Finite Group Databasebeta

← #6471415

Subgroups — #6471415

Order 512.

2522 conjugacy classes of subgroups.

The subgroup lattice has 745 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35014C211yesmaximal, characteristic, complemented
2C2×#2339C211yesmaximal, characteristic
3#32953C211yesmaximal, characteristic, complemented
4#32905C211yesmaximal, characteristic, complemented
5#32883C212yesmaximal
6#29951C211yesmaximal, characteristic, complemented
7#29894C212yesmaximal, complemented
8#29715C211yesmaximal, characteristic, complemented
9#27533C214yesmaximal, complemented
10#27237C214yesmaximal, complemented
11#25154C214yesmaximal, complemented
12#24907C214yesmaximal, complemented
13#22044C212yesmaximal, complemented
14#21902C211yesmaximal, characteristic, complemented
15C4×#450C212yesmaximal, complemented
16C23×#41C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C2×#474C2212yescomplemented
20C2×#461C2212yescomplemented
21C2×#452C2214yes—
22C2×#452C2214yes—
23C2×#451C2212yes—
24C2×#450C2212yes—
25C2×C4×SD16C2214yescomplemented
26#2857C2211yescharacteristic
27#2857C2211yescharacteristic
28#2848C2212yes—
29#2646—21no—
30#2626—21no—
31#2589—22no—
32#2587—22no—
33#2497C2212yescomplemented
34#2492C2212yescomplemented
35#2429C2212yescomplemented
36#2423C2212yescomplemented
37#2339—22no—
38#2339—22no—
39#2339C2214yes—
40#2339C2214yes—
41#2329—21no—
42#2312—21no—
43C4×C22⋊Q8C2212yes—
44C2×#391C2211yescharacteristic
45C2×#391C2211yescharacteristic
46C2×#386—21no—
47C2×#386—21no—
48C2×#381—21no—
49C2×#380C2211yescharacteristic
50C2×C4×C22⋊C4C2211yescharacteristic
51C22×#95—21no—
52#2096C2212yescomplemented
53#2077C2214yescomplemented
54#2075C2214yescomplemented
55#2072C2212yescomplemented
56#2066C2214yescomplemented
57#2056C2214yescomplemented
58#2036C2214yescomplemented
59#2035C2214yescomplemented
60#2024C2212yescomplemented
61#2016C2212yescomplemented
62#1969C2214yescomplemented
63#1933C2214yescomplemented
64#1928C2214yescomplemented
65#1923C2214yescomplemented
66#1923C2214yescomplemented
67#1878C2214yescomplemented
68#1877C2214yescomplemented
69#1877C2214yescomplemented
70#1872C2212yes—
71#1872C2212yes—
72#1864C2214yescomplemented
73#1862C2214yescomplemented
74#1853C2212yes—
75#1852C2212yes—
76#1802C2212yescomplemented
77#1802C2212yescomplemented
78#1797C2211yescharacteristic
79#1797C2211yescharacteristic
80#1788C2214yes—
81#1788C2214yes—
82#1787C2212yes—
83#1786C2212yes—
84#1779C2212yes—
85C4×#102C2214yescomplemented
86C4×C22⋊C8C2212yes—
87C22×#41—24no—
88C22×#41—24no—
89C22×#41—22no—
90C22×#41—44no—
91C22×#41—44no—
92C22×#41C2314yes—
93C23×Q8C2311yescharacteristic
94C23×D4—21no—
95C23×D4C2311yescharacteristic
96C24×C4—21no—
97C24×C4—21no—
98C24×C4—21no—
99C24×C4C2311yescharacteristic
100C22×Q16C2314yes—
