Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
390 conjugacy classes of subgroups; only the 43 normal subgroups are shown.
This group has 390 conjugacy classes of subgroups — too many to draw. Only its 43 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 | Q8×C22⋊H(3,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C4⋊#6258 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #11720736 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C3×#5417 | — | 3 | 3 | no | maximal |
| 5 | C3×#5227 | — | 3 | 1 | no | maximal |
| 6 | C2×#6258 | — | 2 | 1 | no | — |
| 7 | C4×C22⋊H(3,𝔽3) | — | 2 | 1 | no | — |
| 8 | C4×C22⋊H(3,𝔽3) | C22 | 1 | 1 | yes | characteristic |
| 9 | #88549 | — | 4 | 1 | no | maximal |
| 10 | C22×C32×Q8 | D3 | 1 | 1 | yes | characteristic, complemented |
| 11 | C3×Q8×A4 | D3 | 1 | 3 | yes | complemented |
| 12 | C3×C4⋊S4 | — | 3 | 3 | no | — |
| 13 | C2×C3×#215 | — | 3 | 1 | no | — |
| 14 | C2×C3×#215 | — | 3 | 1 | no | — |
| 15 | C3×#930 | — | 3 | 1 | no | — |
| 16 | C3×A4⋊C8 | — | 3 | 3 | no | — |
| 17 | C3×#835 | — | 3 | 1 | no | — |
| 18 | C3×#835 | — | 3 | 1 | no | — |
| 19 | C3×#830 | — | 3 | 1 | no | — |
| 20 | C2×C22⋊H(3,𝔽3) | D4 | 1 | 1 | yes | characteristic |
| 21 | #6258 | — | 4 | 1 | no | — |
| 22 | Q8×H(3,𝔽3) | — | 4 | 1 | no | — |
| 23 | C4⋊#272 | — | 4 | 1 | no | — |
| 24 | #6180 | — | 4 | 1 | no | — |
| 25 | #5417 | — | 3 | 3 | no | — |
| 26 | C3×#448 | — | 9 | 1 | no | — |
| 27 | #5227 | — | 3 | 1 | no | — |
| 28 | C2×C3×S4 | — | 6 | 3 | no | — |
| 29 | C2×C32×Q8 | — | 3 | 1 | no | — |
| 30 | C2×C32×Q8 | — | 3 | 1 | no | — |
| 31 | C2×C32×Q8 | — | 6 | 1 | no | — |
| 32 | C22×C32×C4 | — | 2 | 1 | no | — |
| 33 | C22×C32×C4 | C2×D3 | 1 | 1 | yes | characteristic |
| 34 | C2×C3×#67 | — | 6 | 1 | no | — |
| 35 | C2×C3×D12 | — | 3 | 1 | no | — |
| 36 | C3×C4×A4 | — | 2 | 3 | no | — |
| 37 | C3×C4×A4 | C2×D3 | 1 | 3 | yes | — |
| 38 | C3×#215 | — | 6 | 2 | no | — |
| 39 | C3×#215 | — | 6 | 1 | no | — |
| 40 | C3×#215 | — | 6 | 1 | no | — |
| 41 | C3×#215 | — | 12 | 3 | no | — |
| 42 | C3×(C2×D3)⋊C4 | — | 6 | 1 | no | — |
| 43 | C3×C4⋊Dic3 | — | 3 | 1 | no | — |
| 44 | C2×C3×C3⋊C8 | — | 3 | 1 | no | — |
| 45 | C2×C3×C3⋊C8 | — | 3 | 1 | no | — |
| 46 | C2×#272 | — | 8 | 1 | no | — |
| 47 | C22⋊H(3,𝔽3) | SD16 | 1 | 1 | yes | characteristic, complemented |
| 48 | C4×H(3,𝔽3) | — | 4 | 1 | no | — |
| 49 | C4×H(3,𝔽3) | — | 8 | 1 | no | — |
| 50 | C22×C3×Q8 | — | 3 | 1 | no | — |
| 51 | C22×C3×Q8 | C3⋊D3 | 1 | 1 | yes | characteristic |
| 52 | Q8×A4 | — | 3 | 3 | no | — |
