Finite Group DatabasebetaFinite Group DatabasebetaOrder 1944.
730 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 730 conjugacy classes of subgroups — too many to draw. Only its 53 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 | #11757866 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C32×#638 | — | 3 | 1 | no | maximal |
| 3 | C3×#6258 | — | 3 | 12 | no | maximal |
| 4 | #92338 | — | 4 | 1 | no | maximal |
| 5 | C22×C34 | D3 | 1 | 1 | yes | characteristic, complemented |
| 6 | C2×C32×C3⋊D3 | — | 3 | 1 | no | — |
| 7 | C32×C3⋊Dic3 | — | 3 | 1 | no | — |
| 8 | C3×C22⋊H(3,𝔽3) | D3 | 1 | 12 | yes | complemented |
| 9 | C32⋊H(3,𝔽3) | — | 4 | 1 | no | Sylow |
| 10 | C32×S4 | — | 9 | 9 | no | — |
| 11 | C3×#638 | — | 3 | 4 | no | — |
| 12 | C32×#67 | — | 9 | 4 | no | — |
| 13 | #6258 | — | 3 | 12 | no | — |
| 14 | C2×C34 | — | 3 | 1 | no | — |
| 15 | C32×C3⋊D3 | — | 6 | 1 | no | — |
| 16 | C3×#272 | — | 12 | 12 | no | — |
| 17 | C22×C33 | — | 3 | 4 | no | — |
| 18 | C22×C33 | — | 6 | 4 | no | — |
| 19 | C22×C33 | C3⋊D3 | 1 | 4 | yes | — |
| 20 | C2×C3×C3⋊D3 | — | 3 | 4 | no | — |
| 21 | C2×C32×D3 | — | 9 | 4 | no | — |
| 22 | C32×A4 | C3⋊D3 | 1 | 9 | yes | complemented |
| 23 | C3×C3⋊Dic3 | — | 3 | 4 | no | — |
| 24 | C32×Dic3 | — | 9 | 4 | no | — |
| 25 | C22⋊H(3,𝔽3) | — | 3 | 12 | no | — |
| 26 | C22⋊H(3,𝔽3) | — | 6 | 12 | no | — |
| 27 | C34 | S4 | 1 | 1 | yes | characteristic, complemented |
| 28 | C3×H(3,𝔽3) | — | 4 | 12 | no | — |
| 29 | C3×S4 | — | 9 | 36 | no | — |
| 30 | C32×D4 | — | 27 | 1 | no | — |
| 31 | #638 | — | 3 | 1 | no | — |
| 32 | C3×#67 | — | 9 | 12 | no | — |
| 33 | C3×#67 | — | 9 | 4 | no | — |
| 34 | C2×C33 | — | 3 | 4 | no | — |
| 35 | C2×C33 | — | 3 | 4 | no | — |
| 36 | C2×C33 | — | 6 | 4 | no | — |
| 37 | C2×C33 | — | 6 | 12 | no | — |
| 38 | C3×C3⋊D3 | — | 6 | 4 | no | — |
| 39 | C32×D3 | — | 18 | 4 | no | — |
| 40 | C32×D3 | — | 36 | 9 | no | — |
| 41 | #272 | — | 12 | 12 | no | — |
| 42 | C22×C32 | — | 2 | 4 | no | — |
| 43 | C22×C32 | — | 3 | 1 | no | — |
| 44 | C22×C32 | — | 3 | 12 | no | — |
| 45 | C22×C32 | — | 6 | 6 | no | — |
| 46 | C22×C32 | — | 6 | 4 | no | — |
| 47 | C22×C32 | — | 6 | 3 | no | — |
| 48 | C22×C32 | — | 27 | 1 | no | — |
| 49 | C22×C32 | #272 | 1 | 4 | yes | complemented |
| 50 | C22×C32 | C3⋊(C3⋊D3) | 1 | 1 | yes | characteristic |
| 51 | C2×C3⋊D3 | — | 3 | 1 | no | — |
| 52 | C2×C3×D3 | — | 9 | 12 | no | — |
| 53 | C2×C3×D3 | — | 9 | 4 | no | — |
| 54 | C3×A4 | — | 3 | 36 | no | — |
| 55 | C32×C4 | — | 27 | 1 | no | — |
| 56 | C3⋊Dic3 | — | 3 | 1 | no | — |
| 57 | C3×Dic3 | — | 9 | 12 | no | — |
| 58 | C3×Dic3 | — | 9 | 4 | no | — |
| 59 | C33 | — | 3 | 4 | no | — |
| 60 | C33 | — | 4 | 9 | no | — |
| 61 | C33 | — | 6 | 4 | no | — |
| 62 | C33 | C3⋊S4 | 1 | 4 | yes | — |
