Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
948 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 948 conjugacy classes of subgroups — too many to draw. Only its 119 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 | #11724627 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | C3×#64289 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #11724340 | C2 | 1 | 6 | yes | maximal, complemented |
| 4 | #11724337 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | #11724336 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | #10596522 | — | 3 | 1 | no | maximal |
| 7 | C3×#3832 | — | 2 | 2 | no | — |
| 8 | C3×#3832 | C22 | 1 | 1 | yes | characteristic |
| 9 | C4⋊#1113 | — | 2 | 6 | no | — |
| 10 | C4⋊#1113 | C22 | 1 | 6 | yes | complemented |
| 11 | #89042 | C22 | 1 | 6 | yes | complemented |
| 12 | C3×#3788 | C22 | 1 | 3 | yes | complemented |
| 13 | C3×#3787 | C22 | 1 | 3 | yes | complemented |
| 14 | #88943 | C22 | 1 | 1 | yes | characteristic, complemented |
| 15 | #88942 | C22 | 1 | 6 | yes | complemented |
| 16 | #88941 | C22 | 1 | 3 | yes | complemented |
| 17 | #88941 | C22 | 1 | 6 | yes | complemented |
| 18 | #64320 | — | 3 | 2 | no | — |
| 19 | #64320 | — | 6 | 2 | no | — |
| 20 | #64289 | D3 | 1 | 1 | yes | characteristic, complemented |
| 21 | #64225 | — | 3 | 3 | no | — |
| 22 | #64224 | — | 3 | 6 | no | — |
| 23 | #64223 | — | 3 | 3 | no | — |
| 24 | C2×#1113 | — | 2 | 6 | no | — |
| 25 | C33×Q8 | D4 | 1 | 1 | yes | characteristic |
| 26 | C3×C4⋊(C3⋊D3) | — | 2 | 6 | no | — |
| 27 | C3×C4⋊(C3⋊D3) | D4 | 1 | 3 | yes | — |
| 28 | C3×C4×C3⋊D3 | — | 2 | 6 | no | — |
| 29 | C3×C4×C3⋊D3 | C23 | 1 | 3 | yes | — |
| 30 | #6296 | C23 | 1 | 2 | yes | — |
| 31 | #6295 | — | 2 | 6 | no | — |
| 32 | #6295 | C23 | 1 | 6 | yes | — |
| 33 | C3×#622 | C23 | 1 | 1 | yes | characteristic |
| 34 | C3×#622 | C23 | 1 | 3 | yes | — |
| 35 | #5752 | — | 9 | 1 | no | maximal |
| 36 | #3832 | — | 2 | 2 | no | — |
| 37 | #3832 | C2×D3 | 1 | 1 | yes | characteristic |
| 38 | C3×#239 | — | 6 | 2 | no | — |
| 39 | C3×#239 | — | 6 | 2 | no | — |
| 40 | C3×#239 | — | 6 | 2 | no | — |
| 41 | C3×#239 | — | 12 | 2 | no | — |
| 42 | #3802 | — | 3 | 6 | no | — |
| 43 | #3802 | — | 6 | 6 | no | — |
| 44 | D3×D12 | — | 6 | 6 | no | — |
| 45 | D3×D12 | — | 6 | 6 | no | — |
| 46 | D3×D12 | — | 6 | 6 | no | — |
| 47 | #3798 | — | 3 | 6 | no | — |
| 48 | #3798 | — | 6 | 6 | no | — |
| 49 | #3788 | C2×D3 | 1 | 3 | yes | complemented |
| 50 | #3787 | C2×D3 | 1 | 3 | yes | complemented |
| 51 | #3776 | — | 3 | 1 | no | — |
| 52 | #3775 | — | 3 | 6 | no | — |
| 53 | #3774 | — | 3 | 3 | no | — |
| 54 | #3774 | — | 3 | 6 | no | — |
| 55 | C2×C3×C3⋊D3 | — | 2 | 6 | no | — |
