Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
710 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 710 conjugacy classes of subgroups — too many to draw. Only its 114 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 | C22×C3×#67 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | C22×#625 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C2×#3757 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C2×(C2×D3)⋊Dic3 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #63579 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #5223 | — | 3 | 2 | no | maximal |
| 7 | C24×C32 | C22 | 1 | 1 | yes | characteristic |
| 8 | C23×C3×D3 | C22 | 1 | 2 | yes | — |
| 9 | C22×C3⋊Dic3 | C22 | 1 | 1 | yes | characteristic |
| 10 | C22×C3⋊Dic3 | C22 | 1 | 1 | yes | characteristic |
| 11 | C2×C3×#67 | — | 2 | 4 | no | — |
| 12 | C2×C3×#67 | — | 2 | 4 | no | — |
| 13 | C2×C3×#67 | — | 2 | 4 | no | — |
| 14 | C22×C3×Dic3 | C22 | 1 | 2 | yes | — |
| 15 | C2×#625 | — | 2 | 4 | no | — |
| 16 | C2×#625 | — | 2 | 2 | no | — |
| 17 | #3757 | — | 2 | 2 | no | — |
| 18 | (C2×D3)⋊Dic3 | — | 2 | 4 | no | — |
| 19 | C22×C3×D4 | — | 3 | 2 | no | — |
| 20 | C22×#67 | — | 3 | 2 | no | — |
| 21 | C22×#67 | D3 | 1 | 2 | yes | complemented |
| 22 | C2×#217 | — | 3 | 2 | no | — |
| 23 | C2×#217 | — | 3 | 2 | no | — |
| 24 | C2×#217 | — | 6 | 1 | no | — |
| 25 | C2×(C2×D3)⋊C4 | — | 3 | 2 | no | — |
| 26 | C3⋊#95 | — | 3 | 2 | no | — |
| 27 | C23×C32 | — | 2 | 1 | no | — |
| 28 | C23×C32 | — | 2 | 1 | no | — |
| 29 | C23×C32 | — | 2 | 2 | no | — |
| 30 | C23×C32 | — | 2 | 2 | no | — |
| 31 | C23×C32 | D4 | 1 | 2 | yes | complemented |
| 32 | C23×C32 | C23 | 1 | 1 | yes | characteristic |
| 33 | C22×C3×D3 | — | 2 | 2 | no | — |
| 34 | C22×C3×D3 | — | 2 | 2 | no | — |
| 35 | C22×C3×D3 | — | 2 | 2 | no | — |
| 36 | C22×C3×D3 | — | 2 | 2 | no | — |
| 37 | C22×C3×D3 | — | 2 | 2 | no | — |
| 38 | C22×C3×D3 | — | 2 | 2 | no | — |
| 39 | C22×C3×D3 | D4 | 1 | 4 | yes | complemented |
| 40 | C2×C3⋊Dic3 | — | 2 | 1 | no | — |
| 41 | C2×C3⋊Dic3 | — | 2 | 2 | no | — |
| 42 | C2×C3⋊Dic3 | — | 2 | 2 | no | — |
| 43 | C2×C3⋊Dic3 | D4 | 1 | 2 | yes | complemented |
| 44 | C3×#67 | — | 2 | 16 | no | — |
| 45 | C2×C3×Dic3 | — | 2 | 2 | no | — |
| 46 | C2×C3×Dic3 | — | 2 | 2 | no | — |
| 47 | C2×C3×Dic3 | D4 | 1 | 4 | yes | complemented |
| 48 | #625 | — | 2 | 8 | no | — |
| 49 | #392 | — | 9 | 1 | no | Sylow |
