Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
354 conjugacy classes of subgroups; only the 41 normal subgroups are shown.
This group has 354 conjugacy classes of subgroups — too many to draw. Only its 41 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 | #5590 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C3×B(4,𝔽2) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #5244 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #5241 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #4536 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #4474 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #4471 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #2196 | — | 3 | 1 | no | maximal, Sylow |
| 9 | C3×H(5,𝔽2) | C22 | 1 | 1 | yes | characteristic, complemented |
| 10 | C22×#67 | — | 2 | 1 | no | — |
| 11 | C2×D3×D4 | — | 2 | 1 | no | — |
| 12 | C3×#120 | — | 2 | 1 | no | — |
| 13 | C3×#120 | C22 | 1 | 1 | yes | characteristic |
| 14 | #949 | — | 2 | 1 | no | — |
| 15 | #940 | C22 | 1 | 1 | yes | characteristic |
| 16 | #939 | — | 2 | 1 | no | — |
| 17 | #937 | — | 2 | 1 | no | — |
| 18 | #937 | C22 | 1 | 1 | yes | characteristic |
| 19 | #884 | — | 2 | 1 | no | — |
| 20 | #882 | — | 2 | 1 | no | — |
| 21 | #67⋊C4 | — | 2 | 1 | no | — |
| 22 | C3×#99 | — | 2 | 1 | no | — |
| 23 | C3×#99 | C22 | 1 | 1 | yes | characteristic, complemented |
| 24 | #837 | — | 2 | 1 | no | — |
| 25 | #834 | — | 2 | 1 | no | — |
| 26 | #834 | C22 | 1 | 1 | yes | characteristic, complemented |
| 27 | #833 | C22 | 1 | 1 | yes | characteristic, complemented |
| 28 | D42 | — | 3 | 1 | no | — |
| 29 | B(4,𝔽2) | — | 3 | 1 | no | — |
| 30 | B(4,𝔽2) | D3 | 1 | 1 | yes | characteristic, complemented |
| 31 | #453 | — | 3 | 1 | no | — |
| 32 | #353 | — | 3 | 1 | no | — |
| 33 | AΣL(1,𝔽16) | — | 3 | 1 | no | — |
| 34 | AΣL(1,𝔽16) | — | 3 | 1 | no | — |
| 35 | C24×C3 | D4 | 1 | 1 | yes | characteristic, complemented |
| 36 | C23×D3 | — | 2 | 1 | no | — |
| 37 | C23×D3 | D4 | 1 | 1 | yes | characteristic, complemented |
| 38 | C3×#41 | — | 2 | 1 | no | — |
| 39 | C3×#41 | — | 4 | 1 | no | — |
| 40 | C2×C3×D4 | — | 2 | 1 | no | — |
| 41 | C2×C3×D4 | — | 2 | 1 | no | — |
| 42 | C2×C3×D4 | — | 2 | 1 | no | — |
| 43 | C2×C3×D4 | — | 4 | 1 | no | — |
| 44 | C2×C3×D4 | — | 4 | 1 | no | — |
| 45 | C2×C3×D4 | C23 | 1 | 1 | yes | characteristic |
| 46 | C2×#67 | — | 2 | 1 | no | — |
| 47 | C2×#67 | — | 2 | 1 | no | — |
| 48 | C2×#67 | — | 2 | 1 | no | — |
| 49 | C2×#67 | — | 2 | 1 | no | — |
