Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
412 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 412 conjugacy classes of subgroups — too many to draw. Only its 47 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×#88810 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C3×#63822 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #11723098 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #11722766 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #11722765 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #10593334 | — | 3 | 1 | no | maximal |
| 7 | C2×C3×C4⋊(C3⋊D3) | — | 2 | 1 | no | — |
| 8 | C32×#235 | — | 2 | 1 | no | — |
| 9 | C2×C3×C3⋊(C3⋊C8) | C22 | 1 | 1 | yes | characteristic |
| 10 | #88810 | — | 2 | 1 | no | — |
| 11 | #88810 | C22 | 1 | 2 | yes | — |
| 12 | C3×#3718 | C22 | 1 | 1 | yes | characteristic, complemented |
| 13 | C3×#3715 | — | 2 | 1 | no | — |
| 14 | C3×#3713 | C22 | 1 | 1 | yes | characteristic, complemented |
| 15 | C3⋊#3708 | C22 | 1 | 2 | yes | complemented |
| 16 | C3×#950 | — | 6 | 1 | no | — |
| 17 | C2×#3714 | — | 6 | 1 | no | — |
| 18 | #63822 | D3 | 1 | 1 | yes | characteristic, complemented |
| 19 | C2×#3713 | — | 3 | 1 | no | — |
| 20 | #63773 | — | 3 | 1 | no | — |
| 21 | #63735 | — | 3 | 2 | no | — |
| 22 | #63734 | — | 3 | 2 | no | — |
| 23 | C22×C3×C3⋊D3 | — | 4 | 1 | no | — |
| 24 | C2×C33×C4 | D4 | 1 | 1 | yes | characteristic |
| 25 | C3×C4⋊(C3⋊D3) | — | 2 | 2 | no | — |
| 26 | C3×C4⋊(C3⋊D3) | — | 4 | 1 | no | — |
| 27 | C32×#67 | — | 4 | 1 | no | — |
| 28 | C32×D12 | — | 2 | 1 | no | — |
| 29 | C32×C4×D3 | — | 4 | 1 | no | — |
| 30 | C32×Dic6 | — | 2 | 1 | no | — |
| 31 | C3×C3⋊(C3⋊C8) | D4 | 1 | 1 | yes | characteristic |
| 32 | C3×C3⋊(C3⋊C8) | C23 | 1 | 1 | yes | characteristic |
| 33 | #5193 | — | 9 | 1 | no | maximal |
| 34 | C32×#41 | — | 6 | 1 | no | — |
| 35 | C2×C4⋊(C3⋊D3) | — | 2 | 1 | no | — |
| 36 | C3×#235 | — | 2 | 1 | no | — |
| 37 | C3×#235 | — | 2 | 1 | no | — |
| 38 | C3×#235 | — | 4 | 1 | no | — |
| 39 | C2×C3×D12 | — | 6 | 1 | no | — |
| 40 | C2×C3×D12 | — | 6 | 1 | no | — |
| 41 | C2×C3×D12 | — | 12 | 1 | no | — |
| 42 | C2×C3⋊(C3⋊C8) | C2×D3 | 1 | 1 | yes | characteristic |
| 43 | C3×#216 | — | 6 | 1 | no | — |
| 44 | C3×#215 | — | 6 | 1 | no | — |
| 45 | C3×#214 | — | 6 | 1 | no | — |
