Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
1492 conjugacy classes of subgroups.
The subgroup lattice has 310 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×#1010 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | C3×#537 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #5832 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #5633 | C2 | 1 | 4 | yes | maximal, complemented |
| 5 | #5632 | C2 | 1 | 4 | yes | maximal, complemented |
| 6 | #5599 | C2 | 1 | 4 | yes | maximal, complemented |
| 7 | #5598 | C2 | 1 | 4 | yes | maximal, complemented |
| 8 | #5597 | C2 | 1 | 4 | yes | maximal, complemented |
| 9 | #5596 | C2 | 1 | 4 | yes | maximal, complemented |
| 10 | #5510 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 11 | C2×#887 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 12 | #3497 | — | 3 | 1 | no | maximal, Sylow |
| 13 | C2×C3×#41 | C22 | 1 | 2 | yes | — |
| 14 | #1010 | — | 2 | 16 | no | — |
| 15 | C2×D3×Q8 | C22 | 1 | 4 | yes | — |
| 16 | C2×#237 | C22 | 1 | 8 | yes | — |
| 17 | C2×#237 | C22 | 1 | 4 | yes | — |
| 18 | C2×#235 | C22 | 1 | 4 | yes | — |
| 19 | C2×#235 | C22 | 1 | 2 | yes | — |
| 20 | C22×Dic6 | C22 | 1 | 4 | yes | — |
| 21 | C22×Dic6 | C22 | 1 | 1 | yes | characteristic |
| 22 | C3×C22⋊Q8 | C22 | 1 | 4 | yes | — |
| 23 | C3×#121 | C22 | 1 | 4 | yes | — |
| 24 | C3×#121 | C22 | 1 | 4 | yes | — |
| 25 | C2×C3×C4⋊C4 | C22 | 1 | 1 | yes | characteristic |
| 26 | #947 | C22 | 1 | 2 | yes | — |
| 27 | #944 | C22 | 1 | 2 | yes | — |
| 28 | #940 | C22 | 1 | 2 | yes | — |
| 29 | #940 | C22 | 1 | 4 | yes | — |
| 30 | #939 | C22 | 1 | 4 | yes | — |
| 31 | #939 | C22 | 1 | 4 | yes | — |
| 32 | #939 | C22 | 1 | 8 | yes | — |
| 33 | #936 | C22 | 1 | 2 | yes | — |
| 34 | D4×Dic3 | C22 | 1 | 4 | yes | — |
| 35 | #930 | C22 | 1 | 4 | yes | — |
| 36 | C4×#67 | C22 | 1 | 4 | yes | — |
| 37 | C4×#67 | C22 | 1 | 4 | yes | — |
| 38 | C2×Dic3⋊C4 | C22 | 1 | 2 | yes | — |
| 39 | C2×C4×Dic3 | C22 | 1 | 1 | yes | characteristic |
| 40 | C2×C4×Dic3 | C22 | 1 | 2 | yes | — |
| 41 | #897 | C22 | 1 | 4 | yes | — |
| 42 | #896 | C22 | 1 | 8 | yes | — |
| 43 | #895 | C22 | 1 | 4 | yes | — |
| 44 | #893 | C22 | 1 | 4 | yes | — |
| 45 | #888 | C22 | 1 | 4 | yes | — |
| 46 | #887 | C22 | 1 | 4 | yes | — |
