Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
1316 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 1316 conjugacy classes of subgroups — too many to draw. Only its 91 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 | #11724633 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C3×#64162 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | #11722083 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | #10591725 | — | 3 | 3 | no | maximal |
| 5 | C24×C33 | C22 | 1 | 1 | yes | characteristic, complemented |
| 6 | C2×C3×#638 | — | 2 | 3 | no | — |
| 7 | C2×C3×#638 | C22 | 1 | 3 | yes | — |
| 8 | #89049 | — | 2 | 3 | no | — |
| 9 | #89046 | — | 2 | 3 | no | — |
| 10 | C3×#3757 | — | 2 | 3 | no | — |
| 11 | C3×#3757 | C22 | 1 | 3 | yes | complemented |
| 12 | #64330 | — | 3 | 3 | no | — |
| 13 | #64162 | D3 | 1 | 3 | yes | complemented |
| 14 | C3×#953 | — | 3 | 6 | no | — |
| 15 | C3×#953 | — | 6 | 3 | no | — |
| 16 | #63581 | — | 3 | 3 | no | — |
| 17 | #63580 | — | 3 | 6 | no | — |
| 18 | C23×C33 | — | 2 | 3 | no | — |
| 19 | C23×C33 | — | 4 | 2 | no | — |
| 20 | C23×C33 | C23 | 1 | 1 | yes | characteristic |
| 21 | C22×C3×C3⋊D3 | D4 | 1 | 3 | yes | complemented |
| 22 | C2×#1113 | — | 2 | 1 | no | — |
| 23 | C3×#638 | — | 2 | 6 | no | — |
| 24 | C3×#638 | — | 4 | 6 | no | — |
| 25 | C2×C3×C3⋊Dic3 | — | 2 | 3 | no | — |
| 26 | C2×C3×C3⋊Dic3 | D4 | 1 | 3 | yes | complemented |
| 27 | #6296 | — | 2 | 1 | no | — |
| 28 | #6295 | — | 2 | 3 | no | — |
| 29 | #6294 | — | 2 | 3 | no | — |
| 30 | #5032 | — | 9 | 3 | no | — |
| 31 | C24×C32 | — | 2 | 3 | no | — |
| 32 | C24×C32 | — | 4 | 1 | no | — |
| 33 | C24×C32 | C2×D3 | 1 | 3 | yes | complemented |
| 34 | C2×#638 | — | 2 | 3 | no | — |
| 35 | C2×#638 | C2×D3 | 1 | 3 | yes | — |
| 36 | C2×C3×#67 | — | 3 | 6 | no | — |
| 37 | C2×C3×#67 | — | 6 | 6 | no | — |
| 38 | C2×C3×#67 | — | 6 | 3 | no | — |
| 39 | C2×C3×#67 | — | 6 | 6 | no | — |
| 40 | #3811 | — | 6 | 3 | no | — |
| 41 | D3×#67 | — | 6 | 6 | no | — |
| 42 | #3805 | — | 6 | 3 | no | — |
| 43 | #3804 | — | 6 | 6 | no | — |
| 44 | #3757 | — | 2 | 3 | no | — |
| 45 | #3757 | C2×D3 | 1 | 3 | yes | complemented |
| 46 | C3×#217 | — | 3 | 6 | no | — |
| 47 | C3×#217 | — | 6 | 6 | no | — |
| 48 | C3×#217 | — | 6 | 3 | no | — |
| 49 | C3×#217 | — | 6 | 6 | no | — |
| 50 | C22×C33 | — | 2 | 6 | no | — |
| 51 | C22×C33 | — | 4 | 1 | no | — |
| 52 | C22×C33 | — | 4 | 3 | no | — |
| 53 | C22×C33 | — | 4 | 1 | no | — |
| 54 | C22×C33 | C2×D4 | 1 | 3 | yes | — |
| 55 | C2×C3×C3⋊D3 | — | 2 | 3 | no | — |
| 56 | C2×C3×C3⋊D3 | — | 4 | 3 | no | — |