| 53 | C4⋊S4 | — | 3 | 3 | no | — |
| 54 | C2×C3×SD16 | — | 9 | 1 | no | — |
| 55 | C2×C3×SD16 | — | 9 | 1 | no | — |
| 56 | C3×#121 | — | 9 | 1 | no | — |
| 57 | C2×#215 | — | 3 | 1 | no | — |
| 58 | C2×#215 | — | 3 | 1 | no | — |
| 59 | #930 | — | 3 | 1 | no | — |
| 60 | A4⋊C8 | — | 3 | 3 | no | — |
| 61 | C3×#103 | — | 9 | 1 | no | — |
| 62 | C3×#103 | — | 9 | 1 | no | — |
| 63 | C3×C22⋊C8 | — | 9 | 1 | no | — |
| 64 | #835 | — | 3 | 1 | no | — |
| 65 | #835 | — | 3 | 1 | no | — |
| 66 | #830 | — | 3 | 1 | no | — |
| 67 | C23×C32 | #67 | 1 | 1 | yes | characteristic |
| 68 | C22×C3×D3 | — | 6 | 1 | no | — |
| 69 | C2×C3×A4 | #67 | 1 | 3 | yes | — |
| 70 | C3×S4 | — | 12 | 3 | no | — |
| 71 | C32×Q8 | — | 3 | 2 | no | — |
| 72 | C32×Q8 | — | 3 | 1 | no | — |
| 73 | C32×Q8 | — | 4 | 3 | no | — |
| 74 | C32×Q8 | — | 6 | 1 | no | — |
| 75 | C32×Q8 | S4 | 1 | 1 | yes | characteristic, complemented |
| 76 | C2×C32×C4 | — | 3 | 1 | no | — |
| 77 | C2×C32×C4 | — | 3 | 1 | no | — |
| 78 | C2×C32×C4 | — | 6 | 1 | no | — |
| 79 | C2×C32×C4 | — | 6 | 1 | no | — |
| 80 | C3×#67 | — | 12 | 1 | no | — |
| 81 | C3×#67 | — | 12 | 1 | no | — |
| 82 | C2×C3×Dic3 | — | 6 | 1 | no | — |
| 83 | C3×D12 | — | 6 | 1 | no | — |
| 84 | C3×D12 | — | 6 | 1 | no | — |
| 85 | C3×D12 | — | 12 | 3 | no | — |
| 86 | C3×C3⋊C8 | — | 6 | 1 | no | — |
| 87 | C3×C3⋊C8 | — | 6 | 1 | no | — |
| 88 | C3×C3⋊C8 | — | 12 | 3 | no | — |
| 89 | #448 | — | 9 | 1 | no | Sylow |
| 90 | C2×H(3,𝔽3) | — | 4 | 1 | no | — |
| 91 | #272 | — | 16 | 1 | no | — |
| 92 | C2×S4 | — | 6 | 3 | no | — |
| 93 | C2×C3×Q8 | — | 3 | 1 | no | — |
| 94 | C2×C3×Q8 | — | 3 | 1 | no | — |
| 95 | C2×C3×Q8 | — | 3 | 1 | no | — |
| 96 | C2×C3×Q8 | — | 3 | 1 | no | — |
| 97 | C2×C3×Q8 | — | 6 | 1 | no | — |
| 98 | C2×C3×Q8 | — | 6 | 1 | no | — |
| 99 | C2×C3×Q8 | — | 6 | 1 | no | — |
| 100 | C2×C3×Q8 | — | 6 | 1 | no | — |
| 101 | C2×C3×Q8 | — | 6 | 2 | no | — |
| 102 | C2×C3×D4 | — | 9 | 1 | no | — |
| 103 | C2×C3×D4 | — | 18 | 1 | no | — |
| 104 | C22×C3×C4 | — | 2 | 1 | no | — |
| 105 | C22×C3×C4 | — | 3 | 1 | no | — |
| 106 | C22×C3×C4 | — | 6 | 1 | no | — |
| 107 | C22×C3×C4 | C2×C3⋊D3 | 1 | 1 | yes | characteristic |
| 108 | C2×#67 | — | 6 | 1 | no | — |
| 109 | C2×D12 | — | 3 | 1 | no | — |
| 110 | C4×A4 | — | 3 | 3 | no | — |
| 111 | C4×A4 | — | 6 | 3 | no | — |
| 112 | C3×SD16 | — | 18 | 2 | no | — |
| 113 | C3×SD16 | — | 18 | 1 | no | — |
| 114 | C3×SD16 | — | 18 | 1 | no | — |
| 115 | C2×C3×C8 | — | 9 | 1 | no | — |
| 116 | C2×C3×C8 | — | 9 | 1 | no | — |