| 56 | C2×C3×C3⋊D3 | C2×D4 | 1 | 3 | yes | — |
| 57 | #1113 | — | 8 | 6 | no | — |
| 58 | C33×C4 | C2×D4 | 1 | 3 | yes | — |
| 59 | C3×C3⋊Dic3 | — | 2 | 2 | no | — |
| 60 | C3×C3⋊Dic3 | C24 | 1 | 1 | yes | characteristic |
| 61 | #1009 | — | 9 | 2 | no | — |
| 62 | #1009 | — | 18 | 2 | no | — |
| 63 | C3×#131 | — | 9 | 1 | no | — |
| 64 | #908 | — | 9 | 6 | no | — |
| 65 | #904 | — | 9 | 3 | no | — |
| 66 | C2×D24 | — | 9 | 3 | no | — |
| 67 | C2×D32 | — | 6 | 12 | no | — |
| 68 | C2×D32 | — | 6 | 6 | no | — |
| 69 | C32×Q8 | — | 2 | 2 | no | — |
| 70 | C32×Q8 | — | 4 | 2 | no | — |
| 71 | C32×Q8 | D12 | 1 | 1 | yes | characteristic |
| 72 | C4⋊(C3⋊D3) | — | 2 | 6 | no | — |
| 73 | C4⋊(C3⋊D3) | D12 | 1 | 3 | yes | — |
| 74 | C4×C3⋊D3 | — | 2 | 6 | no | — |
| 75 | C4×C3⋊D3 | C22×D3 | 1 | 3 | yes | — |
| 76 | C3×D12 | — | 6 | 6 | no | — |
| 77 | C3×D12 | — | 6 | 6 | no | — |
| 78 | C3×D12 | — | 6 | 6 | no | — |
| 79 | C3×D12 | — | 12 | 6 | no | — |
| 80 | C3×C4×D3 | — | 6 | 6 | no | — |
| 81 | C3×C4×D3 | — | 6 | 6 | no | — |
| 82 | C3×C4×D3 | — | 6 | 6 | no | — |
| 83 | C3×C4×D3 | — | 12 | 6 | no | — |
| 84 | #627 | — | 3 | 2 | no | — |
| 85 | #627 | — | 6 | 2 | no | — |
| 86 | #626 | — | 6 | 6 | no | — |
| 87 | #626 | — | 6 | 6 | no | — |
| 88 | #626 | — | 6 | 6 | no | — |
| 89 | #625 | — | 3 | 6 | no | — |
| 90 | #625 | — | 6 | 6 | no | — |
| 91 | #622 | C22×D3 | 1 | 1 | yes | characteristic |
| 92 | #622 | C22×D3 | 1 | 3 | yes | — |
| 93 | #576 | — | 27 | 1 | no | Sylow |
| 94 | C2×C33 | C22×D4 | 1 | 1 | yes | characteristic |
| 95 | C3×C3⋊D3 | — | 2 | 3 | no | — |
| 96 | C3×C3⋊D3 | — | 4 | 6 | no | — |
| 97 | C3×#41 | — | 9 | 1 | no | — |
| 98 | C3×#41 | — | 18 | 2 | no | — |
| 99 | #239 | — | 6 | 2 | no | — |
| 100 | #239 | — | 6 | 2 | no | — |
| 101 | #239 | — | 6 | 2 | no | — |
| 102 | #239 | — | 12 | 2 | no | — |
| 103 | D3×D4 | — | 18 | 6 | no | — |
| 104 | D3×D4 | — | 18 | 6 | no | — |
| 105 | D3×D4 | — | 18 | 6 | no | — |
| 106 | #235 | — | 9 | 6 | no | — |
| 107 | #235 | — | 18 | 6 | no | — |
| 108 | C2×D12 | — | 9 | 6 | no | — |
| 109 | C2×D12 | — | 18 | 6 | no | — |
| 110 | C3×M16 | — | 9 | 3 | no | — |
| 111 | C2×C3×C8 | — | 9 | 3 | no | — |
| 112 | Dic12 | — | 9 | 1 | no | — |
| 113 | D24 | — | 9 | 3 | no | — |
| 114 | D24 | — | 9 | 6 | no | — |
| 115 | #204 | — | 9 | 6 | no | — |
| 116 | C2×C3⋊D3 | — | 2 | 6 | no | — |
| 117 | C2×C3⋊D3 | C2×D12 | 1 | 3 | yes | — |
| 118 | C2×C3×D3 | — | 6 | 6 | no | — |
| 119 | C2×C3×D3 | — | 6 | 6 | no | — |
| 120 | C2×C3×D3 | — | 6 | 6 | no | — |
| 121 | C2×C3×D3 | — | 12 | 6 | no | — |
| 122 | D32 | — | 24 | 6 | no | — |
| 123 | D32 | — | 24 | 12 | no | — |
| 124 | C32×C4 | — | 2 | 6 | no | — |