| 50 | C2×#67 | — | 2 | 1 | no | — |
| 51 | C2×#67 | — | 4 | 4 | no | — |
| 52 | C2×#67 | — | 4 | 1 | no | — |
| 53 | C22×Dic3 | — | 2 | 1 | no | — |
| 54 | D3×D4 | — | 4 | 4 | no | — |
| 55 | C2×D12 | — | 2 | 1 | no | — |
| 56 | C2×D12 | D4 | 1 | 1 | yes | characteristic, complemented |
| 57 | C2×C4×D3 | — | 2 | 1 | no | — |
| 58 | C3×C22⋊C4 | — | 2 | 1 | no | — |
| 59 | C3×C22⋊C4 | — | 2 | 1 | no | — |
| 60 | C3×C22⋊C4 | — | 4 | 1 | no | — |
| 61 | C3×C22⋊C4 | D4 | 1 | 1 | yes | characteristic, complemented |
| 62 | #217 | — | 2 | 1 | no | — |
| 63 | #217 | D4 | 1 | 1 | yes | characteristic, complemented |
| 64 | #215 | — | 4 | 1 | no | — |
| 65 | #213 | — | 4 | 1 | no | — |
| 66 | (C2×D3)⋊C4 | — | 2 | 1 | no | — |
| 67 | (C2×D3)⋊C4 | — | 2 | 1 | no | — |
| 68 | (C2×D3)⋊C4 | — | 4 | 1 | no | — |
| 69 | Dic3⋊C4 | — | 2 | 1 | no | — |
| 70 | C4×Dic3 | D4 | 1 | 1 | yes | characteristic, complemented |
| 71 | #208 | — | 2 | 1 | no | — |
| 72 | H(5,𝔽2) | C2×D3 | 1 | 1 | yes | characteristic, complemented |
| 73 | C22×D4 | — | 6 | 1 | no | — |
| 74 | C22×D4 | — | 6 | 1 | no | — |
| 75 | AGL(1,ℤ/8) | — | 6 | 1 | no | — |
| 76 | #127 | — | 3 | 1 | no | — |
| 77 | #121 | — | 6 | 1 | no | — |
| 78 | #121 | — | 6 | 1 | no | — |
| 79 | #120 | — | 2 | 1 | no | — |
| 80 | #120 | — | 3 | 1 | no | — |
| 81 | #120 | — | 6 | 1 | no | — |
| 82 | #120 | — | 6 | 1 | no | — |
| 83 | #120 | C2×D3 | 1 | 1 | yes | characteristic |
| 84 | C4×D4 | — | 6 | 1 | no | — |
| 85 | #104 | — | 6 | 1 | no | — |
| 86 | #100 | — | 3 | 1 | no | — |
| 87 | #99 | — | 2 | 1 | no | — |
| 88 | #99 | — | 3 | 1 | no | — |
| 89 | #99 | — | 6 | 1 | no | — |
| 90 | #99 | C2×D3 | 1 | 1 | yes | characteristic, complemented |
| 91 | C23×C3 | — | 2 | 1 | no | — |
| 92 | C23×C3 | — | 2 | 1 | no | — |
| 93 | C23×C3 | — | 2 | 1 | no | — |
| 94 | C23×C3 | — | 4 | 1 | no | — |
| 95 | C23×C3 | — | 4 | 2 | no | — |
| 96 | C23×C3 | C2×D4 | 1 | 1 | yes | characteristic |
| 97 | C23×C3 | C2×D4 | 1 | 1 | yes | characteristic |
| 98 | C22×D3 | — | 2 | 1 | no | — |
| 99 | C22×D3 | — | 2 | 1 | no | — |
| 100 | C22×D3 | — | 4 | 2 | no | — |
| 101 | C22×D3 | — | 4 | 2 | no | — |
| 102 | C22×D3 | — | 4 | 1 | no | — |
| 103 | C22×D3 | — | 4 | 2 | no | — |
| 104 | C22×D3 | — | 8 | 1 | no | — |
| 105 | C3×Q8 | — | 2 | 1 | no | — |
| 106 | C3×D4 | — | 2 | 1 | no | — |
| 107 | C3×D4 | — | 4 | 1 | no | — |
| 108 | C3×D4 | — | 4 | 2 | no | — |
| 109 | C3×D4 | — | 4 | 1 | no | — |
| 110 | C3×D4 | — | 4 | 1 | no | — |
| 111 | C3×D4 | — | 4 | 1 | no | — |
