Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
1018 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 1018 conjugacy classes of subgroups — too many to draw. Only its 135 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 | D3×H(5,𝔽2) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | D3×#120 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | C3×B(4,𝔽2) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #5244 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #5032 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | D3×#99 | C2 | 1 | 3 | yes | maximal, complemented |
| 7 | #4741 | C2 | 1 | 3 | yes | maximal, complemented |
| 8 | C2×B(4,𝔽2) | — | 3 | 1 | no | maximal, Sylow |
| 9 | C24×D3 | C22 | 1 | 1 | yes | characteristic, complemented |
| 10 | C3×H(5,𝔽2) | C22 | 1 | 1 | yes | characteristic, complemented |
| 11 | #1009 | — | 2 | 3 | no | — |
| 12 | D3×#41 | — | 2 | 3 | no | — |
| 13 | #1004 | — | 2 | 3 | no | — |
| 14 | #1004 | C22 | 1 | 3 | yes | complemented |
| 15 | C2×D3×D4 | — | 2 | 3 | no | — |
| 16 | C2×D3×D4 | — | 2 | 3 | no | — |
| 17 | C2×D3×D4 | C22 | 1 | 3 | yes | — |
| 18 | C3×#120 | C22 | 1 | 3 | yes | complemented |
| 19 | #953 | C22 | 1 | 3 | yes | complemented |
| 20 | #937 | — | 2 | 3 | no | — |
| 21 | #937 | C22 | 1 | 3 | yes | complemented |
| 22 | #882 | — | 2 | 3 | no | — |
| 23 | #882 | C22 | 1 | 3 | yes | complemented |
| 24 | D3×C22⋊C4 | — | 2 | 3 | no | — |
| 25 | D3×C22⋊C4 | C22 | 1 | 3 | yes | complemented |
| 26 | C3×#99 | C22 | 1 | 3 | yes | complemented |
| 27 | #834 | C22 | 1 | 3 | yes | complemented |
| 28 | #806 | C22 | 1 | 6 | yes | complemented |
| 29 | C2×H(5,𝔽2) | — | 3 | 1 | no | — |
| 30 | C2×#120 | — | 3 | 3 | no | — |
| 31 | B(4,𝔽2) | — | 3 | 1 | no | — |
| 32 | B(4,𝔽2) | — | 3 | 3 | no | — |
| 33 | B(4,𝔽2) | — | 3 | 3 | no | — |
| 34 | B(4,𝔽2) | D3 | 1 | 1 | yes | characteristic, direct product factor |
| 35 | C2×#99 | — | 3 | 3 | no | — |
| 36 | C24×C3 | C23 | 1 | 1 | yes | characteristic, complemented |
| 37 | C23×D3 | — | 2 | 1 | no | — |
| 38 | C23×D3 | — | 2 | 3 | no | — |
| 39 | C23×D3 | — | 2 | 3 | no | — |
| 40 | C23×D3 | — | 4 | 4 | no | — |
| 41 | C23×D3 | D4 | 1 | 3 | yes | complemented |
| 42 | C23×D3 | C23 | 1 | 1 | yes | characteristic |
| 43 | C23×D3 | C23 | 1 | 1 | yes | characteristic, complemented |
| 44 | C3×#41 | — | 2 | 3 | no | — |
| 45 | C2×C3×D4 | — | 2 | 3 | no | — |
| 46 | C2×C3×D4 | — | 2 | 3 | no | — |
| 47 | C2×C3×D4 | C23 | 1 | 3 | yes | — |
| 48 | C2×#67 | — | 2 | 3 | no | — |
| 49 | C2×#67 | — | 2 | 3 | no | — |
| 50 | C2×#67 | — | 2 | 3 | no | — |
| 51 | C2×#67 | — | 2 | 3 | no | — |
| 52 | C2×#67 | D4 | 1 | 3 | yes | complemented |
| 53 | C2×#67 | C23 | 1 | 3 | yes | — |
| 54 | #239 | — | 2 | 3 | no | — |