| 125 | C32×C4 | — | 4 | 6 | no | — |
| 126 | C32×C4 | C2×D12 | 1 | 3 | yes | — |
| 127 | C3⋊Dic3 | — | 2 | 2 | no | — |
| 128 | C3⋊Dic3 | C23×D3 | 1 | 1 | yes | characteristic |
| 129 | C3×Dic3 | — | 6 | 2 | no | — |
| 130 | C3×Dic3 | — | 6 | 2 | no | — |
| 131 | C3×Dic3 | — | 6 | 2 | no | — |
| 132 | C3×Dic3 | — | 12 | 2 | no | — |
| 133 | H(5,𝔽2) | — | 27 | 2 | no | — |
| 134 | AGL(1,ℤ/8) | — | 27 | 6 | no | — |
| 135 | #135 | — | 27 | 3 | no | — |
| 136 | C2×D8 | — | 27 | 3 | no | — |
| 137 | #131 | — | 9 | 1 | no | — |
| 138 | C33 | #576 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 139 | C22×D3 | — | 18 | 6 | no | — |
| 140 | C22×D3 | — | 18 | 12 | no | — |
| 141 | C3×Q8 | — | 2 | 2 | no | — |
| 142 | C3×Q8 | — | 4 | 2 | no | — |
| 143 | C3×Q8 | GO+(4,𝔽2) | 1 | 1 | yes | characteristic |
| 144 | C3×D4 | — | 9 | 3 | no | — |
| 145 | C3×D4 | — | 18 | 6 | no | — |
| 146 | C2×C3×C4 | — | 9 | 3 | no | — |
| 147 | C2×C3×C4 | — | 18 | 6 | no | — |
| 148 | #67 | — | 18 | 6 | no | — |
| 149 | #67 | — | 18 | 6 | no | — |
| 150 | #67 | — | 18 | 6 | no | — |
| 151 | D12 | — | 6 | 6 | no | — |
| 152 | D12 | — | 6 | 6 | no | — |
| 153 | D12 | — | 6 | 6 | no | — |
| 154 | D12 | — | 9 | 6 | no | — |
| 155 | D12 | — | 12 | 6 | no | — |
| 156 | D12 | — | 18 | 6 | no | — |
| 157 | C4×D3 | — | 6 | 6 | no | — |
| 158 | C4×D3 | — | 6 | 6 | no | — |
| 159 | C4×D3 | — | 6 | 6 | no | — |
| 160 | C4×D3 | — | 12 | 6 | no | — |
| 161 | Dic6 | — | 9 | 2 | no | — |
| 162 | Dic6 | — | 18 | 2 | no | — |
| 163 | C3×C8 | — | 9 | 3 | no | — |
| 164 | C3×C8 | — | 9 | 1 | no | — |
| 165 | C2×C32 | — | 2 | 2 | no | — |
| 166 | C2×C32 | — | 4 | 2 | no | — |
| 167 | C2×C32 | C22×D12 | 1 | 1 | yes | characteristic |
| 168 | C3⋊D3 | — | 2 | 3 | no | — |
| 169 | C3⋊D3 | — | 4 | 6 | no | — |
| 170 | C3×D3 | — | 12 | 6 | no | — |
| 171 | C3×D3 | — | 12 | 6 | no | — |
| 172 | C3×D3 | — | 12 | 6 | no | — |
| 173 | C3×D3 | — | 24 | 6 | no | — |
| 174 | #41 | — | 9 | 1 | no | — |
| 175 | #41 | — | 18 | 2 | no | — |
| 176 | #41 | — | 27 | 6 | no | — |
| 177 | C2×D4 | — | 27 | 6 | no | — |
| 178 | C2×D4 | — | 54 | 6 | no | — |
| 179 | Q16 | — | 27 | 1 | no | — |
| 180 | SD16 | — | 27 | 6 | no | — |
| 181 | D8 | — | 27 | 3 | no | — |
| 182 | D8 | — | 27 | 6 | no | — |
| 183 | M16 | — | 9 | 3 | no | — |
| 184 | C2×C8 | — | 9 | 3 | no | — |
| 185 | C22×C3 | — | 9 | 3 | no | — |
| 186 | C22×C3 | — | 18 | 6 | no | — |
| 187 | C2×D3 | — | 6 | 6 | no | — |
| 188 | C2×D3 | — | 6 | 6 | no | — |
| 189 | C2×D3 | — | 6 | 6 | no | — |
| 190 | C2×D3 | — | 12 | 6 | no | — |
| 191 | C2×D3 | — | 72 | 6 | no | — |
| 192 | C2×D3 | — | 72 | 12 | no | — |
| 193 | C3×C4 | — | 2 | 6 | no | — |
| 194 | C3×C4 | — | 4 | 6 | no | — |
