Finite Group DatabasebetaFinite Group DatabasebetaOrder 1944.
978 conjugacy classes of subgroups; only the 22 normal subgroups are shown.
This group has 978 conjugacy classes of subgroups — too many to draw. Only its 22 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 | C3×C3⋊D32 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | C3×#66472 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #10620206 | C3 | 1 | 1 | yes | maximal, characteristic, direct product factor |
| 4 | C3×C3⋊(C3⋊(C3⋊D3)) | C22 | 1 | 1 | yes | characteristic |
| 5 | C33×C3⋊D3 | — | 2 | 2 | no | — |
| 6 | C3⋊D32 | C2×C3 | 1 | 2 | yes | complemented |
| 7 | C3×D3×C3⋊D3 | — | 6 | 8 | no | — |
| 8 | #66472 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 9 | C35 | D4 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 10 | C3×GO+(4,𝔽2) | — | 9 | 4 | no | maximal |
| 11 | C3⋊(C3⋊(C3⋊D3)) | C22×C3 | 1 | 1 | yes | characteristic |
| 12 | C3×C3⋊(C3⋊D3) | — | 6 | 8 | no | — |
| 13 | C3×C3⋊(C3⋊D3) | — | 12 | 6 | no | — |
| 14 | C32×C3⋊D3 | — | 2 | 2 | no | — |
| 15 | C32×C3⋊D3 | — | 2 | 8 | no | — |
| 16 | C32×C3⋊D3 | — | 4 | 8 | no | — |
| 17 | C33×D3 | — | 6 | 8 | no | — |
| 18 | C2×C3×C3⋊D3 | — | 18 | 2 | no | — |
| 19 | D3×C3⋊D3 | — | 6 | 8 | no | — |
| 20 | C3×D32 | — | 9 | 8 | no | — |
| 21 | C3×D32 | — | 18 | 12 | no | — |
| 22 | C3×#157 | — | 9 | 4 | no | — |
| 23 | C3×#157 | — | 18 | 3 | no | — |
| 24 | C34 | — | 2 | 8 | no | — |
| 25 | C34 | — | 4 | 6 | no | — |
| 26 | C34 | — | 4 | 8 | no | — |
| 27 | C34 | — | 8 | 6 | no | — |
| 28 | C34 | C3×D4 | 1 | 1 | yes | characteristic, complemented |
| 29 | GO+(4,𝔽2) | — | 9 | 4 | no | — |
| 30 | C2×C33 | — | 18 | 2 | no | — |
| 31 | C3⋊(C3⋊D3) | — | 6 | 8 | no | — |
| 32 | C3⋊(C3⋊D3) | — | 12 | 6 | no | — |
| 33 | C3×C3⋊D3 | — | 2 | 8 | no | — |
| 34 | C3×C3⋊D3 | — | 2 | 2 | no | — |
| 35 | C3×C3⋊D3 | — | 4 | 8 | no | — |
| 36 | C3×C3⋊D3 | — | 9 | 4 | no | — |
| 37 | C3×C3⋊D3 | — | 18 | 2 | no | — |
| 38 | C3×C3⋊D3 | — | 18 | 12 | no | — |
| 39 | C3×C3⋊D3 | — | 18 | 3 | no | — |
| 40 | C3×C3⋊D3 | — | 36 | 12 | no | — |
| 41 | C3×C3⋊D3 | — | 36 | 8 | no | — |
| 42 | C3×C3⋊D3 | — | 36 | 3 | no | — |
| 43 | C32×D3 | — | 6 | 8 | no | — |
| 44 | C32×D3 | — | 6 | 24 | no | — |
| 45 | C32×D3 | — | 6 | 8 | no | — |
| 46 | C32×D3 | — | 12 | 24 | no | — |
| 47 | C32×D3 | — | 12 | 8 | no | — |
| 48 | C2×C3⋊D3 | — | 18 | 2 | no | — |
| 49 | C2×C3×D3 | — | 54 | 8 | no | — |
| 50 | D32 | — | 9 | 8 | no | — |
| 51 | D32 | — | 18 | 12 | no | — |
| 52 | #157 | — | 9 | 4 | no | — |
| 53 | #157 | — | 18 | 3 | no | — |
| 54 | C33 | — | 2 | 2 | no | — |
| 55 | C33 | — | 2 | 8 | no | — |
| 56 | C33 | — | 2 | 12 | no | — |
| 57 | C33 | — | 2 | 3 | no | — |
| 58 | C33 | — | 4 | 6 | no | — |
| 59 | C33 | — | 4 | 12 | no | — |
| 60 | C33 | — | 4 | 24 | no | — |
| 61 | C33 | — | 4 | 8 | no | — |
| 62 | C33 | — | 4 | 8 | no | — |
| 63 | C33 | — | 4 | 3 | no | — |
| 64 | C33 | — | 4 | 8 | no | — |
| 65 | C33 | — | 8 | 12 | no | — |
| 66 | C33 | — | 8 | 6 | no | — |
| 67 | C33 | — | 8 | 24 | no | — |
| 68 | C33 | — | 8 | 8 | no | — |
