Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
442 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 442 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 | #11722412 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #11722372 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C9×#985 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #11722236 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #11722231 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #10596447 | — | 3 | 1 | no | maximal |
| 7 | #10589872 | — | 3 | 1 | no | maximal |
| 8 | C9×#41⋊C3 | C22 | 1 | 1 | yes | characteristic |
| 9 | D3×Dic18 | — | 4 | 1 | no | maximal |
| 10 | #88615 | C22 | 1 | 2 | yes | — |
| 11 | C9⋊#226 | C22 | 1 | 2 | yes | complemented |
| 12 | C9×GL(2,𝔽3) | C22 | 1 | 1 | yes | characteristic, complemented |
| 13 | C9×#226 | C22 | 1 | 1 | yes | characteristic, complemented |
| 14 | C3⋊#994 | — | 3 | 1 | no | — |
| 15 | #64265 | — | 3 | 1 | no | — |
| 16 | C3×#985 | D3 | 1 | 1 | yes | characteristic |
| 17 | #64197 | — | 3 | 2 | no | — |
| 18 | #64196 | — | 3 | 2 | no | — |
| 19 | #63716 | — | 3 | 1 | no | — |
| 20 | #63715 | D3 | 1 | 1 | yes | characteristic, complemented |
| 21 | #63690 | — | 6 | 1 | no | — |
| 22 | C9×#135 | — | 3 | 1 | no | — |
| 23 | #63512 | — | 3 | 1 | no | — |
| 24 | #63511 | — | 3 | 1 | no | — |
| 25 | #63480 | — | 3 | 2 | no | — |
| 26 | #63478 | — | 3 | 2 | no | — |
| 27 | SD16×D9 | — | 3 | 2 | no | — |
| 28 | #63473 | — | 3 | 2 | no | — |
| 29 | C2×#3664 | — | 3 | 1 | no | — |
| 30 | #63464 | — | 3 | 1 | no | — |
| 31 | #6226 | — | 4 | 1 | no | — |
| 32 | C4×D3×C9 | — | 4 | 1 | no | — |
| 33 | C3×Dic18 | — | 4 | 1 | no | — |
| 34 | C9×SL(2,𝔽3) | C23 | 1 | 1 | yes | characteristic |
| 35 | D3×Dic9 | — | 4 | 2 | no | — |
| 36 | #6189 | — | 4 | 2 | no | — |
| 37 | #5924 | — | 9 | 1 | no | — |
| 38 | #5770 | — | 9 | 1 | no | — |
| 39 | C3×#41⋊C3 | C2×D3 | 1 | 1 | yes | characteristic |
| 40 | D3×Dic6 | — | 12 | 1 | no | — |
| 41 | C3⋊(#41⋊C3) | — | 3 | 2 | no | — |
| 42 | C3⋊#226 | — | 3 | 2 | no | — |
| 43 | C3×GL(2,𝔽3) | C2×D3 | 1 | 1 | yes | characteristic |
| 44 | C3×#226 | C2×D3 | 1 | 1 | yes | characteristic |
| 45 | C9×#41 | — | 3 | 1 | no | — |
| 46 | C9×#41 | C2×D3 | 1 | 1 | yes | characteristic |
| 47 | #3701 | C2×D3 | 1 | 2 | yes | — |
| 48 | Q8×D9 | — | 3 | 2 | no | — |
| 49 | #3699 | — | 3 | 2 | no | — |
| 50 | #3699 | — | 6 | 2 | no | — |
| 51 | D4×D9 | — | 3 | 2 | no | — |
| 52 | #3697 | — | 3 | 1 | no | — |
| 53 | #3697 | — | 6 | 1 | no | — |
| 54 | C2×D36 | — | 3 | 1 | no | — |
| 55 | C2×Dic18 | — | 3 | 1 | no | — |
| 56 | C2×Dic18 | — | 6 | 1 | no | — |
| 57 | #41⋊C9 | — | 2 | 1 | no | — |
| 58 | #3688 | — | 6 | 2 | no | — |
