254C2×C42C2×D412yes—
255C2×C42#4112yes—
256#103—48no—
257#103—48no—
258#95—42no—
259#95—42no—
260#95—44no—
261#95—42no—
262#95—42no—
263#95—44no—
264#95—42no—
265#95—42no—
266#95—42no—
267#95—44no—
268#95—42no—
269#95—42no—
270C24—21no—
271C24—21no—
272C24—21no—
273C24—21no—
274C24—21no—
275C24—21no—
276C24—21no—
277C24—21no—
278C24#12112yes—
279C24#12312yes—
280C24#12412yes—
281C24C2×SD1612yes—
282C24C2×Q1612yes—
283C24AGL(1,ℤ/8)12yes—
284C24#13712yes—
285C24C22×D411yescharacteristic
286C2×Q8—24no—
287C2×Q8—22no—
288C2×Q8—24no—
289C2×Q8—22no—
290C2×Q8—42no—
291C2×Q8—41no—
292C2×Q8—41no—
293C2×Q8—42no—
294C2×Q8—42no—
295C2×Q8—42no—
296C2×Q8—42no—
297C2×Q8—42no—
298C2×Q8—42no—
299C2×Q8—42no—
300C2×Q8—42no—
301C2×Q8—42no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—28no—
341C22×C4—28no—
342C22×C4—28no—
343C22×C4—28no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—44no—
347C22×C4—44no—
348C22×C4—44no—
349C22×C4—44no—
350C22×C4—44no—
351C22×C4—44no—
352C22×C4—44no—
353C22×C4—44no—
354C22×C4—44no—
355C22×C4—44no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—42no—
376C22×C4—41no—
377C22×C4—42no—
378C22×C4—41no—
379C22×C4—42no—
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—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4#12112yes—
397C22×C4#12112yes—
398C22×C4#12112yes—
399C22×C4#12318yes—
400C22×C4#12312yes—
401C22×C4#12412yes—
402C22×C4#12618yes—
403C22×C4C2×SD1612yes—
404C22×C4C2×SD1612yes—
405C22×C4C2×Q1612yes—
406C22×C4C2×Q1612yes—
407C22×C4#13512yes—
408C22×C4#13512yes—
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D411yescharacteristic
411C22×C4C2×#4112yes—
412C22×C4C2×#4112yes—
413C22×C4C2×#4112yes—
414C22×C4C2×#4111yescharacteristic
415C22×C4C2×#4111yescharacteristic
416C22×C4H(5,𝔽2)11yescharacteristic
417C22×C4H(5,𝔽2)12yes—
418C22×C42−1+411yescharacteristic
419C2×C8—24no—
420C2×C8—24no—
421C2×C8—24no—
422C2×C8—24no—
423C2×C8—24no—
424C2×C8—24no—
425C2×C8—432no—
426C2×C8—42no—
427C2×C8—42no—
428C2×C8—42no—
429C2×C8—42no—
430C2×C8—42no—
431C2×C8—42no—
432C2×C8—42no—
433C2×C8—42no—
434C4⋊C4—24no—
435C4⋊C4—42no—
436C4⋊C4—48no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C4⋊C4—84no—
450C4⋊C4—84no—
451C4⋊C4—84no—
452C4⋊C4—84no—
453C22⋊C4—48no—
454C22⋊C4—88no—
455C22⋊C4—84no—
456C42—24no—
457C42—24no—
458C42—44no—
459C42—44no—
460C42—42no—
461C42—42no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23#39714yes—
491C23#46014yes—
492C23#46214yes—
493C23#48314yes—
494C23#48414yes—
495C23#48714yes—
496C23#48914yes—
497C23C2×#12111yescharacteristic
498C23C2×#12111yescharacteristic
499C23C2×#12311yescharacteristic
500C23C2×#12411yescharacteristic
501C23#53211yescharacteristic
502C23#53411yescharacteristic
503C23#53711yescharacteristic
504C23C22×SD1611yescharacteristic
505C23C22×Q1611yescharacteristic
506C23C2×#13511yescharacteristic
507C23C2×#13511yescharacteristic
508C23C2×AGL(1,ℤ/8)11yescharacteristic
509C23C2×#13711yescharacteristic
510C23#57511yescharacteristic
511C23#57511yescharacteristic
512Q8—44no—
513Q8—44no—
514Q8—82no—
515Q8—82no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—28no—
535C2×C4—28no—
536C2×C4—28no—
537C2×C4—28no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—416no—
545C2×C4—416no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—81no—
561C2×C4—81no—
562C2×C4—81no—
