Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
1095 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 1095 conjugacy classes of subgroups — too many to draw. Only its 78 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 | #11722577 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #11722437 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | #11722436 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | C3⋊D3×H(5,𝔽2) | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | D3×#994 | — | 3 | 1 | no | maximal |
| 6 | Q8×(C3⋊D3)⋊C3 | — | 4 | 1 | no | maximal |
| 7 | #41⋊H(3,𝔽3) | C22 | 1 | 3 | yes | complemented |
| 8 | #88623 | C22 | 1 | 1 | yes | characteristic, complemented |
| 9 | #88622 | C22 | 1 | 3 | yes | complemented |
| 10 | C32×H(5,𝔽2) | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 11 | #64366 | — | 3 | 1 | no | — |
| 12 | #64366 | C2×C3 | 1 | 3 | yes | complemented |
| 13 | #41×C3⋊D3 | — | 3 | 1 | no | — |
| 14 | #41×C3⋊D3 | C2×C3 | 1 | 3 | yes | — |
| 15 | #64361 | — | 3 | 3 | no | — |
| 16 | C2×D4×C3⋊D3 | — | 3 | 3 | no | — |
| 17 | C3×#994 | — | 2 | 1 | no | — |
| 18 | C3×#994 | D3 | 1 | 1 | yes | characteristic, complemented |
| 19 | D3×#41⋊C3 | — | 3 | 3 | no | — |
| 20 | C3⋊#994 | — | 3 | 3 | no | — |
| 21 | Q8×H(3,𝔽3) | — | 4 | 1 | no | — |
| 22 | C4×(C3⋊D3)⋊C3 | — | 4 | 3 | no | — |
| 23 | #634⋊C3 | — | 4 | 3 | no | — |
| 24 | Q8⋊H(3,𝔽3) | C23 | 1 | 1 | yes | characteristic, complemented |
| 25 | D3×H(5,𝔽2) | — | 3 | 1 | no | — |
| 26 | D3×H(5,𝔽2) | — | 9 | 1 | no | — |
| 27 | C2×#994 | — | 9 | 1 | no | — |
| 28 | C23×C3⋊D3 | — | 3 | 2 | no | — |
| 29 | C32×#41 | — | 3 | 1 | no | — |
| 30 | C32×#41 | C22×C3 | 1 | 3 | yes | — |
| 31 | C2×C32×D4 | — | 3 | 3 | no | — |
| 32 | C2×#638 | — | 3 | 6 | no | — |
| 33 | #3832 | — | 3 | 1 | no | — |
| 34 | #3832 | C22×C3 | 1 | 3 | yes | — |
| 35 | Q8×C3⋊D3 | A4 | 1 | 1 | yes | characteristic |
| 36 | Q8×C3⋊D3 | C22×C3 | 1 | 1 | yes | characteristic |
| 37 | #3830 | — | 3 | 3 | no | — |
| 38 | #3830 | — | 3 | 3 | no | — |
| 39 | D4×C3⋊D3 | — | 3 | 6 | no | — |
| 40 | D4×C3⋊D3 | — | 3 | 3 | no | — |
| 41 | D4×C3⋊D3 | — | 3 | 6 | no | — |
| 42 | D4×C3⋊D3 | — | 3 | 3 | no | — |
| 43 | #3828 | — | 3 | 3 | no | — |
| 44 | #3828 | — | 3 | 3 | no | — |
| 45 | C2×C4⋊(C3⋊D3) | — | 3 | 3 | no | — |
