Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
804 conjugacy classes of subgroups; only the 149 normal subgroups are shown.
This group has 804 conjugacy classes of subgroups — too many to draw. Only its 149 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×#939 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | C2×#938 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | C2×#884 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | C3×#392 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #5223 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #4960 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #4948 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | #4946 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 9 | #4668 | C2 | 1 | 2 | yes | maximal, complemented |
| 10 | #2675 | — | 3 | 1 | no | maximal, Sylow |
| 11 | C22×C3×D4 | C22 | 1 | 2 | yes | — |
| 12 | C22×C3×D4 | C22 | 1 | 1 | yes | characteristic |
| 13 | C22×#67 | C22 | 1 | 2 | yes | — |
| 14 | C22×#67 | C22 | 1 | 2 | yes | — |
| 15 | C22×#67 | C22 | 1 | 2 | yes | — |
| 16 | C23×Dic3 | C22 | 1 | 1 | yes | characteristic |
| 17 | C22×D12 | C22 | 1 | 1 | yes | characteristic |
| 18 | C22×C4×D3 | C22 | 1 | 2 | yes | — |
| 19 | C2×C3×C22⋊C4 | C22 | 1 | 2 | yes | — |
| 20 | C2×C3×C22⋊C4 | C22 | 1 | 1 | yes | characteristic |
| 21 | C2×#217 | C22 | 1 | 2 | yes | — |
| 22 | C2×#217 | C22 | 1 | 2 | yes | — |
| 23 | C2×#217 | C22 | 1 | 1 | yes | characteristic |
| 24 | #939 | — | 2 | 8 | no | — |
| 25 | #938 | — | 2 | 8 | no | — |
| 26 | C2×(C2×D3)⋊C4 | C22 | 1 | 2 | yes | — |
| 27 | C2×(C2×D3)⋊C4 | C22 | 1 | 2 | yes | — |
| 28 | C2×C4⋊Dic3 | C22 | 1 | 2 | yes | — |
| 29 | C2×Dic3⋊C4 | C22 | 1 | 2 | yes | — |
| 30 | C2×Dic3⋊C4 | C22 | 1 | 2 | yes | — |
| 31 | #884 | — | 2 | 8 | no | — |
| 32 | C3×#95 | C22 | 1 | 1 | yes | characteristic |
| 33 | C3⋊#95 | C22 | 1 | 2 | yes | — |
| 34 | C3⋊#95 | C22 | 1 | 1 | yes | characteristic |
| 35 | C2×#121 | — | 3 | 2 | no | — |
| 36 | C2×#121 | — | 3 | 2 | no | — |
| 37 | C2×#121 | — | 3 | 2 | no | — |
| 38 | #396 | — | 3 | 2 | no | — |
| 39 | #392 | — | 3 | 2 | no | — |
| 40 | #392 | — | 3 | 1 | no | — |
| 41 | #392 | D3 | 1 | 1 | yes | characteristic, complemented |
| 42 | #385 | — | 3 | 1 | no | — |
| 43 | #385 | — | 3 | 2 | no | — |
| 44 | C24×C3 | C23 | 1 | 2 | yes | — |
| 45 | C24×C3 | C23 | 1 | 1 | yes | characteristic |
| 46 | C23×D3 | C23 | 1 | 2 | yes | — |
| 47 | C2×C3×D4 | — | 2 | 4 | no | — |
| 48 | C2×C3×D4 | — | 2 | 4 | no | — |
| 49 | C2×C3×D4 | — | 4 | 2 | no | — |
| 50 | C2×C3×D4 | — | 4 | 2 | no | — |
| 51 | C2×C3×D4 | — | 4 | 1 | no | — |
| 52 | C22×C3×C4 | C23 | 1 | 1 | yes | characteristic |
| 53 | C22×C3×C4 | C23 | 1 | 1 | yes | characteristic |
