Finite Group Databasebeta

← #6469841

Subgroups — #6469841

Order 512.

1816 conjugacy classes of subgroups.

The subgroup lattice has 732 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36640C211yesmaximal, characteristic, complemented
2C4×#532C211yesmaximal, characteristic
3#29574C212yesmaximal, complemented
4#29572C212yesmaximal, complemented
5#29223C211yesmaximal, characteristic, complemented
6#28722C211yesmaximal, characteristic, complemented
7#28692C211yesmaximal, characteristic, complemented
8#28516C211yesmaximal, characteristic, complemented
9#27881C212yesmaximal, complemented
10#25676C212yesmaximal
11#24704C214yesmaximal, complemented
12#23761C212yesmaximal, complemented
13#23760C212yesmaximal, complemented
14#21731C212yesmaximal, complemented
15#21656C212yesmaximal, complemented
16#21617C214yesmaximal, complemented
17#20463C211yesmaximal, characteristic, complemented
18C2×#532C2211yescharacteristic
19C2×C4×#41—21no—
20C2×#501C2212yes—
21C2×#490C2212yes—
22C2×#478C2214yes—
23C2×#431C2212yes—
24#2864—21no—
25#2864—21no—
26#2859C2211yescharacteristic
27#2859C2211yescharacteristic
28#2859C2211yescharacteristic
29#2859C2211yescharacteristic
30#2851—21no—
31#2851—21no—
32#2851C2212yes—
33#2851C2212yes—
34C4×C4⋊Q8C2211yescharacteristic
35C4×C4⋊Q8C2211yescharacteristic
36C4×#127C2211yescharacteristic
37C4×#127C2211yescharacteristic
38C4×#125—21no—
39C4×#125—21no—
40C4×#125C2212yes—
41C4×#125C2212yes—
42C4×#124—21no—
43C4×#124—21no—
44C4×#121—21no—
45C4×#121—21no—
46C4×#121C2212yes—
47C4×#121C2212yes—
48#2297—21no—
49#2296—21no—
50#2296—21no—
51#2293—21no—
52#2293C2212yes—
53#2285C2211yescharacteristic
54#2285C2211yescharacteristic
55#2285C2211yescharacteristic
56C42×D4—21no—
57C42×D4C2212yes—
58C2×C43C2211yescharacteristic
59#2100C2212yes—
60#2099C2212yes—
61#2094C2212yes—
62#2093C2212yes—
63#2084C2212yes—
64#2083C2212yes—
65#2077C2214yes—
66#2076C2214yes—
67#2071C2212yes—
68#2071C2212yes—
69#2034C2214yes—
70#2033C2214yes—
71#1993C2212yes—
72#1992C2212yes—
73#1992C2212yes—
74#1989C2214yes—
75#1961C2212yes—
76#1949C2214yes—
77#1942C2214yes—
78#1941C2214yes—
79#1934C2214yes—
80#1934C2214yes—
81#1920C2214yes—
82#1918C2214yes—
83#1864C2214yes—
84#1862C2214yes—
85#1847C2212yes—
86#1836C2212yes—
87#1804C2214yes—
88#1776C2214yes—
89#1776C2214yes—
90#1767C2211yescharacteristic
91#1767C2211yescharacteristic
92#1767C2212yes—
93#1762C2214yes—
94#1762C2214yes—
95#1749C2212yes—
96#1749C2212yes—
97#1744C2212yes—
98C4×#97C2212yes—
99C22×#41—21no—
100#532—24no—
101#532—24no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×#127C2311yescharacteristic
104C2×#125—21no—
105C2×#125—21no—
106C2×#125C2312yes—
107C2×#125C2312yes—
108C2×#124—21no—
109C2×#121—21no—
110C2×#121C2312yes—
111C4×#41—48no—
112C2×C4×Q8—21no—
113C2×C4×Q8—21no—
114C2×C4×D4—22no—
115C2×C4×D4—22no—
116C2×C4×D4—21no—
117C2×C4×D4—21no—
118C2×C4×D4—21no—
119C2×C4×D4—21no—
120C2×C4×D4—21no—
121C2×C4×D4C2312yes—
122C2×C4×D4C2312yes—
123C2×C4×D4C2312yes—
124C2×#117—21no—
125C2×#117—21no—