| 117 | C3×C4⋊C4 | — | 9 | 1 | no | — |
| 118 | C3×C22⋊C4 | — | 18 | 1 | no | — |
| 119 | #215 | — | 6 | 2 | no | — |
| 120 | #215 | — | 6 | 1 | no | — |
| 121 | #215 | — | 6 | 1 | no | — |
| 122 | #215 | — | 12 | 3 | no | — |
| 123 | (C2×D3)⋊C4 | — | 6 | 1 | no | — |
| 124 | C4⋊Dic3 | — | 3 | 1 | no | — |
| 125 | C2×C3⋊C8 | — | 3 | 1 | no | — |
| 126 | C2×C3⋊C8 | — | 3 | 1 | no | — |
| 127 | C22×C32 | — | 3 | 1 | no | — |
| 128 | C22×C32 | — | 3 | 1 | no | — |
| 129 | C22×C32 | #215 | 1 | 1 | yes | characteristic, complemented |
| 130 | C2×C3×D3 | — | 12 | 1 | no | — |
| 131 | C2×C3×D3 | — | 12 | 1 | no | — |
| 132 | C2×C3×D3 | — | 12 | 1 | no | — |
| 133 | C2×C3×D3 | — | 24 | 3 | no | — |
| 134 | C3×A4 | #215 | 1 | 3 | yes | complemented |
| 135 | C32×C4 | — | 2 | 1 | no | — |
| 136 | C32×C4 | — | 3 | 1 | no | — |
| 137 | C32×C4 | — | 4 | 3 | no | — |
| 138 | C32×C4 | — | 6 | 1 | no | — |
| 139 | C32×C4 | — | 8 | 3 | no | — |
| 140 | C32×C4 | C2×S4 | 1 | 1 | yes | characteristic |
| 141 | C3×Dic3 | — | 12 | 1 | no | — |
| 142 | C22×Q8 | #272 | 1 | 1 | yes | characteristic, complemented |
| 143 | C2×SD16 | — | 9 | 1 | no | — |
| 144 | C2×SD16 | — | 9 | 1 | no | — |
| 145 | #121 | — | 9 | 1 | no | — |
| 146 | #103 | — | 9 | 1 | no | — |
| 147 | #103 | — | 9 | 1 | no | — |
| 148 | C22⋊C8 | — | 9 | 1 | no | — |
| 149 | H(3,𝔽3) | — | 4 | 1 | no | Sylow |
| 150 | C23×C3 | — | 3 | 1 | no | — |
| 151 | C23×C3 | — | 18 | 1 | no | — |
| 152 | C23×C3 | #638 | 1 | 1 | yes | characteristic |
| 153 | C22×D3 | — | 6 | 1 | no | — |
| 154 | C2×A4 | — | 3 | 3 | no | — |
| 155 | S4 | — | 12 | 3 | no | — |
| 156 | C3×Q8 | — | 3 | 2 | no | — |
| 157 | C3×Q8 | — | 3 | 1 | no | — |
| 158 | C3×Q8 | — | 3 | 2 | no | — |
| 159 | C3×Q8 | — | 3 | 1 | no | — |
| 160 | C3×Q8 | — | 3 | 1 | no | — |
| 161 | C3×Q8 | — | 6 | 1 | no | — |
| 162 | C3×Q8 | — | 6 | 2 | no | — |
| 163 | C3×Q8 | — | 6 | 1 | no | — |
| 164 | C3×Q8 | — | 6 | 1 | no | — |
| 165 | C3×Q8 | — | 6 | 2 | no | — |
| 166 | C3×Q8 | — | 12 | 3 | no | — |
| 167 | C3×Q8 | C3⋊S4 | 1 | 1 | yes | characteristic |
| 168 | C3×D4 | — | 18 | 1 | no | — |
| 169 | C3×D4 | — | 18 | 1 | no | — |
| 170 | C3×D4 | — | 36 | 1 | no | — |
| 171 | C3×D4 | — | 36 | 1 | no | — |
| 172 | C2×C3×C4 | — | 3 | 1 | no | — |
| 173 | C2×C3×C4 | — | 3 | 1 | no | — |
| 174 | C2×C3×C4 | — | 3 | 1 | no | — |
| 175 | C2×C3×C4 | — | 3 | 1 | no | — |
| 176 | C2×C3×C4 | — | 6 | 1 | no | — |
| 177 | C2×C3×C4 | — | 6 | 1 | no | — |
| 178 | C2×C3×C4 | — | 6 | 1 | no | — |
| 179 | C2×C3×C4 | — | 6 | 1 | no | — |
