Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
1256 conjugacy classes of subgroups; only the 239 normal subgroups are shown.
This group has 1256 conjugacy classes of subgroups — too many to draw. Only its 239 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×#89042 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C3×M16×C3⋊D3 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | #11723656 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | #11723652 | C2 | 1 | 6 | yes | maximal, complemented |
| 5 | #11723651 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #10594722 | — | 3 | 3 | no | maximal |
| 7 | C2×C3×C4×C3⋊D3 | C22 | 1 | 3 | yes | — |
| 8 | C2×#6296 | C4 | 1 | 1 | yes | characteristic |
| 9 | C2×#6295 | C4 | 1 | 3 | yes | — |
| 10 | #89042 | C4 | 1 | 3 | yes | — |
| 11 | #89042 | C4 | 1 | 1 | yes | characteristic |
| 12 | #89042 | C22 | 1 | 1 | yes | characteristic |
| 13 | #89042 | C22 | 1 | 3 | yes | — |
| 14 | C33×M16 | C22 | 1 | 1 | yes | characteristic |
| 15 | C3×#3748 | C22 | 1 | 3 | yes | — |
| 16 | C3×#3743 | C22 | 1 | 6 | yes | complemented |
| 17 | C3×C8×C3⋊D3 | C22 | 1 | 6 | yes | complemented |
| 18 | #88809 | C22 | 1 | 6 | yes | complemented |
| 19 | #88808 | C22 | 1 | 6 | yes | complemented |
| 20 | C2×#3798 | — | 3 | 3 | no | — |
| 21 | M16×C3⋊D3 | D3 | 1 | 3 | yes | complemented |
| 22 | C3×D3×M16 | — | 3 | 6 | no | — |
| 23 | C3×D3×M16 | — | 6 | 3 | no | — |
| 24 | #63818 | — | 3 | 3 | no | — |
| 25 | D3×#208 | — | 3 | 6 | no | — |
| 26 | #63805 | — | 3 | 6 | no | — |
| 27 | #63804 | — | 3 | 12 | no | — |
| 28 | #63803 | — | 3 | 6 | no | — |
| 29 | C22×C3×C3⋊D3 | C2×C4 | 1 | 3 | yes | — |
| 30 | C2×C33×C4 | C23 | 1 | 1 | yes | characteristic |
| 31 | C2×C3×C3⋊Dic3 | C2×C4 | 1 | 3 | yes | — |
| 32 | C3×C4×C3⋊D3 | C2×C4 | 1 | 3 | yes | — |
| 33 | C3×C4×C3⋊D3 | C2×C4 | 1 | 3 | yes | — |
| 34 | C3×C4×C3⋊D3 | C23 | 1 | 3 | yes | — |
| 35 | C3×C4×C3⋊D3 | C23 | 1 | 3 | yes | — |
| 36 | #6296 | C2×C4 | 1 | 1 | yes | characteristic |
| 37 | #6296 | C2×C4 | 1 | 3 | yes | — |
| 38 | #6295 | C2×C4 | 1 | 3 | yes | — |
| 39 | #6295 | C2×C4 | 1 | 3 | yes | — |
| 40 | #6295 | C2×C4 | 1 | 6 | yes | — |
| 41 | C33×C8 | C23 | 1 | 2 | yes | — |
| 42 | C3×C3⋊(C3⋊C8) | C23 | 1 | 6 | yes | — |
| 43 | #5745 | — | 9 | 3 | no | — |
| 44 | C2×C4×C3⋊D3 | C2×D3 | 1 | 3 | yes | — |
