Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
2255 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 2255 conjugacy classes of subgroups — too many to draw. Only its 95 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 | C24×#272 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×C22⋊#1088 | C2 | 1 | 6 | yes | maximal, complemented |
| 3 | C24×H(3,𝔽3) | C4 | 1 | 1 | yes | characteristic, complemented |
| 4 | C23×#272 | — | 2 | 12 | no | — |
| 5 | C23×#272 | C4 | 1 | 3 | yes | complemented |
| 6 | C23×#272 | C22 | 1 | 3 | yes | — |
| 7 | C22×#1088 | C22 | 1 | 4 | yes | complemented |
| 8 | C22⋊#1088 | — | 2 | 12 | no | — |
| 9 | C24×C3×D3 | — | 6 | 2 | no | — |
| 10 | C23×H(3,𝔽3) | — | 2 | 6 | no | — |
| 11 | C23×H(3,𝔽3) | C2×C4 | 1 | 3 | yes | — |
| 12 | C22×#272 | — | 2 | 32 | no | — |
| 13 | C22×#272 | — | 2 | 6 | no | — |
| 14 | C22×#272 | — | 2 | 12 | no | — |
| 15 | C22×#272 | — | 2 | 12 | no | — |
| 16 | C22×#272 | C2×C4 | 1 | 3 | yes | — |
| 17 | C22×#272 | D4 | 1 | 12 | yes | — |
| 18 | C22×#272 | C23 | 1 | 1 | yes | characteristic |
| 19 | C2×#1088 | — | 2 | 12 | no | — |
| 20 | C3×#379 | — | 9 | 1 | no | maximal |
| 21 | C24×C32 | — | 2 | 2 | no | — |
| 22 | C23×C3×D3 | — | 6 | 6 | no | — |
| 23 | C23×C3×D3 | — | 6 | 6 | no | — |
| 24 | C23×C3×D3 | — | 6 | 48 | no | — |
| 25 | C22×H(3,𝔽3) | — | 2 | 8 | no | — |
| 26 | C22×H(3,𝔽3) | — | 2 | 6 | no | — |
| 27 | C22×H(3,𝔽3) | C22⋊C4 | 1 | 6 | yes | complemented |
| 28 | C22×H(3,𝔽3) | C22×C4 | 1 | 1 | yes | characteristic |
| 29 | C2×#272 | — | 2 | 24 | no | — |
| 30 | C2×#272 | — | 2 | 6 | no | — |
| 31 | C2×#272 | — | 2 | 24 | no | — |
| 32 | C2×#272 | C22⋊C4 | 1 | 6 | yes | complemented |
| 33 | C2×#272 | C2×D4 | 1 | 6 | yes | — |
| 34 | #1088 | — | 4 | 4 | no | — |
| 35 | C25×C3 | — | 9 | 1 | no | — |
| 36 | C24×D3 | — | 6 | 2 | no | — |
| 37 | C2×C3×C22⋊C4 | — | 9 | 6 | no | — |
| 38 | C23×C32 | — | 2 | 24 | no | — |
| 39 | C23×C32 | — | 2 | 6 | no | — |
| 40 | C22×C3×D3 | — | 6 | 6 | no | — |
| 41 | C22×C3×D3 | — | 6 | 24 | no | — |
| 42 | C22×C3×D3 | — | 6 | 2 | no | — |
| 43 | C22×C3×D3 | — | 6 | 48 | no | — |
| 44 | C22×C3×D3 | — | 6 | 48 | no | — |
| 45 | C22×C3×D3 | — | 6 | 128 | no | — |
| 46 | C22×C3×D3 | — | 6 | 24 | no | — |
| 47 | #379 | — | 9 | 1 | no | Sylow |
| 48 | C2×H(3,𝔽3) | — | 2 | 6 | no | — |