| 47 | #887 | C22 | 1 | 4 | yes | — |
| 48 | #885 | C22 | 1 | 8 | yes | — |
| 49 | #884 | C22 | 1 | 8 | yes | — |
| 50 | #67⋊C4 | C22 | 1 | 8 | yes | — |
| 51 | #878 | C22 | 1 | 8 | yes | — |
| 52 | C2×2−1+4 | — | 3 | 2 | no | — |
| 53 | #550 | — | 3 | 4 | no | — |
| 54 | D4×Q8 | — | 3 | 4 | no | — |
| 55 | #548 | — | 3 | 4 | no | — |
| 56 | #548 | — | 3 | 4 | no | — |
| 57 | #537 | — | 3 | 4 | no | — |
| 58 | #537 | — | 3 | 1 | no | — |
| 59 | #537 | D3 | 1 | 1 | yes | characteristic, complemented |
| 60 | #532 | — | 3 | 2 | no | — |
| 61 | #532 | — | 3 | 4 | no | — |
| 62 | C2×C4×Q8 | — | 3 | 1 | no | — |
| 63 | C3×#41 | — | 2 | 8 | no | — |
| 64 | C2×C3×Q8 | C23 | 1 | 2 | yes | — |
| 65 | C2×C3×D4 | C23 | 1 | 2 | yes | — |
| 66 | C2×C3×D4 | C23 | 1 | 4 | yes | — |
| 67 | C2×C3×D4 | C23 | 1 | 4 | yes | — |
| 68 | C22×C3×C4 | C23 | 1 | 4 | yes | — |
| 69 | C22×C3×C4 | C23 | 1 | 2 | yes | — |
| 70 | C22×C3×C4 | C23 | 1 | 1 | yes | characteristic |
| 71 | C2×#67 | C23 | 1 | 4 | yes | — |
| 72 | C2×#67 | C23 | 1 | 8 | yes | — |
| 73 | C2×#67 | C23 | 1 | 4 | yes | — |
| 74 | C2×#67 | C23 | 1 | 4 | yes | — |
| 75 | C22×Dic3 | C23 | 1 | 8 | yes | — |
| 76 | C22×Dic3 | C23 | 1 | 1 | yes | characteristic |
| 77 | C22×Dic3 | C23 | 1 | 1 | yes | characteristic |
| 78 | C22×Dic3 | C23 | 1 | 2 | yes | — |
| 79 | D3×Q8 | — | 2 | 16 | no | — |
| 80 | #237 | — | 2 | 32 | no | — |
| 81 | #237 | — | 2 | 16 | no | — |
| 82 | #235 | — | 2 | 8 | no | — |
| 83 | #235 | — | 2 | 16 | no | — |
| 84 | C2×D12 | C23 | 1 | 2 | yes | — |
| 85 | C2×D12 | C23 | 1 | 4 | yes | — |
| 86 | C2×C4×D3 | C23 | 1 | 8 | yes | — |
| 87 | C2×C4×D3 | C23 | 1 | 4 | yes | — |
| 88 | C2×Dic6 | — | 2 | 16 | no | — |
| 89 | C2×Dic6 | D4 | 1 | 8 | yes | complemented |
| 90 | C2×Dic6 | C23 | 1 | 8 | yes | — |
| 91 | C2×Dic6 | C23 | 1 | 2 | yes | — |
| 92 | C2×Dic6 | C23 | 1 | 4 | yes | — |
| 93 | C3×C4⋊C4 | C23 | 1 | 4 | yes | — |
| 94 | C3×C4⋊C4 | C23 | 1 | 4 | yes | — |
| 95 | C3×C22⋊C4 | C23 | 1 | 8 | yes | — |
| 96 | #217 | C23 | 1 | 4 | yes | — |
| 97 | #217 | C23 | 1 | 4 | yes | — |
| 98 | (C2×D3)⋊C4 | C23 | 1 | 8 | yes | — |
| 99 | (C2×D3)⋊C4 | C23 | 1 | 4 | yes | — |
| 100 | (C2×D3)⋊C4 | C23 | 1 | 4 | yes | — |
| 101 | C4⋊Dic3 | C23 | 1 | 4 | yes | — |
