Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
3028 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 3028 conjugacy classes of subgroups — too many to draw. Only its 72 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 | D9×C22⋊A4 | C2 | 1 | 2 | yes | maximal, direct product factor |
| 3 | C2×D3×C22⋊A4 | — | 3 | 1 | no | maximal |
| 4 | C25×D9 | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | C9×C22⋊A4 | C22 | 1 | 1 | yes | characteristic, complemented |
| 6 | C2×A4×D9 | — | 4 | 5 | no | maximal |
| 7 | C2×C3×C22⋊A4 | D3 | 1 | 1 | yes | characteristic |
| 8 | D3×C22⋊A4 | — | 3 | 2 | no | — |
| 9 | C25×C9 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 10 | C24×D9 | — | 3 | 5 | no | — |
| 11 | C24×D9 | — | 3 | 5 | no | — |
| 12 | C24×D9 | — | 3 | 10 | no | — |
| 13 | C24×D9 | C2×C3 | 1 | 2 | yes | complemented |
| 14 | C2×C22⋊(C22⋊C9) | — | 2 | 1 | no | — |
| 15 | C2×C9×A4 | — | 4 | 5 | no | — |
| 16 | A4×D9 | — | 4 | 10 | no | — |
| 17 | C25×D3 | — | 3 | 1 | no | — |
| 18 | C22×C22⋊A4 | — | 9 | 1 | no | — |
| 19 | C3×C22⋊A4 | C2×D3 | 1 | 1 | yes | characteristic |
| 20 | C2×D3×A4 | — | 12 | 5 | no | — |
| 21 | C24×C9 | — | 3 | 5 | no | — |
| 22 | C24×C9 | — | 3 | 5 | no | — |
| 23 | C24×C9 | C22×C3 | 1 | 1 | yes | characteristic, complemented |
| 24 | C23×D9 | — | 3 | 60 | no | — |
| 25 | C23×D9 | — | 3 | 60 | no | — |
| 26 | C23×D9 | — | 3 | 30 | no | — |
| 27 | C23×D9 | — | 3 | 10 | no | — |
| 28 | C23×D9 | — | 3 | 10 | no | — |
| 29 | C23×D9 | — | 3 | 5 | no | — |
| 30 | C23×D9 | — | 3 | 10 | no | — |
| 31 | C23×D9 | — | 3 | 10 | no | — |
| 32 | C23×D9 | — | 3 | 10 | no | — |
| 33 | C23×D9 | A4 | 1 | 5 | yes | complemented |
| 34 | C22⋊(C22⋊C9) | — | 2 | 1 | no | — |
| 35 | C2×C3×D9 | — | 16 | 1 | no | — |
| 36 | C9×A4 | — | 4 | 5 | no | — |
| 37 | C25×C3 | C3×D3 | 1 | 1 | yes | characteristic |
| 38 | C24×D3 | — | 3 | 2 | no | — |
| 39 | C24×D3 | — | 9 | 5 | no | — |
| 40 | C24×D3 | — | 9 | 10 | no | — |
| 41 | C24×D3 | — | 9 | 5 | no | — |
| 42 | C2×C22⋊A4 | — | 2 | 1 | no | — |
| 43 | C2×C22⋊A4 | — | 9 | 2 | no | — |
| 44 | C2×C22⋊A4 | D9 | 1 | 1 | yes | characteristic, direct product factor |
| 45 | C2×C3×A4 | — | 4 | 5 | no | — |
| 46 | D3×A4 | — | 12 | 10 | no | — |
| 47 | C23×C9 | — | 3 | 30 | no | — |
| 48 | C23×C9 | — | 3 | 10 | no | — |
| 49 | C23×C9 | — | 3 | 5 | no | — |
| 50 | C23×C9 | — | 3 | 5 | no | — |
| 51 | C23×C9 | C2×A4 | 1 | 5 | yes | complemented |
| 52 | C22×D9 | — | 3 | 30 | no | — |