| 46 | C3×#213 | — | 6 | 1 | no | — |
| 47 | C2×C3×C3⋊C8 | — | 6 | 1 | no | — |
| 48 | C2×C3×C3⋊C8 | — | 6 | 1 | no | — |
| 49 | #3718 | C2×D3 | 1 | 1 | yes | characteristic, complemented |
| 50 | #3715 | — | 2 | 1 | no | — |
| 51 | #3714 | — | 6 | 2 | no | — |
| 52 | #3714 | — | 6 | 2 | no | — |
| 53 | #3713 | — | 3 | 2 | no | — |
| 54 | #3713 | — | 6 | 1 | no | — |
| 55 | #3713 | C2×D3 | 1 | 1 | yes | characteristic, complemented |
| 56 | #3708 | — | 3 | 2 | no | — |
| 57 | C22×C33 | D8 | 1 | 1 | yes | characteristic |
| 58 | C2×C3×C3⋊D3 | — | 4 | 2 | no | — |
| 59 | C2×C3×C3⋊D3 | — | 8 | 2 | no | — |
| 60 | C2×C32×D3 | — | 4 | 1 | no | — |
| 61 | C33×C4 | D8 | 1 | 1 | yes | characteristic |
| 62 | C33×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 63 | C32×Dic3 | — | 4 | 1 | no | — |
| 64 | C3×#135 | — | 9 | 1 | no | — |
| 65 | #950 | — | 6 | 1 | no | — |
| 66 | C2×#213 | — | 18 | 1 | no | — |
| 67 | C2×D24 | — | 9 | 1 | no | — |
| 68 | #828 | — | 9 | 2 | no | — |
| 69 | #826 | — | 9 | 2 | no | — |
| 70 | #812 | — | 9 | 1 | no | — |
| 71 | C22×C3⋊D3 | — | 4 | 1 | no | — |
| 72 | C22×C3×D3 | — | 12 | 1 | no | — |
| 73 | C22×C3×D3 | — | 12 | 2 | no | — |
| 74 | C22×C3×D3 | — | 12 | 1 | no | — |
| 75 | C32×Q8 | — | 6 | 1 | no | — |
| 76 | C32×D4 | — | 6 | 1 | no | — |
| 77 | C32×D4 | — | 12 | 1 | no | — |
| 78 | C2×C32×C4 | — | 2 | 1 | no | — |
| 79 | C2×C32×C4 | — | 2 | 1 | no | — |
| 80 | C2×C32×C4 | — | 4 | 1 | no | — |
| 81 | C2×C32×C4 | — | 4 | 1 | no | — |
| 82 | C2×C32×C4 | — | 12 | 1 | no | — |
| 83 | C2×C32×C4 | #67 | 1 | 1 | yes | characteristic |
| 84 | C4⋊(C3⋊D3) | — | 2 | 2 | no | — |
| 85 | C4⋊(C3⋊D3) | — | 4 | 1 | no | — |
| 86 | C3×#67 | — | 4 | 1 | no | — |
| 87 | C3×#67 | — | 4 | 2 | no | — |
| 88 | C3×#67 | — | 4 | 1 | no | — |
| 89 | C3×D12 | — | 2 | 1 | no | — |
| 90 | C3×D12 | — | 2 | 1 | no | — |
| 91 | C3×D12 | — | 4 | 1 | no | — |
| 92 | C3×D12 | — | 6 | 2 | no | — |
| 93 | C3×D12 | — | 6 | 2 | no | — |
| 94 | C3×D12 | — | 12 | 2 | no | — |
| 95 | C3×D12 | — | 12 | 1 | no | — |
| 96 | C3×D12 | — | 12 | 2 | no | — |
| 97 | C3×D12 | — | 12 | 1 | no | — |
| 98 | C3×C4×D3 | — | 4 | 1 | no | — |
| 99 | C3×C4×D3 | — | 4 | 2 | no | — |
| 100 | C3×C4×D3 | — | 4 | 1 | no | — |
| 101 | C3×Dic6 | — | 2 | 1 | no | — |
| 102 | C3×Dic6 | — | 2 | 1 | no | — |
