Finite Group DatabasebetaFinite Group DatabasebetaOrder 1944.
1013 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 1013 conjugacy classes of subgroups — too many to draw. Only its 90 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×#92404 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #11758012 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #10619885 | — | 3 | 8 | no | maximal |
| 4 | C2×H(5,𝔽3) | C22 | 1 | 1 | yes | characteristic |
| 5 | #92404 | C4 | 1 | 2 | yes | complemented |
| 6 | C2×C3⋊#272 | — | 3 | 8 | no | — |
| 7 | C2×C3⋊#272 | — | 6 | 12 | no | — |
| 8 | C2×C3⋊#272 | — | 6 | 4 | no | — |
| 9 | C3⋊((C3⋊Dic3)⋊C3) | D3 | 1 | 8 | yes | complemented |
| 10 | C3×(C3⋊Dic3)⋊C3 | — | 3 | 8 | no | — |
| 11 | H(5,𝔽3) | C2×C4 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 12 | #6294 | — | 9 | 4 | no | — |
| 13 | Dic3×C3⋊D3 | — | 9 | 2 | no | — |
| 14 | #6199 | — | 9 | 12 | no | — |
| 15 | C2×C3×H(3,𝔽3) | — | 2 | 4 | no | — |
| 16 | C2×C3×H(3,𝔽3) | — | 2 | 12 | no | — |
| 17 | C2×C3×H(3,𝔽3) | C2×D3 | 1 | 8 | yes | — |
| 18 | C3⋊#272 | — | 3 | 16 | no | — |
| 19 | C3⋊#272 | — | 6 | 24 | no | — |
| 20 | C3⋊#272 | — | 6 | 8 | no | — |
| 21 | C2×C3×C3⋊D3 | — | 9 | 4 | no | — |
| 22 | C2×C3×C3⋊D3 | — | 9 | 2 | no | — |
| 23 | C2×C3×C3⋊D3 | — | 18 | 8 | no | — |
| 24 | C2×C3×C3⋊D3 | — | 18 | 3 | no | — |
| 25 | C2×C3×C3⋊D3 | — | 18 | 6 | no | — |
| 26 | C3⋊(C3⋊Dic3) | C3⋊D3 | 1 | 2 | yes | complemented |
| 27 | C3×C3⋊Dic3 | — | 3 | 8 | no | — |
| 28 | C32×Dic3 | — | 9 | 2 | no | — |
| 29 | C2×#272 | — | 9 | 12 | no | — |
| 30 | C2×#272 | — | 18 | 24 | no | — |
| 31 | C2×#272 | — | 18 | 6 | no | — |
| 32 | C2×#272 | — | 18 | 8 | no | — |
| 33 | C2×#272 | — | 18 | 1 | no | — |
| 34 | (C3⋊Dic3)⋊C3 | — | 3 | 24 | no | — |
| 35 | C3×H(3,𝔽3) | — | 2 | 4 | no | — |
| 36 | C3×H(3,𝔽3) | — | 2 | 12 | no | — |
| 37 | C3×H(3,𝔽3) | C4×D3 | 1 | 8 | yes | complemented |
| 38 | C4×C3⋊D3 | — | 27 | 2 | no | — |
| 39 | #624 | — | 27 | 4 | no | — |
| 40 | D3×Dic3 | — | 27 | 8 | no | — |
| 41 | C2×C33 | — | 2 | 6 | no | — |
| 42 | C2×C33 | — | 2 | 3 | no | — |
| 43 | C2×C33 | — | 2 | 8 | no | — |
| 44 | C2×C33 | D32 | 1 | 4 | yes | — |
| 45 | C2×C33 | C2×C3⋊D3 | 1 | 2 | yes | — |
| 46 | C3×C3⋊D3 | — | 9 | 4 | no | — |
| 47 | C3×C3⋊D3 | — | 9 | 8 | no | — |
| 48 | C3×C3⋊D3 | — | 18 | 16 | no | — |
| 49 | C3×C3⋊D3 | — | 18 | 6 | no | — |
| 50 | C3×C3⋊D3 | — | 18 | 12 | no | — |
| 51 | C2×H(3,𝔽3) | — | 2 | 1 | no | — |
| 52 | C2×H(3,𝔽3) | — | 2 | 8 | no | — |
| 53 | C2×H(3,𝔽3) | — | 2 | 6 | no | — |
| 54 | C2×H(3,𝔽3) | — | 2 | 24 | no | — |
| 55 | C2×H(3,𝔽3) | D32 | 1 | 12 | yes | — |
| 56 | #272 | — | 9 | 24 | no | — |
| 57 | #272 | — | 18 | 48 | no | — |
| 58 | #272 | — | 18 | 12 | no | — |
| 59 | #272 | — | 18 | 16 | no | — |
| 60 | #272 | — | 18 | 2 | no | — |
| 61 | C2×C3⋊D3 | — | 9 | 2 | no | — |
| 62 | C2×C3⋊D3 | — | 9 | 4 | no | — |
| 63 | C2×C3⋊D3 | — | 18 | 8 | no | — |
| 64 | C2×C3⋊D3 | — | 18 | 6 | no | — |
| 65 | C2×C3⋊D3 | — | 18 | 3 | no | — |
| 66 | C2×C3×D3 | — | 27 | 8 | no | — |
| 67 | C2×C3×D3 | — | 54 | 12 | no | — |
| 68 | C2×C3×D3 | — | 54 | 4 | no | — |
| 69 | C32×C4 | — | 27 | 2 | no | — |
