Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
261 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 261 conjugacy classes of subgroups — too many to draw. Only its 35 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×C9⋊GL(2,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×C9×#226 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | SL(2,𝔽3)×Dic9 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #10593208 | — | 3 | 1 | no | maximal |
| 5 | #10588865 | — | 3 | 1 | no | maximal |
| 6 | C2×C9×SL(2,𝔽3) | C22 | 1 | 1 | yes | characteristic |
| 7 | C9⋊GL(2,𝔽3) | — | 2 | 1 | no | — |
| 8 | C9×#226 | — | 2 | 1 | no | — |
| 9 | #88445 | — | 4 | 1 | no | maximal |
| 10 | C2×C3⋊GL(2,𝔽3) | — | 3 | 1 | no | — |
| 11 | C2×C3×#226 | D3 | 1 | 1 | yes | characteristic |
| 12 | Dic3×SL(2,𝔽3) | — | 3 | 1 | no | — |
| 13 | C2×#3689 | — | 6 | 1 | no | — |
| 14 | C2×C9×Q16 | — | 3 | 1 | no | — |
| 15 | #63509 | — | 3 | 1 | no | — |
| 16 | Q8×Dic9 | D3 | 1 | 1 | yes | characteristic, complemented |
| 17 | C2×#3675 | — | 3 | 1 | no | — |
| 18 | #63395 | — | 3 | 1 | no | — |
| 19 | #63380 | — | 3 | 1 | no | — |
| 20 | Dic9⋊C8 | — | 3 | 1 | no | — |
| 21 | C22×C3⋊D9 | — | 4 | 1 | no | — |
| 22 | C2×C9×Dic3 | — | 4 | 1 | no | — |
| 23 | C2×C3×Dic9 | — | 4 | 1 | no | — |
| 24 | C9×SL(2,𝔽3) | D4 | 1 | 1 | yes | characteristic |
| 25 | #5395 | — | 9 | 1 | no | — |
| 26 | #5185 | — | 9 | 1 | no | — |
| 27 | C2×C3×SL(2,𝔽3) | C2×D3 | 1 | 1 | yes | characteristic |
| 28 | C3⋊GL(2,𝔽3) | — | 6 | 1 | no | — |
| 29 | C3×#226 | — | 2 | 1 | no | — |
| 30 | #3722 | — | 12 | 1 | no | — |
| 31 | C2×Q8×C9 | — | 3 | 1 | no | — |
| 32 | C2×Q8×C9 | C2×D3 | 1 | 1 | yes | characteristic |
| 33 | C2×D36 | — | 3 | 1 | no | — |
| 34 | C2×Q8⋊C9 | — | 2 | 1 | no | — |
| 35 | #3689 | — | 6 | 2 | no | — |
| 36 | C9×Q16 | — | 6 | 1 | no | — |
| 37 | C9×Q16 | — | 6 | 1 | no | — |
| 38 | C2×C8×C9 | — | 3 | 1 | no | — |
| 39 | #3675 | — | 6 | 1 | no | — |
| 40 | #3675 | — | 6 | 1 | no | — |
| 41 | (C2×D9)⋊C4 | — | 6 | 1 | no | — |
| 42 | (C2×D9)⋊C4 | — | 6 | 1 | no | — |
| 43 | C4⋊Dic9 | — | 3 | 1 | no | — |
| 44 | C4×Dic9 | — | 3 | 1 | no | — |
| 45 | C2×C9⋊C8 | — | 3 | 1 | no | — |
| 46 | C22×C3×C9 | — | 4 | 1 | no | — |
| 47 | C2×C3⋊D9 | — | 4 | 2 | no | — |
| 48 | C2×C3⋊D9 | — | 8 | 1 | no | — |
| 49 | C2×C3⋊D9 | — | 8 | 1 | no | — |
| 50 | C9×Dic3 | — | 8 | 1 | no | — |
| 51 | C3×Dic9 | — | 8 | 1 | no | — |
| 52 | C2×GL(2,𝔽3) | — | 9 | 1 | no | — |
| 53 | C2×GL(2,𝔽3) | — | 18 | 1 | no | — |
| 54 | C2×#226 | D9 | 1 | 1 | yes | characteristic, complemented |
