Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
2494 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×#2104 | C2 | 1 | 24 | yes | maximal, direct product factor |
| 2 | C22×#342 | C2 | 1 | 7 | yes | maximal |
| 3 | C23×C22⋊C4 | C22 | 1 | 7 | yes | — |
| 4 | #2104 | C22 | 1 | 64 | yes | direct product factor |
| 5 | C2×#342 | C22 | 1 | 84 | yes | — |
| 6 | C26 | C23 | 1 | 1 | yes | characteristic |
| 7 | C24×C4 | — | 2 | 7 | no | — |
| 8 | C22×C22⋊C4 | — | 2 | 14 | no | — |
| 9 | C22×C22⋊C4 | — | 2 | 42 | no | — |
| 10 | C22×C22⋊C4 | — | 2 | 21 | no | — |
| 11 | C22×C22⋊C4 | C23 | 1 | 42 | yes | — |
| 12 | #342 | C23 | 1 | 112 | yes | — |
| 13 | C25 | — | 8 | 4 | no | — |
| 14 | C25 | #41 | 1 | 7 | yes | — |
| 15 | C25 | #41 | 1 | 21 | yes | — |
| 16 | C25 | C24 | 1 | 3 | yes | — |
| 17 | C23×C4 | — | 2 | 42 | no | — |
| 18 | C23×C4 | — | 4 | 7 | no | — |
| 19 | C23×C4 | — | 4 | 21 | no | — |
| 20 | C23×C4 | — | 8 | 28 | no | — |
| 21 | C2×C22⋊C4 | — | 2 | 84 | no | — |
| 22 | C2×C22⋊C4 | — | 2 | 84 | no | — |
| 23 | C2×C22⋊C4 | — | 2 | 42 | no | — |
| 24 | C2×C22⋊C4 | — | 4 | 42 | no | — |
| 25 | C2×C22⋊C4 | — | 4 | 42 | no | — |
| 26 | C2×C22⋊C4 | C24 | 1 | 28 | yes | — |
| 27 | C24 | — | 4 | 28 | no | — |
| 28 | C24 | — | 8 | 42 | no | — |
| 29 | C24 | — | 8 | 6 | no | — |
| 30 | C24 | #126 | 1 | 7 | yes | — |
| 31 | C24 | #126 | 1 | 42 | yes | — |
| 32 | C24 | #126 | 1 | 21 | yes | — |
| 33 | C24 | #126 | 1 | 42 | yes | — |
| 34 | C24 | C2×#41 | 1 | 21 | yes | — |
| 35 | C24 | C2×#41 | 1 | 21 | yes | — |
| 36 | C24 | C25 | 1 | 1 | yes | characteristic |
| 37 | C22×C4 | — | 2 | 28 | no | — |
| 38 | C22×C4 | — | 4 | 42 | no | — |
| 39 | C22×C4 | — | 4 | 42 | no | — |
| 40 | C22×C4 | — | 8 | 28 | no | — |
| 41 | C22×C4 | — | 8 | 84 | no | — |
| 42 | C22×C4 | — | 8 | 84 | no | — |
| 43 | C22⋊C4 | — | 2 | 56 | no | — |
| 44 | C22⋊C4 | — | 4 | 84 | no | — |
| 45 | C23 | — | 2 | 28 | no | — |
| 46 | C23 | — | 4 | 42 | no | — |
| 47 | C23 | — | 4 | 42 | no | — |
| 48 | C23 | — | 4 | 84 | no | — |
| 49 | C23 | — | 8 | 42 | no | — |
| 50 | C23 | — | 8 | 21 | no | — |
| 51 | C23 | — | 8 | 1 | no | — |
| 52 | C23 | #401 | 1 | 21 | yes | — |
| 53 | C23 | #401 | 1 | 42 | yes | — |
| 54 | C23 | #401 | 1 | 1 | yes | characteristic |
| 55 | C23 | C2×#126 | 1 | 21 | yes | — |
| 56 | C23 | C2×#126 | 1 | 42 | yes | — |
| 57 | C23 | C2×#126 | 1 | 21 | yes | — |
| 58 | C23 | C22×#41 | 1 | 7 | yes | — |
| 59 | C2×C4 | — | 4 | 28 | no | — |
| 60 | C2×C4 | — | 8 | 84 | no | — |
| 61 | C2×C4 | — | 8 | 84 | no | — |
| 62 | C2×C4 | — | 8 | 28 | no | — |
| 63 | C22 | — | 2 | 42 | no | — |
| 64 | C22 | — | 2 | 42 | no | — |
| 65 | C22 | — | 4 | 21 | no | — |
| 66 | C22 | — | 4 | 7 | no | — |
| 67 | C22 | — | 4 | 42 | no | — |
| 68 | C22 | — | 4 | 42 | no | — |
| 69 | C22 | #2104 | 1 | 4 | yes | direct product factor |
| 70 | C22 | C2×#401 | 1 | 21 | yes | — |
| 71 | C22 | C2×#401 | 1 | 3 | yes | — |
| 72 | C22 | C22×#126 | 1 | 7 | yes | — |
| 73 | C4 | — | 8 | 28 | no | — |
| 74 | C2 | — | 2 | 7 | no | — |
| 75 | C2 | — | 2 | 21 | no | — |
| 76 | C2 | C2×#2104 | 1 | 6 | yes | direct product factor |
| 77 | C2 | C22×#401 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #401 |
| commutator, Frattini, cosocle | C24 | C25 |
| prime radical | C26 | C23 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C26 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#2104 | C2 | direct (N × H) |
| 4. #2104 | C22 | direct (N × H) |
| 69. C22 | #2104 | direct (N × H) |
| 76. C2 | C2×#2104 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2104 | 2 |
| C22×#342 | 2 |