Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
898 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 898 conjugacy classes of subgroups — too many to draw. Only its 77 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 | C3⋊(C3⋊#1020) | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | C32×#120⋊C3 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #10593630 | — | 3 | 12 | no | maximal |
| 4 | #10592731 | — | 3 | 1 | no | maximal |
| 5 | C32×C22⋊A4 | C22 | 1 | 1 | yes | characteristic, complemented |
| 6 | C2×C3⋊(C3⋊S4) | — | 4 | 1 | no | maximal |
| 7 | C3⋊#1020 | — | 3 | 24 | no | — |
| 8 | #64361 | — | 3 | 1 | no | — |
| 9 | C32×#120 | D3 | 1 | 1 | yes | characteristic, complemented |
| 10 | #64162 | — | 3 | 2 | no | — |
| 11 | C3×#120⋊C3 | D3 | 1 | 12 | yes | complemented |
| 12 | #63661 | — | 3 | 1 | no | — |
| 13 | #63633 | — | 3 | 2 | no | — |
| 14 | C2×C32×A4 | — | 4 | 1 | no | — |
| 15 | C3⋊(C3⋊S4) | — | 4 | 2 | no | — |
| 16 | C3⋊(C3⋊S4) | — | 8 | 2 | no | — |
| 17 | #1020⋊C2 | — | 9 | 9 | no | — |
| 18 | #5032 | — | 9 | 4 | no | — |
| 19 | C24×C32 | C2×D3 | 1 | 1 | yes | characteristic, complemented |
| 20 | C3×C22⋊A4 | C2×D3 | 1 | 12 | yes | complemented |
| 21 | C2×C3⋊S4 | — | 12 | 12 | no | — |
| 22 | C2×C32×D4 | — | 3 | 1 | no | — |
| 23 | C2×#638 | — | 3 | 2 | no | — |
| 24 | C2×#638 | — | 6 | 1 | no | — |
| 25 | C2×#638 | — | 6 | 2 | no | — |
| 26 | #3830 | — | 6 | 2 | no | — |
| 27 | D4×C3⋊D3 | — | 6 | 2 | no | — |
| 28 | #3828 | — | 6 | 1 | no | — |
| 29 | C32×C22⋊C4 | — | 3 | 1 | no | — |
| 30 | #3757 | — | 3 | 2 | no | — |
| 31 | #3757 | — | 6 | 2 | no | — |
| 32 | C2×C3⋊(C3⋊D3) | — | 16 | 1 | no | — |
| 33 | C32×A4 | — | 4 | 1 | no | — |
| 34 | C32×A4 | — | 8 | 2 | no | — |
| 35 | #1020 | — | 9 | 18 | no | — |
| 36 | #1004 | — | 9 | 4 | no | — |
| 37 | C3×#120 | C3⋊D3 | 1 | 4 | yes | complemented |
| 38 | #953 | — | 9 | 8 | no | — |
| 39 | #120⋊C3 | C3⋊D3 | 1 | 9 | yes | complemented |
| 40 | #834 | — | 9 | 4 | no | — |
| 41 | #806 | — | 9 | 8 | no | — |
| 42 | C23×C32 | — | 3 | 1 | no | — |
| 43 | C23×C32 | — | 6 | 2 | no | — |
| 44 | C23×C32 | S4 | 1 | 1 | yes | characteristic, complemented |
| 45 | C22×C3⋊D3 | — | 3 | 2 | no | — |
| 46 | C22×C3⋊D3 | — | 6 | 1 | no | — |
| 47 | C2×C3×A4 | — | 4 | 12 | no | — |
