Finite Group DatabasebetaFinite Group DatabasebetaOrder 4374.
1921 conjugacy classes of subgroups; only the 89 normal subgroups are shown.
This group has 1921 conjugacy classes of subgroups — too many to draw. Only its 89 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 | #14554265 | C2 | 1 | 1 | yes | maximal, characteristic, complemented, Sylow |
| 2 | C3⋊(C3⋊(C3⋊AGL(1,ℤ/9))) | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C3⋊(#4177⋊C3) | — | 3 | 3 | no | maximal |
| 4 | C3×#92301 | — | 3 | 1 | no | maximal |
| 5 | #13091042 | C3 | 1 | 3 | yes | maximal, complemented |
| 6 | C33×3−1+2 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 7 | C3×#6910 | — | 2 | 3 | no | — |
| 8 | C3×#6910 | D3 | 1 | 3 | yes | complemented |
| 9 | C3×#6891 | — | 2 | 1 | no | — |
| 10 | C3×C32⋊H(3,𝔽3) | D3 | 1 | 1 | yes | characteristic |
| 11 | #10624736 | — | 2 | 9 | no | — |
| 12 | #10624731 | — | 2 | 3 | no | — |
| 13 | #10624731 | C2×C3 | 1 | 3 | yes | complemented |
| 14 | C3×C3⋊(C3⋊(C3⋊D3)) | — | 3 | 1 | no | — |
| 15 | C3⋊(C3⋊(C3⋊D9)) | C32 | 1 | 1 | yes | characteristic, complemented |
| 16 | C3⋊(C3⋊AGL(1,ℤ/9)) | — | 3 | 3 | no | — |
| 17 | C3⋊(C3⋊AGL(1,ℤ/9)) | — | 9 | 9 | no | — |
| 18 | C3⋊(C3⋊AGL(1,ℤ/9)) | — | 9 | 3 | no | — |
| 19 | C3×C3⋊((C3⋊D3)⋊C3) | — | 9 | 3 | no | — |
| 20 | C3×C3⋊((C3⋊D3)⋊C3) | — | 9 | 1 | no | — |
| 21 | C3⋊#4177 | — | 3 | 9 | no | — |
| 22 | #92301 | — | 3 | 3 | no | — |
| 23 | #4177⋊C3 | — | 9 | 9 | no | — |
| 24 | C35 | C3×D3 | 1 | 1 | yes | characteristic |
| 25 | C32×3−1+2 | — | 2 | 3 | no | — |
| 26 | C32×3−1+2 | — | 2 | 3 | no | — |
| 27 | C32×3−1+2 | — | 3 | 3 | no | — |
| 28 | C32×3−1+2 | — | 3 | 9 | no | — |
| 29 | C32×3−1+2 | — | 6 | 9 | no | — |
| 30 | C32×3−1+2 | — | 6 | 3 | no | — |
| 31 | C32×3−1+2 | C3×D3 | 1 | 3 | yes | — |
| 32 | C32×H(3,𝔽3) | — | 2 | 1 | no | — |
| 33 | C32×H(3,𝔽3) | — | 3 | 3 | no | — |
| 34 | C32×H(3,𝔽3) | C3⋊D3 | 1 | 1 | yes | characteristic |
| 35 | C33×C9 | — | 2 | 1 | no | — |
| 36 | C33×C9 | C2×C32 | 1 | 1 | yes | characteristic, complemented |
| 37 | #6910 | — | 3 | 9 | no | — |
| 38 | #6910 | — | 6 | 9 | no | — |
| 39 | #6910 | — | 6 | 9 | no | — |
| 40 | #6910 | — | 6 | 9 | no | — |
| 41 | #6910 | — | 6 | 9 | no | — |
| 42 | C3×#731 | — | 2 | 18 | no | — |
| 43 | C3×#731 | — | 2 | 9 | no | — |
| 44 | C3×#731 | — | 2 | 9 | no | — |
| 45 | C3×#731 | C3×D3 | 1 | 9 | yes | complemented |
| 46 | #6891 | — | 2 | 6 | no | — |
| 47 | #6891 | — | 2 | 3 | no | — |
