Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
811 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 811 conjugacy classes of subgroups — too many to draw. Only its 150 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 | #21797 | C2 | 1 | 3 | yes | maximal |
| 2 | #17338 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #8122 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | C22×#127 | C22 | 1 | 1 | yes | characteristic |
| 5 | #2920 | C22 | 1 | 3 | yes | — |
| 6 | #2920 | C22 | 1 | 3 | yes | — |
| 7 | #1965 | — | 2 | 6 | no | — |
| 8 | #1965 | — | 2 | 6 | no | — |
| 9 | #1783 | C4 | 1 | 12 | yes | — |
| 10 | #1453 | — | 2 | 6 | no | — |
| 11 | #1307 | — | 2 | 6 | no | — |
| 12 | C23×D4 | C2×C4 | 1 | 6 | yes | — |
| 13 | C2×#127 | — | 2 | 3 | no | — |
| 14 | C2×#127 | — | 2 | 3 | no | — |
| 15 | C2×#127 | — | 2 | 6 | no | — |
| 16 | C22×C42 | C23 | 1 | 1 | yes | characteristic |
| 17 | #432 | — | 2 | 6 | no | — |
| 18 | #432 | — | 2 | 6 | no | — |
| 19 | #431 | — | 2 | 6 | no | — |
| 20 | #431 | — | 2 | 6 | no | — |
| 21 | #423 | — | 2 | 3 | no | — |
| 22 | #423 | D4 | 1 | 6 | yes | — |
| 23 | C2×#100 | — | 2 | 24 | no | — |
| 24 | #407 | C2×C4 | 1 | 6 | yes | — |
| 25 | #407 | C2×C4 | 1 | 6 | yes | — |
| 26 | C4×M16 | — | 2 | 3 | no | — |
| 27 | C4×M16 | D4 | 1 | 6 | yes | — |
| 28 | #322 | — | 2 | 6 | no | — |
| 29 | C82 | — | 2 | 2 | no | — |
| 30 | C25 | C42 | 1 | 4 | yes | — |
| 31 | C22×D4 | — | 2 | 6 | no | — |
| 32 | C22×D4 | — | 2 | 6 | no | — |
| 33 | C22×D4 | — | 2 | 12 | no | — |
| 34 | C22×D4 | — | 2 | 12 | no | — |
| 35 | C22×D4 | — | 8 | 12 | no | — |
| 36 | C23×C4 | C22×C4 | 1 | 3 | yes | — |
| 37 | C2×M16 | — | 2 | 6 | no | — |
| 38 | C2×M16 | C22⋊C4 | 1 | 12 | yes | — |
| 39 | #127 | — | 4 | 1 | no | — |
| 40 | #127 | — | 4 | 6 | no | — |
| 41 | #127 | — | 4 | 3 | no | — |
| 42 | #127 | — | 4 | 3 | no | — |
| 43 | #127 | — | 4 | 2 | no | — |
| 44 | #127 | — | 4 | 1 | no | — |
| 45 | C2×C42 | C2×D4 | 1 | 3 | yes | — |
| 46 | C2×C42 | C2×D4 | 1 | 3 | yes | — |
| 47 | C2×C42 | #41 | 1 | 6 | yes | — |
| 48 | C4⋊C8 | — | 2 | 12 | no | — |
| 49 | C4⋊C8 | — | 4 | 6 | no | — |
| 50 | #100 | — | 8 | 24 | no | — |
| 51 | C22⋊C8 | — | 2 | 12 | no | — |
| 52 | C22⋊C8 | — | 2 | 12 | no | — |
| 53 | #97 | — | 2 | 6 | no | — |
| 54 | #97 | — | 4 | 3 | no | — |
| 55 | C4×C8 | — | 2 | 6 | no | — |
| 56 | C4×C8 | — | 4 | 3 | no | — |
| 57 | C24 | — | 2 | 12 | no | — |
| 58 | C24 | — | 8 | 12 | no | — |
| 59 | C24 | C2×C42 | 1 | 1 | yes | characteristic |
| 60 | C2×D4 | — | 4 | 24 | no | — |
| 61 | C2×D4 | — | 4 | 6 | no | — |
| 62 | C2×D4 | — | 4 | 12 | no | — |
| 63 | C2×D4 | — | 4 | 6 | no | — |
| 64 | C2×D4 | — | 8 | 24 | no | — |
| 65 | C2×D4 | — | 8 | 12 | no | — |
| 66 | C2×D4 | — | 8 | 24 | no | — |
| 67 | C22×C4 | — | 4 | 6 | no | — |
| 68 | C22×C4 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 69 | C22×C4 | #117 | 1 | 3 | yes | — |
| 70 | C22×C4 | C4×D4 | 1 | 6 | yes | — |
| 71 | C22×C4 | C4×D4 | 1 | 6 | yes | — |
| 72 | M16 | — | 4 | 12 | no | — |
| 73 | M16 | — | 4 | 12 | no | — |
| 74 | C2×C8 | — | 2 | 12 | no | — |
| 75 | C2×C8 | — | 4 | 6 | no | — |
| 76 | C42 | #120 | 1 | 1 | yes | characteristic |
| 77 | C42 | #121 | 1 | 6 | yes | — |
| 78 | C42 | #123 | 1 | 3 | yes | — |
| 79 | C42 | #124 | 1 | 3 | yes | — |
| 80 | C42 | #126 | 1 | 2 | yes | — |
| 81 | C42 | #127 | 1 | 1 | yes | characteristic |
| 82 | C23 | — | 4 | 3 | no | — |
| 83 | C23 | — | 4 | 24 | no | — |
| 84 | C23 | — | 4 | 4 | no | — |
| 85 | C23 | — | 8 | 24 | no | — |
| 86 | C23 | — | 16 | 16 | no | — |
| 87 | C23 | C4×C22⋊C4 | 1 | 3 | yes | — |
| 88 | D4 | — | 8 | 12 | no | — |
| 89 | D4 | — | 8 | 24 | no | — |
| 90 | D4 | — | 8 | 12 | no | — |
| 91 | C2×C4 | — | 2 | 6 | no | — |
| 92 | C2×C4 | — | 2 | 12 | no | — |
| 93 | C2×C4 | — | 4 | 3 | no | — |
| 94 | C2×C4 | — | 4 | 6 | no | — |
| 95 | C2×C4 | #386 | 1 | 3 | yes | — |
| 96 | C2×C4 | #388 | 1 | 6 | yes | — |
| 97 | C2×C4 | #390 | 1 | 3 | yes | — |
| 98 | C8 | — | 4 | 12 | no | — |
| 99 | C8 | — | 8 | 6 | no | — |
| 100 | C22 | — | 2 | 3 | no | — |
| 101 | C22 | — | 2 | 6 | no | — |
| 102 | C22 | — | 4 | 4 | no | — |
| 103 | C22 | — | 8 | 12 | no | — |
| 104 | C22 | — | 16 | 24 | no | — |
| 105 | C22 | #1437 | 1 | 1 | yes | characteristic |
| 106 | C4 | — | 2 | 3 | no | — |
| 107 | C4 | — | 2 | 6 | no | — |
| 108 | C4 | #1914 | 1 | 6 | yes | — |
| 109 | C2 | — | 4 | 3 | no | — |
| 110 | C2 | — | 16 | 4 | no | — |
| 111 | C2 | #10939 | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1437 |
| commutator | C24 | C2×C42 |
| Frattini, cosocle | C22×C42 | C23 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #17338 | C2 | semidirect |
| 3. #8122 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #21797 | 2 |
| #17338 | 2 |
| #8122 | 2 |