| 46 | C2×C4×C3⋊D3 | — | 3 | 3 | no | — |
| 47 | C3×D3×Q8 | — | 12 | 1 | no | — |
| 48 | C3×#41⋊C3 | — | 2 | 3 | no | — |
| 49 | C3×#41⋊C3 | C2×D3 | 1 | 3 | yes | complemented |
| 50 | D3×SL(2,𝔽3) | — | 3 | 1 | no | — |
| 51 | C3⋊(#41⋊C3) | — | 3 | 3 | no | — |
| 52 | C2×(C3⋊D3)⋊C3 | — | 4 | 1 | no | — |
| 53 | C4×H(3,𝔽3) | — | 4 | 3 | no | — |
| 54 | (C3⋊Dic3)⋊C3 | — | 4 | 3 | no | — |
| 55 | C3×H(5,𝔽2) | — | 3 | 1 | no | — |
| 56 | C3×H(5,𝔽2) | C3×D3 | 1 | 1 | yes | characteristic |
| 57 | #1009 | — | 3 | 3 | no | — |
| 58 | #1009 | — | 9 | 3 | no | — |
| 59 | #1009 | — | 9 | 3 | no | — |
| 60 | #1009 | — | 9 | 1 | no | — |
| 61 | D3×#41 | — | 3 | 3 | no | — |
| 62 | D3×#41 | — | 9 | 3 | no | — |
| 63 | D3×#41 | — | 9 | 1 | no | — |
| 64 | D3×#41 | — | 9 | 3 | no | — |
| 65 | #1004 | — | 9 | 9 | no | — |
| 66 | #1004 | — | 9 | 3 | no | — |
| 67 | C2×D3×D4 | — | 9 | 9 | no | — |
| 68 | C2×D3×D4 | — | 9 | 3 | no | — |
| 69 | #994 | — | 3 | 1 | no | — |
| 70 | #994 | — | 6 | 1 | no | — |
| 71 | #994 | — | 9 | 3 | no | — |
| 72 | C2×#41⋊C3 | — | 9 | 3 | no | — |
| 73 | C23×C32 | — | 3 | 2 | no | — |
| 74 | C22×C3⋊D3 | — | 3 | 6 | no | — |
| 75 | C22×C3⋊D3 | — | 3 | 3 | no | — |
| 76 | C22×C3⋊D3 | — | 12 | 4 | no | — |
| 77 | C32×Q8 | — | 4 | 1 | no | — |
| 78 | C32×Q8 | — | 8 | 1 | no | — |
| 79 | C32×Q8 | C2×A4 | 1 | 1 | yes | characteristic |
| 80 | C32×Q8 | C23×C3 | 1 | 1 | yes | characteristic |
| 81 | C32×D4 | — | 3 | 3 | no | — |
| 82 | C32×D4 | — | 3 | 3 | no | — |
| 83 | C2×C32×C4 | — | 3 | 3 | no | — |
| 84 | #638 | — | 3 | 6 | no | — |
| 85 | #638 | — | 3 | 6 | no | — |
| 86 | C2×C3⋊Dic3 | — | 3 | 3 | no | — |
| 87 | C4⋊(C3⋊D3) | — | 3 | 3 | no | — |
| 88 | C4⋊(C3⋊D3) | — | 3 | 3 | no | — |
| 89 | C4×C3⋊D3 | — | 3 | 1 | no | — |
| 90 | C4×C3⋊D3 | — | 3 | 3 | no | — |
| 91 | C4×C3⋊D3 | — | 3 | 3 | no | — |
| 92 | C4×C3⋊D3 | C2×A4 | 1 | 3 | yes | — |
| 93 | #634 | — | 3 | 1 | no | — |
| 94 | #634 | C2×A4 | 1 | 3 | yes | — |
| 95 | C3×C4×D3 | — | 12 | 3 | no | — |
| 96 | C3×Dic6 | — | 12 | 3 | no | — |
| 97 | C3×SL(2,𝔽3) | — | 2 | 1 | no | — |
| 98 | C3×SL(2,𝔽3) | C22×D3 | 1 | 1 | yes | characteristic, complemented |
| 99 | C2×H(5,𝔽2) | — | 9 | 1 | no | Sylow |
| 100 | C2×H(3,𝔽3) | — | 4 | 1 | no | — |
| 101 | (C3⋊D3)⋊C3 | — | 4 | 2 | no | — |