| 48 | #6891 | — | 6 | 3 | no | — |
| 49 | #6891 | — | 6 | 3 | no | — |
| 50 | C32⋊H(3,𝔽3) | — | 2 | 3 | no | — |
| 51 | C32⋊H(3,𝔽3) | — | 6 | 3 | no | — |
| 52 | C32⋊H(3,𝔽3) | C3×D3 | 1 | 3 | yes | — |
| 53 | C3×C32⋊C9 | — | 6 | 3 | no | — |
| 54 | C3⋊(C3⋊(C3⋊D3)) | — | 3 | 1 | no | — |
| 55 | C3×C3⋊(C3⋊D3) | — | 9 | 3 | no | — |
| 56 | C3×C3⋊(C3⋊D3) | — | 9 | 1 | no | — |
| 57 | C3×C3⋊(C3⋊D3) | — | 27 | 3 | no | — |
| 58 | C3×C3⋊(C3⋊D3) | — | 27 | 9 | no | — |
| 59 | C32×C3⋊D3 | — | 27 | 1 | no | — |
| 60 | C3⋊(C3⋊D9) | — | 3 | 3 | no | — |
| 61 | C3⋊(C3⋊D9) | — | 9 | 9 | no | — |
| 62 | C3⋊(C3⋊D9) | — | 9 | 3 | no | — |
| 63 | C3⋊AGL(1,ℤ/9) | — | 9 | 9 | no | — |
| 64 | C3⋊AGL(1,ℤ/9) | — | 27 | 3 | no | — |
| 65 | C3⋊AGL(1,ℤ/9) | — | 27 | 6 | no | — |
| 66 | C3⋊AGL(1,ℤ/9) | — | 27 | 27 | no | — |
| 67 | C3⋊((C3⋊D3)⋊C3) | — | 9 | 9 | no | — |
| 68 | C3⋊((C3⋊D3)⋊C3) | — | 9 | 3 | no | — |
| 69 | C3×(C3⋊D3)⋊C3 | — | 27 | 3 | no | — |
| 70 | C3×(C3⋊D3)⋊C3 | — | 27 | 9 | no | — |
| 71 | #4177 | — | 9 | 27 | no | — |
| 72 | C34 | — | 2 | 3 | no | — |
| 73 | C34 | — | 2 | 1 | no | — |
| 74 | C34 | — | 3 | 3 | no | — |
| 75 | C34 | — | 3 | 1 | no | — |
| 76 | C34 | — | 6 | 3 | no | — |
| 77 | C34 | — | 6 | 1 | no | — |
| 78 | C34 | — | 9 | 9 | no | — |
| 79 | C34 | (C3⋊D3)⋊C3 | 1 | 3 | yes | — |
| 80 | C34 | C32×D3 | 1 | 1 | yes | characteristic |
| 81 | C34 | C3×C3⋊D3 | 1 | 1 | yes | characteristic |
| 82 | C3×3−1+2 | — | 2 | 9 | no | — |
| 83 | C3×3−1+2 | — | 2 | 9 | no | — |
| 84 | C3×3−1+2 | — | 3 | 27 | no | — |
| 85 | C3×3−1+2 | — | 3 | 3 | no | — |
| 86 | C3×3−1+2 | — | 3 | 6 | no | — |
| 87 | C3×3−1+2 | — | 6 | 18 | no | — |
| 88 | C3×3−1+2 | — | 6 | 9 | no | — |
| 89 | C3×3−1+2 | — | 6 | 27 | no | — |
| 90 | C3×3−1+2 | — | 6 | 81 | no | — |
| 91 | C3×3−1+2 | — | 6 | 3 | no | — |
| 92 | C3×3−1+2 | — | 6 | 6 | no | — |
| 93 | C3×3−1+2 | — | 6 | 3 | no | — |
| 94 | C3×3−1+2 | — | 6 | 3 | no | — |
| 95 | C3×3−1+2 | — | 6 | 9 | no | — |
| 96 | C3×3−1+2 | — | 6 | 6 | no | — |
| 97 | C3×3−1+2 | — | 6 | 6 | no | — |
| 98 | C3×3−1+2 | — | 6 | 6 | no | — |
| 99 | C3×3−1+2 | — | 6 | 3 | no | — |
| 100 | C3×3−1+2 | — | 6 | 6 | no | — |
| 101 | C3×3−1+2 | (C3⋊D3)⋊C3 | 1 | 9 | yes | complemented |
| 102 | C3×H(3,𝔽3) | — | 2 | 3 | no | — |
| 103 | C3×H(3,𝔽3) | — | 3 | 3 | no | — |
| 104 | C3×H(3,𝔽3) | — | 3 | 9 | no | — |
| 105 | C3×H(3,𝔽3) | — | 6 | 3 | no | — |
| 106 | C3×H(3,𝔽3) | — | 6 | 3 | no | — |