| 180 | C2×C3×C4 | — | 6 | 1 | no | — |
| 181 | C2×C3×C4 | — | 6 | 2 | no | — |
| 182 | C2×C3×C4 | — | 6 | 2 | no | — |
| 183 | C2×C3×C4 | — | 6 | 1 | no | — |
| 184 | C2×C3×C4 | — | 18 | 1 | no | — |
| 185 | #67 | — | 12 | 1 | no | — |
| 186 | #67 | — | 12 | 1 | no | — |
| 187 | C2×Dic3 | — | 6 | 1 | no | — |
| 188 | D12 | — | 6 | 1 | no | — |
| 189 | D12 | — | 6 | 1 | no | — |
| 190 | D12 | — | 12 | 3 | no | — |
| 191 | C3×C8 | — | 18 | 1 | no | — |
| 192 | C3×C8 | — | 18 | 1 | no | — |
| 193 | C3⋊C8 | — | 6 | 1 | no | — |
| 194 | C3⋊C8 | — | 6 | 1 | no | — |
| 195 | C3⋊C8 | — | 12 | 3 | no | — |
| 196 | C2×C32 | — | 3 | 1 | no | — |
| 197 | C2×C32 | — | 3 | 1 | no | — |
| 198 | C2×C32 | — | 4 | 3 | no | — |
| 199 | C2×C32 | GL(2,ℤ/4) | 1 | 1 | yes | characteristic |
| 200 | C3×D3 | — | 24 | 1 | no | — |
| 201 | C3×D3 | — | 48 | 3 | no | — |
| 202 | C2×Q8 | — | 3 | 1 | no | — |
| 203 | C2×Q8 | — | 3 | 1 | no | — |
| 204 | C2×Q8 | — | 6 | 1 | no | — |
| 205 | C2×D4 | — | 9 | 1 | no | — |
| 206 | C2×D4 | — | 18 | 1 | no | — |
| 207 | C22×C4 | — | 2 | 1 | no | — |
| 208 | C22×C4 | C2×#272 | 1 | 1 | yes | characteristic |
| 209 | SD16 | — | 18 | 2 | no | — |
| 210 | SD16 | — | 18 | 1 | no | — |
| 211 | SD16 | — | 18 | 1 | no | — |
| 212 | C2×C8 | — | 9 | 1 | no | — |
| 213 | C2×C8 | — | 9 | 1 | no | — |
| 214 | C4⋊C4 | — | 9 | 1 | no | — |
| 215 | C22⋊C4 | — | 18 | 1 | no | — |
| 216 | C22×C3 | — | 3 | 1 | no | — |
| 217 | C22×C3 | — | 3 | 1 | no | — |
| 218 | C22×C3 | — | 3 | 1 | no | — |
| 219 | C22×C3 | — | 3 | 1 | no | — |
| 220 | C22×C3 | — | 3 | 1 | no | — |
| 221 | C22×C3 | — | 6 | 1 | no | — |
| 222 | C22×C3 | — | 6 | 1 | no | — |
| 223 | C22×C3 | — | 36 | 1 | no | — |
| 224 | C22×C3 | — | 36 | 1 | no | — |
| 225 | C22×C3 | — | 36 | 1 | no | — |
| 226 | C22×C3 | #3755 | 1 | 1 | yes | characteristic |
| 227 | C2×D3 | — | 12 | 1 | no | — |
| 228 | C2×D3 | — | 12 | 1 | no | — |
| 229 | C2×D3 | — | 12 | 1 | no | — |
| 230 | C2×D3 | — | 24 | 3 | no | — |
| 231 | A4 | — | 3 | 3 | no | — |
| 232 | C3×C4 | — | 2 | 1 | no | — |
| 233 | C3×C4 | — | 3 | 1 | no | — |
| 234 | C3×C4 | — | 3 | 1 | no | — |
| 235 | C3×C4 | — | 3 | 1 | no | — |
| 236 | C3×C4 | — | 6 | 1 | no | — |
| 237 | C3×C4 | — | 6 | 1 | no | — |
| 238 | C3×C4 | — | 6 | 2 | no | — |
| 239 | C3×C4 | — | 6 | 1 | no | — |
| 240 | C3×C4 | — | 6 | 1 | no | — |
| 241 | C3×C4 | — | 12 | 3 | no | — |
| 242 | C3×C4 | — | 24 | 3 | no | — |
| 243 | C3×C4 | — | 36 | 1 | no | — |
| 244 | C3×C4 | C2×C3⋊S4 | 1 | 1 | yes | characteristic |
| 245 | Dic3 | — | 12 | 1 | no | — |
