Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
464 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 464 conjugacy classes of subgroups — too many to draw. Only its 71 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 | C3×#64121 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #11723649 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C3×#63654 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #11722069 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #11722066 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #10591030 | — | 3 | 1 | no | maximal |
| 7 | #10591028 | — | 3 | 2 | no | maximal |
| 8 | C3×#3830 | — | 2 | 1 | no | — |
| 9 | #89040 | — | 2 | 1 | no | — |
| 10 | C33×D8 | C22 | 1 | 1 | yes | characteristic |
| 11 | C3×#3754 | — | 2 | 1 | no | — |
| 12 | C3×#3746 | C22 | 1 | 1 | yes | characteristic |
| 13 | C3×#3746 | C22 | 1 | 2 | yes | — |
| 14 | C3×C8×C3⋊D3 | C22 | 1 | 1 | yes | characteristic |
| 15 | #88812 | — | 2 | 1 | no | — |
| 16 | C3×C3⋊(C3⋊C16) | C22 | 1 | 2 | yes | — |
| 17 | #64121 | D3 | 1 | 1 | yes | characteristic |
| 18 | C3×#912 | — | 3 | 2 | no | — |
| 19 | C3×#912 | — | 6 | 1 | no | — |
| 20 | #63802 | — | 3 | 1 | no | — |
| 21 | D3×Dic12 | — | 3 | 2 | no | — |
| 22 | #63654 | D3 | 1 | 2 | yes | complemented |
| 23 | C3×#827 | — | 3 | 2 | no | — |
| 24 | C3×#827 | — | 3 | 2 | no | — |
| 25 | C3×#827 | — | 6 | 2 | no | — |
| 26 | #63551 | — | 3 | 2 | no | — |
| 27 | #63551 | — | 3 | 2 | no | — |
| 28 | #63550 | — | 3 | 2 | no | — |
| 29 | #63544 | — | 3 | 1 | no | — |
| 30 | #63543 | — | 3 | 2 | no | — |
| 31 | C33×D4 | — | 2 | 1 | no | — |
| 32 | C3×#638 | — | 4 | 1 | no | — |
| 33 | C2×C3×C3⋊Dic3 | — | 4 | 1 | no | — |
| 34 | C3×C4×C3⋊D3 | D4 | 1 | 1 | yes | characteristic |
| 35 | C3×#634 | — | 2 | 1 | no | — |
| 36 | C3×#634 | — | 2 | 2 | no | — |
| 37 | #6296 | — | 4 | 1 | no | — |
| 38 | #6294 | — | 4 | 1 | no | — |
| 39 | C33×C8 | C23 | 1 | 1 | yes | characteristic |
| 40 | C3×C3⋊(C3⋊C8) | D4 | 1 | 1 | yes | characteristic |
| 41 | #5195 | — | 9 | 1 | no | — |
| 42 | #4915 | — | 9 | 2 | no | — |
| 43 | #3830 | — | 2 | 1 | no | — |
| 44 | C3×#237 | — | 6 | 2 | no | — |
| 45 | C3×#237 | — | 6 | 2 | no | — |
| 46 | #3797 | — | 6 | 1 | no | — |
| 47 | D3×Dic6 | — | 6 | 2 | no | — |
| 48 | C32×D8 | — | 2 | 1 | no | — |
| 49 | C32×D8 | — | 2 | 2 | no | — |
| 50 | C32×D8 | — | 4 | 1 | no | — |
| 51 | C32×D8 | C2×D3 | 1 | 2 | yes | — |
| 52 | C32×D8 | C2×D3 | 1 | 1 | yes | characteristic |
| 53 | #3754 | — | 2 | 1 | no | — |
| 54 | #3746 | C2×D3 | 1 | 2 | yes | — |
| 55 | #3746 | C2×D3 | 1 | 1 | yes | characteristic |