| 53 | C22×D9 | — | 3 | 120 | no | — |
| 54 | C22×D9 | — | 3 | 60 | no | — |
| 55 | C22×D9 | — | 3 | 60 | no | — |
| 56 | C22×D9 | — | 3 | 30 | no | — |
| 57 | C22×D9 | — | 3 | 60 | no | — |
| 58 | C22×D9 | — | 3 | 20 | no | — |
| 59 | C22×D9 | — | 3 | 10 | no | — |
| 60 | C22×D9 | — | 3 | 5 | no | — |
| 61 | C22×D9 | — | 3 | 5 | no | — |
| 62 | C22×D9 | — | 3 | 10 | no | — |
| 63 | C22×D9 | C2×A4 | 1 | 10 | yes | complemented |
| 64 | C2×C22⋊C9 | — | 8 | 5 | no | — |
| 65 | C26 | — | 9 | 1 | no | Sylow |
| 66 | C2×C3×C9 | — | 16 | 1 | no | — |
| 67 | C3×D9 | — | 16 | 2 | no | — |
| 68 | C24×C3 | — | 3 | 5 | no | — |
| 69 | C24×C3 | — | 3 | 5 | no | — |
| 70 | C24×C3 | C2×C3×D3 | 1 | 1 | yes | characteristic |
| 71 | C23×D3 | — | 3 | 5 | no | — |
| 72 | C23×D3 | — | 9 | 10 | no | — |
| 73 | C23×D3 | — | 9 | 10 | no | — |
| 74 | C23×D3 | — | 9 | 10 | no | — |
| 75 | C23×D3 | — | 9 | 60 | no | — |
| 76 | C23×D3 | — | 9 | 60 | no | — |
| 77 | C23×D3 | — | 9 | 30 | no | — |
| 78 | C23×D3 | — | 9 | 10 | no | — |
| 79 | C23×D3 | — | 9 | 5 | no | — |
| 80 | C23×D3 | — | 9 | 10 | no | — |
| 81 | C22⋊A4 | — | 2 | 1 | no | — |
| 82 | C22⋊A4 | C2×D9 | 1 | 1 | yes | characteristic, direct product factor |
| 83 | C22×A4 | — | 36 | 5 | no | — |
| 84 | C2×C3×D3 | — | 48 | 1 | no | — |
| 85 | C3×A4 | — | 4 | 5 | no | — |
| 86 | C22×C9 | — | 3 | 5 | no | — |
| 87 | C22×C9 | — | 3 | 30 | no | — |
| 88 | C22×C9 | — | 3 | 10 | no | — |
| 89 | C22×C9 | — | 3 | 5 | no | — |
| 90 | C22×C9 | C22×A4 | 1 | 5 | yes | complemented |
| 91 | C2×D9 | — | 3 | 60 | no | — |
| 92 | C2×D9 | — | 3 | 10 | no | — |
| 93 | C2×D9 | — | 3 | 60 | no | — |
| 94 | C2×D9 | — | 3 | 5 | no | — |
| 95 | C2×D9 | — | 3 | 10 | no | — |
| 96 | C2×D9 | — | 3 | 10 | no | — |
| 97 | C2×D9 | — | 3 | 10 | no | — |
| 98 | C2×D9 | C22⋊A4 | 1 | 1 | yes | characteristic, direct product factor |
| 99 | C22⋊C9 | — | 8 | 5 | no | — |
| 100 | C25 | — | 9 | 2 | no | — |
| 101 | C25 | — | 27 | 5 | no | — |
| 102 | C25 | — | 27 | 10 | no | — |
| 103 | C25 | — | 27 | 5 | no | — |
| 104 | C25 | C3×D9 | 1 | 1 | yes | characteristic, complemented |
| 105 | C3×C9 | — | 16 | 1 | no | Sylow |
| 106 | C23×C3 | — | 3 | 30 | no | — |
| 107 | C23×C3 | — | 3 | 10 | no | — |
| 108 | C23×C3 | — | 3 | 5 | no | — |
| 109 | C23×C3 | — | 3 | 5 | no | — |
| 110 | C23×C3 | D3×A4 | 1 | 5 | yes | — |
| 111 | C22×D3 | — | 3 | 10 | no | — |
| 112 | C22×D3 | — | 9 | 5 | no | — |
| 113 | C22×D3 | — | 9 | 20 | no | — |
| 114 | C22×D3 | — | 9 | 60 | no | — |
