Finite Group Databasebeta

← #6644833

Subgroups — #6644833

Order 512.

1853 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35031C211yesmaximal, characteristic
2C2×#2547C211yesmaximal, characteristic, complemented
3#33099C211yesmaximal, characteristic, complemented
4#33095C211yesmaximal, characteristic, complemented
5#32267C214yesmaximal, complemented
6#31333C214yesmaximal, complemented
7#31331C214yesmaximal, complemented
8#31269C214yesmaximal, complemented
9#29641C211yesmaximal, characteristic, complemented
10#29638C211yesmaximal, characteristic, complemented
11#27602C212yesmaximal, complemented
12#24551C212yesmaximal, complemented
13#24550C212yesmaximal, complemented
14#24004C212yesmaximal, complemented
15#21738C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C22×#124—21no—
18C2×#525C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C2×#483C2212yes—
21C2×#480C2212yes—
22C22×C22⋊C8C2212yes—
23#2855C2211yescharacteristic
24#2854C2211yescharacteristic
25#2848C2211yescharacteristic
26#2847C2211yescharacteristic
27#2622—21no—
28#2613—21no—
29#2602—22no—
30#2600—22no—
31#2547—22no—
32#2547—22no—
33#2547C2214yes—
34#2547C2214yes—
35#2485C2212yes—
36#2415C2212yes—
37#2414C2212yes—
38#2407C2212yes—
39C2×#397—21no—
40C2×#397—21no—
41#2322—21no—
42#2321—21no—
43C2×#391—21no—
44#2280C2211yescharacteristic
45C2×#380C2212yes—
46C2×#380C2212yes—
47C2×C4×C4⋊C4C2211yescharacteristic
48#2077C2214yescomplemented
49#2077C2214yescomplemented
50#2076C2214yescomplemented
51#2075C2214yescomplemented
52#2058C2214yescomplemented
53#2057C2214yescomplemented
54#2036C2214yescomplemented
55#2035C2214yescomplemented
56#2034C2214yescomplemented
57#2033C2214yescomplemented
58#2024C2214yescomplemented
59#2023C2214yescomplemented
60#1928C2214yescomplemented
61#1926C2214yescomplemented
62#1921C2214yescomplemented
63#1919C2214yescomplemented
64#1893C2214yescomplemented
65#1892C2214yescomplemented
66#1874C2212yes—
67#1872C2212yes—
68#1869C2214yes—
69#1869C2214yes—
70#1854C2214yescomplemented
71#1854C2214yescomplemented
72#1853C2212yes—
73#1852C2212yes—
74#1825C2214yescomplemented
75#1823C2212yes—
76#1823C2212yes—
77#1817C2211yescharacteristic
78#1817C2211yescharacteristic
79#1808C2214yescomplemented
80#1807C2214yescomplemented
81#1788C2214yescomplemented
82#1779C2212yes—
83C22×#41C2311yescharacteristic
84C23×Q8—21no—
85C23×D4D411yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C23×C8C2314yes—
89C2×C4⋊Q8—22no—
90C2×C4⋊Q8—44no—
91C2×C4⋊Q8C2314yes—
92C2×#124—24no—
93C2×#124—24no—
94C2×#124—48no—
95#525—44no—
96C2×#123—21no—
97C2×#123—21no—
98C2×C22⋊Q8C2311yescharacteristic
99C2×C22⋊Q8C2312yes—
100C2×C22⋊Q8C2311yescharacteristic
101C2×#121C2311yescharacteristic
102C2×#121C2312yes—
103C2×#121C2311yescharacteristic
104C2×#120—21no—
105C2×#120—21no—
106C2×C4×D4—21no—
107C2×C4×D4—21no—
108C2×C4×D4—44no—
109C2×C4×D4D412yes—
110C2×C4×D4D412yes—
111C2×C4×D4C2312yes—
112C2×C4×D4C2312yes—
113C2×#117C2312yes—
114C2×#117C2311yescharacteristic
115C22×C4⋊C4—21no—
116C22×C4⋊C4D411yescharacteristic
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C22⋊C4—22no—
120C22×C22⋊C4—22no—
121C22×C22⋊C4D411yescharacteristic
122C22×C22⋊C4D411yescharacteristic
123C22×C42—21no—
124C22×C42C2311yescharacteristic
125#483—44no—
126#480—44no—
127C2×#107C2314yes—
128C2×#107C2314yes—
129C2×#106C2314yes—
130C2×#106C2314yes—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×#102C2314yes—
134C2×#102C2314yes—
135C2×C22⋊C8—44no—
136C2×C22⋊C8C2312yes—
137C2×C22⋊C8C2312yes—
138C2×C22⋊C8C2314yes—
139C2×C22⋊C8C2314yes—
140C2×C22⋊C8C2314yes—
141#397—22no—
142#397—22no—
143#397—24no—
144#397—24no—
145#397—24no—
146#397—24no—
147#394—22no—
148#394—22no—
149#394—22no—
150#394—22no—
151#392—22no—
