← C2×#72258201 #72274288
Subgroups — C2×#72258201 #72274288
Order 2048.
229944818 conjugacy classes of subgroups.
The subgroup lattice has 110 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C22×#63103 | C2 | 1 | 255 | yes | maximal, complemented |
| 2 | #72258201 | C2 | 1 | 512 | yes | maximal, direct product factor |
| 3 | C2×2−1+8 | C2 | 1 | 120 | yes | maximal, complemented |
| 4 | C2×H(9,𝔽2) | C2 | 1 | 136 | yes | maximal, complemented |
| 5 | 2−1+8 | C22 | 1 | 30720 | yes | complemented |
| 6 | H(9,𝔽2) | C22 | 1 | 34816 | yes | complemented |
| 7 | C2×#63103 | C22 | 1 | 5440 | yes | complemented |
| 8 | C2×#63103 | C22 | 1 | 16320 | yes | complemented |
| 9 | C2×#63103 | C22 | 1 | 65280 | yes | complemented |
| 10 | C22×2−1+6 | C22 | 1 | 7140 | yes | complemented |
| 11 | C22×H(7,𝔽2) | C22 | 1 | 9180 | yes | complemented |
| 12 | C23×#585 | C22 | 1 | 5355 | yes | complemented |
| 13 | #63103 | C23 | 1 | 696320 | yes | complemented |
| 14 | #63103 | C23 | 1 | 2088960 | yes | complemented |
| 15 | C2×2−1+6 | C23 | 1 | 152320 | yes | complemented |
| 16 | C2×2−1+6 | C23 | 1 | 913920 | yes | complemented |
| 17 | C2×H(7,𝔽2) | C23 | 1 | 195840 | yes | complemented |
| 18 | C2×H(7,𝔽2) | C23 | 1 | 1175040 | yes | complemented |
| 19 | C22×#585 | C23 | 1 | 85680 | yes | complemented |
| 20 | C22×#585 | C23 | 1 | 257040 | yes | complemented |
| 21 | C22×#585 | C23 | 1 | 685440 | yes | complemented |
| 22 | C23×2−1+4 | C23 | 1 | 32130 | yes | complemented |
| 23 | C23×H(5,𝔽2) | C23 | 1 | 53550 | yes | complemented |
| 24 | C24×#41 | C23 | 1 | 11475 | yes | complemented |
| 25 | 2−1+6 | C24 | 1 | 9748480 | yes | complemented |
| 26 | H(7,𝔽2) | C24 | 1 | 12533760 | yes | complemented |
| 27 | C2×#585 | C24 | 1 | 91392 | yes | complemented |
| 28 | C2×#585 | C24 | 1 | 1370880 | yes | complemented |
| 29 | C2×#585 | C24 | 1 | 5483520 | yes | complemented |
| 30 | C2×#585 | C24 | 1 | 16450560 | yes | complemented |
| 31 | C22×2−1+4 | C24 | 1 | 514080 | yes | complemented |
| 32 | C22×2−1+4 | C24 | 1 | 2056320 | yes | complemented |
| 33 | C22×H(5,𝔽2) | C24 | 1 | 856800 | yes | complemented |
| 34 | C22×H(5,𝔽2) | C24 | 1 | 3427200 | yes | complemented |
| 35 | C23×#41 | C24 | 1 | 107100 | yes | complemented |
| 36 | C23×#41 | C24 | 1 | 321300 | yes | complemented |
| 37 | C23×#41 | C24 | 1 | 734400 | yes | complemented |
| 38 | C24×Q8 | C24 | 1 | 11475 | yes | complemented |
| 39 | C24×D4 | C24 | 1 | 34425 | yes | complemented |
| 40 | C25×C4 | C24 | 1 | 2295 | yes | complemented |