| 115 | C22×D3 | — | 9 | 30 | no | — |
| 116 | C22×D3 | — | 9 | 120 | no | — |
| 117 | C22×D3 | — | 9 | 60 | no | — |
| 118 | C22×D3 | — | 9 | 60 | no | — |
| 119 | C22×D3 | — | 9 | 30 | no | — |
| 120 | C22×D3 | — | 9 | 10 | no | — |
| 121 | C22×D3 | — | 9 | 10 | no | — |
| 122 | C22×D3 | — | 9 | 5 | no | — |
| 123 | C2×A4 | — | 4 | 5 | no | — |
| 124 | C2×A4 | — | 8 | 5 | no | — |
| 125 | C2×A4 | — | 36 | 10 | no | — |
| 126 | C2×C32 | — | 16 | 1 | no | — |
| 127 | C3×D3 | — | 48 | 2 | no | — |
| 128 | C2×C9 | — | 3 | 5 | no | — |
| 129 | C2×C9 | — | 3 | 5 | no | — |
| 130 | C2×C9 | — | 32 | 1 | no | — |
| 131 | C2×C9 | C2×C22⋊A4 | 1 | 1 | yes | characteristic, complemented |
| 132 | D9 | — | 3 | 10 | no | — |
| 133 | D9 | C2×C22⋊A4 | 1 | 2 | yes | direct product factor |
| 134 | C24 | — | 3 | 5 | no | — |
| 135 | C24 | — | 3 | 5 | no | — |
| 136 | C24 | — | 9 | 5 | no | — |
| 137 | C24 | — | 27 | 10 | no | — |
| 138 | C24 | — | 27 | 10 | no | — |
| 139 | C24 | — | 27 | 10 | no | — |
| 140 | C24 | — | 27 | 10 | no | — |
| 141 | C24 | — | 27 | 60 | no | — |
| 142 | C24 | — | 27 | 30 | no | — |
| 143 | C24 | — | 27 | 60 | no | — |
| 144 | C24 | — | 27 | 10 | no | — |
| 145 | C24 | — | 27 | 5 | no | — |
| 146 | C24 | C2×C3×D9 | 1 | 1 | yes | characteristic, complemented |
| 147 | C22×C3 | — | 3 | 5 | no | — |
| 148 | C22×C3 | — | 3 | 10 | no | — |
| 149 | C22×C3 | — | 3 | 30 | no | — |
| 150 | C22×C3 | — | 3 | 5 | no | — |
| 151 | C22×C3 | — | 144 | 1 | no | — |
| 152 | C22×C3 | C2×D3×A4 | 1 | 5 | yes | — |
| 153 | C2×D3 | — | 3 | 1 | no | — |
| 154 | C2×D3 | — | 9 | 10 | no | — |
| 155 | C2×D3 | — | 9 | 10 | no | — |
| 156 | C2×D3 | — | 9 | 60 | no | — |
| 157 | C2×D3 | — | 9 | 60 | no | — |
| 158 | C2×D3 | — | 9 | 10 | no | — |
| 159 | C2×D3 | — | 9 | 5 | no | — |
| 160 | C2×D3 | — | 9 | 10 | no | — |
| 161 | A4 | — | 4 | 5 | no | — |
| 162 | A4 | — | 8 | 5 | no | — |
| 163 | C32 | — | 16 | 1 | no | — |
| 164 | C9 | — | 32 | 1 | no | — |
| 165 | C9 | C22×C22⋊A4 | 1 | 1 | yes | characteristic, complemented |
| 166 | C23 | — | 3 | 30 | no | — |
| 167 | C23 | — | 3 | 10 | no | — |
| 168 | C23 | — | 3 | 5 | no | — |
| 169 | C23 | — | 3 | 5 | no | — |
| 170 | C23 | — | 9 | 10 | no | — |
| 171 | C23 | — | 27 | 5 | no | — |
| 172 | C23 | — | 27 | 10 | no | — |
| 173 | C23 | — | 27 | 5 | no | — |
| 174 | C23 | — | 27 | 30 | no | — |
| 175 | C23 | — | 27 | 20 | no | — |
| 176 | C23 | — | 27 | 60 | no | — |
| 177 | C23 | — | 27 | 60 | no | — |
