Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
440 conjugacy classes of subgroups; only the 31 normal subgroups are shown.
This group has 440 conjugacy classes of subgroups — too many to draw. Only its 31 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 | C2×C9⋊(C22⋊A4) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #11720760 | C2 | 1 | 2 | yes | maximal, direct product factor |
| 3 | C2×C3⋊(#120⋊C3) | — | 3 | 1 | no | maximal |
| 4 | C2×#63515 | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | C9⋊(C22⋊A4) | C22 | 1 | 1 | yes | characteristic, complemented |
| 6 | C2×D9⋊A4 | — | 4 | 1 | no | maximal |
| 7 | C2×C3×C22⋊A4 | D3 | 1 | 1 | yes | characteristic |
| 8 | C25×C9 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 9 | C2×C22⋊(C22⋊C9) | — | 2 | 1 | no | — |
| 10 | C3⋊(#120⋊C3) | — | 3 | 2 | no | — |
| 11 | C22×#611 | — | 3 | 1 | no | — |
| 12 | #63515 | — | 3 | 2 | no | — |
| 13 | #63515 | C2×C3 | 1 | 2 | yes | complemented |
| 14 | C2×#3676 | — | 3 | 1 | no | — |
| 15 | C2×C9⋊A4 | — | 4 | 1 | no | — |
| 16 | C2×C9⋊A4 | — | 8 | 2 | no | — |
| 17 | D9⋊A4 | — | 4 | 2 | no | — |
| 18 | C2×#953 | — | 3 | 1 | no | — |
| 19 | C2×#120⋊C3 | — | 9 | 1 | no | — |
| 20 | C3×C22⋊A4 | C2×D3 | 1 | 1 | yes | characteristic |
| 21 | C2×D3×A4 | — | 12 | 1 | no | — |
| 22 | C24×C9 | — | 3 | 1 | no | — |
| 23 | C24×C9 | — | 3 | 1 | no | — |
| 24 | C24×C9 | — | 6 | 2 | no | — |
| 25 | C24×C9 | — | 6 | 2 | no | — |
| 26 | C24×C9 | C22×C3 | 1 | 1 | yes | characteristic, complemented |
| 27 | C23×D9 | A4 | 1 | 1 | yes | characteristic, complemented |
| 28 | C22⋊(C22⋊C9) | — | 2 | 1 | no | — |
| 29 | C2×#611 | — | 3 | 2 | no | — |
| 30 | C2×#611 | — | 3 | 2 | no | — |
| 31 | C2×#611 | — | 6 | 2 | no | — |
| 32 | C2×#611 | — | 6 | 2 | no | — |
| 33 | C22×Dic9 | — | 3 | 1 | no | — |
| 34 | #3676 | — | 3 | 2 | no | — |
| 35 | #3676 | — | 3 | 2 | no | — |
| 36 | C2×AGL(1,ℤ/9) | — | 16 | 1 | no | — |
| 37 | C9⋊A4 | — | 4 | 1 | no | — |
| 38 | C9⋊A4 | — | 8 | 2 | no | — |
| 39 | C25×C3 | C3×D3 | 1 | 1 | yes | characteristic |
| 40 | C2×C22⋊A4 | — | 3 | 1 | no | — |
| 41 | C22×#67 | — | 9 | 1 | no | — |
| 42 | #953 | — | 3 | 2 | no | — |
| 43 | #953 | — | 9 | 2 | no | — |
| 44 | C2×#217 | — | 9 | 1 | no | — |
| 45 | #120⋊C3 | — | 9 | 2 | no | — |
| 46 | C2×C3×A4 | — | 4 | 1 | no | — |
| 47 | C2×C3×A4 | — | 8 | 2 | no | — |
| 48 | D3×A4 | — | 12 | 2 | no | — |
| 49 | C23×C9 | — | 2 | 2 | no | — |
| 50 | C23×C9 | — | 3 | 2 | no | — |
| 51 | C23×C9 | — | 3 | 2 | no | — |
| 52 | C23×C9 | — | 3 | 1 | no | — |
| 53 | C23×C9 | — | 3 | 1 | no | — |
| 54 | C23×C9 | — | 6 | 2 | no | — |
| 55 | C23×C9 | — | 6 | 2 | no | — |
| 56 | C23×C9 | — | 6 | 2 | no | — |
| 57 | C23×C9 | — | 6 | 2 | no | — |
| 58 | C23×C9 | — | 6 | 2 | no | — |
| 59 | C23×C9 | — | 6 | 4 | no | — |