| 112 | C3×D4 | — | 8 | 2 | no | — |
| 113 | C2×C3×C4 | — | 2 | 1 | no | — |
| 114 | C2×C3×C4 | — | 2 | 1 | no | — |
| 115 | C2×C3×C4 | — | 2 | 1 | no | — |
| 116 | C2×C3×C4 | — | 4 | 1 | no | — |
| 117 | C2×C3×C4 | — | 4 | 1 | no | — |
| 118 | C2×C3×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 119 | #67 | — | 2 | 4 | no | — |
| 120 | #67 | — | 4 | 2 | no | — |
| 121 | #67 | — | 4 | 2 | no | — |
| 122 | #67 | — | 4 | 2 | no | — |
| 123 | #67 | — | 4 | 2 | no | — |
| 124 | #67 | — | 8 | 2 | no | — |
| 125 | C2×Dic3 | — | 2 | 1 | no | — |
| 126 | C2×Dic3 | — | 2 | 1 | no | — |
| 127 | C2×Dic3 | — | 4 | 2 | no | — |
| 128 | C2×Dic3 | — | 4 | 1 | no | — |
| 129 | D12 | — | 4 | 1 | no | — |
| 130 | D12 | — | 4 | 2 | no | — |
| 131 | C4×D3 | — | 4 | 2 | no | — |
| 132 | C3⋊C8 | — | 4 | 1 | no | — |
| 133 | C24 | — | 3 | 1 | no | — |
| 134 | C24 | — | 6 | 1 | no | — |
| 135 | C24 | #67 | 1 | 1 | yes | characteristic, complemented |
| 136 | #41 | — | 2 | 1 | no | — |
| 137 | #41 | — | 4 | 1 | no | — |
| 138 | C2×D4 | — | 2 | 1 | no | — |
| 139 | C2×D4 | — | 2 | 1 | no | — |
| 140 | C2×D4 | — | 2 | 1 | no | — |
| 141 | C2×D4 | — | 3 | 1 | no | — |
| 142 | C2×D4 | — | 4 | 1 | no | — |
| 143 | C2×D4 | — | 4 | 1 | no | — |
| 144 | C2×D4 | — | 6 | 1 | no | — |
| 145 | C2×D4 | — | 6 | 1 | no | — |
| 146 | C2×D4 | — | 6 | 1 | no | — |
| 147 | C2×D4 | — | 6 | 1 | no | — |
| 148 | C2×D4 | — | 6 | 1 | no | — |
| 149 | C2×D4 | — | 6 | 1 | no | — |
| 150 | C2×D4 | — | 12 | 4 | no | — |
| 151 | C2×D4 | — | 12 | 4 | no | — |
| 152 | C2×D4 | — | 12 | 1 | no | — |
| 153 | C2×D4 | C22×D3 | 1 | 1 | yes | characteristic |
| 154 | C22×C4 | — | 6 | 1 | no | — |
| 155 | C22×C4 | — | 6 | 1 | no | — |
| 156 | SD16 | — | 12 | 1 | no | — |
| 157 | D8 | — | 12 | 1 | no | — |
| 158 | M16 | — | 6 | 1 | no | — |
| 159 | C4⋊C4 | — | 6 | 1 | no | — |
| 160 | C22⋊C4 | — | 2 | 1 | no | — |
| 161 | C22⋊C4 | — | 2 | 1 | no | — |
| 162 | C22⋊C4 | — | 3 | 1 | no | — |
| 163 | C22⋊C4 | — | 4 | 1 | no | — |
| 164 | C22⋊C4 | — | 6 | 1 | no | — |
| 165 | C22⋊C4 | — | 6 | 1 | no | — |
| 166 | C22⋊C4 | — | 6 | 1 | no | — |
| 167 | C22⋊C4 | — | 12 | 1 | no | — |
| 168 | C22⋊C4 | #67 | 1 | 1 | yes | characteristic, complemented |
| 169 | C42 | — | 3 | 1 | no | — |
| 170 | C22×C3 | — | 2 | 2 | no | — |
| 171 | C22×C3 | — | 2 | 1 | no | — |
| 172 | C22×C3 | — | 2 | 1 | no | — |
| 173 | C22×C3 | — | 4 | 1 | no | — |
| 174 | C22×C3 | — | 4 | 1 | no | — |
| 175 | C22×C3 | — | 4 | 1 | no | — |
| 176 | C22×C3 | — | 4 | 2 | no | — |