| 178 | C23 | — | 27 | 60 | no | — |
| 179 | C23 | — | 27 | 120 | no | — |
| 180 | C23 | — | 27 | 30 | no | — |
| 181 | C23 | — | 27 | 10 | no | — |
| 182 | C23 | A4×D9 | 1 | 5 | yes | complemented |
| 183 | C2×C3 | — | 3 | 5 | no | — |
| 184 | C2×C3 | — | 3 | 5 | no | — |
| 185 | C2×C3 | — | 16 | 1 | no | — |
| 186 | C2×C3 | — | 32 | 1 | no | — |
| 187 | C2×C3 | — | 144 | 2 | no | — |
| 188 | C2×C3 | D3×C22⋊A4 | 1 | 1 | yes | characteristic |
| 189 | D3 | — | 3 | 2 | no | — |
| 190 | D3 | — | 9 | 10 | no | — |
| 191 | C22 | — | 3 | 5 | no | — |
| 192 | C22 | — | 3 | 10 | no | — |
| 193 | C22 | — | 3 | 30 | no | — |
| 194 | C22 | — | 3 | 5 | no | — |
| 195 | C22 | — | 9 | 1 | no | — |
| 196 | C22 | — | 27 | 10 | no | — |
| 197 | C22 | — | 27 | 10 | no | — |
| 198 | C22 | — | 27 | 10 | no | — |
| 199 | C22 | — | 27 | 60 | no | — |
| 200 | C22 | — | 27 | 10 | no | — |
| 201 | C22 | — | 27 | 60 | no | — |
| 202 | C22 | — | 27 | 5 | no | — |
| 203 | C22 | C2×A4×D9 | 1 | 5 | yes | complemented |
| 204 | C3 | — | 16 | 1 | no | — |
| 205 | C3 | — | 32 | 1 | no | — |
| 206 | C3 | C2×D3×C22⋊A4 | 1 | 1 | yes | characteristic |
| 207 | C2 | — | 3 | 5 | no | — |
| 208 | C2 | — | 3 | 5 | no | — |
| 209 | C2 | — | 9 | 2 | no | — |
| 210 | C2 | — | 27 | 10 | no | — |
| 211 | C2 | D9×C22⋊A4 | 1 | 1 | yes | characteristic, direct product factor |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D9×C22⋊A4 |
| commutator, cosocle, hypocenter | C24×C9 | C22×C3 |
| Frattini | C3 | C2×D3×C22⋊A4 |
| Fitting | C25×C9 | C2×C3 |
| socle | C25×C3 | C3×D3 |
| 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. D9×C22⋊A4 | C2 | direct (N × H) |
| 4. C25×D9 | C3 | semidirect |
| 5. C9×C22⋊A4 | C22 | semidirect |
| 9. C25×C9 | C2×C3 | semidirect |
| 13. C24×D9 | C2×C3 | semidirect |
| 23. C24×C9 | C22×C3 | semidirect |
| 33. C23×D9 | A4 | semidirect |
| 44. C2×C22⋊A4 | D9 | direct (N × H) |
| 51. C23×C9 | C2×A4 | semidirect |
| 63. C22×D9 | C2×A4 | semidirect |
| 82. C22⋊A4 | C2×D9 | direct (N × H) |
| 90. C22×C9 | C22×A4 | semidirect |
| 98. C2×D9 | C22⋊A4 | direct (N × H) |
| 104. C25 | C3×D9 | semidirect |
| 131. C2×C9 | C2×C22⋊A4 | semidirect |
| 133. D9 | C2×C22⋊A4 | direct (N × H) |
| 146. C24 | C2×C3×D9 | semidirect |
| 165. C9 | C22×C22⋊A4 | semidirect |
| 182. C23 | A4×D9 | semidirect |
| 203. C22 | C2×A4×D9 | semidirect |
| 211. C2 | D9×C22⋊A4 | 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 |
| D9×C22⋊A4 | 2 |
| C2×D3×C22⋊A4 | 3 |
| C25×D9 | 3 |
| C2×A4×D9 | 4 |