Finite Group Databasebeta

← #6640542

Subgroups — #6640542

Order 512.

1875 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35102C211yesmaximal, characteristic
2C2×#2486C211yesmaximal, characteristic, complemented
3#32982C211yesmaximal, characteristic, complemented
4#32978C211yesmaximal, characteristic, complemented
5#32264C214yesmaximal, complemented
6#32032C214yesmaximal, complemented
7#32006C214yesmaximal, complemented
8#32002C214yesmaximal, complemented
9#29895C211yesmaximal, characteristic, complemented
10#29892C211yesmaximal, characteristic, complemented
11#27616C212yesmaximal, complemented
12#24551C212yesmaximal, complemented
13#24550C212yesmaximal, complemented
14#24066C212yesmaximal, complemented
15#21738C211yesmaximal, characteristic, complemented
16C2×#529C2211yescharacteristic
17C22×#123—21no—
18C22×C4×D4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C2×#483C2212yes—
21C2×#480C2212yes—
22C22×C22⋊C8C2212yes—
23#2848C2211yescharacteristic
24#2848C2211yescharacteristic
25#2847C2211yescharacteristic
26#2847C2211yescharacteristic
27#2623—21no—
28#2613—21no—
29#2603—22no—
30#2598—22no—
31#2562C2212yes—
32#2540C2212yes—
33#2536C2212yes—
34#2535C2212yes—
35#2486—22no—
36#2486—22no—
37#2486C2214yes—
38#2486C2214yes—
39C2×#398—21no—
40C2×#397—21no—
41#2321—21no—
42#2321—21no—
43C2×#389C2212yes—
44#2283C2211yescharacteristic
45C2×#381—21no—
46C2×#380C2211yescharacteristic
47C2×#380C2212yes—
48#2101C2214yescomplemented
49#2085C2214yescomplemented
50#2077C2214yescomplemented
51#2077C2214yescomplemented
52#2076C2214yescomplemented
53#2075C2214yescomplemented
54#2059C2214yescomplemented
55#2042C2214yescomplemented
56#2036C2214yescomplemented
57#2035C2214yescomplemented
58#2034C2214yescomplemented
59#2033C2214yescomplemented
60#1921C2214yescomplemented
61#1920C2214yescomplemented
62#1893C2214yescomplemented
63#1892C2214yescomplemented
64#1874C2212yes—
65#1874C2214yescomplemented
66#1872C2212yes—
67#1872C2214yescomplemented
68#1869C2214yes—
69#1869C2214yes—
70#1864C2214yescomplemented
71#1863C2214yescomplemented
72#1853C2212yes—
73#1853C2214yescomplemented
74#1852C2212yes—
75#1852C2214yescomplemented
76#1823C2212yes—
77#1823C2212yes—
78#1817C2212yes—
79#1808C2214yescomplemented
80#1807C2214yescomplemented
81#1779C2211yescharacteristic
82#1779C2211yescharacteristic
83C23×D4D411yescharacteristic
84C24×C4—21no—
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C23×C8C2314yes—
88#529—44no—
89C2×#126—21no—
90C2×#125C2311yescharacteristic
91C2×#124C2311yescharacteristic
92C2×#123—24no—
93C2×#123—24no—
94C2×#123—21no—
95C2×#123—48no—
96C2×C22⋊Q8C2311yescharacteristic
97C2×C22⋊Q8C2312yes—
98C2×C22⋊Q8C2312yes—
99C2×#121C2311yescharacteristic
100C2×#121C2312yes—
101C2×#121C2312yes—
102C2×#120—21no—
103C2×#120—21no—
104C2×C4×Q8C2311yescharacteristic
105C2×C4×D4—21no—
106C2×C4×D4—44no—
107C2×C4×D4D412yes—
108C2×C4×D4D412yes—
109C2×C4×D4C2311yescharacteristic
110C2×C4×D4C2312yes—
111C2×C4×D4C2312yes—
