← C2×H(3,𝔽3)×#108 #13145046
Order 1728.
583 conjugacy classes of subgroups.
The subgroup lattice has 165 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×M16×H(3,𝔽3) | C2 | 1 | 2 | yes | maximal |
| 2 | C22×C8×H(3,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | H(3,𝔽3)×#108 | C2 | 1 | 4 | yes | maximal, direct product factor |
| 4 | C2×C32×#108 | C3 | 1 | 4 | yes | maximal, complemented |
| 5 | C22×C4×H(3,𝔽3) | C22 | 1 | 1 | yes | characteristic |
| 6 | M16×H(3,𝔽3) | — | 2 | 2 | no | — |
| 7 | M16×H(3,𝔽3) | C22 | 1 | 4 | yes | — |
| 8 | C2×C8×H(3,𝔽3) | — | 2 | 2 | no | — |
| 9 | C2×C8×H(3,𝔽3) | C4 | 1 | 2 | yes | — |
| 10 | C2×C8×H(3,𝔽3) | C4 | 1 | 2 | yes | — |
| 11 | C2×C8×H(3,𝔽3) | C22 | 1 | 2 | yes | — |
| 12 | C2×C32×M16 | C2×C3 | 1 | 8 | yes | — |
| 13 | C22×C32×C8 | C2×C3 | 1 | 4 | yes | — |
| 14 | C32×#108 | C2×C3 | 1 | 16 | yes | complemented |
| 15 | C23×H(3,𝔽3) | Q8 | 1 | 1 | yes | characteristic |
| 16 | C2×C4×H(3,𝔽3) | C2×C4 | 1 | 1 | yes | characteristic |
| 17 | C2×C4×H(3,𝔽3) | C2×C4 | 1 | 1 | yes | characteristic |
| 18 | C2×C4×H(3,𝔽3) | D4 | 1 | 1 | yes | characteristic |
| 19 | C2×C4×H(3,𝔽3) | D4 | 1 | 1 | yes | characteristic |
| 20 | C2×C4×H(3,𝔽3) | Q8 | 1 | 1 | yes | characteristic |
| 21 | C2×C4×H(3,𝔽3) | C23 | 1 | 1 | yes | characteristic |
| 22 | C8×H(3,𝔽3) | — | 2 | 4 | no | — |
| 23 | C8×H(3,𝔽3) | C2×C4 | 1 | 4 | yes | — |
| 24 | C2×C3×#108 | — | 3 | 4 | no | — |
| 25 | C2×C3×#108 | C32 | 1 | 1 | yes | characteristic |
| 26 | C22×C32×C4 | C22×C3 | 1 | 4 | yes | — |
| 27 | C32×M16 | — | 2 | 8 | no | — |
| 28 | C32×M16 | C22×C3 | 1 | 16 | yes | — |
| 29 | C2×C32×C8 | — | 2 | 8 | no | — |
| 30 | C2×C32×C8 | C3×C4 | 1 | 8 | yes | — |
| 31 | C2×C32×C8 | C3×C4 | 1 | 8 | yes | — |
| 32 | C2×C32×C8 | C22×C3 | 1 | 8 | yes | — |
| 33 | C22×H(3,𝔽3) | — | 2 | 2 | no | — |
| 34 | C22×H(3,𝔽3) | C4⋊C4 | 1 | 1 | yes | characteristic |
| 35 | C22×H(3,𝔽3) | C4⋊C4 | 1 | 1 | yes | characteristic |
| 36 | C22×H(3,𝔽3) | C2×Q8 | 1 | 1 | yes | characteristic |
| 37 | C4×H(3,𝔽3) | C4⋊C4 | 1 | 1 | yes | characteristic |
| 38 | C4×H(3,𝔽3) | C4⋊C4 | 1 | 1 | yes | characteristic |
| 39 | C4×H(3,𝔽3) | C22×C4 | 1 | 1 | yes | characteristic |
| 40 | C4×H(3,𝔽3) | C2×D4 | 1 | 1 | yes | characteristic |
| 41 | C2×C3×M16 | — | 3 | 8 | no | — |
| 42 | C2×C3×M16 | C2×C32 | 1 | 2 | yes | — |
| 43 | C22×C3×C8 | — | 3 | 4 | no | — |
| 44 | C22×C3×C8 | C2×C32 | 1 | 1 | yes | characteristic |
| 45 | C3×#108 | — | 3 | 16 | no | — |
| 46 | C3×#108 | C2×C32 | 1 | 4 | yes | — |
| 47 | C23×C32 | C3×Q8 | 1 | 4 | yes | — |
| 48 | C2×C32×C4 | C2×C3×C4 | 1 | 4 | yes | — |
| 49 | C2×C32×C4 | C2×C3×C4 | 1 | 4 | yes | — |
| 50 | C2×C32×C4 | C3×D4 | 1 | 4 | yes | — |
| 51 | C2×C32×C4 | C3×D4 | 1 | 4 | yes | — |
| 52 | C2×C32×C4 | C3×Q8 | 1 | 4 | yes | — |
| 53 | C2×C32×C4 | C23×C3 | 1 | 4 | yes | — |
