Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
244 conjugacy classes of subgroups; only the 29 normal subgroups are shown.
This group has 244 conjugacy classes of subgroups — too many to draw. Only its 29 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 | C9×C22⋊SL(2,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C3⋊#5930 | — | 3 | 1 | no | maximal |
| 3 | #10593105 | — | 3 | 2 | no | maximal |
| 4 | #10588907 | — | 3 | 1 | no | maximal |
| 5 | C9⋊SL(2,ℤ/4) | — | 4 | 1 | no | maximal |
| 6 | C9⋊#226 | — | 4 | 2 | no | maximal |
| 7 | C3×C22⋊SL(2,𝔽3) | D3 | 1 | 1 | yes | characteristic |
| 8 | C22×Q8×C9 | D3 | 1 | 1 | yes | characteristic |
| 9 | C22⋊(Q8⋊C9) | D3 | 1 | 2 | yes | — |
| 10 | C2×#3674 | — | 3 | 2 | no | — |
| 11 | #63486 | — | 3 | 1 | no | — |
| 12 | #63395 | — | 3 | 2 | no | — |
| 13 | #63389 | — | 3 | 1 | no | — |
| 14 | C2×C9×A4 | — | 4 | 1 | no | — |
| 15 | C9×SL(2,𝔽3) | — | 4 | 2 | no | — |
| 16 | C9×SL(2,𝔽3) | — | 8 | 1 | no | — |
| 17 | #5930 | — | 9 | 3 | no | — |
| 18 | #5228 | — | 9 | 1 | no | — |
| 19 | C3⋊SL(2,ℤ/4) | — | 12 | 1 | no | — |
| 20 | C3⋊#226 | — | 12 | 2 | no | — |
| 21 | C2×Q8×C9 | — | 3 | 2 | no | — |
| 22 | C2×Q8×C9 | — | 6 | 1 | no | — |
| 23 | C22×C4×C9 | — | 3 | 1 | no | — |
| 24 | C2×Dic18 | — | 3 | 1 | no | — |
| 25 | (C22⋊C9)⋊C4 | — | 12 | 2 | no | — |
| 26 | #3688 | — | 12 | 4 | no | — |
| 27 | #3676 | — | 6 | 1 | no | — |
| 28 | #3674 | — | 6 | 2 | no | — |
| 29 | #3674 | — | 6 | 2 | no | — |
| 30 | C4⋊Dic9 | — | 3 | 1 | no | — |
| 31 | Dic9⋊C4 | — | 6 | 1 | no | — |
| 32 | C2×C9⋊C8 | — | 3 | 2 | no | — |
| 33 | C9×A4 | — | 4 | 1 | no | — |
| 34 | C3⋊Dic9 | — | 16 | 1 | no | — |
| 35 | C22×C3×Q8 | C3⋊D3 | 1 | 1 | yes | characteristic |
| 36 | C22⋊SL(2,𝔽3) | D9 | 1 | 3 | yes | — |
| 37 | C2×#216 | — | 9 | 2 | no | — |
| 38 | #924 | — | 9 | 1 | no | — |
| 39 | #835 | — | 9 | 2 | no | — |
| 40 | #830 | — | 9 | 1 | no | — |
| 41 | C2×C3×A4 | — | 4 | 1 | no | — |
| 42 | C3×SL(2,𝔽3) | — | 4 | 2 | no | — |
| 43 | C3×SL(2,𝔽3) | — | 8 | 1 | no | — |
| 44 | C23×C9 | S4 | 1 | 1 | yes | characteristic |
| 45 | C2×C22⋊C9 | — | 4 | 2 | no | — |
| 46 | Q8×C9 | — | 2 | 1 | no | — |
| 47 | Q8×C9 | — | 3 | 2 | no | — |
| 48 | Q8×C9 | — | 6 | 1 | no | — |
| 49 | Q8×C9 | S4 | 1 | 2 | yes | — |