152#392—22no—
153#391—24no—
154#391—24no—
155#391C2312yes—
156#390—22no—
157#390C2312yes—
158#389—22no—
159#389C2312yes—
160#389C2312yes—
161#388—22no—
162#388—22no—
163#388—22no—
164#388—22no—
165#387—22no—
166#387—22no—
167#386—21no—
168#386—21no—
169#386—22no—
170#386—22no—
171#385—21no—
172#385—21no—
173#385—22no—
174#385—22no—
175#381—21no—
176#381—21no—
177#380—22no—
178#380—22no—
179#380—22no—
180#380—22no—
181#380C2312yes—
182#380C2314yes—
183#380C2312yes—
184#380C2314yes—
185#380C2312yes—
186#380C2314yes—
187#380C2311yescharacteristic
188#380C2311yescharacteristic
189#380C2314yes—
190#379—21no—
191#379—21no—
192C4×C4⋊C4—22no—
193C4×C4⋊C4C2314yes—
194C4×C22⋊C4—21no—
195C4×C22⋊C4—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
211C2×#41—42no—
212C2×#41—44no—
213C22×Q8—24no—
214C22×Q8—44no—
215C22×Q8—44no—
216C22×Q8—44no—
217C22×Q8C2411yescharacteristic
218C22×D4—24no—
219C22×D4—21no—
220C22×D4—21no—
221C22×D4—22no—
222C22×D4—21no—
223C22×D4—24no—
224C22×D4—42no—
225C22×D4C2×D412yes—
226C22×D4C2×D412yes—
227C22×D4C2411yescharacteristic
228C23×C4—24no—
229C23×C4—24no—
230C23×C4—22no—
231C23×C4—22no—
232C23×C4—21no—
233C23×C4—44no—
234C23×C4—44no—
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D412yes—
242C23×C4C2411yescharacteristic
243C23×C4C2411yescharacteristic
244C23×C4C2412yes—
245C23×C4C2411yescharacteristic
246C22×C8—44no—
247C22×C8—44no—
248C22×C8—44no—
249C22×C8C2414yes—
250C22×C8C2414yes—
251C4⋊Q8—44no—
252C4⋊Q8—44no—
253C4⋊Q8—48no—
254#124—416no—
255#124—416no—
256#123—44no—
257#123—44no—
258C22⋊Q8—24no—
259C22⋊Q8—42no—
260C22⋊Q8—44no—
261#121—24no—
262#121—42no—
263#121—44no—
264#120—44no—
265#120—44no—
266C4×D4—44no—
267C4×D4—44no—
268C4×D4—42no—
269C4×D4—42no—
270C4×D4—42no—
271C4×D4—42no—
272C4×D4—44no—
273C4×D4—44no—
274#117—42no—
275#117—44no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—42no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4C2×D412yes—
290C2×C4⋊C4C2×D412yes—
291C2×C4⋊C4C2414yes—
292C2×C4⋊C4C2414yes—
293C2×C4⋊C4C2411yescharacteristic
294C2×C4⋊C4C2411yescharacteristic
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2412yes—
297C2×C22⋊C4—24no—
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—21no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—21no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—21no—
309C2×C22⋊C4—22no—
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—48no—
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—24no—
327C2×C42—22no—
328C2×C42—44no—
329C2×C42C2×D412yes—
330C2×C42C2412yes—
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—22no—
351#95—21no—
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—22no—
363C24—21no—
364C24—42no—
365C24—42no—
366C24—42no—
367C24—42no—
368C24AGL(1,ℤ/8)14yes—
369C24C22×D411yescharacteristic
370C24C22×D411yescharacteristic
371C24C22×D412yes—
372C24C22×D411yescharacteristic
373#41—88no—
374C2×Q8—22no—
375C2×Q8—24no—
376C2×Q8—48no—
377C2×Q8—48no—
378C2×Q8—48no—
379C2×Q8—816no—
380C2×D4—24no—
381C2×D4—24no—
382C2×D4—24no—
383C2×D4—24no—
384C2×D4—22no—
385C2×D4—24no—
386C2×D4—42no—
387C2×D4—42no—
388C2×D4—416no—
389C2×D4—42no—
390C2×D4—42no—
391C2×D4—42no—
392C2×D4—42no—
393C2×D4—42no—
394C2×D4—44no—
395C2×D4—82no—
396C2×D4—82no—
397C2×D4—82no—
398C2×D4—82no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—24no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—88no—
438C22×C4—88no—
439C22×C4—82no—
440C22×C4#13514yes—
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D412yes—
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D412yes—
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D412yes—
447C22×C4H(5,𝔽2)14yes—
448C22×C4C2511yescharacteristic
449C2×C8—24no—
450C2×C8—24no—