| 50 | C24×C3 | — | 2 | 1 | no | — |
| 51 | C24×C3 | — | 3 | 2 | no | — |
| 52 | C24×C3 | C2×D3 | 1 | 2 | yes | — |
| 53 | C23×D3 | C2×D3 | 1 | 2 | yes | — |
| 54 | C2×C3×D4 | — | 6 | 4 | no | — |
| 55 | C2×C3×D4 | — | 6 | 4 | no | — |
| 56 | C2×C3×D4 | — | 6 | 4 | no | — |
| 57 | C22×C3×C4 | — | 3 | 2 | no | — |
| 58 | C2×#67 | — | 2 | 4 | no | — |
| 59 | C2×#67 | — | 2 | 4 | no | — |
| 60 | C2×#67 | — | 2 | 4 | no | — |
| 61 | C2×#67 | — | 6 | 4 | no | — |
| 62 | C2×#67 | — | 6 | 4 | no | — |
| 63 | C2×#67 | — | 6 | 4 | no | — |
| 64 | C22×Dic3 | — | 3 | 2 | no | — |
| 65 | C22×Dic3 | — | 3 | 2 | no | — |
| 66 | C22×Dic3 | — | 6 | 1 | no | — |
| 67 | C22×Dic3 | — | 6 | 1 | no | — |
| 68 | C22×Dic3 | C2×D3 | 1 | 2 | yes | — |
| 69 | #217 | — | 6 | 4 | no | — |
| 70 | #217 | — | 6 | 4 | no | — |
| 71 | #217 | — | 6 | 4 | no | — |
| 72 | (C2×D3)⋊C4 | — | 6 | 4 | no | — |
| 73 | C22×C32 | — | 2 | 2 | no | — |
| 74 | C22×C32 | — | 2 | 4 | no | — |
| 75 | C22×C32 | — | 4 | 1 | no | — |
| 76 | C22×C32 | — | 4 | 2 | no | — |
| 77 | C22×C32 | — | 4 | 1 | no | — |
| 78 | C22×C32 | C2×D4 | 1 | 1 | yes | characteristic |
| 79 | C22×C32 | C2×D4 | 1 | 2 | yes | — |
| 80 | C22×C32 | C2×D4 | 1 | 2 | yes | — |
| 81 | C22×C32 | C2×D4 | 1 | 1 | yes | characteristic |
| 82 | C22×C32 | #41 | 1 | 1 | yes | characteristic |
| 83 | C2×C3×D3 | — | 2 | 4 | no | — |
| 84 | C2×C3×D3 | — | 2 | 4 | no | — |
| 85 | C2×C3×D3 | — | 2 | 4 | no | — |
| 86 | C2×C3×D3 | — | 4 | 2 | no | — |
| 87 | C2×C3×D3 | — | 4 | 2 | no | — |
| 88 | C2×C3×D3 | — | 4 | 2 | no | — |
| 89 | C2×C3×D3 | — | 4 | 2 | no | — |
| 90 | C3⋊Dic3 | — | 2 | 2 | no | — |
| 91 | C3⋊Dic3 | — | 4 | 1 | no | — |
| 92 | C3×Dic3 | — | 2 | 4 | no | — |
| 93 | C22×D4 | — | 3 | 2 | no | — |
| 94 | C22×D4 | — | 9 | 1 | no | — |
| 95 | C2×C22⋊C4 | — | 9 | 2 | no | — |
| 96 | C2×C22⋊C4 | — | 9 | 1 | no | — |
| 97 | #95 | — | 9 | 1 | no | — |
| 98 | C23×C3 | — | 2 | 2 | no | — |
| 99 | C23×C3 | — | 2 | 2 | no | — |
| 100 | C23×C3 | — | 2 | 2 | no | — |
| 101 | C23×C3 | — | 2 | 2 | no | — |
| 102 | C23×C3 | — | 2 | 2 | no | — |
| 103 | C23×C3 | — | 2 | 2 | no | — |
| 104 | C23×C3 | — | 2 | 2 | no | — |
| 105 | C23×C3 | — | 2 | 2 | no | — |
| 106 | C23×C3 | — | 2 | 2 | no | — |
| 107 | C23×C3 | — | 2 | 1 | no | — |
| 108 | C23×C3 | — | 3 | 4 | no | — |
| 109 | C23×C3 | — | 4 | 2 | no | — |
| 110 | C23×C3 | — | 4 | 2 | no | — |
| 111 | C23×C3 | — | 6 | 2 | no | — |
| 112 | C23×C3 | — | 6 | 2 | no | — |