126C2×#117—21no—
127C2×#117—22no—
128C22×C42—22no—
129C22×C42D411yescharacteristic
130C22×C42D411yescharacteristic
131C22×C42D411yescharacteristic
132C22×C42D411yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136#501—28no—
137#490—28no—
138#478—216no—
139#431—28no—
140C2×#107C2314yes—
141C2×#107C2314yes—
142C2×#106C2314yes—
143C2×#106C2314yes—
144C2×C4⋊C8C2312yes—
145C2×C4⋊C8C2312yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×C22⋊C8C2314yes—
151C2×#97C2312yes—
152C2×#97C2312yes—
153C2×#97C2314yes—
154C2×C4×C8C2314yes—
155#391—21no—
156#391—21no—
157#390—21no—
158#390—21no—
159#390—21no—
160#390—21no—
161#390C2312yes—
162#390C2312yes—
163#389—21no—
164#389—21no—
165#389—21no—
166#389—21no—
167#389—21no—
168#389—21no—
169#389—21no—
170#389—21no—
171#389—21no—
172#389—21no—
173#389—21no—
174#389—21no—
175#389—21no—
176#389—21no—
177#389C2312yes—
178#389C2312yes—
179#389C2312yes—
180#389C2312yes—
181#389C2312yes—
182#389C2312yes—
183#388—22no—
184#388—22no—
185#388—22no—
186#388—22no—
187#387—22no—
188#387—22no—
189#387—22no—
190#387—22no—
191#387—22no—
192#387—22no—
193#387—22no—
194#387—22no—
195#386—22no—
196#386—22no—
197#385—22no—
198#385—22no—
199#385—22no—
200#385—22no—
201#384C2311yescharacteristic
202#384C2311yescharacteristic
203#384C2311yescharacteristic
204#384C2311yescharacteristic
205#384C2311yescharacteristic
206#384C2311yescharacteristic
207#382—21no—
208#382—21no—
209#382—21no—
210#382—21no—
211#382—21no—
212#382C2312yes—
213#382C2312yes—
214#380—21no—
215#380—21no—
216#380—21no—
217#380C2312yes—
218#380C2312yes—
219#380C2312yes—
220C4×C4⋊C4—21no—
221C4×C4⋊C4—21no—
222C4×C4⋊C4—22no—
223C4×C4⋊C4—22no—
224C4×C4⋊C4—22no—
225C4×C4⋊C4—22no—
226C4×C4⋊C4—21no—
227C4×C4⋊C4—21no—
228C4×C4⋊C4—21no—
229C4×C4⋊C4C2312yes—
230C4×C4⋊C4C2312yes—
231C4×C4⋊C4C2312yes—
232C4×C4⋊C4C2312yes—
233C4×C4⋊C4C2312yes—
234C4×C22⋊C4—22no—
235C4×C22⋊C4—22no—
236C4×C22⋊C4—22no—
237C4×C22⋊C4—22no—
238C4×C22⋊C4—22no—
239C4×C22⋊C4—22no—
240C43D411yescharacteristic
241C43D411yescharacteristic
242C43D411yescharacteristic
243C43D411yescharacteristic
244C43C2311yescharacteristic
245C43C2311yescharacteristic
246C43C2311yescharacteristic
247C43C2311yescharacteristic
248#340C2314yes—
249#340C2314yes—
250#340C2314yes—
251#340C2314yes—
252#340C2314yes—
253#340C2314yes—
254#340C2314yes—
255#340C2314yes—
256#336C2312yes—
257#336C2312yes—
258#336C2314yes—
259C2×#41—28no—
260C2×#41—44no—
261C2×#41—44no—
262C22×Q8—21no—
263C22×D4—22no—
264C22×D4—21no—
265C22×D4—21no—
266C22×D4C2412yes—
267C23×C4—22no—
268C23×C4—22no—
269C23×C4—22no—
270C23×C4C2×D411yescharacteristic
271C23×C4C2×D411yescharacteristic
272C23×C4C2×D411yescharacteristic
273C23×C4C2×D411yescharacteristic
274C23×C4C2×D411yescharacteristic
275C23×C4C2×D411yescharacteristic
276C23×C4C2411yescharacteristic
277C22×C8C2414yes—
278C22×C8C2414yes—
279C4⋊Q8—22no—
280C4⋊Q8#4114yes—
281#127—22no—
282#127#4114yes—
283#125—28no—
284#125—24no—
285#125—24no—
286#125—44no—
287#125—44no—
288#124—28no—
