Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
739 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 739 conjugacy classes of subgroups — too many to draw. Only its 70 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×C3⋊D3×SL(2,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×C3×C3⋊#226 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #11724209 | C2 | 1 | 4 | yes | maximal, direct product factor |
| 4 | C2×D3×#226 | — | 3 | 2 | no | maximal |
| 5 | C2×#63944 | — | 3 | 1 | no | maximal |
| 6 | C2×C32×SL(2,𝔽3) | C22 | 1 | 1 | yes | characteristic |
| 7 | C2×#6294 | — | 4 | 1 | no | maximal |
| 8 | C3⋊D3×SL(2,𝔽3) | C22 | 1 | 2 | yes | — |
| 9 | C3×C3⋊#226 | C22 | 1 | 4 | yes | complemented |
| 10 | C2×Q8×C3⋊D3 | D3 | 1 | 1 | yes | characteristic |
| 11 | C2×#3797 | — | 3 | 1 | no | — |
| 12 | C2×D3×SL(2,𝔽3) | — | 3 | 2 | no | — |
| 13 | C2×D3×SL(2,𝔽3) | — | 6 | 1 | no | — |
| 14 | C2×C3⋊#226 | D3 | 1 | 2 | yes | complemented |
| 15 | C2×C3×#226 | — | 3 | 2 | no | — |
| 16 | C2×C3×#226 | — | 6 | 2 | no | — |
| 17 | D3×#226 | — | 3 | 8 | no | — |
| 18 | C2×C3×#216 | — | 3 | 2 | no | — |
| 19 | #63944 | — | 3 | 4 | no | — |
| 20 | #63944 | — | 3 | 4 | no | — |
| 21 | C2×#3719 | — | 3 | 2 | no | — |
| 22 | C2×#3710 | — | 3 | 1 | no | — |
| 23 | C22×C3×C3⋊D3 | — | 4 | 1 | no | — |
| 24 | C2×C3×C3⋊Dic3 | — | 4 | 2 | no | — |
| 25 | C32×SL(2,𝔽3) | C23 | 1 | 1 | yes | characteristic |
| 26 | #6294 | — | 4 | 4 | no | — |
| 27 | C22×#226 | — | 9 | 1 | no | — |
| 28 | C2×D3×Q16 | — | 9 | 2 | no | — |
| 29 | C2×C32×Q8 | C2×D3 | 1 | 1 | yes | characteristic |
| 30 | Q8×C3⋊D3 | — | 3 | 2 | no | — |
| 31 | Q8×C3⋊D3 | C2×D3 | 1 | 2 | yes | — |
| 32 | C2×C4×C3⋊D3 | — | 3 | 1 | no | — |
| 33 | C2×#634 | — | 3 | 1 | no | — |
| 34 | C2×C3×Dic6 | — | 3 | 2 | no | — |
| 35 | C2×C3×SL(2,𝔽3) | — | 2 | 2 | no | — |
| 36 | C2×C3×SL(2,𝔽3) | — | 2 | 1 | no | — |
| 37 | C2×C3×SL(2,𝔽3) | — | 4 | 1 | no | — |
| 38 | C2×C3×SL(2,𝔽3) | C2×D3 | 1 | 2 | yes | — |
| 39 | C2×#627 | — | 6 | 1 | no | — |
| 40 | C2×#624 | — | 6 | 1 | no | — |
| 41 | C2×D3×Dic3 | — | 12 | 2 | no | — |
| 42 | #3797 | — | 3 | 4 | no | — |
| 43 | #3797 | — | 6 | 2 | no | — |
| 44 | D3×SL(2,𝔽3) | — | 3 | 4 | no | — |
| 45 | D3×SL(2,𝔽3) | — | 6 | 2 | no | — |
| 46 | C3⋊#226 | C2×D3 | 1 | 4 | yes | complemented |
| 47 | C3×#226 | — | 3 | 4 | no | — |
| 48 | C3×#226 | — | 6 | 4 | no | — |
| 49 | C3×#216 | — | 3 | 4 | no | — |
| 50 | C3×#216 | — | 3 | 4 | no | — |
| 51 | C2×C3×C3⋊C8 | — | 3 | 2 | no | — |
