Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
510 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 510 conjugacy classes of subgroups — too many to draw. Only its 66 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 | #11723343 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #11722069 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #11722000 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C3×#63548 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | C3×#63545 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #10591035 | — | 3 | 2 | no | maximal |
| 7 | #10590245 | — | 3 | 1 | no | maximal |
| 8 | #89015 | — | 2 | 2 | no | — |
| 9 | C8×C3⋊(C3⋊D3) | C22 | 1 | 1 | yes | characteristic |
| 10 | C3×#3746 | C22 | 1 | 2 | yes | — |
| 11 | C32×D24 | C22 | 1 | 2 | yes | — |
| 12 | #88792 | — | 2 | 2 | no | — |
| 13 | C3×C3⋊(C3⋊C16) | C22 | 1 | 1 | yes | characteristic |
| 14 | C32×C3⋊C16 | C22 | 1 | 1 | yes | characteristic |
| 15 | #64121 | — | 3 | 2 | no | — |
| 16 | #63810 | — | 3 | 2 | no | — |
| 17 | #63810 | — | 3 | 2 | no | — |
| 18 | #63810 | — | 6 | 2 | no | — |
| 19 | C16×C3⋊D3 | — | 3 | 1 | no | — |
| 20 | C3×#827 | — | 3 | 2 | no | — |
| 21 | C3×#826 | — | 3 | 2 | no | — |
| 22 | C3×#800 | — | 3 | 2 | no | — |
| 23 | #63551 | — | 3 | 2 | no | — |
| 24 | #63550 | — | 3 | 1 | no | — |
| 25 | #63548 | D3 | 1 | 2 | yes | complemented |
| 26 | #63545 | D3 | 1 | 1 | yes | characteristic, complemented |
| 27 | #63542 | — | 3 | 2 | no | — |
| 28 | #63542 | — | 6 | 1 | no | — |
| 29 | C4×C3⋊(C3⋊D3) | D4 | 1 | 1 | yes | characteristic |
| 30 | C3×#634 | — | 2 | 2 | no | — |
| 31 | C32×D12 | — | 2 | 2 | no | — |
| 32 | #6291 | — | 4 | 2 | no | — |
| 33 | D3×C3⋊Dic3 | — | 4 | 2 | no | — |
| 34 | C33×C8 | C23 | 1 | 1 | yes | characteristic |
| 35 | C3⋊(C3⋊(C3⋊C8)) | D4 | 1 | 1 | yes | characteristic |
| 36 | #4916 | — | 9 | 2 | no | — |
| 37 | #4912 | — | 9 | 1 | no | — |
| 38 | #3830 | — | 6 | 2 | no | — |
| 39 | #3796 | — | 6 | 2 | no | — |
| 40 | #3796 | — | 6 | 2 | no | — |
| 41 | #3796 | — | 6 | 4 | no | — |
| 42 | C32×D8 | — | 3 | 2 | no | — |
| 43 | #3754 | — | 6 | 2 | no | — |
| 44 | #3746 | C2×D3 | 1 | 2 | yes | — |
| 45 | C8×C3⋊D3 | — | 3 | 2 | no | — |
| 46 | C8×C3⋊D3 | — | 3 | 1 | no | — |
| 47 | C8×C3⋊D3 | — | 6 | 1 | no | — |
| 48 | C8×C3⋊D3 | — | 6 | 2 | no | — |
| 49 | C8×C3⋊D3 | — | 12 | 1 | no | — |
| 50 | C3×Dic12 | — | 3 | 2 | no | — |
| 51 | C3×Dic12 | — | 3 | 2 | no | — |
| 52 | C3×Dic12 | — | 6 | 2 | no | — |
| 53 | C3×D24 | — | 2 | 2 | no | — |