| 107 | C3×H(3,𝔽3) | — | 6 | 3 | no | — |
| 108 | C3×H(3,𝔽3) | — | 6 | 3 | no | — |
| 109 | C3×H(3,𝔽3) | — | 6 | 1 | no | — |
| 110 | C3×H(3,𝔽3) | — | 6 | 9 | no | — |
| 111 | C3×H(3,𝔽3) | — | 6 | 2 | no | — |
| 112 | C3×H(3,𝔽3) | — | 9 | 9 | no | — |
| 113 | C3×H(3,𝔽3) | — | 18 | 9 | no | — |
| 114 | C3×H(3,𝔽3) | C3×C3⋊D3 | 1 | 3 | yes | — |
| 115 | C32×C9 | — | 2 | 3 | no | — |
| 116 | C32×C9 | — | 3 | 3 | no | — |
| 117 | C32×C9 | — | 3 | 9 | no | — |
| 118 | C32×C9 | — | 6 | 9 | no | — |
| 119 | C32×C9 | — | 6 | 3 | no | — |
| 120 | C32×C9 | — | 6 | 3 | no | — |
| 121 | C32×C9 | C32×D3 | 1 | 3 | yes | — |
| 122 | #731 | — | 3 | 27 | no | — |
| 123 | #731 | — | 6 | 27 | no | — |
| 124 | #731 | — | 6 | 54 | no | — |
| 125 | #731 | — | 6 | 27 | no | — |
| 126 | C32⋊C9 | — | 6 | 18 | no | — |
| 127 | C32⋊C9 | — | 6 | 9 | no | — |
| 128 | C3⋊(C3⋊D3) | — | 9 | 3 | no | — |
| 129 | C3⋊(C3⋊D3) | — | 9 | 1 | no | — |
| 130 | C3⋊(C3⋊D3) | — | 27 | 3 | no | — |
| 131 | C3⋊(C3⋊D3) | — | 27 | 9 | no | — |
| 132 | C3×C3⋊D3 | — | 27 | 3 | no | — |
| 133 | C3×C3⋊D3 | — | 27 | 1 | no | — |
| 134 | C3×C3⋊D3 | — | 27 | 9 | no | — |
| 135 | C3×C3⋊D3 | — | 27 | 3 | no | — |
| 136 | C3×C3⋊D3 | — | 81 | 3 | no | — |
| 137 | C3×C3⋊D3 | — | 81 | 6 | no | — |
| 138 | C3×C3⋊D3 | — | 81 | 3 | no | — |
| 139 | C3×C3⋊D3 | — | 81 | 27 | no | — |
| 140 | C32×D3 | — | 81 | 3 | no | — |
| 141 | C32×D3 | — | 81 | 1 | no | — |
| 142 | C3⋊D9 | — | 9 | 9 | no | — |
| 143 | C3⋊D9 | — | 27 | 3 | no | — |
| 144 | C3⋊D9 | — | 27 | 6 | no | — |
| 145 | C3⋊D9 | — | 27 | 27 | no | — |
| 146 | AGL(1,ℤ/9) | — | 81 | 9 | no | — |
| 147 | (C3⋊D3)⋊C3 | — | 27 | 9 | no | — |
| 148 | (C3⋊D3)⋊C3 | — | 27 | 27 | no | — |
| 149 | C33 | — | 2 | 1 | no | — |
| 150 | C33 | — | 2 | 3 | no | — |
| 151 | C33 | — | 2 | 3 | no | — |
| 152 | C33 | — | 3 | 9 | no | — |
| 153 | C33 | — | 3 | 3 | no | — |
| 154 | C33 | — | 3 | 3 | no | — |
| 155 | C33 | — | 3 | 9 | no | — |
| 156 | C33 | — | 3 | 3 | no | — |
| 157 | C33 | — | 6 | 9 | no | — |
| 158 | C33 | — | 6 | 3 | no | — |
| 159 | C33 | — | 6 | 9 | no | — |
| 160 | C33 | — | 6 | 3 | no | — |
| 161 | C33 | — | 6 | 3 | no | — |
| 162 | C33 | — | 9 | 6 | no | — |
| 163 | C33 | — | 9 | 3 | no | — |
| 164 | C33 | — | 9 | 27 | no | — |
| 165 | C33 | — | 9 | 3 | no | — |
| 166 | C33 | — | 9 | 1 | no | — |
| 167 | C33 | — | 18 | 1 | no | — |
| 168 | C33 | — | 18 | 1 | no | — |
| 169 | C33 | — | 18 | 1 | no | — |
| 170 | C33 | — | 18 | 6 | no | — |
