Finite Group Databasebeta

← #6506497

Subgroups — #6506497

Order 512.

2508 conjugacy classes of subgroups.

The subgroup lattice has 643 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37425C211yesmaximal, characteristic, complemented
2#35527C211yesmaximal, characteristic
3#29115C212yesmaximal, complemented
4#29114C212yesmaximal, complemented
5#27279C214yesmaximal, complemented
6#27276C214yesmaximal, complemented
7#27264C214yesmaximal, complemented
8#27263C214yesmaximal, complemented
9#25867C212yesmaximal, complemented
10#25860C212yesmaximal, complemented
11#21770C212yesmaximal, complemented
12#20602C212yesmaximal, complemented
13#20532C211yesmaximal, characteristic, complemented
14C22×H(5,𝔽2)C2211yescharacteristic
15C2×#535C2211yescharacteristic
16C2×#535C2211yescharacteristic
17C22×#127C2211yescharacteristic
18C2×#490C2214yescomplemented
19C2×#490C2212yescomplemented
20C2×#488C2212yescomplemented
21C2×#417C2212yescomplemented
22C2×#418C2212yescomplemented
23C2×#416C2212yescomplemented
24C22×#102C2212yescomplemented
25C2×#408C2212yes—
26#2851C2212yescomplemented
27#2809—22no—
28#2780—24no—
29#2748—22no—
30#2629C2212yescomplemented
31#2442C2214yescomplemented
32#2441C2214yescomplemented
33#2433—22no—
34#2433C2214yes—
35#2431—22no—
36#2431C2214yes—
37#2403—24no—
38#2311C2212yes—
39#2309C2211yescharacteristic
40#2297C2211yescharacteristic
41#2297C2211yescharacteristic
42#2074C2212yescomplemented
43#2073C2212yescomplemented
44#2071C2214yescomplemented
45#2042C2214yescomplemented
46#2041C2214yescomplemented
47#2041C2214yescomplemented
48#2041C2214yescomplemented
49#2026C2212yescomplemented
50#2025C2214yescomplemented
51#2025C2212yescomplemented
52#1961C2214yescomplemented
53#1959C2212yescomplemented
54#1957C2212yescomplemented
55#1936C2214yescomplemented
56#1936C2214yescomplemented
57#1935C2214yescomplemented
58#1934C2214yescomplemented
59#1895C2214yescomplemented
60#1893C2214yescomplemented
61#1875C2214yescomplemented
62#1874C2214yescomplemented
63#1825C2214yescomplemented
64#1788C2214yes—
65#1788C2214yescomplemented
66#1786C2214yes—
67#1786C2214yescomplemented
68#1786C2214yescomplemented
69#1753C2214yescomplemented
70#1752C2214yescomplemented
71#1750C2212yescomplemented
72#1749C2212yescomplemented
73C2×H(5,𝔽2)—28no—
74C2×H(5,𝔽2)—42no—
75C2×H(5,𝔽2)—44no—
76C2×H(5,𝔽2)—42no—
77C22×#41—21no—
78C22×#41C2312yes—
79C22×#41C2311yescharacteristic
80C22×#41C2311yescharacteristic
81C23×D4—22no—
82C23×D4—21no—
83C23×D4—22no—
84C23×D4C2311yescharacteristic
85C23×D4C2312yes—
86C23×D4C2311yescharacteristic
87C22×M16C2312yes—
88C23×C8C2312yes—
89#535—28no—
90#535—42no—
91#535—42no—
92C2×#127—21no—
93C2×#127—22no—
94C2×#127—44no—
95C2×#127C2314yes—
96C2×#127C2312yes—
97C2×#125C2312yes—
98C2×#125C2312yes—
99C2×#124—21no—
100C2×#124C2312yes—
101C2×#121—22no—
102C2×#121C2314yes—
103C2×#120—22no—
104C2×#120—22no—
105C2×#120—24no—
106C2×#120—24no—
107C2×#117C2311yescharacteristic
108C2×#117C2312yes—
109C2×#117C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C42C2311yescharacteristic
112#490—44no—
113#490—48no—
114#488—44no—
115#418—44no—
116#417—44no—