| 102 | C23×D3 | — | 9 | 2 | no | — |
| 103 | C23×D3 | — | 9 | 6 | no | — |
| 104 | C3×#41 | — | 3 | 3 | no | — |
| 105 | C3×#41 | — | 3 | 1 | no | — |
| 106 | C3×#41 | — | 3 | 3 | no | — |
| 107 | C3×#41 | C2×C3×D3 | 1 | 3 | yes | — |
| 108 | C2×C3×Q8 | — | 36 | 1 | no | — |
| 109 | C2×C3×D4 | — | 3 | 9 | no | — |
| 110 | C2×C3×D4 | — | 3 | 3 | no | — |
| 111 | C2×#67 | — | 9 | 6 | no | — |
| 112 | C2×#67 | — | 9 | 18 | no | — |
| 113 | #239 | — | 3 | 3 | no | — |
| 114 | #239 | — | 9 | 3 | no | — |
| 115 | #239 | — | 9 | 3 | no | — |
| 116 | #239 | — | 9 | 1 | no | — |
| 117 | D3×Q8 | — | 3 | 1 | no | — |
| 118 | D3×Q8 | — | 3 | 1 | no | — |
| 119 | D3×Q8 | — | 9 | 1 | no | — |
| 120 | D3×Q8 | — | 9 | 1 | no | — |
| 121 | #237 | — | 9 | 3 | no | — |
| 122 | #237 | — | 9 | 9 | no | — |
| 123 | #237 | — | 9 | 9 | no | — |
| 124 | #237 | — | 9 | 3 | no | — |
| 125 | D3×D4 | — | 9 | 3 | no | — |
| 126 | D3×D4 | — | 9 | 3 | no | — |
| 127 | D3×D4 | — | 9 | 9 | no | — |
| 128 | D3×D4 | — | 9 | 9 | no | — |
| 129 | D3×D4 | — | 9 | 18 | no | — |
| 130 | D3×D4 | — | 9 | 6 | no | — |
| 131 | D3×D4 | — | 9 | 18 | no | — |
| 132 | D3×D4 | — | 9 | 6 | no | — |
| 133 | #235 | — | 9 | 3 | no | — |
| 134 | #235 | — | 9 | 9 | no | — |
| 135 | #235 | — | 9 | 9 | no | — |
| 136 | #235 | — | 9 | 3 | no | — |
| 137 | C2×D12 | — | 9 | 9 | no | — |
| 138 | C2×D12 | — | 9 | 3 | no | — |
| 139 | C2×C4×D3 | — | 9 | 3 | no | — |
| 140 | C2×C4×D3 | — | 9 | 9 | no | — |
| 141 | #41⋊C3 | — | 3 | 3 | no | — |
| 142 | #41⋊C3 | — | 6 | 3 | no | — |
| 143 | #41⋊C3 | — | 9 | 3 | no | — |
| 144 | C2×SL(2,𝔽3) | — | 9 | 1 | no | — |
| 145 | C22×C32 | — | 3 | 3 | no | — |
| 146 | C22×C32 | — | 12 | 2 | no | — |
| 147 | C2×C3⋊D3 | — | 3 | 3 | no | — |
| 148 | C2×C3⋊D3 | — | 6 | 6 | no | — |
| 149 | C2×C3⋊D3 | — | 12 | 6 | no | — |
| 150 | C2×C3⋊D3 | C22×A4 | 1 | 1 | yes | characteristic |
| 151 | C2×C3×D3 | — | 12 | 1 | no | — |
| 152 | C32×C4 | — | 3 | 1 | no | — |
| 153 | C32×C4 | — | 4 | 3 | no | — |
| 154 | C32×C4 | — | 8 | 3 | no | — |
| 155 | C32×C4 | C22×A4 | 1 | 3 | yes | — |
| 156 | C3⋊Dic3 | — | 3 | 1 | no | — |
| 157 | C3⋊Dic3 | C22×A4 | 1 | 3 | yes | — |
| 158 | C3×Dic3 | — | 12 | 3 | no | — |
| 159 | H(5,𝔽2) | — | 9 | 3 | no | — |
| 160 | H(5,𝔽2) | — | 27 | 3 | no | — |
| 161 | H(5,𝔽2) | — | 27 | 1 | no | — |
