← H(3,𝔽3)×#405 #13144974
Subgroups — H(3,𝔽3)×#405 #13144974
Order 1728.
715 conjugacy classes of subgroups.
The subgroup lattice has 175 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 | 1 | yes | maximal, characteristic |
| 2 | C22×C8×H(3,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | H(3,𝔽3)×#117 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | H(3,𝔽3)×#97 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | C4×C8×H(3,𝔽3) | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | C32×#405 | C3 | 1 | 4 | yes | maximal, complemented |
| 7 | C22×C4×H(3,𝔽3) | C22 | 1 | 1 | yes | characteristic |
| 8 | M16×H(3,𝔽3) | C4 | 1 | 4 | yes | complemented |
| 9 | C2×C8×H(3,𝔽3) | C4 | 1 | 4 | yes | complemented |
| 10 | C2×C8×H(3,𝔽3) | C22 | 1 | 1 | yes | characteristic |
| 11 | C2×C8×H(3,𝔽3) | C22 | 1 | 2 | yes | — |
| 12 | C2×C8×H(3,𝔽3) | C22 | 1 | 1 | yes | characteristic |
| 13 | C4⋊C4×H(3,𝔽3) | C4 | 1 | 2 | yes | — |
| 14 | C22⋊C4×H(3,𝔽3) | C4 | 1 | 2 | yes | — |
| 15 | C42×H(3,𝔽3) | C22 | 1 | 2 | yes | — |
| 16 | C2×C32×M16 | C2×C3 | 1 | 4 | yes | — |
| 17 | C22×C32×C8 | C2×C3 | 1 | 4 | yes | — |
| 18 | C32×#117 | C2×C3 | 1 | 4 | yes | — |
| 19 | C32×#97 | C2×C3 | 1 | 8 | yes | complemented |
| 20 | C32×C4×C8 | C2×C3 | 1 | 8 | yes | complemented |
| 21 | C23×H(3,𝔽3) | C2×C4 | 1 | 1 | yes | characteristic |
| 22 | C2×C4×H(3,𝔽3) | C2×C4 | 1 | 4 | yes | — |
| 23 | C2×C4×H(3,𝔽3) | C2×C4 | 1 | 1 | yes | characteristic |
| 24 | C2×C4×H(3,𝔽3) | C2×C4 | 1 | 2 | yes | — |
| 25 | C2×C4×H(3,𝔽3) | C2×C4 | 1 | 2 | yes | — |
| 26 | C2×C4×H(3,𝔽3) | C23 | 1 | 1 | yes | characteristic |
| 27 | C8×H(3,𝔽3) | C2×C4 | 1 | 2 | yes | — |
| 28 | C8×H(3,𝔽3) | C2×C4 | 1 | 2 | yes | — |
| 29 | C8×H(3,𝔽3) | C2×C4 | 1 | 4 | yes | — |
| 30 | C3×#405 | — | 3 | 4 | no | — |
| 31 | C3×#405 | C32 | 1 | 1 | yes | characteristic |
| 32 | C22×C32×C4 | C22×C3 | 1 | 4 | yes | — |
| 33 | C32×M16 | C3×C4 | 1 | 16 | yes | complemented |
| 34 | C2×C32×C8 | C3×C4 | 1 | 16 | yes | complemented |
| 35 | C2×C32×C8 | C22×C3 | 1 | 8 | yes | — |
| 36 | C2×C32×C8 | C22×C3 | 1 | 4 | yes | — |
| 37 | C2×C32×C8 | C22×C3 | 1 | 4 | yes | — |
| 38 | C32×C4⋊C4 | C3×C4 | 1 | 8 | yes | — |
| 39 | C32×C22⋊C4 | C3×C4 | 1 | 8 | yes | — |
| 40 | C32×C42 | C22×C3 | 1 | 8 | yes | — |
| 41 | C22×H(3,𝔽3) | — | 2 | 2 | no | — |
| 42 | C22×H(3,𝔽3) | C42 | 1 | 2 | yes | — |
| 43 | C22×H(3,𝔽3) | C22×C4 | 1 | 1 | yes | characteristic |
| 44 | C4×H(3,𝔽3) | — | 2 | 4 | no | — |
| 45 | C4×H(3,𝔽3) | C42 | 1 | 2 | yes | — |
| 46 | C4×H(3,𝔽3) | C22×C4 | 1 | 1 | yes | characteristic |
| 47 | C4×H(3,𝔽3) | C22×C4 | 1 | 1 | yes | characteristic |
| 48 | C2×C3×M16 | — | 3 | 4 | no | — |
| 49 | C2×C3×M16 | C2×C32 | 1 | 1 | yes | characteristic |
| 50 | C22×C3×C8 | — | 3 | 4 | no | — |
| 51 | C22×C3×C8 | C2×C32 | 1 | 1 | yes | characteristic |
| 52 | C3×#117 | — | 3 | 4 | no | — |