| 55 | D3×Q8 | — | 2 | 1 | no | — |
| 56 | #237 | — | 2 | 3 | no | — |
| 57 | #237 | — | 2 | 6 | no | — |
| 58 | D3×D4 | — | 2 | 3 | no | — |
| 59 | D3×D4 | — | 2 | 6 | no | — |
| 60 | D3×D4 | — | 2 | 6 | no | — |
| 61 | D3×D4 | — | 2 | 6 | no | — |
| 62 | D3×D4 | — | 2 | 6 | no | — |
| 63 | D3×D4 | — | 4 | 12 | no | — |
| 64 | #235 | — | 2 | 6 | no | — |
| 65 | #235 | — | 2 | 3 | no | — |
| 66 | C2×D12 | — | 2 | 3 | no | — |
| 67 | C2×D12 | — | 2 | 3 | no | — |
| 68 | C2×D12 | D4 | 1 | 3 | yes | complemented |
| 69 | C2×C4×D3 | — | 2 | 3 | no | — |
| 70 | C2×C4×D3 | — | 2 | 3 | no | — |
| 71 | C2×C4×D3 | D4 | 1 | 3 | yes | complemented |
| 72 | C3×C22⋊C4 | — | 2 | 3 | no | — |
| 73 | C3×C22⋊C4 | C23 | 1 | 3 | yes | complemented |
| 74 | #217 | — | 2 | 3 | no | — |
| 75 | #217 | C23 | 1 | 3 | yes | complemented |
| 76 | (C2×D3)⋊C4 | — | 2 | 3 | no | — |
| 77 | (C2×D3)⋊C4 | — | 2 | 3 | no | — |
| 78 | (C2×D3)⋊C4 | — | 2 | 3 | no | — |
| 79 | C25 | — | 3 | 1 | no | — |
| 80 | H(5,𝔽2) | — | 3 | 3 | no | — |
| 81 | H(5,𝔽2) | — | 6 | 3 | no | — |
| 82 | H(5,𝔽2) | — | 6 | 3 | no | — |
| 83 | H(5,𝔽2) | C2×D3 | 1 | 1 | yes | characteristic, complemented |
| 84 | C2×#41 | — | 6 | 3 | no | — |
| 85 | C22×D4 | — | 3 | 3 | no | — |
| 86 | C22×D4 | — | 6 | 3 | no | — |
| 87 | C22×D4 | — | 6 | 3 | no | — |
| 88 | #120 | — | 3 | 3 | no | — |
| 89 | #120 | — | 3 | 3 | no | — |
| 90 | #120 | — | 3 | 3 | no | — |
| 91 | #120 | — | 6 | 3 | no | — |
| 92 | #120 | — | 6 | 3 | no | — |
| 93 | #120 | C2×D3 | 1 | 3 | yes | complemented |
| 94 | C2×C22⋊C4 | — | 3 | 3 | no | — |
| 95 | C2×C22⋊C4 | — | 6 | 3 | no | — |
| 96 | #99 | — | 3 | 3 | no | — |
| 97 | #99 | — | 3 | 6 | no | — |
| 98 | #99 | C2×D3 | 1 | 3 | yes | complemented |
| 99 | C23×C3 | — | 2 | 1 | no | — |
| 100 | C23×C3 | — | 2 | 3 | no | — |
| 101 | C23×C3 | — | 4 | 2 | no | — |
| 102 | C23×C3 | C2×D4 | 1 | 3 | yes | complemented |
| 103 | C23×C3 | C24 | 1 | 1 | yes | characteristic |
| 104 | C22×D3 | — | 2 | 12 | no | — |
| 105 | C22×D3 | — | 2 | 3 | no | — |
| 106 | C22×D3 | — | 2 | 3 | no | — |
| 107 | C22×D3 | — | 2 | 3 | no | — |
| 108 | C22×D3 | — | 2 | 3 | no | — |
| 109 | C22×D3 | — | 4 | 2 | no | — |
| 110 | C22×D3 | — | 4 | 2 | no | — |
| 111 | C22×D3 | — | 4 | 6 | no | — |
| 112 | C22×D3 | — | 4 | 6 | no | — |
| 113 | C22×D3 | — | 4 | 6 | no | — |
| 114 | C22×D3 | — | 4 | 6 | no | — |
| 115 | C22×D3 | — | 4 | 3 | no | — |
| 116 | C22×D3 | — | 4 | 1 | no | — |
| 117 | C22×D3 | — | 8 | 2 | no | — |
| 118 | C22×D3 | C2×D4 | 1 | 3 | yes | — |
| 119 | C22×D3 | C2×D4 | 1 | 3 | yes | — |
| 120 | C22×D3 | C2×D4 | 1 | 3 | yes | complemented |
| 121 | C3×Q8 | — | 2 | 1 | no | — |
| 122 | C3×D4 | — | 2 | 3 | no | — |
| 123 | C3×D4 | — | 2 | 6 | no | — |