| 102 | Dic3⋊C4 | C23 | 1 | 4 | yes | — |
| 103 | Dic3⋊C4 | C23 | 1 | 8 | yes | — |
| 104 | Dic3⋊C4 | C23 | 1 | 4 | yes | — |
| 105 | C4×Dic3 | C23 | 1 | 4 | yes | — |
| 106 | C4×Dic3 | C23 | 1 | 4 | yes | — |
| 107 | C4×Dic3 | C23 | 1 | 2 | yes | — |
| 108 | C4×Dic3 | C23 | 1 | 2 | yes | — |
| 109 | 2−1+4 | — | 6 | 16 | no | — |
| 110 | C2×#41 | — | 3 | 8 | no | — |
| 111 | C2×#41 | — | 3 | 4 | no | — |
| 112 | C2×#41 | — | 3 | 2 | no | — |
| 113 | C2×#41 | — | 3 | 4 | no | — |
| 114 | C2×#41 | C2×D3 | 1 | 2 | yes | — |
| 115 | C22×Q8 | — | 3 | 4 | no | — |
| 116 | C22×Q8 | — | 3 | 1 | no | — |
| 117 | C22×Q8 | — | 3 | 4 | no | — |
| 118 | C4⋊Q8 | — | 3 | 2 | no | — |
| 119 | C4⋊Q8 | — | 3 | 4 | no | — |
| 120 | #127 | — | 3 | 4 | no | — |
| 121 | #127 | — | 3 | 2 | no | — |
| 122 | #124 | — | 3 | 2 | no | — |
| 123 | #124 | — | 3 | 2 | no | — |
| 124 | #124 | — | 3 | 8 | no | — |
| 125 | C22⋊Q8 | — | 3 | 8 | no | — |
| 126 | C22⋊Q8 | — | 3 | 4 | no | — |
| 127 | C22⋊Q8 | — | 3 | 8 | no | — |
| 128 | C22⋊Q8 | C2×D3 | 1 | 4 | yes | — |
| 129 | #121 | — | 3 | 4 | no | — |
| 130 | #121 | — | 3 | 4 | no | — |
| 131 | #121 | — | 3 | 8 | no | — |
| 132 | #121 | — | 3 | 4 | no | — |
| 133 | #121 | — | 3 | 4 | no | — |
| 134 | #121 | — | 3 | 8 | no | — |
| 135 | #121 | C2×D3 | 1 | 4 | yes | — |
| 136 | #121 | C2×D3 | 1 | 4 | yes | — |
| 137 | C4×Q8 | — | 3 | 4 | no | — |
| 138 | C4×Q8 | — | 3 | 4 | no | — |
| 139 | C4×D4 | — | 3 | 4 | no | — |
| 140 | C4×D4 | — | 3 | 4 | no | — |
| 141 | C4×D4 | — | 3 | 4 | no | — |
| 142 | C4×D4 | — | 3 | 4 | no | — |
| 143 | C4×D4 | — | 3 | 8 | no | — |
| 144 | C2×C4⋊C4 | — | 3 | 2 | no | — |
| 145 | C2×C4⋊C4 | C2×D3 | 1 | 1 | yes | characteristic |
| 146 | C2×C42 | — | 3 | 2 | no | — |
| 147 | C2×C42 | — | 3 | 1 | no | — |
| 148 | C23×C3 | C24 | 1 | 4 | yes | — |
| 149 | C23×C3 | C24 | 1 | 1 | yes | characteristic |
| 150 | C22×D3 | C24 | 1 | 4 | yes | — |
| 151 | C3×Q8 | — | 2 | 4 | no | — |
| 152 | C3×D4 | — | 2 | 8 | no | — |
| 153 | C3×D4 | — | 2 | 4 | no | — |
| 154 | C3×D4 | — | 4 | 4 | no | — |
| 155 | C2×C3×C4 | — | 2 | 8 | no | — |
| 156 | C2×C3×C4 | — | 4 | 2 | no | — |
| 157 | C2×C3×C4 | C2×D4 | 1 | 4 | yes | — |
| 158 | C2×C3×C4 | C24 | 1 | 4 | yes | — |
| 159 | C2×C3×C4 | C24 | 1 | 4 | yes | — |