| 52 | #3719 | — | 3 | 8 | no | — |
| 53 | #3710 | — | 3 | 4 | no | — |
| 54 | C22×C33 | — | 4 | 1 | no | — |
| 55 | C2×C3×C3⋊D3 | — | 4 | 4 | no | — |
| 56 | C2×C3×C3⋊D3 | — | 4 | 2 | no | — |
| 57 | C3×C3⋊Dic3 | — | 4 | 4 | no | — |
| 58 | C2×D3×Q8 | — | 3 | 2 | no | — |
| 59 | C2×D3×Q8 | — | 6 | 1 | no | — |
| 60 | C2×D3×Q8 | — | 9 | 2 | no | — |
| 61 | C22×SL(2,𝔽3) | — | 9 | 1 | no | — |
| 62 | C2×#226 | — | 3 | 2 | no | — |
| 63 | C2×#226 | — | 6 | 2 | no | — |
| 64 | C2×#226 | — | 9 | 4 | no | — |
| 65 | C2×C3×Q16 | — | 9 | 2 | no | — |
| 66 | C2×#216 | — | 3 | 2 | no | — |
| 67 | C2×#216 | — | 9 | 2 | no | — |
| 68 | D3×Q16 | — | 9 | 8 | no | — |
| 69 | D3×Q16 | — | 9 | 8 | no | — |
| 70 | C2×Dic12 | — | 9 | 2 | no | — |
| 71 | C2×D3×C8 | — | 9 | 2 | no | — |
| 72 | C22×C3⋊D3 | S4 | 1 | 1 | yes | characteristic |
| 73 | C22×C3×D3 | — | 12 | 2 | no | — |
| 74 | C22×C3×D3 | — | 24 | 1 | no | — |
| 75 | C32×Q8 | — | 3 | 1 | no | — |
| 76 | C32×Q8 | C22×D3 | 1 | 1 | yes | characteristic |
| 77 | C2×C32×C4 | — | 3 | 1 | no | — |
| 78 | C2×C3⋊Dic3 | — | 3 | 1 | no | — |
| 79 | C2×C3⋊Dic3 | — | 4 | 2 | no | — |
| 80 | C4×C3⋊D3 | — | 3 | 2 | no | — |
| 81 | C4×C3⋊D3 | — | 3 | 2 | no | — |
| 82 | #634 | — | 3 | 2 | no | — |
| 83 | #634 | — | 6 | 1 | no | — |
| 84 | C2×C3×Dic3 | — | 6 | 2 | no | — |
| 85 | C2×C3×Dic3 | — | 12 | 2 | no | — |
| 86 | C2×C3×Dic3 | — | 24 | 2 | no | — |
| 87 | C3×Dic6 | — | 3 | 4 | no | — |
| 88 | C3×Dic6 | — | 6 | 2 | no | — |
| 89 | C3×SL(2,𝔽3) | — | 2 | 2 | no | — |
| 90 | C3×SL(2,𝔽3) | — | 2 | 1 | no | — |
| 91 | C3×SL(2,𝔽3) | — | 4 | 1 | no | — |
| 92 | C3×SL(2,𝔽3) | C22×D3 | 1 | 2 | yes | — |
| 93 | #627 | — | 6 | 4 | no | — |
| 94 | #624 | — | 6 | 4 | no | — |
| 95 | D3×Dic3 | — | 12 | 8 | no | — |
| 96 | C3×C3⋊C8 | — | 3 | 4 | no | — |
| 97 | C22×Q16 | — | 27 | 1 | no | Sylow |
| 98 | C2×C33 | — | 4 | 2 | no | — |
| 99 | C2×C33 | — | 4 | 1 | no | — |
| 100 | C3×C3⋊D3 | — | 4 | 4 | no | — |
| 101 | C2×C3×Q8 | — | 2 | 1 | no | — |
| 102 | C2×C3×Q8 | — | 9 | 2 | no | — |
| 103 | C2×C3×Q8 | D32 | 1 | 2 | yes | — |
| 104 | C22×Dic3 | — | 36 | 1 | no | — |
| 105 | D3×Q8 | — | 3 | 4 | no | — |
| 106 | D3×Q8 | — | 6 | 2 | no | — |
| 107 | D3×Q8 | — | 9 | 8 | no | — |
| 108 | D3×Q8 | — | 9 | 4 | no | — |
| 109 | D3×Q8 | — | 18 | 2 | no | — |
| 110 | D3×Q8 | — | 18 | 4 | no | — |
| 111 | C2×C4×D3 | — | 9 | 2 | no | — |
| 112 | C2×C4×D3 | — | 18 | 1 | no | — |
| 113 | C2×C4×D3 | — | 18 | 2 | no | — |
| 114 | C2×Dic6 | — | 3 | 2 | no | — |
