Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
232 conjugacy classes of subgroups; only the 21 normal subgroups are shown.
This group has 232 conjugacy classes of subgroups — too many to draw. Only its 21 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 | #11724209 | C2 | 1 | 2 | yes | maximal |
| 2 | SL(2,𝔽3)×#157 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #10593477 | — | 3 | 1 | no | maximal |
| 4 | C3⋊D3×SL(2,𝔽3) | C22 | 1 | 1 | yes | characteristic |
| 5 | C3×C3⋊#226 | — | 2 | 2 | no | — |
| 6 | #88940 | — | 4 | 1 | no | maximal |
| 7 | Q8×#157 | D3 | 1 | 1 | yes | characteristic |
| 8 | #64222 | — | 3 | 1 | no | — |
| 9 | D3×#226 | — | 6 | 2 | no | — |
| 10 | #63944 | — | 3 | 2 | no | — |
| 11 | #63732 | — | 3 | 1 | no | — |
| 12 | #63730 | — | 3 | 2 | no | — |
| 13 | C2×C3×#157 | — | 4 | 1 | no | — |
| 14 | C32×SL(2,𝔽3) | D4 | 1 | 1 | yes | characteristic |
| 15 | #6294 | — | 4 | 2 | no | — |
| 16 | C4⋊#226 | — | 9 | 1 | no | maximal |
| 17 | Q8×C3⋊D3 | C2×D3 | 1 | 1 | yes | characteristic |
| 18 | #3797 | — | 3 | 2 | no | — |
| 19 | C4⋊#157 | — | 3 | 1 | no | — |
| 20 | C4×#157 | — | 3 | 1 | no | — |
| 21 | D3×SL(2,𝔽3) | — | 6 | 2 | no | — |
| 22 | C3⋊#226 | — | 2 | 2 | no | — |
| 23 | C3×#226 | — | 6 | 2 | no | — |
| 24 | C3×#226 | — | 12 | 2 | no | — |
| 25 | #3773 | — | 6 | 1 | no | — |
| 26 | #3773 | — | 6 | 1 | no | — |
| 27 | C3×#216 | — | 6 | 2 | no | — |
| 28 | #3719 | — | 6 | 2 | no | — |
| 29 | #3710 | — | 3 | 2 | no | — |
| 30 | C2×C3×C3⋊D3 | — | 4 | 1 | no | — |
| 31 | C3×#157 | — | 4 | 2 | no | — |
| 32 | C3×C3⋊Dic3 | — | 8 | 2 | no | — |
| 33 | C2×#226 | — | 9 | 2 | no | — |
| 34 | D3×Q16 | — | 18 | 2 | no | — |
| 35 | C4×SL(2,𝔽3) | — | 9 | 1 | no | — |
| 36 | C2×#157 | — | 3 | 1 | no | — |
| 37 | C2×#157 | S4 | 1 | 1 | yes | characteristic |
| 38 | C32×Q8 | D12 | 1 | 1 | yes | characteristic |
| 39 | C4×C3⋊D3 | — | 3 | 1 | no | — |
| 40 | #634 | — | 3 | 1 | no | — |
| 41 | C3×Dic6 | — | 6 | 2 | no | — |
| 42 | C3×SL(2,𝔽3) | — | 2 | 2 | no | — |
| 43 | C3×SL(2,𝔽3) | — | 4 | 2 | no | — |
| 44 | #627 | — | 6 | 2 | no | — |
| 45 | #624 | — | 6 | 2 | no | — |
| 46 | D3×Dic3 | — | 24 | 2 | no | — |
| 47 | C3×C3⋊C8 | — | 6 | 2 | no | — |
| 48 | #462 | — | 27 | 1 | no | Sylow |
| 49 | C2×C33 | — | 4 | 1 | no | — |
| 50 | C3×C3⋊D3 | — | 4 | 1 | no | — |
| 51 | C3×C3⋊D3 | — | 4 | 1 | no | — |
| 52 | D3×Q8 | — | 6 | 2 | no | — |
| 53 | D3×Q8 | — | 18 | 2 | no | — |
| 54 | C2×SL(2,𝔽3) | — | 9 | 1 | no | — |
| 55 | #226 | — | 6 | 2 | no | — |
| 56 | #226 | — | 12 | 2 | no | — |
| 57 | C3×Q16 | — | 18 | 2 | no | — |
| 58 | #216 | — | 6 | 2 | no | — |
| 59 | #216 | — | 18 | 2 | no | — |
| 60 | C4⋊Dic3 | — | 36 | 1 | no | — |
| 61 | Dic12 | — | 18 | 2 | no | — |
| 62 | D3×C8 | — | 18 | 2 | no | — |
| 63 | C2×C3⋊D3 | C2×S4 | 1 | 1 | yes | characteristic |
| 64 | C2×C3×D3 | — | 24 | 2 | no | — |
| 65 | #157 | — | 6 | 1 | no | — |
| 66 | #157 | #226 | 1 | 2 | yes | complemented |
| 67 | C32×C4 | — | 3 | 1 | no | — |
| 68 | C3⋊Dic3 | — | 3 | 1 | no | — |
| 69 | C3⋊Dic3 | — | 8 | 2 | no | — |
| 70 | C3×Dic3 | — | 12 | 2 | no | — |
| 71 | C3×Dic3 | — | 24 | 2 | no | — |