| 45 | C2×C3×C4×D3 | — | 3 | 6 | no | — |
| 46 | C2×C3×C4×D3 | — | 6 | 3 | no | — |
| 47 | C2×#627 | — | 3 | 3 | no | — |
| 48 | C2×#626 | — | 3 | 6 | no | — |
| 49 | C2×#625 | — | 3 | 3 | no | — |
| 50 | #3798 | — | 3 | 3 | no | — |
| 51 | #3798 | — | 3 | 6 | no | — |
| 52 | #3798 | — | 3 | 3 | no | — |
| 53 | #3798 | — | 3 | 3 | no | — |
| 54 | #3798 | — | 3 | 6 | no | — |
| 55 | #3798 | — | 3 | 3 | no | — |
| 56 | C32×M16 | — | 2 | 3 | no | — |
| 57 | C32×M16 | — | 4 | 1 | no | — |
| 58 | C32×M16 | C2×D3 | 1 | 3 | yes | — |
| 59 | #3748 | C2×D3 | 1 | 3 | yes | — |
| 60 | #3743 | C2×D3 | 1 | 6 | yes | complemented |
| 61 | C8×C3⋊D3 | C2×D3 | 1 | 6 | yes | complemented |
| 62 | C3×#208 | — | 3 | 6 | no | — |
| 63 | C3×#208 | — | 6 | 3 | no | — |
| 64 | C3×ΓL(1,𝔽25) | — | 3 | 12 | no | — |
| 65 | C3×ΓL(1,𝔽25) | — | 6 | 6 | no | — |
| 66 | C3×D3×C8 | — | 3 | 12 | no | — |
| 67 | C3×D3×C8 | — | 6 | 6 | no | — |
| 68 | #3712 | — | 3 | 6 | no | — |
| 69 | #3711 | — | 3 | 12 | no | — |
| 70 | #3710 | — | 3 | 6 | no | — |
| 71 | D3×C3⋊C8 | — | 3 | 12 | no | — |
| 72 | C22×C33 | C22×C4 | 1 | 1 | yes | characteristic |
| 73 | C2×C3×C3⋊D3 | — | 4 | 3 | no | — |
| 74 | C2×C3×C3⋊D3 | C22×C4 | 1 | 3 | yes | — |
| 75 | C2×C3×C3⋊D3 | C22×C4 | 1 | 3 | yes | — |
| 76 | C33×C4 | C22×C4 | 1 | 1 | yes | characteristic |
| 77 | C33×C4 | C24 | 1 | 1 | yes | characteristic |
| 78 | C3×C3⋊Dic3 | C22×C4 | 1 | 3 | yes | — |
| 79 | C3×C3⋊Dic3 | C22×C4 | 1 | 3 | yes | — |
| 80 | C2×#235 | — | 9 | 3 | no | — |
| 81 | C2×C3×M16 | — | 9 | 3 | no | — |
| 82 | C2×#208 | — | 9 | 3 | no | — |
| 83 | #907 | — | 9 | 12 | no | — |
| 84 | D3×M16 | — | 3 | 6 | no | — |
| 85 | D3×M16 | — | 6 | 3 | no | — |
| 86 | D3×M16 | — | 9 | 6 | no | — |
| 87 | #901 | — | 9 | 6 | no | — |
| 88 | #901 | — | 9 | 6 | no | — |
| 89 | C22×C3⋊D3 | C4×D3 | 1 | 3 | yes | — |
| 90 | C22×C3×D3 | — | 3 | 6 | no | — |
| 91 | C22×C3×D3 | — | 6 | 3 | no | — |
| 92 | C2×C32×C4 | — | 2 | 3 | no | — |
| 93 | C2×C32×C4 | — | 4 | 1 | no | — |
| 94 | C2×C32×C4 | C22×D3 | 1 | 3 | yes | — |
| 95 | C2×C3⋊Dic3 | C4×D3 | 1 | 3 | yes | — |
| 96 | C4×C3⋊D3 | C4×D3 | 1 | 3 | yes | — |
| 97 | C4×C3⋊D3 | C4×D3 | 1 | 3 | yes | — |
| 98 | C4×C3⋊D3 | C22×D3 | 1 | 3 | yes | — |
| 99 | C4×C3⋊D3 | C22×D3 | 1 | 3 | yes | — |
| 100 | C2×C3×Dic3 | — | 3 | 6 | no | — |
| 101 | C2×C3×Dic3 | — | 6 | 3 | no | — |