| 115 | C2×Dic6 | — | 9 | 2 | no | — |
| 116 | C2×Dic6 | — | 18 | 1 | no | — |
| 117 | C2×Dic6 | — | 18 | 2 | no | — |
| 118 | C2×SL(2,𝔽3) | — | 2 | 2 | no | — |
| 119 | C2×SL(2,𝔽3) | — | 4 | 1 | no | — |
| 120 | C2×SL(2,𝔽3) | — | 9 | 2 | no | — |
| 121 | C2×SL(2,𝔽3) | D32 | 1 | 1 | yes | characteristic |
| 122 | #226 | — | 3 | 4 | no | — |
| 123 | #226 | — | 6 | 4 | no | — |
| 124 | C3×Q16 | — | 9 | 4 | no | — |
| 125 | C3×Q16 | — | 9 | 4 | no | — |
| 126 | C2×C3×C8 | — | 9 | 2 | no | — |
| 127 | #216 | — | 3 | 4 | no | — |
| 128 | #216 | — | 3 | 4 | no | — |
| 129 | #216 | — | 9 | 8 | no | — |
| 130 | C2×C3⋊C8 | — | 3 | 2 | no | — |
| 131 | Dic12 | — | 9 | 8 | no | — |
| 132 | D3×C8 | — | 9 | 8 | no | — |
| 133 | C22×C32 | — | 4 | 2 | no | — |
| 134 | C22×C32 | — | 8 | 1 | no | — |
| 135 | C22×C32 | — | 8 | 2 | no | — |
| 136 | C22×C32 | — | 16 | 1 | no | — |
| 137 | C22×C32 | C2×S4 | 1 | 1 | yes | characteristic |
| 138 | C2×C3⋊D3 | #226 | 1 | 4 | yes | complemented |
| 139 | C2×C3⋊D3 | C2×S4 | 1 | 2 | yes | — |
| 140 | C2×C3×D3 | — | 12 | 4 | no | — |
| 141 | C2×C3×D3 | — | 12 | 8 | no | — |
| 142 | C2×C3×D3 | — | 24 | 4 | no | — |
| 143 | C2×C3×D3 | — | 24 | 2 | no | — |
| 144 | C32×C4 | — | 3 | 1 | no | — |
| 145 | C32×C4 | — | 3 | 1 | no | — |
| 146 | C3⋊Dic3 | — | 3 | 2 | no | — |
| 147 | C3⋊Dic3 | — | 4 | 4 | no | — |
| 148 | C3×Dic3 | — | 6 | 4 | no | — |
| 149 | C3×Dic3 | — | 12 | 4 | no | — |
| 150 | C3×Dic3 | — | 24 | 4 | no | — |
| 151 | C22×Q8 | — | 9 | 1 | no | — |
| 152 | C22×Q8 | — | 27 | 1 | no | — |
| 153 | C2×Q16 | — | 9 | 2 | no | — |
| 154 | C2×Q16 | — | 27 | 2 | no | — |
| 155 | C2×Q16 | — | 27 | 4 | no | — |
| 156 | C2×Q16 | — | 27 | 4 | no | — |
| 157 | C22×C8 | — | 27 | 1 | no | — |
| 158 | C33 | — | 4 | 1 | no | Sylow |
| 159 | C23×C3 | — | 36 | 1 | no | — |
| 160 | C22×D3 | — | 3 | 2 | no | — |
| 161 | C22×D3 | — | 6 | 1 | no | — |
| 162 | C3×Q8 | — | 2 | 1 | no | — |
| 163 | C3×Q8 | — | 3 | 2 | no | — |
| 164 | C3×Q8 | — | 6 | 1 | no | — |
| 165 | C3×Q8 | — | 9 | 4 | no | — |
| 166 | C3×Q8 | — | 18 | 2 | no | — |
| 167 | C3×Q8 | C2×D32 | 1 | 2 | yes | — |
| 168 | C2×C3×C4 | — | 3 | 2 | no | — |
| 169 | C2×C3×C4 | — | 6 | 1 | no | — |
| 170 | C2×C3×C4 | — | 18 | 2 | no | — |
| 171 | C2×Dic3 | — | 6 | 2 | no | — |
| 172 | C2×Dic3 | — | 9 | 2 | no | — |
| 173 | C2×Dic3 | — | 12 | 2 | no | — |
| 174 | C2×Dic3 | — | 18 | 1 | no | — |
| 175 | C2×Dic3 | — | 24 | 2 | no | — |
| 176 | C2×Dic3 | — | 36 | 4 | no | — |
| 177 | C4×D3 | — | 9 | 4 | no | — |