| 50 | C2×C4×C9 | — | 3 | 1 | no | — |
| 51 | C2×C4×C9 | — | 3 | 1 | no | — |
| 52 | C2×C4×C9 | — | 6 | 2 | no | — |
| 53 | C2×Dic9 | — | 6 | 1 | no | — |
| 54 | C2×Dic9 | — | 6 | 1 | no | — |
| 55 | Dic18 | — | 6 | 2 | no | — |
| 56 | Q8⋊C9 | — | 4 | 4 | no | — |
| 57 | Q8⋊C9 | — | 8 | 2 | no | — |
| 58 | C9⋊C8 | — | 6 | 2 | no | — |
| 59 | #451 | — | 27 | 1 | no | Sylow |
| 60 | C2×C3×C9 | — | 16 | 1 | no | — |
| 61 | C2×C3×Q8 | — | 3 | 2 | no | — |
| 62 | C2×C3×Q8 | — | 6 | 1 | no | — |
| 63 | C22×C3×C4 | — | 3 | 1 | no | — |
| 64 | C2×Dic6 | — | 9 | 1 | no | — |
| 65 | SL(2,ℤ/4) | — | 36 | 3 | no | — |
| 66 | #226 | — | 36 | 6 | no | — |
| 67 | #217 | — | 18 | 1 | no | — |
| 68 | #216 | — | 18 | 2 | no | — |
| 69 | #216 | — | 18 | 2 | no | — |
| 70 | C4⋊Dic3 | — | 9 | 1 | no | — |
| 71 | Dic3⋊C4 | — | 18 | 1 | no | — |
| 72 | C2×C3⋊C8 | — | 9 | 2 | no | — |
| 73 | C3×A4 | — | 4 | 1 | no | — |
| 74 | C3⋊Dic3 | — | 48 | 1 | no | — |
| 75 | C22×C9 | — | 3 | 1 | no | — |
| 76 | C22×C9 | — | 3 | 1 | no | — |
| 77 | C22×C9 | #226 | 1 | 1 | yes | characteristic, complemented |
| 78 | C22⋊C9 | — | 4 | 2 | no | — |
| 79 | C4×C9 | — | 3 | 2 | no | — |
| 80 | C4×C9 | — | 6 | 1 | no | — |
| 81 | Dic9 | — | 12 | 1 | no | — |
| 82 | Dic9 | — | 12 | 1 | no | — |
| 83 | Dic9 | — | 48 | 2 | no | — |
| 84 | C22×Q8 | C3⋊D9 | 1 | 1 | yes | characteristic |
| 85 | C2×Q16 | — | 27 | 2 | no | — |
| 86 | C22⋊Q8 | — | 27 | 1 | no | — |
| 87 | #103 | — | 27 | 2 | no | — |
| 88 | C22⋊C8 | — | 27 | 1 | no | — |
| 89 | C3×C9 | — | 16 | 1 | no | Sylow |
| 90 | C23×C3 | C3⋊S4 | 1 | 1 | yes | characteristic |
| 91 | C2×A4 | — | 4 | 3 | no | — |
| 92 | C3×Q8 | — | 2 | 1 | no | — |
| 93 | C3×Q8 | — | 3 | 2 | no | — |
| 94 | C3×Q8 | — | 6 | 1 | no | — |
| 95 | C3×Q8 | C3⋊S4 | 1 | 2 | yes | — |
| 96 | C2×C3×C4 | — | 3 | 1 | no | — |
| 97 | C2×C3×C4 | — | 3 | 1 | no | — |
| 98 | C2×C3×C4 | — | 6 | 2 | no | — |
| 99 | C2×Dic3 | — | 18 | 1 | no | — |
| 100 | C2×Dic3 | — | 18 | 1 | no | — |
| 101 | Dic6 | — | 18 | 2 | no | — |
| 102 | SL(2,𝔽3) | — | 4 | 6 | no | — |
| 103 | SL(2,𝔽3) | — | 8 | 3 | no | — |
| 104 | C3⋊C8 | — | 18 | 2 | no | — |
| 105 | C2×C32 | — | 16 | 1 | no | — |
| 106 | C2×C9 | — | 3 | 1 | no | — |
| 107 | C2×C9 | — | 3 | 1 | no | — |
| 108 | C2×C9 | — | 16 | 2 | no | — |