| 102 | C3×C4×D3 | — | 3 | 6 | no | — |
| 103 | C3×C4×D3 | — | 3 | 6 | no | — |
| 104 | C3×C4×D3 | — | 3 | 6 | no | — |
| 105 | C3×C4×D3 | — | 3 | 6 | no | — |
| 106 | C3×C4×D3 | — | 6 | 3 | no | — |
| 107 | C3×C4×D3 | — | 6 | 3 | no | — |
| 108 | C3×C4×D3 | — | 6 | 3 | no | — |
| 109 | C3×C4×D3 | — | 6 | 3 | no | — |
| 110 | #627 | — | 3 | 3 | no | — |
| 111 | #627 | — | 3 | 6 | no | — |
| 112 | #627 | — | 3 | 3 | no | — |
| 113 | #626 | — | 3 | 6 | no | — |
| 114 | #626 | — | 3 | 6 | no | — |
| 115 | #626 | — | 3 | 6 | no | — |
| 116 | #626 | — | 3 | 6 | no | — |
| 117 | #625 | — | 3 | 3 | no | — |
| 118 | #625 | — | 3 | 6 | no | — |
| 119 | #625 | — | 3 | 3 | no | — |
| 120 | C32×C8 | — | 2 | 6 | no | — |
| 121 | C32×C8 | — | 4 | 2 | no | — |
| 122 | C32×C8 | C22×D3 | 1 | 6 | yes | — |
| 123 | C3⋊(C3⋊C8) | C22×D3 | 1 | 6 | yes | — |
| 124 | C3×C3⋊C8 | — | 3 | 12 | no | — |
| 125 | C3×C3⋊C8 | — | 6 | 6 | no | — |
| 126 | #568 | — | 27 | 1 | no | Sylow |
| 127 | C2×C33 | — | 2 | 1 | no | — |
| 128 | C2×C33 | C23×C4 | 1 | 1 | yes | characteristic |
| 129 | C3×C3⋊D3 | — | 2 | 3 | no | — |
| 130 | C3×C3⋊D3 | — | 2 | 3 | no | — |
| 131 | C22×C3×C4 | — | 9 | 3 | no | — |
| 132 | C2×#67 | — | 9 | 6 | no | — |
| 133 | #235 | — | 9 | 3 | no | — |
| 134 | #235 | — | 9 | 3 | no | — |
| 135 | #235 | — | 9 | 6 | no | — |
| 136 | #235 | — | 9 | 3 | no | — |
| 137 | #235 | — | 9 | 3 | no | — |
| 138 | #235 | — | 9 | 6 | no | — |
| 139 | C2×D12 | — | 9 | 3 | no | — |
| 140 | C2×C4×D3 | — | 3 | 6 | no | — |
| 141 | C2×C4×D3 | — | 6 | 3 | no | — |
| 142 | C2×Dic6 | — | 9 | 3 | no | — |
| 143 | C3×M16 | — | 2 | 3 | no | — |
| 144 | C3×M16 | — | 4 | 1 | no | — |
| 145 | C3×M16 | — | 9 | 3 | no | — |
| 146 | C3×M16 | — | 9 | 6 | no | — |
| 147 | C3×M16 | D32 | 1 | 3 | yes | — |
| 148 | C2×C3×C8 | — | 9 | 6 | no | — |
| 149 | #208 | — | 3 | 6 | no | — |
| 150 | #208 | — | 6 | 3 | no | — |
| 151 | #208 | — | 9 | 6 | no | — |
| 152 | C2×C3⋊C8 | — | 9 | 6 | no | — |
| 153 | ΓL(1,𝔽25) | — | 3 | 12 | no | — |
| 154 | ΓL(1,𝔽25) | — | 6 | 6 | no | — |
| 155 | ΓL(1,𝔽25) | — | 9 | 12 | no | — |
| 156 | D3×C8 | — | 3 | 12 | no | — |
| 157 | D3×C8 | — | 6 | 6 | no | — |
| 158 | D3×C8 | — | 9 | 12 | no | — |
| 159 | C22×C32 | — | 2 | 3 | no | — |
| 160 | C22×C32 | — | 4 | 1 | no | — |
| 161 | C22×C32 | C2×C4×D3 | 1 | 3 | yes | — |
