Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
115 conjugacy classes of subgroups; only the 15 normal subgroups are shown.
This group has 115 conjugacy classes of subgroups — too many to draw. Only its 15 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 | A4×SL(2,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #5930 | — | 3 | 2 | no | maximal |
| 3 | #5419 | — | 3 | 1 | no | maximal |
| 4 | #5416 | — | 3 | 1 | no | maximal |
| 5 | C3⋊SL(2,ℤ/4) | — | 4 | 1 | no | maximal |
| 6 | C3⋊#226 | — | 4 | 1 | no | maximal |
| 7 | C22⋊SL(2,𝔽3) | D3 | 1 | 2 | yes | — |
| 8 | Q8×A4 | D3 | 1 | 1 | yes | characteristic |
| 9 | C22×SL(2,𝔽3) | D3 | 1 | 1 | yes | characteristic |
| 10 | C2×#226 | — | 3 | 1 | no | — |
| 11 | A4⋊Q8 | — | 3 | 1 | no | — |
| 12 | #859 | — | 3 | 1 | no | — |
| 13 | A4⋊C8 | — | 3 | 1 | no | — |
| 14 | C2×C3×A4 | — | 4 | 1 | no | — |
| 15 | C3×SL(2,𝔽3) | — | 4 | 1 | no | — |
| 16 | #451 | — | 9 | 1 | no | Sylow |
| 17 | C2×SL(2,𝔽3) | — | 3 | 1 | no | — |
| 18 | C4×A4 | — | 3 | 1 | no | — |
| 19 | SL(2,ℤ/4) | — | 6 | 1 | no | — |
| 20 | SL(2,ℤ/4) | — | 12 | 2 | no | — |
| 21 | #226 | — | 6 | 1 | no | — |
| 22 | #226 | — | 12 | 2 | no | — |
| 23 | #226 | — | 12 | 2 | no | — |
| 24 | #217 | — | 12 | 1 | no | — |
| 25 | #216 | — | 12 | 1 | no | — |
| 26 | C3×A4 | — | 4 | 1 | no | — |
| 27 | C3⋊Dic3 | — | 16 | 1 | no | — |
| 28 | C22×Q8 | C3⋊D3 | 1 | 1 | yes | characteristic |
| 29 | C2×Q16 | — | 9 | 1 | no | — |
| 30 | C2×Q16 | — | 9 | 1 | no | — |
| 31 | C22⋊Q8 | — | 9 | 1 | no | — |
| 32 | #103 | — | 9 | 1 | no | — |
| 33 | #103 | — | 9 | 1 | no | — |
| 34 | C22⋊C8 | — | 9 | 1 | no | — |
| 35 | C23×C3 | — | 4 | 1 | no | — |
| 36 | C2×A4 | — | 4 | 2 | no | — |
| 37 | C2×A4 | S4 | 1 | 1 | yes | characteristic |
| 38 | C3×Q8 | — | 4 | 1 | no | — |
| 39 | C2×Dic3 | — | 12 | 1 | no | — |
| 40 | C2×Dic3 | — | 12 | 1 | no | — |
| 41 | Dic6 | — | 12 | 1 | no | — |
| 42 | SL(2,𝔽3) | — | 4 | 2 | no | — |
| 43 | SL(2,𝔽3) | — | 12 | 2 | no | — |
| 44 | SL(2,𝔽3) | S4 | 1 | 1 | yes | characteristic |
| 45 | C3⋊C8 | — | 12 | 1 | no | — |
| 46 | C2×C32 | — | 16 | 1 | no | — |
| 47 | C2×Q8 | — | 3 | 1 | no | — |
| 48 | C2×Q8 | — | 9 | 1 | no | — |
| 49 | C2×Q8 | — | 9 | 1 | no | — |
| 50 | C22×C4 | — | 3 | 1 | no | — |
| 51 | Q16 | — | 18 | 2 | no | — |
| 52 | Q16 | — | 18 | 1 | no | — |
| 53 | Q16 | — | 18 | 1 | no | — |
| 54 | C2×C8 | — | 9 | 1 | no | — |
| 55 | C2×C8 | — | 9 | 1 | no | — |
| 56 | C4⋊C4 | — | 9 | 1 | no | — |
| 57 | C4⋊C4 | — | 18 | 1 | no | — |
| 58 | C22⋊C4 | — | 18 | 1 | no | — |
| 59 | C22×C3 | — | 4 | 1 | no | — |
| 60 | C22×C3 | — | 12 | 1 | no | — |
| 61 | C22×C3 | — | 12 | 1 | no | — |
| 62 | A4 | — | 4 | 2 | no | — |
| 63 | A4 | #226 | 1 | 1 | yes | characteristic, complemented |
| 64 | C3×C4 | — | 12 | 1 | no | — |
| 65 | Dic3 | — | 24 | 1 | no | — |
| 66 | Dic3 | — | 24 | 1 | no | — |
| 67 | Dic3 | — | 24 | 1 | no | — |
| 68 | Dic3 | — | 48 | 2 | no | — |
| 69 | C32 | — | 16 | 1 | no | Sylow |
| 70 | C23 | C3⋊S4 | 1 | 1 | yes | characteristic |
| 71 | Q8 | — | 3 | 2 | no | — |
| 72 | Q8 | — | 9 | 1 | no | — |
| 73 | Q8 | — | 18 | 1 | no | — |
| 74 | Q8 | — | 18 | 1 | no | — |
| 75 | Q8 | C3⋊S4 | 1 | 1 | yes | characteristic |
| 76 | C2×C4 | — | 9 | 1 | no | — |
| 77 | C2×C4 | — | 9 | 1 | no | — |
| 78 | C2×C4 | — | 18 | 1 | no | — |
| 79 | C2×C4 | — | 18 | 1 | no | — |
| 80 | C8 | — | 18 | 1 | no | — |
| 81 | C8 | — | 18 | 1 | no | — |
| 82 | C2×C3 | — | 4 | 1 | no | — |
| 83 | C2×C3 | — | 4 | 1 | no | — |
| 84 | C2×C3 | — | 12 | 1 | no | — |
| 85 | C2×C3 | — | 12 | 1 | no | — |
| 86 | C2×C3 | — | 16 | 2 | no | — |
| 87 | C22 | — | 3 | 1 | no | — |
| 88 | C22 | — | 3 | 1 | no | — |
| 89 | C22 | C3⋊#226 | 1 | 1 | yes | characteristic, complemented |
| 90 | C4 | — | 3 | 1 | no | — |
| 91 | C4 | — | 9 | 1 | no | — |
| 92 | C4 | — | 36 | 1 | no | — |
| 93 | C4 | — | 36 | 1 | no | — |
| 94 | C3 | — | 4 | 1 | no | — |
| 95 | C3 | — | 4 | 1 | no | — |
| 96 | C3 | — | 16 | 2 | no | — |
| 97 | C2 | — | 3 | 1 | no | — |
| 98 | C2 | — | 3 | 1 | no | — |
| 99 | C2 | A4⋊S4 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | A4⋊S4 |
| commutator, prime radical, cosocle, hypocenter | A4×SL(2,𝔽3) | C2 |
| Fitting | C22×Q8 | C3⋊D3 |
| socle | C23 | C3⋊S4 |
| LCS | G ▷ A4×SL(2,𝔽3) | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 63. A4 | #226 | semidirect |
| 89. C22 | C3⋊#226 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| A4×SL(2,𝔽3) | 2 |
| #5930 | 3 |
| #5419 | 3 |
| #5416 | 3 |
| C3⋊SL(2,ℤ/4) | 4 |
| C3⋊#226 | 4 |