Finite Group Databasebeta

← #6481177

Subgroups — #6481177

Order 512.

1644 conjugacy classes of subgroups.

The subgroup lattice has 727 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2863C211yesmaximal, characteristic, complemented
2#36631C211yesmaximal, characteristic
3#33265C214yesmaximal, complemented
4#30377C212yesmaximal, complemented
5#30337C212yesmaximal, complemented
6#29512C212yesmaximal, complemented
7#29492C212yesmaximal, complemented
8#29432C214yesmaximal, complemented
9#29328C211yesmaximal, characteristic
10#29297C211yesmaximal, characteristic, complemented
11#29105C211yesmaximal, characteristic, complemented
12#28305C211yesmaximal, characteristic
13#24603C212yesmaximal, complemented
14#24503C212yesmaximal, complemented
15#22076C212yesmaximal, complemented
16#22041C212yesmaximal, complemented
17C4×#431C211yesmaximal, characteristic, complemented
18C2×#529—21no—
19C2×#481C2212yes—
20C2×#448C2212yes—
21C2×#431C2212yes—
22#2863—21no—
23#2863—21no—
24#2863—21no—
25#2863—21no—
26#2863C2212yes—
27#2863C2212yes—
28#2863C2212yes—
29#2863C2212yes—
30#2861—21no—
31#2860—21no—
32#2860C2212yes—
33#2859C2211yescharacteristic
34#2858C2211yescharacteristic
35#2857C2211yescharacteristic
36#2856C2211yescharacteristic
37#2854—21no—
38#2854C2212yes—
39#2850—21no—
40C4×C4⋊Q8C2211yescharacteristic
41C4×#127C2211yescharacteristic
42C4×#124—21no—
43C4×#121—21no—
44C4×#121C2212yes—
45C2×#391C2211yescharacteristic
46C2×#391C2211yescharacteristic
47C2×#391C2211yescharacteristic
48#2293—21no—
49#2293C2212yes—
50C2×#389C2211yescharacteristic
51C2×#389C2211yescharacteristic
52C2×#389C2211yescharacteristic
53C2×#387—21no—
54C2×#387—21no—
55C2×#387—21no—
56C2×#383—21no—
57C2×C43C2211yescharacteristic
58#2099C2212yes—
59#2096C2212yes—
60#2088C2214yes—
61#2084C2212yes—
62#2072C2212yes—
63#2066C2214yes—
64#2065C2214yes—
65#2058C2212yes—
66#2055C2214yes—
67#2041C2212yes—
68#2024C2212yes—
69#2014C2212yes—
70#1968C2214yes—
71#1968C2214yes—
72#1968C2214yes—
73#1941C2214yes—
74#1941C2214yes—
75#1941C2214yes—
76#1934C2214yes—
77#1934C2214yes—
78#1934C2214yes—
79#1933C2214yes—
80#1933C2214yes—
81#1933C2214yes—
82#1927C2214yes—
83#1920C2214yes—
84#1877C2214yes—
85#1874C2212yes—
86#1874C2212yes—
87#1864C2214yes—
88#1853C2212yes—
89#1853C2212yes—
90#1824C2212yes—
91#1787C2212yes—
92C4×C4⋊C8C2212yes—
93C4×C4⋊C8C2212yes—
94C4×C22⋊C8C2212yes—
95C4×C22⋊C8C2211yescharacteristic
96C4×C22⋊C8C2211yescharacteristic
97#1744C2212yes—
98C42×C8C2212yes—
99C23×Q8C2311yescharacteristic
100C22×SD16C2314yes—
101C2×C4⋊Q8C2311yescharacteristic
102C2×#127C2311yescharacteristic
103#529—48no—
104C2×#126—22no—
105C2×#125—21no—
106C2×#124—21no—
107C2×#124—21no—
108C2×#123—22no—
109C2×C22⋊Q8—21no—
110C2×#121—21no—
111C2×#121—21no—
112C2×#121C2312yes—
113C2×C4×Q8—21no—
114C2×C4×D4—22no—
115C2×C4×D4—21no—
116C2×C4×D4—21no—
117C2×C4×D4C2312yes—
118C2×#117—22no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4—21no—
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C42D411yescharacteristic