| 55 | C2×C3×Q16 | — | 3 | 1 | no | — |
| 56 | #947 | — | 9 | 1 | no | — |
| 57 | Q8×Dic3 | — | 3 | 1 | no | — |
| 58 | C2×#215 | — | 9 | 1 | no | — |
| 59 | C4×SL(2,𝔽3) | — | 9 | 1 | no | — |
| 60 | #835 | — | 9 | 1 | no | — |
| 61 | #821 | — | 9 | 1 | no | — |
| 62 | Dic3⋊C8 | — | 9 | 1 | no | — |
| 63 | C22×C3⋊D3 | — | 12 | 1 | no | — |
| 64 | C2×C3×Dic3 | — | 4 | 1 | no | — |
| 65 | C2×C3×Dic3 | — | 12 | 1 | no | — |
| 66 | C3×SL(2,𝔽3) | #67 | 1 | 1 | yes | characteristic |
| 67 | C22×D9 | — | 6 | 1 | no | — |
| 68 | C22×D9 | — | 24 | 1 | no | — |
| 69 | Q8×C9 | — | 3 | 1 | no | — |
| 70 | Q8×C9 | — | 6 | 1 | no | — |
| 71 | Q8×C9 | — | 6 | 1 | no | — |
| 72 | Q8×C9 | D12 | 1 | 1 | yes | characteristic |
| 73 | C2×C4×C9 | — | 3 | 1 | no | — |
| 74 | C2×C4×C9 | — | 6 | 1 | no | — |
| 75 | C2×Dic9 | — | 3 | 1 | no | — |
| 76 | C2×Dic9 | S4 | 1 | 1 | yes | characteristic |
| 77 | D36 | — | 6 | 1 | no | — |
| 78 | D36 | — | 6 | 1 | no | — |
| 79 | Q8⋊C9 | — | 2 | 1 | no | — |
| 80 | C8×C9 | — | 6 | 1 | no | — |
| 81 | C9⋊C8 | — | 6 | 1 | no | — |
| 82 | #464 | — | 27 | 1 | no | Sylow |
| 83 | C2×C3×C9 | — | 4 | 1 | no | — |
| 84 | C2×C3×C9 | — | 4 | 1 | no | — |
| 85 | C2×C3×C9 | — | 4 | 1 | no | — |
| 86 | C3⋊D9 | — | 8 | 2 | no | — |
| 87 | C2×C3×Q8 | — | 3 | 1 | no | — |
| 88 | C2×C3×Q8 | D32 | 1 | 1 | yes | characteristic |
| 89 | C2×D12 | — | 9 | 1 | no | — |
| 90 | C2×SL(2,𝔽3) | — | 2 | 1 | no | — |
| 91 | C2×SL(2,𝔽3) | C2×D9 | 1 | 1 | yes | characteristic |
| 92 | GL(2,𝔽3) | — | 18 | 2 | no | — |
| 93 | GL(2,𝔽3) | — | 18 | 1 | no | — |
| 94 | #226 | — | 2 | 1 | no | — |
| 95 | C3×Q16 | — | 6 | 1 | no | — |
| 96 | C3×Q16 | — | 6 | 1 | no | — |
| 97 | C2×C3×C8 | — | 3 | 1 | no | — |
| 98 | #215 | — | 18 | 1 | no | — |
| 99 | #215 | — | 18 | 1 | no | — |
| 100 | (C2×D3)⋊C4 | — | 18 | 1 | no | — |
| 101 | (C2×D3)⋊C4 | — | 18 | 1 | no | — |
| 102 | (C2×D3)⋊C4 | — | 36 | 1 | no | — |
| 103 | C4⋊Dic3 | — | 9 | 1 | no | — |
| 104 | C4×Dic3 | — | 9 | 1 | no | — |
| 105 | C2×C3⋊C8 | — | 9 | 1 | no | — |
| 106 | C22×C32 | — | 4 | 1 | no | — |
| 107 | C2×C3⋊D3 | — | 12 | 2 | no | — |
| 108 | C2×C3⋊D3 | — | 24 | 1 | no | — |
| 109 | C2×C3⋊D3 | — | 24 | 1 | no | — |
| 110 | C3×Dic3 | — | 8 | 1 | no | — |
| 111 | C3×Dic3 | — | 24 | 1 | no | — |
| 112 | C22×C9 | — | 8 | 1 | no | — |
| 113 | C22×C9 | C2×S4 | 1 | 1 | yes | characteristic |
| 114 | C2×D9 | — | 12 | 1 | no | — |
| 115 | C2×D9 | — | 12 | 1 | no | — |
| 116 | C2×D9 | — | 12 | 1 | no | — |
| 117 | C2×D9 | — | 24 | 2 | no | — |
| 118 | C2×D9 | — | 24 | 2 | no | — |