| 109 | C2×C9 | #1020 | 1 | 1 | yes | characteristic |
| 110 | C2×Q8 | — | 3 | 2 | no | — |
| 111 | C2×Q8 | — | 6 | 1 | no | — |
| 112 | C2×Q8 | — | 27 | 1 | no | — |
| 113 | C22×C4 | — | 3 | 1 | no | — |
| 114 | Q16 | — | 54 | 2 | no | — |
| 115 | Q16 | — | 54 | 2 | no | — |
| 116 | C2×C8 | — | 27 | 2 | no | — |
| 117 | C4⋊C4 | — | 27 | 1 | no | — |
| 118 | C4⋊C4 | — | 54 | 1 | no | — |
| 119 | C22⋊C4 | — | 54 | 1 | no | — |
| 120 | C22×C3 | — | 3 | 1 | no | — |
| 121 | C22×C3 | — | 3 | 1 | no | — |
| 122 | C22×C3 | C3⋊#226 | 1 | 1 | yes | characteristic |
| 123 | A4 | — | 4 | 3 | no | — |
| 124 | C3×C4 | — | 3 | 2 | no | — |
| 125 | C3×C4 | — | 6 | 1 | no | — |
| 126 | Dic3 | — | 36 | 1 | no | — |
| 127 | Dic3 | — | 36 | 1 | no | — |
| 128 | Dic3 | — | 144 | 3 | no | — |
| 129 | C32 | — | 16 | 1 | no | — |
| 130 | C9 | — | 16 | 2 | no | — |
| 131 | C9 | #5930 | 1 | 1 | yes | characteristic, complemented |
| 132 | C23 | C9⋊S4 | 1 | 1 | yes | characteristic |
| 133 | Q8 | — | 2 | 1 | no | — |
| 134 | Q8 | — | 3 | 2 | no | — |
| 135 | Q8 | — | 6 | 1 | no | — |
| 136 | Q8 | — | 54 | 2 | no | — |
| 137 | Q8 | C9⋊S4 | 1 | 2 | yes | — |
| 138 | C2×C4 | — | 3 | 1 | no | — |
| 139 | C2×C4 | — | 3 | 1 | no | — |
| 140 | C2×C4 | — | 6 | 2 | no | — |
| 141 | C2×C4 | — | 54 | 1 | no | — |
| 142 | C2×C4 | — | 54 | 1 | no | — |
| 143 | C8 | — | 54 | 2 | no | — |
| 144 | C2×C3 | — | 3 | 1 | no | — |
| 145 | C2×C3 | — | 3 | 1 | no | — |
| 146 | C2×C3 | — | 16 | 3 | no | — |
| 147 | C2×C3 | C3⋊#1020 | 1 | 1 | yes | characteristic |
| 148 | C22 | — | 3 | 1 | no | — |
| 149 | C22 | — | 3 | 1 | no | — |
| 150 | C22 | C9⋊#226 | 1 | 1 | yes | characteristic, complemented |
| 151 | C4 | — | 3 | 2 | no | — |
| 152 | C4 | — | 6 | 1 | no | — |
| 153 | C4 | — | 108 | 1 | no | — |
| 154 | C4 | — | 108 | 1 | no | — |
| 155 | C3 | — | 16 | 3 | no | — |
| 156 | C3 | C3⋊#5930 | 1 | 1 | yes | characteristic |
| 157 | C2 | — | 3 | 1 | no | — |
| 158 | C2 | — | 3 | 1 | no | — |
| 159 | C2 | C9⋊#1020 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 77. C22×C9 | #226 | semidirect |
| 131. C9 | #5930 | semidirect |
| 150. C22 | C9⋊#226 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C9×C22⋊SL(2,𝔽3) | 2 |
| C3⋊#5930 | 3 |
| #10593105 | 3 |
| #10588907 | 3 |
| C9⋊SL(2,ℤ/4) | 4 |
| C9⋊#226 | 4 |