Finite Group Databasebeta

← #404847

Subgroups — #404847

Order 512.

1634 conjugacy classes of subgroups.

The subgroup lattice has 703 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2285C211yesmaximal, characteristic
2C2×#2096C211yesmaximal, characteristic, complemented
3C2×#2072C211yesmaximal, characteristic, complemented
4#14702C211yesmaximal, characteristic
5#11373C211yesmaximal, characteristic, complemented
6#11357C211yesmaximal, characteristic, complemented
7#11095C212yesmaximal
8#10970C212yesmaximal
9#10932C212yesmaximal, complemented
10#10908C212yesmaximal, complemented
11#10712C211yesmaximal, characteristic
12C22×C22⋊Q8C2211yescharacteristic
13C22×#117—21no—
14C23×C42C2211yescharacteristic
15C22×#106C2212yes—
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×C22⋊C8C2212yes—
19C2×#391C2211yescharacteristic
20C2×#391C2211yescharacteristic
21#2285—24no—
22#2285—24no—
23C2×#384C2211yescharacteristic
24C2×#384C2211yescharacteristic
25C2×#382—21no—
26C2×#380—21no—
27C2×#380C2212yes—
28C2×#380C2211yescharacteristic
29C2×C4×C4⋊C4—21no—
30C2×C4×C4⋊C4C2212yes—
31#2096—24no—
32#2072—24no—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#336C2211yescharacteristic
39C2×#336C2211yescharacteristic
40C2×#336C2212yes—
41#1436—21no—
42#1436—21no—
43C4⋊#95C2211yescharacteristic
44C4⋊#95C2211yescharacteristic
45C22⋊#95—21no—
46C22⋊#95—21no—
47C23×Q8C2311yescharacteristic
48C24×C4D411yescharacteristic
49C24×C4D411yescharacteristic
50C24×C4C2311yescharacteristic
51C23×C8C2312yes—
52C23×C8C2312yes—
53C2×C22⋊Q8—22no—
54C2×C22⋊Q8—22no—
55C2×C22⋊Q8—24no—
56C2×C22⋊Q8—42no—
57C2×#117—28no—
58C2×#117—44no—
59C2×#117—44no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2311yescharacteristic
70C22×C22⋊C4—22no—
71C22×C22⋊C4—21no—
72C22×C22⋊C4—21no—
73C22×C42—24no—
74C22×C42—22no—
75C22×C42—22no—
76C22×C42—24no—
77C22×C42—22no—
78C22×C42D411yescharacteristic
79C22×C42D411yescharacteristic
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82C2×#106—24no—
83C2×#106—24no—
84C2×#106—42no—
85C2×#103—24no—
86C2×#103—24no—
87C2×#103—24no—
88C2×#103—24no—
89C2×#103—42no—
90C2×#103—42no—
91C2×C22⋊C8—24no—
92C2×C22⋊C8—24no—
93C2×C22⋊C8—42no—
94#391—22no—
95#391—22no—
96#391—24no—
97#384—22no—
98#384—22no—
99#384D414yes—
100#384D414yes—
101#382—28no—
102#380—28no—
103#380—24no—
104#380—24no—
105#380—42no—
106C4×C4⋊C4—28no—
107C4×C4⋊C4—24no—
108C4×C4⋊C4—24no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—21no—
120#340—24no—
121#340—24no—
122#340—42no—
123#340—42no—
124#340—42no—
125#336—22no—
126#336—22no—
127#336—24no—
128C25C2×D411yescharacteristic
129C22×Q8—21no—
130C22×Q8—21no—
131C22×Q8—21no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—21no—
137C22×Q8—21no—
138C22×Q8#4112yes—
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—22no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—41no—
166C23×C4—41no—
167C23×C4SD1614yes—
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4#4114yes—
174C23×C4C2411yescharacteristic
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8#4114yes—
189C22⋊Q8—48no—
190C22⋊Q8—48no—
191#117—432no—
192C2×C4⋊C4—22no—
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—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—21no—
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⋊C4C2×D414yes—
229C2×C4⋊C4C2×D414yes—
230C2×C4⋊C4#4112yes—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—42no—