101C22×SD16C2314yes—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×C22⋊Q8—21no—
105C2×C22⋊Q8C2312yes—
106C2×C22⋊Q8C2312yes—
107C2×C22⋊Q8C2312yes—
108C2×C4×Q8C2312yes—
109C2×C4×D4—22no—
110C2×C4×D4—21no—
111C2×C4×D4—21no—
112C2×C4×D4—44no—
113C2×C4×D4—44no—
114C2×C4×D4C2312yes—
115C2×C4×D4C2312yes—
116C2×C4×D4C2314yes—
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C22⋊C4—21no—
120C22×C22⋊C4—21no—
121C22×C22⋊C4—21no—
122C22×C22⋊C4D411yescharacteristic
123C22×C22⋊C4D411yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126#474—44no—
127#461—44no—
128#452—48no—
129#452—48no—
130#451—28no—
131#450—28no—
132C4×SD16—48no—
133C2×#107C2314yes—
134C2×#106C2314yes—
135C2×C4⋊C8C2312yes—
136C2×C4⋊C8C2312yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×C22⋊C8C2312yes—
146C2×C22⋊C8C2312yes—
147C2×C22⋊C8C2314yes—
148C2×C22⋊C8C2314yes—
149C2×C4×C8C2314yes—
150#400—22no—
151#400—22no—
152#399—22no—
153#398—22no—
154#398—22no—
155#396—22no—
156#396—22no—
157#393—22no—
158#391—21no—
159#391—22no—
160#391—21no—
161#391—21no—
162#391—22no—
163#391C2312yes—
164#391C2312yes—
165#391C2312yes—
166#391C2314yes—
167#391C2312yes—
168#389—21no—
169#389—21no—
170#389—21no—
171#389—22no—
172#389C2312yes—
173#389C2312yes—
174#388—22no—
175#388—22no—
176#388—22no—
177#388—22no—
178#387—22no—
179#387—22no—
180#387—22no—
181#386—24no—
182#386—24no—
183#386—24no—
184#386—24no—
185#385—22no—
186#385—22no—
187#385—22no—
188#385—22no—
189#384C2311yescharacteristic
190#384C2311yescharacteristic
191#381—24no—
192#381—24no—
193#380—21no—
194#380—21no—
195#380—21no—
196#380—22no—
197#380C2312yes—
198#380C2314yes—
199#380C2312yes—
200#380C2312yes—
201#379—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4C2312yes—
204C4×C22⋊C4—22no—
205C4×C22⋊C4D412yes—
206C4×C22⋊C4D412yes—
207C4×C22⋊C4C2312yes—
208C4×C22⋊C4C2312yes—
209C2×#95—21no—
210C2×#95—21no—
211C2×#95—21no—
212C2×#95—21no—
213C2×#95—21no—
214C2×#95—24no—
215C2×#95—24no—
216C2×#95D411yescharacteristic
217C2×#95D411yescharacteristic
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#336C2312yes—
223#336C2312yes—
224#336C2314yes—
225C25—21no—
226C25C2×D411yescharacteristic
227C2×#41—216no—
228C2×#41—28no—
229C2×#41—28no—
230C2×#41—28no—
231C2×#41—28no—
232C2×#41—44no—
233C2×#41—44no—
234C2×#41—48no—
235C2×#41—48no—
236C2×#41—432no—
237C2×#41—432no—
238C22×Q8—21no—
239C22×Q8—21no—
240C22×Q8—22no—
241C22×Q8—42no—
242C22×Q8—42no—
243C22×Q8#4114yes—
244C22×Q8C2412yes—
245C22×Q8C2412yes—
246C22×D4—24no—
247C22×D4—24no—
248C22×D4—28no—
249C22×D4—24no—
250C22×D4—21no—
251C22×D4—22no—
252C22×D4—44no—
253C22×D4—44no—
254C22×D4#4114yes—
255C22×D4C2412yes—
256C23×C4—28no—
257C23×C4—21no—
258C23×C4—24no—
259C23×C4—22no—
260C23×C4—22no—
261C23×C4—21no—
262C23×C4—21no—
263C23×C4—22no—