563C2×C4—81no—
564C2×C4—81no—
565C2×C4—81no—
566C2×C4—81no—
567C2×C4—81no—
568C2×C4—81no—
569C2×C4—81no—
570C2×C4—81no—
571C2×C4—81no—
572C2×C4—81no—
573C2×C4—81no—
574C2×C4—82no—
575C2×C4—81no—
576C2×C4—82no—
577C2×C4—81no—
578C2×C4—82no—
579C2×C4—82no—
580C2×C4—82no—
581C2×C4—81no—
582C2×C4—81no—
583C2×C4—82no—
584C2×C4—82no—
585C2×C4—81no—
586C2×C4—81no—
587C2×C4—82no—
588C2×C4—82no—
589C2×C4—82no—
590C2×C4—82no—
591C2×C4—82no—
592C2×C4—82no—
593C2×C4—82no—
594C2×C4—81no—
595C2×C4—81no—
596C2×C4#39714yes—
597C2×C4#46514yes—
598C2×C4#46714yes—
599C2×C4C2×#12312yes—
600C2×C4#52512yes—
601C2×C4C2×#12612yes—
602C2×C4#52912yes—
603C2×C4#53812yes—
604C2×C4#53912yes—
605C2×C4D4211yescharacteristic
606C2×C4#54612yes—
607C2×C4#54612yes—
608C2×C4#54711yescharacteristic
609C2×C4#54811yescharacteristic
610C2×C4#55112yes—
611C2×C4#55112yes—
612C2×C4#55212yes—
613C2×C4#55312yes—
614C2×C4#55311yescharacteristic
615C8—44no—
616C8—44no—
617C22—24no—
618C22—24no—
619C22—24no—
620C22—24no—
621C22—24no—
622C22—24no—
623C22—24no—
624C22—24no—
625C22—24no—
626C22—24no—
627C22—24no—
628C22—24no—
629C22—24no—
630C22—24no—
631C22#207218yescomplemented
632C22C2×#39711yescharacteristic
633C22#241211yescharacteristic
634C22#241611yescharacteristic
635C22#249811yescharacteristic
636C22#254311yescharacteristic
637C22#263911yescharacteristic
638C22#285011yescharacteristic
639C22C2×#46011yescharacteristic
640C22C2×#46211yescharacteristic
641C22#303511yescharacteristic
642C22#303511yescharacteristic
643C22C2×#46511yescharacteristic
644C22C2×#46711yescharacteristic
645C22#305511yescharacteristic
646C22#305611yescharacteristic
647C22C2×#48311yescharacteristic
648C22C2×#48411yescharacteristic
649C22#309111yescharacteristic
650C22#309111yescharacteristic
651C22C2×#48711yescharacteristic
652C22#313111yescharacteristic
653C22C2×#48911yescharacteristic
654C22#313511yescharacteristic
655C22#316711yescharacteristic
656C22#317011yescharacteristic
657C22#318011yescharacteristic
658C22#318111yescharacteristic
659C22#318711yescharacteristic
660C22#318811yescharacteristic
661C22#318911yescharacteristic
662C22#319011yescharacteristic
663C22#324611yescharacteristic
664C22#324711yescharacteristic
665C22#324811yescharacteristic
666C22#324911yescharacteristic
667C4—24no—
668C4—24no—
669C4—24no—
670C4—42no—
671C4—42no—
672C4—82no—
673C4—82no—
674C4—161no—
675C4—162no—
676C4—162no—
677C4—161no—
678C4#241812yes—
679C4#241912yes—
680C4#248112yes—
681C4#251012yes—
682C2—28no—
683C2#1473511yescharacteristic
684C2C2×#207211yescharacteristic
685C2#2362711yescharacteristic
686C2#2455111yescharacteristic
687C2#2455211yescharacteristic
688C2#2475211yescharacteristic
689C2#2475311yescharacteristic
690C2#2785511yescharacteristic
691C2#2785811yescharacteristic
692C2#2857411yescharacteristic
693C2#2857511yescharacteristic
694C2#3018711yescharacteristic
695C2#3018811yescharacteristic
696C2#3304311yescharacteristic
697C2#3305111yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2072C2semidirect
6. #11357C2semidirect
7. #11316C2semidirect
8. #11267C2semidirect
9. #11190C2semidirect
10. #11050C2semidirect
631. C22#2072semidirect

Maximal subgroups

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

subgroupindex
C2×#20722
C2×#18922
C22×#3362
#147022
#146862
#113572
#113162
#112672
#111902
#110502
#107212