| 162 | C2×C3⋊D3 | — | 4 | 3 | no | — |
| 163 | C2×C3⋊D3 | C2×C4×D3 | 1 | 3 | yes | — |
| 164 | C2×C3⋊D3 | C2×C4×D3 | 1 | 3 | yes | — |
| 165 | C2×C3×D3 | — | 3 | 6 | no | — |
| 166 | C2×C3×D3 | — | 3 | 6 | no | — |
| 167 | C2×C3×D3 | — | 6 | 3 | no | — |
| 168 | C2×C3×D3 | — | 6 | 3 | no | — |
| 169 | C2×C3×D3 | — | 12 | 6 | no | — |
| 170 | C2×C3×D3 | — | 12 | 6 | no | — |
| 171 | C32×C4 | — | 2 | 3 | no | — |
| 172 | C32×C4 | — | 2 | 3 | no | — |
| 173 | C32×C4 | — | 4 | 1 | no | — |
| 174 | C32×C4 | — | 4 | 1 | no | — |
| 175 | C32×C4 | C2×C4×D3 | 1 | 3 | yes | — |
| 176 | C32×C4 | C23×D3 | 1 | 3 | yes | — |
| 177 | C3⋊Dic3 | C2×C4×D3 | 1 | 3 | yes | — |
| 178 | C3⋊Dic3 | C2×C4×D3 | 1 | 3 | yes | — |
| 179 | C3×Dic3 | — | 3 | 6 | no | — |
| 180 | C3×Dic3 | — | 3 | 6 | no | — |
| 181 | C3×Dic3 | — | 6 | 3 | no | — |
| 182 | C3×Dic3 | — | 6 | 3 | no | — |
| 183 | C2×#41 | — | 27 | 1 | no | — |
| 184 | #131 | — | 27 | 2 | no | — |
| 185 | #131 | — | 27 | 6 | no | — |
| 186 | C2×M16 | — | 9 | 3 | no | — |
| 187 | C2×M16 | — | 27 | 3 | no | — |
| 188 | C33 | #568 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 189 | C23×C3 | — | 9 | 3 | no | — |
| 190 | C22×D3 | — | 3 | 6 | no | — |
| 191 | C22×D3 | — | 6 | 3 | no | — |
| 192 | C2×C3×C4 | — | 2 | 3 | no | — |
| 193 | C2×C3×C4 | — | 4 | 1 | no | — |
| 194 | C2×C3×C4 | — | 9 | 3 | no | — |
| 195 | C2×C3×C4 | — | 9 | 3 | no | — |
| 196 | C2×C3×C4 | — | 9 | 3 | no | — |
| 197 | C2×C3×C4 | — | 9 | 3 | no | — |
| 198 | C2×C3×C4 | — | 9 | 3 | no | — |
| 199 | C2×C3×C4 | C2×D32 | 1 | 3 | yes | — |
| 200 | #67 | — | 9 | 6 | no | — |
| 201 | #67 | — | 9 | 6 | no | — |
| 202 | #67 | — | 9 | 6 | no | — |
| 203 | #67 | — | 9 | 6 | no | — |
| 204 | C2×Dic3 | — | 3 | 6 | no | — |
| 205 | C2×Dic3 | — | 6 | 3 | no | — |
| 206 | D12 | — | 9 | 3 | no | — |
| 207 | D12 | — | 9 | 3 | no | — |
| 208 | D12 | — | 9 | 6 | no | — |
| 209 | C4×D3 | — | 3 | 6 | no | — |
| 210 | C4×D3 | — | 3 | 6 | no | — |
| 211 | C4×D3 | — | 3 | 6 | no | — |
| 212 | C4×D3 | — | 3 | 6 | no | — |
| 213 | C4×D3 | — | 6 | 3 | no | — |
| 214 | C4×D3 | — | 6 | 3 | no | — |
| 215 | C4×D3 | — | 6 | 3 | no | — |
| 216 | C4×D3 | — | 6 | 3 | no | — |
| 217 | Dic6 | — | 9 | 3 | no | — |
| 218 | Dic6 | — | 9 | 3 | no | — |
| 219 | Dic6 | — | 9 | 6 | no | — |
| 220 | C3×C8 | — | 2 | 6 | no | — |
| 221 | C3×C8 | — | 4 | 2 | no | — |
