Finite Group Databasebeta

← #6630461

Subgroups — #6630461

Order 512.

2133 conjugacy classes of subgroups.

The subgroup lattice has 665 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36621C211yesmaximal, characteristic
2C2×C4×#117C211yesmaximal, characteristic, complemented
3#33152C214yesmaximal, complemented
4#33112C214yesmaximal, complemented
5#33100C212yesmaximal, complemented
6#33096C212yesmaximal, complemented
7#32985C212yesmaximal, complemented
8#32981C212yesmaximal
9#21583C214yesmaximal, complemented
10#21582C214yesmaximal, complemented
11#21391C211yesmaximal, characteristic, complemented
12#21391C211yesmaximal, characteristic, complemented
13#21363C211yesmaximal, characteristic, complemented
14#21363C211yesmaximal, characteristic, complemented
15#20715C211yesmaximal, characteristic, complemented
16C2×#529C2211yescharacteristic
17C22×#117C2212yes—
18C23×C42C2211yescharacteristic
19C2×#485C2214yes—
20C2×#482C2214yes—
21C2×#432C2211yescharacteristic
22C2×#432C2211yescharacteristic
23#2855C2211yescharacteristic
24#2854C2211yescharacteristic
25#2853—21no—
26#2852—22no—
27#2851C2212yes—
28#2850C2212yes—
29#2849—21no—
30#2848C2211yescharacteristic
31#2847C2211yescharacteristic
32C4×C22⋊Q8C2212yes—
33C4×#123—21no—
34C4×#121C2212yes—
35C4×#120—21no—
36#2290—22no—
37#2283C2211yescharacteristic
38C4×#117—22no—
39C4×#117—22no—
40C4×#117C2214yes—
41C4×#117C2214yes—
42C2×C4×C4⋊C4C2212yes—
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C22⋊C4—21no—
46C2×C4×C22⋊C4—21no—
47C2×#376—21no—
48C2×C43—21no—
49#1989C2212yescomplemented
50#1989C2212yescomplemented
51#1983C2212yescomplemented
52#1983C2212yescomplemented
53#1949C2214yescomplemented
54#1949C2214yescomplemented
55#1928C2214yescomplemented
56#1927C2214yescomplemented
57#1926C2214yescomplemented
58#1925C2214yescomplemented
59#1921C2214yescomplemented
60#1920C2214yescomplemented
61#1919C2214yescomplemented
62#1918C2214yescomplemented
63#1875C2214yes—
64#1874C2212yes—
65#1874C2212yes—
66#1873C2214yes—
67#1872C2212yes—
68#1872C2212yes—
69#1854C2214yes—
70#1854C2214yes—
71#1853C2212yes—
72#1853C2212yes—
73#1852C2212yes—
74#1852C2212yes—
75#1825C2214yes—
76#1825C2214yes—
77#1817C2212yes—
78#1797C2214yes—
79#1779C2212yes—
80#1776C2214yescomplemented
81#1776C2214yescomplemented
82#1763C2212yescomplemented
83#1763C2212yescomplemented
84#1762C2212yescomplemented
85#1762C2212yescomplemented
86#1749C2214yes—
87C24×C4D411yescharacteristic
88C24×C4D411yescharacteristic
89C24×C4C2311yescharacteristic
90#529—24no—
91#529—24no—
92C2×#126—21no—
93C2×#125C2311yescharacteristic
94C2×#124C2311yescharacteristic
95C2×#123—21no—
96C2×#123—21no—
97C2×C22⋊Q8C2312yes—
98C2×C22⋊Q8C2311yescharacteristic
99C2×#121C2312yes—
100C2×#121C2311yescharacteristic
101C2×#120—21no—
102C2×C4×Q8C2311yescharacteristic
103C2×C4×D4—21no—
104C2×C4×D4—22no—
105C2×C4×D4C2311yescharacteristic
106C2×#117—22no—
107C2×#117—22no—
108C2×#117—216no—
109C2×#117—21no—
110C2×#117C2314yes—