| 246 | C32 | — | 4 | 3 | no | — |
| 247 | C32 | #5417 | 1 | 1 | yes | characteristic, complemented |
| 248 | C23 | — | 18 | 1 | no | — |
| 249 | C23 | #6235 | 1 | 1 | yes | characteristic |
| 250 | Q8 | — | 3 | 2 | no | — |
| 251 | Q8 | — | 3 | 1 | no | — |
| 252 | Q8 | — | 6 | 1 | no | — |
| 253 | Q8 | #6258 | 1 | 1 | yes | characteristic, complemented |
| 254 | D4 | — | 18 | 1 | no | — |
| 255 | D4 | — | 18 | 1 | no | — |
| 256 | D4 | — | 36 | 1 | no | — |
| 257 | D4 | — | 36 | 1 | no | — |
| 258 | C2×C4 | — | 3 | 1 | no | — |
| 259 | C2×C4 | — | 3 | 1 | no | — |
| 260 | C2×C4 | — | 6 | 1 | no | — |
| 261 | C2×C4 | — | 6 | 1 | no | — |
| 262 | C2×C4 | — | 18 | 1 | no | — |
| 263 | C8 | — | 18 | 1 | no | — |
| 264 | C8 | — | 18 | 1 | no | — |
| 265 | C2×C3 | — | 3 | 1 | no | — |
| 266 | C2×C3 | — | 3 | 1 | no | — |
| 267 | C2×C3 | — | 3 | 1 | no | — |
| 268 | C2×C3 | — | 3 | 1 | no | — |
| 269 | C2×C3 | — | 3 | 1 | no | — |
| 270 | C2×C3 | — | 6 | 1 | no | — |
| 271 | C2×C3 | — | 6 | 1 | no | — |
| 272 | C2×C3 | — | 12 | 3 | no | — |
| 273 | C2×C3 | — | 72 | 1 | no | — |
| 274 | C2×C3 | #64269 | 1 | 1 | yes | characteristic |
| 275 | D3 | — | 24 | 1 | no | — |
| 276 | D3 | — | 48 | 3 | no | — |
| 277 | C22 | — | 3 | 1 | no | — |
| 278 | C22 | — | 3 | 1 | no | — |
| 279 | C22 | — | 36 | 1 | no | — |
| 280 | C22 | — | 36 | 1 | no | — |
| 281 | C22 | — | 36 | 1 | no | — |
| 282 | C22 | #88549 | 1 | 1 | yes | characteristic, complemented |
| 283 | C4 | — | 2 | 1 | no | — |
| 284 | C4 | — | 3 | 1 | no | — |
| 285 | C4 | — | 6 | 1 | no | — |
| 286 | C4 | — | 36 | 1 | no | — |
| 287 | C4 | C2×#6258 | 1 | 1 | yes | characteristic |
| 288 | C3 | — | 3 | 1 | no | — |
| 289 | C3 | — | 12 | 3 | no | — |
| 290 | C3 | #10593651 | 1 | 1 | yes | characteristic |
| 291 | C2 | — | 3 | 1 | no | — |
| 292 | C2 | — | 3 | 1 | no | — |
| 293 | C2 | — | 72 | 1 | no | — |
| 294 | C2 | #11722398 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. Q8×C22⋊H(3,𝔽3) | C2 | semidirect |
| 3. #11720736 | C2 | semidirect |
| 10. C22×C32×Q8 | D3 | semidirect |
| 11. C3×Q8×A4 | D3 | semidirect |
| 47. C22⋊H(3,𝔽3) | SD16 | semidirect |
| 75. C32×Q8 | S4 | semidirect |
| 129. C22×C32 | #215 | semidirect |
| 134. C3×A4 | #215 | semidirect |
| 142. C22×Q8 | #272 | semidirect |
| 247. C32 | #5417 | semidirect |
| 253. Q8 | #6258 | semidirect |
| 282. C22 | #88549 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| Q8×C22⋊H(3,𝔽3) | 2 |
| C4⋊#6258 | 2 |
| #11720736 | 2 |
| C3×#5417 | 3 |
| C3×#5227 | 3 |
| #88549 | 4 |