| 57 | #1113 | — | 8 | 1 | no | — |
| 58 | C3×C3⋊Dic3 | — | 2 | 3 | no | — |
| 59 | C3×C3⋊Dic3 | — | 4 | 3 | no | — |
| 60 | #1004 | — | 9 | 3 | no | — |
| 61 | C3×#120 | — | 9 | 3 | no | — |
| 62 | #953 | — | 3 | 6 | no | — |
| 63 | #953 | — | 6 | 3 | no | — |
| 64 | #834 | — | 9 | 3 | no | — |
| 65 | #806 | — | 9 | 6 | no | — |
| 66 | C23×C32 | — | 2 | 3 | no | — |
| 67 | C23×C32 | — | 2 | 6 | no | — |
| 68 | C23×C32 | — | 2 | 6 | no | — |
| 69 | C23×C32 | — | 2 | 3 | no | — |
| 70 | C23×C32 | — | 4 | 1 | no | — |
| 71 | C23×C32 | — | 4 | 6 | no | — |
| 72 | C23×C32 | — | 4 | 6 | no | — |
| 73 | C23×C32 | — | 4 | 6 | no | — |
| 74 | C23×C32 | — | 4 | 12 | no | — |
| 75 | C23×C32 | — | 4 | 8 | no | — |
| 76 | C23×C32 | C22×D3 | 1 | 3 | yes | — |
| 77 | C22×C3⋊D3 | #67 | 1 | 3 | yes | complemented |
| 78 | C22×C3×D3 | — | 3 | 6 | no | — |
| 79 | C22×C3×D3 | — | 6 | 3 | no | — |
| 80 | C2×D32 | — | 6 | 3 | no | — |
| 81 | #638 | — | 2 | 6 | no | — |
| 82 | #638 | — | 4 | 6 | no | — |
| 83 | C2×C3⋊Dic3 | — | 2 | 3 | no | — |
| 84 | C2×C3⋊Dic3 | #67 | 1 | 3 | yes | complemented |
| 85 | C3×#67 | — | 6 | 6 | no | — |
| 86 | C3×#67 | — | 6 | 6 | no | — |
| 87 | C3×#67 | — | 12 | 6 | no | — |
| 88 | C3×#67 | — | 12 | 12 | no | — |
| 89 | C3×#67 | — | 12 | 12 | no | — |
| 90 | C2×C3×Dic3 | — | 3 | 6 | no | — |
| 91 | C2×C3×Dic3 | — | 6 | 3 | no | — |
| 92 | C2×C3×Dic3 | — | 6 | 6 | no | — |
| 93 | C2×C3×Dic3 | — | 6 | 6 | no | — |
| 94 | #627 | — | 6 | 3 | no | — |
| 95 | #626 | — | 6 | 6 | no | — |
| 96 | #625 | — | 6 | 3 | no | — |
| 97 | #624 | — | 6 | 3 | no | — |
| 98 | D3×Dic3 | — | 6 | 6 | no | — |
| 99 | B(4,𝔽2) | — | 27 | 1 | no | Sylow |
| 100 | C2×C33 | — | 2 | 3 | no | — |
| 101 | C2×C33 | — | 4 | 2 | no | — |
| 102 | C2×C33 | #120 | 1 | 1 | yes | characteristic |
| 103 | C3×C3⋊D3 | — | 4 | 3 | no | — |
| 104 | C24×C3 | — | 2 | 3 | no | — |
| 105 | C24×C3 | — | 4 | 1 | no | — |
| 106 | C24×C3 | D32 | 1 | 3 | yes | complemented |
| 107 | C2×C3×D4 | — | 9 | 3 | no | — |
| 108 | C2×C3×D4 | — | 18 | 3 | no | — |
| 109 | C2×#67 | — | 3 | 6 | no | — |
| 110 | C2×#67 | — | 6 | 3 | no | — |
| 111 | C2×#67 | — | 6 | 6 | no | — |
| 112 | C2×#67 | — | 6 | 6 | no | — |
| 113 | C2×#67 | — | 18 | 3 | no | — |
| 114 | #237 | — | 18 | 6 | no | — |
| 115 | D3×D4 | — | 18 | 6 | no | — |
| 116 | #235 | — | 18 | 3 | no | — |
| 117 | C3×C22⋊C4 | — | 9 | 3 | no | — |
| 118 | C3×C22⋊C4 | — | 18 | 3 | no | — |
| 119 | #217 | — | 3 | 6 | no | — |
| 120 | #217 | — | 6 | 3 | no | — |
| 121 | #217 | — | 6 | 6 | no | — |
