Finite Group Databasebeta

← #416351

Subgroups — #416351

Order 512.

1346 conjugacy classes of subgroups.

The subgroup lattice has 695 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1959C211yesmaximal, characteristic
2C2×#1959C211yesmaximal, characteristic
3C2×#1836C211yesmaximal, characteristic, complemented
4C2×#1767C211yesmaximal, characteristic, complemented
5#14712C211yesmaximal, characteristic, complemented
6#14708C211yesmaximal, characteristic
7#11077C212yesmaximal, complemented
8#11076C212yesmaximal
9#11067C212yesmaximal, complemented
10#11066C212yesmaximal
11#10843C211yesmaximal, characteristic, complemented
12C22×#124C2211yescharacteristic
13C22×C4⋊C8C2212yes—
14C22×#103C2212yes—
15C22×#103C2212yes—
16C22×#102C2212yes—
17C22×#102C2212yes—
18C22×#97C2211yescharacteristic
19C22×#97C2211yescharacteristic
20C2×#391C2211yescharacteristic
21C2×#391C2211yescharacteristic
22C2×#390C2211yescharacteristic
23C2×#390C2211yescharacteristic
24C2×#389—21no—
25C2×#389—21no—
26C2×#389C2212yes—
27C2×#384C2211yescharacteristic
28C2×#384C2211yescharacteristic
29C2×#382C2211yescharacteristic
30C2×C43C2211yescharacteristic
31#1959—24no—
32#1959—24no—
33#1836—24no—
34#1767—24no—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2212yes—
40#1437—21no—
41#1437—21no—
42#1437—21no—
43#1437—21no—
44C4⋊#95—21no—
45C4⋊#95—21no—
46C4⋊#95C2212yes—
47C23×Q8C2311yescharacteristic
48C23×D4C2311yescharacteristic
49C23×C8C2312yes—
50C23×C8C2312yes—
51C2×#124—22no—
52C2×#124—22no—
53C2×#124—22no—
54C2×#124—22no—
55C2×#124—22no—
56C2×#124Q814yes—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2312yes—
66C22×C22⋊C4—21no—
67C22×C22⋊C4—21no—
68C22×C22⋊C4—21no—
69C22×C22⋊C4—21no—
70C22×C22⋊C4—21no—
71C22×C22⋊C4—21no—
72C22×C42D411yescharacteristic
73C22×C42D411yescharacteristic
74C22×C42D411yescharacteristic
75C22×C42D411yescharacteristic
76C22×C42C2311yescharacteristic
77C22×C42C2311yescharacteristic
78C22×C42C2311yescharacteristic
79C2×C4⋊C8—24no—
80C2×C4⋊C8—24no—
81C2×C4⋊C8—24no—
82C2×#103—24no—
83C2×#103—24no—
84C2×#103—24no—
85C2×#103—42no—
86C2×#103—42no—
87C2×#103—42no—
88C2×#102—24no—
89C2×#102—24no—
90C2×#102—24no—
91C2×#102—42no—
92C2×#102—42no—
93C2×#102—42no—
94C2×#97—22no—
95C2×#97—22no—
96C2×#97—22no—
97C2×#97—22no—
98C2×#97—22no—
99C2×#97D414yes—
100#391—24no—
101#391—42no—
102#390—24no—
103#390—42no—
104#389—42no—
105#389—42no—
106#389—44no—
107#389—44no—
108#384—22no—
109#384—22no—
110#384—22no—
111#384D414yes—
112#382—22no—
113#382—22no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—21no—
120C2×#95—21no—
121C2×#95—21no—
122C2×#95—21no—
123C43—22no—
124C43—22no—
125#340—24no—
126#340—24no—
127#340—42no—
128#340—42no—
129#336—24no—
130C25—21no—
131C22×Q8—21no—
132C22×Q8—21no—
133C22×Q8—21no—
134C22×Q8—21no—
135C22×Q8—21no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8C2×Q812yes—
141C22×D4—21no—
142C22×D4—21no—
143C22×D4—21no—
144C22×D4—21no—
145C22×D4—22no—
146C22×D4—22no—
147C22×D4—21no—
148C22×D4—22no—
149C22×D4—22no—
150C22×D4C2×Q812yes—
151C23×C4—21no—
152C23×C4—22no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—22no—
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2411yescharacteristic
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8C2×D414yes—
176C22×C8#4114yes—
177#124—28no—
178#124—28no—
179#124—44no—
180#124—44no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4C2×D414yes—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—42no—
219C2×C22⋊C4—42no—
220C2×C22⋊C4—42no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4—42no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—42no—
231C2×C22⋊C4—42no—