126C22×C42D411yescharacteristic
127C22×C42D411yescharacteristic
128C22×C42D411yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132#481—28no—
133#448—28no—
134#431—28no—
135C2×#106C2314yes—
136C2×#106C2314yes—
137C2×#106C2314yes—
138C2×C4⋊C8C2314yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×C22⋊C8C2312yes—
146C2×C22⋊C8C2312yes—
147C2×C4×C8C2314yes—
148C2×C4×C8C2312yes—
149C2×C4×C8C2314yes—
150C2×C4×C8C2312yes—
151#391—21no—
152#391—21no—
153#391—21no—
154#391—21no—
155#391—21no—
156#391—21no—
157#391—21no—
158#391—21no—
159#391C2312yes—
160#391C2312yes—
161#391C2312yes—
162#391C2312yes—
163#391C2312yes—
164#391C2312yes—
165#390—21no—
166#390—21no—
167#390—21no—
168#390—21no—
169#390—21no—
170#390—21no—
171#390C2312yes—
172#390C2312yes—
173#390C2312yes—
174#390C2312yes—
175#389—21no—
176#389—21no—
177#389—21no—
178#389—21no—
179#389—21no—
180#389—21no—
181#389—21no—
182#389—21no—
183#389—21no—
184#389—21no—
185#389C2312yes—
186#389C2312yes—
187#389C2312yes—
188#389C2312yes—
189#389C2312yes—
190#389C2312yes—
191#389C2312yes—
192#389C2312yes—
193#389C2312yes—
194#389C2312yes—
195#388—22no—
196#388—22no—
197#388—22no—
198#388—22no—
199#388—22no—
200#388—22no—
201#388—22no—
202#388—22no—
203#387—22no—
204#387—22no—
205#387—22no—
206#387—22no—
207#387—22no—
208#387—22no—
209#387—22no—
210#387—22no—
211#387—22no—
212#387—22no—
213#387—22no—
214#387—22no—
215#386—22no—
216#386—22no—
217#384C2311yescharacteristic
218#384C2311yescharacteristic
219#383—22no—
220#383—22no—
221#383—22no—
222#383—22no—
223#382—21no—
224#380—21no—
225#380C2312yes—
226C4×C4⋊C4—21no—
227C4×C4⋊C4C2312yes—
228C4×C22⋊C4—22no—
229C4×C22⋊C4—22no—
230C2×#95—21no—
231C2×#95—21no—
232C2×#95—21no—
233C2×#95—21no—
234C2×#95—21no—
235C2×#95—21no—
236C2×#95—21no—
237C2×#95—21no—
238C2×#95—21no—
239C43D411yescharacteristic
240C43D411yescharacteristic
241C43D411yescharacteristic
242C43D411yescharacteristic
243C43C2311yescharacteristic
244C43C2311yescharacteristic
245C43C2311yescharacteristic
246C43C2311yescharacteristic
247#340C2314yes—
248#340C2314yes—
249#340C2314yes—
250#340C2314yes—
251#340C2314yes—
252#340C2314yes—
253#336C2312yes—
254#336C2312yes—
255#336C2314yes—
256C22×Q8—21no—
257C22×Q8—21no—
258C22×Q8—24no—
259C22×Q8—42no—
260C22×Q8—42no—
261C22×Q8C2412yes—
262C22×D4—22no—
263C22×D4—21no—
264C22×D4—21no—
265C22×D4C2412yes—
266C23×C4—21no—
267C23×C4—21no—
268C23×C4—21no—
269C23×C4—21no—
270C23×C4—22no—
271C23×C4C2×D411yescharacteristic
272C23×C4C2×D411yescharacteristic
273C23×C4C2×D411yescharacteristic
274C23×C4C2×D411yescharacteristic
275C23×C4C2×D411yescharacteristic
276C23×C4C2×D411yescharacteristic
277C23×C4C2411yescharacteristic
278C2×SD16—28no—
279C2×SD16—44no—
280C2×SD16—44no—
281C22×C8C2414yes—
282C22×C8C2414yes—
283C4⋊Q8—42no—
284#127—42no—
285#126—48no—
286#125—44no—
287#124—44no—
288#124—44no—
289#123—48no—
290C22⋊Q8—44no—
291#121—24no—
292#121—24no—
293#121—44no—
294#121—44no—
295C4×Q8—44no—
296C4×D4—44no—
297C4×D4—44no—
298C4×D4—48no—