| 41 | C26 | C25 | 1 | 2295 | yes | — |
| 42 | #585 | C25 | 1 | 2924544 | yes | complemented |
| 43 | #585 | C25 | 1 | 43868160 | yes | complemented |
| 44 | C2×2−1+4 | C25 | 1 | 548352 | yes | — |
| 45 | C2×2−1+4 | C25 | 1 | 16450560 | yes | complemented |
| 46 | C2×H(5,𝔽2) | C25 | 1 | 913920 | yes | — |
| 47 | C2×H(5,𝔽2) | C25 | 1 | 27417600 | yes | complemented |
| 48 | C22×#41 | C25 | 1 | 85680 | yes | — |
| 49 | C22×#41 | C25 | 1 | 1285200 | yes | — |
| 50 | C22×#41 | C25 | 1 | 3427200 | yes | complemented |
| 51 | C22×#41 | C25 | 1 | 10281600 | yes | complemented |
| 52 | C23×Q8 | C25 | 1 | 107100 | yes | — |
| 53 | C23×Q8 | C25 | 1 | 367200 | yes | complemented |
| 54 | C23×D4 | C25 | 1 | 321300 | yes | — |
| 55 | C23×D4 | C25 | 1 | 1101600 | yes | complemented |
| 56 | C24×C4 | C25 | 1 | 11475 | yes | — |
| 57 | C24×C4 | C25 | 1 | 34425 | yes | — |
| 58 | C24×C4 | C25 | 1 | 73440 | yes | complemented |
| 59 | C25 | — | 16 | 4590 | no | — |
| 60 | C25 | C26 | 1 | 11475 | yes | — |
| 61 | C25 | C26 | 1 | 36720 | yes | — |
| 62 | 2−1+4 | C26 | 1 | 8773632 | yes | — |
| 63 | H(5,𝔽2) | C26 | 1 | 14622720 | yes | — |
| 64 | C2×#41 | C26 | 1 | 5440 | yes | — |
| 65 | C2×#41 | C26 | 1 | 342720 | yes | — |
| 66 | C2×#41 | C26 | 1 | 1370880 | yes | — |
| 67 | C2×#41 | C26 | 1 | 20563200 | yes | — |
| 68 | C22×Q8 | C26 | 1 | 85680 | yes | — |
| 69 | C22×Q8 | C26 | 1 | 1713600 | yes | — |
| 70 | C22×D4 | C26 | 1 | 257040 | yes | — |
| 71 | C22×D4 | C26 | 1 | 5140800 | yes | — |
| 72 | C23×C4 | C26 | 1 | 5355 | yes | — |
| 73 | C23×C4 | C26 | 1 | 80325 | yes | — |
| 74 | C23×C4 | C26 | 1 | 183600 | yes | — |
| 75 | C23×C4 | C26 | 1 | 550800 | yes | — |
| 76 | C24 | — | 8 | 22950 | no | — |
| 77 | C24 | — | 16 | 36720 | no | — |
| 78 | C24 | C27 | 1 | 5355 | yes | — |
| 79 | C24 | C27 | 1 | 91800 | yes | — |
| 80 | #41 | C27 | 1 | 43520 | yes | — |
| 81 | #41 | C27 | 1 | 2741760 | yes | — |
| 82 | C2×Q8 | C27 | 1 | 5440 | yes | — |
| 83 | C2×Q8 | C27 | 1 | 685440 | yes | — |
| 84 | C2×D4 | C27 | 1 | 16320 | yes | — |
| 85 | C2×D4 | C27 | 1 | 2056320 | yes | — |
| 86 | C22×C4 | C27 | 1 | 16065 | yes | — |
| 87 | C22×C4 | C27 | 1 | 42840 | yes | — |
| 88 | C22×C4 | C27 | 1 | 642600 | yes | — |
| 89 | C22×C4 | C27 | 1 | 255 | yes | — |
| 90 | C23 | — | 4 | 10710 | no | — |
| 91 | C23 | — | 8 | 91800 | no | — |
| 92 | C23 | C28 | 1 | 255 | yes | — |
| 93 | C23 | C28 | 1 | 21420 | yes | — |
| 94 | Q8 | C28 | 1 | 21760 | yes | — |
| 95 | D4 | C28 | 1 | 65280 | yes | — |