| 56 | C8×C3⋊D3 | C2×D3 | 1 | 1 | yes | characteristic |
| 57 | C3×#214 | — | 6 | 2 | no | — |
| 58 | C3×#214 | — | 6 | 2 | no | — |
| 59 | C3×Dic12 | — | 3 | 2 | no | — |
| 60 | C3×Dic12 | — | 3 | 2 | no | — |
| 61 | C3×Dic12 | — | 3 | 2 | no | — |
| 62 | C3×Dic12 | — | 6 | 1 | no | — |
| 63 | C3×Dic12 | — | 6 | 2 | no | — |
| 64 | C3×D3×C8 | — | 3 | 2 | no | — |
| 65 | C3×D3×C8 | — | 6 | 1 | no | — |
| 66 | #3719 | — | 6 | 2 | no | — |
| 67 | #3718 | — | 6 | 1 | no | — |
| 68 | C3⋊(C3⋊C16) | C2×D3 | 1 | 2 | yes | — |
| 69 | C3×C3⋊C16 | — | 3 | 2 | no | — |
| 70 | C3×C3⋊C16 | — | 3 | 2 | no | — |
| 71 | C3×C3⋊C16 | — | 6 | 2 | no | — |
| 72 | C22×C33 | — | 4 | 1 | no | — |
| 73 | C2×C3×C3⋊D3 | D8 | 1 | 1 | yes | characteristic |
| 74 | C33×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 75 | C3×C3⋊Dic3 | — | 4 | 1 | no | — |
| 76 | C3×C3⋊Dic3 | — | 4 | 2 | no | — |
| 77 | C3×C3⋊Dic3 | D8 | 1 | 1 | yes | characteristic |
| 78 | C3×#135 | — | 9 | 1 | no | — |
| 79 | D3×Q16 | — | 9 | 2 | no | — |
| 80 | #912 | — | 3 | 2 | no | — |
| 81 | #912 | — | 6 | 1 | no | — |
| 82 | C2×Dic12 | — | 9 | 1 | no | — |
| 83 | C3×SD32 | — | 9 | 2 | no | — |
| 84 | #829 | — | 9 | 2 | no | — |
| 85 | #829 | — | 9 | 2 | no | — |
| 86 | #827 | — | 3 | 2 | no | — |
| 87 | #827 | — | 3 | 2 | no | — |
| 88 | #827 | — | 6 | 2 | no | — |
| 89 | #812 | — | 9 | 1 | no | — |
| 90 | Dic24 | — | 9 | 2 | no | — |
| 91 | #798 | — | 9 | 2 | no | — |
| 92 | C32×D4 | — | 2 | 1 | no | — |
| 93 | C32×D4 | — | 2 | 2 | no | — |
| 94 | C32×D4 | — | 2 | 2 | no | — |
| 95 | C32×D4 | — | 4 | 2 | no | — |
| 96 | C32×D4 | — | 4 | 2 | no | — |
| 97 | #638 | — | 4 | 1 | no | — |
| 98 | C2×C3⋊Dic3 | — | 4 | 1 | no | — |
| 99 | C4×C3⋊D3 | #67 | 1 | 1 | yes | characteristic |
| 100 | #634 | — | 2 | 1 | no | — |
| 101 | #634 | — | 2 | 2 | no | — |
| 102 | C3×#67 | — | 12 | 2 | no | — |
| 103 | C3×#67 | — | 12 | 2 | no | — |
| 104 | C2×C3×Dic3 | — | 12 | 2 | no | — |
| 105 | C2×C3×Dic3 | — | 12 | 2 | no | — |
| 106 | C3×C4×D3 | — | 3 | 2 | no | — |
| 107 | C3×C4×D3 | — | 6 | 1 | no | — |
| 108 | C3×Dic6 | — | 6 | 2 | no | — |
| 109 | C3×Dic6 | — | 6 | 2 | no | — |
| 110 | C3×Dic6 | — | 6 | 2 | no | — |
| 111 | C3×Dic6 | — | 6 | 2 | no | — |
| 112 | C3×Dic6 | — | 12 | 2 | no | — |
| 113 | #627 | — | 12 | 2 | no | — |
| 114 | #627 | — | 12 | 1 | no | — |
| 115 | #624 | — | 12 | 1 | no | — |
| 116 | D3×Dic3 | — | 12 | 2 | no | — |
| 117 | C32×C8 | — | 2 | 2 | no | — |
| 118 | C32×C8 | — | 2 | 1 | no | — |
| 119 | C32×C8 | — | 4 | 1 | no | — |
| 120 | C32×C8 | C22×D3 | 1 | 2 | yes | — |
| 121 | C32×C8 | C22×D3 | 1 | 1 | yes | characteristic |