| 60 | C23×C9 | — | 6 | 2 | no | — |
| 61 | C23×C9 | — | 6 | 2 | no | — |
| 62 | C23×C9 | — | 6 | 4 | no | — |
| 63 | C23×C9 | C2×A4 | 1 | 1 | yes | characteristic |
| 64 | C22×D9 | — | 6 | 1 | no | — |
| 65 | C22×D9 | — | 6 | 1 | no | — |
| 66 | C22×D9 | C2×A4 | 1 | 2 | yes | complemented |
| 67 | C2×C22⋊C9 | — | 8 | 4 | no | — |
| 68 | C2×C22⋊C9 | — | 8 | 1 | no | — |
| 69 | #611 | — | 6 | 4 | no | — |
| 70 | #611 | — | 6 | 4 | no | — |
| 71 | C2×Dic9 | — | 3 | 2 | no | — |
| 72 | C2×Dic9 | — | 6 | 1 | no | — |
| 73 | C2×Dic9 | — | 6 | 1 | no | — |
| 74 | C2×#120 | — | 9 | 1 | no | Sylow |
| 75 | C2×3−1+2 | — | 16 | 1 | no | — |
| 76 | AGL(1,ℤ/9) | — | 16 | 2 | no | — |
| 77 | C24×C3 | — | 3 | 1 | no | — |
| 78 | C24×C3 | — | 3 | 1 | no | — |
| 79 | C24×C3 | — | 6 | 2 | no | — |
| 80 | C24×C3 | — | 6 | 2 | no | — |
| 81 | C24×C3 | C2×C3×D3 | 1 | 1 | yes | characteristic |
| 82 | C23×D3 | — | 3 | 1 | no | — |
| 83 | C22⋊A4 | — | 3 | 1 | no | — |
| 84 | C22×A4 | — | 36 | 1 | no | — |
| 85 | C2×#67 | — | 9 | 2 | no | — |
| 86 | C2×#67 | — | 9 | 2 | no | — |
| 87 | C2×#67 | — | 18 | 2 | no | — |
| 88 | C2×#67 | — | 18 | 2 | no | — |
| 89 | C22×Dic3 | — | 9 | 1 | no | — |
| 90 | #217 | — | 9 | 2 | no | — |
| 91 | #217 | — | 9 | 2 | no | — |
| 92 | C2×C3×D3 | — | 48 | 1 | no | — |
| 93 | C3×A4 | — | 4 | 1 | no | — |
| 94 | C3×A4 | — | 8 | 2 | no | — |
| 95 | C22×C9 | — | 2 | 2 | no | — |
| 96 | C22×C9 | — | 3 | 1 | no | — |
| 97 | C22×C9 | — | 3 | 1 | no | — |
| 98 | C22×C9 | — | 3 | 2 | no | — |
| 99 | C22×C9 | — | 3 | 2 | no | — |
| 100 | C22×C9 | — | 6 | 2 | no | — |
| 101 | C22×C9 | — | 6 | 2 | no | — |
| 102 | C22×C9 | — | 6 | 4 | no | — |
| 103 | C22×C9 | — | 6 | 2 | no | — |
| 104 | C22×C9 | — | 6 | 2 | no | — |
| 105 | C22×C9 | — | 6 | 2 | no | — |
| 106 | C22×C9 | — | 6 | 2 | no | — |
| 107 | C22×C9 | — | 6 | 4 | no | — |
| 108 | C22×C9 | — | 6 | 2 | no | — |
| 109 | C22×C9 | C22×A4 | 1 | 1 | yes | characteristic |
| 110 | C2×D9 | — | 4 | 1 | no | — |
| 111 | C2×D9 | — | 6 | 2 | no | — |
| 112 | C2×D9 | — | 12 | 1 | no | — |
| 113 | C22⋊C9 | — | 8 | 4 | no | — |
| 114 | C22⋊C9 | — | 8 | 1 | no | — |
| 115 | Dic9 | — | 6 | 2 | no | — |
| 116 | C25 | AGL(1,ℤ/9) | 1 | 1 | yes | characteristic, complemented |
| 117 | C22×D4 | — | 27 | 1 | no | — |
| 118 | #120 | — | 9 | 2 | no | — |
| 119 | #120 | — | 27 | 2 | no | — |
| 120 | C2×C22⋊C4 | — | 27 | 1 | no | — |
| 121 | 3−1+2 | — | 16 | 1 | no | Sylow |
| 122 | C23×C3 | — | 2 | 2 | no | — |
| 123 | C23×C3 | — | 3 | 2 | no | — |
| 124 | C23×C3 | — | 3 | 2 | no | — |
| 125 | C23×C3 | — | 3 | 1 | no | — |
| 126 | C23×C3 | — | 3 | 1 | no | — |
| 127 | C23×C3 | — | 6 | 2 | no | — |
| 128 | C23×C3 | — | 6 | 2 | no | — |
| 129 | C23×C3 | — | 6 | 2 | no | — |
| 130 | C23×C3 | — | 6 | 2 | no | — |
| 131 | C23×C3 | — | 6 | 2 | no | — |
