Finite Group Databasebeta

← #6593449

Subgroups — #6593449

Order 512.

1821 conjugacy classes of subgroups.

The subgroup lattice has 718 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36194C211yesmaximal, characteristic, complemented
2C4×#547C211yesmaximal, characteristic
3#33195C211yesmaximal, characteristic, complemented
4#32907C212yesmaximal, complemented
5#29141C211yesmaximal, characteristic, complemented
6#28255C211yesmaximal, characteristic, complemented
7#27631C212yesmaximal, complemented
8#27237C214yesmaximal, complemented
9#26735C214yesmaximal, complemented
10#24970C212yesmaximal, complemented
11#24652C212yesmaximal, complemented
12#24241C212yesmaximal
13#23096C212yesmaximal, complemented
14#22795C211yesmaximal, characteristic, complemented
15#22770C212yesmaximal, complemented
16C4×#461C211yesmaximal, characteristic, complemented
17#20737C212yesmaximal, complemented
18C2×#547C2211yescharacteristic
19C2×C4×#41C2212yes—
20C2×#474C2212yes—
21C2×#462C2212yes—
22C2×#462C2212yes—
23C2×#461C2211yescharacteristic
24C2×#452C2214yes—
25C2×C4×SD16C2212yes—
26C2×#431C2212yes—
27#2868C2211yescharacteristic
28#2857C2212yes—
29#2571—21no—
30#2563—21no—
31#2538C2212yes—
32#2497—22no—
33#2492C2214yes—
34#2439—22no—
35#2369—21no—
36#2365C2211yescharacteristic
37#2362C2212yes—
38#2361—21no—
39#2359—22no—
40#2355C2212yes—
41#2354—21no—
42#2354C2211yescharacteristic
43D4×C4⋊C4C2211yescharacteristic
44#2344—21no—
45#2339C2212yes—
46C4×C4⋊Q8C2211yescharacteristic
47C4×C4⋊Q8C2211yescharacteristic
48C4×C22⋊Q8C2212yes—
49C4×#123—22no—
50C4×#121—21no—
51#2285C2212yes—
52C42×D4C2211yescharacteristic
53C42×D4C2211yescharacteristic
54C2×C4×C4⋊C4—21no—
55#2075C2214yes—
56#2056C2214yes—
57#2036C2214yes—
58#2035C2214yes—
59#2024C2214yes—
60#2001C2214yes—
61#1988C2212yes—
62#1981C2211yescharacteristic
63#1969C2214yes—
64#1956C2212yes—
65#1954C2212yes—
66#1948C2211yescharacteristic
67#1947C2212yes—
68#1947C2212yes—
69#1933C2214yes—
70#1928C2214yes—
71#1927C2212yes—
72#1923C2214yes—
73#1919C2214yes—
74#1878C2214yes—
75#1877C2212yes—
76#1877C2214yes—
77#1873C2214yes—
78#1842C2211yescharacteristic
79#1834C2212yes—
80#1825C2214yes—
81#1803C2212yes—
82#1803C2212yes—
83#1803C2212yes—
84#1803C2212yes—
85#1802C2211yescharacteristic
86#1802C2212yes—
87#1802C2212yes—
88#1797C2212yes—
89#1788C2214yes—
90C8×C4⋊C4C2212yes—
91C4×C4⋊C8C2211yescharacteristic
92C4×#103C2214yes—
93C4×#103C2212yes—
94C4×#102C2214yes—
95C22×#41C2312yes—
96C22×Q16C2314yes—
97C22×SD16C2312yes—
98#547—24no—
99#547—24no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×#123—22no—
106C2×C22⋊Q8C2312yes—
107C2×C22⋊Q8C2314yes—
108C2×#121—21no—
109C4×#41—28no—
110C4×#41—44no—
111C2×C4×Q8C2312yes—
112C2×C4×Q8C2312yes—
113C2×C4×Q8C2312yes—
114C2×C4×D4—22no—
115C2×C4×D4—22no—
116C2×C4×D4D411yescharacteristic
117C2×C4×D4D411yescharacteristic
118C2×C4×D4C2311yescharacteristic
119C2×C4×D4C2312yes—
120C2×C4×D4C2311yescharacteristic
121C2×C4×D4C2311yescharacteristic
122C2×C4×D4C2311yescharacteristic
123C2×#117—22no—
124C2×#117—22no—
125C2×#117C2312yes—
126C22×C4⋊C4—21no—
127C22×C4⋊C4—21no—
128C22×C42—22no—
129C22×C42—21no—
130C22×C42C2312yes—
131#474—44no—