| 48 | C3⋊S4 | — | 12 | 24 | no | — |
| 49 | C3⋊S4 | — | 24 | 24 | no | — |
| 50 | C32×D4 | — | 6 | 2 | no | — |
| 51 | C2×C32×C4 | — | 3 | 1 | no | — |
| 52 | #638 | — | 6 | 2 | no | — |
| 53 | #638 | — | 6 | 2 | no | — |
| 54 | #638 | — | 6 | 2 | no | — |
| 55 | #638 | — | 12 | 2 | no | — |
| 56 | #638 | — | 12 | 2 | no | — |
| 57 | C2×C3⋊Dic3 | — | 3 | 2 | no | — |
| 58 | C2×C3⋊Dic3 | — | 6 | 1 | no | — |
| 59 | C2×C3⋊Dic3 | — | 6 | 2 | no | — |
| 60 | C4⋊(C3⋊D3) | — | 6 | 1 | no | — |
| 61 | C4×C3⋊D3 | — | 6 | 2 | no | — |
| 62 | #634 | — | 6 | 1 | no | — |
| 63 | B(4,𝔽2) | — | 27 | 1 | no | Sylow |
| 64 | C2×C33 | — | 16 | 1 | no | — |
| 65 | C3⋊(C3⋊D3) | — | 16 | 2 | no | — |
| 66 | C24×C3 | C2×C3⋊D3 | 1 | 4 | yes | complemented |
| 67 | C22⋊A4 | C2×C3⋊D3 | 1 | 9 | yes | complemented |
| 68 | C2×S4 | — | 36 | 9 | no | — |
| 69 | C2×C3×D4 | — | 3 | 4 | no | — |
| 70 | C2×#67 | — | 9 | 8 | no | — |
| 71 | C2×#67 | — | 18 | 4 | no | — |
| 72 | C2×#67 | — | 18 | 8 | no | — |
| 73 | #237 | — | 18 | 8 | no | — |
| 74 | D3×D4 | — | 18 | 8 | no | — |
| 75 | #235 | — | 18 | 4 | no | — |
| 76 | C3×C22⋊C4 | — | 3 | 4 | no | — |
| 77 | #217 | — | 9 | 8 | no | — |
| 78 | #217 | — | 18 | 8 | no | — |
| 79 | C22×C32 | — | 2 | 2 | no | — |
| 80 | C22×C32 | — | 3 | 2 | no | — |
| 81 | C22×C32 | — | 6 | 1 | no | — |
| 82 | C22×C32 | — | 6 | 2 | no | — |
| 83 | C22×C32 | — | 12 | 1 | no | — |
| 84 | C22×C32 | C2×S4 | 1 | 1 | yes | characteristic |
| 85 | C2×C3⋊D3 | — | 6 | 2 | no | — |
| 86 | C2×C3⋊D3 | — | 12 | 2 | no | — |
| 87 | C2×C3⋊D3 | — | 24 | 1 | no | — |
| 88 | C2×C3⋊D3 | — | 48 | 12 | no | — |
| 89 | C3×A4 | — | 4 | 12 | no | — |
| 90 | C3×A4 | — | 8 | 24 | no | — |
| 91 | C32×C4 | — | 6 | 1 | no | — |
| 92 | C3⋊Dic3 | — | 6 | 2 | no | — |
| 93 | C3⋊Dic3 | — | 12 | 2 | no | — |
| 94 | H(5,𝔽2) | — | 27 | 1 | no | — |
| 95 | #120 | — | 27 | 2 | no | — |
| 96 | #120 | C3⋊(C3⋊D3) | 1 | 1 | yes | characteristic, complemented |
| 97 | #99 | — | 27 | 2 | no | — |
| 98 | #99 | — | 27 | 1 | no | — |
| 99 | C33 | — | 16 | 1 | no | Sylow |
| 100 | C23×C3 | — | 3 | 4 | no | — |
| 101 | C23×C3 | — | 6 | 8 | no | — |
| 102 | C23×C3 | C3⋊S4 | 1 | 4 | yes | complemented |
| 103 | C22×D3 | — | 9 | 8 | no | — |
| 104 | C22×D3 | — | 18 | 4 | no | — |