117#416—44no—
118C2×#103C2314yes—
119C2×#103C2314yes—
120C2×#102—44no—
121C2×#102C2314yes—
122C2×#102C2314yes—
123C2×#102C2314yes—
124C2×#102C2314yes—
125C2×#102C2314yes—
126C2×#102C2314yes—
127C2×#102C2314yes—
128C2×#102C2314yes—
129C2×#102C2314yes—
130C2×#102C2314yes—
131#408—28no—
132C2×C22⋊C8C2314yes—
133C2×C22⋊C8C2314yes—
134C2×#97C2314yes—
135C2×#97C2312yes—
136C2×C4×C8C2312yes—
137#400—24no—
138#399—24no—
139#398—24no—
140#398—24no—
141#396—24no—
142#396—24no—
143#392—24no—
144#392—24no—
145#392—24no—
146#392—24no—
147#391—22no—
148#391D412yes—
149#391C2312yes—
150#390—22no—
151#390—22no—
152#390D412yes—
153#390C2314yes—
154#390C2312yes—
155#389—24no—
156#389C2314yes—
157#389C2314yes—
158#387—24no—
159#387—24no—
160#386—22no—
161#386—22no—
162#386—22no—
163#386—24no—
164#386—24no—
165#386D414yes—
166#385—24no—
167#385—24no—
168#384C2312yes—
169#382C2312yes—
170#380—24no—
171#380C2314yes—
172#380C2314yes—
173#379—22no—
174C4×C22⋊C4—22no—
175C4×C22⋊C4C2314yes—
176#340C2314yes—
177#340C2314yes—
178#340C2314yes—
179#340C2314yes—
180#336C2314yes—
181#336C2312yes—
182#336C2312yes—
183C25—21no—
184C25—22no—
185C25—21no—
186H(5,𝔽2)—432no—
187H(5,𝔽2)—432no—
188C2×#41—22no—
189C2×#41—22no—
190C2×#41—24no—
191C2×#41—24no—
192C2×#41—24no—
193C2×#41—24no—
194C2×#41—24no—
195C2×#41—42no—
196C2×#41—41no—
197C2×#41—41no—
198C2×#41—42no—
199C2×#41—42no—
200C2×#41—42no—
201C2×#41—44no—
202C2×#41—44no—
203C2×#41—44no—
204C2×#41#4118yes—
205C22×Q8C2×D411yescharacteristic
206C22×Q8C2411yescharacteristic
207C22×D4—24no—
208C22×D4—24no—
209C22×D4—21no—
210C22×D4—24no—
211C22×D4—22no—
212C22×D4—24no—
213C22×D4—21no—
214C22×D4—21no—
215C22×D4—22no—
216C22×D4—24no—
217C22×D4—24no—
218C22×D4—24no—
219C22×D4—24no—
220C22×D4—24no—
221C22×D4—24no—
222C22×D4—24no—
223C22×D4—42no—
224C22×D4—42no—
225C22×D4—41no—
226C22×D4—41no—
227C22×D4—42no—
228C22×D4—41no—
229C22×D4—41no—
230C22×D4—44no—
231C22×D4—48no—
232C22×D4—44no—
233C22×D4—48no—
234C22×D4—48no—
235C22×D4—44no—
236C22×D4—48no—
237C22×D4—88no—
238C22×D4C2×D412yes—
239C22×D4C2×D411yescharacteristic
240C22×D4C2412yes—
241C22×D4C2411yescharacteristic
242C22×D4C2412yes—
243C23×C4—21no—
244C23×C4—22no—
245C23×C4—22no—
246C23×C4C2×D412yes—
247C23×C4C2412yes—
248C23×C4C2411yescharacteristic
249C2×M16—24no—
250C2×M16—42no—
251C2×M16—42no—
252C22×C8—24no—
253C22×C8—42no—
254C22×C8—42no—
255C22×C8C2414yes—
256C22×C8C2414yes—
257#127—28no—
258#127—24no—
259#127—44no—
260#127—42no—
261#127—48no—
262#127—44no—
263#125—42no—
264#125—44no—
265#125—42no—
266#124—24no—
267#124—42no—
268#124—44no—
269#121—28no—
270#121—28no—
271#121—48no—
272#120—48no—
273#120—48no—
274#120—88no—
275#120—88no—
276#117—24no—
277#117—41no—
278#117—41no—
279#117—42no—
280#117—42no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—42no—
288C2×C4⋊C4—42no—
289C2×C4⋊C4C2×D412yes—
290C2×C4⋊C4C2414yes—
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2414yes—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4C2×D412yes—
303C2×C22⋊C4C2×D414yes—
304C2×C22⋊C4C2×D414yes—
305C2×C42—21no—
306C2×C42—22no—
307C2×C42—42no—
308C2×C42C2×D412yes—