| 119 | C2×D9 | — | 24 | 2 | no | — |
| 120 | C4×C9 | — | 3 | 1 | no | — |
| 121 | C4×C9 | — | 3 | 1 | no | — |
| 122 | C4×C9 | — | 12 | 1 | no | — |
| 123 | Dic9 | — | 2 | 1 | no | — |
| 124 | Dic9 | — | 6 | 1 | no | — |
| 125 | C2×Q16 | — | 3 | 1 | no | — |
| 126 | C2×SD16 | — | 27 | 1 | no | — |
| 127 | #124 | — | 27 | 1 | no | — |
| 128 | C4×Q8 | — | 9 | 1 | no | — |
| 129 | C4⋊C8 | — | 27 | 1 | no | — |
| 130 | #103 | — | 27 | 1 | no | — |
| 131 | #102 | — | 27 | 1 | no | — |
| 132 | C3×C9 | — | 4 | 1 | no | Sylow |
| 133 | C22×D3 | — | 18 | 1 | no | — |
| 134 | C22×D3 | — | 36 | 1 | no | — |
| 135 | C22×D3 | — | 72 | 1 | no | — |
| 136 | C3×Q8 | — | 3 | 1 | no | — |
| 137 | C3×Q8 | — | 6 | 1 | no | — |
| 138 | C3×Q8 | — | 6 | 1 | no | — |
| 139 | C3×Q8 | #626 | 1 | 1 | yes | characteristic |
| 140 | C2×C3×C4 | — | 3 | 1 | no | — |
| 141 | C2×C3×C4 | — | 6 | 1 | no | — |
| 142 | C2×C3×C4 | — | 36 | 1 | no | — |
| 143 | C2×Dic3 | — | 3 | 1 | no | — |
| 144 | C2×Dic3 | — | 4 | 1 | no | — |
| 145 | C2×Dic3 | — | 9 | 1 | no | — |
| 146 | D12 | — | 18 | 1 | no | — |
| 147 | D12 | — | 18 | 1 | no | — |
| 148 | SL(2,𝔽3) | — | 2 | 1 | no | — |
| 149 | SL(2,𝔽3) | #611 | 1 | 1 | yes | characteristic |
| 150 | C3×C8 | — | 6 | 1 | no | — |
| 151 | C3⋊C8 | — | 18 | 1 | no | — |
| 152 | C2×C32 | — | 4 | 1 | no | — |
| 153 | C2×C32 | — | 4 | 1 | no | — |
| 154 | C2×C32 | — | 4 | 1 | no | — |
| 155 | C3⋊D3 | — | 24 | 2 | no | — |
| 156 | C2×C9 | — | 8 | 1 | no | — |
| 157 | C2×C9 | — | 8 | 1 | no | — |
| 158 | C2×C9 | — | 8 | 1 | no | — |
| 159 | C2×C9 | C4⋊S4 | 1 | 1 | yes | characteristic |
| 160 | C2×C9 | #983 | 1 | 1 | yes | characteristic |
| 161 | C2×C9 | #985 | 1 | 1 | yes | characteristic |
| 162 | D9 | — | 24 | 4 | no | — |
| 163 | D9 | — | 24 | 1 | no | — |
| 164 | C2×Q8 | — | 3 | 1 | no | — |
| 165 | C2×Q8 | D3×D9 | 1 | 1 | yes | characteristic |
| 166 | C2×D4 | — | 27 | 1 | no | — |
| 167 | Q16 | — | 6 | 1 | no | — |
| 168 | Q16 | — | 6 | 1 | no | — |
| 169 | SD16 | — | 54 | 1 | no | — |
| 170 | SD16 | — | 54 | 1 | no | — |
| 171 | C2×C8 | — | 3 | 1 | no | — |
| 172 | C2×C8 | — | 27 | 1 | no | — |
| 173 | C4⋊C4 | — | 27 | 1 | no | — |
| 174 | C22⋊C4 | — | 54 | 1 | no | — |
| 175 | C22⋊C4 | — | 54 | 1 | no | — |
| 176 | C42 | — | 27 | 1 | no | — |
| 177 | C22×C3 | — | 4 | 1 | no | — |
| 178 | C22×C3 | — | 8 | 1 | no | — |
| 179 | C22×C3 | D3×S4 | 1 | 1 | yes | characteristic |
| 180 | C2×D3 | — | 36 | 2 | no | — |
| 181 | C2×D3 | — | 36 | 1 | no | — |
| 182 | C2×D3 | — | 36 | 1 | no | — |
| 183 | C2×D3 | — | 36 | 1 | no | — |