| 124 | C3×D4 | — | 4 | 6 | no | — |
| 125 | C2×C3×C4 | — | 2 | 3 | no | — |
| 126 | C2×C3×C4 | — | 2 | 3 | no | — |
| 127 | C2×C3×C4 | C2×D4 | 1 | 3 | yes | complemented |
| 128 | #67 | — | 2 | 6 | no | — |
| 129 | #67 | — | 2 | 6 | no | — |
| 130 | #67 | — | 2 | 6 | no | — |
| 131 | #67 | — | 4 | 6 | no | — |
| 132 | #67 | — | 4 | 6 | no | — |
| 133 | C2×Dic3 | — | 2 | 3 | no | — |
| 134 | C2×Dic3 | — | 2 | 3 | no | — |
| 135 | C2×Dic3 | C2×D4 | 1 | 3 | yes | complemented |
| 136 | D12 | — | 2 | 3 | no | — |
| 137 | D12 | — | 2 | 6 | no | — |
| 138 | D12 | — | 4 | 6 | no | — |
| 139 | C4×D3 | — | 2 | 6 | no | — |
| 140 | C4×D3 | — | 2 | 3 | no | — |
| 141 | C4×D3 | — | 2 | 3 | no | — |
| 142 | C4×D3 | — | 4 | 6 | no | — |
| 143 | Dic6 | — | 2 | 3 | no | — |
| 144 | C24 | — | 3 | 1 | no | — |
| 145 | C24 | — | 3 | 3 | no | — |
| 146 | C24 | — | 3 | 1 | no | — |
| 147 | C24 | — | 6 | 3 | no | — |
| 148 | C24 | — | 6 | 3 | no | — |
| 149 | C24 | — | 6 | 1 | no | — |
| 150 | C24 | — | 12 | 4 | no | — |
| 151 | C24 | C22×D3 | 1 | 1 | yes | characteristic, complemented |
| 152 | #41 | — | 2 | 3 | no | — |
| 153 | #41 | — | 6 | 6 | no | — |
| 154 | #41 | — | 6 | 6 | no | — |
| 155 | #41 | — | 6 | 3 | no | — |
| 156 | #41 | — | 6 | 3 | no | — |
| 157 | #41 | — | 6 | 3 | no | — |
| 158 | C2×Q8 | — | 6 | 1 | no | — |
| 159 | C2×D4 | — | 2 | 3 | no | — |
| 160 | C2×D4 | — | 2 | 3 | no | — |
| 161 | C2×D4 | — | 3 | 3 | no | — |
| 162 | C2×D4 | — | 3 | 3 | no | — |
| 163 | C2×D4 | — | 3 | 3 | no | — |
| 164 | C2×D4 | — | 6 | 3 | no | — |
| 165 | C2×D4 | — | 6 | 3 | no | — |
| 166 | C2×D4 | — | 6 | 6 | no | — |
| 167 | C2×D4 | — | 6 | 6 | no | — |
| 168 | C2×D4 | — | 6 | 6 | no | — |
| 169 | C2×D4 | — | 6 | 3 | no | — |
| 170 | C2×D4 | — | 6 | 6 | no | — |
| 171 | C2×D4 | — | 6 | 3 | no | — |
| 172 | C2×D4 | — | 6 | 3 | no | — |
| 173 | C2×D4 | — | 6 | 3 | no | — |
| 174 | C2×D4 | — | 6 | 3 | no | — |
| 175 | C2×D4 | — | 12 | 12 | no | — |
| 176 | C2×D4 | C22×D3 | 1 | 3 | yes | — |
| 177 | C22×C4 | — | 3 | 3 | no | — |
| 178 | C22×C4 | — | 6 | 3 | no | — |
| 179 | C22×C4 | — | 6 | 3 | no | — |
| 180 | C22⋊C4 | — | 2 | 3 | no | — |
| 181 | C22⋊C4 | — | 3 | 3 | no | — |
| 182 | C22⋊C4 | — | 6 | 3 | no | — |
| 183 | C22⋊C4 | — | 6 | 3 | no | — |
| 184 | C22⋊C4 | — | 6 | 3 | no | — |
| 185 | C22⋊C4 | — | 6 | 3 | no | — |
| 186 | C22⋊C4 | C22×D3 | 1 | 3 | yes | complemented |
| 187 | C22×C3 | — | 2 | 6 | no | — |
| 188 | C22×C3 | — | 2 | 3 | no | — |
| 189 | C22×C3 | — | 4 | 1 | no | — |
| 190 | C22×C3 | — | 4 | 3 | no | — |
| 191 | C22×C3 | — | 4 | 3 | no | — |
| 192 | C22×C3 | — | 4 | 1 | no | — |
| 193 | C22×C3 | — | 8 | 1 | no | — |
| 194 | C22×C3 | C22×D4 | 1 | 3 | yes | — |
| 195 | C2×D3 | — | 2 | 6 | no | — |