| 178 | C4×D3 | — | 9 | 4 | no | — |
| 179 | C4×D3 | — | 18 | 2 | no | — |
| 180 | C4×D3 | — | 18 | 2 | no | — |
| 181 | C4×D3 | — | 18 | 8 | no | — |
| 182 | Dic6 | — | 3 | 4 | no | — |
| 183 | Dic6 | — | 6 | 2 | no | — |
| 184 | Dic6 | — | 9 | 4 | no | — |
| 185 | Dic6 | — | 18 | 2 | no | — |
| 186 | Dic6 | — | 18 | 8 | no | — |
| 187 | Dic6 | — | 18 | 2 | no | — |
| 188 | Dic6 | — | 18 | 2 | no | — |
| 189 | SL(2,𝔽3) | — | 2 | 2 | no | — |
| 190 | SL(2,𝔽3) | — | 4 | 1 | no | — |
| 191 | SL(2,𝔽3) | C2×D32 | 1 | 1 | yes | characteristic |
| 192 | C3×C8 | — | 9 | 4 | no | — |
| 193 | C3⋊C8 | — | 3 | 4 | no | — |
| 194 | C2×C32 | — | 4 | 2 | no | — |
| 195 | C2×C32 | — | 4 | 4 | no | — |
| 196 | C2×C32 | — | 8 | 1 | no | — |
| 197 | C2×C32 | — | 8 | 2 | no | — |
| 198 | C2×C32 | — | 8 | 2 | no | — |
| 199 | C2×C32 | — | 8 | 4 | no | — |
| 200 | C2×C32 | — | 16 | 1 | no | — |
| 201 | C2×C32 | — | 16 | 2 | no | — |
| 202 | C2×C32 | C2×#226 | 1 | 2 | yes | complemented |
| 203 | C2×C32 | C22×S4 | 1 | 1 | yes | characteristic |
| 204 | C3⋊D3 | C2×#226 | 1 | 4 | yes | complemented |
| 205 | C3×D3 | — | 12 | 8 | no | — |
| 206 | C3×D3 | — | 24 | 4 | no | — |
| 207 | C2×Q8 | — | 9 | 2 | no | — |
| 208 | C2×Q8 | — | 9 | 2 | no | — |
| 209 | C2×Q8 | — | 27 | 1 | no | — |
| 210 | C2×Q8 | — | 27 | 2 | no | — |
| 211 | C2×Q8 | — | 27 | 4 | no | — |
| 212 | C2×Q8 | — | 54 | 1 | no | — |
| 213 | C2×Q8 | — | 54 | 2 | no | — |
| 214 | C2×Q8 | #1113 | 1 | 1 | yes | characteristic |
| 215 | C22×C4 | — | 27 | 1 | no | — |
| 216 | C22×C4 | — | 54 | 1 | no | — |
| 217 | Q16 | — | 9 | 4 | no | — |
| 218 | Q16 | — | 9 | 4 | no | — |
| 219 | Q16 | — | 27 | 8 | no | — |
| 220 | C2×C8 | — | 9 | 2 | no | — |
| 221 | C2×C8 | — | 27 | 4 | no | — |
| 222 | C22×C3 | — | 2 | 1 | no | — |
| 223 | C22×C3 | — | 4 | 1 | no | — |
| 224 | C22×C3 | — | 8 | 2 | no | — |
| 225 | C22×C3 | — | 16 | 1 | no | — |
| 226 | C22×C3 | — | 36 | 2 | no | — |
| 227 | C22×C3 | — | 36 | 4 | no | — |
| 228 | C22×C3 | D3×S4 | 1 | 2 | yes | — |
| 229 | C2×D3 | — | 3 | 4 | no | — |
| 230 | C2×D3 | — | 3 | 8 | no | — |
| 231 | C2×D3 | — | 6 | 2 | no | — |
| 232 | C2×D3 | — | 6 | 4 | no | — |
| 233 | C3×C4 | — | 3 | 2 | no | — |
| 234 | C3×C4 | — | 3 | 2 | no | — |
| 235 | C3×C4 | — | 6 | 1 | no | — |
| 236 | C3×C4 | — | 6 | 1 | no | — |
| 237 | C3×C4 | — | 18 | 4 | no | — |
| 238 | Dic3 | — | 6 | 4 | no | — |
| 239 | Dic3 | — | 9 | 4 | no | — |
| 240 | Dic3 | — | 12 | 4 | no | — |
| 241 | Dic3 | — | 18 | 2 | no | — |
| 242 | Dic3 | — | 24 | 4 | no | — |