| 69 | C33 | — | 8 | 24 | no | — |
| 70 | C33 | — | 8 | 24 | no | — |
| 71 | C33 | — | 8 | 12 | no | — |
| 72 | C33 | GO+(4,𝔽2) | 1 | 4 | yes | complemented |
| 73 | C3×D4 | — | 81 | 1 | no | — |
| 74 | C2×C32 | — | 18 | 2 | no | — |
| 75 | C2×C32 | — | 18 | 8 | no | — |
| 76 | C2×C32 | — | 36 | 8 | no | — |
| 77 | C3⋊D3 | — | 2 | 2 | no | — |
| 78 | C3⋊D3 | — | 9 | 4 | no | — |
| 79 | C3⋊D3 | — | 18 | 2 | no | — |
| 80 | C3⋊D3 | — | 18 | 12 | no | — |
| 81 | C3⋊D3 | — | 18 | 3 | no | — |
| 82 | C3⋊D3 | — | 36 | 12 | no | — |
| 83 | C3⋊D3 | — | 36 | 8 | no | — |
| 84 | C3⋊D3 | — | 36 | 3 | no | — |
| 85 | C3×D3 | — | 6 | 8 | no | — |
| 86 | C3×D3 | — | 6 | 24 | no | — |
| 87 | C3×D3 | — | 6 | 8 | no | — |
| 88 | C3×D3 | — | 12 | 8 | no | — |
| 89 | C3×D3 | — | 12 | 24 | no | — |
| 90 | C3×D3 | — | 54 | 8 | no | — |
| 91 | C3×D3 | — | 108 | 6 | no | — |
| 92 | C22×C3 | — | 81 | 2 | no | — |
| 93 | C2×D3 | — | 54 | 8 | no | — |
| 94 | C3×C4 | — | 81 | 1 | no | — |
| 95 | C32 | — | 2 | 2 | no | — |
| 96 | C32 | — | 2 | 12 | no | — |
| 97 | C32 | — | 2 | 8 | no | — |
| 98 | C32 | — | 2 | 3 | no | — |
| 99 | C32 | — | 4 | 12 | no | — |
| 100 | C32 | — | 4 | 8 | no | — |
| 101 | C32 | — | 4 | 8 | no | — |
| 102 | C32 | — | 4 | 8 | no | — |
| 103 | C32 | — | 4 | 24 | no | — |
| 104 | C32 | — | 4 | 3 | no | — |
| 105 | C32 | — | 4 | 6 | no | — |
| 106 | C32 | — | 8 | 24 | no | — |
| 107 | C32 | — | 8 | 8 | no | — |
| 108 | C32 | — | 8 | 12 | no | — |
| 109 | C32 | — | 8 | 24 | no | — |
| 110 | C32 | — | 8 | 24 | no | — |
| 111 | C32 | — | 8 | 6 | no | — |
| 112 | C32 | — | 8 | 12 | no | — |
| 113 | C32 | C3×GO+(4,𝔽2) | 1 | 4 | yes | complemented |
| 114 | D4 | — | 81 | 1 | no | Sylow |
| 115 | C2×C3 | — | 18 | 8 | no | — |
| 116 | C2×C3 | — | 18 | 2 | no | — |
| 117 | C2×C3 | — | 36 | 8 | no | — |
| 118 | C2×C3 | — | 81 | 1 | no | — |
| 119 | D3 | — | 6 | 8 | no | — |
| 120 | D3 | — | 54 | 8 | no | — |
| 121 | D3 | — | 108 | 6 | no | — |
| 122 | C22 | — | 81 | 2 | no | — |
| 123 | C4 | — | 81 | 1 | no | — |
| 124 | C3 | — | 2 | 8 | no | — |
| 125 | C3 | — | 4 | 6 | no | — |
| 126 | C3 | — | 4 | 8 | no | — |
| 127 | C3 | — | 8 | 6 | no | — |
| 128 | C3 | #10620206 | 1 | 1 | yes | characteristic, direct product factor |
| 129 | C2 | — | 18 | 2 | no | — |
| 130 | C2 | — | 81 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | #10620206 |
| commutator, cosocle | C3⋊(C3⋊(C3⋊D3)) | C22×C3 |
| Frattini | C1 | G |
| Fitting, socle | C35 | D4 |
| hypocenter | C34 | C3×D4 |
| LCS | G ▷ C3⋊(C3⋊(C3⋊D3)) ▷ C34 | |
| UCS | C1 ◁ C3 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×C3⋊D32 | C2 | semidirect |
| 2. C3×#66472 | C2 | semidirect |
| 3. #10620206 | C3 | direct (N × H) |
| 6. C3⋊D32 | C2×C3 | semidirect |
| 8. #66472 | C2×C3 | semidirect |
| 9. C35 | D4 | semidirect |
| 28. C34 | C3×D4 | semidirect |
| 72. C33 | GO+(4,𝔽2) | semidirect |
| 113. C32 | C3×GO+(4,𝔽2) | semidirect |
| 128. C3 | #10620206 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×C3⋊D32 | 2 |
| C3×#66472 | 2 |
| #10620206 | 3 |
| C3×GO+(4,𝔽2) | 9 |