112C2×#117—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—22no—
115C22×C4⋊C4—22no—
116C22×C4⋊C4—44no—
117C22×C4⋊C4D411yescharacteristic
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2314yes—
121C22×C22⋊C4—22no—
122C22×C22⋊C4—21no—
123C22×C22⋊C4D411yescharacteristic
124C22×C22⋊C4D411yescharacteristic
125C22×C42C2311yescharacteristic
126#483—44no—
127#480—44no—
128C2×#107C2314yes—
129C2×#107C2314yes—
130C2×#106C2314yes—
131C2×#106C2314yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#102C2314yes—
135C2×#102C2314yes—
136C2×C22⋊C8—44no—
137C2×C22⋊C8C2312yes—
138C2×C22⋊C8C2312yes—
139C2×C22⋊C8C2314yes—
140C2×C22⋊C8C2314yes—
141C2×C22⋊C8C2314yes—
142#400—22no—
143#399—22no—
144#398—22no—
145#398—24no—
146#398—24no—
147#397—22no—
148#397—24no—
149#397—24no—
150#394—22no—
151#394—22no—
152#392—22no—
153#392—22no—
154#390—22no—
155#390—22no—
156#389—22no—
157#389—22no—
158#389C2314yes—
159#389C2314yes—
160#388—22no—
161#388—22no—
162#388—22no—
163#388—22no—
164#386—21no—
165#386—21no—
166#386—22no—
167#386—22no—
168#386—22no—
169#386—22no—
170#385—21no—
171#385—21no—
172#385—22no—
173#385—22no—
174#383—21no—
175#382C2312yes—
176#381—21no—
177#381—24no—
178#381—24no—
179#380—22no—
180#380—22no—
181#380—22no—
182#380C2312yes—
183#380C2312yes—
184#380C2312yes—
185#380C2312yes—
186#380C2314yes—
187#380C2314yes—
188#380C2314yes—
189#380C2311yescharacteristic
190#380C2311yescharacteristic
191#379—21no—
192#379—21no—
193C4×C4⋊C4C2312yes—
194C4×C22⋊C4—21no—
195#376—21no—
196C2×#95—21no—
197C2×#95—22no—
198C2×#95—22no—
199#340C2314yes—
200#340C2314yes—
201#340C2314yes—
202#340C2314yes—
203#340C2314yes—
204#340C2314yes—
205#340C2314yes—
206#340C2314yes—
207#336C2314yes—
208#336C2314yes—
209C25C2×D411yescharacteristic
210C25C2×D411yescharacteristic
211C22×Q8C2411yescharacteristic
212C22×D4—24no—
213C22×D4—24no—
214C22×D4—21no—
215C22×D4—22no—
216C22×D4—22no—
217C22×D4—42no—
218C22×D4C2×D412yes—
219C22×D4C2×D412yes—
220C22×D4C2411yescharacteristic
221C23×C4—24no—
222C23×C4—24no—
223C23×C4—22no—
224C23×C4—22no—
225C23×C4—24no—
226C23×C4—21no—
227C23×C4—44no—
228C23×C4—44no—
229C23×C4—44no—
230C23×C4—44no—
231C23×C4—44no—
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D412yes—
237C23×C4C2×D412yes—
238C23×C4C2412yes—
239C23×C4C2412yes—
240C23×C4C2411yescharacteristic
241C22×C8—44no—
242C22×C8—44no—
243C22×C8—44no—
244C22×C8C2414yes—
245C22×C8C2414yes—
246#126—44no—
247#125—42no—
248#124—42no—
249#123—44no—
250#123—416no—
251#123—416no—
252C22⋊Q8—24no—
253C22⋊Q8—44no—
254C22⋊Q8—44no—
255#121—24no—
256#121—44no—
257#121—44no—
258#120—44no—
259#120—44no—
260C4×Q8—42no—
261C4×D4—42no—
262C4×D4—44no—
263C4×D4—44no—
264C4×D4—42no—
265C4×D4—42no—
266C4×D4—42no—
267C4×D4—42no—
268C4×D4—44no—
269#117—44no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—42no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—48no—
289C2×C4⋊C4—48no—
290C2×C4⋊C4C2×D412yes—
291C2×C4⋊C4C2×D412yes—
292C2×C4⋊C4C2414yes—