| 160 | C2×C3×C4 | C24 | 1 | 2 | yes | — |
| 161 | #67 | — | 2 | 8 | no | — |
| 162 | #67 | — | 2 | 16 | no | — |
| 163 | #67 | — | 2 | 8 | no | — |
| 164 | #67 | — | 4 | 4 | no | — |
| 165 | C2×Dic3 | — | 2 | 16 | no | — |
| 166 | C2×Dic3 | — | 2 | 2 | no | — |
| 167 | C2×Dic3 | — | 4 | 1 | no | — |
| 168 | C2×Dic3 | C2×D4 | 1 | 8 | yes | — |
| 169 | C2×Dic3 | C24 | 1 | 1 | yes | characteristic |
| 170 | C2×Dic3 | C24 | 1 | 2 | yes | — |
| 171 | C2×Dic3 | C24 | 1 | 4 | yes | — |
| 172 | C2×Dic3 | C24 | 1 | 4 | yes | — |
| 173 | C2×Dic3 | C24 | 1 | 1 | yes | characteristic |
| 174 | D12 | — | 2 | 4 | no | — |
| 175 | D12 | — | 4 | 4 | no | — |
| 176 | C4×D3 | — | 2 | 8 | no | — |
| 177 | C4×D3 | — | 2 | 16 | no | — |
| 178 | Dic6 | — | 2 | 4 | no | — |
| 179 | Dic6 | — | 2 | 16 | no | — |
| 180 | Dic6 | C2×D4 | 1 | 16 | yes | — |
| 181 | #41 | — | 2 | 8 | no | — |
| 182 | #41 | — | 6 | 16 | no | — |
| 183 | #41 | — | 6 | 32 | no | — |
| 184 | #41 | — | 6 | 16 | no | — |
| 185 | #41 | — | 6 | 8 | no | — |
| 186 | C2×Q8 | — | 3 | 8 | no | — |
| 187 | C2×Q8 | — | 3 | 4 | no | — |
| 188 | C2×Q8 | — | 3 | 8 | no | — |
| 189 | C2×Q8 | — | 3 | 2 | no | — |
| 190 | C2×Q8 | — | 6 | 16 | no | — |
| 191 | C2×Q8 | — | 6 | 16 | no | — |
| 192 | C2×Q8 | C22×D3 | 1 | 2 | yes | — |
| 193 | C2×D4 | — | 3 | 4 | no | — |
| 194 | C2×D4 | — | 3 | 4 | no | — |
| 195 | C2×D4 | — | 3 | 8 | no | — |
| 196 | C2×D4 | — | 3 | 4 | no | — |
| 197 | C2×D4 | — | 3 | 2 | no | — |
| 198 | C2×D4 | — | 3 | 4 | no | — |
| 199 | C2×D4 | C22×D3 | 1 | 4 | yes | — |
| 200 | C2×D4 | C22×D3 | 1 | 2 | yes | — |
| 201 | C2×D4 | C22×D3 | 1 | 4 | yes | — |
| 202 | C22×C4 | — | 3 | 2 | no | — |
| 203 | C22×C4 | — | 3 | 1 | no | — |
| 204 | C22×C4 | — | 3 | 8 | no | — |
| 205 | C22×C4 | — | 3 | 1 | no | — |
| 206 | C22×C4 | — | 3 | 8 | no | — |
| 207 | C22×C4 | — | 3 | 4 | no | — |
| 208 | C22×C4 | C22×D3 | 1 | 4 | yes | — |
| 209 | C22×C4 | C22×D3 | 1 | 2 | yes | — |
| 210 | C22×C4 | C22×D3 | 1 | 1 | yes | characteristic |
| 211 | C4⋊C4 | — | 3 | 8 | no | — |
| 212 | C4⋊C4 | — | 3 | 4 | no | — |
| 213 | C4⋊C4 | — | 3 | 4 | no | — |
| 214 | C4⋊C4 | — | 3 | 4 | no | — |
| 215 | C4⋊C4 | C22×D3 | 1 | 4 | yes | — |
| 216 | C4⋊C4 | C22×D3 | 1 | 4 | yes | — |