| 49 | C2×H(3,𝔽3) | C2×C22⋊C4 | 1 | 3 | yes | — |
| 50 | #272 | — | 2 | 6 | no | — |
| 51 | #272 | #120 | 1 | 4 | yes | — |
| 52 | C24×C3 | — | 6 | 2 | no | — |
| 53 | C24×C3 | — | 9 | 3 | no | — |
| 54 | C24×C3 | — | 9 | 3 | no | — |
| 55 | C24×C3 | — | 18 | 12 | no | — |
| 56 | C24×C3 | #157 | 1 | 1 | yes | characteristic |
| 57 | C23×D3 | — | 6 | 6 | no | — |
| 58 | C23×D3 | — | 6 | 6 | no | — |
| 59 | C23×D3 | — | 6 | 48 | no | — |
| 60 | C22×C3×C4 | — | 9 | 4 | no | — |
| 61 | C3×C22⋊C4 | — | 18 | 12 | no | — |
| 62 | C22×C32 | — | 2 | 32 | no | — |
| 63 | C22×C32 | — | 2 | 24 | no | — |
| 64 | C22×C32 | — | 2 | 2 | no | — |
| 65 | C22×C32 | — | 2 | 12 | no | — |
| 66 | C2×C3×D3 | — | 6 | 12 | no | — |
| 67 | C2×C3×D3 | — | 6 | 12 | no | — |
| 68 | C2×C3×D3 | — | 6 | 96 | no | — |
| 69 | C2×C3×D3 | — | 6 | 96 | no | — |
| 70 | C2×C3×D3 | — | 6 | 24 | no | — |
| 71 | C25 | — | 9 | 1 | no | — |
| 72 | C2×C22⋊C4 | — | 9 | 6 | no | — |
| 73 | H(3,𝔽3) | #379 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 74 | C23×C3 | — | 2 | 6 | no | — |
| 75 | C23×C3 | — | 6 | 24 | no | — |
| 76 | C23×C3 | — | 6 | 6 | no | — |
| 77 | C23×C3 | — | 9 | 1 | no | — |
| 78 | C23×C3 | — | 9 | 3 | no | — |
| 79 | C23×C3 | — | 9 | 12 | no | — |
| 80 | C23×C3 | — | 18 | 12 | no | — |
| 81 | C23×C3 | — | 18 | 12 | no | — |
| 82 | C23×C3 | — | 18 | 32 | no | — |
| 83 | C23×C3 | — | 18 | 6 | no | — |
| 84 | C23×C3 | C2×#157 | 1 | 3 | yes | — |
| 85 | C22×D3 | — | 6 | 2 | no | — |
| 86 | C22×D3 | — | 6 | 24 | no | — |
| 87 | C22×D3 | — | 6 | 6 | no | — |
| 88 | C22×D3 | — | 6 | 48 | no | — |
| 89 | C22×D3 | — | 6 | 48 | no | — |
| 90 | C22×D3 | — | 6 | 128 | no | — |
| 91 | C22×D3 | — | 6 | 24 | no | — |
| 92 | C2×C3×C4 | — | 18 | 12 | no | — |
| 93 | C2×C32 | — | 2 | 24 | no | — |
| 94 | C2×C32 | — | 2 | 6 | no | — |
| 95 | C3×D3 | — | 6 | 8 | no | — |
| 96 | C3×D3 | — | 6 | 24 | no | — |
| 97 | C24 | — | 9 | 3 | no | — |
| 98 | C24 | — | 9 | 3 | no | — |
| 99 | C24 | — | 18 | 12 | no | — |
| 100 | C24 | #1088 | 1 | 1 | yes | characteristic, complemented |
| 101 | C22×C4 | — | 9 | 4 | no | — |
| 102 | C22⋊C4 | — | 18 | 12 | no | — |
| 103 | C22×C3 | — | 2 | 8 | no | — |
| 104 | C22×C3 | — | 2 | 6 | no | — |
| 105 | C22×C3 | — | 6 | 32 | no | — |
| 106 | C22×C3 | — | 6 | 24 | no | — |
| 107 | C22×C3 | — | 6 | 12 | no | — |
| 108 | C22×C3 | — | 6 | 2 | no | — |