| 113 | C23×C3 | — | 6 | 2 | no | — |
| 114 | C23×C3 | — | 6 | 2 | no | — |
| 115 | C23×C3 | — | 6 | 2 | no | — |
| 116 | C23×C3 | — | 6 | 2 | no | — |
| 117 | C23×C3 | #67 | 1 | 4 | yes | complemented |
| 118 | C23×C3 | C22×D3 | 1 | 2 | yes | — |
| 119 | C22×D3 | — | 2 | 2 | no | — |
| 120 | C22×D3 | — | 2 | 2 | no | — |
| 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 | #67 | 1 | 4 | yes | complemented |
| 126 | C3×D4 | — | 6 | 16 | no | — |
| 127 | C2×C3×C4 | — | 3 | 4 | no | — |
| 128 | C2×C3×C4 | — | 6 | 2 | no | — |
| 129 | C2×C3×C4 | — | 6 | 2 | no | — |
| 130 | #67 | — | 2 | 16 | no | — |
| 131 | #67 | — | 6 | 16 | no | — |
| 132 | C2×Dic3 | — | 2 | 2 | no | — |
| 133 | C2×Dic3 | — | 2 | 2 | no | — |
| 134 | C2×Dic3 | — | 3 | 4 | no | — |
| 135 | C2×Dic3 | — | 6 | 2 | no | — |
| 136 | C2×Dic3 | — | 6 | 2 | no | — |
| 137 | C2×Dic3 | — | 6 | 2 | no | — |
| 138 | C2×Dic3 | — | 6 | 2 | no | — |
| 139 | C2×Dic3 | — | 6 | 2 | no | — |
| 140 | C2×Dic3 | — | 6 | 2 | no | — |
| 141 | C2×Dic3 | — | 6 | 2 | no | — |
| 142 | C2×Dic3 | — | 12 | 2 | no | — |
| 143 | C2×Dic3 | — | 12 | 2 | no | — |
| 144 | C2×Dic3 | #67 | 1 | 4 | yes | complemented |
| 145 | C2×C32 | — | 4 | 2 | no | — |
| 146 | C2×C32 | #120 | 1 | 1 | yes | characteristic |
| 147 | C2×C32 | #120 | 1 | 2 | yes | — |
| 148 | C2×C32 | #121 | 1 | 2 | yes | — |
| 149 | C2×C32 | #121 | 1 | 1 | yes | characteristic |
| 150 | C2×C32 | #127 | 1 | 1 | yes | characteristic |
| 151 | C3×D3 | — | 4 | 4 | no | — |
| 152 | C24 | — | 3 | 2 | no | — |
| 153 | C24 | D32 | 1 | 1 | yes | characteristic |
| 154 | C2×D4 | — | 6 | 4 | no | — |
| 155 | C2×D4 | — | 6 | 4 | no | — |
| 156 | C2×D4 | — | 6 | 4 | no | — |
| 157 | C2×D4 | — | 18 | 2 | no | — |
| 158 | C2×D4 | — | 18 | 4 | no | — |
| 159 | C22×C4 | — | 3 | 2 | no | — |
| 160 | C22×C4 | — | 9 | 1 | no | — |
| 161 | C22×C4 | — | 9 | 1 | no | — |
| 162 | C22⋊C4 | — | 18 | 4 | no | — |
| 163 | C22⋊C4 | — | 18 | 2 | no | — |
| 164 | C22×C3 | — | 2 | 4 | no | — |
| 165 | C22×C3 | — | 2 | 4 | no | — |
| 166 | C22×C3 | — | 2 | 4 | no | — |
| 167 | C22×C3 | — | 2 | 4 | no | — |
| 168 | C22×C3 | — | 2 | 1 | no | — |
| 169 | C22×C3 | — | 2 | 1 | no | — |
| 170 | C22×C3 | — | 2 | 2 | no | — |
| 171 | C22×C3 | — | 2 | 2 | no | — |
| 172 | C22×C3 | — | 2 | 1 | no | — |
| 173 | C22×C3 | — | 4 | 4 | no | — |
| 174 | C22×C3 | — | 4 | 4 | no | — |
| 175 | C22×C3 | — | 4 | 2 | no | — |
