Finite Group Databasebeta

← #461741

Subgroups — #461741

Order 512.

1398 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2072C211yesmaximal, characteristic, complemented
2C2×#2035C212yesmaximal
3C2×#1797C211yesmaximal, characteristic
4#14702C211yesmaximal, characteristic
5#14686C211yesmaximal, characteristic
6#11357C211yesmaximal, characteristic, complemented
7#11323C211yesmaximal, characteristic, complemented
8#11231C212yesmaximal, complemented
9#11199C212yesmaximal
10#11077C212yesmaximal, complemented
11#11050C211yesmaximal, characteristic, complemented
12C22×C22⋊Q8C2211yescharacteristic
13C23×C42C2211yescharacteristic
14C22×#106C2212yes—
15C22×C4⋊C8C2212yes—
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×C22⋊C8C2212yes—
19C2×#391C2211yescharacteristic
20C2×#391C2211yescharacteristic
21C2×#384C2211yescharacteristic
22C2×#384C2211yescharacteristic
23C2×#380—21no—
24C2×#380—21no—
25C2×#380C2212yes—
26C2×#380C2211yescharacteristic
27#2072—24no—
28#2035—28no—
29#1797—24no—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#336C2211yescharacteristic
36C2×#336C2211yescharacteristic
37#1436—21no—
38#1436—21no—
39C4⋊#95—21no—
40C4⋊#95—21no—
41C4⋊#95C2211yescharacteristic
42C4⋊#95C2212yes—
43C4⋊#95C2211yescharacteristic
44C22⋊#95—21no—
45C22⋊#95—21no—
46C23×Q8C2311yescharacteristic
47C24×C4D411yescharacteristic
48C24×C4D411yescharacteristic
49C24×C4C2311yescharacteristic
50C23×C8C2312yes—
51C23×C8C2312yes—
52C2×C22⋊Q8—22no—
53C2×C22⋊Q8—42no—
54C2×C22⋊Q8—42no—
55C2×C22⋊Q8D414yes—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C22⋊C4—21no—
67C22×C22⋊C4—21no—
68C22×C22⋊C4—22no—
69C22×C42—22no—
70C22×C42—22no—
71C22×C42—24no—
72C22×C42—42no—
73C22×C42D411yescharacteristic
74C22×C42D411yescharacteristic
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77C2×#106—24no—
78C2×#106—42no—
79C2×#106—42no—
80C2×C4⋊C8—24no—
81C2×C4⋊C8—24no—
82C2×C4⋊C8—42no—
83C2×#103—24no—
84C2×#103—24no—
85C2×#103—24no—
86C2×#103—42no—
87C2×#103—42no—
88C2×#103—42no—
89C2×C22⋊C8—24no—
90C2×C22⋊C8—24no—
91C2×C22⋊C8—42no—
92#391—22no—
93#391—22no—
94#391—24no—
95#384—22no—
96#384—22no—
97#384—42no—
98#380—24no—
99#380—24no—
100#380—42no—
101#380—44no—
102#380—44no—
103C2×#95—21no—
104C2×#95—21no—
105C2×#95—21no—
106C2×#95—21no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—21no—
115#340—24no—
116#340—24no—
117#340—42no—
118#340—42no—
119#340—42no—
120#336—22no—
121#336D414yes—
122C25C2×D411yescharacteristic
123C22×Q8—21no—
124C22×Q8—21no—
125C22×Q8—21no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—21no—
130C22×Q8—22no—
131C22×Q8—21no—
132C22×Q8C2×D412yes—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—21no—
146C23×C4—22no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—41no—
157C23×C4—41no—
158C23×C4—41no—
159C23×C4—41no—
160C23×C4SD1614yes—
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D412yes—
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4#4112yes—
168C23×C4C2411yescharacteristic
169C22×C8—22no—
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×C8C2×D414yes—
183C22⋊Q8—48no—
184C22⋊Q8—48no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
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—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
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⋊C4C2×D412yes—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—42no—
232C2×C22⋊C4—44no—
233C2×C42—21no—