| 63 | H(3,𝔽3) | — | 12 | 12 | no | — |
| 64 | H(3,𝔽3) | — | 24 | 12 | no | — |
| 65 | S4 | — | 9 | 9 | no | — |
| 66 | C3×D4 | — | 27 | 4 | no | — |
| 67 | #67 | — | 9 | 4 | no | — |
| 68 | C2×C32 | — | 3 | 1 | no | — |
| 69 | C2×C32 | — | 3 | 1 | no | — |
| 70 | C2×C32 | — | 3 | 12 | no | — |
| 71 | C2×C32 | — | 3 | 4 | no | — |
| 72 | C2×C32 | — | 6 | 4 | no | — |
| 73 | C2×C32 | — | 6 | 12 | no | — |
| 74 | C2×C32 | — | 6 | 12 | no | — |
| 75 | C2×C32 | — | 6 | 4 | no | — |
| 76 | C2×C32 | — | 6 | 1 | no | — |
| 77 | C2×C32 | — | 6 | 6 | no | — |
| 78 | C2×C32 | — | 6 | 8 | no | — |
| 79 | C2×C32 | — | 6 | 9 | no | — |
| 80 | C2×C32 | — | 54 | 1 | no | — |
| 81 | C3⋊D3 | — | 6 | 1 | no | — |
| 82 | C3×D3 | — | 18 | 12 | no | — |
| 83 | C3×D3 | — | 18 | 4 | no | — |
| 84 | C3×D3 | — | 36 | 36 | no | — |
| 85 | C22×C3 | — | 3 | 4 | no | — |
| 86 | C22×C3 | — | 6 | 4 | no | — |
| 87 | C22×C3 | — | 27 | 4 | no | — |
| 88 | C22×C3 | C3⋊#272 | 1 | 4 | yes | — |
| 89 | C2×D3 | — | 9 | 4 | no | — |
| 90 | A4 | — | 9 | 9 | no | — |
| 91 | C3×C4 | — | 27 | 4 | no | — |
| 92 | Dic3 | — | 9 | 4 | no | — |
| 93 | C32 | — | 2 | 4 | no | — |
| 94 | C32 | — | 3 | 1 | no | — |
| 95 | C32 | — | 3 | 12 | no | — |
| 96 | C32 | — | 6 | 6 | no | — |
| 97 | C32 | — | 6 | 4 | no | — |
| 98 | C32 | — | 6 | 3 | no | — |
| 99 | C32 | — | 12 | 36 | no | — |
| 100 | C32 | #6258 | 1 | 4 | yes | complemented |
| 101 | C32 | C3⋊(C3⋊S4) | 1 | 1 | yes | characteristic |
| 102 | D4 | — | 27 | 1 | no | Sylow |
| 103 | C2×C3 | — | 3 | 4 | no | — |
| 104 | C2×C3 | — | 3 | 4 | no | — |
| 105 | C2×C3 | — | 6 | 4 | no | — |
| 106 | C2×C3 | — | 6 | 12 | no | — |
| 107 | C2×C3 | — | 54 | 4 | no | — |
| 108 | D3 | — | 18 | 4 | no | — |
| 109 | D3 | — | 36 | 9 | no | — |
| 110 | C22 | — | 27 | 1 | no | — |
| 111 | C22 | #92338 | 1 | 1 | yes | characteristic, complemented |
| 112 | C4 | — | 27 | 1 | no | — |
| 113 | C3 | — | 3 | 4 | no | — |
| 114 | C3 | — | 6 | 4 | no | — |
| 115 | C3 | — | 36 | 9 | no | — |
| 116 | C3 | C3⋊#6258 | 1 | 4 | yes | — |
| 117 | C2 | — | 3 | 1 | no | — |
| 118 | C2 | — | 54 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C32 | C3⋊(C3⋊S4) |
| commutator, cosocle, hypocenter | #11757866 | C2 |
| Fitting | C22×C34 | D3 |
| socle | C22×C32 | C3⋊(C3⋊D3) |
| LCS | G ▷ #11757866 | |
| UCS | C1 ◁ C32 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #11757866 | C2 | semidirect |
| 5. C22×C34 | D3 | semidirect |
| 8. C3×C22⋊H(3,𝔽3) | D3 | semidirect |
| 22. C32×A4 | C3⋊D3 | semidirect |
| 27. C34 | S4 | semidirect |
| 49. C22×C32 | #272 | semidirect |
| 100. C32 | #6258 | semidirect |
| 111. C22 | #92338 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11757866 | 2 |
| C32×#638 | 3 |
| C3×#6258 | 3 |
| #92338 | 4 |