| 103 | C3×Dic6 | — | 4 | 1 | no | — |
| 104 | C3⋊(C3⋊C8) | #67 | 1 | 1 | yes | characteristic |
| 105 | C3⋊(C3⋊C8) | C22×D3 | 1 | 1 | yes | characteristic |
| 106 | C3×C3⋊C8 | — | 6 | 1 | no | — |
| 107 | C3×C3⋊C8 | — | 6 | 1 | no | — |
| 108 | C3×C3⋊C8 | — | 6 | 1 | no | — |
| 109 | C3×C3⋊C8 | — | 6 | 1 | no | — |
| 110 | #509 | — | 27 | 1 | no | Sylow |
| 111 | C2×C33 | — | 2 | 1 | no | — |
| 112 | C2×C33 | C2×D8 | 1 | 1 | yes | characteristic |
| 113 | C3×C3⋊D3 | — | 8 | 2 | no | — |
| 114 | C32×D3 | — | 8 | 1 | no | — |
| 115 | C3×#41 | — | 6 | 1 | no | — |
| 116 | C3×#41 | — | 6 | 1 | no | — |
| 117 | C3×#41 | — | 12 | 1 | no | — |
| 118 | C2×C3×D4 | — | 18 | 1 | no | — |
| 119 | #235 | — | 2 | 1 | no | — |
| 120 | C2×D12 | — | 6 | 1 | no | — |
| 121 | C2×D12 | — | 6 | 1 | no | — |
| 122 | C2×D12 | — | 12 | 1 | no | — |
| 123 | C3×Q16 | — | 9 | 1 | no | — |
| 124 | C3×SD16 | — | 18 | 1 | no | — |
| 125 | C3×D8 | — | 9 | 1 | no | — |
| 126 | C2×C3×C8 | — | 9 | 1 | no | — |
| 127 | #216 | — | 6 | 1 | no | — |
| 128 | #215 | — | 6 | 1 | no | — |
| 129 | #214 | — | 6 | 1 | no | — |
| 130 | #213 | — | 6 | 1 | no | — |
| 131 | #213 | — | 18 | 2 | no | — |
| 132 | #213 | — | 18 | 2 | no | — |
| 133 | C2×C3⋊C8 | — | 6 | 1 | no | — |
| 134 | C2×C3⋊C8 | — | 6 | 1 | no | — |
| 135 | D24 | — | 9 | 2 | no | — |
| 136 | D24 | — | 18 | 1 | no | — |
| 137 | C3⋊C16 | — | 9 | 2 | no | — |
| 138 | C22×C32 | — | 2 | 1 | no | — |
| 139 | C22×C32 | — | 2 | 1 | no | — |
| 140 | C22×C32 | — | 4 | 1 | no | — |
| 141 | C22×C32 | — | 4 | 1 | no | — |
| 142 | C22×C32 | — | 12 | 1 | no | — |
| 143 | C22×C32 | #213 | 1 | 1 | yes | characteristic |
| 144 | C2×C3⋊D3 | — | 4 | 2 | no | — |
| 145 | C2×C3⋊D3 | — | 8 | 2 | no | — |
| 146 | C2×C3×D3 | — | 4 | 1 | no | — |
| 147 | C2×C3×D3 | — | 4 | 2 | no | — |
| 148 | C2×C3×D3 | — | 4 | 1 | no | — |
| 149 | C2×C3×D3 | — | 12 | 2 | no | — |
| 150 | C2×C3×D3 | — | 12 | 4 | no | — |
| 151 | C2×C3×D3 | — | 12 | 2 | no | — |
| 152 | C2×C3×D3 | — | 24 | 2 | no | — |
| 153 | C2×C3×D3 | — | 24 | 2 | no | — |
| 154 | C2×C3×D3 | — | 24 | 4 | no | — |
| 155 | C32×C4 | — | 2 | 1 | no | — |
| 156 | C32×C4 | — | 2 | 1 | no | — |
| 157 | C32×C4 | — | 2 | 1 | no | — |
| 158 | C32×C4 | — | 2 | 1 | no | — |
| 159 | C32×C4 | — | 4 | 1 | no | — |
| 160 | C32×C4 | — | 4 | 1 | no | — |