132#462—28no—
133#462—44no—
134#461—24no—
135#452—48no—
136C4×SD16—28no—
137#431—44no—
138C2×#107C2314yes—
139C2×#106C2312yes—
140C2×C4⋊C8C2311yescharacteristic
141C2×C4⋊C8C2312yes—
142C2×C4⋊C8C2311yescharacteristic
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2312yes—
145C2×#103C2314yes—
146C2×#103C2312yes—
147C2×#103C2312yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2312yes—
152C2×#102C2314yes—
153C2×#102C2312yes—
154C2×C22⋊C8C2314yes—
155C2×C22⋊C8C2314yes—
156C2×C4×C8C2312yes—
157C2×C4×C8C2314yes—
158#400—24no—
159#396—24no—
160#393—24no—
161#391—24no—
162#391C2312yes—
163#391C2312yes—
164#391C2312yes—
165#391C2314yes—
166#390—22no—
167#390—21no—
168#389—22no—
169#389—21no—
170#389—21no—
171#389—21no—
172#389—21no—
173#389—21no—
174#389—22no—
175#389—22no—
176#389C2312yes—
177#388—24no—
178#388—22no—
179#388—22no—
180#388—22no—
181#387—22no—
182#387—22no—
183#387—24no—
184#387—22no—
185#386—22no—
186#386—22no—
187#385—22no—
188#385—22no—
189#384—21no—
190#384—21no—
191#384C2311yescharacteristic
192#384C2311yescharacteristic
193#384C2312yes—
194#384C2312yes—
195#382—22no—
196#382—22no—
197#381—22no—
198#380—22no—
199#380—22no—
200#380—22no—
201#380—21no—
202#380C2312yes—
203#380C2314yes—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4D411yescharacteristic
215C4×C4⋊C4D411yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2314yes—
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2311yescharacteristic
221C4×C22⋊C4—22no—
222C4×C22⋊C4—21no—
223C4×C22⋊C4—21no—
224C4×C22⋊C4—21no—
225C4×C22⋊C4D412yes—
226C4×C22⋊C4D412yes—
227C4×C22⋊C4C2312yes—
228C2×#95—22no—
229C43C2311yescharacteristic
230#340C2314yes—
231#340C2312yes—
232#340C2314yes—
233#340C2314yes—
234#340C2314yes—
235#340C2314yes—
236#340C2312yes—
237#336C2314yes—
238#336C2312yes—
239C2×#41—24no—
240C2×#41—24no—
241C2×#41—24no—
242C2×#41—24no—
243C2×#41—42no—
244C2×#41—42no—
245C22×Q8—22no—
246C22×Q8C2412yes—
247C22×Q8C2412yes—
248C22×D4—22no—
249C22×D4C2×D411yescharacteristic
250C22×D4C2411yescharacteristic
251C23×C4—21no—
252C23×C4—22no—
253C23×C4—21no—
254C23×C4—22no—
255C23×C4—21no—
256C23×C4C2×D412yes—
257C23×C4C2×D412yes—
258C23×C4C2412yes—
259C2×Q16—28no—
260C2×Q16—44no—
261C2×Q16—44no—
262C2×SD16—24no—
263C2×SD16—24no—
264C2×SD16—24no—
265C22×C8C2412yes—
266C22×C8C2414yes—
267C22×C8C2412yes—
268C4⋊Q8—22no—
269C4⋊Q8—22no—
270C4⋊Q8—22no—
271C4⋊Q8—42no—
272C4⋊Q8—42no—
273C4⋊Q8—44no—
274C4⋊Q8—82no—
275C4⋊Q8#4114yes—
276#123—216no—
277C22⋊Q8—24no—
278C22⋊Q8—24no—
279C22⋊Q8—44no—
280C22⋊Q8—44no—
281#121—28no—
282C4×Q8—24no—
283C4×Q8—24no—
284C4×Q8—24no—
285C4×Q8—24no—
286C4×Q8—24no—
287C4×Q8—42no—
288C4×D4—24no—
289C4×D4—24no—
290C4×D4—22no—
291C4×D4—24no—
292C4×D4—22no—
293C4×D4—22no—
294C4×D4—22no—
295C4×D4—22no—
296C4×D4—41no—
297C4×D4—42no—
298C4×D4—48no—
299C4×D4—48no—
300C4×D4#4114yes—
301C4×D4#4114yes—
302#117—24no—
303#117—48no—
304#117—42no—
305#117—84no—
306C2×C4⋊C4—28no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—44no—
327C2×C4⋊C4C2×D411yescharacteristic
328C2×C4⋊C4C2×D411yescharacteristic
329C2×C4⋊C4C2×D411yescharacteristic
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2411yescharacteristic