| 54 | C3×D24 | C2×D3 | 1 | 2 | yes | — |
| 55 | C3×D24 | C2×D3 | 1 | 2 | yes | — |
| 56 | #3715 | — | 6 | 2 | no | — |
| 57 | #3715 | — | 6 | 2 | no | — |
| 58 | #3715 | — | 6 | 4 | no | — |
| 59 | C32×C16 | — | 3 | 1 | no | — |
| 60 | C3⋊(C3⋊C16) | C2×D3 | 1 | 1 | yes | characteristic |
| 61 | C3×C3⋊C16 | — | 2 | 1 | no | — |
| 62 | C3×C3⋊C16 | — | 3 | 2 | no | — |
| 63 | C3×C3⋊C16 | — | 6 | 1 | no | — |
| 64 | C3×C3⋊C16 | C2×D3 | 1 | 2 | yes | — |
| 65 | C2×C3⋊(C3⋊D3) | D8 | 1 | 1 | yes | characteristic |
| 66 | C2×C32×D3 | — | 4 | 2 | no | — |
| 67 | C33×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 68 | C3⋊(C3⋊Dic3) | D8 | 1 | 1 | yes | characteristic |
| 69 | C3×C3⋊Dic3 | — | 4 | 2 | no | — |
| 70 | #919 | — | 9 | 2 | no | — |
| 71 | #912 | — | 9 | 2 | no | — |
| 72 | #912 | — | 9 | 2 | no | — |
| 73 | #912 | — | 18 | 2 | no | — |
| 74 | C3×SD32 | — | 9 | 2 | no | — |
| 75 | C3×D16 | — | 9 | 1 | no | — |
| 76 | #829 | — | 9 | 2 | no | — |
| 77 | #827 | — | 3 | 2 | no | — |
| 78 | #826 | — | 3 | 2 | no | — |
| 79 | Dic24 | — | 9 | 1 | no | — |
| 80 | #800 | — | 3 | 2 | no | — |
| 81 | D3×C16 | — | 9 | 2 | no | — |
| 82 | D3×C16 | — | 9 | 1 | no | — |
| 83 | D3×C16 | — | 18 | 1 | no | — |
| 84 | C32×D4 | — | 6 | 2 | no | — |
| 85 | #638 | — | 12 | 2 | no | — |
| 86 | C2×C3⋊Dic3 | — | 12 | 2 | no | — |
| 87 | C4×C3⋊D3 | — | 3 | 1 | no | — |
| 88 | C4×C3⋊D3 | — | 3 | 2 | no | — |
| 89 | C4×C3⋊D3 | — | 6 | 1 | no | — |
| 90 | C4×C3⋊D3 | — | 6 | 2 | no | — |
| 91 | C4×C3⋊D3 | — | 12 | 1 | no | — |
| 92 | #634 | — | 2 | 2 | no | — |
| 93 | C3×D12 | — | 2 | 4 | no | — |
| 94 | C3×D12 | — | 2 | 2 | no | — |
| 95 | C3×D12 | — | 2 | 2 | no | — |
| 96 | C3×Dic6 | — | 6 | 4 | no | — |
| 97 | C3×Dic6 | — | 6 | 2 | no | — |
| 98 | C3×Dic6 | — | 6 | 2 | no | — |
| 99 | #626 | — | 12 | 2 | no | — |
| 100 | #626 | — | 12 | 2 | no | — |
| 101 | #626 | — | 12 | 4 | no | — |
| 102 | D3×Dic3 | — | 12 | 2 | no | — |
| 103 | D3×Dic3 | — | 12 | 2 | no | — |
| 104 | D3×Dic3 | — | 12 | 4 | no | — |
| 105 | C32×C8 | — | 2 | 1 | no | — |
| 106 | C32×C8 | — | 2 | 2 | no | — |
| 107 | C32×C8 | — | 4 | 1 | no | — |
| 108 | C32×C8 | C22×D3 | 1 | 2 | yes | — |
| 109 | C32×C8 | C22×D3 | 1 | 1 | yes | characteristic |
| 110 | C3⋊(C3⋊C8) | — | 3 | 1 | no | — |
| 111 | C3⋊(C3⋊C8) | — | 3 | 2 | no | — |
| 112 | C3⋊(C3⋊C8) | — | 6 | 2 | no | — |
| 113 | C3⋊(C3⋊C8) | — | 6 | 1 | no | — |
| 114 | C3⋊(C3⋊C8) | — | 12 | 1 | no | — |
| 115 | #508 | — | 27 | 1 | no | Sylow |