289#121—28no—
290#121—24no—
291#121—24no—
292C4×Q8—44no—
293C4×Q8—44no—
294C4×D4—28no—
295C4×D4—24no—
296C4×D4—24no—
297C4×D4—28no—
298C4×D4—28no—
299C4×D4—44no—
300C4×D4—44no—
301C4×D4—44no—
302C4×D4—44no—
303C4×D4—48no—
304C4×D4—48no—
305#117—44no—
306#117—44no—
307#117—44no—
308#117—48no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2412yes—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C42—24no—
351C2×C42—22no—
352C2×C42—22no—
353C2×C42—24no—
354C2×C42—24no—
355C2×C42—22no—
356C2×C42—22no—
357C2×C42—22no—
358C2×C42—22no—
359C2×C42—22no—
360C2×C42—42no—
361C2×C42—42no—
362C2×C42—48no—
363C2×C42—42no—
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×C42C2×D411yescharacteristic
373C2×C42C2×D411yescharacteristic
374C2×C42C2×D411yescharacteristic
375C2×C42C2×D411yescharacteristic
376C2×C42C2×D411yescharacteristic
377C2×C42C2×D411yescharacteristic
378C2×C42C2×D411yescharacteristic
379C2×C42C2×D411yescharacteristic
380C2×C42C2×D411yescharacteristic
381C2×C42C2×D411yescharacteristic
382C2×C42C2×D411yescharacteristic
383C2×C42C2×D411yescharacteristic
384C2×C42C2×D411yescharacteristic
385C2×C42C2×D411yescharacteristic
386C2×C42C2411yescharacteristic
387C2×C42C2411yescharacteristic
388C2×C42C2411yescharacteristic
389C2×C42C2411yescharacteristic
390C2×C42C2411yescharacteristic
391C2×C42C2411yescharacteristic
392#107—28no—
393#107—28no—
394#106—28no—
395#106—28no—
396C4⋊C8—24no—
397C4⋊C8#4118yescomplemented
398#102—28no—
399#102—28no—
400#102—28no—
401#102—28no—
402C22⋊C8—28no—
403#97—24no—
404#97—24no—
405#97—28no—
406C4×C8—28no—
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—
418#95—22no—
419#95—22no—
420#95—22no—
421#95—22no—
422#95—22no—
423#95—22no—
424#95—22no—
425C24—22no—
426C24C22×D411yescharacteristic
427#41—432no—
428C2×Q8—24no—
429C2×Q8—42no—
430C2×Q8—42no—
431C2×D4—28no—
432C2×D4—24no—
433C2×D4—24no—
434C2×D4—24no—
435C2×D4—22no—
436C2×D4—24no—
437C2×D4—42no—
438C2×D4—42no—
439C2×D4—42no—
440C2×D4—42no—
441C2×D4—44no—
442C2×D4—44no—
443C2×D4C2×#4114yes—
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—28no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—22no—
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—44no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4—44no—
478C22×C4—44no—
479C22×C4—44no—
480C22×C4—44no—
481C22×C4—44no—
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
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×C4C2511yescharacteristic
496C2×C8—24no—
497C2×C8—24no—
498C2×C8—24no—
499C2×C8—24no—
500C2×C8—24no—
501C2×C8C2×#4118yes—
502C4⋊C4—24no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—24no—
507C4⋊C4—24no—
508C4⋊C4—22no—
509C4⋊C4—24no—
510C4⋊C4—22no—
511C4⋊C4—24no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—44no—
531C4⋊C4—42no—
532C4⋊C4—42no—
533C4⋊C4—44no—
534C4⋊C4C2×#4114yes—
535C22⋊C4—28no—
536C22⋊C4—28no—
537C22⋊C4—44no—
538C22⋊C4—44no—
539C22⋊C4—44no—
540C22⋊C4—44no—
541C22⋊C4—44no—
542C22⋊C4—44no—
543C22⋊C4—44no—
544C22⋊C4—44no—
