Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
1002 conjugacy classes of subgroups.
The subgroup lattice has 266 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C3×Q82 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #5735 | C2 | 1 | 4 | yes | maximal, complemented |
| 3 | #5733 | C2 | 1 | 4 | yes | maximal, complemented |
| 4 | #5727 | C2 | 1 | 4 | yes | maximal, complemented |
| 5 | #5725 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | #5724 | C2 | 1 | 4 | yes | maximal, complemented |
| 7 | #5722 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #5585 | C2 | 1 | 2 | yes | maximal, complemented |
| 9 | #5582 | C2 | 1 | 4 | yes | maximal, complemented |
| 10 | #5578 | C2 | 1 | 2 | yes | maximal, complemented |
| 11 | #5568 | C2 | 1 | 4 | yes | maximal, complemented |
| 12 | #3550 | — | 3 | 1 | no | maximal, Sylow |
| 13 | C2×#239 | C22 | 1 | 2 | yes | — |
| 14 | C2×D3×Q8 | C22 | 1 | 4 | yes | — |
| 15 | C3×C4⋊Q8 | C22 | 1 | 1 | yes | characteristic |
| 16 | C3×C4⋊Q8 | C22 | 1 | 4 | yes | — |
| 17 | C3×C4⋊Q8 | C22 | 1 | 4 | yes | — |
| 18 | C3×C4×Q8 | C22 | 1 | 4 | yes | — |
| 19 | C3×C4×Q8 | C22 | 1 | 2 | yes | — |
| 20 | #947 | C22 | 1 | 4 | yes | — |
| 21 | #947 | C22 | 1 | 2 | yes | — |
| 22 | #947 | C22 | 1 | 4 | yes | — |
| 23 | #946 | C22 | 1 | 8 | yes | — |
| 24 | #946 | C22 | 1 | 2 | yes | — |
| 25 | #944 | C22 | 1 | 4 | yes | — |
| 26 | #944 | C22 | 1 | 2 | yes | — |
| 27 | #898 | C22 | 1 | 8 | yes | — |
| 28 | #897 | C22 | 1 | 2 | yes | — |
| 29 | #896 | C22 | 1 | 4 | yes | — |
| 30 | #896 | C22 | 1 | 8 | yes | — |
| 31 | #896 | C22 | 1 | 8 | yes | — |
| 32 | #895 | C22 | 1 | 4 | yes | — |
| 33 | #894 | C22 | 1 | 8 | yes | — |
| 34 | #894 | C22 | 1 | 8 | yes | — |
| 35 | #893 | C22 | 1 | 4 | yes | — |
| 36 | #893 | C22 | 1 | 4 | yes | — |
| 37 | #889 | C22 | 1 | 4 | yes | — |
| 38 | #888 | C22 | 1 | 4 | yes | — |
| 39 | #888 | C22 | 1 | 2 | yes | — |
| 40 | #887 | C22 | 1 | 4 | yes | — |
| 41 | #876 | C22 | 1 | 8 | yes | — |
| 42 | #875 | C22 | 1 | 4 | yes | — |
| 43 | #875 | C22 | 1 | 4 | yes | — |
| 44 | #874 | C22 | 1 | 1 | yes | characteristic |
| 45 | C4×D12 | C22 | 1 | 4 | yes | — |
| 46 | #872 | C22 | 1 | 4 | yes | — |
| 47 | #872 | C22 | 1 | 4 | yes | — |
| 48 | C42×D3 | C22 | 1 | 1 | yes | characteristic |
| 49 | C4×Dic6 | C22 | 1 | 2 | yes | — |