| 176 | C22×C3 | — | 4 | 2 | no | — |
| 177 | C22×C3 | — | 4 | 2 | no | — |
| 178 | C22×C3 | — | 4 | 2 | no | — |
| 179 | C22×C3 | — | 4 | 2 | no | — |
| 180 | C22×C3 | — | 4 | 2 | no | — |
| 181 | C22×C3 | — | 6 | 4 | no | — |
| 182 | C22×C3 | — | 6 | 4 | no | — |
| 183 | C22×C3 | — | 6 | 4 | no | — |
| 184 | C22×C3 | — | 12 | 2 | no | — |
| 185 | C22×C3 | — | 12 | 2 | no | — |
| 186 | C22×C3 | — | 12 | 2 | no | — |
| 187 | C22×C3 | — | 12 | 2 | no | — |
| 188 | C22×C3 | D3×D4 | 1 | 2 | yes | — |
| 189 | C22×C3 | D3×D4 | 1 | 2 | yes | — |
| 190 | C22×C3 | D3×D4 | 1 | 2 | yes | — |
| 191 | C22×C3 | #237 | 1 | 2 | yes | — |
| 192 | C22×C3 | C2×#67 | 1 | 2 | yes | — |
| 193 | C22×C3 | C2×#67 | 1 | 2 | yes | — |
| 194 | C22×C3 | C2×#67 | 1 | 2 | yes | — |
| 195 | C2×D3 | — | 2 | 4 | no | — |
| 196 | C2×D3 | — | 2 | 4 | no | — |
| 197 | C2×D3 | — | 2 | 4 | no | — |
| 198 | C2×D3 | — | 4 | 2 | no | — |
| 199 | C2×D3 | — | 4 | 2 | no | — |
| 200 | C2×D3 | — | 4 | 2 | no | — |
| 201 | C2×D3 | — | 4 | 2 | no | — |
| 202 | C3×C4 | — | 6 | 4 | no | — |
| 203 | Dic3 | — | 2 | 4 | no | — |
| 204 | Dic3 | — | 6 | 4 | no | — |
| 205 | Dic3 | — | 12 | 2 | no | — |
| 206 | Dic3 | — | 12 | 2 | no | — |
| 207 | Dic3 | — | 12 | 2 | no | — |
| 208 | C32 | #392 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 209 | C23 | — | 2 | 1 | no | — |
| 210 | C23 | — | 2 | 2 | no | — |
| 211 | C23 | — | 2 | 2 | no | — |
| 212 | C23 | — | 2 | 1 | no | — |
| 213 | C23 | — | 3 | 4 | no | — |
| 214 | C23 | — | 6 | 2 | no | — |
| 215 | C23 | — | 6 | 2 | no | — |
| 216 | C23 | — | 6 | 2 | no | — |
| 217 | C23 | — | 6 | 2 | no | — |
| 218 | C23 | — | 6 | 2 | no | — |
| 219 | C23 | — | 6 | 2 | no | — |
| 220 | C23 | #625 | 1 | 2 | yes | complemented |
| 221 | C23 | C2×D32 | 1 | 1 | yes | characteristic |
| 222 | D4 | — | 6 | 16 | no | — |
| 223 | D4 | — | 18 | 8 | no | — |
| 224 | C2×C4 | — | 3 | 4 | no | — |
| 225 | C2×C4 | — | 6 | 2 | no | — |
| 226 | C2×C4 | — | 6 | 2 | no | — |
| 227 | C2×C4 | — | 9 | 2 | no | — |
| 228 | C2×C4 | — | 18 | 2 | no | — |
| 229 | C2×C4 | — | 18 | 2 | no | — |
| 230 | C2×C4 | — | 18 | 1 | no | — |
| 231 | C2×C3 | — | 2 | 2 | no | — |
| 232 | C2×C3 | — | 2 | 2 | no | — |
| 233 | C2×C3 | — | 2 | 1 | no | — |
| 234 | C2×C3 | — | 2 | 1 | no | — |
| 235 | C2×C3 | — | 2 | 1 | no | — |
| 236 | C2×C3 | — | 4 | 4 | no | — |
| 237 | C2×C3 | — | 4 | 4 | no | — |
| 238 | C2×C3 | — | 12 | 4 | no | — |