111C2×#117C2314yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C22⋊C4—22no—
115C22×C42—22no—
116C22×C42—24no—
117C22×C42—24no—
118C22×C42—22no—
119C22×C42—21no—
120C22×C42—21no—
121C22×C42D412yes—
122C22×C42D411yescharacteristic
123C22×C42D411yescharacteristic
124C22×C42D412yes—
125C22×C42C2312yes—
126C22×C42C2311yescharacteristic
127C22×C42C2312yes—
128C22×C42C2311yescharacteristic
129#485—216no—
130#482—216no—
131#432—24no—
132#432—24no—
133C2×#107C2314yes—
134C2×#107C2314yes—
135C2×#106C2314yes—
136C2×#106C2314yes—
137C2×C4⋊C8C2312yes—
138C2×C4⋊C8C2312yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×C22⋊C8C2314yes—
144C2×C22⋊C8C2314yes—
145C2×C22⋊C8C2312yes—
146C2×C22⋊C8C2312yes—
147C2×#97C2312yes—
148C2×#97C2312yes—
149#391C2312yes—
150#390C2312yes—
151#389C2312yes—
152#389C2312yes—
153#388—22no—
154#388—22no—
155#388—22no—
156#388—22no—
157#387—22no—
158#387—22no—
159#387—22no—
160#387—22no—
161#386—22no—
162#386—21no—
163#386—22no—
164#386—22no—
165#386—22no—
166#386—21no—
167#385—22no—
168#385—21no—
169#385—22no—
170#385—22no—
171#385—22no—
172#385—21no—
173#383—21no—
174#382C2312yes—
175#381—21no—
176#381—21no—
177#380C2312yes—
178#380C2311yescharacteristic
179#380C2312yes—
180#380C2311yescharacteristic
181#379—21no—
182C4×C4⋊C4—22no—
183C4×C4⋊C4—22no—
184C4×C4⋊C4—21no—
185C4×C4⋊C4—21no—
186C4×C4⋊C4—21no—
187C4×C4⋊C4—21no—
188C4×C4⋊C4—22no—
189C4×C4⋊C4C2314yes—
190C4×C4⋊C4C2314yes—
191C4×C4⋊C4C2312yes—
192C4×C4⋊C4C2312yes—
193C4×C4⋊C4C2312yes—
194C4×C4⋊C4C2312yes—
195C4×C4⋊C4C2312yes—
196C4×C22⋊C4—24no—
197C4×C22⋊C4—24no—
198C4×C22⋊C4—24no—
199C4×C22⋊C4—24no—
200C4×C22⋊C4—21no—
201C4×C22⋊C4—22no—
202C4×C22⋊C4—22no—
203#376—24no—
204#376—24no—
205#376—21no—
206C2×#95—22no—
207C2×#95—21no—
208C2×#95—21no—
209C43—24no—
210C43—24no—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#336C2314yes—
220#336C2314yes—
221C25C2×D411yescharacteristic
222C22×Q8C2411yescharacteristic
223C22×D4—21no—
224C22×D4—22no—
225C22×D4C2411yescharacteristic
226C23×C4—28no—
227C23×C4—22no—
228C23×C4—22no—
229C23×C4—22no—
230C23×C4—22no—
231C23×C4—24no—
232C23×C4—24no—
233C23×C4—21no—
234C23×C4—44no—
235C23×C4—42no—
236C23×C4—42no—
237C23×C4C2×D412yes—
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D412yes—
242C23×C4C2×D412yes—
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D412yes—
245C23×C4C2×D412yes—
246C23×C4C2×D411yescharacteristic
247C23×C4C2411yescharacteristic
248C23×C4C2412yes—
249C22×C8C2414yes—
250C22×C8C2414yes—
251#126—44no—
252#125—22no—
253#125—22no—
254#124—22no—
255#124—22no—
256#123—44no—
257#123—44no—
258C22⋊Q8—22no—
259C22⋊Q8—22no—
260C22⋊Q8—28no—
261#121—22no—
262#121—22no—
263#121—28no—
264#120—44no—
265C4×Q8—22no—
266C4×Q8—22no—
267C4×D4—22no—
268C4×D4—22no—
269C4×D4—48no—
270C4×D4—44no—
271#117—216no—
272#117—216no—
273#117—28no—
274#117—28no—
275#117—28no—
276#117—44no—
277#117#41116yes—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—28no—