| 116 | C2×C33 | C2×D8 | 1 | 1 | yes | characteristic |
| 117 | C3⋊(C3⋊D3) | — | 2 | 1 | no | — |
| 118 | C32×D3 | — | 8 | 2 | no | — |
| 119 | #239 | — | 18 | 2 | no | — |
| 120 | #237 | — | 18 | 2 | no | — |
| 121 | #237 | — | 18 | 2 | no | — |
| 122 | #237 | — | 18 | 4 | no | — |
| 123 | C3×Q16 | — | 9 | 2 | no | — |
| 124 | C3×D8 | — | 3 | 2 | no | — |
| 125 | C3×D8 | — | 3 | 2 | no | — |
| 126 | C3×D8 | — | 6 | 2 | no | — |
| 127 | #215 | — | 18 | 2 | no | — |
| 128 | #214 | — | 18 | 2 | no | — |
| 129 | #214 | — | 18 | 2 | no | — |
| 130 | #214 | — | 18 | 4 | no | — |
| 131 | Dic12 | — | 3 | 2 | no | — |
| 132 | Dic12 | — | 3 | 2 | no | — |
| 133 | Dic12 | — | 6 | 2 | no | — |
| 134 | D24 | D32 | 1 | 2 | yes | — |
| 135 | D3×C8 | — | 9 | 1 | no | — |
| 136 | D3×C8 | — | 9 | 2 | no | — |
| 137 | D3×C8 | — | 18 | 1 | no | — |
| 138 | D3×C8 | — | 18 | 2 | no | — |
| 139 | D3×C8 | — | 36 | 1 | no | — |
| 140 | C3×C16 | — | 3 | 2 | no | — |
| 141 | C3×C16 | — | 6 | 1 | no | — |
| 142 | C3×C16 | — | 9 | 1 | no | — |
| 143 | C3⋊C16 | — | 3 | 2 | no | — |
| 144 | C3⋊C16 | — | 6 | 1 | no | — |
| 145 | C3⋊C16 | D32 | 1 | 1 | yes | characteristic |
| 146 | C22×C32 | — | 12 | 2 | no | — |
| 147 | C2×C3⋊D3 | — | 3 | 1 | no | — |
| 148 | C2×C3⋊D3 | — | 3 | 2 | no | — |
| 149 | C2×C3⋊D3 | — | 6 | 1 | no | — |
| 150 | C2×C3⋊D3 | — | 6 | 2 | no | — |
| 151 | C2×C3⋊D3 | — | 12 | 1 | no | — |
| 152 | C2×C3×D3 | — | 4 | 4 | no | — |
| 153 | C2×C3×D3 | — | 4 | 2 | no | — |
| 154 | C2×C3×D3 | — | 4 | 2 | no | — |
| 155 | C32×C4 | — | 2 | 1 | no | — |
| 156 | C32×C4 | — | 2 | 2 | no | — |
| 157 | C32×C4 | — | 4 | 1 | no | — |
| 158 | C32×C4 | D3×D4 | 1 | 2 | yes | — |
| 159 | C32×C4 | D3×D4 | 1 | 1 | yes | characteristic |
| 160 | C3⋊Dic3 | — | 3 | 1 | no | — |
| 161 | C3⋊Dic3 | — | 3 | 2 | no | — |
| 162 | C3⋊Dic3 | — | 4 | 2 | no | — |
| 163 | C3⋊Dic3 | — | 6 | 2 | no | — |
| 164 | C3⋊Dic3 | — | 6 | 1 | no | — |
| 165 | C3⋊Dic3 | — | 12 | 1 | no | — |
| 166 | C3×Dic3 | — | 12 | 4 | no | — |
| 167 | C3×Dic3 | — | 12 | 2 | no | — |
| 168 | C3×Dic3 | — | 12 | 2 | no | — |
| 169 | #135 | — | 27 | 2 | no | — |
| 170 | Q32 | — | 27 | 1 | no | — |
| 171 | SD32 | — | 9 | 2 | no | — |
| 172 | D16 | — | 9 | 1 | no | — |
| 173 | C2×C16 | — | 27 | 1 | no | — |
| 174 | C33 | #508 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 175 | C3×Q8 | — | 18 | 2 | no | — |
| 176 | C3×D4 | — | 6 | 4 | no | — |
| 177 | C3×D4 | — | 6 | 2 | no | — |
| 178 | C3×D4 | — | 6 | 2 | no | — |
| 179 | #67 | — | 36 | 2 | no | — |
| 180 | #67 | — | 36 | 2 | no | — |