| 122 | C3⋊(C3⋊C8) | #67 | 1 | 1 | yes | characteristic |
| 123 | C3×C3⋊C8 | — | 3 | 2 | no | — |
| 124 | C3×C3⋊C8 | — | 6 | 1 | no | — |
| 125 | #510 | — | 27 | 1 | no | Sylow |
| 126 | C2×C33 | — | 8 | 1 | no | — |
| 127 | C2×C33 | C2×D8 | 1 | 1 | yes | characteristic |
| 128 | C3×C3⋊D3 | — | 2 | 1 | no | — |
| 129 | C3×#41 | — | 18 | 1 | no | — |
| 130 | D3×Q8 | — | 18 | 2 | no | — |
| 131 | #237 | — | 6 | 2 | no | — |
| 132 | #237 | — | 6 | 2 | no | — |
| 133 | C2×Dic6 | — | 18 | 1 | no | — |
| 134 | C3×Q16 | — | 9 | 1 | no | — |
| 135 | C3×Q16 | — | 9 | 2 | no | — |
| 136 | C3×SD16 | — | 18 | 1 | no | — |
| 137 | C3×D8 | — | 2 | 1 | no | — |
| 138 | C3×D8 | — | 2 | 2 | no | — |
| 139 | C3×D8 | — | 4 | 1 | no | — |
| 140 | C3×D8 | D32 | 1 | 1 | yes | characteristic |
| 141 | C3×D8 | D32 | 1 | 2 | yes | — |
| 142 | C2×C3×C8 | — | 9 | 1 | no | — |
| 143 | #216 | — | 18 | 2 | no | — |
| 144 | #214 | — | 6 | 2 | no | — |
| 145 | #214 | — | 6 | 2 | no | — |
| 146 | Dic12 | — | 3 | 2 | no | — |
| 147 | Dic12 | — | 3 | 2 | no | — |
| 148 | Dic12 | — | 3 | 2 | no | — |
| 149 | Dic12 | — | 6 | 1 | no | — |
| 150 | Dic12 | — | 6 | 2 | no | — |
| 151 | Dic12 | — | 18 | 1 | no | — |
| 152 | D3×C8 | — | 3 | 2 | no | — |
| 153 | D3×C8 | — | 6 | 1 | no | — |
| 154 | C3×C16 | — | 9 | 2 | no | — |
| 155 | C3⋊C16 | — | 3 | 2 | no | — |
| 156 | C3⋊C16 | — | 3 | 2 | no | — |
| 157 | C3⋊C16 | — | 6 | 2 | no | — |
| 158 | C22×C32 | — | 4 | 1 | no | — |
| 159 | C22×C32 | — | 4 | 2 | no | — |
| 160 | C22×C32 | — | 4 | 2 | no | — |
| 161 | C22×C32 | — | 8 | 2 | no | — |
| 162 | C22×C32 | — | 8 | 2 | no | — |
| 163 | C2×C3⋊D3 | #213 | 1 | 1 | yes | characteristic |
| 164 | C2×C3×D3 | — | 3 | 2 | no | — |
| 165 | C2×C3×D3 | — | 6 | 1 | no | — |
| 166 | C32×C4 | — | 2 | 2 | no | — |
| 167 | C32×C4 | — | 2 | 1 | no | — |
| 168 | C32×C4 | — | 4 | 1 | no | — |
| 169 | C32×C4 | D3×D4 | 1 | 2 | yes | — |
| 170 | C32×C4 | C2×#67 | 1 | 1 | yes | characteristic |
| 171 | C3⋊Dic3 | — | 4 | 1 | no | — |
| 172 | C3⋊Dic3 | — | 4 | 2 | no | — |
| 173 | C3⋊Dic3 | #213 | 1 | 1 | yes | characteristic |
| 174 | C3×Dic3 | — | 3 | 2 | no | — |
| 175 | C3×Dic3 | — | 6 | 1 | no | — |
| 176 | C3×Dic3 | — | 12 | 2 | no | — |
| 177 | C3×Dic3 | — | 12 | 2 | no | — |
| 178 | C3×Dic3 | — | 12 | 2 | no | — |
| 179 | C3×Dic3 | — | 12 | 2 | no | — |
| 180 | C3×Dic3 | — | 24 | 2 | no | — |
| 181 | #135 | — | 9 | 1 | no | — |
| 182 | C2×Q16 | — | 27 | 1 | no | — |
| 183 | Q32 | — | 27 | 2 | no | — |
| 184 | SD32 | — | 9 | 2 | no | — |
| 185 | M32 | — | 27 | 1 | no | — |
