Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
2870 conjugacy classes of subgroups; only the 618 normal subgroups are shown.
This group has 2870 conjugacy classes of subgroups — too many to draw. Only its 618 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 | C22×#564 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C22×#490 | C2 | 1 | 3 | yes | maximal |
| 3 | C2×#1658 | C2 | 1 | 24 | yes | maximal, direct product factor |
| 4 | C22×#330 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | C2×#564 | — | 2 | 18 | no | — |
| 6 | C2×#564 | C22 | 1 | 12 | yes | complemented |
| 7 | C22×#127 | C22 | 1 | 1 | yes | characteristic |
| 8 | C22×#125 | — | 2 | 6 | no | — |
| 9 | C22×#125 | C22 | 1 | 3 | yes | — |
| 10 | C2×#490 | — | 2 | 18 | no | — |
| 11 | C2×#490 | C22 | 1 | 36 | yes | — |
| 12 | C22×#102 | — | 2 | 6 | no | — |
| 13 | C22×#97 | C22 | 1 | 3 | yes | — |
| 14 | #1658 | C22 | 1 | 64 | yes | direct product factor |
| 15 | C2×#330 | C22 | 1 | 36 | yes | complemented |
| 16 | C23×D4 | — | 2 | 3 | no | — |
| 17 | C23×C8 | — | 2 | 3 | no | — |
| 18 | #564 | — | 2 | 48 | no | — |
| 19 | #564 | — | 2 | 72 | no | — |
| 20 | #564 | C23 | 1 | 16 | yes | complemented |
| 21 | C2×#127 | — | 2 | 9 | no | — |
| 22 | C2×#127 | C23 | 1 | 6 | yes | — |
| 23 | C2×#125 | — | 2 | 72 | no | — |
| 24 | C2×#125 | — | 2 | 72 | no | — |
| 25 | C2×#125 | — | 2 | 18 | no | — |
| 26 | C2×#125 | D4 | 1 | 18 | yes | — |
| 27 | C2×#125 | C23 | 1 | 18 | yes | — |
| 28 | C22×C4⋊C4 | — | 2 | 6 | no | — |
| 29 | C22×C4⋊C4 | — | 2 | 6 | no | — |
| 30 | C22×C4⋊C4 | D4 | 1 | 6 | yes | — |
| 31 | C22×C42 | — | 2 | 2 | no | — |
| 32 | C22×C42 | C23 | 1 | 1 | yes | characteristic |
| 33 | #490 | — | 2 | 72 | no | — |
| 34 | #490 | C23 | 1 | 48 | yes | — |
| 35 | C2×#102 | — | 2 | 72 | no | — |
| 36 | C2×#97 | — | 2 | 9 | no | — |
| 37 | C2×#97 | C23 | 1 | 18 | yes | — |
| 38 | #330 | C23 | 1 | 48 | yes | complemented |
| 39 | C25 | — | 4 | 1 | no | — |
| 40 | C22×D4 | — | 2 | 18 | no | — |
| 41 | C22×D4 | — | 4 | 24 | no | — |
| 42 | C22×D4 | — | 4 | 9 | no | — |
| 43 | C23×C4 | — | 2 | 6 | no | — |
| 44 | C23×C4 | C2×D4 | 1 | 3 | yes | — |
| 45 | C22×C8 | — | 2 | 18 | no | — |
| 46 | C22×C8 | — | 4 | 12 | no | — |
| 47 | #127 | — | 2 | 18 | no | — |
| 48 | #127 | — | 4 | 6 | no | — |
| 49 | #127 | C24 | 1 | 4 | yes | — |
| 50 | #125 | — | 2 | 288 | no | — |
| 51 | #125 | — | 2 | 96 | no | — |
| 52 | #125 | — | 2 | 36 | no | — |
| 53 | #125 | — | 2 | 36 | no | — |
| 54 | #125 | C2×D4 | 1 | 36 | yes | — |
| 55 | #125 | C24 | 1 | 12 | yes | — |
| 56 | C2×C4⋊C4 | — | 2 | 72 | no | — |
| 57 | C2×C4⋊C4 | — | 2 | 36 | no | — |
| 58 | C2×C4⋊C4 | — | 2 | 18 | no | — |
| 59 | C2×C4⋊C4 | — | 2 | 36 | no | — |