| 50 | C4×Dic6 | C22 | 1 | 4 | yes | — |
| 51 | #563 | — | 3 | 4 | no | — |
| 52 | #559 | — | 3 | 2 | no | — |
| 53 | Q82 | — | 3 | 1 | no | — |
| 54 | Q82 | D3 | 1 | 1 | yes | characteristic, complemented |
| 55 | #556 | — | 3 | 4 | no | — |
| 56 | D4×Q8 | — | 3 | 4 | no | — |
| 57 | #542 | — | 3 | 4 | no | — |
| 58 | #542 | — | 3 | 4 | no | — |
| 59 | #536 | — | 3 | 4 | no | — |
| 60 | #533 | — | 3 | 2 | no | — |
| 61 | #532 | — | 3 | 1 | no | — |
| 62 | C2×C3×Q8 | C23 | 1 | 2 | yes | — |
| 63 | C2×C3×Q8 | C23 | 1 | 2 | yes | — |
| 64 | C2×C3×Q8 | C23 | 1 | 4 | yes | — |
| 65 | #239 | — | 4 | 4 | no | — |
| 66 | D3×Q8 | — | 4 | 8 | no | — |
| 67 | C2×D12 | C23 | 1 | 4 | yes | — |
| 68 | C2×D12 | C23 | 1 | 2 | yes | — |
| 69 | C2×D12 | C23 | 1 | 4 | yes | — |
| 70 | C2×C4×D3 | C23 | 1 | 2 | yes | — |
| 71 | C2×C4×D3 | C23 | 1 | 1 | yes | characteristic |
| 72 | C2×C4×D3 | C23 | 1 | 4 | yes | — |
| 73 | C2×C4×D3 | C23 | 1 | 8 | yes | — |
| 74 | C2×Dic6 | C23 | 1 | 2 | yes | — |
| 75 | C2×Dic6 | C23 | 1 | 4 | yes | — |
| 76 | C2×Dic6 | C23 | 1 | 4 | yes | — |
| 77 | C3×C4⋊C4 | C23 | 1 | 4 | yes | — |
| 78 | C3×C4⋊C4 | C23 | 1 | 2 | yes | — |
| 79 | C3×C4⋊C4 | C23 | 1 | 4 | yes | — |
| 80 | C3×C4⋊C4 | C23 | 1 | 8 | yes | — |
| 81 | C3×C42 | C23 | 1 | 1 | yes | characteristic |
| 82 | C3×C42 | C23 | 1 | 4 | yes | — |
| 83 | C3×C42 | C23 | 1 | 4 | yes | — |
| 84 | (C2×D3)⋊C4 | C23 | 1 | 4 | yes | — |
| 85 | (C2×D3)⋊C4 | C23 | 1 | 8 | yes | — |
| 86 | (C2×D3)⋊C4 | C23 | 1 | 4 | yes | — |
| 87 | (C2×D3)⋊C4 | C23 | 1 | 8 | yes | — |
| 88 | (C2×D3)⋊C4 | C23 | 1 | 8 | yes | — |
| 89 | (C2×D3)⋊C4 | C23 | 1 | 8 | yes | — |
| 90 | C4⋊Dic3 | C23 | 1 | 2 | yes | — |
| 91 | C4⋊Dic3 | C23 | 1 | 4 | yes | — |
| 92 | Dic3⋊C4 | C23 | 1 | 4 | yes | — |
| 93 | Dic3⋊C4 | C23 | 1 | 4 | yes | — |
| 94 | Dic3⋊C4 | C23 | 1 | 4 | yes | — |
| 95 | Dic3⋊C4 | C23 | 1 | 4 | yes | — |
| 96 | Dic3⋊C4 | C23 | 1 | 4 | yes | — |
| 97 | Dic3⋊C4 | C23 | 1 | 8 | yes | — |
| 98 | C4×Dic3 | C23 | 1 | 2 | yes | — |
| 99 | C4×Dic3 | C23 | 1 | 4 | yes | — |
| 100 | C4×Dic3 | C23 | 1 | 1 | yes | characteristic |
| 101 | C4×Dic3 | C23 | 1 | 4 | yes | — |
| 102 | C2×#41 | — | 3 | 2 | no | — |
| 103 | C22×Q8 | — | 3 | 4 | no | — |
| 104 | C4⋊Q8 | — | 3 | 2 | no | — |
| 105 | C4⋊Q8 | — | 3 | 4 | no | — |
| 106 | C4⋊Q8 | — | 3 | 2 | no | — |