| 186 | C33 | #510 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 187 | C3×Q8 | — | 18 | 2 | no | — |
| 188 | C3×Q8 | — | 18 | 1 | no | — |
| 189 | C3×D4 | — | 2 | 2 | no | — |
| 190 | C3×D4 | — | 2 | 2 | no | — |
| 191 | C3×D4 | — | 2 | 1 | no | — |
| 192 | C3×D4 | — | 4 | 2 | no | — |
| 193 | C3×D4 | — | 4 | 2 | no | — |
| 194 | C3×D4 | — | 36 | 1 | no | — |
| 195 | C2×C3×C4 | — | 9 | 1 | no | — |
| 196 | C2×C3×C4 | — | 36 | 1 | no | — |
| 197 | #67 | — | 12 | 2 | no | — |
| 198 | #67 | — | 12 | 2 | no | — |
| 199 | C2×Dic3 | — | 12 | 2 | no | — |
| 200 | C2×Dic3 | — | 12 | 2 | no | — |
| 201 | C2×Dic3 | — | 36 | 1 | no | — |
| 202 | C4×D3 | — | 3 | 2 | no | — |
| 203 | C4×D3 | — | 6 | 1 | no | — |
| 204 | C4×D3 | — | 36 | 2 | no | — |
| 205 | Dic6 | — | 6 | 2 | no | — |
| 206 | Dic6 | — | 6 | 2 | no | — |
| 207 | Dic6 | — | 6 | 2 | no | — |
| 208 | Dic6 | — | 6 | 2 | no | — |
| 209 | Dic6 | — | 12 | 2 | no | — |
| 210 | Dic6 | — | 36 | 2 | no | — |
| 211 | Dic6 | — | 36 | 1 | no | — |
| 212 | C3×C8 | — | 2 | 2 | no | — |
| 213 | C3×C8 | — | 2 | 1 | no | — |
| 214 | C3×C8 | — | 4 | 1 | no | — |
| 215 | C3×C8 | — | 9 | 1 | no | — |
| 216 | C3×C8 | C2×D32 | 1 | 2 | yes | — |
| 217 | C3×C8 | C2×D32 | 1 | 1 | yes | characteristic |
| 218 | C3⋊C8 | — | 3 | 2 | no | — |
| 219 | C3⋊C8 | — | 6 | 1 | no | — |
| 220 | C2×C32 | — | 2 | 2 | no | — |
| 221 | C2×C32 | — | 2 | 1 | no | — |
| 222 | C2×C32 | — | 4 | 1 | no | — |
| 223 | C2×C32 | — | 8 | 1 | no | — |
| 224 | C2×C32 | — | 8 | 2 | no | — |
| 225 | C2×C32 | — | 8 | 2 | no | — |
| 226 | C2×C32 | — | 16 | 2 | no | — |
| 227 | C2×C32 | — | 16 | 2 | no | — |
| 228 | C2×C32 | D3×D8 | 1 | 2 | yes | — |
| 229 | C2×C32 | C2×#213 | 1 | 1 | yes | characteristic |
| 230 | C3⋊D3 | — | 2 | 1 | no | — |
| 231 | C3×D3 | — | 6 | 2 | no | — |
| 232 | C3×D3 | — | 12 | 1 | no | — |
| 233 | #41 | — | 18 | 1 | no | — |
| 234 | C2×Q8 | — | 54 | 1 | no | — |
| 235 | Q16 | — | 9 | 2 | no | — |
| 236 | Q16 | — | 9 | 1 | no | — |
| 237 | Q16 | — | 54 | 1 | no | — |
| 238 | SD16 | — | 18 | 1 | no | — |
| 239 | D8 | #1113 | 1 | 1 | yes | characteristic |
| 240 | C2×C8 | — | 9 | 1 | no | — |
| 241 | C16 | — | 9 | 2 | no | — |
| 242 | C22×C3 | — | 4 | 2 | no | — |
| 243 | C22×C3 | — | 4 | 2 | no | — |
| 244 | C22×C3 | — | 4 | 1 | no | — |
| 245 | C22×C3 | — | 8 | 2 | no | — |
| 246 | C22×C3 | — | 8 | 2 | no | — |
| 247 | C22×C3 | — | 9 | 1 | no | — |
| 248 | C2×D3 | — | 3 | 2 | no | — |
| 249 | C2×D3 | — | 6 | 1 | no | — |
| 250 | C3×C4 | — | 2 | 1 | no | — |
| 251 | C3×C4 | — | 2 | 2 | no | — |
| 252 | C3×C4 | — | 4 | 1 | no | — |
| 253 | C3×C4 | — | 9 | 1 | no | — |