299C4×D4—44no—
300#117—48no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—42no—
321C2×C4⋊C4—42no—
322C2×C4⋊C4—44no—
323C2×C4⋊C4—44no—
324C2×C4⋊C4—44no—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2412yes—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C42—24no—
343C2×C42—24no—
344C2×C42—22no—
345C2×C42—42no—
346C2×C42—42no—
347C2×C42—42no—
348C2×C42—42no—
349C2×C42SD1618yescomplemented
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D411yescharacteristic
360C2×C42C2×D411yescharacteristic
361C2×C42C2×D411yescharacteristic
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D411yescharacteristic
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D411yescharacteristic
366C2×C42C2×D411yescharacteristic
367C2×C42C2×D411yescharacteristic
368C2×C42C2×D411yescharacteristic
369C2×C42C2×D411yescharacteristic
370C2×C42C2×D411yescharacteristic
371C2×C42C2×D411yescharacteristic
372C2×C42C2411yescharacteristic
373C2×C42C2411yescharacteristic
374C2×C42C2411yescharacteristic
375C2×C42C2411yescharacteristic
376C2×C42C2411yescharacteristic
377C2×C42C2411yescharacteristic
378#106—28no—
379#106—44no—
380#106—44no—
381C4⋊C8—28no—
382#103—28no—
383#103—44no—
384#103—44no—
385#102—28no—
386#102—44no—
387#102—44no—
388C22⋊C8—24no—
389C22⋊C8#4118yes—
390C4×C8—28no—
391C4×C8—42no—
392C4×C8—44no—
393C4×C8—42no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—22no—
402#95—22no—
403#95—22no—
404#95—22no—
405#95—22no—
406#95—22no—
407#95—22no—
408#95—22no—
409#95—22no—
410#95—22no—
411#95—22no—
412#95—22no—
413#95—22no—
414#95—22no—
415#95—22no—
416#95—22no—
417#95—22no—
418C24—22no—
419C24C22×D411yescharacteristic
420C2×Q8—24no—
421C2×Q8—22no—
422C2×Q8—24no—
423C2×Q8—22no—
424C2×Q8—42no—
425C2×Q8—42no—
426C2×Q8—42no—
427C2×Q8—42no—
428C2×Q8—42no—
429C2×Q8—42no—
430C2×Q8—88no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—24no—
434C2×D4—24no—
435C2×D4—42no—
436C2×D4—42no—
437C2×D4—42no—
438C2×D4—42no—
439C2×D4—44no—
440C2×D4—44no—
441C2×D4—44no—
442C2×D4—42no—
443C2×D4—42no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—41no—
460C22×C4—41no—
461C22×C4—41no—
462C22×C4—41no—
463C22×C4—41no—
464C22×C4—41no—
465C22×C4—41no—
466C22×C4—41no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—44no—
482C22×C4C2×SD1614yes—
483C22×C4C2×SD1614yes—
484C22×C4C2×SD1614yes—
485C22×C4#13514yes—
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D411yescharacteristic
493C22×C4C22×D411yescharacteristic
494C22×C4C22×D411yescharacteristic
495C22×C4C22×D411yescharacteristic
496C22×C4C22×D411yescharacteristic
497C22×C4C22×D411yescharacteristic
498C22×C4C22×D411yescharacteristic
499C22×C4C2×#4114yes—
500C22×C4C2511yescharacteristic
501SD16—88no—
502C2×C8—24no—
503C2×C8—24no—
504C2×C8—24no—
505C2×C8—24no—
506C2×C8—24no—
507C2×C8C2×#4118yes—
508C4⋊C4—24no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—24no—
513C4⋊C4—24no—
514C4⋊C4—24no—
515C4⋊C4—22no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—42no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—42no—