| 239 | C2×C3 | #937 | 1 | 2 | yes | — |
| 240 | C2×C3 | #937 | 1 | 2 | yes | — |
| 241 | C2×C3 | #938 | 1 | 2 | yes | — |
| 242 | C2×C3 | #939 | 1 | 2 | yes | — |
| 243 | C2×C3 | #939 | 1 | 2 | yes | — |
| 244 | C2×C3 | #940 | 1 | 2 | yes | — |
| 245 | C2×C3 | #953 | 1 | 2 | yes | — |
| 246 | D3 | — | 4 | 4 | no | — |
| 247 | C22 | — | 2 | 4 | no | — |
| 248 | C22 | — | 2 | 2 | no | — |
| 249 | C22 | — | 4 | 2 | no | — |
| 250 | C22 | — | 4 | 1 | no | — |
| 251 | C22 | — | 4 | 1 | no | — |
| 252 | C22 | — | 6 | 4 | no | — |
| 253 | C22 | — | 6 | 4 | no | — |
| 254 | C22 | — | 6 | 4 | no | — |
| 255 | C22 | — | 12 | 2 | no | — |
| 256 | C22 | — | 12 | 2 | no | — |
| 257 | C22 | — | 12 | 2 | no | — |
| 258 | C22 | — | 12 | 2 | no | — |
| 259 | C22 | #3805 | 1 | 1 | yes | characteristic |
| 260 | C22 | C2×#625 | 1 | 1 | yes | characteristic |
| 261 | C22 | D3×#67 | 1 | 2 | yes | — |
| 262 | C22 | D3×#67 | 1 | 2 | yes | — |
| 263 | C22 | #3811 | 1 | 1 | yes | characteristic |
| 264 | C4 | — | 6 | 4 | no | — |
| 265 | C4 | — | 18 | 2 | no | — |
| 266 | C4 | — | 36 | 1 | no | — |
| 267 | C3 | — | 2 | 1 | no | — |
| 268 | C3 | #5223 | 1 | 2 | yes | complemented |
| 269 | C2 | — | 4 | 2 | no | — |
| 270 | C2 | — | 12 | 4 | no | — |
| 271 | C2 | #63970 | 1 | 2 | yes | — |
| 272 | C2 | #63976 | 1 | 2 | yes | — |
| 273 | C2 | #63979 | 1 | 1 | yes | characteristic |
| 274 | C2 | #63980 | 1 | 1 | yes | characteristic |
| 275 | C2 | #63983 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C2×D32 |
| commutator, socle, cosocle | C23×C32 | C23 |
| Fitting | C24×C32 | C22 |
| hypercenter | C24 | D32 |
| hypocenter | C32 | #392 |
| LCS | G ▷ C23×C32 ▷ C32 | |
| UCS | C1 ◁ C23 ◁ C24 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C3×#67 | C2 | semidirect |
| 2. C22×#625 | C2 | semidirect |
| 3. C2×#3757 | C2 | semidirect |
| 4. C2×(C2×D3)⋊Dic3 | C2 | semidirect |
| 5. #63579 | C2 | semidirect |
| 21. C22×#67 | D3 | semidirect |
| 31. C23×C32 | D4 | semidirect |
| 39. C22×C3×D3 | D4 | semidirect |
| 43. C2×C3⋊Dic3 | D4 | semidirect |
| 47. C2×C3×Dic3 | D4 | semidirect |
| 117. C23×C3 | #67 | semidirect |
| 125. C22×D3 | #67 | semidirect |
| 144. C2×Dic3 | #67 | semidirect |
| 208. C32 | #392 | semidirect |
| 220. C23 | #625 | semidirect |
| 268. C3 | #5223 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C3×#67 | 2 |
| C22×#625 | 2 |
| C2×#3757 | 2 |
| C2×(C2×D3)⋊Dic3 | 2 |
| #63579 | 2 |
| #5223 | 3 |