| 132 | C23×C3 | — | 6 | 4 | no | — |
| 133 | C23×C3 | — | 6 | 2 | no | — |
| 134 | C23×C3 | — | 6 | 2 | no | — |
| 135 | C23×C3 | — | 6 | 4 | no | — |
| 136 | C23×C3 | D3×A4 | 1 | 1 | yes | characteristic |
| 137 | C22×D3 | — | 3 | 2 | no | — |
| 138 | C22×D3 | — | 18 | 1 | no | — |
| 139 | C22×D3 | — | 18 | 1 | no | — |
| 140 | C2×A4 | — | 12 | 1 | no | — |
| 141 | C2×A4 | — | 24 | 2 | no | — |
| 142 | C2×A4 | — | 36 | 2 | no | — |
| 143 | #67 | — | 18 | 4 | no | — |
| 144 | #67 | — | 18 | 4 | no | — |
| 145 | C2×Dic3 | — | 9 | 2 | no | — |
| 146 | C2×Dic3 | — | 18 | 1 | no | — |
| 147 | C2×Dic3 | — | 18 | 1 | no | — |
| 148 | C2×C32 | — | 16 | 1 | no | — |
| 149 | C3×D3 | — | 48 | 2 | no | — |
| 150 | C2×C9 | — | 3 | 1 | no | — |
| 151 | C2×C9 | — | 3 | 1 | no | — |
| 152 | C2×C9 | — | 6 | 2 | no | — |
| 153 | C2×C9 | — | 6 | 2 | no | — |
| 154 | C2×C9 | — | 32 | 1 | no | — |
| 155 | C2×C9 | #120⋊C3 | 1 | 1 | yes | characteristic, complemented |
| 156 | D9 | — | 4 | 2 | no | — |
| 157 | C24 | — | 3 | 1 | no | — |
| 158 | C24 | — | 3 | 1 | no | — |
| 159 | C24 | — | 6 | 2 | no | — |
| 160 | C24 | — | 6 | 2 | no | — |
| 161 | C24 | — | 9 | 1 | no | — |
| 162 | C24 | C2×AGL(1,ℤ/9) | 1 | 1 | yes | characteristic, complemented |
| 163 | C2×D4 | — | 27 | 2 | no | — |
| 164 | C2×D4 | — | 27 | 2 | no | — |
| 165 | C2×D4 | — | 54 | 2 | no | — |
| 166 | C2×D4 | — | 54 | 2 | no | — |
| 167 | C22×C4 | — | 27 | 1 | no | — |
| 168 | C22⋊C4 | — | 27 | 2 | no | — |
| 169 | C22⋊C4 | — | 27 | 2 | no | — |
| 170 | C22×C3 | — | 2 | 2 | no | — |
| 171 | C22×C3 | — | 3 | 1 | no | — |
| 172 | C22×C3 | — | 3 | 1 | no | — |
| 173 | C22×C3 | — | 3 | 2 | no | — |
| 174 | C22×C3 | — | 3 | 2 | no | — |
| 175 | C22×C3 | — | 6 | 2 | no | — |
| 176 | C22×C3 | — | 6 | 2 | no | — |
| 177 | C22×C3 | — | 6 | 4 | no | — |
| 178 | C22×C3 | — | 6 | 2 | no | — |
| 179 | C22×C3 | — | 6 | 2 | no | — |
| 180 | C22×C3 | — | 6 | 2 | no | — |
| 181 | C22×C3 | — | 6 | 2 | no | — |
| 182 | C22×C3 | — | 6 | 4 | no | — |
| 183 | C22×C3 | — | 6 | 2 | no | — |
| 184 | C22×C3 | — | 144 | 1 | no | — |
| 185 | C22×C3 | C2×D3×A4 | 1 | 1 | yes | characteristic |
| 186 | C2×D3 | — | 12 | 1 | no | — |
| 187 | C2×D3 | — | 18 | 2 | no | — |
| 188 | C2×D3 | — | 36 | 1 | no | — |
| 189 | A4 | — | 12 | 1 | no | — |
| 190 | A4 | — | 24 | 2 | no | — |
| 191 | Dic3 | — | 18 | 2 | no | — |
| 192 | C32 | — | 16 | 1 | no | — |
| 193 | C9 | — | 32 | 1 | no | — |
| 194 | C9 | C2×#120⋊C3 | 1 | 1 | yes | characteristic, complemented |
| 195 | C23 | — | 2 | 2 | no | — |
| 196 | C23 | — | 3 | 2 | no | — |
| 197 | C23 | — | 3 | 2 | no | — |
| 198 | C23 | — | 3 | 1 | no | — |
| 199 | C23 | — | 3 | 1 | no | — |
| 200 | C23 | — | 6 | 2 | no | — |
| 201 | C23 | — | 6 | 2 | no | — |
| 202 | C23 | — | 6 | 2 | no | — |
| 203 | C23 | — | 6 | 2 | no | — |