451C2×C8—24no—
452C2×C8—24no—
453C2×C8—24no—
454C2×C8—24no—
455C2×C8—88no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—24no—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—44no—
467C4⋊C4—48no—
468C4⋊C4—42no—
469C4⋊C4—44no—
470C4⋊C4—42no—
471C4⋊C4—44no—
472C4⋊C4—82no—
473C4⋊C4—82no—
474C22⋊C4—42no—
475C22⋊C4—42no—
476C22⋊C4—42no—
477C22⋊C4—42no—
478C22⋊C4—42no—
479C22⋊C4—48no—
480C22⋊C4—48no—
481C22⋊C4—42no—
482C22⋊C4—42no—
483C22⋊C4—42no—
484C22⋊C4—42no—
485C22⋊C4—48no—
486C22⋊C4—48no—
487C22⋊C4—44no—
488C22⋊C4—42no—
489C22⋊C4—42no—
490C22⋊C4—44no—
491C22⋊C4—82no—
492C22⋊C4—82no—
493C22⋊C4—82no—
494C22⋊C4—82no—
495C22⋊C4—82no—
496C22⋊C4—82no—
497C42—24no—
498C42—22no—
499C42—22no—
500C42—48no—
501C42—42no—
502C42—82no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—28no—
512C23—22no—
513C23—22no—
514C23—21no—
515C23—21no—
516C23—22no—
517C23—22no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—48no—
523C23—48no—
524C23—44no—
525C23—42no—
526C23—41no—
527C23—42no—
528C23—41no—
529C23—41no—
530C23—82no—
531C23—82no—
532C23—84no—
533C23—84no—
534C23—82no—
535C23—82no—
536C23#53412yes—
537C23C2×#13512yes—
538C23C2×AGL(1,ℤ/8)14yes—
539C23C2×AGL(1,ℤ/8)12yes—
540C23C23×D411yescharacteristic
541Q8—44no—
542Q8—816no—
543D4—48no—
544D4—48no—
545D4—44no—
546D4—84no—
547D4—84no—
548D4—84no—
549D4—164no—
550C2×C4—28no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—28no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—84no—
590C2×C4—82no—
591C2×C4—82no—
592C2×C4—82no—
593C2×C4—82no—
594C2×C4—84no—
595C2×C4—88no—
596C2×C4—88no—
597C2×C4#53412yes—
598C2×C4C2×#13514yes—
599C2×C4C2×H(5,𝔽2)12yes—
600C2×C4#58514yes—
601C8—44no—
602C8—44no—
603C22—24no—
604C22—24no—
605C22—22no—
606C22—22no—
607C22—24no—
608C22—24no—
609C22—24no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22—41no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22—41no—
624C22—88no—
625C22—88no—
626C22—82no—
627C22—82no—
628C22—81no—
629C22—81no—
630C22—81no—
631C22#318814yes—
632C22C2×#53411yescharacteristic
633C22#348512yes—
634C22C22×#13511yescharacteristic
635C22C22×AGL(1,ℤ/8)11yescharacteristic
636C22#358612yes—
637C4—24no—
638C4—22no—
639C4—42no—
640C4—41no—
641C4—42no—
642C4—41no—
643C4—82no—
644C4—84no—
645C4—82no—
646C4—84no—
647C4—81no—
648C4—81no—
649C4—81no—
650C4#354514yes—
651C2—24no—
652C2—44no—
653C2—81no—
654C2—81no—
655C2—161no—
656C2#3711911yescharacteristic
657C2C2×#318811yescharacteristic
658C2#6075411yescharacteristic
659C2#6170112yes—
660C2#6170612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35031C2
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×#2547C2semidirect
3. #33099C2semidirect
4. #33095C2semidirect
5. #32267C2semidirect
6. #31333C2semidirect
7. #31331C2semidirect
8. #31269C2semidirect
9. #29641C2semidirect
10. #29638C2semidirect
11. #27602C2semidirect
12. #24551C2semidirect
13. #24550C2semidirect
14. #24004C2semidirect
15. #21738C2semidirect
48. #2077C22semidirect
49. #2077C22semidirect
50. #2076C22semidirect
51. #2075C22semidirect
52. #2058C22semidirect
53. #2057C22semidirect
54. #2036C22semidirect
55. #2035C22semidirect
56. #2034C22semidirect
57. #2033C22semidirect
58. #2024C22semidirect
59. #2023C22semidirect
60. #1928C22semidirect
61. #1926C22semidirect
62. #1921C22semidirect
63. #1919C22semidirect
64. #1893C22semidirect
65. #1892C22semidirect
70. #1854C22semidirect
71. #1854C22semidirect
74. #1825C22semidirect
79. #1808C22semidirect
80. #1807C22semidirect
81. #1788C22semidirect

Maximal subgroups

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

subgroupindex
#350312
C2×#25472
#330992
#330952
#322672
#313332
#313312
#312692
#296412
#296382
#276022
#245512
#245502
#240042
#217382