234C2×C42—22no—
235C2×C42—21no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—48no—
243C2×C42—48no—
244C2×C42—41no—
245C2×C42—41no—
246C2×C42C2×D412yes—
247C2×C42#4112yes—
248C2×C42#4114yes—
249#106—48no—
250C4⋊C8—48no—
251#103—48no—
252#103—48no—
253C22⋊C8—48no—
254#95—42no—
255#95—42no—
256#95—44no—
257#95—44no—
258#95—42no—
259#95—42no—
260#95—42no—
261#95—42no—
262#95—44no—
263#95—42no—
264#95—42no—
265#95—42no—
266C24—21no—
267C24—21no—
268C24—21no—
269C24—21no—
270C24—21no—
271C24—21no—
272C24—21no—
273C24—21no—
274C24—21no—
275C24—21no—
276C24—21no—
277C24—21no—
278C24#12112yes—
279C24C2×SD1612yes—
280C24AGL(1,ℤ/8)12yes—
281C24C22×D411yescharacteristic
282C2×Q8—24no—
283C2×Q8—22no—
284C2×Q8—22no—
285C2×Q8—24no—
286C2×Q8—42no—
287C2×Q8—41no—
288C2×Q8—42no—
289C2×Q8—41no—
290C2×Q8—42no—
291C2×Q8—42no—
292C2×Q8—42no—
293C2×Q8—42no—
294C2×Q8—42no—
295C2×Q8—42no—
296C2×Q8—42no—
297C2×Q8—42no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
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—28no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—28no—
332C22×C4—28no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—44no—
336C22×C4—44no—
337C22×C4—44no—
338C22×C4—44no—
339C22×C4—44no—
340C22×C4—44no—
341C22×C4—44no—
342C22×C4—44no—
343C22×C4—44no—
344C22×C4—44no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—44no—
354C22×C4—41no—
355C22×C4—41no—
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—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—41no—
371C22×C4—42no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—42no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—41no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—42no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4#12112yes—
395C22×C4#12112yes—
396C22×C4#12112yes—
397C22×C4#12312yes—
398C22×C4#12312yes—
399C22×C4C2×SD1612yes—
400C22×C4C2×SD1612yes—
401C22×C4#13512yes—
402C22×C4#13712yes—
403C22×C4#13712yes—
404C22×C4#13712yes—
405C22×C4C22×D411yescharacteristic
406C22×C4C22×D411yescharacteristic
407C22×C4C2×#4111yescharacteristic
408C22×C4C2×#4112yes—
409C22×C4C2×#4111yescharacteristic
410C22×C4H(5,𝔽2)12yes—
411C22×C4H(5,𝔽2)12yes—
412C22×C4H(5,𝔽2)11yescharacteristic
413C22×C42−1+412yes—
414C22×C42−1+411yescharacteristic
415C2×C8—24no—
416C2×C8—24no—
417C2×C8—24no—
418C2×C8—42no—
419C2×C8—42no—
420C2×C8—42no—
421C2×C8—42no—
422C2×C8—42no—
423C2×C8—42no—
424C2×C8—42no—
425C2×C8—42no—
426C2×C8—42no—
427C2×C8—42no—
428C2×C8—42no—
429C4⋊C4—24no—
430C4⋊C4—48no—
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—42no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—84no—
445C4⋊C4—84no—
446C4⋊C4—84no—
447C4⋊C4—84no—
448C22⋊C4—48no—
449C22⋊C4—84no—
450C22⋊C4—88no—
451C42—24no—
452C42—24no—
453C42—24no—
454C42—24no—
455C42—44no—
456C42—44no—
457C23—22no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—22no—
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—41no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23#46014yes—
492C23C2×#12111yescharacteristic
493C23C2×#12111yescharacteristic
494C23C2×#12311yescharacteristic
495C23#53411yescharacteristic
496C23#53511yescharacteristic
497C23#53611yescharacteristic
498C23#53711yescharacteristic
499C23C22×SD1611yescharacteristic
500C23C2×#13511yescharacteristic
501C23C2×AGL(1,ℤ/8)11yescharacteristic
502C23C2×#13711yescharacteristic
503C23C2×#13711yescharacteristic
504C23C2×#13711yescharacteristic