| 217 | C22⋊C4 | — | 3 | 4 | no | — |
| 218 | C22⋊C4 | — | 3 | 4 | no | — |
| 219 | C22⋊C4 | — | 3 | 4 | no | — |
| 220 | C22⋊C4 | — | 3 | 4 | no | — |
| 221 | C22⋊C4 | — | 3 | 8 | no | — |
| 222 | C22⋊C4 | C22×D3 | 1 | 8 | yes | — |
| 223 | C42 | — | 3 | 4 | no | — |
| 224 | C42 | — | 3 | 2 | no | — |
| 225 | C42 | — | 3 | 4 | no | — |
| 226 | C42 | — | 3 | 2 | no | — |
| 227 | C22×C3 | — | 2 | 1 | no | — |
| 228 | C22×C3 | — | 2 | 1 | no | — |
| 229 | C22×C3 | — | 2 | 4 | no | — |
| 230 | C22×C3 | — | 2 | 4 | no | — |
| 231 | C22×C3 | — | 2 | 4 | no | — |
| 232 | C22×C3 | C22×D4 | 1 | 2 | yes | — |
| 233 | C22×C3 | C25 | 1 | 1 | yes | characteristic |
| 234 | C2×D3 | — | 2 | 4 | no | — |
| 235 | C2×D3 | — | 2 | 4 | no | — |
| 236 | C2×D3 | — | 2 | 4 | no | — |
| 237 | C3×C4 | — | 2 | 4 | no | — |
| 238 | C3×C4 | — | 2 | 4 | no | — |
| 239 | C3×C4 | C22×D4 | 1 | 4 | yes | — |
| 240 | Dic3 | — | 2 | 4 | no | — |
| 241 | Dic3 | — | 2 | 1 | no | — |
| 242 | Dic3 | — | 2 | 2 | no | — |
| 243 | Dic3 | C22×D4 | 1 | 8 | yes | — |
| 244 | Dic3 | 2−1+4 | 1 | 2 | yes | complemented |
| 245 | C23 | — | 3 | 4 | no | — |
| 246 | C23 | C23×D3 | 1 | 4 | yes | — |
| 247 | C23 | C23×D3 | 1 | 1 | yes | characteristic |
| 248 | Q8 | — | 2 | 4 | no | — |
| 249 | Q8 | — | 3 | 16 | no | — |
| 250 | Q8 | — | 6 | 16 | no | — |
| 251 | Q8 | — | 6 | 4 | no | — |
| 252 | D4 | — | 2 | 8 | no | — |
| 253 | D4 | — | 2 | 4 | no | — |
| 254 | D4 | — | 4 | 4 | no | — |
| 255 | D4 | — | 6 | 16 | no | — |
| 256 | D4 | — | 6 | 8 | no | — |
| 257 | D4 | — | 6 | 8 | no | — |
| 258 | D4 | — | 6 | 4 | no | — |
| 259 | D4 | — | 12 | 4 | no | — |
| 260 | D4 | — | 12 | 4 | no | — |
| 261 | C2×C4 | — | 2 | 8 | no | — |
| 262 | C2×C4 | — | 3 | 8 | no | — |
| 263 | C2×C4 | — | 3 | 1 | no | — |
| 264 | C2×C4 | — | 3 | 4 | no | — |
| 265 | C2×C4 | — | 3 | 1 | no | — |
| 266 | C2×C4 | — | 3 | 2 | no | — |
| 267 | C2×C4 | — | 3 | 4 | no | — |
| 268 | C2×C4 | — | 4 | 2 | no | — |
| 269 | C2×C4 | — | 6 | 16 | no | — |
| 270 | C2×C4 | — | 6 | 16 | no | — |
| 271 | C2×C4 | — | 6 | 2 | no | — |
| 272 | C2×C4 | — | 6 | 8 | no | — |
| 273 | C2×C4 | — | 12 | 1 | no | — |
| 274 | C2×C4 | D3×D4 | 1 | 4 | yes | — |
| 275 | C2×C4 | C23×D3 | 1 | 4 | yes | — |
| 276 | C2×C4 | C23×D3 | 1 | 4 | yes | — |