526C4⋊C4—44no—
527C4⋊C4—42no—
528C4⋊C4—44no—
529C4⋊C4—44no—
530C4⋊C4—44no—
531C4⋊C4—42no—
532C4⋊C4—44no—
533C4⋊C4—44no—
534C22⋊C4—44no—
535C22⋊C4—44no—
536C22⋊C4—44no—
537C22⋊C4—44no—
538C22⋊C4—44no—
539C22⋊C4—44no—
540C22⋊C4—44no—
541C22⋊C4—44no—
542C22⋊C4—44no—
543C22⋊C4—44no—
544C22⋊C4—44no—
545C42—22no—
546C42—22no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—22no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—22no—
560C42—22no—
561C42—41no—
562C42—41no—
563C42—41no—
564C42—41no—
565C42—41no—
566C42—41no—
567C42—41no—
568C42—41no—
569C42—44no—
570C42C2×SD1614yes—
571C42C2×SD1614yes—
572C42C2×SD1614yes—
573C42C2×SD1614yes—
574C23—21no—
575C23—21no—
576C23—21no—
577C23—21no—
578C23—42no—
579C23—42no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23#53212yes—
586C23C22×SD1612yes—
587C23C2×#13512yes—
588C23C23×D411yescharacteristic
589Q8—44no—
590Q8—44no—
591Q8—84no—
592Q8—84no—
593D4—44no—
594D4—44no—
595D4—84no—
596D4—84no—
597D4—88no—
598C2×C4—28no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—28no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—88no—
655C2×C4—84no—
656C2×C4—84no—
657C2×C4—84no—
658C2×C4—84no—
659C2×C4#53212yes—
660C2×C4C22×SD1612yes—
661C2×C4C22×SD1612yes—
662C2×C4C22×SD1612yes—
663C2×C4C22×SD1612yes—
664C2×C4C22×SD1612yes—
665C2×C4C22×SD1612yes—
666C2×C4C2×#13512yes—
667C2×C4C2×#13512yes—
668C2×C4#57512yes—
669C2×C4#57512yes—
670C2×C4C22×#4112yes—
671C2×C4#58512yes—
672C2×C4#58512yes—
673C8—28no—
674C8—44no—
675C22—22no—
676C22—22no—
677C22—22no—
678C22—22no—
679C22—22no—
680C22—22no—
681C22—82no—
682C22—82no—
683C22—82no—
684C22—82no—
685C22—82no—
686C22—82no—
687C22—82no—
688C22#316714yes—
689C22C2×#53211yescharacteristic
690C22#344911yescharacteristic
691C22#344911yescharacteristic
692C22C23×SD1611yescharacteristic
693C22C22×#13511yescharacteristic
694C22C2×#57511yescharacteristic
695C22C2×#57511yescharacteristic
696C4—22no—
697C4—22no—
698C4—22no—
699C4—22no—
700C4—22no—
701C4—22no—
702C4—22no—
703C4—22no—
704C4—22no—
705C4—22no—
706C4—22no—
707C4—41no—
708C4—41no—
709C4—82no—
710C4—82no—
711C4—82no—
712C4—82no—
713C4—82no—
714C4—82no—
715C4—82no—
716C4#352114yes—
717C2—24no—
718C2—162no—
719C2#3514511yescharacteristic
720C2C2×#316711yescharacteristic
721C2#6070211yescharacteristic
722C2#6094111yescharacteristic
723C2#6094111yescharacteristic
724C2#6094211yescharacteristic
725C2#6094511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2863C2semidirect
3. #33265C2semidirect
4. #30377C2semidirect
5. #30337C2semidirect
6. #29512C2semidirect
7. #29492C2semidirect
8. #29432C2semidirect
10. #29297C2semidirect
11. #29105C2semidirect
13. #24603C2semidirect
14. #24503C2semidirect
15. #22076C2semidirect
16. #22041C2semidirect
17. C4×#431C2semidirect
349. C2×C42SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#28632
#366312
#332652
#303772
#303372
#295122
#294922
#294322
#293282
#292972
#291052
#283052
#246032
#245032
#220762
#220412
C4×#4312