240C2×C42—216no—
241C2×C42—216no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—28no—
254C2×C42—44no—
255C2×C42—44no—
256C2×C42C2×D412yes—
257C2×C42C2×D412yes—
258C2×C42#4114yes—
259#106—48no—
260#103—48no—
261#103—48no—
262C22⋊C8—48no—
263#95—28no—
264#95—28no—
265#95—42no—
266#95—42no—
267#95—42no—
268#95—42no—
269#95—42no—
270#95—42no—
271#95—42no—
272#95—42no—
273#95—42no—
274C24—21no—
275C24—21no—
276C24—21no—
277C24—21no—
278C24—21no—
279C24—21no—
280C24—21no—
281C24—21no—
282C24—21no—
283C24—21no—
284C24—21no—
285C24—21no—
286C24#12312yes—
287C24C2×SD1612yes—
288C24#13712yes—
289C24C22×D411yescharacteristic
290C2×Q8—24no—
291C2×Q8—22no—
292C2×Q8—22no—
293C2×Q8—24no—
294C2×Q8—42no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×Q8—42no—
298C2×Q8—42no—
299C2×Q8—42no—
300C2×Q8—42no—
301C2×Q8—42no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×Q8—42no—
305C2×Q8—42no—
306C22×C4—28no—
307C22×C4—28no—
308C22×C4—28no—
309C22×C4—28no—
310C22×C4—21no—
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—28no—
323C22×C4—28no—
324C22×C4—28no—
325C22×C4—28no—
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—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—44no—
349C22×C4—44no—
350C22×C4—44no—
351C22×C4—44no—
352C22×C4—44no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—41no—
358C22×C4—42no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—42no—
363C22×C4—44no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—41no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—41no—
379C22×C4—42no—
380C22×C4—41no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—41no—
384C22×C4—42no—
385C22×C4—41no—
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—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4#12012yes—
403C22×C4#12012yes—
404C22×C4#12312yes—
405C22×C4#12312yes—
406C22×C4#12312yes—
407C22×C4C2×SD1612yes—
408C22×C4C2×SD1612yes—
409C22×C4#13512yes—
410C22×C4AGL(1,ℤ/8)12yes—
411C22×C4#13712yes—
412C22×C4#13712yes—
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D411yescharacteristic
415C22×C4C2×#4112yes—
416C22×C4C2×#4112yes—
417C22×C4C2×#4111yescharacteristic
418C22×C4C2×#4112yes—
419C22×C4C2×#4111yescharacteristic
420C22×C4H(5,𝔽2)11yescharacteristic
421C22×C4H(5,𝔽2)12yes—
422C22×C42−1+411yescharacteristic
423C2×C8—24no—
424C2×C8—24no—
425C2×C8—24no—
426C2×C8—42no—
427C2×C8—42no—
428C2×C8—42no—
429C2×C8—42no—
430C2×C8—42no—
431C2×C8—42no—
432C2×C8—42no—
433C2×C8—42no—
434C2×C8—42no—
435C2×C8—42no—
436C2×C8—42no—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C4⋊C4—48no—
443C4⋊C4—48no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C4⋊C4—48no—
450C4⋊C4—48no—
451C4⋊C4—48no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C22⋊C4—416no—
457C22⋊C4—48no—
458C22⋊C4—84no—
459C42—28no—
460C42—28no—
461C42—24no—
462C42—24no—
463C42—24no—
464C42—24no—
465C42—416no—
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—41no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23#48314yes—
501C23C2×#12011yescharacteristic
502C23C2×#12311yescharacteristic
503C23C2×#12311yescharacteristic
504C23#52511yescharacteristic
505C23#53211yescharacteristic
506C23#53511yescharacteristic
507C23#53711yescharacteristic
508C23C22×SD1611yescharacteristic
509C23C2×#13511yescharacteristic
510C23C2×AGL(1,ℤ/8)11yescharacteristic
511C23C2×#13711yescharacteristic
512C23C2×#13711yescharacteristic
513C23C2×#13711yescharacteristic