| 53 | C3×#117 | C2×C32 | 1 | 1 | yes | characteristic |
| 54 | C3×#97 | — | 3 | 8 | no | — |
| 55 | C3×#97 | C2×C32 | 1 | 2 | yes | — |
| 56 | C3×C4×C8 | — | 3 | 8 | no | — |
| 57 | C3×C4×C8 | C2×C32 | 1 | 2 | yes | — |
| 58 | C23×C32 | C2×C3×C4 | 1 | 4 | yes | — |
| 59 | C2×C32×C4 | C2×C3×C4 | 1 | 16 | yes | — |
| 60 | C2×C32×C4 | C2×C3×C4 | 1 | 4 | yes | — |
| 61 | C2×C32×C4 | C2×C3×C4 | 1 | 8 | yes | — |
| 62 | C2×C32×C4 | C2×C3×C4 | 1 | 8 | yes | — |
| 63 | C2×C32×C4 | C23×C3 | 1 | 4 | yes | — |
| 64 | C32×C8 | C2×C3×C4 | 1 | 8 | yes | — |
| 65 | C32×C8 | C2×C3×C4 | 1 | 8 | yes | — |
| 66 | C32×C8 | C2×C3×C4 | 1 | 16 | yes | — |
| 67 | #405 | H(3,𝔽3) | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 68 | C2×H(3,𝔽3) | — | 2 | 2 | no | — |
| 69 | C2×H(3,𝔽3) | C2×C42 | 1 | 1 | yes | characteristic |
| 70 | C2×H(3,𝔽3) | #131 | 1 | 2 | yes | — |
| 71 | C22×C3×C4 | — | 3 | 4 | no | — |
| 72 | C22×C3×C4 | C22×C32 | 1 | 1 | yes | characteristic |
| 73 | C3×M16 | — | 3 | 16 | no | — |
| 74 | C3×M16 | C32×C4 | 1 | 4 | yes | — |
| 75 | C2×C3×C8 | — | 3 | 16 | no | — |
| 76 | C2×C3×C8 | — | 3 | 4 | no | — |
| 77 | C2×C3×C8 | — | 3 | 8 | no | — |
| 78 | C2×C3×C8 | — | 3 | 4 | no | — |
| 79 | C2×C3×C8 | C32×C4 | 1 | 4 | yes | — |
| 80 | C2×C3×C8 | C22×C32 | 1 | 2 | yes | — |
| 81 | C2×C3×C8 | C22×C32 | 1 | 1 | yes | characteristic |
| 82 | C2×C3×C8 | C22×C32 | 1 | 1 | yes | characteristic |
| 83 | C3×C4⋊C4 | — | 3 | 8 | no | — |
| 84 | C3×C4⋊C4 | C32×C4 | 1 | 2 | yes | — |
| 85 | C3×C22⋊C4 | — | 3 | 8 | no | — |
| 86 | C3×C22⋊C4 | C32×C4 | 1 | 2 | yes | — |
| 87 | C3×C42 | — | 3 | 8 | no | — |
| 88 | C3×C42 | C22×C32 | 1 | 2 | yes | — |
| 89 | C22×C32 | — | 2 | 8 | no | — |
| 90 | C22×C32 | C3×C42 | 1 | 8 | yes | — |
| 91 | C22×C32 | C22×C3×C4 | 1 | 4 | yes | — |
| 92 | C32×C4 | — | 2 | 16 | no | — |
| 93 | C32×C4 | C3×C42 | 1 | 8 | yes | — |
| 94 | C32×C4 | C22×C3×C4 | 1 | 4 | yes | — |
| 95 | C32×C4 | C22×C3×C4 | 1 | 4 | yes | — |
| 96 | C2×M16 | C2×H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 97 | C22×C8 | C2×H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 98 | #117 | C2×H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 99 | #97 | C2×H(3,𝔽3) | 1 | 2 | yes | complemented |
| 100 | C4×C8 | C2×H(3,𝔽3) | 1 | 2 | yes | complemented |
| 101 | H(3,𝔽3) | #405 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 102 | C23×C3 | — | 3 | 4 | no | — |
| 103 | C23×C3 | C2×C32×C4 | 1 | 1 | yes | characteristic |
| 104 | C2×C3×C4 | — | 3 | 4 | no | — |
| 105 | C2×C3×C4 | — | 3 | 8 | no | — |
| 106 | C2×C3×C4 | — | 3 | 8 | no | — |
| 107 | C2×C3×C4 | — | 3 | 16 | no | — |
| 108 | C2×C3×C4 | — | 3 | 4 | no | — |
| 109 | C2×C3×C4 | C2×C32×C4 | 1 | 4 | yes | — |
| 110 | C2×C3×C4 | C2×C32×C4 | 1 | 1 | yes | characteristic |
| 111 | C2×C3×C4 | C2×C32×C4 | 1 | 2 | yes | — |
| 112 | C2×C3×C4 | C2×C32×C4 | 1 | 2 | yes | — |