293C2×C4⋊C4C2414yes—
294C2×C4⋊C4C2411yescharacteristic
295C2×C4⋊C4C2411yescharacteristic
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2412yes—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—21no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—21no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—21no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—21no—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—42no—
317C2×C22⋊C4—42no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—48no—
320C2×C22⋊C4C2×D412yes—
321C2×C22⋊C4C2×D412yes—
322C2×C22⋊C4C2×D412yes—
323C2×C22⋊C4C2×D412yes—
324C2×C42—24no—
325C2×C42—21no—
326C2×C42—22no—
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D411yescharacteristic
329C2×C42C2411yescharacteristic
330C2×C42C2411yescharacteristic
331#107—44no—
332#107—44no—
333#106—44no—
334#106—44no—
335#103—44no—
336#103—44no—
337#102—44no—
338#102—44no—
339C22⋊C8—24no—
340C22⋊C8—24no—
341C22⋊C8—44no—
342C22⋊C8—44no—
343C22⋊C8—44no—
344#95—24no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95—24no—
349#95—21no—
350#95—21no—
351#95—22no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—22no—
358C24—22no—
359C24—22no—
360C24—22no—
361C24—22no—
362C24—21no—
363C24—42no—
364C24—42no—
365C24—42no—
366C24—42no—
367C24#13514yes—
368C24AGL(1,ℤ/8)14yes—
369C24C22×D412yes—
370C24C22×D412yes—
371C24C22×D411yescharacteristic
372C2×Q8—22no—
373C2×Q8—24no—
374C2×D4—24no—
375C2×D4—24no—
376C2×D4—24no—
377C2×D4—24no—
378C2×D4—22no—
379C2×D4—24no—
380C2×D4—42no—
381C2×D4—42no—
382C2×D4—416no—
383C2×D4—42no—
384C2×D4—42no—
385C2×D4—42no—
386C2×D4—44no—
387C2×D4—44no—
388C2×D4—82no—
389C2×D4—82no—
390C2×D4—82no—
391C2×D4—82no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—24no—
415C22×C4—21no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—416no—
424C22×C4—42no—
425C22×C4—48no—
426C22×C4—44no—
427C22×C4—48no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—48no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—88no—
435C22×C4—82no—
436C22×C4—816no—
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D412yes—
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D412yes—
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4H(5,𝔽2)14yes—
445C22×C4C2511yescharacteristic
446C2×C8—24no—
447C2×C8—24no—
448C2×C8—24no—
449C2×C8—24no—
450C2×C8—24no—
451C2×C8—24no—
452C2×C8—88no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—24no—
458C4⋊C4—24no—
459C4⋊C4—24no—
460C4⋊C4—24no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—44no—
464C4⋊C4—48no—
465C4⋊C4—48no—
466C4⋊C4—48no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—44no—
471C4⋊C4—42no—
472C4⋊C4—82no—
473C4⋊C4—82no—
474C22⋊C4—42no—
475C22⋊C4—42no—
476C22⋊C4—42no—
477C22⋊C4—42no—
478C22⋊C4—42no—
479C22⋊C4—42no—
480C22⋊C4—42no—
481C22⋊C4—48no—
482C22⋊C4—42no—
483C22⋊C4—44no—
484C22⋊C4—42no—
485C22⋊C4—48no—
486C22⋊C4—48no—
487C22⋊C4—44no—
488C22⋊C4—42no—
489C22⋊C4—42no—
490C22⋊C4—82no—
491C22⋊C4—82no—