505C23#57511yescharacteristic
506C23#57511yescharacteristic
507Q8—44no—
508Q8—44no—
509Q8—82no—
510Q8—82no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
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—22no—
535C2×C4—416no—
536C2×C4—416no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—81no—
556C2×C4—81no—
557C2×C4—81no—
558C2×C4—81no—
559C2×C4—81no—
560C2×C4—81no—
561C2×C4—81no—
562C2×C4—81no—
563C2×C4—81no—
564C2×C4—81no—
565C2×C4—82no—
566C2×C4—81no—
567C2×C4—81no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4—82no—
572C2×C4—81no—
573C2×C4—81no—
574C2×C4—81no—
575C2×C4—82no—
576C2×C4—81no—
577C2×C4—81no—
578C2×C4—81no—
579C2×C4—82no—
580C2×C4—82no—
581C2×C4—81no—
582C2×C4—81no—
583C2×C4—82no—
584C2×C4—82no—
585C2×C4—82no—
586C2×C4—82no—
587C2×C4—82no—
588C2×C4—82no—
589C2×C4—81no—
590C2×C4—81no—
591C2×C4#46514yes—
592C2×C4#47014yes—
593C2×C4#48314yes—
594C2×C4#53412yes—
595C2×C4#53712yes—
596C2×C4#53812yes—
597C2×C4#54012yes—
598C2×C4#54212yes—
599C2×C4#54212yes—
600C2×C4D4211yescharacteristic
601C2×C4#54612yes—
602C2×C4#54712yes—
603C2×C4#54711yescharacteristic
604C2×C4#54811yescharacteristic
605C2×C4#55112yes—
606C2×C4#55212yes—
607C2×C4#55311yescharacteristic
608C2×C4#56212yes—
609C2×C4#56312yes—
610C8—44no—
611C8—82no—
612C22—24no—
613C22—24no—
614C22—24no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22#241211yescharacteristic
628C22#241611yescharacteristic
629C22#248611yescharacteristic
630C22#261711yescharacteristic
631C22#262011yescharacteristic
632C22#262411yescharacteristic
633C22#281411yescharacteristic
634C22C2×#46011yescharacteristic
635C22#303511yescharacteristic
636C22#303711yescharacteristic
637C22#303911yescharacteristic
638C22C2×#46511yescharacteristic
639C22C2×#47011yescharacteristic
640C22#305511yescharacteristic
641C22#305611yescharacteristic
642C22C2×#48311yescharacteristic
643C22#309111yescharacteristic
644C22#309211yescharacteristic
645C22#309511yescharacteristic
646C22#318711yescharacteristic
647C22#319011yescharacteristic
648C22#319311yescharacteristic
649C22#319611yescharacteristic
650C22#320611yescharacteristic
651C22#321011yescharacteristic
652C22#321311yescharacteristic
653C22#321611yescharacteristic
654C22#322911yescharacteristic
655C22#323111yescharacteristic
656C22#323811yescharacteristic
657C22#324011yescharacteristic
658C22#324611yescharacteristic
659C22#324711yescharacteristic
660C22#325211yescharacteristic
661C22#325511yescharacteristic
662C4—24no—
663C4—24no—
664C4—24no—
665C4—42no—
666C4—42no—
667C4—82no—
668C4—82no—
669C4—162no—
670C4—161no—
671C4—162no—
672C4—161no—
673C4#267112yes—
674C4#270212yes—
675C4#271712yes—
676C4#274112yes—
677C2—44no—
678C2#1640111yescharacteristic
679C2#2455111yescharacteristic
680C2#2455611yescharacteristic
681C2#2475311yescharacteristic
682C2#2476011yescharacteristic
683C2#2764711yescharacteristic
684C2#2765211yescharacteristic
685C2#2971311yescharacteristic
686C2#2971711yescharacteristic
687C2#2984511yescharacteristic
688C2#2985311yescharacteristic
689C2#2992211yescharacteristic
690C2#2992811yescharacteristic
691C2#3274011yescharacteristic
692C2#3274511yescharacteristic
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. #11323C2semidirect
8. #11231C2semidirect
10. #11077C2semidirect
11. #11050C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20722
C2×#20352
C2×#17972
#147022
#146862
#113572
#113232
#112312
#111992
#110772
#110502