| 122 | #217 | — | 6 | 6 | no | — |
| 123 | C22×C32 | — | 2 | 6 | no | — |
| 124 | C22×C32 | — | 2 | 12 | no | — |
| 125 | C22×C32 | — | 2 | 12 | no | — |
| 126 | C22×C32 | — | 2 | 6 | no | — |
| 127 | C22×C32 | — | 2 | 3 | no | — |
| 128 | C22×C32 | — | 4 | 3 | no | — |
| 129 | C22×C32 | — | 4 | 12 | no | — |
| 130 | C22×C32 | — | 4 | 12 | no | — |
| 131 | C22×C32 | — | 4 | 3 | no | — |
| 132 | C22×C32 | — | 4 | 6 | no | — |
| 133 | C22×C32 | — | 4 | 3 | no | — |
| 134 | C22×C32 | — | 4 | 6 | no | — |
| 135 | C22×C32 | — | 4 | 6 | no | — |
| 136 | C22×C32 | — | 4 | 6 | no | — |
| 137 | C22×C32 | — | 4 | 4 | no | — |
| 138 | C22×C32 | — | 4 | 4 | no | — |
| 139 | C22×C32 | — | 4 | 12 | no | — |
| 140 | C22×C32 | — | 4 | 12 | no | — |
| 141 | C22×C32 | — | 4 | 3 | no | — |
| 142 | C22×C32 | D3×D4 | 1 | 6 | yes | — |
| 143 | C22×C32 | C2×#67 | 1 | 3 | yes | — |
| 144 | C2×C3⋊D3 | — | 2 | 3 | no | — |
| 145 | C2×C3⋊D3 | — | 4 | 3 | no | — |
| 146 | C2×C3×D3 | — | 6 | 6 | no | — |
| 147 | C2×C3×D3 | — | 12 | 3 | no | — |
| 148 | C2×C3×D3 | — | 12 | 6 | no | — |
| 149 | C2×C3×D3 | — | 12 | 6 | no | — |
| 150 | D32 | — | 24 | 3 | no | — |
| 151 | C3⋊Dic3 | — | 2 | 3 | no | — |
| 152 | C3⋊Dic3 | — | 4 | 3 | no | — |
| 153 | C3×Dic3 | — | 6 | 6 | no | — |
| 154 | C3×Dic3 | — | 12 | 3 | no | — |
| 155 | C3×Dic3 | — | 12 | 6 | no | — |
| 156 | C3×Dic3 | — | 12 | 6 | no | — |
| 157 | H(5,𝔽2) | — | 27 | 1 | no | — |
| 158 | #120 | — | 9 | 3 | no | — |
| 159 | #99 | — | 27 | 3 | no | — |
| 160 | C33 | B(4,𝔽2) | 1 | 1 | yes | characteristic, complemented, Sylow |
| 161 | C23×C3 | — | 2 | 6 | no | — |
| 162 | C23×C3 | — | 2 | 6 | no | — |
| 163 | C23×C3 | — | 2 | 3 | no | — |
| 164 | C23×C3 | — | 2 | 3 | no | — |
| 165 | C23×C3 | — | 4 | 6 | no | — |
| 166 | C23×C3 | — | 4 | 6 | no | — |
| 167 | C23×C3 | — | 4 | 6 | no | — |
| 168 | C23×C3 | — | 4 | 12 | no | — |
| 169 | C23×C3 | — | 4 | 8 | no | — |
| 170 | C23×C3 | — | 4 | 1 | no | — |
| 171 | C23×C3 | — | 9 | 3 | no | — |
| 172 | C23×C3 | C2×D32 | 1 | 3 | yes | — |
| 173 | C22×D3 | — | 3 | 6 | no | — |
| 174 | C22×D3 | — | 6 | 3 | no | — |
| 175 | C22×D3 | — | 18 | 3 | no | — |
| 176 | C3×D4 | — | 18 | 6 | no | — |
| 177 | C3×D4 | — | 36 | 6 | no | — |
| 178 | C2×C3×C4 | — | 9 | 3 | no | — |
| 179 | C2×C3×C4 | — | 18 | 3 | no | — |
| 180 | #67 | — | 6 | 6 | no | — |
| 181 | #67 | — | 6 | 6 | no | — |
| 182 | #67 | — | 12 | 6 | no | — |
| 183 | #67 | — | 12 | 12 | no | — |
| 184 | #67 | — | 12 | 12 | no | — |
| 185 | #67 | — | 18 | 6 | no | — |
| 186 | C2×Dic3 | — | 3 | 6 | no | — |
| 187 | C2×Dic3 | — | 6 | 3 | no | — |