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2414yes—
334C2×C4⋊C4C2411yescharacteristic
335C2×C4⋊C4C2411yescharacteristic
336C2×C22⋊C4—22no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—24no—
343C2×C22⋊C4—24no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4C2×D412yes—
346C2×C22⋊C4C2×D412yes—
347C2×C22⋊C4C2×D412yes—
348C2×C42—24no—
349C2×C42—24no—
350C2×C42—22no—
351C2×C42—22no—
352C2×C42—22no—
353C2×C42—22no—
354C2×C42—21no—
355C2×C42—21no—
356C2×C42—21no—
357C2×C42—24no—
358C2×C42—48no—
359C2×C42—44no—
360C2×C42C2×D412yes—
361C2×C42C2×D411yescharacteristic
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D412yes—
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D411yescharacteristic
366C2×C42C2412yes—
367C2×C42C2411yescharacteristic
368C2×C42C2411yescharacteristic
369C2×C42C2411yescharacteristic
370#107—44no—
371#106—24no—
372C4⋊C8—22no—
373C4⋊C8—24no—
374C4⋊C8—22no—
375C4⋊C8—24no—
376C4⋊C8—42no—
377#103—28no—
378#103—24no—
379#103—24no—
380#103—42no—
381#103—44no—
382#103—44no—
383#103—44no—
384#102—24no—
385#102—28no—
386C22⋊C8—44no—
387C22⋊C8—44no—
388C4×C8—24no—
389C4×C8—44no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—21no—
394#95—21no—
395#95—21no—
396#95—22no—
397#95—22no—
398#95—24no—
399#95—24no—
400#95—22no—
401#95—22no—
402#95—22no—
403#95—24no—
404#95—24no—
405#95C2×D412yes—
406#95C2×D412yes—
407#95C2×D412yes—
408C24—21no—
409C24C22×D412yes—
410#41—48no—
411#41—48no—
412#41—48no—
413#41—48no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—44no—
426C2×Q8—82no—
427C2×Q8—82no—
428C2×Q8C2×#4114yes—
429C2×D4—28no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4—44no—
437C2×D4—44no—
438C2×D4C2×#4112yes—
439C2×D4C2×#4112yes—
440C2×D4H(5,𝔽2)12yes—
441C22×C4—28no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—42no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—44no—
474C22×C4—81no—
475C22×C4—82no—
476C22×C4C22×D412yes—
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D412yes—
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D412yes—
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C2511yescharacteristic
486Q16—88no—
487SD16—48no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—24no—
492C2×C8—22no—
493C2×C8—24no—
494C2×C8—22no—
495C2×C8—24no—
496C2×C8—22no—
497C4⋊C4—28no—
498C4⋊C4—22no—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—22no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—24no—
508C4⋊C4—22no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—24no—
512C4⋊C4—44no—
513C4⋊C4—42no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—81no—
525C4⋊C4—84no—
526C4⋊C4—82no—
527C4⋊C4—82no—
528C4⋊C4#13714yes—
529C4⋊C4C2×#4112yes—
530C4⋊C4C2×#4112yes—
531C22⋊C4—28no—
532C22⋊C4—24no—
533C22⋊C4—24no—
534C22⋊C4—24no—
535C22⋊C4—24no—
536C22⋊C4—24no—
537C22⋊C4—44no—
538C22⋊C4—44no—
539C22⋊C4—42no—
540C22⋊C4—82no—
541C22⋊C4—84no—
542C22⋊C4—84no—
543C42—24no—
544C42—24no—
545C42—22no—
546C42—22no—
547C42—22no—
548C42—22no—
549C42—21no—
550C42—21no—
551C42—41no—
552C42—44no—
553C42—44no—
554C42—44no—
555C42—44no—
556C42—42no—
557C42—42no—
558C42—42no—
559C42—84no—
560C42#13514yes—
561C42C2×#4112yes—