| 277 | C2×C4 | C23×D3 | 1 | 2 | yes | — |
| 278 | C2×C3 | — | 2 | 2 | no | — |
| 279 | C2×C3 | — | 4 | 4 | no | — |
| 280 | C2×C3 | C23×D4 | 1 | 1 | yes | characteristic |
| 281 | C2×C3 | C2×2−1+4 | 1 | 1 | yes | characteristic |
| 282 | C2×C3 | #585 | 1 | 1 | yes | characteristic |
| 283 | D3 | — | 4 | 4 | no | — |
| 284 | C22 | — | 2 | 1 | no | — |
| 285 | C22 | — | 2 | 1 | no | — |
| 286 | C22 | — | 2 | 4 | no | — |
| 287 | C22 | — | 2 | 4 | no | — |
| 288 | C22 | — | 2 | 4 | no | — |
| 289 | C22 | — | 6 | 4 | no | — |
| 290 | C22 | — | 6 | 4 | no | — |
| 291 | C22 | — | 6 | 4 | no | — |
| 292 | C22 | C2×D3×D4 | 1 | 2 | yes | — |
| 293 | C22 | C24×D3 | 1 | 1 | yes | characteristic |
| 294 | C4 | — | 2 | 4 | no | — |
| 295 | C4 | — | 2 | 4 | no | — |
| 296 | C4 | — | 3 | 8 | no | — |
| 297 | C4 | — | 3 | 2 | no | — |
| 298 | C4 | — | 6 | 4 | no | — |
| 299 | C4 | — | 6 | 2 | no | — |
| 300 | C4 | — | 6 | 1 | no | — |
| 301 | C4 | C2×D3×D4 | 1 | 4 | yes | — |
| 302 | C3 | #3497 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 303 | C2 | — | 2 | 2 | no | — |
| 304 | C2 | — | 4 | 4 | no | — |
| 305 | C2 | — | 12 | 4 | no | — |
| 306 | C2 | C22×D3×D4 | 1 | 1 | yes | characteristic |
| 307 | C2 | #5966 | 1 | 1 | yes | characteristic |
| 308 | C2 | D3×2−1+4 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D3 |
| commutator, socle, cosocle | C22×C3 | C25 |
| Fitting | C3×#537 | C2 |
| hypercenter | #537 | D3 |
| hypocenter | C3 | #3497 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ #537 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#1010 | C2 | semidirect |
| 2. C3×#537 | C2 | semidirect |
| 3. #5832 | C2 | semidirect |
| 4. #5633 | C2 | semidirect |
| 5. #5632 | C2 | semidirect |
| 6. #5599 | C2 | semidirect |
| 7. #5598 | C2 | semidirect |
| 8. #5597 | C2 | semidirect |
| 9. #5596 | C2 | semidirect |
| 10. #5510 | C2 | semidirect |
| 11. C2×#887 | C2 | semidirect |
| 59. #537 | D3 | semidirect |
| 89. C2×Dic6 | D4 | semidirect |
| 244. Dic3 | 2−1+4 | semidirect |
| 302. C3 | #3497 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#1010 | 2 |
| C3×#537 | 2 |
| #5832 | 2 |
| #5633 | 2 |
| #5632 | 2 |
| #5599 | 2 |
| #5598 | 2 |
| #5597 | 2 |
| #5596 | 2 |
| #5510 | 2 |
| C2×#887 | 2 |
| #3497 | 3 |