| 59 | C9×Q16 | — | 3 | 1 | no | — |
| 60 | C9×SD16 | — | 3 | 1 | no | — |
| 61 | C9×SD16 | — | 3 | 1 | no | — |
| 62 | C9×D8 | — | 3 | 1 | no | — |
| 63 | C2×C8×C9 | — | 3 | 1 | no | — |
| 64 | #3675 | — | 3 | 2 | no | — |
| 65 | #3674 | — | 3 | 2 | no | — |
| 66 | #3673 | — | 3 | 2 | no | — |
| 67 | #3672 | — | 3 | 2 | no | — |
| 68 | #3667 | — | 3 | 1 | no | — |
| 69 | #3664 | — | 3 | 2 | no | — |
| 70 | #3664 | — | 3 | 2 | no | — |
| 71 | #3663 | — | 3 | 2 | no | — |
| 72 | C8×D9 | — | 3 | 2 | no | — |
| 73 | C2×D3×C9 | — | 4 | 1 | no | — |
| 74 | C3×C4×C9 | — | 4 | 1 | no | — |
| 75 | C3⋊Dic9 | — | 4 | 2 | no | — |
| 76 | C9×Dic3 | — | 4 | 1 | no | — |
| 77 | C3×Dic9 | — | 4 | 2 | no | — |
| 78 | #1010 | — | 9 | 1 | no | — |
| 79 | #1009 | — | 3 | 1 | no | — |
| 80 | #994 | — | 9 | 1 | no | — |
| 81 | #985 | — | 9 | 2 | no | — |
| 82 | #985 | D9 | 1 | 1 | yes | characteristic, complemented |
| 83 | #984 | — | 9 | 1 | no | — |
| 84 | #984 | — | 9 | 2 | no | — |
| 85 | #984 | — | 18 | 1 | no | — |
| 86 | C3×#135 | — | 3 | 1 | no | — |
| 87 | #951 | — | 9 | 1 | no | — |
| 88 | #949 | — | 9 | 1 | no | — |
| 89 | #918 | — | 9 | 2 | no | — |
| 90 | #916 | — | 9 | 2 | no | — |
| 91 | D3×SD16 | — | 9 | 2 | no | — |
| 92 | #911 | — | 9 | 2 | no | — |
| 93 | C2×#204 | — | 9 | 1 | no | — |
| 94 | #901 | — | 9 | 1 | no | — |
| 95 | #634 | — | 12 | 1 | no | — |
| 96 | C3×C4×D3 | — | 4 | 1 | no | — |
| 97 | C3×Dic6 | — | 12 | 1 | no | — |
| 98 | C3×SL(2,𝔽3) | C22×D3 | 1 | 1 | yes | characteristic |
| 99 | #627 | — | 12 | 2 | no | — |
| 100 | D3×Dic3 | — | 12 | 2 | no | — |
| 101 | C22×D9 | — | 6 | 1 | no | — |
| 102 | Q8×C9 | — | 3 | 1 | no | — |
| 103 | Q8×C9 | C22×D3 | 1 | 1 | yes | characteristic |
| 104 | D4×C9 | — | 3 | 1 | no | — |
| 105 | D4×C9 | — | 3 | 1 | no | — |
| 106 | D4×C9 | — | 6 | 1 | no | — |
| 107 | C2×C4×C9 | — | 3 | 1 | no | — |
| 108 | C2×C4×C9 | — | 6 | 1 | no | — |
| 109 | #611 | — | 3 | 2 | no | — |
| 110 | #611 | — | 6 | 2 | no | — |
| 111 | C2×Dic9 | — | 6 | 1 | no | — |
| 112 | C2×Dic9 | — | 6 | 2 | no | — |
| 113 | D36 | — | 3 | 2 | no | — |
| 114 | D36 | — | 3 | 1 | no | — |
| 115 | C4×D9 | — | 3 | 2 | no | — |
| 116 | C4×D9 | — | 6 | 2 | no | — |
| 117 | Dic18 | — | 3 | 2 | no | — |
| 118 | Dic18 | — | 6 | 1 | no | — |
| 119 | Dic18 | — | 6 | 2 | no | — |
| 120 | Dic18 | — | 24 | 1 | no | — |
| 121 | Dic18 | S4 | 1 | 1 | yes | characteristic |
| 122 | Q8⋊C9 | — | 2 | 1 | no | — |
| 123 | C8×C9 | — | 3 | 1 | no | — |
| 124 | C8×C9 | — | 3 | 1 | no | — |
| 125 | C9⋊C8 | — | 3 | 2 | no | — |
| 126 | #577 | — | 27 | 1 | no | Sylow |
| 127 | C2×C3×C9 | — | 4 | 1 | no | — |
| 128 | D3×C9 | — | 4 | 2 | no | — |
| 129 | C3×#41 | — | 3 | 1 | no | — |
| 130 | C3×#41 | D32 | 1 | 1 | yes | characteristic |