| 162 | H(5,𝔽2) | (C3⋊D3)⋊C3 | 1 | 1 | yes | characteristic, complemented |
| 163 | C2×#41 | — | 9 | 3 | no | — |
| 164 | C2×#41 | — | 27 | 1 | no | — |
| 165 | C22×D4 | — | 27 | 3 | no | — |
| 166 | H(3,𝔽3) | — | 4 | 1 | no | Sylow |
| 167 | C23×C3 | — | 3 | 6 | no | — |
| 168 | C23×C3 | — | 3 | 2 | no | — |
| 169 | C22×D3 | — | 9 | 3 | no | — |
| 170 | C22×D3 | — | 9 | 9 | no | — |
| 171 | C22×D3 | — | 9 | 18 | no | — |
| 172 | C22×D3 | — | 9 | 6 | no | — |
| 173 | C22×D3 | — | 36 | 4 | no | — |
| 174 | C22×D3 | — | 36 | 12 | no | — |
| 175 | C3×Q8 | — | 3 | 1 | no | — |
| 176 | C3×Q8 | — | 3 | 1 | no | — |
| 177 | C3×Q8 | — | 12 | 1 | no | — |
| 178 | C3×Q8 | — | 24 | 1 | no | — |
| 179 | C3×Q8 | — | 36 | 3 | no | — |
| 180 | C3×Q8 | D3×A4 | 1 | 1 | yes | characteristic |
| 181 | C3×Q8 | C22×C3×D3 | 1 | 1 | yes | characteristic |
| 182 | C3×D4 | — | 3 | 3 | no | — |
| 183 | C3×D4 | — | 3 | 9 | no | — |
| 184 | C3×D4 | — | 3 | 9 | no | — |
| 185 | C3×D4 | — | 3 | 3 | no | — |
| 186 | C2×C3×C4 | — | 3 | 9 | no | — |
| 187 | C2×C3×C4 | — | 3 | 3 | no | — |
| 188 | C2×C3×C4 | — | 36 | 3 | no | — |
| 189 | #67 | — | 9 | 6 | no | — |
| 190 | #67 | — | 9 | 18 | no | — |
| 191 | #67 | — | 9 | 18 | no | — |
| 192 | #67 | — | 9 | 6 | no | — |
| 193 | C2×Dic3 | — | 9 | 3 | no | — |
| 194 | C2×Dic3 | — | 9 | 9 | no | — |
| 195 | D12 | — | 9 | 3 | no | — |
| 196 | D12 | — | 9 | 9 | no | — |
| 197 | D12 | — | 9 | 9 | no | — |
| 198 | D12 | — | 9 | 3 | no | — |
| 199 | C4×D3 | — | 3 | 3 | no | — |
| 200 | C4×D3 | — | 9 | 3 | no | — |
| 201 | C4×D3 | — | 9 | 3 | no | — |
| 202 | C4×D3 | — | 9 | 1 | no | — |
| 203 | C4×D3 | — | 9 | 3 | no | — |
| 204 | C4×D3 | — | 9 | 9 | no | — |
| 205 | C4×D3 | — | 9 | 9 | no | — |
| 206 | C4×D3 | — | 9 | 3 | no | — |
| 207 | Dic6 | — | 3 | 3 | no | — |
| 208 | Dic6 | — | 9 | 3 | no | — |
| 209 | Dic6 | — | 9 | 3 | no | — |
| 210 | Dic6 | — | 9 | 1 | no | — |
| 211 | SL(2,𝔽3) | — | 3 | 1 | no | — |
| 212 | SL(2,𝔽3) | — | 6 | 1 | no | — |
| 213 | C2×C32 | — | 4 | 1 | no | — |
| 214 | C2×C32 | — | 6 | 3 | no | — |
| 215 | C2×C32 | — | 8 | 1 | no | — |
| 216 | C2×C32 | C23×A4 | 1 | 1 | yes | characteristic |
| 217 | C3⋊D3 | — | 6 | 3 | no | — |
| 218 | C3⋊D3 | #994 | 1 | 2 | yes | complemented |
| 219 | C3×D3 | — | 12 | 2 | no | — |