| 181 | #67 | — | 36 | 4 | no | — |
| 182 | C2×Dic3 | — | 36 | 2 | no | — |
| 183 | C2×Dic3 | — | 36 | 2 | no | — |
| 184 | C2×Dic3 | — | 36 | 4 | no | — |
| 185 | D12 | — | 2 | 2 | no | — |
| 186 | D12 | — | 36 | 2 | no | — |
| 187 | C4×D3 | — | 9 | 1 | no | — |
| 188 | C4×D3 | — | 9 | 2 | no | — |
| 189 | C4×D3 | — | 18 | 1 | no | — |
| 190 | C4×D3 | — | 18 | 2 | no | — |
| 191 | C4×D3 | — | 36 | 1 | no | — |
| 192 | C4×D3 | — | 36 | 2 | no | — |
| 193 | Dic6 | — | 6 | 4 | no | — |
| 194 | Dic6 | — | 6 | 2 | no | — |
| 195 | Dic6 | — | 6 | 2 | no | — |
| 196 | C3×C8 | — | 2 | 1 | no | — |
| 197 | C3×C8 | — | 2 | 2 | no | — |
| 198 | C3×C8 | — | 4 | 1 | no | — |
| 199 | C3×C8 | C2×D32 | 1 | 1 | yes | characteristic |
| 200 | C3×C8 | C2×D32 | 1 | 2 | yes | — |
| 201 | C3⋊C8 | — | 9 | 1 | no | — |
| 202 | C3⋊C8 | — | 9 | 2 | no | — |
| 203 | C3⋊C8 | — | 18 | 1 | no | — |
| 204 | C3⋊C8 | — | 18 | 2 | no | — |
| 205 | C3⋊C8 | — | 36 | 1 | no | — |
| 206 | C2×C32 | — | 2 | 1 | no | — |
| 207 | C2×C32 | — | 2 | 2 | no | — |
| 208 | C2×C32 | — | 4 | 1 | no | — |
| 209 | C2×C32 | — | 24 | 2 | no | — |
| 210 | C2×C32 | D3×D8 | 1 | 2 | yes | — |
| 211 | C2×C32 | D3×D8 | 1 | 1 | yes | characteristic |
| 212 | C3⋊D3 | — | 6 | 1 | no | — |
| 213 | C3⋊D3 | — | 6 | 2 | no | — |
| 214 | C3⋊D3 | — | 6 | 2 | no | — |
| 215 | C3⋊D3 | — | 12 | 2 | no | — |
| 216 | C3⋊D3 | — | 12 | 2 | no | — |
| 217 | C3×D3 | — | 8 | 4 | no | — |
| 218 | C3×D3 | — | 8 | 2 | no | — |
| 219 | C3×D3 | — | 8 | 2 | no | — |
| 220 | #41 | — | 54 | 2 | no | — |
| 221 | Q16 | — | 9 | 2 | no | — |
| 222 | SD16 | — | 54 | 2 | no | — |
| 223 | D8 | — | 3 | 2 | no | — |
| 224 | C2×C8 | — | 27 | 1 | no | — |
| 225 | C16 | — | 3 | 1 | no | — |
| 226 | C16 | — | 9 | 1 | no | — |
| 227 | C22×C3 | — | 12 | 4 | no | — |
| 228 | C22×C3 | — | 12 | 2 | no | — |
| 229 | C22×C3 | — | 12 | 2 | no | — |
| 230 | C2×D3 | — | 4 | 2 | no | — |
| 231 | C2×D3 | — | 9 | 1 | no | — |
| 232 | C2×D3 | — | 9 | 2 | no | — |
| 233 | C2×D3 | — | 18 | 1 | no | — |
| 234 | C2×D3 | — | 18 | 2 | no | — |
| 235 | C2×D3 | — | 36 | 1 | no | — |
| 236 | C3×C4 | — | 2 | 2 | no | — |
| 237 | C3×C4 | — | 2 | 1 | no | — |
| 238 | C3×C4 | — | 4 | 1 | no | — |
| 239 | C3×C4 | — | 36 | 2 | no | — |
| 240 | C3×C4 | #3802 | 1 | 1 | yes | characteristic |
| 241 | C3×C4 | #3811 | 1 | 2 | yes | — |
| 242 | Dic3 | — | 9 | 1 | no | — |
| 243 | Dic3 | — | 9 | 2 | no | — |
| 244 | Dic3 | — | 12 | 4 | no | — |
| 245 | Dic3 | — | 12 | 2 | no | — |