| 188 | C2×Dic3 | — | 6 | 6 | no | — |
| 189 | C2×Dic3 | — | 6 | 6 | no | — |
| 190 | C2×Dic3 | — | 18 | 3 | no | — |
| 191 | D12 | — | 18 | 3 | no | — |
| 192 | C4×D3 | — | 18 | 6 | no | — |
| 193 | Dic6 | — | 18 | 3 | no | — |
| 194 | C2×C32 | — | 2 | 3 | no | — |
| 195 | C2×C32 | — | 2 | 6 | no | — |
| 196 | C2×C32 | — | 2 | 6 | no | — |
| 197 | C2×C32 | — | 2 | 3 | no | — |
| 198 | C2×C32 | — | 4 | 1 | no | — |
| 199 | C2×C32 | — | 4 | 6 | no | — |
| 200 | C2×C32 | — | 4 | 6 | no | — |
| 201 | C2×C32 | — | 4 | 12 | no | — |
| 202 | C2×C32 | — | 4 | 8 | no | — |
| 203 | C2×C32 | — | 4 | 6 | no | — |
| 204 | C2×C32 | #937 | 1 | 3 | yes | — |
| 205 | C3⋊D3 | — | 4 | 3 | no | — |
| 206 | C3×D3 | — | 12 | 6 | no | — |
| 207 | C3×D3 | — | 24 | 3 | no | — |
| 208 | C24 | #1113 | 1 | 1 | yes | characteristic, complemented |
| 209 | #41 | — | 54 | 3 | no | — |
| 210 | C2×D4 | — | 9 | 3 | no | — |
| 211 | C2×D4 | — | 18 | 3 | no | — |
| 212 | C2×D4 | — | 54 | 3 | no | — |
| 213 | C22⋊C4 | — | 9 | 3 | no | — |
| 214 | C22⋊C4 | — | 18 | 3 | no | — |
| 215 | C22×C3 | — | 2 | 12 | no | — |
| 216 | C22×C3 | — | 2 | 12 | no | — |
| 217 | C22×C3 | — | 2 | 6 | no | — |
| 218 | C22×C3 | — | 2 | 3 | no | — |
| 219 | C22×C3 | — | 2 | 6 | no | — |
| 220 | C22×C3 | — | 4 | 12 | no | — |
| 221 | C22×C3 | — | 4 | 12 | no | — |
| 222 | C22×C3 | — | 4 | 3 | no | — |
| 223 | C22×C3 | — | 4 | 6 | no | — |
| 224 | C22×C3 | — | 4 | 6 | no | — |
| 225 | C22×C3 | — | 4 | 6 | no | — |
| 226 | C22×C3 | — | 4 | 3 | no | — |
| 227 | C22×C3 | — | 4 | 6 | no | — |
| 228 | C22×C3 | — | 4 | 4 | no | — |
| 229 | C22×C3 | — | 4 | 12 | no | — |
| 230 | C22×C3 | — | 4 | 12 | no | — |
| 231 | C22×C3 | — | 4 | 4 | no | — |
| 232 | C22×C3 | — | 4 | 3 | no | — |
| 233 | C22×C3 | — | 4 | 3 | no | — |
| 234 | C22×C3 | — | 18 | 3 | no | — |
| 235 | C22×C3 | — | 36 | 3 | no | — |
| 236 | C22×C3 | D3×#67 | 1 | 6 | yes | — |
| 237 | C22×C3 | #3811 | 1 | 3 | yes | — |
| 238 | C2×D3 | — | 6 | 6 | no | — |
| 239 | C2×D3 | — | 12 | 3 | no | — |
| 240 | C2×D3 | — | 12 | 6 | no | — |
| 241 | C2×D3 | — | 12 | 6 | no | — |
| 242 | C2×D3 | — | 72 | 3 | no | — |
| 243 | C3×C4 | — | 18 | 3 | no | — |
| 244 | C3×C4 | — | 36 | 3 | no | — |
| 245 | Dic3 | — | 6 | 6 | no | — |
| 246 | Dic3 | — | 12 | 3 | no | — |
| 247 | Dic3 | — | 12 | 6 | no | — |
| 248 | Dic3 | — | 12 | 6 | no | — |
| 249 | C32 | — | 2 | 3 | no | — |
| 250 | C32 | — | 4 | 1 | no | — |
| 251 | C32 | #5032 | 1 | 3 | yes | complemented |
| 252 | C23 | — | 2 | 3 | no | — |
| 253 | C23 | — | 4 | 2 | no | — |
| 254 | C23 | — | 9 | 3 | no | — |