| 220 | C24 | — | 27 | 2 | no | — |
| 221 | #41 | — | 3 | 1 | no | — |
| 222 | #41 | — | 9 | 3 | no | — |
| 223 | #41 | — | 27 | 3 | no | — |
| 224 | #41 | — | 27 | 3 | no | — |
| 225 | #41 | — | 27 | 3 | no | — |
| 226 | #41 | — | 27 | 1 | no | — |
| 227 | #41 | — | 27 | 3 | no | — |
| 228 | #41 | C2×(C3⋊D3)⋊C3 | 1 | 3 | yes | — |
| 229 | C2×Q8 | — | 9 | 1 | no | — |
| 230 | C2×Q8 | — | 9 | 1 | no | — |
| 231 | C2×D4 | — | 3 | 3 | no | — |
| 232 | C2×D4 | — | 27 | 3 | no | — |
| 233 | C2×D4 | — | 27 | 3 | no | — |
| 234 | C2×D4 | — | 27 | 3 | no | — |
| 235 | C2×D4 | — | 27 | 6 | no | — |
| 236 | C2×D4 | — | 27 | 6 | no | — |
| 237 | C2×D4 | — | 27 | 6 | no | — |
| 238 | C22×C4 | — | 27 | 3 | no | — |
| 239 | C22×C3 | — | 3 | 9 | no | — |
| 240 | C22×C3 | — | 3 | 3 | no | — |
| 241 | C22×C3 | — | 12 | 6 | no | — |
| 242 | C22×C3 | — | 12 | 2 | no | — |
| 243 | C22×C3 | — | 36 | 1 | no | — |
| 244 | C2×D3 | — | 3 | 1 | no | — |
| 245 | C2×D3 | — | 9 | 1 | no | — |
| 246 | C2×D3 | — | 9 | 9 | no | — |
| 247 | C2×D3 | — | 9 | 3 | no | — |
| 248 | C2×D3 | — | 18 | 6 | no | — |
| 249 | C2×D3 | — | 18 | 18 | no | — |
| 250 | C2×D3 | — | 36 | 18 | no | — |
| 251 | C2×D3 | — | 36 | 6 | no | — |
| 252 | C3×C4 | — | 3 | 3 | no | — |
| 253 | C3×C4 | — | 3 | 1 | no | — |
| 254 | C3×C4 | — | 3 | 3 | no | — |
| 255 | C3×C4 | — | 12 | 3 | no | — |
| 256 | C3×C4 | — | 24 | 3 | no | — |
| 257 | C3×C4 | — | 36 | 3 | no | — |
| 258 | C3×C4 | C2×D3×A4 | 1 | 3 | yes | — |
| 259 | Dic3 | — | 3 | 3 | no | — |
| 260 | Dic3 | — | 9 | 3 | no | — |
| 261 | Dic3 | — | 9 | 3 | no | — |
| 262 | Dic3 | — | 9 | 1 | no | — |
| 263 | C32 | — | 4 | 1 | no | — |
| 264 | C32 | — | 8 | 1 | no | — |
| 265 | C32 | C2×#994 | 1 | 1 | yes | characteristic, complemented |
| 266 | C23 | — | 3 | 2 | no | — |
| 267 | C23 | — | 27 | 3 | no | — |
| 268 | C23 | — | 27 | 6 | no | — |
| 269 | C23 | — | 108 | 4 | no | — |
| 270 | Q8 | — | 9 | 3 | no | — |
| 271 | Q8 | — | 27 | 1 | no | — |
| 272 | Q8 | (C3⋊D3)⋊A4 | 1 | 1 | yes | characteristic |
| 273 | Q8 | C22×(C3⋊D3)⋊C3 | 1 | 1 | yes | characteristic |
| 274 | D4 | — | 3 | 3 | no | — |
| 275 | D4 | — | 3 | 3 | no | — |
| 276 | D4 | — | 27 | 3 | no | — |
| 277 | D4 | — | 27 | 6 | no | — |
| 278 | D4 | — | 27 | 6 | no | — |
| 279 | D4 | — | 27 | 3 | no | — |
| 280 | C2×C4 | — | 3 | 3 | no | — |