| 96 | C2×C4 | C28 | 1 | 255 | yes | — |
| 97 | C2×C4 | C28 | 1 | 1020 | yes | — |
| 98 | C2×C4 | C28 | 1 | 64260 | yes | — |
| 99 | C2×C4 | C28 | 1 | 1 | yes | characteristic |
| 100 | C22 | — | 2 | 510 | no | — |
| 101 | C22 | — | 4 | 21420 | no | — |
| 102 | C22 | C29 | 1 | 510 | yes | — |
| 103 | C22 | C29 | 1 | 1 | yes | characteristic |
| 104 | C4 | C29 | 1 | 2 | yes | — |
| 105 | C4 | C29 | 1 | 510 | yes | — |
| 106 | C2 | — | 2 | 510 | no | — |
| 107 | C2 | C210 | 1 | 1 | yes | characteristic |
| 108 | C2 | #72258201 | 1 | 2 | yes | direct product factor |
Complements
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|
| 1. C22×#63103 | C2 | semidirect |
| 2. #72258201 | C2 | direct (N × H) |
| 3. C2×2−1+8 | C2 | semidirect |
| 4. C2×H(9,𝔽2) | C2 | semidirect |
| 5. 2−1+8 | C22 | semidirect |
| 6. H(9,𝔽2) | C22 | semidirect |
| 7. C2×#63103 | C22 | semidirect |
| 8. C2×#63103 | C22 | semidirect |
| 9. C2×#63103 | C22 | semidirect |
| 10. C22×2−1+6 | C22 | semidirect |
| 11. C22×H(7,𝔽2) | C22 | semidirect |
| 12. C23×#585 | C22 | semidirect |
| 13. #63103 | C23 | semidirect |
| 14. #63103 | C23 | semidirect |
| 15. C2×2−1+6 | C23 | semidirect |
| 16. C2×2−1+6 | C23 | semidirect |
| 17. C2×H(7,𝔽2) | C23 | semidirect |
| 18. C2×H(7,𝔽2) | C23 | semidirect |
| 19. C22×#585 | C23 | semidirect |
| 20. C22×#585 | C23 | semidirect |
| 21. C22×#585 | C23 | semidirect |
| 22. C23×2−1+4 | C23 | semidirect |
| 23. C23×H(5,𝔽2) | C23 | semidirect |
| 24. C24×#41 | C23 | semidirect |
| 25. 2−1+6 | C24 | semidirect |
| 26. H(7,𝔽2) | C24 | semidirect |
| 27. C2×#585 | C24 | semidirect |
| 28. C2×#585 | C24 | semidirect |
| 29. C2×#585 | C24 | semidirect |
| 30. C2×#585 | C24 | semidirect |
| 31. C22×2−1+4 | C24 | semidirect |
| 32. C22×2−1+4 | C24 | semidirect |
| 33. C22×H(5,𝔽2) | C24 | semidirect |
| 34. C22×H(5,𝔽2) | C24 | semidirect |
| 35. C23×#41 | C24 | semidirect |
| 36. C23×#41 | C24 | semidirect |
| 37. C23×#41 | C24 | semidirect |
| 38. C24×Q8 | C24 | semidirect |
| 39. C24×D4 | C24 | semidirect |
| 40. C25×C4 | C24 | semidirect |
| 42. #585 | C25 | semidirect |
| 43. #585 | C25 | semidirect |
| 45. C2×2−1+4 | C25 | semidirect |
| 47. C2×H(5,𝔽2) | C25 | semidirect |
| 50. C22×#41 | C25 | semidirect |
| 51. C22×#41 | C25 | semidirect |
| 53. C23×Q8 | C25 | semidirect |
| 55. C23×D4 | C25 | semidirect |
| 58. C24×C4 | C25 | semidirect |
| 108. C2 | #72258201 | direct (N × H) |
Maximal subgroups
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
Finite Group Databasebeta