281C2×C4⋊C4—28no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4C2411yescharacteristic
290C2×C4⋊C4C2411yescharacteristic
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2414yes—
294C2×C4⋊C4C2414yes—
295C2×C22⋊C4—216no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—21no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—21no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—21no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—21no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—21no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C42—216no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—22no—
317C2×C42—22no—
318C2×C42—28no—
319C2×C42—28no—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—24no—
324C2×C42—21no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—24no—
331C2×C42—24no—
332C2×C42—44no—
333C2×C42—44no—
334C2×C42—48no—
335C2×C42C2×D412yes—
336C2×C42C2×D412yes—
337C2×C42C2×D412yes—
338C2×C42C2×D412yes—
339C2×C42C2×D412yes—
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42#4114yes—
343C2×C42#4114yes—
344C2×C42C2412yes—
345C2×C42C2411yescharacteristic
346C2×C42C2411yescharacteristic
347#107—28no—
348#107—28no—
349#106—28no—
350#106—28no—
351C4⋊C8—24no—
352C4⋊C8—24no—
353#103—28no—
354#103—28no—
355#102—28no—
356#102—28no—
357C22⋊C8—24no—
358C22⋊C8—24no—
359C22⋊C8—28no—
360C22⋊C8—28no—
361#97—24no—
362#97—24no—
363#95—24no—
364#95—24no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—21no—
370#95—21no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380C24—22no—
381C24—24no—
382C24—22no—
383C24—22no—
384C24—21no—
385C24—42no—
386C24C22×D412yes—
387C24C22×D411yescharacteristic
388C2×Q8—24no—
389C2×Q8—21no—
390C2×Q8—21no—
391C2×D4—24no—
392C2×D4—21no—
393C2×D4—21no—
394C2×D4—44no—
395C2×D4—44no—
396C2×D4—42no—
397C2×D4—44no—
398C2×D4—42no—
399C2×D4—42no—
400C22×C4—216no—
401C22×C4—216no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—28no—
416C22×C4—28no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—416no—
441C22×C4—416no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—42no—
448C22×C4#12314yes—
449C22×C4#12314yes—
450C22×C4C22×D412yes—
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C2×#4112yes—
459C22×C4C2×#4112yes—
460C22×C4C2×#4118yes—
461C22×C4C2511yescharacteristic
462C2×C8—24no—
463C2×C8—24no—
464C2×C8—24no—
465C2×C8—24no—
466C2×C8—24no—
467C2×C8—24no—
468C4⋊C4—28no—
469C4⋊C4—28no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—44no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4C2×#4118yes—
486C4⋊C4C2×#4118yes—
487C22⋊C4—216no—
488C22⋊C4—216no—
489C22⋊C4—44no—
490C22⋊C4—44no—
491C22⋊C4—44no—
492C22⋊C4—42no—
493C22⋊C4—44no—
494C22⋊C4—42no—
495C22⋊C4—42no—
496C22⋊C4—42no—
497C22⋊C4—42no—
498C22⋊C4—42no—
499C22⋊C4—42no—
500C22⋊C4—44no—
501C22⋊C4—44no—
502C42—216no—
503C42—24no—
504C42—24no—
505C42—24no—
506C42—24no—
507C42—24no—
508C42—216no—
509C42—22no—
510C42—22no—
511C42—44no—
512C42—44no—
513C42—44no—
514C42—44no—
515C42—48no—
516C42—44no—
517C42—42no—