492C22⋊C4—82no—
493C22⋊C4—82no—
494C22⋊C4—82no—
495C22⋊C4—82no—
496C42—22no—
497C42—22no—
498C42—22no—
499C42—22no—
500C42—42no—
501C42—82no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—28no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—28no—
518C23—48no—
519C23—48no—
520C23—44no—
521C23—48no—
522C23—44no—
523C23—42no—
524C23—41no—
525C23—42no—
526C23—41no—
527C23—41no—
528C23—82no—
529C23—84no—
530C23—82no—
531C23#53412yes—
532C23C2×#13514yes—
533C23C2×#13512yes—
534C23C2×AGL(1,ℤ/8)14yes—
535C23C2×AGL(1,ℤ/8)12yes—
536C23C23×D411yescharacteristic
537Q8—44no—
538D4—48no—
539D4—48no—
540D4—44no—
541D4—84no—
542D4—164no—
543D4—164no—
544C2×C4—22no—
545C2×C4—28no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—48no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—84no—
589C2×C4—82no—
590C2×C4—82no—
591C2×C4—84no—
592C2×C4—88no—
593C2×C4—816no—
594C2×C4—88no—
595C2×C4#53412yes—
596C2×C4C2×H(5,𝔽2)12yes—
597C2×C4#58514yes—
598C8—44no—
599C8—44no—
600C22—24no—
601C22—24no—
602C22—22no—
603C22—22no—
604C22—24no—
605C22—24no—
606C22—24no—
607C22—24no—
608C22—24no—
609C22—24no—
610C22—416no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—88no—
620C22—82no—
621C22—82no—
622C22—81no—
623C22—81no—
624C22—81no—
625C22#318814yes—
626C22C2×#53411yescharacteristic
627C22#348512yes—
628C22C22×#13511yescharacteristic
629C22C22×AGL(1,ℤ/8)11yescharacteristic
630C22#358612yes—
631C4—22no—
632C4—41no—
633C4—41no—
634C4—41no—
635C4—41no—
636C4—42no—
637C4—42no—
638C4—82no—
639C4—84no—
640C4—82no—
641C4—84no—
642C4—81no—
643C4—81no—
644C4—81no—
645C4#353314yes—
646C2—24no—
647C2—44no—
648C2—44no—
649C2—161no—
650C2#3695511yescharacteristic
651C2C2×#318811yescharacteristic
652C2#6075411yescharacteristic
653C2#6169612yes—
654C2#6170612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35102C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2486C2semidirect
3. #32982C2semidirect
4. #32978C2semidirect
5. #32264C2semidirect
6. #32032C2semidirect
7. #32006C2semidirect
8. #32002C2semidirect
9. #29895C2semidirect
10. #29892C2semidirect
11. #27616C2semidirect
12. #24551C2semidirect
13. #24550C2semidirect
14. #24066C2semidirect
15. #21738C2semidirect
48. #2101C22semidirect
49. #2085C22semidirect
50. #2077C22semidirect
51. #2077C22semidirect
52. #2076C22semidirect
53. #2075C22semidirect
54. #2059C22semidirect
55. #2042C22semidirect
56. #2036C22semidirect
57. #2035C22semidirect
58. #2034C22semidirect
59. #2033C22semidirect
60. #1921C22semidirect
61. #1920C22semidirect
62. #1893C22semidirect
63. #1892C22semidirect
65. #1874C22semidirect
67. #1872C22semidirect
70. #1864C22semidirect
71. #1863C22semidirect
73. #1853C22semidirect
75. #1852C22semidirect
79. #1808C22semidirect
80. #1807C22semidirect

Maximal subgroups

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

subgroupindex
#351022
C2×#24862
#329822
#329782
#322642
#320322
#320062
#320022
#298952
#298922
#276162
#245512
#245502
#240662
#217382