| 171 | C33 | — | 18 | 3 | no | — |
| 172 | C33 | — | 18 | 3 | no | — |
| 173 | C33 | — | 18 | 27 | no | — |
| 174 | C33 | — | 18 | 3 | no | — |
| 175 | C33 | — | 18 | 6 | no | — |
| 176 | C33 | — | 18 | 1 | no | — |
| 177 | C33 | C3×(C3⋊D3)⋊C3 | 1 | 3 | yes | — |
| 178 | C33 | C3⋊((C3⋊D3)⋊C3) | 1 | 1 | yes | characteristic |
| 179 | C33 | C3⋊((C3⋊D3)⋊C3) | 1 | 3 | yes | — |
| 180 | C33 | C32×C3⋊D3 | 1 | 1 | yes | characteristic |
| 181 | 3−1+2 | — | 3 | 9 | no | — |
| 182 | 3−1+2 | — | 6 | 18 | no | — |
| 183 | 3−1+2 | — | 6 | 9 | no | — |
| 184 | 3−1+2 | — | 6 | 81 | no | — |
| 185 | 3−1+2 | — | 6 | 9 | no | — |
| 186 | 3−1+2 | — | 6 | 9 | no | — |
| 187 | 3−1+2 | — | 6 | 18 | no | — |
| 188 | 3−1+2 | — | 18 | 3 | no | — |
| 189 | 3−1+2 | — | 18 | 6 | no | — |
| 190 | 3−1+2 | — | 18 | 6 | no | — |
| 191 | 3−1+2 | — | 18 | 3 | no | — |
| 192 | H(3,𝔽3) | — | 3 | 9 | no | — |
| 193 | H(3,𝔽3) | — | 6 | 9 | no | — |
| 194 | H(3,𝔽3) | — | 9 | 27 | no | — |
| 195 | H(3,𝔽3) | — | 18 | 3 | no | — |
| 196 | H(3,𝔽3) | — | 18 | 6 | no | — |
| 197 | H(3,𝔽3) | — | 18 | 27 | no | — |
| 198 | C3×C9 | — | 2 | 9 | no | — |
| 199 | C3×C9 | — | 3 | 27 | no | — |
| 200 | C3×C9 | — | 3 | 6 | no | — |
| 201 | C3×C9 | — | 3 | 3 | no | — |
| 202 | C3×C9 | — | 6 | 27 | no | — |
| 203 | C3×C9 | — | 6 | 6 | no | — |
| 204 | C3×C9 | — | 6 | 3 | no | — |
| 205 | C3×C9 | — | 6 | 9 | no | — |
| 206 | C3×C9 | — | 18 | 3 | no | — |
| 207 | C3×C9 | — | 18 | 6 | no | — |
| 208 | C3×C9 | C3×(C3⋊D3)⋊C3 | 1 | 9 | yes | complemented |
| 209 | C2×C32 | — | 243 | 1 | no | — |
| 210 | C3⋊D3 | — | 27 | 1 | no | — |
| 211 | C3⋊D3 | — | 27 | 9 | no | — |
| 212 | C3⋊D3 | — | 27 | 3 | no | — |
| 213 | C3⋊D3 | — | 81 | 3 | no | — |
| 214 | C3⋊D3 | — | 81 | 6 | no | — |
| 215 | C3⋊D3 | — | 81 | 3 | no | — |
| 216 | C3⋊D3 | — | 81 | 27 | no | — |
| 217 | C3×D3 | — | 81 | 3 | no | — |
| 218 | C3×D3 | — | 81 | 9 | no | — |
| 219 | C3×D3 | — | 81 | 3 | no | — |
| 220 | C3×D3 | — | 81 | 1 | no | — |
| 221 | C3×D3 | — | 243 | 3 | no | — |
| 222 | C3×D3 | — | 243 | 9 | no | — |
| 223 | D9 | — | 81 | 9 | no | — |
| 224 | C32 | — | 2 | 3 | no | — |
| 225 | C32 | — | 2 | 1 | no | — |
| 226 | C32 | — | 3 | 27 | no | — |
| 227 | C32 | — | 3 | 6 | no | — |
| 228 | C32 | — | 3 | 3 | no | — |
| 229 | C32 | — | 3 | 3 | no | — |
| 230 | C32 | — | 3 | 3 | no | — |
| 231 | C32 | — | 6 | 3 | no | — |
| 232 | C32 | — | 6 | 3 | no | — |
| 233 | C32 | — | 6 | 6 | no | — |