| 54 | C22×C3×C4 | C23 | 1 | 2 | yes | — |
| 55 | C2×#67 | — | 2 | 4 | no | — |
| 56 | C2×#67 | — | 2 | 4 | no | — |
| 57 | C2×#67 | — | 2 | 4 | no | — |
| 58 | C2×#67 | — | 2 | 4 | no | — |
| 59 | C2×#67 | — | 2 | 4 | no | — |
| 60 | C2×#67 | — | 2 | 4 | no | — |
| 61 | C2×#67 | — | 4 | 2 | no | — |
| 62 | C2×#67 | — | 4 | 2 | no | — |
| 63 | C2×#67 | — | 4 | 2 | no | — |
| 64 | C22×Dic3 | — | 2 | 4 | no | — |
| 65 | C22×Dic3 | — | 4 | 1 | no | — |
| 66 | C22×Dic3 | C23 | 1 | 2 | yes | — |
| 67 | C22×Dic3 | C23 | 1 | 2 | yes | — |
| 68 | C22×Dic3 | C23 | 1 | 2 | yes | — |
| 69 | C2×D12 | — | 2 | 4 | no | — |
| 70 | C2×D12 | — | 4 | 1 | no | — |
| 71 | C2×C4×D3 | — | 2 | 4 | no | — |
| 72 | C2×C4×D3 | — | 2 | 4 | no | — |
| 73 | C2×C4×D3 | — | 4 | 2 | no | — |
| 74 | C3×C22⋊C4 | — | 2 | 4 | no | — |
| 75 | C3×C22⋊C4 | — | 4 | 1 | no | — |
| 76 | #217 | — | 2 | 4 | no | — |
| 77 | #217 | — | 2 | 4 | no | — |
| 78 | #217 | D4 | 1 | 4 | yes | complemented |
| 79 | (C2×D3)⋊C4 | — | 4 | 2 | no | — |
| 80 | (C2×D3)⋊C4 | D4 | 1 | 8 | yes | complemented |
| 81 | C4⋊Dic3 | — | 2 | 4 | no | — |
| 82 | Dic3⋊C4 | — | 2 | 4 | no | — |
| 83 | Dic3⋊C4 | — | 2 | 4 | no | — |
| 84 | C22×D4 | — | 3 | 2 | no | — |
| 85 | C22×D4 | — | 3 | 2 | no | — |
| 86 | C22×D4 | — | 3 | 2 | no | — |
| 87 | C22×D4 | — | 3 | 1 | no | — |
| 88 | C22×D4 | C2×D3 | 1 | 2 | yes | — |
| 89 | C22×D4 | C2×D3 | 1 | 1 | yes | characteristic |
| 90 | C23×C4 | — | 3 | 1 | no | — |
| 91 | C23×C4 | — | 3 | 2 | no | — |
| 92 | #121 | — | 6 | 8 | no | — |
| 93 | #121 | — | 6 | 8 | no | — |
| 94 | #121 | — | 6 | 8 | no | — |
| 95 | C2×C4⋊C4 | — | 3 | 2 | no | — |
| 96 | C2×C4⋊C4 | — | 3 | 2 | no | — |
| 97 | C2×C4⋊C4 | — | 3 | 2 | no | — |
| 98 | C2×C22⋊C4 | — | 3 | 1 | no | — |
| 99 | C2×C22⋊C4 | — | 3 | 2 | no | — |
| 100 | C2×C22⋊C4 | — | 3 | 2 | no | — |
| 101 | C2×C22⋊C4 | — | 3 | 2 | no | — |
| 102 | C2×C22⋊C4 | — | 3 | 2 | no | — |
| 103 | C2×C22⋊C4 | C2×D3 | 1 | 2 | yes | — |
| 104 | C2×C22⋊C4 | C2×D3 | 1 | 1 | yes | characteristic |
| 105 | #95 | — | 3 | 2 | no | — |
| 106 | #95 | — | 3 | 1 | no | — |
| 107 | #95 | C2×D3 | 1 | 1 | yes | characteristic |
| 108 | C23×C3 | — | 2 | 2 | no | — |
| 109 | C23×C3 | — | 2 | 1 | no | — |
| 110 | C23×C3 | — | 2 | 2 | no | — |
| 111 | C23×C3 | — | 2 | 1 | no | — |
| 112 | C23×C3 | — | 2 | 2 | no | — |
| 113 | C23×C3 | — | 2 | 2 | no | — |
| 114 | C23×C3 | — | 2 | 2 | no | — |
| 115 | C23×C3 | — | 2 | 2 | no | — |
| 116 | C23×C3 | — | 2 | 2 | no | — |
| 117 | C23×C3 | — | 2 | 2 | no | — |
| 118 | C23×C3 | — | 2 | 2 | no | — |
| 119 | C23×C3 | C2×D4 | 1 | 2 | yes | — |
| 120 | C23×C3 | C24 | 1 | 1 | yes | characteristic |
| 121 | C22×D3 | — | 2 | 2 | no | — |