| 177 | C22×C3 | — | 4 | 1 | no | — |
| 178 | C22×C3 | — | 4 | 1 | no | — |
| 179 | C22×C3 | — | 8 | 1 | no | — |
| 180 | C22×C3 | — | 8 | 1 | no | — |
| 181 | C22×C3 | — | 8 | 1 | no | — |
| 182 | C22×C3 | #120 | 1 | 1 | yes | characteristic |
| 183 | C2×D3 | — | 2 | 2 | no | — |
| 184 | C2×D3 | — | 4 | 1 | no | — |
| 185 | C2×D3 | — | 4 | 1 | no | — |
| 186 | C2×D3 | — | 4 | 2 | no | — |
| 187 | C2×D3 | — | 4 | 1 | no | — |
| 188 | C2×D3 | — | 4 | 1 | no | — |
| 189 | C2×D3 | — | 4 | 1 | no | — |
| 190 | C2×D3 | — | 8 | 1 | no | — |
| 191 | C2×D3 | — | 8 | 4 | no | — |
| 192 | C3×C4 | — | 2 | 1 | no | — |
| 193 | C3×C4 | — | 4 | 1 | no | — |
| 194 | C3×C4 | — | 4 | 1 | no | — |
| 195 | C3×C4 | — | 8 | 1 | no | — |
| 196 | Dic3 | — | 2 | 2 | no | — |
| 197 | Dic3 | — | 4 | 1 | no | — |
| 198 | Dic3 | — | 8 | 1 | no | — |
| 199 | C23 | — | 2 | 1 | no | — |
| 200 | C23 | — | 2 | 1 | no | — |
| 201 | C23 | — | 2 | 1 | no | — |
| 202 | C23 | — | 4 | 1 | no | — |
| 203 | C23 | — | 4 | 2 | no | — |
| 204 | C23 | — | 6 | 1 | no | — |
| 205 | C23 | — | 6 | 1 | no | — |
| 206 | C23 | — | 12 | 1 | no | — |
| 207 | C23 | — | 12 | 2 | no | — |
| 208 | C23 | — | 12 | 2 | no | — |
| 209 | C23 | — | 12 | 2 | no | — |
| 210 | C23 | — | 24 | 1 | no | — |
| 211 | C23 | D3×D4 | 1 | 1 | yes | characteristic |
| 212 | C23 | C2×#67 | 1 | 1 | yes | characteristic |
| 213 | Q8 | — | 2 | 1 | no | — |
| 214 | D4 | — | 2 | 1 | no | — |
| 215 | D4 | — | 4 | 1 | no | — |
| 216 | D4 | — | 4 | 1 | no | — |
| 217 | D4 | — | 4 | 1 | no | — |
| 218 | D4 | — | 4 | 2 | no | — |
| 219 | D4 | — | 4 | 1 | no | — |
| 220 | D4 | — | 6 | 4 | no | — |
| 221 | D4 | — | 8 | 2 | no | — |
| 222 | D4 | — | 12 | 1 | no | — |
| 223 | D4 | — | 12 | 2 | no | — |
| 224 | D4 | — | 12 | 2 | no | — |
| 225 | D4 | — | 12 | 2 | no | — |
| 226 | D4 | — | 12 | 2 | no | — |
| 227 | D4 | — | 12 | 2 | no | — |
| 228 | D4 | — | 24 | 2 | no | — |
| 229 | C2×C4 | — | 2 | 1 | no | — |
| 230 | C2×C4 | — | 2 | 1 | no | — |
| 231 | C2×C4 | — | 2 | 1 | no | — |
| 232 | C2×C4 | — | 4 | 1 | no | — |
| 233 | C2×C4 | — | 4 | 1 | no | — |
| 234 | C2×C4 | — | 6 | 1 | no | — |
| 235 | C2×C4 | — | 6 | 1 | no | — |
| 236 | C2×C4 | — | 12 | 2 | no | — |
| 237 | C2×C4 | — | 12 | 2 | no | — |
| 238 | C2×C4 | — | 12 | 1 | no | — |
| 239 | C2×C4 | D3×D4 | 1 | 1 | yes | characteristic |
| 240 | C8 | — | 12 | 1 | no | — |
| 241 | C2×C3 | — | 2 | 1 | no | — |
| 242 | C2×C3 | — | 4 | 1 | no | — |
| 243 | C2×C3 | — | 4 | 1 | no | — |