232C2×C22⋊C4—42no—
233C2×C22⋊C4—42no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—41no—
250C2×C42—41no—
251C2×C42—41no—
252C2×C42—41no—
253C2×C42C2×D412yes—
254C2×C42C2×D412yes—
255C2×C42C2×Q812yes—
256C2×C42#4114yes—
257C2×C42#4112yes—
258C2×C42#4112yes—
259C2×C42#4112yes—
260C4⋊C8—216no—
261#103—48no—
262#103—48no—
263#102—48no—
264#102—48no—
265#97—28no—
266#97—44no—
267#95—44no—
268#95—44no—
269#95—42no—
270#95—42no—
271#95—42no—
272#95—42no—
273#95—42no—
274#95—42no—
275#95—44no—
276C24—22no—
277C24—41no—
278C24—41no—
279C24—41no—
280C24—41no—
281C24—41no—
282C24—41no—
283C24—41no—
284C24—41no—
285C24—41no—
286C24—41no—
287C24—41no—
288C24—41no—
289C24—41no—
290C24—41no—
291C24C22×D411yescharacteristic
292C2×Q8—22no—
293C2×Q8—22no—
294C2×Q8—24no—
295C2×Q8—24no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8—42no—
299C2×Q8—42no—
300C2×Q8—42no—
301C2×Q8—42no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×Q8—42no—
305C2×Q8—42no—
306C2×Q8—42no—
307C2×Q8—42no—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—24no—
311C2×D4—24no—
312C2×D4—41no—
313C2×D4—42no—
314C2×D4—41no—
315C2×D4—42no—
316C2×D4—42no—
317C2×D4—42no—
318C2×D4—42no—
319C2×D4—42no—
320C2×D4—42no—
321C2×D4—42no—
322C2×D4—42no—
323C2×D4—42no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
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—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—42no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—42no—
363C22×C4—41no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—42no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—42no—
375C22×C4—41no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4#12112yes—
390C22×C4#12112yes—
391C22×C4#12112yes—
392C22×C4#12112yes—
393C22×C4C22⋊Q812yes—
394C22×C4C22⋊Q812yes—
395C22×C4#12412yes—
396C22×C4#12412yes—
397C22×C4AGL(1,ℤ/8)12yes—
398C22×C4AGL(1,ℤ/8)12yes—
399C22×C4#13712yes—
400C22×C4#13712yes—
401C22×C4C22×D411yescharacteristic
402C22×C4C22×D411yescharacteristic
403C22×C4C22×Q811yescharacteristic
404C22×C4C2×#4111yescharacteristic
405C22×C4C2×#4111yescharacteristic
406C22×C4C2×#4111yescharacteristic
407C22×C4C2×#4112yes—
408C22×C4C2×#4112yes—
409C22×C4H(5,𝔽2)12yes—
410C22×C4H(5,𝔽2)12yes—
411C2×C8—24no—
412C2×C8—24no—
413C2×C8—24no—
414C2×C8—24no—
415C2×C8—24no—
416C2×C8—24no—
417C2×C8—42no—
418C2×C8—42no—
419C2×C8—42no—
420C2×C8—42no—
421C2×C8—42no—
422C2×C8—42no—
423C2×C8—42no—
424C2×C8—42no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—48no—
428C4⋊C4—48no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—84no—
440C4⋊C4—84no—
441C4⋊C4—88no—
442C22⋊C4—48no—
443C22⋊C4—48no—
444C22⋊C4—84no—
445C22⋊C4—84no—
446C22⋊C4—84no—
447C22⋊C4—84no—
448C42—24no—
449C42—24no—
450C42—44no—
451C42—44no—
452C42—44no—
453C42—44no—
454C42—42no—
455C42—42no—
456C42#12114yes—
457C42C22⋊Q814yes—
458C42C22⋊Q814yes—
459C42#12614yes—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—81no—
468C23—81no—
469C23—81no—
470C23—81no—
471C23—81no—
472C23—81no—
473C23—81no—
474C23—81no—
475C23—81no—
476C23—81no—
477C23—81no—
478C23—81no—
479C23—81no—
480C23—81no—
481C23—81no—
482C23—81no—
483C23—81no—
484C23—81no—
485C23—81no—
486C23—81no—
487C23—81no—
488C23—81no—
489C23—81no—
490C23—81no—
491C23—81no—
492C23—81no—
493C23—81no—
494C23—81no—
495C23C2×#12111yescharacteristic
496C23C2×#12111yescharacteristic
497C23C2×C22⋊Q811yescharacteristic
498C23C2×#12411yescharacteristic
499C23#53211yescharacteristic
500C23#53511yescharacteristic
501C23#53511yescharacteristic
502C23C2×AGL(1,ℤ/8)11yescharacteristic
503C23C2×AGL(1,ℤ/8)11yescharacteristic
504C23C2×#13711yescharacteristic