309C2×C42C2412yes—
310C2×C42C2412yes—
311#103—28no—
312#103—44no—
313#102—28no—
314#102—28no—
315#102—28no—
316#102—44no—
317#102—44no—
318#102—44no—
319#102—44no—
320#102—44no—
321#102—44no—
322#102—44no—
323C22⋊C8—28no—
324C22⋊C8—28no—
325#97—44no—
326#97—42no—
327C4×C8—42no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95C2×D414yes—
335#95C2×D414yes—
336C24—24no—
337C24—21no—
338C24—22no—
339C24—22no—
340C24—44no—
341C24—44no—
342C24—42no—
343C24—42no—
344C24—42no—
345C24—44no—
346C24—42no—
347C24—42no—
348C24—84no—
349C24—84no—
350C24—84no—
351C24—84no—
352C24C22×D411yescharacteristic
353C24C22×D412yes—
354#41—216no—
355#41—48no—
356#41—48no—
357#41—48no—
358#41—48no—
359#41—48no—
360#41—48no—
361#41—48no—
362#41—432no—
363C2×Q8—21no—
364C2×Q8—22no—
365C2×Q8—22no—
366C2×Q8—41no—
367C2×Q8—41no—
368C2×Q8#12414yes—
369C2×Q8H(5,𝔽2)12yes—
370C2×D4—216no—
371C2×D4—24no—
372C2×D4—24no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—22no—
376C2×D4—24no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—21no—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—24no—
385C2×D4—48no—
386C2×D4—48no—
387C2×D4—48no—
388C2×D4—48no—
389C2×D4—41no—
390C2×D4—42no—
391C2×D4—41no—
392C2×D4—42no—
393C2×D4—44no—
394C2×D4—44no—
395C2×D4—44no—
396C2×D4—42no—
397C2×D4—48no—
398C2×D4—48no—
399C2×D4—48no—
400C2×D4—48no—
401C2×D4—48no—
402C2×D4—42no—
403C2×D4—42no—
404C2×D4—48no—
405C2×D4—44no—
406C2×D4—42no—
407C2×D4—432no—
408C2×D4—464no—
409C2×D4—42no—
410C2×D4—44no—
411C2×D4—48no—
412C2×D4—42no—
413C2×D4—44no—
414C2×D4—42no—
415C2×D4—84no—
416C2×D4—84no—
417C2×D4—84no—
418C2×D4—84no—
419C2×D4—84no—
420C2×D4—84no—
421C2×D4—832no—
422C2×D4—84no—
423C2×D4—84no—
424C2×D4#12418yes—
425C2×D4#12414yes—
426C2×D4H(5,𝔽2)12yes—
427C2×D4H(5,𝔽2)14yes—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—24no—
436C22×C4—21no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—42no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—44no—
447C22×C4—82no—
448C22×C4—84no—
449C22×C4C22×D414yes—
450C22×C4C22×D414yes—
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D412yes—
453C22×C4C2×#4114yes—
454C22×C4C2×#4114yes—
455C22×C4C2×#4114yes—
456C22×C4C2511yescharacteristic
457M16—84no—
458C2×C8—24no—
459C2×C8—24no—
460C2×C8—24no—
461C2×C8—24no—
462C2×C8—24no—
463C2×C8—24no—
464C2×C8—84no—
465C4⋊C4—24no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—42no—
471C4⋊C4—84no—
472C4⋊C4—84no—
473C4⋊C4—84no—
474C4⋊C4—82no—
475C4⋊C4—84no—
476C4⋊C4—84no—
477C22⋊C4—42no—
478C22⋊C4—44no—
479C22⋊C4—44no—
480C22⋊C4—48no—
481C22⋊C4—48no—
482C22⋊C4—42no—
483C22⋊C4—48no—
484C22⋊C4—84no—
485C22⋊C4—84no—
486C22⋊C4—84no—
487C22⋊C4—84no—
488C22⋊C4—82no—
489C42—22no—
490C42—24no—
491C42—22no—
492C42—42no—
493C42—42no—
494C42—82no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—21no—
499C23—22no—
500C23—21no—
501C23—22no—
502C23—22no—
503C23—24no—
504C23—416no—
505C23—48no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—41no—
510C23—42no—
511C23—41no—
512C23—42no—
513C23—44no—
514C23—44no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—84no—
519C23—88no—
