Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1178 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #21690 | C2 | 1 | 6 | yes | maximal, complemented |
| 2 | #10683 | C2 | 1 | 4 | yes | maximal, complemented |
| 3 | #10304 | C2 | 1 | 4 | yes | maximal, complemented |
| 4 | #8136 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | C2×#493 | C22 | 1 | 6 | yes | — |
| 6 | C2×#442 | — | 2 | 12 | no | — |
| 7 | C4×#127 | C22 | 1 | 4 | yes | — |
| 8 | #1972 | — | 2 | 12 | no | — |
| 9 | #1957 | C22 | 1 | 12 | yes | — |
| 10 | #1844 | C22 | 1 | 6 | yes | — |
| 11 | C4×#97 | C22 | 1 | 3 | yes | — |
| 12 | #1273 | C22 | 1 | 4 | yes | — |
| 13 | C22×D8 | — | 2 | 12 | no | — |
| 14 | C2×#127 | C23 | 1 | 4 | yes | — |
| 15 | C2×C4×D4 | — | 2 | 12 | no | — |
| 16 | #493 | — | 2 | 24 | no | — |
| 17 | #442 | — | 4 | 48 | no | — |
| 18 | C2×#106 | — | 2 | 12 | no | — |
| 19 | C2×#102 | — | 2 | 24 | no | — |
| 20 | C2×#97 | D4 | 1 | 12 | yes | — |
| 21 | C2×C4×C8 | C23 | 1 | 6 | yes | — |
| 22 | #390 | — | 2 | 12 | no | — |
| 23 | #384 | C23 | 1 | 4 | yes | — |
| 24 | C43 | C23 | 1 | 1 | yes | characteristic |
| 25 | C22×D4 | — | 2 | 12 | no | — |
| 26 | C23×C4 | — | 4 | 4 | no | — |
| 27 | C2×D8 | — | 2 | 48 | no | — |
| 28 | C2×D8 | — | 4 | 24 | no | — |
| 29 | C2×D8 | — | 4 | 24 | no | — |
| 30 | C22×C8 | C2×D4 | 1 | 12 | yes | — |
| 31 | #127 | — | 2 | 12 | no | — |
| 32 | #127 | #41 | 1 | 8 | yes | — |
| 33 | C4×D4 | — | 4 | 48 | no | — |
| 34 | C2×C4⋊C4 | — | 2 | 12 | no | — |
| 35 | C2×C22⋊C4 | — | 4 | 12 | no | — |
| 36 | C2×C42 | C2×D4 | 1 | 6 | yes | — |
| 37 | C2×C42 | C24 | 1 | 1 | yes | characteristic |
| 38 | #106 | — | 4 | 24 | no | — |
| 39 | #102 | — | 4 | 48 | no | — |
| 40 | #97 | — | 2 | 24 | no | — |
| 41 | C4×C8 | — | 2 | 12 | no | — |
| 42 | C24 | — | 4 | 4 | no | — |
| 43 | C2×D4 | — | 2 | 24 | no | — |
| 44 | C2×D4 | — | 4 | 12 | no | — |
| 45 | C2×D4 | — | 4 | 24 | no | — |
| 46 | C2×D4 | — | 4 | 24 | no | — |
| 47 | C22×C4 | — | 4 | 4 | no | — |
| 48 | C22×C4 | — | 8 | 24 | no | — |
| 49 | C22×C4 | #127 | 1 | 4 | yes | — |
| 50 | C22×C4 | C22×D4 | 1 | 3 | yes | — |
| 51 | D8 | — | 4 | 96 | no | — |
| 52 | C2×C8 | — | 2 | 12 | no | — |
| 53 | C2×C8 | — | 2 | 12 | no | — |
| 54 | C2×C8 | AGL(1,ℤ/8) | 1 | 24 | yes | — |
| 55 | C4⋊C4 | — | 4 | 24 | no | — |
| 56 | C22⋊C4 | — | 8 | 24 | no | — |
| 57 | C42 | — | 2 | 12 | no | — |
| 58 | C42 | C2×#41 | 1 | 4 | yes | — |
| 59 | C23 | — | 4 | 8 | no | — |
| 60 | C23 | — | 8 | 12 | no | — |
| 61 | C23 | — | 8 | 12 | no | — |
| 62 | C23 | C2×#127 | 1 | 1 | yes | characteristic |
| 63 | D4 | — | 4 | 48 | no | — |
| 64 | D4 | — | 8 | 24 | no | — |
| 65 | C2×C4 | — | 2 | 12 | no | — |
| 66 | C2×C4 | — | 8 | 12 | no | — |
| 67 | C2×C4 | — | 16 | 16 | no | — |
| 68 | C2×C4 | #525 | 1 | 6 | yes | — |
| 69 | C2×C4 | C2×AGL(1,ℤ/8) | 1 | 12 | yes | — |
| 70 | C8 | — | 2 | 24 | no | — |
| 71 | C22 | — | 8 | 24 | no | — |
| 72 | C22 | — | 16 | 4 | no | — |
| 73 | C22 | — | 16 | 12 | no | — |
| 74 | C22 | #2397 | 1 | 1 | yes | characteristic |
| 75 | C22 | #3151 | 1 | 6 | yes | — |
| 76 | C4 | — | 4 | 4 | no | — |
| 77 | C4 | — | 16 | 4 | no | — |
| 78 | C4 | #3109 | 1 | 12 | yes | — |
| 79 | C2 | — | 16 | 8 | no | — |
| 80 | C2 | #19745 | 1 | 4 | yes | — |
| 81 | C2 | #23802 | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#127 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #21690 | C2 | semidirect |
| 2. #10683 | C2 | semidirect |
| 3. #10304 | C2 | semidirect |
| 4. #8136 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #21690 | 2 |
| #10683 | 2 |
| #10304 | 2 |
| #8136 | 2 |