| 109 | C22×C3 | — | 9 | 6 | no | — |
| 110 | C22×C3 | — | 9 | 6 | no | — |
| 111 | C22×C3 | — | 18 | 24 | no | — |
| 112 | C22×C3 | — | 18 | 24 | no | — |
| 113 | C22×C3 | — | 18 | 6 | no | — |
| 114 | C22×C3 | C22⋊#157 | 1 | 6 | yes | — |
| 115 | C22×C3 | C22×#157 | 1 | 1 | yes | characteristic |
| 116 | C2×D3 | — | 6 | 12 | no | — |
| 117 | C2×D3 | — | 6 | 12 | no | — |
| 118 | C2×D3 | — | 6 | 96 | no | — |
| 119 | C2×D3 | — | 6 | 96 | no | — |
| 120 | C2×D3 | — | 6 | 24 | no | — |
| 121 | C3×C4 | — | 36 | 4 | no | — |
| 122 | C32 | — | 2 | 2 | no | — |
| 123 | C23 | — | 2 | 6 | no | — |
| 124 | C23 | — | 9 | 1 | no | — |
| 125 | C23 | — | 9 | 12 | no | — |
| 126 | C23 | — | 9 | 3 | no | — |
| 127 | C23 | — | 18 | 12 | no | — |
| 128 | C23 | — | 18 | 12 | no | — |
| 129 | C23 | — | 18 | 32 | no | — |
| 130 | C23 | — | 18 | 6 | no | — |
| 131 | C23 | C2×#1088 | 1 | 3 | yes | — |
| 132 | C2×C4 | — | 18 | 12 | no | — |
| 133 | C2×C3 | — | 2 | 6 | no | — |
| 134 | C2×C3 | — | 6 | 24 | no | — |
| 135 | C2×C3 | — | 6 | 6 | no | — |
| 136 | C2×C3 | — | 9 | 4 | no | — |
| 137 | C2×C3 | — | 18 | 6 | no | — |
| 138 | C2×C3 | C2×C22⋊#157 | 1 | 3 | yes | — |
| 139 | D3 | — | 6 | 8 | no | — |
| 140 | D3 | — | 6 | 24 | no | — |
| 141 | C22 | — | 2 | 8 | no | — |
| 142 | C22 | — | 2 | 6 | no | — |
| 143 | C22 | — | 9 | 6 | no | — |
| 144 | C22 | — | 9 | 6 | no | — |
| 145 | C22 | — | 18 | 24 | no | — |
| 146 | C22 | — | 18 | 24 | no | — |
| 147 | C22 | — | 18 | 6 | no | — |
| 148 | C22 | C22⋊#1088 | 1 | 6 | yes | complemented |
| 149 | C22 | C22×#1088 | 1 | 1 | yes | characteristic |
| 150 | C4 | — | 36 | 4 | no | — |
| 151 | C3 | — | 6 | 2 | no | — |
| 152 | C3 | #10593847 | 1 | 1 | yes | characteristic |
| 153 | C2 | — | 2 | 6 | no | — |
| 154 | C2 | — | 9 | 4 | no | — |
| 155 | C2 | — | 18 | 6 | no | — |
| 156 | C2 | C2×C22⋊#1088 | 1 | 3 | yes | — |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C22⋊#1088 | C2 | semidirect |
| 3. C24×H(3,𝔽3) | C4 | semidirect |
| 5. C23×#272 | C4 | semidirect |
| 7. C22×#1088 | C22 | semidirect |
| 27. C22×H(3,𝔽3) | C22⋊C4 | semidirect |
| 32. C2×#272 | C22⋊C4 | semidirect |
| 73. H(3,𝔽3) | #379 | semidirect |
| 100. C24 | #1088 | semidirect |
| 148. C22 | C22⋊#1088 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×#272 | 2 |
| C2×C22⋊#1088 | 2 |
| C3×#379 | 9 |