| 54 | C32×C8 | — | 2 | 16 | no | — |
| 55 | C32×C8 | C2×C3×C4 | 1 | 16 | yes | — |
| 56 | C2×#108 | H(3,𝔽3) | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 57 | C2×H(3,𝔽3) | — | 2 | 1 | no | — |
| 58 | C2×H(3,𝔽3) | — | 2 | 1 | no | — |
| 59 | C2×H(3,𝔽3) | #108 | 1 | 2 | yes | direct product factor |
| 60 | C2×H(3,𝔽3) | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 61 | C22×C3×C4 | — | 3 | 4 | no | — |
| 62 | C22×C3×C4 | C22×C32 | 1 | 1 | yes | characteristic |
| 63 | C3×M16 | — | 2 | 2 | no | — |
| 64 | C3×M16 | — | 3 | 16 | no | — |
| 65 | C3×M16 | — | 6 | 8 | no | — |
| 66 | C3×M16 | C22×C32 | 1 | 4 | yes | — |
| 67 | C2×C3×C8 | — | 2 | 2 | no | — |
| 68 | C2×C3×C8 | — | 3 | 8 | no | — |
| 69 | C2×C3×C8 | — | 3 | 8 | no | — |
| 70 | C2×C3×C8 | — | 3 | 8 | no | — |
| 71 | C2×C3×C8 | — | 6 | 8 | no | — |
| 72 | C2×C3×C8 | C32×C4 | 1 | 2 | yes | — |
| 73 | C2×C3×C8 | C32×C4 | 1 | 2 | yes | — |
| 74 | C2×C3×C8 | C22×C32 | 1 | 2 | yes | — |
| 75 | C22×C32 | — | 2 | 8 | no | — |
| 76 | C22×C32 | C3×C4⋊C4 | 1 | 4 | yes | — |
| 77 | C22×C32 | C3×C4⋊C4 | 1 | 4 | yes | — |
| 78 | C22×C32 | C2×C3×Q8 | 1 | 4 | yes | — |
| 79 | C32×C4 | C3×C4⋊C4 | 1 | 4 | yes | — |
| 80 | C32×C4 | C3×C4⋊C4 | 1 | 4 | yes | — |
| 81 | C32×C4 | C22×C3×C4 | 1 | 4 | yes | — |
| 82 | C32×C4 | C2×C3×D4 | 1 | 4 | yes | — |
| 83 | C2×M16 | C2×H(3,𝔽3) | 1 | 2 | yes | — |
| 84 | C22×C8 | C2×H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 85 | #108 | C2×H(3,𝔽3) | 1 | 4 | yes | direct product factor |
| 86 | H(3,𝔽3) | C2×#108 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 87 | C23×C3 | — | 3 | 4 | no | — |
| 88 | C23×C3 | C32×Q8 | 1 | 1 | yes | characteristic |
| 89 | C2×C3×C4 | — | 3 | 4 | no | — |
| 90 | C2×C3×C4 | — | 3 | 4 | no | — |
| 91 | C2×C3×C4 | — | 3 | 4 | no | — |
| 92 | C2×C3×C4 | — | 3 | 4 | no | — |
| 93 | C2×C3×C4 | — | 3 | 4 | no | — |
| 94 | C2×C3×C4 | — | 3 | 4 | no | — |
| 95 | C2×C3×C4 | C2×C32×C4 | 1 | 1 | yes | characteristic |
| 96 | C2×C3×C4 | C2×C32×C4 | 1 | 1 | yes | characteristic |
| 97 | C2×C3×C4 | C32×D4 | 1 | 1 | yes | characteristic |
| 98 | C2×C3×C4 | C32×D4 | 1 | 1 | yes | characteristic |
| 99 | C2×C3×C4 | C32×Q8 | 1 | 1 | yes | characteristic |
| 100 | C2×C3×C4 | C23×C32 | 1 | 1 | yes | characteristic |
| 101 | C3×C8 | — | 2 | 4 | no | — |
| 102 | C3×C8 | — | 3 | 16 | no | — |
| 103 | C3×C8 | — | 6 | 16 | no | — |
| 104 | C3×C8 | C2×C32×C4 | 1 | 4 | yes | — |
| 105 | C2×C32 | — | 2 | 4 | no | — |
| 106 | C2×C32 | — | 2 | 4 | no | — |
| 107 | C2×C32 | C3×#108 | 1 | 8 | yes | complemented |
| 108 | C2×C32 | C2×C3×C4⋊C4 | 1 | 4 | yes | — |
| 109 | C22×C4 | C22×H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 110 | M16 | — | 2 | 2 | no | — |
| 111 | M16 | C22×H(3,𝔽3) | 1 | 4 | yes | — |
| 112 | C2×C8 | — | 2 | 2 | no | — |
| 113 | C2×C8 | C4×H(3,𝔽3) | 1 | 2 | yes | — |