| 161 | C32×C4 | — | 4 | 1 | no | — |
| 162 | C32×C4 | — | 4 | 1 | no | — |
| 163 | C32×C4 | — | 12 | 1 | no | — |
| 164 | C32×C4 | #213 | 1 | 1 | yes | characteristic |
| 165 | C32×C4 | C2×#67 | 1 | 1 | yes | characteristic |
| 166 | C3×Dic3 | — | 4 | 1 | no | — |
| 167 | C3×Dic3 | — | 4 | 2 | no | — |
| 168 | C3×Dic3 | — | 4 | 1 | no | — |
| 169 | #135 | — | 9 | 1 | no | — |
| 170 | C2×D8 | — | 27 | 1 | no | — |
| 171 | SD32 | — | 27 | 2 | no | — |
| 172 | D16 | — | 27 | 2 | no | — |
| 173 | M32 | — | 27 | 1 | no | — |
| 174 | C33 | #509 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 175 | C23×C3 | — | 36 | 1 | no | — |
| 176 | C22×D3 | — | 12 | 1 | no | — |
| 177 | C22×D3 | — | 12 | 2 | no | — |
| 178 | C22×D3 | — | 12 | 1 | no | — |
| 179 | C3×Q8 | — | 6 | 1 | no | — |
| 180 | C3×Q8 | — | 6 | 1 | no | — |
| 181 | C3×Q8 | — | 12 | 1 | no | — |
| 182 | C3×D4 | — | 6 | 1 | no | — |
| 183 | C3×D4 | — | 6 | 1 | no | — |
| 184 | C3×D4 | — | 12 | 1 | no | — |
| 185 | C3×D4 | — | 12 | 1 | no | — |
| 186 | C3×D4 | — | 12 | 2 | no | — |
| 187 | C3×D4 | — | 12 | 1 | no | — |
| 188 | C3×D4 | — | 18 | 2 | no | — |
| 189 | C3×D4 | — | 36 | 1 | no | — |
| 190 | C2×C3×C4 | — | 2 | 1 | no | — |
| 191 | C2×C3×C4 | — | 2 | 1 | no | — |
| 192 | C2×C3×C4 | — | 4 | 1 | no | — |
| 193 | C2×C3×C4 | — | 4 | 1 | no | — |
| 194 | C2×C3×C4 | — | 12 | 1 | no | — |
| 195 | C2×C3×C4 | — | 12 | 2 | no | — |
| 196 | C2×C3×C4 | — | 12 | 1 | no | — |
| 197 | C2×C3×C4 | GO+(4,𝔽2) | 1 | 1 | yes | characteristic |
| 198 | #67 | — | 4 | 1 | no | — |
| 199 | D12 | — | 2 | 1 | no | — |
| 200 | D12 | — | 6 | 2 | no | — |
| 201 | D12 | — | 6 | 2 | no | — |
| 202 | D12 | — | 12 | 2 | no | — |
| 203 | D12 | — | 12 | 1 | no | — |
| 204 | D12 | — | 12 | 2 | no | — |
| 205 | D12 | — | 12 | 1 | no | — |
| 206 | C4×D3 | — | 4 | 1 | no | — |
| 207 | Dic6 | — | 2 | 1 | no | — |
| 208 | C3×C8 | — | 9 | 1 | no | — |
| 209 | C3×C8 | — | 9 | 1 | no | — |
| 210 | C3⋊C8 | — | 6 | 1 | no | — |
| 211 | C3⋊C8 | — | 6 | 1 | no | — |
| 212 | C3⋊C8 | — | 6 | 1 | no | — |
| 213 | C3⋊C8 | — | 6 | 1 | no | — |
| 214 | C2×C32 | — | 2 | 2 | no | — |
| 215 | C2×C32 | — | 2 | 1 | no | — |
| 216 | C2×C32 | — | 2 | 1 | no | — |
| 217 | C2×C32 | — | 2 | 1 | no | — |
| 218 | C2×C32 | — | 4 | 1 | no | — |