| 131 | #239 | — | 3 | 2 | no | — |
| 132 | D3×Q8 | — | 9 | 2 | no | — |
| 133 | D3×Q8 | — | 36 | 1 | no | — |
| 134 | #237 | — | 9 | 2 | no | — |
| 135 | #237 | — | 18 | 2 | no | — |
| 136 | D3×D4 | — | 9 | 2 | no | — |
| 137 | #235 | — | 9 | 1 | no | — |
| 138 | #235 | — | 18 | 1 | no | — |
| 139 | C2×D12 | — | 9 | 1 | no | — |
| 140 | C2×Dic6 | — | 9 | 1 | no | — |
| 141 | C2×Dic6 | — | 18 | 1 | no | — |
| 142 | #41⋊C3 | — | 2 | 1 | no | — |
| 143 | #41⋊C3 | — | 9 | 2 | no | — |
| 144 | #41⋊C3 | C2×D9 | 1 | 1 | yes | characteristic |
| 145 | GL(2,𝔽3) | C2×D9 | 1 | 1 | yes | characteristic, complemented |
| 146 | #226 | — | 9 | 2 | no | — |
| 147 | #226 | — | 18 | 2 | no | — |
| 148 | #226 | C2×D9 | 1 | 1 | yes | characteristic, complemented |
| 149 | C3×Q16 | — | 3 | 1 | no | — |
| 150 | C3×SD16 | — | 3 | 1 | no | — |
| 151 | C3×SD16 | — | 3 | 1 | no | — |
| 152 | C3×D8 | — | 3 | 1 | no | — |
| 153 | C2×C3×C8 | — | 3 | 1 | no | — |
| 154 | #216 | — | 9 | 2 | no | — |
| 155 | #215 | — | 9 | 2 | no | — |
| 156 | #214 | — | 9 | 2 | no | — |
| 157 | #213 | — | 9 | 2 | no | — |
| 158 | #208 | — | 9 | 1 | no | — |
| 159 | #204 | — | 9 | 2 | no | — |
| 160 | #204 | — | 9 | 2 | no | — |
| 161 | ΓL(1,𝔽25) | — | 9 | 2 | no | — |
| 162 | D3×C8 | — | 9 | 2 | no | — |
| 163 | C2×C3×D3 | — | 4 | 1 | no | — |
| 164 | C32×C4 | — | 4 | 1 | no | — |
| 165 | C3⋊Dic3 | — | 12 | 2 | no | — |
| 166 | C3×Dic3 | — | 4 | 1 | no | — |
| 167 | C3×Dic3 | — | 12 | 2 | no | — |
| 168 | C22×C9 | — | 3 | 1 | no | — |
| 169 | C22×C9 | — | 6 | 1 | no | — |
| 170 | C2×D9 | — | 3 | 2 | no | — |
| 171 | C2×D9 | — | 24 | 1 | no | — |
| 172 | C4×C9 | — | 3 | 1 | no | — |
| 173 | C4×C9 | — | 6 | 1 | no | — |
| 174 | C4×C9 | — | 8 | 1 | no | — |
| 175 | C4×C9 | C2×S4 | 1 | 1 | yes | characteristic |
| 176 | Dic9 | — | 6 | 2 | no | — |
| 177 | Dic9 | — | 24 | 2 | no | — |
| 178 | Dic9 | C2×S4 | 1 | 2 | yes | — |
| 179 | 2−1+4 | — | 27 | 1 | no | — |
| 180 | H(5,𝔽2) | — | 9 | 1 | no | — |
| 181 | #137 | — | 27 | 2 | no | — |
| 182 | #137 | — | 27 | 1 | no | — |
| 183 | AGL(1,ℤ/8) | — | 27 | 2 | no | — |
| 184 | AGL(1,ℤ/8) | — | 27 | 1 | no | — |
| 185 | #135 | — | 3 | 1 | no | — |
| 186 | #135 | — | 27 | 2 | no | — |
| 187 | C2×SD16 | — | 27 | 1 | no | — |
| 188 | C2×SD16 | — | 27 | 2 | no | — |
| 189 | #131 | — | 27 | 1 | no | — |
| 190 | C3×C9 | — | 4 | 1 | no | Sylow |
| 191 | C22×D3 | — | 18 | 1 | no | — |
| 192 | C3×Q8 | — | 3 | 1 | no | — |
| 193 | C3×Q8 | — | 36 | 1 | no | — |
| 194 | C3×Q8 | C2×D32 | 1 | 1 | yes | characteristic |
| 195 | C3×D4 | — | 3 | 1 | no | — |
| 196 | C3×D4 | — | 3 | 1 | no | — |
| 197 | C3×D4 | — | 6 | 1 | no | — |
| 198 | C2×C3×C4 | — | 3 | 1 | no | — |
| 199 | C2×C3×C4 | — | 6 | 1 | no | — |