520C23—88no—
521C23—168no—
522C23—164no—
523C23—164no—
524C23—168no—
525C23—164no—
526C23—164no—
527C23—164no—
528C23—164no—
529C23C2×#12412yes—
530C23C2×#12414yes—
531C23C23×D411yescharacteristic
532Q8—24no—
533Q8—44no—
534Q8—42no—
535D4—24no—
536D4—24no—
537D4—24no—
538D4—44no—
539D4—48no—
540D4—48no—
541D4—416no—
542D4—48no—
543D4—416no—
544D4—42no—
545D4—42no—
546D4—42no—
547D4—44no—
548D4—44no—
549D4—84no—
550D4—816no—
551D4—816no—
552D4—168no—
553D4—168no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—48no—
572C2×C4—416no—
573C2×C4—48no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—164no—
585C2×C4C2×#12412yes—
586C2×C4C2×#12414yes—
587C2×C4C22×#4112yes—
588C2×C4C2×H(5,𝔽2)11yescharacteristic
589C2×C4C2×H(5,𝔽2)11yescharacteristic
590C2×C4C2×H(5,𝔽2)12yes—
591C8—44no—
592C8—44no—
593C22—22no—
594C22—24no—
595C22—28no—
596C22—24no—
597C22—42no—
598C22—44no—
599C22—42no—
600C22—42no—
601C22—42no—
602C22—42no—
603C22—44no—
604C22—48no—
605C22—44no—
606C22—82no—
607C22—82no—
608C22—84no—
609C22—82no—
610C22—82no—
611C22—82no—
612C22—82no—
613C22—82no—
614C22—84no—
615C22—84no—
616C22—168no—
617C22—1616no—
618C22—168no—
619C22C22×#12411yescharacteristic
620C22#348411yescharacteristic
621C22#348411yescharacteristic
622C22#358514yes—
623C4—22no—
624C4—24no—
625C4—42no—
626C4—44no—
627C4—44no—
628C4—84no—
629C4—82no—
630C4—84no—
631C4#349012yes—
632C4#349012yes—
633C2—44no—
634C2—42no—
635C2—84no—
636C2—82no—
637C2—162no—
638C2#3536711yescharacteristic
639C2#6062612yes—
640C2#6164312yes—
641C2#6167512yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35527C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2297 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #37425C2semidirect
3. #29115C2semidirect
4. #29114C2semidirect
5. #27279C2semidirect
6. #27276C2semidirect
7. #27264C2semidirect
8. #27263C2semidirect
9. #25867C2semidirect
10. #25860C2semidirect
11. #21770C2semidirect
12. #20602C2semidirect
13. #20532C2semidirect
18. C2×#490C22semidirect
19. C2×#490C22semidirect
20. C2×#488C22semidirect
21. C2×#417C22semidirect
22. C2×#418C22semidirect
23. C2×#416C22semidirect
24. C22×#102C22semidirect
26. #2851C22semidirect
30. #2629C22semidirect
31. #2442C22semidirect
32. #2441C22semidirect
42. #2074C22semidirect
43. #2073C22semidirect
44. #2071C22semidirect
45. #2042C22semidirect
46. #2041C22semidirect
47. #2041C22semidirect
48. #2041C22semidirect
49. #2026C22semidirect
50. #2025C22semidirect
51. #2025C22semidirect
52. #1961C22semidirect
53. #1959C22semidirect
54. #1957C22semidirect
55. #1936C22semidirect
56. #1936C22semidirect
57. #1935C22semidirect
58. #1934C22semidirect
59. #1895C22semidirect
60. #1893C22semidirect
61. #1875C22semidirect
62. #1874C22semidirect
63. #1825C22semidirect
65. #1788C22semidirect
67. #1786C22semidirect
68. #1786C22semidirect
69. #1753C22semidirect
70. #1752C22semidirect
71. #1750C22semidirect
72. #1749C22semidirect

Maximal subgroups

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

subgroupindex
#374252
#355272
#291152
#291142
#272792
#272762
#272642
#272632
#258672
#258602
#217702
#206022
#205322