| 122 | C22×D3 | — | 2 | 2 | no | — |
| 123 | C22×D3 | — | 2 | 2 | no | — |
| 124 | C22×D3 | — | 2 | 2 | no | — |
| 125 | C22×D3 | — | 2 | 2 | no | — |
| 126 | C22×D3 | — | 2 | 2 | no | — |
| 127 | C22×D3 | C2×D4 | 1 | 4 | yes | — |
| 128 | C3×D4 | — | 4 | 8 | no | — |
| 129 | C3×D4 | — | 4 | 4 | no | — |
| 130 | C2×C3×C4 | — | 2 | 2 | no | — |
| 131 | C2×C3×C4 | — | 2 | 2 | no | — |
| 132 | C2×C3×C4 | — | 2 | 2 | no | — |
| 133 | C2×C3×C4 | — | 2 | 2 | no | — |
| 134 | C2×C3×C4 | — | 2 | 1 | no | — |
| 135 | C2×C3×C4 | — | 2 | 1 | no | — |
| 136 | C2×C3×C4 | C2×D4 | 1 | 4 | yes | — |
| 137 | #67 | — | 4 | 8 | no | — |
| 138 | #67 | — | 4 | 8 | no | — |
| 139 | #67 | — | 4 | 8 | no | — |
| 140 | C2×Dic3 | — | 2 | 2 | no | — |
| 141 | C2×Dic3 | — | 2 | 2 | no | — |
| 142 | C2×Dic3 | — | 2 | 2 | no | — |
| 143 | C2×Dic3 | — | 2 | 2 | no | — |
| 144 | C2×Dic3 | — | 2 | 2 | no | — |
| 145 | C2×Dic3 | — | 2 | 2 | no | — |
| 146 | C2×Dic3 | — | 2 | 2 | no | — |
| 147 | C2×Dic3 | — | 4 | 4 | no | — |
| 148 | C2×Dic3 | C2×D4 | 1 | 4 | yes | — |
| 149 | C2×Dic3 | C2×D4 | 1 | 4 | yes | — |
| 150 | D12 | — | 4 | 4 | no | — |
| 151 | C4×D3 | — | 4 | 8 | no | — |
| 152 | C24 | — | 3 | 2 | no | — |
| 153 | C24 | C22×D3 | 1 | 2 | yes | — |
| 154 | C24 | C22×D3 | 1 | 1 | yes | characteristic |
| 155 | C2×D4 | — | 2 | 4 | no | — |
| 156 | C2×D4 | — | 2 | 4 | no | — |
| 157 | C2×D4 | — | 4 | 2 | no | — |
| 158 | C2×D4 | — | 4 | 1 | no | — |
| 159 | C2×D4 | — | 4 | 2 | no | — |
| 160 | C2×D4 | — | 6 | 4 | no | — |
| 161 | C2×D4 | — | 6 | 4 | no | — |
| 162 | C2×D4 | — | 6 | 4 | no | — |
| 163 | C2×D4 | — | 6 | 4 | no | — |
| 164 | C2×D4 | — | 6 | 4 | no | — |
| 165 | C2×D4 | — | 6 | 4 | no | — |
| 166 | C2×D4 | — | 6 | 4 | no | — |
| 167 | C2×D4 | — | 12 | 1 | no | — |
| 168 | C2×D4 | — | 12 | 2 | no | — |
| 169 | C2×D4 | — | 12 | 2 | no | — |
| 170 | C2×D4 | — | 12 | 2 | no | — |
| 171 | C22×C4 | — | 3 | 2 | no | — |
| 172 | C22×C4 | — | 3 | 2 | no | — |
| 173 | C22×C4 | — | 3 | 2 | no | — |
| 174 | C22×C4 | — | 6 | 4 | no | — |
| 175 | C22×C4 | — | 6 | 4 | no | — |
| 176 | C22×C4 | — | 6 | 4 | no | — |
| 177 | C22×C4 | — | 12 | 2 | no | — |
| 178 | C22×C4 | — | 12 | 1 | no | — |
| 179 | C22×C4 | C22×D3 | 1 | 1 | yes | characteristic |
| 180 | C22×C4 | C22×D3 | 1 | 1 | yes | characteristic |
| 181 | C22×C4 | C22×D3 | 1 | 2 | yes | — |
| 182 | C4⋊C4 | — | 6 | 4 | no | — |
| 183 | C4⋊C4 | — | 6 | 4 | no | — |
| 184 | C4⋊C4 | — | 6 | 4 | no | — |
| 185 | C22⋊C4 | — | 2 | 4 | no | — |
| 186 | C22⋊C4 | — | 3 | 4 | no | — |
| 187 | C22⋊C4 | — | 3 | 8 | no | — |
| 188 | C22⋊C4 | — | 4 | 1 | no | — |
| 189 | C22⋊C4 | — | 6 | 4 | no | — |
| 190 | C22⋊C4 | — | 6 | 4 | no | — |
| 191 | C22⋊C4 | — | 12 | 2 | no | — |