| 222 | C3×C8 | — | 9 | 6 | no | — |
| 223 | C3×C8 | C2×D32 | 1 | 6 | yes | — |
| 224 | C3⋊C8 | — | 3 | 12 | no | — |
| 225 | C3⋊C8 | — | 6 | 6 | no | — |
| 226 | C2×C32 | — | 2 | 3 | no | — |
| 227 | C2×C32 | — | 2 | 3 | no | — |
| 228 | C2×C32 | — | 4 | 1 | no | — |
| 229 | C2×C32 | — | 4 | 3 | no | — |
| 230 | C2×C32 | — | 8 | 1 | no | — |
| 231 | C2×C32 | C22×C4×D3 | 1 | 3 | yes | — |
| 232 | C3⋊D3 | — | 2 | 3 | no | — |
| 233 | C3⋊D3 | — | 2 | 3 | no | — |
| 234 | C3×D3 | — | 6 | 6 | no | — |
| 235 | C3×D3 | — | 6 | 6 | no | — |
| 236 | C3×D3 | — | 6 | 6 | no | — |
| 237 | C3×D3 | — | 12 | 3 | no | — |
| 238 | #41 | — | 27 | 1 | no | — |
| 239 | #41 | — | 27 | 3 | no | — |
| 240 | #41 | — | 27 | 1 | no | — |
| 241 | #41 | — | 27 | 3 | no | — |
| 242 | C2×Q8 | — | 27 | 1 | no | — |
| 243 | C2×D4 | — | 27 | 3 | no | — |
| 244 | C22×C4 | — | 9 | 3 | no | — |
| 245 | M16 | — | 9 | 3 | no | — |
| 246 | M16 | — | 9 | 6 | no | — |
| 247 | M16 | — | 27 | 6 | no | — |
| 248 | M16 | #1113 | 1 | 1 | yes | characteristic |
| 249 | C2×C8 | — | 9 | 6 | no | — |
| 250 | C2×C8 | — | 27 | 6 | no | — |
| 251 | C22×C3 | — | 2 | 3 | no | — |
| 252 | C22×C3 | — | 4 | 1 | no | — |
| 253 | C22×C3 | — | 9 | 3 | no | — |
| 254 | C22×C3 | — | 9 | 3 | no | — |
| 255 | C22×C3 | — | 36 | 3 | no | — |
| 256 | C22×C3 | C4×D32 | 1 | 3 | yes | — |
| 257 | C2×D3 | — | 3 | 6 | no | — |
| 258 | C2×D3 | — | 3 | 6 | no | — |
| 259 | C2×D3 | — | 6 | 3 | no | — |
| 260 | C2×D3 | — | 6 | 3 | no | — |
| 261 | C2×D3 | — | 12 | 6 | no | — |
| 262 | C2×D3 | — | 12 | 6 | no | — |
| 263 | C3×C4 | — | 2 | 3 | no | — |
| 264 | C3×C4 | — | 2 | 3 | no | — |
| 265 | C3×C4 | — | 4 | 1 | no | — |
| 266 | C3×C4 | — | 4 | 1 | no | — |
| 267 | C3×C4 | — | 9 | 3 | no | — |
| 268 | C3×C4 | — | 9 | 3 | no | — |
| 269 | C3×C4 | C4×D32 | 1 | 3 | yes | — |
| 270 | C3×C4 | C22×D32 | 1 | 3 | yes | — |
| 271 | Dic3 | — | 3 | 6 | no | — |
| 272 | Dic3 | — | 3 | 6 | no | — |
| 273 | Dic3 | — | 6 | 3 | no | — |
| 274 | Dic3 | — | 6 | 3 | no | — |
| 275 | C32 | — | 2 | 3 | no | — |
| 276 | C32 | — | 4 | 1 | no | — |
| 277 | C32 | #5745 | 1 | 3 | yes | complemented |
| 278 | C23 | — | 9 | 3 | no | — |
| 279 | Q8 | — | 27 | 1 | no | — |
| 280 | Q8 | — | 27 | 3 | no | — |
| 281 | D4 | — | 27 | 3 | no | — |
| 282 | D4 | — | 27 | 3 | no | — |
| 283 | D4 | — | 27 | 6 | no | — |