| 196 | C2×D3 | — | 2 | 6 | no | — |
| 197 | C2×D3 | — | 2 | 3 | no | — |
| 198 | C2×D3 | — | 4 | 3 | no | — |
| 199 | C2×D3 | — | 4 | 12 | no | — |
| 200 | C2×D3 | — | 4 | 3 | no | — |
| 201 | C2×D3 | — | 4 | 1 | no | — |
| 202 | C2×D3 | — | 4 | 4 | no | — |
| 203 | C2×D3 | — | 4 | 6 | no | — |
| 204 | C2×D3 | — | 4 | 3 | no | — |
| 205 | C2×D3 | — | 8 | 3 | no | — |
| 206 | C2×D3 | — | 8 | 3 | no | — |
| 207 | C2×D3 | #120 | 1 | 3 | yes | — |
| 208 | C2×D3 | #120 | 1 | 1 | yes | characteristic |
| 209 | C3×C4 | — | 2 | 3 | no | — |
| 210 | C3×C4 | — | 4 | 3 | no | — |
| 211 | Dic3 | — | 2 | 3 | no | — |
| 212 | Dic3 | — | 4 | 3 | no | — |
| 213 | C23 | — | 2 | 1 | no | — |
| 214 | C23 | — | 2 | 3 | no | — |
| 215 | C23 | — | 3 | 3 | no | — |
| 216 | C23 | — | 3 | 3 | no | — |
| 217 | C23 | — | 3 | 3 | no | — |
| 218 | C23 | — | 4 | 2 | no | — |
| 219 | C23 | — | 6 | 3 | no | — |
| 220 | C23 | — | 6 | 3 | no | — |
| 221 | C23 | — | 6 | 12 | no | — |
| 222 | C23 | — | 6 | 3 | no | — |
| 223 | C23 | — | 6 | 3 | no | — |
| 224 | C23 | — | 12 | 6 | no | — |
| 225 | C23 | — | 12 | 1 | no | — |
| 226 | C23 | — | 12 | 3 | no | — |
| 227 | C23 | — | 12 | 2 | no | — |
| 228 | C23 | — | 12 | 2 | no | — |
| 229 | C23 | — | 12 | 6 | no | — |
| 230 | C23 | — | 12 | 6 | no | — |
| 231 | C23 | — | 12 | 6 | no | — |
| 232 | C23 | — | 24 | 2 | no | — |
| 233 | C23 | D3×D4 | 1 | 3 | yes | complemented |
| 234 | C23 | C23×D3 | 1 | 1 | yes | characteristic |
| 235 | Q8 | — | 2 | 1 | no | — |
| 236 | Q8 | — | 6 | 3 | no | — |
| 237 | D4 | — | 2 | 3 | no | — |
| 238 | D4 | — | 2 | 6 | no | — |
| 239 | D4 | — | 4 | 6 | no | — |
| 240 | D4 | — | 6 | 6 | no | — |
| 241 | D4 | — | 6 | 3 | no | — |
| 242 | D4 | — | 6 | 6 | no | — |
| 243 | D4 | — | 6 | 6 | no | — |
| 244 | D4 | — | 6 | 6 | no | — |
| 245 | D4 | — | 12 | 6 | no | — |
| 246 | D4 | — | 12 | 6 | no | — |
| 247 | D4 | — | 12 | 6 | no | — |
| 248 | C2×C4 | — | 2 | 3 | no | — |
| 249 | C2×C4 | — | 2 | 3 | no | — |
| 250 | C2×C4 | — | 3 | 3 | no | — |
| 251 | C2×C4 | — | 6 | 3 | no | — |
| 252 | C2×C4 | — | 6 | 3 | no | — |
| 253 | C2×C4 | — | 6 | 6 | no | — |
| 254 | C2×C4 | — | 6 | 3 | no | — |
| 255 | C2×C4 | — | 6 | 3 | no | — |
| 256 | C2×C4 | — | 12 | 6 | no | — |
| 257 | C2×C4 | D3×D4 | 1 | 3 | yes | complemented |
| 258 | C2×C3 | — | 2 | 3 | no | — |
| 259 | C2×C3 | — | 4 | 2 | no | — |
| 260 | C2×C3 | — | 4 | 3 | no | — |
| 261 | C2×C3 | C2×#120 | 1 | 1 | yes | characteristic |
| 262 | D3 | — | 2 | 3 | no | — |
| 263 | D3 | — | 4 | 2 | no | — |
| 264 | D3 | — | 4 | 3 | no | — |
| 265 | D3 | B(4,𝔽2) | 1 | 2 | yes | direct product factor |
| 266 | C22 | — | 2 | 6 | no | — |
| 267 | C22 | — | 2 | 3 | no | — |
| 268 | C22 | — | 3 | 1 | no | — |