| 246 | Dic3 | — | 12 | 2 | no | — |
| 247 | Dic3 | — | 18 | 2 | no | — |
| 248 | Dic3 | — | 18 | 1 | no | — |
| 249 | Dic3 | — | 36 | 1 | no | — |
| 250 | C32 | — | 2 | 1 | no | — |
| 251 | C32 | — | 2 | 2 | no | — |
| 252 | C32 | — | 4 | 1 | no | — |
| 253 | C32 | #4912 | 1 | 1 | yes | characteristic, complemented |
| 254 | C32 | #4916 | 1 | 2 | yes | complemented |
| 255 | Q8 | — | 18 | 2 | no | — |
| 256 | D4 | — | 6 | 2 | no | — |
| 257 | D4 | — | 108 | 2 | no | — |
| 258 | C2×C4 | — | 27 | 1 | no | — |
| 259 | C2×C4 | — | 108 | 2 | no | — |
| 260 | C8 | — | 27 | 1 | no | — |
| 261 | C8 | D33 | 1 | 1 | yes | characteristic |
| 262 | C2×C3 | — | 2 | 2 | no | — |
| 263 | C2×C3 | — | 2 | 1 | no | — |
| 264 | C2×C3 | — | 4 | 1 | no | — |
| 265 | C2×C3 | — | 24 | 4 | no | — |
| 266 | C2×C3 | — | 24 | 2 | no | — |
| 267 | C2×C3 | — | 24 | 2 | no | — |
| 268 | C2×C3 | #63797 | 1 | 1 | yes | characteristic |
| 269 | C2×C3 | #63926 | 1 | 2 | yes | — |
| 270 | D3 | — | 8 | 2 | no | — |
| 271 | D3 | — | 18 | 2 | no | — |
| 272 | D3 | — | 18 | 1 | no | — |
| 273 | D3 | — | 18 | 2 | no | — |
| 274 | D3 | — | 36 | 2 | no | — |
| 275 | D3 | — | 36 | 2 | no | — |
| 276 | C22 | — | 12 | 2 | no | — |
| 277 | C22 | — | 27 | 1 | no | — |
| 278 | C4 | — | 27 | 1 | no | — |
| 279 | C4 | — | 36 | 2 | no | — |
| 280 | C4 | #88954 | 1 | 1 | yes | characteristic |
| 281 | C3 | — | 2 | 1 | no | — |
| 282 | C3 | — | 2 | 2 | no | — |
| 283 | C3 | — | 4 | 1 | no | — |
| 284 | C3 | #10590245 | 1 | 1 | yes | characteristic, complemented |
| 285 | C3 | #10591035 | 1 | 2 | yes | complemented |
| 286 | C2 | — | 24 | 2 | no | — |
| 287 | C2 | — | 54 | 1 | no | — |
| 288 | C2 | #11722832 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #11722832 |
| commutator, Fitting, cosocle | C33×C8 | C23 |
| Frattini, hypercenter | C8 | D33 |
| socle | C2×C33 | C2×D8 |
| hypocenter | C33 | #508 |
| LCS | G ▷ C33×C8 ▷ C33×C4 ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #11723343 | C2 | semidirect |
| 2. #11722069 | C2 | semidirect |
| 3. #11722000 | C2 | semidirect |
| 4. C3×#63548 | C2 | semidirect |
| 5. C3×#63545 | C2 | semidirect |
| 25. #63548 | D3 | semidirect |
| 26. #63545 | D3 | semidirect |
| 174. C33 | #508 | semidirect |
| 253. C32 | #4912 | semidirect |
| 254. C32 | #4916 | semidirect |
| 284. C3 | #10590245 | semidirect |
| 285. C3 | #10591035 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11723343 | 2 |
| #11722069 | 2 |
| #11722000 | 2 |
| C3×#63548 | 2 |
| C3×#63545 | 2 |
| #10591035 | 3 |
| #10590245 | 3 |