| 200 | #67 | — | 9 | 2 | no | — |
| 201 | #67 | — | 18 | 2 | no | — |
| 202 | C2×Dic3 | — | 18 | 2 | no | — |
| 203 | C2×Dic3 | — | 18 | 1 | no | — |
| 204 | D12 | — | 9 | 2 | no | — |
| 205 | D12 | — | 9 | 1 | no | — |
| 206 | C4×D3 | — | 4 | 1 | no | — |
| 207 | C4×D3 | — | 9 | 2 | no | — |
| 208 | C4×D3 | — | 18 | 2 | no | — |
| 209 | C4×D3 | — | 36 | 2 | no | — |
| 210 | Dic6 | — | 3 | 1 | no | — |
| 211 | Dic6 | — | 9 | 2 | no | — |
| 212 | Dic6 | — | 18 | 2 | no | — |
| 213 | Dic6 | — | 18 | 1 | no | — |
| 214 | Dic6 | — | 36 | 2 | no | — |
| 215 | Dic6 | — | 36 | 1 | no | — |
| 216 | Dic6 | — | 72 | 1 | no | — |
| 217 | SL(2,𝔽3) | — | 2 | 1 | no | — |
| 218 | SL(2,𝔽3) | C22×D9 | 1 | 1 | yes | characteristic |
| 219 | C3×C8 | — | 3 | 1 | no | — |
| 220 | C3×C8 | — | 3 | 1 | no | — |
| 221 | C3⋊C8 | — | 9 | 2 | no | — |
| 222 | C2×C32 | — | 4 | 1 | no | — |
| 223 | C3×D3 | — | 4 | 2 | no | — |
| 224 | C2×C9 | — | 6 | 1 | no | — |
| 225 | C2×C9 | — | 8 | 1 | no | — |
| 226 | C2×C9 | — | 12 | 1 | no | — |
| 227 | C2×C9 | C22×S4 | 1 | 1 | yes | characteristic |
| 228 | D9 | — | 6 | 2 | no | — |
| 229 | #41 | — | 3 | 1 | no | — |
| 230 | #41 | — | 9 | 2 | no | — |
| 231 | #41 | — | 27 | 1 | no | — |
| 232 | #41 | — | 27 | 2 | no | — |
| 233 | #41 | — | 54 | 1 | no | — |
| 234 | #41 | — | 54 | 2 | no | — |
| 235 | #41 | D3×D9 | 1 | 1 | yes | characteristic |
| 236 | C2×Q8 | — | 27 | 1 | no | — |
| 237 | C2×Q8 | — | 27 | 2 | no | — |
| 238 | C2×Q8 | — | 54 | 1 | no | — |
| 239 | C2×D4 | — | 27 | 1 | no | — |
| 240 | C2×D4 | — | 27 | 2 | no | — |
| 241 | Q16 | — | 3 | 1 | no | — |
| 242 | Q16 | — | 27 | 2 | no | — |
| 243 | SD16 | — | 3 | 1 | no | — |
| 244 | SD16 | — | 3 | 1 | no | — |
| 245 | SD16 | — | 27 | 2 | no | — |
| 246 | SD16 | — | 27 | 2 | no | — |
| 247 | SD16 | — | 27 | 2 | no | — |
| 248 | SD16 | — | 27 | 2 | no | — |
| 249 | D8 | — | 3 | 1 | no | — |
| 250 | D8 | — | 27 | 2 | no | — |
| 251 | M16 | — | 27 | 2 | no | — |
| 252 | M16 | — | 27 | 1 | no | — |
| 253 | C2×C8 | — | 3 | 1 | no | — |
| 254 | C2×C8 | — | 27 | 2 | no | — |
| 255 | C22×C3 | — | 3 | 1 | no | — |
| 256 | C22×C3 | — | 6 | 1 | no | — |
| 257 | C2×D3 | — | 4 | 1 | no | — |
| 258 | C2×D3 | — | 9 | 2 | no | — |
| 259 | C2×D3 | — | 72 | 1 | no | — |
| 260 | C3×C4 | — | 3 | 1 | no | — |
| 261 | C3×C4 | — | 4 | 1 | no | — |
| 262 | C3×C4 | — | 6 | 1 | no | — |
| 263 | C3×C4 | — | 8 | 1 | no | — |
| 264 | C3×C4 | — | 36 | 2 | no | — |
| 265 | C3×C4 | D3×S4 | 1 | 1 | yes | characteristic |
| 266 | Dic3 | — | 3 | 2 | no | — |
| 267 | Dic3 | — | 4 | 1 | no | — |
| 268 | Dic3 | — | 18 | 2 | no | — |
| 269 | Dic3 | — | 36 | 2 | no | — |
| 270 | Dic3 | — | 72 | 2 | no | — |
| 271 | C32 | — | 4 | 1 | no | — |