| 284 | C2×C4 | — | 9 | 3 | no | — |
| 285 | C2×C4 | — | 9 | 3 | no | — |
| 286 | C2×C4 | — | 9 | 3 | no | — |
| 287 | C2×C4 | — | 9 | 3 | no | — |
| 288 | C2×C4 | — | 9 | 3 | no | — |
| 289 | C2×C4 | C2×#1113 | 1 | 1 | yes | characteristic |
| 290 | C8 | — | 9 | 6 | no | — |
| 291 | C8 | C2×#1113 | 1 | 2 | yes | — |
| 292 | C2×C3 | — | 2 | 3 | no | — |
| 293 | C2×C3 | — | 2 | 3 | no | — |
| 294 | C2×C3 | — | 4 | 3 | no | — |
| 295 | C2×C3 | — | 4 | 1 | no | — |
| 296 | C2×C3 | — | 8 | 1 | no | — |
| 297 | C2×C3 | — | 18 | 3 | no | — |
| 298 | C2×C3 | — | 18 | 3 | no | — |
| 299 | C2×C3 | C2×C4×D32 | 1 | 3 | yes | — |
| 300 | D3 | — | 6 | 6 | no | — |
| 301 | D3 | — | 6 | 6 | no | — |
| 302 | D3 | — | 6 | 6 | no | — |
| 303 | D3 | — | 12 | 3 | no | — |
| 304 | C22 | — | 9 | 3 | no | — |
| 305 | C22 | — | 9 | 3 | no | — |
| 306 | C22 | — | 36 | 3 | no | — |
| 307 | C22 | C4×#1113 | 1 | 1 | yes | characteristic |
| 308 | C4 | — | 9 | 3 | no | — |
| 309 | C4 | — | 9 | 3 | no | — |
| 310 | C4 | C4×#1113 | 1 | 1 | yes | characteristic |
| 311 | C4 | C22×#1113 | 1 | 1 | yes | characteristic |
| 312 | C3 | — | 2 | 3 | no | — |
| 313 | C3 | — | 4 | 1 | no | — |
| 314 | C3 | #10594722 | 1 | 3 | yes | complemented |
| 315 | C2 | — | 2 | 1 | no | — |
| 316 | C2 | — | 18 | 3 | no | — |
| 317 | C2 | — | 18 | 3 | no | — |
| 318 | C2 | C2×C4×#1113 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C4 | C22×#1113 |
| commutator, socle | C2×C33 | C23×C4 |
| Fitting | C33×M16 | C22 |
| prime radical | C2×#89042 | C2 |
| cosocle | C33×C4 | C24 |
| hypercenter | M16 | #1113 |
| hypocenter | C33 | #568 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C4 ◁ M16 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3×M16×C3⋊D3 | C2 | semidirect |
| 3. #11723656 | C2 | semidirect |
| 4. #11723652 | C2 | semidirect |
| 5. #11723651 | C2 | semidirect |
| 16. C3×#3743 | C22 | semidirect |
| 17. C3×C8×C3⋊D3 | C22 | semidirect |
| 18. #88809 | C22 | semidirect |
| 19. #88808 | C22 | semidirect |
| 21. M16×C3⋊D3 | D3 | semidirect |
| 60. #3743 | C2×D3 | semidirect |
| 61. C8×C3⋊D3 | C2×D3 | semidirect |
| 188. C33 | #568 | semidirect |
| 277. C32 | #5745 | semidirect |
| 314. C3 | #10594722 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#89042 | 2 |
| C3×M16×C3⋊D3 | 2 |
| #11723656 | 2 |
| #11723652 | 2 |
| #11723651 | 2 |
| #10594722 | 3 |