| 107 | C4⋊Q8 | — | 3 | 4 | no | — |
| 108 | C4⋊Q8 | C2×D3 | 1 | 1 | yes | characteristic |
| 109 | C4⋊Q8 | C2×D3 | 1 | 4 | yes | — |
| 110 | C4⋊Q8 | C2×D3 | 1 | 4 | yes | — |
| 111 | #127 | — | 3 | 1 | no | — |
| 112 | #126 | — | 3 | 8 | no | — |
| 113 | #126 | — | 3 | 8 | no | — |
| 114 | #125 | — | 3 | 4 | no | — |
| 115 | #124 | — | 3 | 2 | no | — |
| 116 | #124 | — | 3 | 4 | no | — |
| 117 | #124 | — | 3 | 4 | no | — |
| 118 | #124 | — | 3 | 4 | no | — |
| 119 | #124 | — | 3 | 4 | no | — |
| 120 | #123 | — | 3 | 8 | no | — |
| 121 | #123 | — | 3 | 8 | no | — |
| 122 | C22⋊Q8 | — | 3 | 2 | no | — |
| 123 | C22⋊Q8 | — | 3 | 8 | no | — |
| 124 | C22⋊Q8 | — | 3 | 4 | no | — |
| 125 | C22⋊Q8 | — | 3 | 2 | no | — |
| 126 | C22⋊Q8 | — | 3 | 8 | no | — |
| 127 | C22⋊Q8 | — | 3 | 8 | no | — |
| 128 | #121 | — | 3 | 4 | no | — |
| 129 | C4×Q8 | — | 3 | 4 | no | — |
| 130 | C4×Q8 | — | 3 | 4 | no | — |
| 131 | C4×Q8 | — | 3 | 2 | no | — |
| 132 | C4×Q8 | C2×D3 | 1 | 2 | yes | — |
| 133 | C4×Q8 | C2×D3 | 1 | 4 | yes | — |
| 134 | C4×D4 | — | 3 | 4 | no | — |
| 135 | C4×D4 | — | 3 | 4 | no | — |
| 136 | C4×D4 | — | 3 | 4 | no | — |
| 137 | #117 | — | 3 | 4 | no | — |
| 138 | #117 | — | 3 | 4 | no | — |
| 139 | C2×C42 | — | 3 | 1 | no | — |
| 140 | C22×D3 | C24 | 1 | 4 | yes | — |
| 141 | C22×D3 | C24 | 1 | 1 | yes | characteristic |
| 142 | C3×Q8 | — | 2 | 4 | no | — |
| 143 | C3×Q8 | — | 4 | 4 | no | — |
| 144 | C3×Q8 | — | 4 | 2 | no | — |
| 145 | C2×C3×C4 | C24 | 1 | 1 | yes | characteristic |
| 146 | C2×C3×C4 | C24 | 1 | 4 | yes | — |
| 147 | C2×C3×C4 | C24 | 1 | 4 | yes | — |
| 148 | C2×C3×C4 | C24 | 1 | 4 | yes | — |
| 149 | C2×C3×C4 | C24 | 1 | 2 | yes | — |
| 150 | C2×Dic3 | C24 | 1 | 4 | yes | — |
| 151 | C2×Dic3 | C24 | 1 | 2 | yes | — |
| 152 | C2×Dic3 | C24 | 1 | 4 | yes | — |
| 153 | C2×Dic3 | C24 | 1 | 1 | yes | characteristic |
| 154 | D12 | — | 2 | 8 | no | — |
| 155 | D12 | — | 4 | 4 | no | — |
| 156 | D12 | — | 4 | 2 | no | — |
| 157 | C4×D3 | — | 2 | 4 | no | — |
| 158 | C4×D3 | — | 2 | 8 | no | — |
| 159 | C4×D3 | — | 2 | 2 | no | — |
| 160 | C4×D3 | — | 4 | 8 | no | — |
| 161 | Dic6 | — | 2 | 4 | no | — |
| 162 | Dic6 | — | 2 | 8 | no | — |
| 163 | Dic6 | — | 4 | 4 | no | — |
| 164 | #41 | — | 12 | 4 | no | — |
| 165 | C2×Q8 | — | 3 | 2 | no | — |
| 166 | C2×Q8 | — | 3 | 4 | no | — |