264C23×C4—22no—
265C23×C4—24no—
266C23×C4—42no—
267C23×C4—42no—
268C23×C4—44no—
269C23×C4—44no—
270C23×C4—44no—
271C23×C4—44no—
272C23×C4—44no—
273C23×C4—44no—
274C23×C4—82no—
275C23×C4—82no—
276C23×C4C2×D411yescharacteristic
277C23×C4C2×D411yescharacteristic
278C23×C4C2×D411yescharacteristic
279C23×C4C2×D412yes—
280C23×C4C2×D411yescharacteristic
281C23×C4C2×D411yescharacteristic
282C23×C4C2412yes—
283C23×C4C2411yescharacteristic
284C2×Q16—28no—
285C2×Q16—44no—
286C2×Q16—44no—
287C2×SD16—28no—
288C2×SD16—28no—
289C2×SD16—28no—
290C22×C8C2414yes—
291C22×C8C2414yes—
292C4⋊Q8—42no—
293C4⋊Q8—42no—
294C22⋊Q8—24no—
295C22⋊Q8—24no—
296C22⋊Q8—44no—
297C22⋊Q8—44no—
298C22⋊Q8—44no—
299C4×Q8—44no—
300C4×D4—28no—
301C4×D4—24no—
302C4×D4—24no—
303C4×D4—44no—
304C4×D4—42no—
305C4×D4—42no—
306C4×D4—48no—
307C4×D4—44no—
308C4×D4—44no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—42no—
325C2×C4⋊C4—42no—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2412yes—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—44no—
344C2×C22⋊C4—44no—
345C2×C22⋊C4—82no—
346C2×C22⋊C4C2×D412yes—
347C2×C22⋊C4C2×D412yes—
348C2×C22⋊C4C2×D412yes—
349C2×C22⋊C4C2×D412yes—
350C2×C42—22no—
351C2×C42—22no—
352C2×C42—42no—
353C2×C42—42no—
354C2×C42C2×D412yes—
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D411yescharacteristic
357C2×C42C2412yes—
358C2×C42C2411yescharacteristic
359C2×C42C2411yescharacteristic
360#107—44no—
361#106—44no—
362C4⋊C8—42no—
363C4⋊C8—42no—
364#103—28no—
365#103—44no—
366#103—44no—
367#103—44no—
368#103—44no—
369#102—28no—
370#102—28no—
371#102—28no—
372C22⋊C8—24no—
373C22⋊C8—24no—
374C22⋊C8—44no—
375C22⋊C8—44no—
376C4×C8—44no—
377#95—24no—
378#95—22no—
379#95—22no—
380#95—24no—
381#95—22no—
382#95—22no—
383#95—24no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95C2×D412yes—
395#95C2×D412yes—
396#95C2×D412yes—
397#95C2×D412yes—
398C24—24no—
399C24—24no—
400C24—24no—
401C24—22no—
402C24—22no—
403C24—22no—
404C24—22no—
405C24—41no—
406C24—41no—
407C24—41no—
408C24—41no—
409C24—42no—
410C24—42no—
411C24—42no—
412C24—42no—
413C24C22×D412yes—
414C24C22×D411yescharacteristic
415#41—416no—
416#41—416no—
417#41—464no—
418#41—416no—
419#41—416no—
420C2×Q8—24no—
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8—22no—
424C2×Q8—24no—
425C2×Q8—24no—
426C2×Q8—24no—
427C2×Q8—24no—
428C2×Q8—48no—
429C2×Q8—42no—
430C2×Q8—42no—
431C2×Q8—48no—
432C2×Q8—42no—
433C2×Q8—81no—
434C2×Q8—81no—
435C2×Q8C2×#4114yes—
436C2×D4—24no—
437C2×D4—24no—
438C2×D4—216no—
439C2×D4—216no—
440C2×D4—216no—
441C2×D4—24no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—48no—
446C2×D4—48no—
447C2×D4—432no—
448C2×D4C2×#4114yes—
449C2×D4H(5,𝔽2)14yes—
450C2×D4H(5,𝔽2)14yes—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—216no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—41no—