| 272 | C9 | — | 8 | 1 | no | — |
| 273 | C9 | #5924 | 1 | 1 | yes | characteristic, complemented |
| 274 | C23 | — | 54 | 1 | no | — |
| 275 | Q8 | — | 3 | 1 | no | — |
| 276 | Q8 | — | 9 | 1 | no | — |
| 277 | Q8 | — | 27 | 2 | no | — |
| 278 | Q8 | — | 54 | 2 | no | — |
| 279 | Q8 | — | 54 | 1 | no | — |
| 280 | Q8 | C2×D3×D9 | 1 | 1 | yes | characteristic |
| 281 | D4 | — | 3 | 1 | no | — |
| 282 | D4 | — | 3 | 1 | no | — |
| 283 | D4 | — | 6 | 1 | no | — |
| 284 | D4 | — | 27 | 2 | no | — |
| 285 | D4 | — | 27 | 2 | no | — |
| 286 | D4 | — | 27 | 1 | no | — |
| 287 | D4 | — | 54 | 2 | no | — |
| 288 | C2×C4 | — | 3 | 1 | no | — |
| 289 | C2×C4 | — | 6 | 1 | no | — |
| 290 | C2×C4 | — | 27 | 2 | no | — |
| 291 | C2×C4 | — | 54 | 2 | no | — |
| 292 | C2×C4 | — | 54 | 2 | no | — |
| 293 | C2×C4 | — | 54 | 1 | no | — |
| 294 | C8 | — | 3 | 1 | no | — |
| 295 | C8 | — | 3 | 1 | no | — |
| 296 | C8 | — | 27 | 2 | no | — |
| 297 | C2×C3 | — | 4 | 1 | no | — |
| 298 | C2×C3 | — | 6 | 1 | no | — |
| 299 | C2×C3 | — | 8 | 1 | no | — |
| 300 | C2×C3 | — | 12 | 1 | no | — |
| 301 | C2×C3 | C2×D3×S4 | 1 | 1 | yes | characteristic |
| 302 | D3 | — | 4 | 2 | no | — |
| 303 | D3 | — | 18 | 2 | no | — |
| 304 | C22 | — | 3 | 1 | no | — |
| 305 | C22 | — | 6 | 1 | no | — |
| 306 | C22 | — | 27 | 2 | no | — |
| 307 | C22 | — | 216 | 1 | no | — |
| 308 | C4 | — | 3 | 1 | no | — |
| 309 | C4 | — | 6 | 1 | no | — |
| 310 | C4 | — | 9 | 2 | no | — |
| 311 | C4 | — | 54 | 2 | no | — |
| 312 | C4 | D9×S4 | 1 | 1 | yes | characteristic |
| 313 | C3 | — | 4 | 1 | no | — |
| 314 | C3 | — | 8 | 1 | no | — |
| 315 | C3 | #10596447 | 1 | 1 | yes | characteristic |
| 316 | C2 | — | 6 | 1 | no | — |
| 317 | C2 | — | 12 | 1 | no | — |
| 318 | C2 | — | 54 | 2 | no | — |
| 319 | C2 | C2×D9×S4 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×D9×S4 |
| commutator, cosocle, hypocenter | C9×SL(2,𝔽3) | C23 |
| Frattini, socle | C2×C3 | C2×D3×S4 |
| Fitting | C9×#41 | C2×D3 |
| hypercenter | C4 | D9×S4 |
| LCS | G ▷ C9×SL(2,𝔽3) | |
| UCS | C1 ◁ C2 ◁ C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #11722412 | C2 | semidirect |
| 2. #11722372 | C2 | semidirect |
| 3. C9×#985 | C2 | semidirect |
| 4. #11722236 | C2 | semidirect |
| 5. #11722231 | C2 | semidirect |
| 11. C9⋊#226 | C22 | semidirect |
| 12. C9×GL(2,𝔽3) | C22 | semidirect |
| 13. C9×#226 | C22 | semidirect |
| 20. #63715 | D3 | semidirect |
| 82. #985 | D9 | semidirect |
| 145. GL(2,𝔽3) | C2×D9 | semidirect |
| 148. #226 | C2×D9 | semidirect |
| 273. C9 | #5924 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11722412 | 2 |
| #11722372 | 2 |
| C9×#985 | 2 |
| #11722236 | 2 |
| #11722231 | 2 |
| #10596447 | 3 |
| #10589872 | 3 |
| D3×Dic18 | 4 |