Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
3909 conjugacy classes of subgroups; only the 546 normal subgroups are shown.
This group has 3909 conjugacy classes of subgroups — too many to draw. Only its 546 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 | C2×#2867 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×#2865 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #16503 | C2 | 1 | 14 | yes | maximal, complemented |
| 4 | #16502 | C2 | 1 | 14 | yes | maximal, complemented |
| 5 | #14691 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | C22×#127 | C22 | 1 | 14 | yes | — |
| 7 | #2867 | — | 2 | 8 | no | — |
| 8 | #2865 | — | 2 | 8 | no | — |
| 9 | C2×#394 | C22 | 1 | 28 | yes | — |
| 10 | C2×#392 | C22 | 1 | 28 | yes | — |
| 11 | C2×#388 | C22 | 1 | 7 | yes | — |
| 12 | C2×#387 | C22 | 1 | 7 | yes | — |
| 13 | C2×C43 | C22 | 1 | 1 | yes | characteristic |
| 14 | #1438 | C22 | 1 | 56 | yes | — |
| 15 | #1434 | C22 | 1 | 14 | yes | — |
| 16 | C23×D4 | C23 | 1 | 28 | yes | — |
| 17 | C23×D4 | C23 | 1 | 8 | yes | — |
| 18 | C2×#127 | — | 2 | 56 | no | — |
| 19 | C2×#127 | — | 4 | 56 | no | — |
| 20 | C22×C4⋊C4 | C23 | 1 | 7 | yes | — |
| 21 | C22×C22⋊C4 | C23 | 1 | 7 | yes | — |
| 22 | C22×C22⋊C4 | C23 | 1 | 56 | yes | — |
| 23 | C22×C42 | C23 | 1 | 7 | yes | — |
| 24 | #394 | — | 4 | 56 | no | — |
| 25 | #392 | — | 4 | 56 | no | — |
| 26 | #388 | — | 2 | 28 | no | — |
| 27 | #387 | — | 2 | 28 | no | — |
| 28 | C2×#95 | C23 | 1 | 28 | yes | — |
| 29 | C2×#95 | C23 | 1 | 14 | yes | — |
| 30 | C43 | D4 | 1 | 8 | yes | complemented |
| 31 | C25 | C24 | 1 | 1 | yes | characteristic |
| 32 | C25 | C24 | 1 | 8 | yes | — |
| 33 | C22×D4 | — | 2 | 56 | no | — |
| 34 | C22×D4 | — | 2 | 56 | no | — |
| 35 | C22×D4 | — | 2 | 56 | no | — |
| 36 | C22×D4 | — | 4 | 112 | no | — |
| 37 | C22×D4 | — | 4 | 56 | no | — |
| 38 | C23×C4 | C24 | 1 | 7 | yes | — |
| 39 | C23×C4 | C24 | 1 | 8 | yes | — |
| 40 | C23×C4 | C24 | 1 | 7 | yes | — |
| 41 | #127 | — | 4 | 224 | no | — |
| 42 | C2×C4⋊C4 | — | 2 | 14 | no | — |
| 43 | C2×C4⋊C4 | — | 2 | 28 | no | — |
| 44 | C2×C22⋊C4 | — | 2 | 14 | no | — |
| 45 | C2×C22⋊C4 | — | 2 | 28 | no | — |
| 46 | C2×C22⋊C4 | — | 4 | 56 | no | — |
| 47 | C2×C22⋊C4 | — | 4 | 112 | no | — |
| 48 | C2×C42 | — | 2 | 28 | no | — |
| 49 | C2×C42 | C2×D4 | 1 | 28 | yes | — |
| 50 | #95 | — | 2 | 28 | no | — |
| 51 | #95 | — | 2 | 56 | no | — |
| 52 | C24 | — | 2 | 1 | no | — |
| 53 | C24 | — | 2 | 8 | no | — |
| 54 | C24 | — | 2 | 14 | no | — |
| 55 | C24 | — | 2 | 56 | no | — |
| 56 | C24 | — | 2 | 56 | no | — |
| 57 | C24 | C25 | 1 | 1 | yes | characteristic |
| 58 | C2×D4 | — | 4 | 224 | no | — |
| 59 | C2×D4 | — | 4 | 112 | no | — |
| 60 | C2×D4 | — | 8 | 112 | no | — |
| 61 | C2×D4 | — | 8 | 112 | no | — |
| 62 | C2×D4 | — | 8 | 112 | no | — |
| 63 | C22×C4 | — | 2 | 28 | no | — |
| 64 | C22×C4 | — | 2 | 7 | no | — |
| 65 | C22×C4 | — | 2 | 56 | no | — |
| 66 | C22×C4 | — | 2 | 14 | no | — |
| 67 | C22×C4 | — | 2 | 28 | no | — |
| 68 | C22×C4 | C22×D4 | 1 | 14 | yes | — |
| 69 | C22×C4 | H(5,𝔽2) | 1 | 28 | yes | — |
| 70 | C4⋊C4 | — | 4 | 28 | no | — |
| 71 | C22⋊C4 | — | 4 | 28 | no | — |
| 72 | C22⋊C4 | — | 8 | 112 | no | — |
| 73 | C42 | — | 2 | 56 | no | — |
| 74 | C23 | — | 4 | 7 | no | — |
| 75 | C23 | — | 4 | 56 | no | — |
| 76 | C23 | — | 4 | 56 | no | — |
| 77 | C23 | — | 4 | 28 | no | — |
| 78 | C23 | — | 4 | 56 | no | — |
| 79 | C23 | — | 4 | 56 | no | — |
| 80 | C23 | — | 4 | 56 | no | — |
| 81 | C23 | C23×D4 | 1 | 1 | yes | characteristic |
| 82 | C23 | C2×H(5,𝔽2) | 1 | 14 | yes | — |
| 83 | D4 | — | 8 | 224 | no | — |
| 84 | D4 | — | 8 | 64 | no | — |
| 85 | C2×C4 | — | 2 | 28 | no | — |
| 86 | C2×C4 | — | 2 | 56 | no | — |
| 87 | C2×C4 | — | 4 | 7 | no | — |
| 88 | C2×C4 | — | 4 | 14 | no | — |
| 89 | C2×C4 | — | 4 | 28 | no | — |
| 90 | C2×C4 | — | 4 | 56 | no | — |
| 91 | C2×C4 | #535 | 1 | 28 | yes | — |
| 92 | C22 | — | 8 | 8 | no | — |
| 93 | C22 | — | 8 | 8 | no | — |
| 94 | C22 | — | 8 | 56 | no | — |
| 95 | C22 | — | 8 | 56 | no | — |
| 96 | C22 | — | 8 | 7 | no | — |
| 97 | C22 | C2×#535 | 1 | 7 | yes | — |
| 98 | C22 | #3543 | 1 | 28 | yes | — |
| 99 | C4 | — | 2 | 28 | no | — |
| 100 | C4 | — | 8 | 8 | no | — |
| 101 | C4 | — | 8 | 7 | no | — |
| 102 | C2 | — | 16 | 1 | no | — |
| 103 | C2 | — | 16 | 8 | no | — |
| 104 | C2 | #37614 | 1 | 7 | yes | — |
| 105 | C2 | #53386 | 1 | 8 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#2867 | C2 | semidirect |
| 2. C2×#2865 | C2 | semidirect |
| 3. #16503 | C2 | semidirect |
| 4. #16502 | C2 | semidirect |
| 5. #14691 | C2 | semidirect |
| 30. C43 | D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2867 | 2 |
| C2×#2865 | 2 |
| #16503 | 2 |
| #16502 | 2 |
| #14691 | 2 |