| 72 | C3×Dic3 | — | 48 | 2 | no | — |
| 73 | C2×Q16 | — | 27 | 2 | no | — |
| 74 | C4⋊Q8 | — | 27 | 1 | no | — |
| 75 | C4×Q8 | — | 9 | 1 | no | — |
| 76 | C4⋊C8 | — | 27 | 1 | no | — |
| 77 | #103 | — | 27 | 2 | no | — |
| 78 | C33 | — | 4 | 1 | no | Sylow |
| 79 | C3×Q8 | — | 2 | 2 | no | — |
| 80 | C3×Q8 | — | 18 | 2 | no | — |
| 81 | C2×C3×C4 | — | 36 | 1 | no | — |
| 82 | C2×Dic3 | — | 36 | 2 | no | — |
| 83 | C4×D3 | — | 18 | 2 | no | — |
| 84 | C4×D3 | — | 36 | 2 | no | — |
| 85 | Dic6 | — | 6 | 2 | no | — |
| 86 | Dic6 | — | 18 | 2 | no | — |
| 87 | Dic6 | — | 36 | 2 | no | — |
| 88 | SL(2,𝔽3) | — | 4 | 2 | no | — |
| 89 | SL(2,𝔽3) | GO+(4,𝔽2) | 1 | 1 | yes | characteristic |
| 90 | C3×C8 | — | 18 | 2 | no | — |
| 91 | C3⋊C8 | — | 6 | 2 | no | — |
| 92 | C2×C32 | — | 8 | 2 | no | — |
| 93 | C2×C32 | — | 16 | 2 | no | — |
| 94 | C2×C32 | C4⋊S4 | 1 | 1 | yes | characteristic |
| 95 | C3⋊D3 | C2×#226 | 1 | 1 | yes | characteristic |
| 96 | C3⋊D3 | #984 | 1 | 1 | yes | characteristic |
| 97 | C3×D3 | — | 24 | 2 | no | — |
| 98 | C3×D3 | — | 24 | 2 | no | — |
| 99 | C2×Q8 | — | 9 | 1 | no | — |
| 100 | C2×Q8 | — | 27 | 2 | no | — |
| 101 | Q16 | — | 18 | 2 | no | — |
| 102 | Q16 | — | 54 | 2 | no | — |
| 103 | C2×C8 | — | 27 | 2 | no | — |
| 104 | C4⋊C4 | — | 27 | 1 | no | — |
| 105 | C4⋊C4 | — | 54 | 1 | no | — |
| 106 | C4⋊C4 | — | 54 | 1 | no | — |
| 107 | C42 | — | 27 | 1 | no | — |
| 108 | C22×C3 | — | 36 | 1 | no | — |
| 109 | C2×D3 | — | 6 | 2 | no | — |
| 110 | C3×C4 | — | 6 | 2 | no | — |
| 111 | C3×C4 | — | 36 | 2 | no | — |
| 112 | C3×C4 | — | 36 | 2 | no | — |
| 113 | Dic3 | — | 12 | 2 | no | — |
| 114 | Dic3 | — | 18 | 2 | no | — |
| 115 | Dic3 | — | 24 | 2 | no | — |
| 116 | Dic3 | — | 48 | 2 | no | — |
| 117 | C32 | — | 8 | 2 | no | — |
| 118 | C32 | — | 16 | 2 | no | — |
| 119 | C32 | C4⋊#226 | 1 | 1 | yes | characteristic, complemented |
| 120 | Q8 | — | 18 | 2 | no | — |
| 121 | Q8 | — | 27 | 1 | no | — |
| 122 | Q8 | — | 54 | 2 | no | — |
| 123 | Q8 | #157⋊D3 | 1 | 1 | yes | characteristic |
| 124 | C2×C4 | — | 9 | 1 | no | — |
| 125 | C2×C4 | — | 27 | 1 | no | — |
| 126 | C2×C4 | — | 27 | 1 | no | — |
| 127 | C2×C4 | — | 54 | 2 | no | — |
| 128 | C8 | — | 18 | 2 | no | — |
| 129 | C2×C3 | — | 2 | 2 | no | — |
| 130 | C2×C3 | — | 4 | 1 | no | — |
| 131 | C2×C3 | — | 16 | 2 | no | — |
| 132 | C2×C3 | — | 36 | 1 | no | — |
| 133 | C2×C3 | — | 36 | 1 | no | — |
| 134 | D3 | — | 6 | 2 | no | — |
| 135 | D3 | — | 6 | 2 | no | — |
| 136 | C22 | — | 9 | 1 | no | — |
| 137 | C4 | — | 3 | 1 | no | — |
| 138 | C4 | — | 9 | 2 | no | — |
| 139 | C4 | — | 27 | 1 | no | — |
| 140 | C4 | — | 36 | 2 | no | — |
| 141 | C4 | — | 54 | 1 | no | — |
| 142 | C3 | — | 2 | 2 | no | — |
| 143 | C3 | — | 4 | 1 | no | — |
| 144 | C3 | — | 16 | 2 | no | — |
| 145 | C2 | — | 9 | 1 | no | — |
| 146 | C2 | — | 9 | 1 | no | — |
| 147 | C2 | #157⋊S4 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 66. #157 | #226 | semidirect |
| 119. C32 | C4⋊#226 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11724209 | 2 |
| SL(2,𝔽3)×#157 | 2 |
| #10593477 | 3 |
| #88940 | 4 |
| C4⋊#226 | 9 |