| 255 | C23 | — | 54 | 1 | no | — |
| 256 | C23 | C2×#1113 | 1 | 1 | yes | characteristic |
| 257 | Q8 | — | 54 | 1 | no | — |
| 258 | D4 | — | 18 | 6 | no | — |
| 259 | D4 | — | 36 | 6 | no | — |
| 260 | D4 | — | 54 | 3 | no | — |
| 261 | C2×C4 | — | 9 | 3 | no | — |
| 262 | C2×C4 | — | 18 | 3 | no | — |
| 263 | C2×C4 | — | 54 | 3 | no | — |
| 264 | C2×C3 | — | 2 | 6 | no | — |
| 265 | C2×C3 | — | 2 | 6 | no | — |
| 266 | C2×C3 | — | 2 | 3 | no | — |
| 267 | C2×C3 | — | 2 | 3 | no | — |
| 268 | C2×C3 | — | 4 | 6 | no | — |
| 269 | C2×C3 | — | 4 | 6 | no | — |
| 270 | C2×C3 | — | 4 | 12 | no | — |
| 271 | C2×C3 | — | 4 | 8 | no | — |
| 272 | C2×C3 | — | 4 | 6 | no | — |
| 273 | C2×C3 | — | 4 | 1 | no | — |
| 274 | C2×C3 | — | 36 | 3 | no | — |
| 275 | C2×C3 | #63983 | 1 | 3 | yes | — |
| 276 | D3 | — | 12 | 6 | no | — |
| 277 | D3 | — | 24 | 3 | no | — |
| 278 | C22 | — | 2 | 6 | no | — |
| 279 | C22 | — | 4 | 1 | no | — |
| 280 | C22 | — | 4 | 3 | no | — |
| 281 | C22 | — | 4 | 1 | no | — |
| 282 | C22 | — | 18 | 3 | no | — |
| 283 | C22 | — | 36 | 3 | no | — |
| 284 | C22 | — | 216 | 1 | no | — |
| 285 | C22 | #89049 | 1 | 3 | yes | — |
| 286 | C4 | — | 18 | 3 | no | — |
| 287 | C4 | — | 36 | 3 | no | — |
| 288 | C3 | — | 2 | 3 | no | — |
| 289 | C3 | — | 4 | 1 | no | — |
| 290 | C3 | #10591725 | 1 | 3 | yes | complemented |
| 291 | C2 | — | 2 | 3 | no | — |
| 292 | C2 | — | 4 | 2 | no | — |
| 293 | C2 | — | 36 | 3 | no | — |
| 294 | C2 | #11723729 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #11723729 |
| commutator, cosocle | C23×C33 | C23 |
| Frattini | C23 | C2×#1113 |
| Fitting | C24×C33 | C22 |
| socle | C2×C33 | #120 |
| hypercenter | C24 | #1113 |
| hypocenter | C33 | B(4,𝔽2) |
| LCS | G ▷ C23×C33 ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C24 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #11724633 | C2 | semidirect |
| 2. C3×#64162 | C2 | semidirect |
| 3. #11722083 | C2 | semidirect |
| 5. C24×C33 | C22 | semidirect |
| 11. C3×#3757 | C22 | semidirect |
| 13. #64162 | D3 | semidirect |
| 21. C22×C3×C3⋊D3 | D4 | semidirect |
| 26. C2×C3×C3⋊Dic3 | D4 | semidirect |
| 33. C24×C32 | C2×D3 | semidirect |
| 45. #3757 | C2×D3 | semidirect |
| 77. C22×C3⋊D3 | #67 | semidirect |
| 84. C2×C3⋊Dic3 | #67 | semidirect |
| 106. C24×C3 | D32 | semidirect |
| 160. C33 | B(4,𝔽2) | semidirect |
| 208. C24 | #1113 | semidirect |
| 251. C32 | #5032 | semidirect |
| 290. C3 | #10591725 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11724633 | 2 |
| C3×#64162 | 2 |
| #11722083 | 2 |
| #10591725 | 3 |