| 105 | C2×A4 | — | 4 | 9 | no | — |
| 106 | S4 | — | 36 | 18 | no | — |
| 107 | S4 | — | 72 | 18 | no | — |
| 108 | C3×D4 | — | 6 | 8 | no | — |
| 109 | C2×C3×C4 | — | 3 | 4 | no | — |
| 110 | #67 | — | 18 | 8 | no | — |
| 111 | #67 | — | 18 | 8 | no | — |
| 112 | #67 | — | 18 | 8 | no | — |
| 113 | #67 | — | 36 | 8 | no | — |
| 114 | #67 | — | 36 | 8 | no | — |
| 115 | C2×Dic3 | — | 9 | 8 | no | — |
| 116 | C2×Dic3 | — | 18 | 4 | no | — |
| 117 | C2×Dic3 | — | 18 | 8 | no | — |
| 118 | D12 | — | 18 | 4 | no | — |
| 119 | C4×D3 | — | 18 | 8 | no | — |
| 120 | Dic6 | — | 18 | 4 | no | — |
| 121 | C2×C32 | — | 3 | 1 | no | — |
| 122 | C2×C32 | — | 4 | 1 | no | — |
| 123 | C2×C32 | — | 6 | 2 | no | — |
| 124 | C2×C32 | — | 16 | 12 | no | — |
| 125 | C3⋊D3 | — | 12 | 2 | no | — |
| 126 | C3⋊D3 | — | 48 | 24 | no | — |
| 127 | C24 | C2×C3⋊(C3⋊D3) | 1 | 1 | yes | characteristic, complemented |
| 128 | #41 | — | 54 | 1 | no | — |
| 129 | #41 | — | 54 | 2 | no | — |
| 130 | C2×D4 | — | 3 | 1 | no | — |
| 131 | C2×D4 | — | 27 | 2 | no | — |
| 132 | C2×D4 | — | 54 | 2 | no | — |
| 133 | C2×D4 | — | 54 | 1 | no | — |
| 134 | C2×D4 | — | 54 | 2 | no | — |
| 135 | C22⋊C4 | — | 3 | 1 | no | — |
| 136 | C22⋊C4 | — | 27 | 2 | no | — |
| 137 | C22⋊C4 | — | 54 | 2 | no | — |
| 138 | C22×C3 | — | 2 | 8 | no | — |
| 139 | C22×C3 | — | 3 | 8 | no | — |
| 140 | C22×C3 | — | 6 | 4 | no | — |
| 141 | C22×C3 | — | 6 | 8 | no | — |
| 142 | C22×C3 | — | 12 | 4 | no | — |
| 143 | C22×C3 | C2×C3⋊S4 | 1 | 4 | yes | — |
| 144 | C2×D3 | — | 18 | 8 | no | — |
| 145 | C2×D3 | — | 36 | 8 | no | — |
| 146 | C2×D3 | — | 72 | 4 | no | — |
| 147 | C2×D3 | — | 144 | 9 | no | — |
| 148 | A4 | — | 4 | 9 | no | — |
| 149 | A4 | — | 8 | 18 | no | — |
| 150 | C3×C4 | — | 6 | 4 | no | — |
| 151 | Dic3 | — | 18 | 8 | no | — |
| 152 | Dic3 | — | 36 | 8 | no | — |
| 153 | C32 | — | 16 | 12 | no | — |
| 154 | C32 | #1020⋊C2 | 1 | 1 | yes | characteristic, complemented |
| 155 | C23 | — | 3 | 1 | no | — |
| 156 | C23 | — | 6 | 2 | no | — |
| 157 | C23 | — | 27 | 2 | no | — |
| 158 | C23 | — | 54 | 1 | no | — |
| 159 | C23 | C3⋊(C3⋊S4) | 1 | 1 | yes | characteristic, complemented |
| 160 | Q8 | — | 54 | 1 | no | — |
| 161 | D4 | — | 6 | 2 | no | — |
| 162 | D4 | — | 54 | 1 | no | — |