| 114 | C2×C8 | C4×H(3,𝔽3) | 1 | 2 | yes | — |
| 115 | C2×C8 | C22×H(3,𝔽3) | 1 | 2 | yes | — |
| 116 | C22×C3 | — | 2 | 2 | no | — |
| 117 | C22×C3 | — | 3 | 4 | no | — |
| 118 | C22×C3 | — | 3 | 4 | no | — |
| 119 | C22×C3 | — | 3 | 4 | no | — |
| 120 | C22×C3 | — | 6 | 8 | no | — |
| 121 | C22×C3 | C32×C4⋊C4 | 1 | 1 | yes | characteristic |
| 122 | C22×C3 | C32×C4⋊C4 | 1 | 1 | yes | characteristic |
| 123 | C22×C3 | C2×C32×Q8 | 1 | 1 | yes | characteristic |
| 124 | C3×C4 | — | 3 | 4 | no | — |
| 125 | C3×C4 | — | 3 | 4 | no | — |
| 126 | C3×C4 | — | 3 | 4 | no | — |
| 127 | C3×C4 | — | 3 | 4 | no | — |
| 128 | C3×C4 | C32×C4⋊C4 | 1 | 1 | yes | characteristic |
| 129 | C3×C4 | C32×C4⋊C4 | 1 | 1 | yes | characteristic |
| 130 | C3×C4 | C22×C32×C4 | 1 | 1 | yes | characteristic |
| 131 | C3×C4 | C2×C32×D4 | 1 | 1 | yes | characteristic |
| 132 | C32 | C2×C3×#108 | 1 | 4 | yes | complemented |
| 133 | C23 | Q8×H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 134 | C2×C4 | C2×C4×H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 135 | C2×C4 | C2×C4×H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 136 | C2×C4 | D4×H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 137 | C2×C4 | D4×H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 138 | C2×C4 | Q8×H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 139 | C2×C4 | C23×H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 140 | C8 | — | 2 | 4 | no | — |
| 141 | C8 | C2×C4×H(3,𝔽3) | 1 | 4 | yes | — |
| 142 | C2×C3 | — | 2 | 1 | no | — |
| 143 | C2×C3 | — | 2 | 1 | no | — |
| 144 | C2×C3 | — | 3 | 4 | no | — |
| 145 | C2×C3 | — | 3 | 8 | no | — |
| 146 | C2×C3 | — | 6 | 4 | no | — |
| 147 | C2×C3 | — | 6 | 4 | no | — |
| 148 | C2×C3 | C32×#108 | 1 | 2 | yes | — |
| 149 | C2×C3 | C2×C32×C4⋊C4 | 1 | 1 | yes | characteristic |
| 150 | C22 | — | 2 | 2 | no | — |
| 151 | C22 | C4⋊C4×H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 152 | C22 | C4⋊C4×H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 153 | C22 | C2×Q8×H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 154 | C4 | C4⋊C4×H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 155 | C4 | C4⋊C4×H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 156 | C4 | C22×C4×H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 157 | C4 | C2×D4×H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 158 | C3 | — | 3 | 4 | no | — |
| 159 | C3 | C2×C32×#108 | 1 | 1 | yes | characteristic |
| 160 | C2 | — | 2 | 1 | no | — |
| 161 | C2 | — | 2 | 1 | no | — |
| 162 | C2 | H(3,𝔽3)×#108 | 1 | 2 | yes | direct product factor |
| 163 | C2 | C2×C4⋊C4×H(3,𝔽3) | 1 | 1 | yes | characteristic |
Complements
Splittings G = N ⋊ H with N normal; H ≅ G/N.
Maximal subgroups
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
Finite Group Databasebeta