| 269 | C22 | — | 3 | 3 | no | — |
| 270 | C22 | — | 4 | 1 | no | — |
| 271 | C22 | — | 4 | 3 | no | — |
| 272 | C22 | — | 4 | 3 | no | — |
| 273 | C22 | — | 4 | 1 | no | — |
| 274 | C22 | — | 6 | 6 | no | — |
| 275 | C22 | — | 6 | 6 | no | — |
| 276 | C22 | — | 6 | 3 | no | — |
| 277 | C22 | — | 8 | 1 | no | — |
| 278 | C22 | — | 12 | 3 | no | — |
| 279 | C22 | — | 12 | 3 | no | — |
| 280 | C22 | — | 12 | 3 | no | — |
| 281 | C22 | — | 12 | 4 | no | — |
| 282 | C22 | — | 12 | 6 | no | — |
| 283 | C22 | — | 12 | 12 | no | — |
| 284 | C22 | — | 12 | 1 | no | — |
| 285 | C22 | — | 24 | 3 | no | — |
| 286 | C22 | — | 24 | 3 | no | — |
| 287 | C22 | C2×D3×D4 | 1 | 3 | yes | — |
| 288 | C4 | — | 2 | 3 | no | — |
| 289 | C4 | — | 4 | 3 | no | — |
| 290 | C4 | — | 6 | 3 | no | — |
| 291 | C4 | — | 12 | 3 | no | — |
| 292 | C3 | C2×B(4,𝔽2) | 1 | 1 | yes | characteristic, complemented, Sylow |
| 293 | C2 | — | 2 | 3 | no | — |
| 294 | C2 | — | 3 | 2 | no | — |
| 295 | C2 | — | 4 | 2 | no | — |
| 296 | C2 | — | 4 | 3 | no | — |
| 297 | C2 | — | 6 | 3 | no | — |
| 298 | C2 | — | 12 | 2 | no | — |
| 299 | C2 | — | 12 | 3 | no | — |
| 300 | C2 | D3×#120 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. D3×H(5,𝔽2) | C2 | semidirect |
| 2. D3×#120 | C2 | semidirect |
| 3. C3×B(4,𝔽2) | C2 | semidirect |
| 4. #5244 | C2 | semidirect |
| 5. #5032 | C2 | semidirect |
| 6. D3×#99 | C2 | semidirect |
| 7. #4741 | C2 | semidirect |
| 9. C24×D3 | C22 | semidirect |
| 10. C3×H(5,𝔽2) | C22 | semidirect |
| 14. #1004 | C22 | semidirect |
| 18. C3×#120 | C22 | semidirect |
| 19. #953 | C22 | semidirect |
| 21. #937 | C22 | semidirect |
| 23. #882 | C22 | semidirect |
| 25. D3×C22⋊C4 | C22 | semidirect |
| 26. C3×#99 | C22 | semidirect |
| 27. #834 | C22 | semidirect |
| 28. #806 | C22 | semidirect |
| 34. B(4,𝔽2) | D3 | direct (N × H) |
| 36. C24×C3 | C23 | semidirect |
| 41. C23×D3 | D4 | semidirect |
| 43. C23×D3 | C23 | semidirect |
| 52. C2×#67 | D4 | semidirect |
| 68. C2×D12 | D4 | semidirect |
| 71. C2×C4×D3 | D4 | semidirect |
| 73. C3×C22⋊C4 | C23 | semidirect |
| 75. #217 | C23 | semidirect |
| 83. H(5,𝔽2) | C2×D3 | semidirect |
| 93. #120 | C2×D3 | semidirect |
| 98. #99 | C2×D3 | semidirect |
| 102. C23×C3 | C2×D4 | semidirect |
| 120. C22×D3 | C2×D4 | semidirect |
| 127. C2×C3×C4 | C2×D4 | semidirect |
| 135. C2×Dic3 | C2×D4 | semidirect |
| 151. C24 | C22×D3 | semidirect |
| 186. C22⋊C4 | C22×D3 | semidirect |
| 233. C23 | D3×D4 | semidirect |
| 257. C2×C4 | D3×D4 | semidirect |
| 265. D3 | B(4,𝔽2) | direct (N × H) |
| 292. C3 | C2×B(4,𝔽2) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| D3×H(5,𝔽2) | 2 |
| D3×#120 | 2 |
| C3×B(4,𝔽2) | 2 |
| #5244 | 2 |
| #5032 | 2 |
| D3×#99 | 2 |
| #4741 | 2 |
| C2×B(4,𝔽2) | 3 |