Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
7800 conjugacy classes of subgroups; only the 3440 normal subgroups are shown.
This group has 7800 conjugacy classes of subgroups — too many to draw. Only its 3440 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #63093 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #62637 | C2 | 1 | 30 | yes | maximal, complemented |
| 3 | #62635 | C2 | 1 | 15 | yes | maximal, complemented |
| 4 | C2×#3417 | C2 | 1 | 15 | yes | maximal, complemented |
| 5 | #62624 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | 2−1+6 | — | 2 | 2 | no | — |
| 7 | #3586 | — | 2 | 30 | no | — |
| 8 | #3584 | C22 | 1 | 30 | yes | complemented |
| 9 | C2×#578 | C22 | 1 | 30 | yes | complemented |
| 10 | #3573 | C22 | 1 | 1 | yes | characteristic |
| 11 | #3417 | C22 | 1 | 60 | yes | complemented |
| 12 | #3417 | C22 | 1 | 120 | yes | complemented |
| 13 | #3416 | C22 | 1 | 120 | yes | complemented |
| 14 | #3415 | C22 | 1 | 20 | yes | complemented |
| 15 | #3414 | C22 | 1 | 90 | yes | complemented |
| 16 | C2×#510 | C22 | 1 | 90 | yes | complemented |
| 17 | C2×#508 | C22 | 1 | 45 | yes | complemented |
| 18 | C22×Q32 | C22 | 1 | 15 | yes | complemented |
| 19 | #3407 | C22 | 1 | 15 | yes | complemented |
| 20 | C2×#504 | C22 | 1 | 15 | yes | complemented |
| 21 | #585 | — | 2 | 30 | no | — |
| 22 | #585 | — | 4 | 20 | no | — |
| 23 | #585 | D4 | 1 | 1 | yes | characteristic |
| 24 | C2×2−1+4 | — | 2 | 30 | no | — |
| 25 | C2×2−1+4 | — | 4 | 12 | no | — |
| 26 | #578 | — | 2 | 120 | no | — |
| 27 | #578 | C23 | 1 | 120 | yes | — |
| 28 | #577 | — | 2 | 120 | no | — |
| 29 | #576 | C23 | 1 | 40 | yes | — |
| 30 | #575 | — | 2 | 90 | no | — |
| 31 | C2×#137 | — | 2 | 90 | no | — |
| 32 | C2×#135 | C23 | 1 | 90 | yes | — |
| 33 | C22×Q16 | C23 | 1 | 30 | yes | — |
| 34 | #568 | D4 | 1 | 15 | yes | — |
| 35 | C2×#131 | C23 | 1 | 15 | yes | — |
| 36 | #510 | C23 | 1 | 360 | yes | — |
| 37 | #509 | C23 | 1 | 120 | yes | — |
| 38 | #508 | C23 | 1 | 60 | yes | — |
| 39 | #508 | C23 | 1 | 180 | yes | — |
| 40 | C2×Q32 | C23 | 1 | 45 | yes | — |
| 41 | C2×Q32 | C23 | 1 | 90 | yes | — |
| 42 | C2×SD32 | C23 | 1 | 90 | yes | — |
| 43 | C2×D16 | C23 | 1 | 15 | yes | — |
| 44 | #504 | C23 | 1 | 20 | yes | — |
| 45 | #504 | C23 | 1 | 60 | yes | — |
| 46 | C2×M32 | C23 | 1 | 45 | yes | — |
| 47 | C22×C16 | C23 | 1 | 15 | yes | — |
| 48 | 2−1+4 | — | 2 | 120 | no | — |
| 49 | 2−1+4 | — | 4 | 120 | no | — |
| 50 | 2−1+4 | — | 4 | 30 | no | — |
| 51 | 2−1+4 | D8 | 1 | 6 | yes | — |
| 52 | H(5,𝔽2) | — | 2 | 40 | no | — |
| 53 | H(5,𝔽2) | — | 4 | 90 | no | — |
| 54 | H(5,𝔽2) | D8 | 1 | 10 | yes | — |
| 55 | C2×#41 | — | 2 | 90 | no | — |
| 56 | C2×#41 | — | 4 | 120 | no | — |
| 57 | C2×#41 | — | 4 | 90 | no | — |
| 58 | C2×#41 | C2×D4 | 1 | 15 | yes | — |
| 59 | C22×Q8 | — | 2 | 30 | no | — |
| 60 | C22×Q8 | — | 4 | 30 | no | — |
| 61 | #137 | — | 2 | 360 | no | — |
| 62 | AGL(1,ℤ/8) | — | 2 | 120 | no | — |
| 63 | #135 | — | 2 | 180 | no | — |
| 64 | #135 | C24 | 1 | 120 | yes | — |
| 65 | C2×Q16 | — | 2 | 90 | no | — |
| 66 | C2×Q16 | C24 | 1 | 90 | yes | — |
| 67 | C2×SD16 | — | 2 | 90 | no | — |
| 68 | C2×D8 | C24 | 1 | 30 | yes | — |
| 69 | #131 | C2×D4 | 1 | 60 | yes | — |
| 70 | #131 | C24 | 1 | 20 | yes | — |