514C23#57511yescharacteristic
515C23#57511yescharacteristic
516Q8—44no—
517Q8—44no—
518Q8—82no—
519Q8—82no—
520C2×C4—232no—
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—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—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—81no—
571C2×C4—81no—
572C2×C4—82no—
573C2×C4—81no—
574C2×C4—82no—
575C2×C4—81no—
576C2×C4—82no—
577C2×C4—82no—
578C2×C4—81no—
579C2×C4—82no—
580C2×C4—81no—
581C2×C4—82no—
582C2×C4—81no—
583C2×C4—82no—
584C2×C4—82no—
585C2×C4—81no—
586C2×C4—81no—
587C2×C4—81no—
588C2×C4—81no—
589C2×C4—81no—
590C2×C4—81no—
591C2×C4—81no—
592C2×C4—81no—
593C2×C4—81no—
594C2×C4—81no—
595C2×C4—81no—
596C2×C4—81no—
597C2×C4—81no—
598C2×C4—81no—
599C2×C4—81no—
600C2×C4#45014yes—
601C2×C4#45214yes—
602C2×C4#48314yes—
603C2×C4#52512yes—
604C2×C4#52512yes—
605C2×C4#52912yes—
606C2×C4#52912yes—
607C2×C4#54012yes—
608C2×C4#54212yes—
609C2×C4#54611yescharacteristic
610C2×C4#54612yes—
611C2×C4#54711yescharacteristic
612C2×C4#54811yescharacteristic
613C2×C4#55011yescharacteristic
614C2×C4#55012yes—
615C2×C4#55112yes—
616C2×C4#55112yes—
617C2×C4#55112yes—
618C2×C4#55212yes—
619C8—44no—
620C8—82no—
621C22—24no—
622C22—24no—
623C22—24no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22#239511yescharacteristic
637C22#240711yescharacteristic
638C22#243111yescharacteristic
639C22#248211yescharacteristic
640C22#249811yescharacteristic
641C22#262411yescharacteristic
642C22#284911yescharacteristic
643C22C2×#45011yescharacteristic
644C22C2×#45211yescharacteristic
645C22#300411yescharacteristic
646C22#300511yescharacteristic
647C22C2×#48311yescharacteristic
648C22C2×#48311yescharacteristic
649C22#309111yescharacteristic
650C22#309111yescharacteristic
651C22#309211yescharacteristic
652C22#309211yescharacteristic
653C22#309511yescharacteristic
654C22#309511yescharacteristic
655C22#310311yescharacteristic
656C22#310711yescharacteristic
657C22#311311yescharacteristic
658C22#311511yescharacteristic
659C22#316711yescharacteristic
660C22#317211yescharacteristic
661C22#318111yescharacteristic
662C22#318211yescharacteristic
663C22#320611yescharacteristic
664C22#320711yescharacteristic
665C22#321511yescharacteristic
666C22#321611yescharacteristic
667C22#324611yescharacteristic
668C22#324811yescharacteristic
669C22#325111yescharacteristic
670C22#325511yescharacteristic
671C4—24no—
672C4—24no—
673C4—24no—
674C4—42no—
675C4—42no—
676C4—82no—
677C4—84no—
678C4—82no—
679C4—84no—
680C4—161no—
681C4—161no—
682C4#247112yes—
683C4#247112yes—
684C4#247712yes—
685C4#247912yes—
686C2—44no—
687C2#1490511yescharacteristic
688C2#2375611yescharacteristic
689C2#2376311yescharacteristic
690C2#2415411yescharacteristic
691C2#2415711yescharacteristic
692C2#2559711yescharacteristic
693C2#2560311yescharacteristic
694C2#2758311yescharacteristic
695C2#2758711yescharacteristic
696C2#2785511yescharacteristic
697C2#2786111yescharacteristic
698C2#2992211yescharacteristic
699C2#2992711yescharacteristic
700C2#3298211yescharacteristic
701C2#3298411yescharacteristic
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
2. C2×#2096C2semidirect
3. C2×#2072C2semidirect
5. #11373C2semidirect
6. #11357C2semidirect
9. #10932C2semidirect
10. #10908C2semidirect

Maximal subgroups

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

subgroupindex
C2×#22852
C2×#20962
C2×#20722
#147022
#113732
#113572
#110952
#109702
#109322
#109082
#107122