| 234 | C32 | — | 6 | 3 | no | — |
| 235 | C32 | — | 9 | 3 | no | — |
| 236 | C32 | — | 9 | 9 | no | — |
| 237 | C32 | — | 9 | 9 | no | — |
| 238 | C32 | — | 9 | 3 | no | — |
| 239 | C32 | — | 18 | 3 | no | — |
| 240 | C32 | — | 18 | 9 | no | — |
| 241 | C32 | — | 18 | 27 | no | — |
| 242 | C32 | — | 18 | 9 | no | — |
| 243 | C32 | — | 18 | 9 | no | — |
| 244 | C32 | — | 18 | 3 | no | — |
| 245 | C32 | — | 27 | 1 | no | — |
| 246 | C32 | — | 54 | 1 | no | — |
| 247 | C32 | — | 54 | 3 | no | — |
| 248 | C32 | — | 54 | 3 | no | — |
| 249 | C32 | — | 54 | 1 | no | — |
| 250 | C32 | — | 54 | 3 | no | — |
| 251 | C32 | — | 54 | 2 | no | — |
| 252 | C32 | #4177⋊C3 | 1 | 9 | yes | complemented |
| 253 | C32 | #92301 | 1 | 1 | yes | characteristic |
| 254 | C32 | C3×C3⋊((C3⋊D3)⋊C3) | 1 | 3 | yes | — |
| 255 | C32 | C3×C3⋊((C3⋊D3)⋊C3) | 1 | 1 | yes | characteristic |
| 256 | C9 | — | 3 | 9 | no | — |
| 257 | C9 | — | 6 | 9 | no | — |
| 258 | C9 | — | 18 | 9 | no | — |
| 259 | C2×C3 | — | 243 | 1 | no | — |
| 260 | C2×C3 | — | 243 | 3 | no | — |
| 261 | D3 | — | 81 | 3 | no | — |
| 262 | D3 | — | 81 | 1 | no | — |
| 263 | D3 | — | 243 | 9 | no | — |
| 264 | D3 | — | 243 | 3 | no | — |
| 265 | C3 | — | 3 | 9 | no | — |
| 266 | C3 | — | 3 | 3 | no | — |
| 267 | C3 | — | 3 | 1 | no | — |
| 268 | C3 | — | 6 | 1 | no | — |
| 269 | C3 | — | 6 | 3 | no | — |
| 270 | C3 | — | 18 | 3 | no | — |
| 271 | C3 | — | 27 | 3 | no | — |
| 272 | C3 | — | 54 | 3 | no | — |
| 273 | C3 | C3×#92301 | 1 | 1 | yes | characteristic |
| 274 | C3 | C3⋊(#4177⋊C3) | 1 | 3 | yes | — |
| 275 | C2 | — | 243 | 1 | no | Sylow |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C33×C9 | C2×C32 |
| Frattini | C33 | C32×C3⋊D3 |
| Fitting | #14554265 | C2 |
| socle | C32 | C3×C3⋊((C3⋊D3)⋊C3) |
| LCS | G ▷ C33×C9 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #14554265 | C2 | semidirect |
| 2. C3⋊(C3⋊(C3⋊AGL(1,ℤ/9))) | C3 | semidirect |
| 5. #13091042 | C3 | semidirect |
| 6. C33×3−1+2 | C2×C3 | semidirect |
| 8. C3×#6910 | D3 | semidirect |
| 13. #10624731 | C2×C3 | semidirect |
| 15. C3⋊(C3⋊(C3⋊D9)) | C32 | semidirect |
| 36. C33×C9 | C2×C32 | semidirect |
| 45. C3×#731 | C3×D3 | semidirect |
| 101. C3×3−1+2 | (C3⋊D3)⋊C3 | semidirect |
| 208. C3×C9 | C3×(C3⋊D3)⋊C3 | semidirect |
| 252. C32 | #4177⋊C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #14554265 | 2 |
| C3⋊(C3⋊(C3⋊AGL(1,ℤ/9))) | 3 |
| C3⋊(#4177⋊C3) | 3 |
| C3×#92301 | 3 |
| #13091042 | 3 |