| 243 | C32 | — | 4 | 2 | no | — |
| 244 | C32 | — | 8 | 1 | no | — |
| 245 | C32 | — | 8 | 2 | no | — |
| 246 | C32 | — | 16 | 1 | no | — |
| 247 | C32 | C22×#226 | 1 | 1 | yes | characteristic, complemented |
| 248 | C23 | — | 9 | 1 | no | — |
| 249 | Q8 | — | 3 | 1 | no | — |
| 250 | Q8 | — | 9 | 4 | no | — |
| 251 | Q8 | — | 18 | 2 | no | — |
| 252 | Q8 | — | 27 | 2 | no | — |
| 253 | Q8 | — | 54 | 1 | no | — |
| 254 | Q8 | — | 54 | 4 | no | — |
| 255 | Q8 | C2×#1113 | 1 | 1 | yes | characteristic |
| 256 | C2×C4 | — | 3 | 1 | no | — |
| 257 | C2×C4 | — | 18 | 2 | no | — |
| 258 | C2×C4 | — | 27 | 2 | no | — |
| 259 | C2×C4 | — | 27 | 2 | no | — |
| 260 | C2×C4 | — | 27 | 1 | no | — |
| 261 | C2×C4 | — | 54 | 4 | no | — |
| 262 | C8 | — | 9 | 4 | no | — |
| 263 | C2×C3 | — | 2 | 2 | no | — |
| 264 | C2×C3 | — | 2 | 1 | no | — |
| 265 | C2×C3 | — | 4 | 2 | no | — |
| 266 | C2×C3 | — | 4 | 1 | no | — |
| 267 | C2×C3 | — | 8 | 4 | no | — |
| 268 | C2×C3 | — | 8 | 2 | no | — |
| 269 | C2×C3 | — | 16 | 2 | no | — |
| 270 | C2×C3 | — | 16 | 1 | no | — |
| 271 | C2×C3 | — | 36 | 4 | no | — |
| 272 | C2×C3 | D3×#226 | 1 | 4 | yes | complemented |
| 273 | C2×C3 | C2×D3×S4 | 1 | 2 | yes | — |
| 274 | D3 | — | 3 | 8 | no | — |
| 275 | D3 | — | 6 | 4 | no | — |
| 276 | C22 | — | 9 | 4 | no | — |
| 277 | C22 | — | 9 | 2 | no | — |
| 278 | C22 | #89101 | 1 | 1 | yes | characteristic |
| 279 | C4 | — | 3 | 1 | no | — |
| 280 | C4 | — | 3 | 1 | no | — |
| 281 | C4 | — | 18 | 4 | no | — |
| 282 | C4 | — | 27 | 2 | no | — |
| 283 | C3 | — | 2 | 1 | no | — |
| 284 | C3 | — | 4 | 1 | no | — |
| 285 | C3 | — | 8 | 2 | no | — |
| 286 | C3 | — | 16 | 1 | no | — |
| 287 | C3 | C2×D3×#226 | 1 | 2 | yes | complemented |
| 288 | C2 | — | 9 | 4 | no | — |
| 289 | C2 | #11724209 | 1 | 2 | yes | direct product factor |
| 290 | C2 | C2×#89101 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C3×C3⋊#226 | C2 | semidirect |
| 3. #11724209 | C2 | direct (N × H) |
| 9. C3×C3⋊#226 | C22 | semidirect |
| 14. C2×C3⋊#226 | D3 | semidirect |
| 46. C3⋊#226 | C2×D3 | semidirect |
| 138. C2×C3⋊D3 | #226 | semidirect |
| 202. C2×C32 | C2×#226 | semidirect |
| 204. C3⋊D3 | C2×#226 | semidirect |
| 247. C32 | C22×#226 | semidirect |
| 272. C2×C3 | D3×#226 | semidirect |
| 287. C3 | C2×D3×#226 | semidirect |
| 289. C2 | #11724209 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3⋊D3×SL(2,𝔽3) | 2 |
| C2×C3×C3⋊#226 | 2 |
| #11724209 | 2 |
| C2×D3×#226 | 3 |
| C2×#63944 | 3 |
| C2×#6294 | 4 |