471C22×C4—41no—
472C22×C4—41no—
473C22×C4—41no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—41no—
477C22×C4—41no—
478C22×C4—41no—
479C22×C4—41no—
480C22×C4—41no—
481C22×C4—41no—
482C22×C4—48no—
483C22×C4—48no—
484C22×C4—48no—
485C22×C4—48no—
486C22×C4—48no—
487C22×C4—44no—
488C22×C4—44no—
489C22×C4—42no—
490C22×C4—42no—
491C22×C4—432no—
492C22×C4—42no—
493C22×C4—42no—
494C22×C4—42no—
495C22×C4—42no—
496C22×C4—42no—
497C22×C4—42no—
498C22×C4—44no—
499C22×C4—44no—
500C22×C4—48no—
501C22×C4—84no—
502C22×C4—82no—
503C22×C4—82no—
504C22×C4—816no—
505C22×C4—816no—
506C22×C4#13514yes—
507C22×C4#13714yes—
508C22×C4C22×D412yes—
509C22×C4C22×D412yes—
510C22×C4C22×D411yescharacteristic
511C22×C4C22×D411yescharacteristic
512C22×C4C22×D411yescharacteristic
513C22×C4C22×D411yescharacteristic
514C22×C4C22×D411yescharacteristic
515C22×C4C22×D412yes—
516C22×C4C2×#4112yes—
517C22×C4C2×#4112yes—
518C22×C4C2511yescharacteristic
519Q16—88no—
520SD16—416no—
521C2×C8—24no—
522C2×C8—24no—
523C2×C8—24no—
524C2×C8—24no—
525C2×C8—24no—
526C2×C8—24no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—24no—
530C4⋊C4—24no—
531C4⋊C4—22no—
532C4⋊C4—24no—
533C4⋊C4—22no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—44no—
537C4⋊C4—42no—
538C4⋊C4—44no—
539C4⋊C4—81no—
540C4⋊C4—82no—
541C4⋊C4—81no—
542C4⋊C4—81no—
543C4⋊C4—82no—
544C4⋊C4—82no—
545C4⋊C4—82no—
546C4⋊C4—81no—
547C4⋊C4—81no—
548C4⋊C4—81no—
549C22⋊C4—24no—
550C22⋊C4—24no—
551C22⋊C4—24no—
552C22⋊C4—24no—
553C22⋊C4—48no—
554C22⋊C4—44no—
555C22⋊C4—44no—
556C22⋊C4—44no—
557C22⋊C4—44no—
558C22⋊C4—82no—
559C22⋊C4—82no—
560C22⋊C4—84no—
561C22⋊C4—82no—
562C22⋊C4—82no—
563C42—24no—
564C42—22no—
565C42—22no—
566C42—42no—
567C42—41no—
568C42—41no—
569C42—82no—
570C42—82no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—216no—
576C23—24no—
577C23—24no—
578C23—24no—
579C23—22no—
580C23—22no—
581C23—22no—
582C23—22no—
583C23—22no—
584C23—22no—
585C23—22no—
586C23—48no—
587C23—48no—
588C23—48no—
589C23—48no—
590C23—48no—
591C23—44no—
592C23—44no—
593C23—42no—
594C23—42no—
595C23—48no—
596C23—42no—
597C23—42no—
598C23—44no—
599C23—82no—
600C23—82no—
601C23—82no—
602C23—82no—
603C23#52512yes—
604C23C2×#13512yes—
605C23C2×#13712yes—
606C23C23×D411yescharacteristic
607Q8—28no—
608Q8—48no—
609Q8—44no—
610Q8—44no—
611Q8—44no—
612Q8—162no—
613D4—216no—
614D4—24no—
615D4—24no—
616D4—24no—
617D4—432no—
618D4#53818yes—
619C2×C4—22no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—22no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—24no—
628C2×C4—24no—
629C2×C4—22no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—22no—
633C2×C4—24no—
634C2×C4—22no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—22no—
638C2×C4—22no—