| 219 | C2×C32 | — | 4 | 1 | no | — |
| 220 | C2×C32 | — | 4 | 1 | no | — |
| 221 | C2×C32 | — | 4 | 2 | no | — |
| 222 | C2×C32 | — | 8 | 1 | no | — |
| 223 | C2×C32 | — | 24 | 1 | no | — |
| 224 | C2×C32 | C2×#213 | 1 | 1 | yes | characteristic |
| 225 | C3⋊D3 | — | 8 | 2 | no | — |
| 226 | C3×D3 | — | 8 | 1 | no | — |
| 227 | C3×D3 | — | 8 | 2 | no | — |
| 228 | C3×D3 | — | 8 | 1 | no | — |
| 229 | C3×D3 | — | 24 | 2 | no | — |
| 230 | C3×D3 | — | 24 | 4 | no | — |
| 231 | C3×D3 | — | 24 | 2 | no | — |
| 232 | #41 | — | 6 | 1 | no | — |
| 233 | C2×D4 | — | 18 | 1 | no | — |
| 234 | Q16 | — | 9 | 1 | no | — |
| 235 | SD16 | — | 18 | 1 | no | — |
| 236 | D8 | — | 9 | 1 | no | — |
| 237 | D8 | — | 27 | 2 | no | — |
| 238 | D8 | — | 54 | 1 | no | — |
| 239 | C2×C8 | — | 9 | 1 | no | — |
| 240 | C16 | — | 27 | 2 | no | — |
| 241 | C22×C3 | — | 2 | 1 | no | — |
| 242 | C22×C3 | — | 2 | 1 | no | — |
| 243 | C22×C3 | — | 4 | 1 | no | — |
| 244 | C22×C3 | — | 4 | 1 | no | — |
| 245 | C22×C3 | — | 12 | 1 | no | — |
| 246 | C22×C3 | — | 12 | 2 | no | — |
| 247 | C22×C3 | — | 12 | 1 | no | — |
| 248 | C22×C3 | — | 36 | 2 | no | — |
| 249 | C22×C3 | — | 72 | 2 | no | — |
| 250 | C22×C3 | #3774 | 1 | 1 | yes | characteristic |
| 251 | C2×D3 | — | 4 | 1 | no | — |
| 252 | C2×D3 | — | 12 | 2 | no | — |
| 253 | C2×D3 | — | 12 | 4 | no | — |
| 254 | C2×D3 | — | 12 | 2 | no | — |
| 255 | C2×D3 | — | 24 | 2 | no | — |
| 256 | C2×D3 | — | 24 | 2 | no | — |
| 257 | C2×D3 | — | 24 | 4 | no | — |
| 258 | C3×C4 | — | 2 | 1 | no | — |
| 259 | C3×C4 | — | 2 | 1 | no | — |
| 260 | C3×C4 | — | 2 | 1 | no | — |
| 261 | C3×C4 | — | 2 | 1 | no | — |
| 262 | C3×C4 | — | 4 | 1 | no | — |
| 263 | C3×C4 | — | 4 | 1 | no | — |
| 264 | C3×C4 | — | 4 | 1 | no | — |
| 265 | C3×C4 | — | 4 | 1 | no | — |
| 266 | C3×C4 | — | 12 | 1 | no | — |
| 267 | C3×C4 | — | 12 | 2 | no | — |
| 268 | C3×C4 | — | 12 | 1 | no | — |
| 269 | C3×C4 | #3774 | 1 | 1 | yes | characteristic |
| 270 | C3×C4 | C2×GO+(4,𝔽2) | 1 | 1 | yes | characteristic |
| 271 | Dic3 | — | 4 | 1 | no | — |
| 272 | C32 | — | 2 | 1 | no | — |
| 273 | C32 | — | 2 | 1 | no | — |
| 274 | C32 | — | 4 | 1 | no | — |
| 275 | C32 | — | 4 | 1 | no | — |
| 276 | C32 | #5193 | 1 | 1 | yes | characteristic, complemented |
| 277 | C23 | — | 36 | 1 | no | — |
| 278 | Q8 | — | 6 | 1 | no | — |