562C42C2×#4112yes—
563C42C2×#4112yes—
564C42H(5,𝔽2)12yes—
565C23—22no—
566C23—22no—
567C23—22no—
568C23—22no—
569C23—22no—
570C23—22no—
571C23—22no—
572C23—22no—
573C23—22no—
574C23—22no—
575C23—41no—
576C23—41no—
577C23—41no—
578C23—41no—
579C23C23×D411yescharacteristic
580Q8—24no—
581Q8—24no—
582Q8—42no—
583Q8—42no—
584Q8—42no—
585Q8—42no—
586Q8—42no—
587Q8—42no—
588Q8—164no—
589D4—22no—
590D4—22no—
591D4—22no—
592D4—416no—
593D4—44no—
594D4#55514yes—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—24no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—21no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—22no—
615C2×C4—48no—
616C2×C4—416no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—48no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—48no—
629C2×C4—48no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—41no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—41no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—41no—
642C2×C4—44no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—88no—
646C2×C4—84no—
647C2×C4#52512yes—
648C2×C4#52512yes—
649C2×C4#54612yes—
650C2×C4#54612yes—
651C2×C4C2×#13512yes—
652C2×C4C2×#13512yes—
653C2×C4C2×#13512yes—
654C2×C4C2×#13714yes—
655C2×C4C2×#13712yes—
656C2×C4#57812yes—
657C2×C4C22×#4111yescharacteristic
658C2×C4C22×#4111yescharacteristic
659C2×C4C22×#4111yescharacteristic
660C2×C4C2×H(5,𝔽2)11yescharacteristic
661C2×C4C2×H(5,𝔽2)11yescharacteristic
662C2×C4#58511yescharacteristic
663C8—42no—
664C8—42no—
665C8—44no—
666C22—24no—
667C22—42no—
668C22—42no—
669C22—42no—
670C22—42no—
671C22—42no—
672C22—42no—
673C22—42no—
674C22—42no—
675C22—42no—
676C22—42no—
677C22—42no—
678C22—42no—
679C22—42no—
680C22C2×#52511yescharacteristic
681C22C2×#54611yescharacteristic
682C22#348811yescharacteristic
683C22C22×#13511yescharacteristic
684C22C22×#13711yescharacteristic
685C22C2×#57811yescharacteristic
686C22#358511yescharacteristic
687C4—22no—
688C4—22no—
689C4—24no—
690C4—22no—
691C4—41no—
692C4—42no—
693C4—44no—
694C4—42no—
695C4—42no—
696C4—42no—
697C4—82no—
698C4—82no—
699C4—82no—
700C4—81no—
701C4—84no—
702C4#310714yes—
703C4#330214yes—
704C4C2×#55511yescharacteristic
705C4#347511yescharacteristic
706C4#350611yescharacteristic
707C4#351911yescharacteristic
708C2—44no—
709C2—82no—
710C2#3616011yescharacteristic
711C2C2×#310711yescharacteristic
712C2#6060011yescharacteristic
713C2C2×#330211yescharacteristic
714C2#6116211yescharacteristic
715C2#6194911yescharacteristic
716C2#6200011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#547C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36194C2semidirect
3. #33195C2semidirect
4. #32907C2semidirect
5. #29141C2semidirect
6. #28255C2semidirect
7. #27631C2semidirect
8. #27237C2semidirect
9. #26735C2semidirect
10. #24970C2semidirect
11. #24652C2semidirect
13. #23096C2semidirect
14. #22795C2semidirect
15. #22770C2semidirect
16. C4×#461C2semidirect
17. #20737C2semidirect

Maximal subgroups

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

subgroupindex
#361942
C4×#5472
#331952
#329072
#291412
#282552
#276312
#272372
#267352
#249702
#246522
#242412
#230962
#227952
#227702
C4×#4612
#207372