| 281 | C2×C4 | — | 9 | 3 | no | — |
| 282 | C2×C4 | — | 27 | 3 | no | — |
| 283 | C2×C4 | — | 27 | 1 | no | — |
| 284 | C2×C4 | — | 27 | 3 | no | — |
| 285 | C2×C4 | — | 27 | 3 | no | — |
| 286 | C2×C3 | — | 3 | 1 | no | — |
| 287 | C2×C3 | — | 6 | 9 | no | — |
| 288 | C2×C3 | — | 6 | 3 | no | — |
| 289 | C2×C3 | — | 12 | 1 | no | — |
| 290 | C2×C3 | — | 24 | 1 | no | — |
| 291 | C2×C3 | — | 36 | 2 | no | — |
| 292 | C2×C3 | C22×D3×A4 | 1 | 1 | yes | characteristic |
| 293 | D3 | — | 3 | 2 | no | — |
| 294 | D3 | — | 9 | 2 | no | — |
| 295 | D3 | — | 18 | 9 | no | — |
| 296 | D3 | — | 18 | 3 | no | — |
| 297 | C22 | — | 3 | 3 | no | — |
| 298 | C22 | — | 9 | 1 | no | — |
| 299 | C22 | — | 12 | 2 | no | — |
| 300 | C22 | — | 27 | 3 | no | — |
| 301 | C22 | — | 54 | 6 | no | — |
| 302 | C22 | — | 108 | 6 | no | — |
| 303 | C4 | — | 3 | 1 | no | — |
| 304 | C4 | — | 9 | 3 | no | — |
| 305 | C4 | — | 27 | 1 | no | — |
| 306 | C4 | C2×(C3⋊D3)⋊A4 | 1 | 3 | yes | — |
| 307 | C3 | — | 3 | 1 | no | — |
| 308 | C3 | — | 12 | 1 | no | — |
| 309 | C3 | — | 24 | 1 | no | — |
| 310 | C3 | D3×#994 | 1 | 1 | yes | characteristic |
| 311 | C2 | — | 6 | 3 | no | — |
| 312 | C2 | — | 9 | 2 | no | — |
| 313 | C2 | — | 54 | 3 | no | — |
| 314 | C2 | C22×(C3⋊D3)⋊A4 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×(C3⋊D3)⋊A4 |
| commutator, cosocle, hypocenter | C32×Q8 | C23×C3 |
| Frattini, socle | C2×C3 | C22×D3×A4 |
| Fitting | C32×H(5,𝔽2) | C2×C3 |
| hypercenter | Q8 | (C3⋊D3)⋊A4 |
| LCS | G ▷ C32×Q8 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #11722577 | C2 | semidirect |
| 2. #11722437 | C2 | semidirect |
| 3. #11722436 | C2 | semidirect |
| 4. C3⋊D3×H(5,𝔽2) | C3 | semidirect |
| 7. #41⋊H(3,𝔽3) | C22 | semidirect |
| 8. #88623 | C22 | semidirect |
| 9. #88622 | C22 | semidirect |
| 10. C32×H(5,𝔽2) | C2×C3 | semidirect |
| 12. #64366 | C2×C3 | semidirect |
| 18. C3×#994 | D3 | semidirect |
| 24. Q8⋊H(3,𝔽3) | C23 | semidirect |
| 49. C3×#41⋊C3 | C2×D3 | semidirect |
| 98. C3×SL(2,𝔽3) | C22×D3 | semidirect |
| 162. H(5,𝔽2) | (C3⋊D3)⋊C3 | semidirect |
| 218. C3⋊D3 | #994 | semidirect |
| 265. C32 | C2×#994 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11722577 | 2 |
| #11722437 | 2 |
| #11722436 | 2 |
| C3⋊D3×H(5,𝔽2) | 3 |
| D3×#994 | 3 |
| Q8×(C3⋊D3)⋊C3 | 4 |