505C23C2×#13711yescharacteristic
506C23#57511yescharacteristic
507C23#57511yescharacteristic
508C23#57511yescharacteristic
509C23#57511yescharacteristic
510Q8—44no—
511Q8—44no—
512Q8—82no—
513Q8—82no—
514D4—44no—
515D4—44no—
516D4—82no—
517D4—82no—
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—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—41no—
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—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—81no—
564C2×C4—81no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—81no—
570C2×C4—81no—
571C2×C4—82no—
572C2×C4—81no—
573C2×C4—81no—
574C2×C4—81no—
575C2×C4—81no—
576C2×C4—81no—
577C2×C4—82no—
578C2×C4—81no—
579C2×C4—82no—
580C2×C4—81no—
581C2×C4—81no—
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—81no—
591C2×C4—81no—
592C2×C4#39614yes—
593C2×C4#39614yes—
594C2×C4#48914yes—
595C2×C4C2×#12111yescharacteristic
596C2×C4C2×C22⋊Q811yescharacteristic
597C2×C4C2×C22⋊Q811yescharacteristic
598C2×C4#52512yes—
599C2×C4C2×#12611yescharacteristic
600C2×C4#52912yes—
601C2×C4#53512yes—
602C2×C4#54012yes—
603C2×C4#54612yes—
604C2×C4#55012yes—
605C2×C4#55312yes—
606C2×C4#55412yes—
607C2×C4#55412yes—
608C2×C4#55512yes—
609C2×C4#55912yes—
610C2×C4#55912yes—
611C8—44no—
612C8—44no—
613C22—82no—
614C22—82no—
615C22—82no—
616C22—82no—
617C22—82no—
618C22—82no—
619C22—82no—
620C22—161no—
621C22—161no—
622C22—161no—
623C22—161no—
624C22—161no—
625C22—161no—
626C22—161no—
627C22—161no—
628C22C2×#39611yescharacteristic
629C22#254411yescharacteristic
630C22#254811yescharacteristic
631C22#256311yescharacteristic
632C22#256611yescharacteristic
633C22#261911yescharacteristic
634C22#285011yescharacteristic
635C22#303611yescharacteristic
636C22#303611yescharacteristic
637C22#303711yescharacteristic
638C22#303711yescharacteristic
639C22#305511yescharacteristic
640C22#305511yescharacteristic
641C22#305611yescharacteristic
642C22#305611yescharacteristic
643C22#307911yescharacteristic
644C22#307911yescharacteristic
645C22#308011yescharacteristic
646C22#308011yescharacteristic
647C22C2×#48911yescharacteristic
648C22#313511yescharacteristic
649C22#314011yescharacteristic
650C22#314111yescharacteristic
651C22#317111yescharacteristic
652C22#317211yescharacteristic
653C22#318011yescharacteristic
654C22#318111yescharacteristic
655C22#321311yescharacteristic
656C22#321411yescharacteristic
657C22#321511yescharacteristic
658C22#321611yescharacteristic
659C22#321711yescharacteristic
660C22#321811yescharacteristic
661C22#321911yescharacteristic
662C22#321911yescharacteristic
663C4—24no—
664C4—24no—
665C4—24no—
666C4—42no—
667C4—42no—
668C4—42no—
669C4—82no—
670C4—84no—
671C4—161no—
672C4—161no—
673C4—162no—
674C4#241412yes—
675C4#245412yes—
676C4#251312yes—
677C4#258012yes—
678C2—162no—
679C2#1522211yescharacteristic
680C2#2361711yescharacteristic
681C2#2361711yescharacteristic
682C2#2861811yescharacteristic
683C2#2861911yescharacteristic
684C2#2875311yescharacteristic
685C2#2875411yescharacteristic
686C2#2914611yescharacteristic
687C2#2914811yescharacteristic
688C2#2919711yescharacteristic
689C2#2919811yescharacteristic
690C2#2981411yescharacteristic
691C2#2981511yescharacteristic
692C2#3305111yescharacteristic
693C2#3305711yescharacteristic
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
3. C2×#1836C2semidirect
4. C2×#1767C2semidirect
5. #14712C2semidirect
7. #11077C2semidirect
9. #11067C2semidirect
11. #10843C2semidirect

Maximal subgroups

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

subgroupindex
C2×#19592
C2×#18362
C2×#17672
#147122
#147082
#110772
#110762
#110672
#110662
#108432