518C42—48no—
519C42#12314yes—
520C42#12314yes—
521C42C2×#4112yes—
522C42C2×#4112yes—
523C42C2×#4112yes—
524C42C2×#4112yes—
525C23—22no—
526C23—22no—
527C23—28no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—48no—
535C23—48no—
536C23—44no—
537C23—44no—
538C23—41no—
539C23—42no—
540C23—41no—
541C23—42no—
542C23—41no—
543C23C2×#12312yes—
544C23#52514yes—
545C23#57512yes—
546C23#57512yes—
547C23C23×D411yescharacteristic
548Q8—42no—
549Q8—42no—
550D4—42no—
551D4—42no—
552D4—88no—
553D4—84no—
554C2×C4—216no—
555C2×C4—28no—
556C2×C4—28no—
557C2×C4—28no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—28no—
572C2×C4—28no—
573C2×C4—22no—
574C2×C4—28no—
575C2×C4—28no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—44no—
579C2×C4—48no—
580C2×C4—48no—
581C2×C4—48no—
582C2×C4—48no—
583C2×C4—416no—
584C2×C4—416no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—84no—
604C2×C4C2×#12312yes—
605C2×C4C2×#12312yes—
606C2×C4C2×#12312yes—
607C2×C4C2×#12312yes—
608C2×C4C2×#12312yes—
609C2×C4#52514yes—
610C2×C4#57514yes—
611C2×C4C22×#4111yescharacteristic
612C2×C4C22×#4111yescharacteristic
613C2×C4C22×#4114yes—
614C8—44no—
615C8—44no—
616C22—22no—
617C22—22no—
618C22—24no—
619C22—24no—
620C22—24no—
621C22—24no—
622C22—24no—
623C22—416no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—81no—
629C22—81no—
630C22—81no—
631C22—82no—
632C22—82no—
633C22#309314yes—
634C22C22×#12311yescharacteristic
635C22C2×#52512yes—
636C22C2×#57511yescharacteristic
637C22C2×#57511yescharacteristic
638C22C2×#57512yes—
639C4—22no—
640C4—24no—
641C4—22no—
642C4—22no—
643C4—24no—
644C4—22no—
645C4—22no—
646C4—42no—
647C4—48no—
648C4—48no—
649C4—81no—
650C4—81no—
651C4—81no—
652C4—82no—
653C4—82no—
654C4#344412yes—
655C4#344412yes—
656C2—24no—
657C2—44no—
658C2—161no—
659C2#3662611yescharacteristic
660C2C2×#309311yescharacteristic
661C2#6051111yescharacteristic
662C2#6054612yes—
663C2#6056312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2847C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×C4×#117C2semidirect
3. #33152C2semidirect
4. #33112C2semidirect
5. #33100C2semidirect
6. #33096C2semidirect
7. #32985C2semidirect
9. #21583C2semidirect
10. #21582C2semidirect
11. #21391C2semidirect
12. #21391C2semidirect
13. #21363C2semidirect
14. #21363C2semidirect
15. #20715C2semidirect
49. #1989C22semidirect
50. #1989C22semidirect
51. #1983C22semidirect
52. #1983C22semidirect
53. #1949C22semidirect
54. #1949C22semidirect
55. #1928C22semidirect
56. #1927C22semidirect
57. #1926C22semidirect
58. #1925C22semidirect
59. #1921C22semidirect
60. #1920C22semidirect
61. #1919C22semidirect
62. #1918C22semidirect
80. #1776C22semidirect
81. #1776C22semidirect
82. #1763C22semidirect
83. #1763C22semidirect
84. #1762C22semidirect
85. #1762C22semidirect

Maximal subgroups

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

subgroupindex
#366212
C2×C4×#1172
#331522
#331122
#331002
#330962
#329852
#329812
#215832
#215822
#213912
#213632
#207152