| 167 | C2×Q8 | — | 3 | 4 | no | — |
| 168 | C2×Q8 | — | 12 | 8 | no | — |
| 169 | C2×Q8 | C22×D3 | 1 | 2 | yes | — |
| 170 | C2×Q8 | C22×D3 | 1 | 4 | yes | — |
| 171 | C2×Q8 | C22×D3 | 1 | 2 | yes | — |
| 172 | C2×D4 | — | 3 | 4 | no | — |
| 173 | C2×D4 | — | 3 | 4 | no | — |
| 174 | C2×D4 | — | 3 | 2 | no | — |
| 175 | C22×C4 | — | 3 | 2 | no | — |
| 176 | C22×C4 | — | 3 | 1 | no | — |
| 177 | C22×C4 | — | 3 | 4 | no | — |
| 178 | C22×C4 | — | 3 | 8 | no | — |
| 179 | C4⋊C4 | — | 3 | 4 | no | — |
| 180 | C4⋊C4 | — | 3 | 4 | no | — |
| 181 | C4⋊C4 | — | 3 | 4 | no | — |
| 182 | C4⋊C4 | — | 3 | 2 | no | — |
| 183 | C4⋊C4 | — | 3 | 4 | no | — |
| 184 | C4⋊C4 | — | 3 | 4 | no | — |
| 185 | C4⋊C4 | — | 3 | 8 | no | — |
| 186 | C4⋊C4 | — | 3 | 4 | no | — |
| 187 | C4⋊C4 | C22×D3 | 1 | 2 | yes | — |
| 188 | C4⋊C4 | C22×D3 | 1 | 4 | yes | — |
| 189 | C4⋊C4 | C22×D3 | 1 | 8 | yes | — |
| 190 | C4⋊C4 | C22×D3 | 1 | 4 | yes | — |
| 191 | C22⋊C4 | — | 3 | 4 | no | — |
| 192 | C22⋊C4 | — | 3 | 4 | no | — |
| 193 | C22⋊C4 | — | 3 | 8 | no | — |
| 194 | C22⋊C4 | — | 3 | 8 | no | — |
| 195 | C22⋊C4 | — | 3 | 8 | no | — |
| 196 | C22⋊C4 | — | 3 | 8 | no | — |
| 197 | C42 | — | 3 | 2 | no | — |
| 198 | C42 | — | 3 | 1 | no | — |
| 199 | C42 | — | 3 | 4 | no | — |
| 200 | C42 | — | 3 | 4 | no | — |
| 201 | C42 | C22×D3 | 1 | 1 | yes | characteristic |
| 202 | C42 | C22×D3 | 1 | 4 | yes | — |
| 203 | C42 | C22×D3 | 1 | 4 | yes | — |
| 204 | C22×C3 | C25 | 1 | 1 | yes | characteristic |
| 205 | C2×D3 | — | 2 | 4 | no | — |
| 206 | C2×D3 | — | 2 | 1 | no | — |
| 207 | C2×D3 | — | 2 | 2 | no | — |
| 208 | C2×D3 | — | 2 | 8 | no | — |
| 209 | C3×C4 | — | 2 | 4 | no | — |
| 210 | C3×C4 | — | 2 | 4 | no | — |
| 211 | C3×C4 | — | 2 | 1 | no | — |
| 212 | C3×C4 | — | 2 | 4 | no | — |
| 213 | C3×C4 | 2−1+4 | 1 | 4 | yes | — |
| 214 | Dic3 | — | 2 | 4 | no | — |
| 215 | Dic3 | — | 2 | 4 | no | — |
| 216 | Dic3 | — | 2 | 2 | no | — |
| 217 | Dic3 | H(5,𝔽2) | 1 | 2 | yes | — |
| 218 | C23 | — | 3 | 1 | no | — |
| 219 | C23 | — | 3 | 4 | no | — |
| 220 | Q8 | — | 2 | 4 | no | — |
| 221 | Q8 | — | 4 | 4 | no | — |
| 222 | Q8 | — | 4 | 2 | no | — |
| 223 | Q8 | — | 6 | 8 | no | — |
| 224 | Q8 | — | 6 | 4 | no | — |
| 225 | Q8 | — | 12 | 4 | no | — |
| 226 | D4 | — | 6 | 8 | no | — |
| 227 | D4 | — | 12 | 2 | no | — |