| 71 | C2×M16 | C2×D4 | 1 | 45 | yes | — |
| 72 | C22×C8 | C24 | 1 | 15 | yes | — |
| 73 | Q32 | C24 | 1 | 10 | yes | — |
| 74 | Q32 | C24 | 1 | 90 | yes | — |
| 75 | SD32 | C24 | 1 | 120 | yes | — |
| 76 | D16 | C24 | 1 | 6 | yes | — |
| 77 | D16 | C24 | 1 | 30 | yes | — |
| 78 | M32 | C24 | 1 | 60 | yes | — |
| 79 | C2×C16 | C24 | 1 | 15 | yes | — |
| 80 | C2×C16 | C24 | 1 | 45 | yes | — |
| 81 | #41 | — | 2 | 120 | no | — |
| 82 | #41 | — | 4 | 40 | no | — |
| 83 | #41 | — | 4 | 360 | no | — |
| 84 | #41 | — | 4 | 180 | no | — |
| 85 | #41 | C2×D8 | 1 | 60 | yes | — |
| 86 | #41 | C22×D4 | 1 | 20 | yes | — |
| 87 | C2×Q8 | — | 2 | 90 | no | — |
| 88 | C2×Q8 | — | 4 | 120 | no | — |
| 89 | C2×Q8 | — | 4 | 90 | no | — |
| 90 | C2×Q8 | C2×D8 | 1 | 15 | yes | — |
| 91 | C2×D4 | — | 2 | 30 | no | — |
| 92 | C2×D4 | — | 4 | 90 | no | — |
| 93 | C2×D4 | — | 4 | 120 | no | — |
| 94 | C2×D4 | C2×D8 | 1 | 45 | yes | — |
| 95 | C22×C4 | — | 4 | 30 | no | — |
| 96 | C22×C4 | — | 4 | 90 | no | — |
| 97 | C22×C4 | C22×D4 | 1 | 15 | yes | — |
| 98 | Q16 | — | 2 | 90 | no | — |
| 99 | Q16 | C25 | 1 | 20 | yes | — |
| 100 | SD16 | — | 2 | 120 | no | — |
| 101 | D8 | — | 2 | 30 | no | — |
| 102 | D8 | C25 | 1 | 12 | yes | — |
| 103 | M16 | C22×D4 | 1 | 60 | yes | — |
| 104 | C2×C8 | C22×D4 | 1 | 45 | yes | — |
| 105 | C2×C8 | C25 | 1 | 15 | yes | — |
| 106 | C16 | C25 | 1 | 1 | yes | characteristic |
| 107 | C16 | C25 | 1 | 15 | yes | — |
| 108 | C23 | — | 4 | 30 | no | — |
| 109 | C23 | C22×D8 | 1 | 15 | yes | — |
| 110 | Q8 | — | 2 | 20 | no | — |
| 111 | Q8 | — | 4 | 90 | no | — |
| 112 | Q8 | C22×D8 | 1 | 20 | yes | — |
| 113 | D4 | — | 2 | 12 | no | — |
| 114 | D4 | — | 4 | 120 | no | — |
| 115 | D4 | — | 4 | 30 | no | — |
| 116 | D4 | C22×D8 | 1 | 60 | yes | — |
| 117 | C2×C4 | — | 4 | 90 | no | — |
| 118 | C2×C4 | — | 4 | 120 | no | — |
| 119 | C2×C4 | C22×D8 | 1 | 45 | yes | — |
| 120 | C2×C4 | C23×D4 | 1 | 15 | yes | — |
| 121 | C8 | C23×D4 | 1 | 15 | yes | — |
| 122 | C8 | C26 | 1 | 1 | yes | characteristic |
| 123 | C22 | — | 4 | 15 | no | — |
| 124 | C22 | — | 4 | 12 | no | — |
| 125 | C22 | — | 16 | 30 | no | — |
| 126 | C22 | C23×D8 | 1 | 15 | yes | — |
| 127 | C4 | — | 4 | 20 | no | — |
| 128 | C4 | C23×D8 | 1 | 15 | yes | — |
| 129 | C4 | C24×D4 | 1 | 1 | yes | characteristic |
| 130 | C2 | — | 2 | 15 | no | — |
| 131 | C2 | — | 8 | 12 | no | — |
| 132 | C2 | C24×D8 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C24×D8 |
| commutator, Frattini, cosocle | C8 | C26 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ #585 ◁ #3573 ◁ G | |
| LECS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #63093 | C2 | semidirect |
| 2. #62637 | C2 | semidirect |
| 3. #62635 | C2 | semidirect |
| 4. C2×#3417 | C2 | semidirect |
| 5. #62624 | C2 | semidirect |
| 8. #3584 | C22 | semidirect |
| 9. C2×#578 | C22 | semidirect |
| 11. #3417 | C22 | semidirect |
| 12. #3417 | C22 | semidirect |
| 13. #3416 | C22 | semidirect |
| 14. #3415 | C22 | semidirect |
| 15. #3414 | C22 | semidirect |
| 16. C2×#510 | C22 | semidirect |
| 17. C2×#508 | C22 | semidirect |
| 18. C22×Q32 | C22 | semidirect |
| 19. #3407 | C22 | semidirect |
| 20. C2×#504 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #63093 | 2 |
| #62637 | 2 |
| #62635 | 2 |
| C2×#3417 | 2 |
| #62624 | 2 |