639C2×C4—22no—
640C2×C4—21no—
641C2×C4—21no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—48no—
645C2×C4—44no—
646C2×C4—48no—
647C2×C4—432no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—88no—
671C2×C4—88no—
672C2×C4—816no—
673C2×C4—82no—
674C2×C4—82no—
675C2×C4—84no—
676C2×C4—84no—
677C2×C4#52512yes—
678C2×C4C2×#13512yes—
679C2×C4C2×#13512yes—
680C2×C4C2×#13714yes—
681C2×C4C22×#4111yescharacteristic
682C2×C4C22×#4111yescharacteristic
683C2×C4C2×H(5,𝔽2)12yes—
684C2×C4C2×H(5,𝔽2)12yes—
685C8—44no—
686C8—44no—
687C22—22no—
688C22—22no—
689C22—22no—
690C22—22no—
691C22—22no—
692C22—22no—
693C22—28no—
694C22—432no—
695C22—44no—
696C22—44no—
697C22—44no—
698C22—44no—
699C22—44no—
700C22—44no—
701C22—42no—
702C22—42no—
703C22—42no—
704C22—42no—
705C22—42no—
706C22—42no—
707C22—42no—
708C22—88no—
709C22#310714yes—
710C22C2×#52511yescharacteristic
711C22#348411yescharacteristic
712C22#348411yescharacteristic
713C22C22×#13511yescharacteristic
714C22C22×#13711yescharacteristic
715C22#358511yescharacteristic
716C22#358611yescharacteristic
717C4—22no—
718C4—22no—
719C4—24no—
720C4—22no—
721C4—42no—
722C4—41no—
723C4—41no—
724C4—42no—
725C4—44no—
726C4—82no—
727C4—82no—
728C4—82no—
729C4—82no—
730C4—82no—
731C4—82no—
732C4C2×#53812yes—
733C4#349012yes—
734C2—24no—
735C2—48no—
736C2—82no—
737C2#3498811yescharacteristic
738C2C2×#310711yescharacteristic
739C2#6056511yescharacteristic
740C2#6156111yescharacteristic
741C2#6156511yescharacteristic
742C2#6157011yescharacteristic
743C2#6158611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35014C2semidirect
3. #32953C2semidirect
4. #32905C2semidirect
6. #29951C2semidirect
7. #29894C2semidirect
8. #29715C2semidirect
9. #27533C2semidirect
10. #27237C2semidirect
11. #25154C2semidirect
12. #24907C2semidirect
13. #22044C2semidirect
14. #21902C2semidirect
15. C4×#450C2semidirect
19. C2×#474C22semidirect
20. C2×#461C22semidirect
25. C2×C4×SD16C22semidirect
33. #2497C22semidirect
34. #2492C22semidirect
35. #2429C22semidirect
36. #2423C22semidirect
52. #2096C22semidirect
53. #2077C22semidirect
54. #2075C22semidirect
55. #2072C22semidirect
56. #2066C22semidirect
57. #2056C22semidirect
58. #2036C22semidirect
59. #2035C22semidirect
60. #2024C22semidirect
61. #2016C22semidirect
62. #1969C22semidirect
63. #1933C22semidirect
64. #1928C22semidirect
65. #1923C22semidirect
66. #1923C22semidirect
67. #1878C22semidirect
68. #1877C22semidirect
69. #1877C22semidirect
72. #1864C22semidirect
73. #1862C22semidirect
76. #1802C22semidirect
77. #1802C22semidirect
85. C4×#102C22semidirect

Maximal subgroups

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

subgroupindex
#350142
C2×#23392
#329532
#329052
#328832
#299512
#298942
#297152
#275332
#272372
#251542
#249072
#220442
#219022
C4×#4502