| 204 | C23 | — | 6 | 2 | no | — |
| 205 | C23 | — | 6 | 4 | no | — |
| 206 | C23 | — | 6 | 2 | no | — |
| 207 | C23 | — | 6 | 2 | no | — |
| 208 | C23 | — | 6 | 4 | no | — |
| 209 | C23 | — | 9 | 2 | no | — |
| 210 | C23 | — | 54 | 1 | no | — |
| 211 | C23 | — | 54 | 1 | no | — |
| 212 | C23 | D9⋊A4 | 1 | 1 | yes | characteristic |
| 213 | D4 | — | 54 | 4 | no | — |
| 214 | D4 | — | 54 | 4 | no | — |
| 215 | C2×C4 | — | 27 | 2 | no | — |
| 216 | C2×C4 | — | 54 | 1 | no | — |
| 217 | C2×C4 | — | 54 | 1 | no | — |
| 218 | C2×C3 | — | 3 | 1 | no | — |
| 219 | C2×C3 | — | 3 | 1 | no | — |
| 220 | C2×C3 | — | 6 | 2 | no | — |
| 221 | C2×C3 | — | 6 | 2 | no | — |
| 222 | C2×C3 | — | 48 | 1 | no | — |
| 223 | C2×C3 | — | 144 | 2 | no | — |
| 224 | C2×C3 | C3⋊(#120⋊C3) | 1 | 1 | yes | characteristic |
| 225 | D3 | — | 12 | 2 | no | — |
| 226 | C22 | — | 2 | 2 | no | — |
| 227 | C22 | — | 3 | 1 | no | — |
| 228 | C22 | — | 3 | 1 | no | — |
| 229 | C22 | — | 3 | 2 | no | — |
| 230 | C22 | — | 3 | 2 | no | — |
| 231 | C22 | — | 6 | 2 | no | — |
| 232 | C22 | — | 6 | 2 | no | — |
| 233 | C22 | — | 6 | 4 | no | — |
| 234 | C22 | — | 6 | 2 | no | — |
| 235 | C22 | — | 6 | 2 | no | — |
| 236 | C22 | — | 6 | 2 | no | — |
| 237 | C22 | — | 6 | 2 | no | — |
| 238 | C22 | — | 6 | 4 | no | — |
| 239 | C22 | — | 6 | 2 | no | — |
| 240 | C22 | — | 36 | 1 | no | — |
| 241 | C22 | — | 54 | 2 | no | — |
| 242 | C22 | — | 108 | 1 | no | — |
| 243 | C22 | C2×D9⋊A4 | 1 | 1 | yes | characteristic |
| 244 | C4 | — | 54 | 2 | no | — |
| 245 | C3 | — | 48 | 1 | no | — |
| 246 | C3 | C2×C3⋊(#120⋊C3) | 1 | 1 | yes | characteristic |
| 247 | C2 | — | 3 | 1 | no | — |
| 248 | C2 | — | 3 | 1 | no | — |
| 249 | C2 | — | 6 | 2 | no | — |
| 250 | C2 | — | 6 | 2 | no | — |
| 251 | C2 | — | 36 | 2 | no | — |
| 252 | C2 | #11720760 | 1 | 1 | yes | characteristic, direct product factor |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #11720760 |
| commutator, cosocle, hypocenter | C24×C9 | C22×C3 |
| Frattini | C22×C3 | C2×D3×A4 |
| Fitting | C25×C9 | C2×C3 |
| socle | C23×C3 | D3×A4 |
| LCS | G ▷ C24×C9 | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C9⋊(C22⋊A4) | C2 | semidirect |
| 2. #11720760 | C2 | direct (N × H) |
| 4. C2×#63515 | C3 | semidirect |
| 5. C9⋊(C22⋊A4) | C22 | semidirect |
| 8. C25×C9 | C2×C3 | semidirect |
| 13. #63515 | C2×C3 | semidirect |
| 26. C24×C9 | C22×C3 | semidirect |
| 27. C23×D9 | A4 | semidirect |
| 66. C22×D9 | C2×A4 | semidirect |
| 116. C25 | AGL(1,ℤ/9) | semidirect |
| 155. C2×C9 | #120⋊C3 | semidirect |
| 162. C24 | C2×AGL(1,ℤ/9) | semidirect |
| 194. C9 | C2×#120⋊C3 | semidirect |
| 252. C2 | #11720760 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C9⋊(C22⋊A4) | 2 |
| #11720760 | 2 |
| C2×C3⋊(#120⋊C3) | 3 |
| C2×#63515 | 3 |
| C2×D9⋊A4 | 4 |