| 192 | C22×C3 | — | 2 | 2 | no | — |
| 193 | C22×C3 | — | 2 | 4 | no | — |
| 194 | C22×C3 | — | 4 | 2 | no | — |
| 195 | C22×C3 | — | 4 | 1 | no | — |
| 196 | C22×C3 | — | 4 | 1 | no | — |
| 197 | C22×C3 | — | 4 | 2 | no | — |
| 198 | C22×C3 | — | 4 | 2 | no | — |
| 199 | C22×C3 | — | 4 | 2 | no | — |
| 200 | C22×C3 | — | 4 | 2 | no | — |
| 201 | C22×C3 | — | 4 | 2 | no | — |
| 202 | C22×C3 | — | 4 | 2 | no | — |
| 203 | C22×C3 | — | 4 | 2 | no | — |
| 204 | C22×C3 | C22×D4 | 1 | 2 | yes | — |
| 205 | C22×C3 | C22×D4 | 1 | 1 | yes | characteristic |
| 206 | C22×C3 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 207 | C22×C3 | H(5,𝔽2) | 1 | 2 | yes | — |
| 208 | C22×C3 | 2−1+4 | 1 | 1 | yes | characteristic |
| 209 | C2×D3 | — | 2 | 4 | no | — |
| 210 | C2×D3 | — | 2 | 4 | no | — |
| 211 | C2×D3 | — | 2 | 4 | no | — |
| 212 | C2×D3 | — | 4 | 2 | no | — |
| 213 | C2×D3 | — | 4 | 2 | no | — |
| 214 | C2×D3 | — | 4 | 2 | no | — |
| 215 | C2×D3 | — | 4 | 2 | no | — |
| 216 | C3×C4 | — | 2 | 4 | no | — |
| 217 | C3×C4 | — | 4 | 1 | no | — |
| 218 | C3×C4 | — | 4 | 1 | no | — |
| 219 | Dic3 | — | 2 | 4 | no | — |
| 220 | Dic3 | — | 2 | 4 | no | — |
| 221 | Dic3 | — | 4 | 2 | no | — |
| 222 | C23 | — | 2 | 2 | no | — |
| 223 | C23 | — | 2 | 1 | no | — |
| 224 | C23 | — | 2 | 2 | no | — |
| 225 | C23 | — | 2 | 1 | no | — |
| 226 | C23 | — | 2 | 2 | no | — |
| 227 | C23 | — | 2 | 2 | no | — |
| 228 | C23 | — | 2 | 2 | no | — |
| 229 | C23 | — | 2 | 2 | no | — |
| 230 | C23 | — | 2 | 2 | no | — |
| 231 | C23 | — | 2 | 2 | no | — |
| 232 | C23 | — | 2 | 2 | no | — |
| 233 | C23 | — | 3 | 4 | no | — |
| 234 | C23 | — | 6 | 2 | no | — |
| 235 | C23 | — | 6 | 2 | no | — |
| 236 | C23 | — | 6 | 2 | no | — |
| 237 | C23 | — | 6 | 2 | no | — |
| 238 | C23 | — | 6 | 2 | no | — |
| 239 | C23 | — | 6 | 2 | no | — |
| 240 | C23 | D3×D4 | 1 | 2 | yes | — |
| 241 | C23 | C23×D3 | 1 | 1 | yes | characteristic |
| 242 | D4 | — | 4 | 8 | no | — |
| 243 | D4 | — | 4 | 4 | no | — |
| 244 | D4 | — | 12 | 4 | no | — |
| 245 | D4 | — | 12 | 8 | no | — |
| 246 | D4 | — | 12 | 8 | no | — |
| 247 | D4 | — | 12 | 8 | no | — |
| 248 | C2×C4 | — | 2 | 2 | no | — |
| 249 | C2×C4 | — | 2 | 2 | no | — |
| 250 | C2×C4 | — | 2 | 2 | no | — |
| 251 | C2×C4 | — | 2 | 1 | no | — |
| 252 | C2×C4 | — | 2 | 1 | no | — |
| 253 | C2×C4 | — | 2 | 2 | no | — |
| 254 | C2×C4 | — | 3 | 4 | no | — |
| 255 | C2×C4 | — | 3 | 4 | no | — |
| 256 | C2×C4 | — | 6 | 2 | no | — |
| 257 | C2×C4 | — | 6 | 2 | no | — |
| 258 | C2×C4 | — | 6 | 2 | no | — |
| 259 | C2×C4 | — | 6 | 2 | no | — |
| 260 | C2×C4 | — | 6 | 2 | no | — |
| 261 | C2×C4 | — | 6 | 2 | no | — |
| 262 | C2×C4 | — | 6 | 2 | no | — |
| 263 | C2×C4 | — | 12 | 8 | no | — |