| 184 | C2×D3 | — | 72 | 1 | no | — |
| 185 | C2×D3 | — | 72 | 1 | no | — |
| 186 | C2×D3 | — | 72 | 2 | no | — |
| 187 | C2×D3 | — | 72 | 2 | no | — |
| 188 | C2×D3 | — | 72 | 2 | no | — |
| 189 | C3×C4 | — | 3 | 1 | no | — |
| 190 | C3×C4 | — | 3 | 1 | no | — |
| 191 | C3×C4 | — | 12 | 1 | no | — |
| 192 | C3×C4 | — | 72 | 1 | no | — |
| 193 | Dic3 | — | 6 | 1 | no | — |
| 194 | Dic3 | — | 8 | 1 | no | — |
| 195 | Dic3 | — | 18 | 1 | no | — |
| 196 | C32 | — | 4 | 1 | no | — |
| 197 | C9 | — | 8 | 1 | no | — |
| 198 | C9 | #5395 | 1 | 1 | yes | characteristic, complemented |
| 199 | C23 | — | 54 | 1 | no | — |
| 200 | Q8 | — | 3 | 1 | no | — |
| 201 | Q8 | — | 6 | 1 | no | — |
| 202 | Q8 | — | 6 | 1 | no | — |
| 203 | Q8 | #6195 | 1 | 1 | yes | characteristic |
| 204 | D4 | — | 54 | 1 | no | — |
| 205 | D4 | — | 54 | 1 | no | — |
| 206 | C2×C4 | — | 3 | 1 | no | — |
| 207 | C2×C4 | — | 6 | 1 | no | — |
| 208 | C2×C4 | — | 9 | 1 | no | — |
| 209 | C2×C4 | — | 27 | 1 | no | — |
| 210 | C8 | — | 6 | 1 | no | — |
| 211 | C8 | — | 54 | 1 | no | — |
| 212 | C2×C3 | — | 4 | 1 | no | — |
| 213 | C2×C3 | — | 4 | 1 | no | — |
| 214 | C2×C3 | — | 4 | 1 | no | — |
| 215 | C2×C3 | — | 8 | 1 | no | — |
| 216 | C2×C3 | — | 8 | 1 | no | — |
| 217 | C2×C3 | — | 8 | 1 | no | — |
| 218 | C2×C3 | #64198 | 1 | 1 | yes | characteristic |
| 219 | C2×C3 | #64202 | 1 | 1 | yes | characteristic |
| 220 | C2×C3 | #64207 | 1 | 1 | yes | characteristic |
| 221 | D3 | — | 72 | 2 | no | — |
| 222 | D3 | — | 72 | 4 | no | — |
| 223 | D3 | — | 72 | 1 | no | — |
| 224 | C22 | — | 108 | 1 | no | — |
| 225 | C22 | — | 108 | 1 | no | — |
| 226 | C22 | — | 108 | 1 | no | — |
| 227 | C22 | D9×S4 | 1 | 1 | yes | characteristic |
| 228 | C4 | — | 3 | 1 | no | — |
| 229 | C4 | — | 3 | 1 | no | — |
| 230 | C4 | — | 12 | 1 | no | — |
| 231 | C4 | — | 18 | 1 | no | — |
| 232 | C4 | — | 54 | 1 | no | — |
| 233 | C3 | — | 4 | 1 | no | — |
| 234 | C3 | — | 8 | 1 | no | — |
| 235 | C3 | #10593208 | 1 | 1 | yes | characteristic |
| 236 | C2 | — | 216 | 1 | no | — |
| 237 | C2 | #11722232 | 1 | 1 | yes | characteristic |
| 238 | C2 | #11722234 | 1 | 1 | yes | characteristic |
| 239 | C2 | #11722243 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C9×#226 | C2 | semidirect |
| 3. SL(2,𝔽3)×Dic9 | C2 | semidirect |
| 16. Q8×Dic9 | D3 | semidirect |
| 54. C2×#226 | D9 | semidirect |
| 198. C9 | #5395 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C9⋊GL(2,𝔽3) | 2 |
| C2×C9×#226 | 2 |
| SL(2,𝔽3)×Dic9 | 2 |
| #10593208 | 3 |
| #10588865 | 3 |
| #88445 | 4 |