| 70 | C3⋊Dic3 | — | 3 | 8 | no | — |
| 71 | C3⋊Dic3 | — | 27 | 2 | no | — |
| 72 | C3×Dic3 | — | 9 | 8 | no | — |
| 73 | C3×Dic3 | — | 27 | 8 | no | — |
| 74 | C33 | — | 2 | 6 | no | — |
| 75 | C33 | — | 2 | 3 | no | — |
| 76 | C33 | — | 2 | 8 | no | — |
| 77 | C33 | #624 | 1 | 4 | yes | complemented |
| 78 | C33 | C4×C3⋊D3 | 1 | 2 | yes | complemented |
| 79 | H(3,𝔽3) | — | 2 | 1 | no | — |
| 80 | H(3,𝔽3) | — | 2 | 8 | no | — |
| 81 | H(3,𝔽3) | — | 2 | 6 | no | — |
| 82 | H(3,𝔽3) | — | 2 | 24 | no | — |
| 83 | H(3,𝔽3) | #624 | 1 | 12 | yes | complemented |
| 84 | C2×Dic3 | — | 81 | 1 | no | — |
| 85 | C4×D3 | — | 81 | 8 | no | — |
| 86 | C2×C32 | — | 2 | 4 | no | — |
| 87 | C2×C32 | — | 2 | 12 | no | — |
| 88 | C2×C32 | — | 9 | 2 | no | — |
| 89 | C2×C32 | — | 9 | 4 | no | — |
| 90 | C2×C32 | — | 18 | 8 | no | — |
| 91 | C2×C32 | — | 18 | 6 | no | — |
| 92 | C2×C32 | — | 18 | 3 | no | — |
| 93 | C2×C32 | D3×C3⋊D3 | 1 | 8 | yes | — |
| 94 | C3⋊D3 | — | 9 | 4 | no | — |
| 95 | C3⋊D3 | — | 9 | 8 | no | — |
| 96 | C3⋊D3 | — | 18 | 16 | no | — |
| 97 | C3⋊D3 | — | 18 | 12 | no | — |
| 98 | C3⋊D3 | — | 18 | 6 | no | — |
| 99 | C3×D3 | — | 27 | 16 | no | — |
| 100 | C3×D3 | — | 54 | 24 | no | — |
| 101 | C3×D3 | — | 54 | 8 | no | — |
| 102 | C22×C3 | — | 81 | 1 | no | — |
| 103 | C2×D3 | — | 27 | 8 | no | — |
| 104 | C2×D3 | — | 54 | 4 | no | — |
| 105 | C2×D3 | — | 54 | 12 | no | — |
| 106 | C3×C4 | — | 27 | 8 | no | — |
| 107 | Dic3 | — | 9 | 2 | no | — |
| 108 | Dic3 | — | 27 | 8 | no | — |
| 109 | C32 | — | 2 | 4 | no | — |
| 110 | C32 | — | 2 | 12 | no | — |
| 111 | C32 | — | 9 | 2 | no | — |
| 112 | C32 | — | 9 | 4 | no | — |
| 113 | C32 | — | 18 | 8 | no | — |
| 114 | C32 | — | 18 | 6 | no | — |
| 115 | C32 | — | 18 | 3 | no | — |
| 116 | C32 | #6289 | 1 | 8 | yes | — |
| 117 | C2×C4 | — | 243 | 1 | no | Sylow |
| 118 | C2×C3 | — | 3 | 8 | no | — |
| 119 | C2×C3 | — | 6 | 4 | no | — |
| 120 | C2×C3 | — | 6 | 12 | no | — |
| 121 | C2×C3 | — | 81 | 2 | no | — |
| 122 | C2×C3 | C3⋊D32 | 1 | 1 | yes | characteristic |
| 123 | D3 | — | 27 | 16 | no | — |
| 124 | D3 | — | 54 | 8 | no | — |
| 125 | D3 | — | 54 | 24 | no | — |
| 126 | C22 | — | 81 | 1 | no | — |
| 127 | C4 | — | 27 | 2 | no | — |
| 128 | C3 | — | 3 | 8 | no | — |
| 129 | C3 | — | 6 | 4 | no | — |
| 130 | C3 | — | 6 | 12 | no | — |
| 131 | C3 | #10620150 | 1 | 1 | yes | characteristic |
| 132 | C2 | — | 81 | 2 | no | — |
| 133 | C2 | #11758086 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #11758086 |
| commutator, hypocenter | H(5,𝔽3) | C2×C4 |
| Frattini, socle | C2×C3 | C3⋊D32 |
| Fitting, cosocle | C2×H(5,𝔽3) | C22 |
| prime radical | C2×#92404 | C2 |
| LCS | G ▷ H(5,𝔽3) | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #11758012 | C2 | semidirect |
| 5. #92404 | C4 | semidirect |
| 9. C3⋊((C3⋊Dic3)⋊C3) | D3 | semidirect |
| 11. H(5,𝔽3) | C2×C4 | semidirect |
| 26. C3⋊(C3⋊Dic3) | C3⋊D3 | semidirect |
| 37. C3×H(3,𝔽3) | C4×D3 | semidirect |
| 77. C33 | #624 | semidirect |
| 78. C33 | C4×C3⋊D3 | semidirect |
| 83. H(3,𝔽3) | #624 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#92404 | 2 |
| #11758012 | 2 |
| #10619885 | 3 |