| 279 | D4 | — | 6 | 1 | no | — |
| 280 | D4 | — | 12 | 1 | no | — |
| 281 | D4 | — | 18 | 2 | no | — |
| 282 | D4 | — | 36 | 1 | no | — |
| 283 | C2×C4 | — | 12 | 1 | no | — |
| 284 | C2×C4 | D32⋊D3 | 1 | 1 | yes | characteristic |
| 285 | C8 | — | 9 | 1 | no | — |
| 286 | C8 | — | 9 | 1 | no | — |
| 287 | C2×C3 | — | 2 | 2 | no | — |
| 288 | C2×C3 | — | 2 | 1 | no | — |
| 289 | C2×C3 | — | 2 | 1 | no | — |
| 290 | C2×C3 | — | 2 | 1 | no | — |
| 291 | C2×C3 | — | 4 | 1 | no | — |
| 292 | C2×C3 | — | 4 | 2 | no | — |
| 293 | C2×C3 | — | 4 | 1 | no | — |
| 294 | C2×C3 | — | 4 | 1 | no | — |
| 295 | C2×C3 | — | 8 | 1 | no | — |
| 296 | C2×C3 | — | 24 | 1 | no | — |
| 297 | C2×C3 | — | 24 | 2 | no | — |
| 298 | C2×C3 | — | 24 | 1 | no | — |
| 299 | C2×C3 | — | 72 | 2 | no | — |
| 300 | C2×C3 | C2×#3774 | 1 | 1 | yes | characteristic |
| 301 | D3 | — | 8 | 1 | no | — |
| 302 | D3 | — | 24 | 2 | no | — |
| 303 | D3 | — | 24 | 4 | no | — |
| 304 | D3 | — | 24 | 2 | no | — |
| 305 | C22 | — | 12 | 1 | no | — |
| 306 | C22 | — | 36 | 2 | no | — |
| 307 | C22 | — | 72 | 2 | no | — |
| 308 | C22 | #88935 | 1 | 1 | yes | characteristic |
| 309 | C4 | — | 12 | 1 | no | — |
| 310 | C4 | #88935 | 1 | 1 | yes | characteristic |
| 311 | C4 | C2×D32⋊D3 | 1 | 1 | yes | characteristic |
| 312 | C3 | — | 2 | 1 | no | — |
| 313 | C3 | — | 2 | 1 | no | — |
| 314 | C3 | — | 4 | 1 | no | — |
| 315 | C3 | — | 4 | 1 | no | — |
| 316 | C3 | #10593334 | 1 | 1 | yes | characteristic, complemented |
| 317 | C2 | — | 2 | 1 | no | — |
| 318 | C2 | — | 24 | 1 | no | — |
| 319 | C2 | — | 72 | 2 | no | — |
| 320 | C2 | C2×#88935 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#88810 | C2 | semidirect |
| 2. C3×#63822 | C2 | semidirect |
| 3. #11723098 | C2 | semidirect |
| 4. #11722766 | C2 | semidirect |
| 5. #11722765 | C2 | semidirect |
| 12. C3×#3718 | C22 | semidirect |
| 14. C3×#3713 | C22 | semidirect |
| 15. C3⋊#3708 | C22 | semidirect |
| 18. #63822 | D3 | semidirect |
| 49. #3718 | C2×D3 | semidirect |
| 55. #3713 | C2×D3 | semidirect |
| 174. C33 | #509 | semidirect |
| 276. C32 | #5193 | semidirect |
| 316. C3 | #10593334 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#88810 | 2 |
| C3×#63822 | 2 |
| #11723098 | 2 |
| #11722766 | 2 |
| #11722765 | 2 |
| #10593334 | 3 |
| #5193 | 9 |