| 244 | C2×C3 | — | 4 | 2 | no | — |
| 245 | C2×C3 | — | 8 | 1 | no | — |
| 246 | C2×C3 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 247 | D3 | — | 4 | 2 | no | — |
| 248 | D3 | — | 8 | 1 | no | — |
| 249 | C22 | — | 2 | 2 | no | — |
| 250 | C22 | — | 2 | 1 | no | — |
| 251 | C22 | — | 2 | 1 | no | — |
| 252 | C22 | — | 4 | 1 | no | — |
| 253 | C22 | — | 4 | 1 | no | — |
| 254 | C22 | — | 4 | 1 | no | — |
| 255 | C22 | — | 4 | 2 | no | — |
| 256 | C22 | — | 4 | 1 | no | — |
| 257 | C22 | — | 4 | 1 | no | — |
| 258 | C22 | — | 6 | 2 | no | — |
| 259 | C22 | — | 8 | 1 | no | — |
| 260 | C22 | — | 8 | 1 | no | — |
| 261 | C22 | — | 8 | 1 | no | — |
| 262 | C22 | — | 12 | 1 | no | — |
| 263 | C22 | — | 12 | 1 | no | — |
| 264 | C22 | — | 12 | 1 | no | — |
| 265 | C22 | — | 12 | 2 | no | — |
| 266 | C22 | — | 12 | 1 | no | — |
| 267 | C22 | — | 12 | 1 | no | — |
| 268 | C22 | — | 24 | 1 | no | — |
| 269 | C22 | — | 24 | 4 | no | — |
| 270 | C22 | #937 | 1 | 1 | yes | characteristic |
| 271 | C4 | — | 2 | 1 | no | — |
| 272 | C4 | — | 4 | 1 | no | — |
| 273 | C4 | — | 4 | 1 | no | — |
| 274 | C4 | — | 6 | 2 | no | — |
| 275 | C4 | — | 8 | 1 | no | — |
| 276 | C4 | — | 12 | 1 | no | — |
| 277 | C4 | — | 24 | 1 | no | — |
| 278 | C3 | #2196 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 279 | C2 | — | 2 | 1 | no | — |
| 280 | C2 | — | 4 | 1 | no | — |
| 281 | C2 | — | 4 | 1 | no | — |
| 282 | C2 | — | 4 | 2 | no | — |
| 283 | C2 | — | 8 | 1 | no | — |
| 284 | C2 | — | 12 | 2 | no | — |
| 285 | C2 | — | 24 | 1 | no | — |
| 286 | C2 | #5032 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #5590 | C2 | semidirect |
| 2. C3×B(4,𝔽2) | C2 | semidirect |
| 3. #5244 | C2 | semidirect |
| 4. #5241 | C2 | semidirect |
| 5. #4536 | C2 | semidirect |
| 6. #4474 | C2 | semidirect |
| 7. #4471 | C2 | semidirect |
| 9. C3×H(5,𝔽2) | C22 | semidirect |
| 23. C3×#99 | C22 | semidirect |
| 26. #834 | C22 | semidirect |
| 27. #833 | C22 | semidirect |
| 30. B(4,𝔽2) | D3 | semidirect |
| 35. C24×C3 | D4 | semidirect |
| 37. C23×D3 | D4 | semidirect |
| 56. C2×D12 | D4 | semidirect |
| 61. C3×C22⋊C4 | D4 | semidirect |
| 63. #217 | D4 | semidirect |
| 70. C4×Dic3 | D4 | semidirect |
| 72. H(5,𝔽2) | C2×D3 | semidirect |
| 90. #99 | C2×D3 | semidirect |
| 135. C24 | #67 | semidirect |
| 168. C22⋊C4 | #67 | semidirect |
| 278. C3 | #2196 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #5590 | 2 |
| C3×B(4,𝔽2) | 2 |
| #5244 | 2 |
| #5241 | 2 |
| #4536 | 2 |
| #4474 | 2 |
| #4471 | 2 |
| #2196 | 3 |