| 60 | C2×C4⋊C4 | C2×D4 | 1 | 36 | yes | — |
| 61 | C2×C42 | — | 2 | 24 | no | — |
| 62 | C2×C42 | C2×D4 | 1 | 9 | yes | — |
| 63 | C2×C42 | C24 | 1 | 3 | yes | — |
| 64 | #102 | — | 2 | 96 | no | — |
| 65 | #97 | — | 2 | 18 | no | — |
| 66 | #97 | C24 | 1 | 12 | yes | — |
| 67 | C24 | — | 4 | 6 | no | — |
| 68 | C24 | — | 8 | 12 | no | — |
| 69 | C24 | #120 | 1 | 1 | yes | characteristic |
| 70 | C2×D4 | — | 2 | 12 | no | — |
| 71 | C2×D4 | — | 4 | 72 | no | — |
| 72 | C2×D4 | — | 4 | 18 | no | — |
| 73 | C2×D4 | — | 8 | 36 | no | — |
| 74 | C22×C4 | — | 2 | 12 | no | — |
| 75 | C22×C4 | — | 2 | 36 | no | — |
| 76 | C22×C4 | — | 4 | 24 | no | — |
| 77 | C22×C4 | #120 | 1 | 9 | yes | — |
| 78 | C22×C4 | C22×D4 | 1 | 9 | yes | — |
| 79 | C2×C8 | — | 2 | 12 | no | — |
| 80 | C2×C8 | — | 4 | 36 | no | — |
| 81 | C4⋊C4 | — | 2 | 96 | no | — |
| 82 | C4⋊C4 | — | 2 | 72 | no | — |
| 83 | C4⋊C4 | — | 2 | 36 | no | — |
| 84 | C4⋊C4 | — | 2 | 24 | no | — |
| 85 | C4⋊C4 | C22×D4 | 1 | 24 | yes | — |
| 86 | C42 | — | 2 | 32 | no | — |
| 87 | C42 | #120 | 1 | 6 | yes | — |
| 88 | C42 | C22×D4 | 1 | 9 | yes | — |
| 89 | C42 | C25 | 1 | 1 | yes | characteristic |
| 90 | C23 | — | 4 | 4 | no | — |
| 91 | C23 | — | 8 | 36 | no | — |
| 92 | C23 | — | 16 | 16 | no | — |
| 93 | C23 | #454 | 1 | 12 | yes | — |
| 94 | C23 | C2×#120 | 1 | 3 | yes | — |
| 95 | D4 | — | 4 | 24 | no | — |
| 96 | D4 | — | 8 | 36 | no | — |
| 97 | C2×C4 | — | 2 | 18 | no | — |
| 98 | C2×C4 | — | 2 | 18 | no | — |
| 99 | C2×C4 | — | 2 | 24 | no | — |
| 100 | C2×C4 | — | 4 | 72 | no | — |
| 101 | C2×C4 | C2×#120 | 1 | 9 | yes | — |
| 102 | C2×C4 | C23×D4 | 1 | 3 | yes | — |
| 103 | C8 | — | 4 | 12 | no | — |
| 104 | C22 | — | 8 | 12 | no | — |
| 105 | C22 | — | 16 | 24 | no | — |
| 106 | C22 | #1658 | 1 | 16 | yes | direct product factor |
| 107 | C22 | C2×#454 | 1 | 18 | yes | — |
| 108 | C22 | C22×#120 | 1 | 1 | yes | characteristic |
| 109 | C4 | — | 2 | 3 | no | — |
| 110 | C4 | — | 2 | 9 | no | — |
| 111 | C4 | — | 4 | 24 | no | — |
| 112 | C2 | — | 16 | 4 | no | — |
| 113 | C2 | C2×#1658 | 1 | 12 | yes | direct product factor |
| 114 | C2 | C22×#454 | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | #120 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#564 | C2 | semidirect |
| 3. C2×#1658 | C2 | direct (N × H) |
| 4. C22×#330 | C2 | semidirect |
| 6. C2×#564 | C22 | semidirect |
| 14. #1658 | C22 | direct (N × H) |
| 15. C2×#330 | C22 | semidirect |
| 20. #564 | C23 | semidirect |
| 38. #330 | C23 | semidirect |
| 106. C22 | #1658 | direct (N × H) |
| 113. C2 | C2×#1658 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#564 | 2 |
| C22×#490 | 2 |
| C2×#1658 | 2 |
| C22×#330 | 2 |