545C22⋊C4—44no—
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—21no—
561C42—21no—
562C42—22no—
563C42—41no—
564C42—41no—
565C42—41no—
566C42—41no—
567C42—41no—
568C42—41no—
569C42—41no—
570C42—41no—
571C42—44no—
572C42—44no—
573C42—44no—
574C42—44no—
575C42—44no—
576C42AGL(1,ℤ/8)14yes—
577C42C2×#4112yes—
578C42C2×#4112yes—
579C42C2×#4114yes—
580C422−1+414yes—
581C23—21no—
582C23—21no—
583C23—21no—
584C23—21no—
585C23—21no—
586C23—21no—
587C23—24no—
588C23—42no—
589C23—42no—
590C23—42no—
591C23—42no—
592C23—42no—
593C23—42no—
594C23#57512yes—
595C23C23×D411yescharacteristic
596Q8—48no—
597D4—28no—
598D4—48no—
599D4—48no—
600D4—416no—
601D4—44no—
602C2×C4—28no—
603C2×C4—28no—
604C2×C4—28no—
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—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—24no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—416no—
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—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—88no—
661C2×C4—88no—
662C2×C4#52512yes—
663C2×C4#52512yes—
664C2×C4#53212yes—
665C2×C4#53212yes—
666C2×C4#53612yes—
667C2×C4#53612yes—
668C2×C4C2×AGL(1,ℤ/8)12yes—
669C2×C4C2×AGL(1,ℤ/8)12yes—
670C2×C4C2×AGL(1,ℤ/8)12yes—
671C2×C4#57512yes—
672C2×C4#57512yes—
673C2×C4C22×#4111yescharacteristic
674C2×C4C22×#4111yescharacteristic
675C2×C4C22×#4112yes—
676C2×C4C2×2−1+412yes—
677C8—28no—
678C8—44no—
679C22—22no—
680C22—22no—
681C22—22no—
682C22—41no—
683C22—41no—
684C22—41no—
685C22—41no—
686C22—44no—
687C22—44no—
688C22—44no—
689C22—82no—
690C22—82no—
691C22—82no—
692C22—82no—
693C22C2×#52511yescharacteristic
694C22C2×#53211yescharacteristic
695C22C2×#53611yescharacteristic
696C22C22×AGL(1,ℤ/8)11yescharacteristic
697C22C2×#57511yescharacteristic
698C22C2×#57511yescharacteristic
699C22C2×#57511yescharacteristic
700C4—22no—
701C4—22no—
702C4—22no—
703C4—22no—
704C4—22no—
705C4—22no—
706C4—22no—
707C4—22no—
708C4—22no—
709C4—22no—
710C4—22no—
711C4—44no—
712C4—41no—
713C4—41no—
714C4—82no—
715C4—82no—
716C4—82no—
717C4—82no—
718C4—82no—
719C4—82no—
720C4#348212yes—
721C4#348212yes—
722C2—42no—
723C2—84no—
724C2#3497911yescharacteristic
725C2#6055811yescharacteristic
726C2#6055911yescharacteristic
727C2#6068811yescharacteristic
728C2#6070011yescharacteristic
729C2#6086111yescharacteristic
730C2#6086411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
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. #36640C2semidirect
3. #29574C2semidirect
4. #29572C2semidirect
5. #29223C2semidirect
6. #28722C2semidirect
7. #28692C2semidirect
8. #28516C2semidirect
9. #27881C2semidirect
11. #24704C2semidirect
12. #23761C2semidirect
13. #23760C2semidirect
14. #21731C2semidirect
15. #21656C2semidirect
16. #21617C2semidirect
17. #20463C2semidirect
397. C4⋊C8#41semidirect

Maximal subgroups

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

subgroupindex
#366402
C4×#5322
#295742
#295722
#292232
#287222
#286922
#285162
#278812
#256762
#247042
#237612
#237602
#217312
#216562
#216172
#204632