Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
162 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #1273⋊C3 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #7852 | C3 | 1 | 1 | yes | maximal, characteristic, complemented, Sylow |
| 3 | C22×C42⋊C3 | — | 4 | 1 | no | maximal |
| 4 | C2×C4×M16 | — | 3 | 1 | no | — |
| 5 | #1273 | — | 3 | 1 | no | — |
| 6 | #1273 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 7 | C2×C42⋊C3 | — | 4 | 2 | no | — |
| 8 | C2×C42⋊C3 | — | 4 | 1 | no | — |
| 9 | C22×M16 | — | 3 | 2 | no | — |
| 10 | C22×C42 | A4 | 1 | 1 | yes | characteristic |
| 11 | C4×M16 | — | 6 | 4 | no | — |
| 12 | C2×#97 | — | 3 | 2 | no | — |
| 13 | C2×C4×C8 | — | 3 | 1 | no | — |
| 14 | C2×C4×C8 | — | 3 | 1 | no | — |
| 15 | C22×A4 | — | 16 | 1 | no | — |
| 16 | C42⋊C3 | — | 4 | 1 | no | — |
| 17 | C23×C4 | — | 3 | 1 | no | — |
| 18 | C2×M16 | — | 6 | 4 | no | — |
| 19 | C2×M16 | — | 6 | 4 | no | — |
| 20 | C2×M16 | — | 6 | 4 | no | — |
| 21 | C22×C8 | — | 3 | 2 | no | — |
| 22 | C22×C8 | — | 3 | 2 | no | — |
| 23 | C2×C42 | — | 3 | 1 | no | — |
| 24 | C2×C42 | — | 3 | 2 | no | — |
| 25 | C2×C42 | SL(2,𝔽3) | 1 | 2 | yes | — |
| 26 | C2×C42 | C2×A4 | 1 | 1 | yes | characteristic |
| 27 | #97 | — | 6 | 4 | no | — |
| 28 | C4×C8 | — | 6 | 2 | no | — |
| 29 | C4×C8 | — | 6 | 2 | no | — |
| 30 | C2×A4 | — | 16 | 2 | no | — |
| 31 | C2×A4 | — | 16 | 1 | no | — |
| 32 | C24 | C42⋊C3 | 1 | 1 | yes | characteristic |
| 33 | C22×C4 | — | 3 | 1 | no | — |
| 34 | C22×C4 | — | 3 | 2 | no | — |
| 35 | C22×C4 | — | 3 | 2 | no | — |
| 36 | C22×C4 | — | 3 | 1 | no | — |
| 37 | C22×C4 | — | 6 | 2 | no | — |
| 38 | C22×C4 | — | 6 | 2 | no | — |
| 39 | M16 | — | 6 | 16 | no | — |
| 40 | C2×C8 | — | 3 | 4 | no | — |
| 41 | C2×C8 | — | 3 | 4 | no | — |
| 42 | C2×C8 | — | 6 | 2 | no | — |
| 43 | C2×C8 | — | 6 | 2 | no | — |
| 44 | C2×C8 | — | 6 | 2 | no | — |
| 45 | C2×C8 | — | 6 | 2 | no | — |
| 46 | C42 | — | 3 | 1 | no | — |
| 47 | C42 | — | 3 | 2 | no | — |
| 48 | C42 | — | 3 | 2 | no | — |
| 49 | C42 | C2×SL(2,𝔽3) | 1 | 1 | yes | characteristic |
| 50 | C22×C3 | — | 64 | 1 | no | — |
| 51 | A4 | — | 16 | 1 | no | — |
| 52 | C23 | — | 6 | 1 | no | — |
| 53 | C23 | — | 6 | 1 | no | — |
| 54 | C23 | #95⋊C3 | 1 | 2 | yes | — |
| 55 | C23 | C2×C42⋊C3 | 1 | 1 | yes | characteristic |
| 56 | C2×C4 | — | 3 | 2 | no | — |
| 57 | C2×C4 | — | 3 | 2 | no | — |
| 58 | C2×C4 | — | 3 | 1 | no | — |
| 59 | C2×C4 | — | 3 | 1 | no | — |
| 60 | C2×C4 | — | 3 | 2 | no | — |
| 61 | C2×C4 | — | 6 | 4 | no | — |
| 62 | C2×C4 | — | 6 | 4 | no | — |
| 63 | C2×C4 | — | 6 | 1 | no | — |
| 64 | C2×C4 | — | 6 | 1 | no | — |
| 65 | C8 | — | 6 | 4 | no | — |
| 66 | C8 | — | 6 | 4 | no | — |
| 67 | C2×C3 | — | 64 | 2 | no | — |
| 68 | C2×C3 | — | 64 | 1 | no | — |
| 69 | C22 | — | 3 | 1 | no | — |
| 70 | C22 | — | 3 | 1 | no | — |
| 71 | C22 | — | 4 | 1 | no | — |
| 72 | C22 | — | 6 | 2 | no | — |
| 73 | C22 | — | 12 | 1 | no | — |
| 74 | C22 | C2×#95⋊C3 | 1 | 1 | yes | characteristic |
| 75 | C4 | — | 3 | 2 | no | — |
| 76 | C4 | — | 3 | 2 | no | — |
| 77 | C4 | — | 6 | 2 | no | — |
| 78 | C3 | — | 64 | 1 | no | Sylow |
| 79 | C2 | — | 3 | 1 | no | — |
| 80 | C2 | — | 3 | 1 | no | — |
| 81 | C2 | — | 4 | 2 | no | — |
| 82 | C2 | #1455⋊C3 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #1455⋊C3 |
| commutator, cosocle, hypocenter | #1273 | C2×C3 |
| Frattini | C2×C42 | C2×A4 |
| Fitting | #7852 | C3 |
| socle | C23 | C2×C42⋊C3 |
| LCS | G ▷ #1273 | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #1273⋊C3 | C2 | semidirect |
| 2. #7852 | C3 | semidirect |
| 6. #1273 | C2×C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #1273⋊C3 | 2 |
| #7852 | 3 |
| C22×C42⋊C3 | 4 |