| 254 | C3×C4 | — | 36 | 2 | no | — |
| 255 | C3×C4 | — | 36 | 1 | no | — |
| 256 | C3×C4 | D3×#67 | 1 | 2 | yes | — |
| 257 | C3×C4 | #3811 | 1 | 1 | yes | characteristic |
| 258 | Dic3 | — | 3 | 2 | no | — |
| 259 | Dic3 | — | 6 | 1 | no | — |
| 260 | Dic3 | — | 12 | 2 | no | — |
| 261 | Dic3 | — | 12 | 2 | no | — |
| 262 | Dic3 | — | 12 | 2 | no | — |
| 263 | Dic3 | — | 12 | 2 | no | — |
| 264 | Dic3 | — | 24 | 2 | no | — |
| 265 | C32 | — | 2 | 2 | no | — |
| 266 | C32 | — | 2 | 1 | no | — |
| 267 | C32 | — | 4 | 1 | no | — |
| 268 | C32 | #4915 | 1 | 2 | yes | complemented |
| 269 | C32 | #5195 | 1 | 1 | yes | characteristic, complemented |
| 270 | Q8 | — | 18 | 1 | no | — |
| 271 | Q8 | — | 18 | 2 | no | — |
| 272 | Q8 | — | 108 | 1 | no | — |
| 273 | D4 | — | 2 | 1 | no | — |
| 274 | D4 | — | 36 | 1 | no | — |
| 275 | C2×C4 | — | 9 | 1 | no | — |
| 276 | C2×C4 | — | 36 | 1 | no | — |
| 277 | C2×C4 | — | 108 | 1 | no | — |
| 278 | C8 | — | 9 | 1 | no | — |
| 279 | C8 | C2×#1113 | 1 | 1 | yes | characteristic |
| 280 | C2×C3 | — | 2 | 1 | no | — |
| 281 | C2×C3 | — | 2 | 2 | no | — |
| 282 | C2×C3 | — | 4 | 1 | no | — |
| 283 | C2×C3 | — | 8 | 2 | no | — |
| 284 | C2×C3 | — | 8 | 2 | no | — |
| 285 | C2×C3 | — | 8 | 1 | no | — |
| 286 | C2×C3 | — | 16 | 2 | no | — |
| 287 | C2×C3 | — | 16 | 2 | no | — |
| 288 | C2×C3 | — | 18 | 1 | no | — |
| 289 | C2×C3 | D3×#213 | 1 | 2 | yes | — |
| 290 | C2×C3 | #63926 | 1 | 1 | yes | characteristic |
| 291 | D3 | — | 6 | 2 | no | — |
| 292 | D3 | — | 12 | 1 | no | — |
| 293 | C22 | — | 4 | 1 | no | — |
| 294 | C22 | — | 9 | 1 | no | — |
| 295 | C4 | — | 9 | 1 | no | — |
| 296 | C4 | — | 36 | 1 | no | — |
| 297 | C4 | — | 36 | 2 | no | — |
| 298 | C4 | #89049 | 1 | 1 | yes | characteristic |
| 299 | C3 | — | 2 | 1 | no | — |
| 300 | C3 | — | 2 | 2 | no | — |
| 301 | C3 | — | 4 | 1 | no | — |
| 302 | C3 | #10591028 | 1 | 2 | yes | complemented |
| 303 | C3 | #10591030 | 1 | 1 | yes | characteristic, complemented |
| 304 | C2 | — | 8 | 1 | no | — |
| 305 | C2 | — | 18 | 1 | no | — |
| 306 | C2 | #11723702 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #11723649 | C2 | semidirect |
| 3. C3×#63654 | C2 | semidirect |
| 4. #11722069 | C2 | semidirect |
| 5. #11722066 | C2 | semidirect |
| 22. #63654 | D3 | semidirect |
| 186. C33 | #510 | semidirect |
| 268. C32 | #4915 | semidirect |
| 269. C32 | #5195 | semidirect |
| 302. C3 | #10591028 | semidirect |
| 303. C3 | #10591030 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#64121 | 2 |
| #11723649 | 2 |
| C3×#63654 | 2 |
| #11722069 | 2 |
| #11722066 | 2 |
| #10591030 | 3 |
| #10591028 | 3 |