| 264 | C2×C4 | — | 12 | 4 | no | — |
| 265 | C2×C4 | D3×D4 | 1 | 4 | yes | — |
| 266 | C2×C3 | — | 4 | 2 | no | — |
| 267 | C2×C3 | — | 8 | 2 | no | — |
| 268 | C2×C3 | #537 | 1 | 1 | yes | characteristic |
| 269 | C2×C3 | #537 | 1 | 2 | yes | — |
| 270 | C2×C3 | D42 | 1 | 1 | yes | characteristic |
| 271 | C2×C3 | D42 | 1 | 2 | yes | — |
| 272 | C2×C3 | #560 | 1 | 1 | yes | characteristic |
| 273 | D3 | — | 4 | 4 | no | — |
| 274 | C22 | — | 2 | 4 | no | — |
| 275 | C22 | — | 2 | 2 | no | — |
| 276 | C22 | — | 4 | 1 | no | — |
| 277 | C22 | — | 4 | 1 | no | — |
| 278 | C22 | — | 4 | 2 | no | — |
| 279 | C22 | — | 4 | 2 | no | — |
| 280 | C22 | — | 4 | 2 | no | — |
| 281 | C22 | — | 4 | 2 | no | — |
| 282 | C22 | — | 4 | 2 | no | — |
| 283 | C22 | — | 4 | 2 | no | — |
| 284 | C22 | — | 4 | 2 | no | — |
| 285 | C22 | — | 4 | 2 | no | — |
| 286 | C22 | — | 6 | 4 | no | — |
| 287 | C22 | — | 6 | 4 | no | — |
| 288 | C22 | — | 6 | 4 | no | — |
| 289 | C22 | — | 12 | 2 | no | — |
| 290 | C22 | — | 12 | 2 | no | — |
| 291 | C22 | — | 12 | 2 | no | — |
| 292 | C22 | — | 12 | 2 | no | — |
| 293 | C22 | C2×D3×D4 | 1 | 2 | yes | — |
| 294 | C22 | C2×D3×D4 | 1 | 1 | yes | characteristic |
| 295 | C22 | #1004 | 1 | 1 | yes | characteristic |
| 296 | C22 | #1004 | 1 | 2 | yes | — |
| 297 | C22 | #1010 | 1 | 1 | yes | characteristic |
| 298 | C4 | — | 2 | 4 | no | — |
| 299 | C4 | — | 4 | 1 | no | — |
| 300 | C4 | — | 4 | 1 | no | — |
| 301 | C4 | — | 6 | 4 | no | — |
| 302 | C4 | — | 6 | 4 | no | — |
| 303 | C4 | — | 12 | 2 | no | — |
| 304 | C3 | #2675 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 305 | C2 | — | 4 | 2 | no | — |
| 306 | C2 | — | 8 | 2 | no | — |
| 307 | C2 | — | 12 | 4 | no | — |
| 308 | C2 | #5590 | 1 | 2 | yes | — |
| 309 | C2 | #5595 | 1 | 1 | yes | characteristic |
| 310 | C2 | #5597 | 1 | 1 | yes | characteristic |
| 311 | C2 | #5608 | 1 | 2 | yes | — |
| 312 | C2 | #5717 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C3×#392 | C2 |
| hypercenter | #392 | D3 |
| hypocenter | C3 | #2675 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #392 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#939 | C2 | semidirect |
| 2. C2×#938 | C2 | semidirect |
| 3. C2×#884 | C2 | semidirect |
| 4. C3×#392 | C2 | semidirect |
| 5. #5223 | C2 | semidirect |
| 6. #4960 | C2 | semidirect |
| 7. #4948 | C2 | semidirect |
| 8. #4946 | C2 | semidirect |
| 9. #4668 | C2 | semidirect |
| 41. #392 | D3 | semidirect |
| 78. #217 | D4 | semidirect |
| 80. (C2×D3)⋊C4 | D4 | semidirect |
| 304. C3 | #2675 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#939 | 2 |
| C2×#938 | 2 |
| C2×#884 | 2 |
| C3×#392 | 2 |
| #5223 | 2 |
| #4960 | 2 |
| #4948 | 2 |
| #4946 | 2 |
| #4668 | 2 |
| #2675 | 3 |