| 228 | D4 | — | 12 | 4 | no | — |
| 229 | C2×C4 | — | 3 | 1 | no | — |
| 230 | C2×C4 | — | 3 | 4 | no | — |
| 231 | C2×C4 | — | 3 | 2 | no | — |
| 232 | C2×C4 | — | 3 | 4 | no | — |
| 233 | C2×C4 | — | 6 | 8 | no | — |
| 234 | C2×C4 | — | 6 | 2 | no | — |
| 235 | C2×C4 | — | 6 | 4 | no | — |
| 236 | C2×C4 | — | 12 | 8 | no | — |
| 237 | C2×C4 | C23×D3 | 1 | 4 | yes | — |
| 238 | C2×C4 | C23×D3 | 1 | 1 | yes | characteristic |
| 239 | C2×C4 | C23×D3 | 1 | 4 | yes | — |
| 240 | C2×C4 | C23×D3 | 1 | 4 | yes | — |
| 241 | C2×C4 | C23×D3 | 1 | 2 | yes | — |
| 242 | C2×C3 | C2×H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 243 | C2×C3 | C2×2−1+4 | 1 | 2 | yes | — |
| 244 | D3 | — | 4 | 4 | no | — |
| 245 | D3 | — | 4 | 1 | no | — |
| 246 | C22 | — | 6 | 2 | no | — |
| 247 | C22 | — | 6 | 8 | no | — |
| 248 | C22 | — | 6 | 4 | no | — |
| 249 | C22 | — | 6 | 1 | no | — |
| 250 | C22 | C24×D3 | 1 | 1 | yes | characteristic |
| 251 | C4 | — | 2 | 4 | no | — |
| 252 | C4 | — | 2 | 1 | no | — |
| 253 | C4 | — | 2 | 4 | no | — |
| 254 | C4 | — | 2 | 4 | no | — |
| 255 | C4 | — | 3 | 2 | no | — |
| 256 | C4 | — | 6 | 2 | no | — |
| 257 | C4 | — | 6 | 4 | no | — |
| 258 | C4 | — | 6 | 4 | no | — |
| 259 | C4 | #1007 | 1 | 4 | yes | — |
| 260 | C3 | #3550 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 261 | C2 | — | 12 | 1 | no | — |
| 262 | C2 | — | 12 | 4 | no | — |
| 263 | C2 | C2×#1007 | 1 | 2 | yes | — |
| 264 | C2 | D3×H(5,𝔽2) | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D3 |
| commutator, socle, cosocle | C22×C3 | C25 |
| Fitting | C3×Q82 | C2 |
| prime radical | #5725 | C2 |
| hypercenter | Q82 | D3 |
| hypocenter | C3 | #3550 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ Q82 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×Q82 | C2 | semidirect |
| 2. #5735 | C2 | semidirect |
| 3. #5733 | C2 | semidirect |
| 4. #5727 | C2 | semidirect |
| 6. #5724 | C2 | semidirect |
| 7. #5722 | C2 | semidirect |
| 8. #5585 | C2 | semidirect |
| 9. #5582 | C2 | semidirect |
| 10. #5578 | C2 | semidirect |
| 11. #5568 | C2 | semidirect |
| 54. Q82 | D3 | semidirect |
| 260. C3 | #3550 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×Q82 | 2 |
| #5735 | 2 |
| #5733 | 2 |
| #5727 | 2 |
| #5725 | 2 |
| #5724 | 2 |
| #5722 | 2 |
| #5585 | 2 |
| #5582 | 2 |
| #5578 | 2 |
| #5568 | 2 |
| #3550 | 3 |