| 195 | C3×C4 | — | 9 | 1 | no | — |
| 196 | C3×C4 | — | 18 | 2 | no | — |
| 197 | C3×C4 | C2×GO+(4,𝔽2) | 1 | 3 | yes | — |
| 198 | Dic3 | — | 6 | 2 | no | — |
| 199 | Dic3 | — | 6 | 2 | no | — |
| 200 | Dic3 | — | 6 | 2 | no | — |
| 201 | Dic3 | — | 12 | 2 | no | — |
| 202 | C32 | — | 2 | 2 | no | — |
| 203 | C32 | — | 4 | 2 | no | — |
| 204 | C32 | #5752 | 1 | 1 | yes | characteristic, complemented |
| 205 | C23 | — | 54 | 6 | no | — |
| 206 | Q8 | — | 27 | 2 | no | — |
| 207 | Q8 | #157⋊D3 | 1 | 1 | yes | characteristic |
| 208 | D4 | — | 9 | 3 | no | — |
| 209 | D4 | — | 18 | 6 | no | — |
| 210 | D4 | — | 27 | 6 | no | — |
| 211 | D4 | — | 54 | 6 | no | — |
| 212 | C2×C4 | — | 9 | 3 | no | — |
| 213 | C2×C4 | — | 18 | 6 | no | — |
| 214 | C8 | — | 9 | 1 | no | — |
| 215 | C8 | — | 9 | 3 | no | — |
| 216 | C2×C3 | — | 2 | 2 | no | — |
| 217 | C2×C3 | — | 4 | 2 | no | — |
| 218 | C2×C3 | — | 18 | 3 | no | — |
| 219 | C2×C3 | — | 36 | 6 | no | — |
| 220 | C2×C3 | C22×GO+(4,𝔽2) | 1 | 1 | yes | characteristic |
| 221 | D3 | — | 12 | 6 | no | — |
| 222 | D3 | — | 12 | 6 | no | — |
| 223 | D3 | — | 12 | 6 | no | — |
| 224 | D3 | — | 24 | 6 | no | — |
| 225 | C22 | — | 9 | 3 | no | — |
| 226 | C22 | — | 18 | 6 | no | — |
| 227 | C22 | — | 216 | 6 | no | — |
| 228 | C4 | — | 9 | 1 | no | — |
| 229 | C4 | — | 18 | 2 | no | — |
| 230 | C4 | C2×#157⋊D3 | 1 | 3 | yes | — |
| 231 | C3 | — | 2 | 2 | no | — |
| 232 | C3 | — | 4 | 2 | no | — |
| 233 | C3 | #10596522 | 1 | 1 | yes | characteristic, complemented |
| 234 | C2 | — | 18 | 3 | no | — |
| 235 | C2 | — | 36 | 6 | no | — |
| 236 | C2 | C22×#157⋊D3 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×#157⋊D3 |
| commutator, cosocle | C3×C3⋊Dic3 | C24 |
| Fitting | C33×Q8 | D4 |
| socle | C2×C33 | C22×D4 |
| hypercenter | Q8 | #157⋊D3 |
| hypocenter | C33 | #576 |
| LCS | G ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #11724627 | C2 | semidirect |
| 2. C3×#64289 | C2 | semidirect |
| 3. #11724340 | C2 | semidirect |
| 4. #11724337 | C2 | semidirect |
| 5. #11724336 | C2 | semidirect |
| 10. C4⋊#1113 | C22 | semidirect |
| 11. #89042 | C22 | semidirect |
| 12. C3×#3788 | C22 | semidirect |
| 13. C3×#3787 | C22 | semidirect |
| 14. #88943 | C22 | semidirect |
| 15. #88942 | C22 | semidirect |
| 16. #88941 | C22 | semidirect |
| 17. #88941 | C22 | semidirect |
| 20. #64289 | D3 | semidirect |
| 49. #3788 | C2×D3 | semidirect |
| 50. #3787 | C2×D3 | semidirect |
| 138. C33 | #576 | semidirect |
| 204. C32 | #5752 | semidirect |
| 233. C3 | #10596522 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11724627 | 2 |
| C3×#64289 | 2 |
| #11724340 | 2 |
| #11724337 | 2 |
| #11724336 | 2 |
| #10596522 | 3 |
| #5752 | 9 |