| 163 | D4 | — | 54 | 2 | no | — |
| 164 | D4 | — | 54 | 2 | no | — |
| 165 | D4 | — | 54 | 2 | no | — |
| 166 | D4 | — | 108 | 2 | no | — |
| 167 | D4 | — | 108 | 2 | no | — |
| 168 | C2×C4 | — | 3 | 1 | no | — |
| 169 | C2×C4 | — | 27 | 2 | no | — |
| 170 | C2×C4 | — | 54 | 2 | no | — |
| 171 | C2×C4 | — | 54 | 1 | no | — |
| 172 | C2×C4 | — | 54 | 2 | no | — |
| 173 | C2×C3 | — | 3 | 4 | no | — |
| 174 | C2×C3 | — | 4 | 4 | no | — |
| 175 | C2×C3 | — | 6 | 8 | no | — |
| 176 | C2×C3 | — | 16 | 9 | no | — |
| 177 | D3 | — | 36 | 8 | no | — |
| 178 | D3 | — | 144 | 18 | no | — |
| 179 | C22 | — | 2 | 2 | no | — |
| 180 | C22 | — | 3 | 2 | no | — |
| 181 | C22 | — | 6 | 1 | no | — |
| 182 | C22 | — | 6 | 2 | no | — |
| 183 | C22 | — | 12 | 1 | no | — |
| 184 | C22 | — | 54 | 2 | no | — |
| 185 | C22 | — | 108 | 2 | no | — |
| 186 | C22 | — | 216 | 1 | no | — |
| 187 | C22 | C2×C3⋊(C3⋊S4) | 1 | 1 | yes | characteristic |
| 188 | C4 | — | 6 | 1 | no | — |
| 189 | C4 | — | 54 | 2 | no | — |
| 190 | C4 | — | 108 | 2 | no | — |
| 191 | C3 | — | 16 | 9 | no | — |
| 192 | C3 | #10593630 | 1 | 4 | yes | complemented |
| 193 | C2 | — | 3 | 1 | no | — |
| 194 | C2 | — | 4 | 1 | no | — |
| 195 | C2 | — | 6 | 2 | no | — |
| 196 | C2 | — | 108 | 2 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C32×C22⋊A4 | C22 |
| Frattini | C22 | C2×C3⋊(C3⋊S4) |
| Fitting | C32×#120 | D3 |
| socle | C22×C32 | C2×S4 |
| LCS | G ▷ C32×C22⋊A4 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3⋊(C3⋊#1020) | C2 | semidirect |
| 2. C32×#120⋊C3 | C2 | semidirect |
| 5. C32×C22⋊A4 | C22 | semidirect |
| 9. C32×#120 | D3 | semidirect |
| 11. C3×#120⋊C3 | D3 | semidirect |
| 19. C24×C32 | C2×D3 | semidirect |
| 20. C3×C22⋊A4 | C2×D3 | semidirect |
| 37. C3×#120 | C3⋊D3 | semidirect |
| 39. #120⋊C3 | C3⋊D3 | semidirect |
| 44. C23×C32 | S4 | semidirect |
| 66. C24×C3 | C2×C3⋊D3 | semidirect |
| 67. C22⋊A4 | C2×C3⋊D3 | semidirect |
| 96. #120 | C3⋊(C3⋊D3) | semidirect |
| 102. C23×C3 | C3⋊S4 | semidirect |
| 127. C24 | C2×C3⋊(C3⋊D3) | semidirect |
| 154. C32 | #1020⋊C2 | semidirect |
| 159. C23 | C3⋊(C3⋊S4) | semidirect |
| 192. C3 | #10593630 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3⋊(C3⋊#1020) | 2 |
| C32×#120⋊C3 | 2 |
| #10593630 | 3 |
| #10592731 | 3 |
| C2×C3⋊(C3⋊S4) | 4 |