| 113 | C2×C3×C4 | C23×C32 | 1 | 1 | yes | characteristic |
| 114 | C3×C8 | — | 3 | 8 | no | — |
| 115 | C3×C8 | — | 3 | 8 | no | — |
| 116 | C3×C8 | — | 3 | 16 | no | — |
| 117 | C3×C8 | C2×C32×C4 | 1 | 2 | yes | — |
| 118 | C3×C8 | C2×C32×C4 | 1 | 2 | yes | — |
| 119 | C3×C8 | C2×C32×C4 | 1 | 4 | yes | — |
| 120 | C2×C32 | — | 2 | 8 | no | — |
| 121 | C2×C32 | C2×C3×C42 | 1 | 4 | yes | — |
| 122 | C2×C32 | C3×#131 | 1 | 8 | yes | — |
| 123 | C22×C4 | C22×H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 124 | M16 | C4×H(3,𝔽3) | 1 | 4 | yes | complemented |
| 125 | C2×C8 | C4×H(3,𝔽3) | 1 | 4 | yes | complemented |
| 126 | C2×C8 | C22×H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 127 | C2×C8 | C22×H(3,𝔽3) | 1 | 2 | yes | — |
| 128 | C2×C8 | C22×H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 129 | C4⋊C4 | C4×H(3,𝔽3) | 1 | 2 | yes | — |
| 130 | C22⋊C4 | C4×H(3,𝔽3) | 1 | 2 | yes | — |
| 131 | C42 | C22×H(3,𝔽3) | 1 | 2 | yes | — |
| 132 | C22×C3 | — | 2 | 2 | no | — |
| 133 | C22×C3 | — | 3 | 4 | no | — |
| 134 | C22×C3 | — | 3 | 8 | no | — |
| 135 | C22×C3 | — | 6 | 8 | no | — |
| 136 | C22×C3 | C32×C42 | 1 | 2 | yes | — |
| 137 | C22×C3 | C22×C32×C4 | 1 | 1 | yes | characteristic |
| 138 | C3×C4 | — | 2 | 4 | no | — |
| 139 | C3×C4 | — | 3 | 4 | no | — |
| 140 | C3×C4 | — | 3 | 4 | no | — |
| 141 | C3×C4 | — | 3 | 8 | no | — |
| 142 | C3×C4 | — | 6 | 16 | no | — |
| 143 | C3×C4 | C32×C42 | 1 | 2 | yes | — |
| 144 | C3×C4 | C22×C32×C4 | 1 | 1 | yes | characteristic |
| 145 | C3×C4 | C22×C32×C4 | 1 | 1 | yes | characteristic |
| 146 | C32 | C3×#405 | 1 | 4 | yes | complemented |
| 147 | C23 | C2×C4×H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 148 | C2×C4 | C2×C4×H(3,𝔽3) | 1 | 4 | yes | — |
| 149 | C2×C4 | C2×C4×H(3,𝔽3) | 1 | 2 | yes | — |
| 150 | C2×C4 | C2×C4×H(3,𝔽3) | 1 | 2 | yes | — |
| 151 | C2×C4 | C2×C4×H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 152 | C2×C4 | C23×H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 153 | C8 | C2×C4×H(3,𝔽3) | 1 | 2 | yes | — |
| 154 | C8 | C2×C4×H(3,𝔽3) | 1 | 2 | yes | — |
| 155 | C8 | C2×C4×H(3,𝔽3) | 1 | 4 | yes | — |
| 156 | C2×C3 | — | 2 | 2 | no | — |
| 157 | C2×C3 | — | 3 | 4 | no | — |
| 158 | C2×C3 | — | 3 | 8 | no | — |
| 159 | C2×C3 | — | 6 | 8 | no | — |
| 160 | C2×C3 | C2×C32×C42 | 1 | 1 | yes | characteristic |
| 161 | C2×C3 | C32×#131 | 1 | 2 | yes | — |
| 162 | C22 | — | 2 | 2 | no | — |
| 163 | C22 | C42×H(3,𝔽3) | 1 | 2 | yes | — |
| 164 | C22 | C22×C4×H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 165 | C4 | — | 2 | 4 | no | — |
| 166 | C4 | C42×H(3,𝔽3) | 1 | 2 | yes | — |
| 167 | C4 | C22×C4×H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 168 | C4 | C22×C4×H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 169 | C3 | — | 3 | 4 | no | — |
| 170 | C3 | C32×#405 | 1 | 1 | yes | characteristic |
| 171 | C2 | — | 2 | 2 | no | — |
| 172 | C2 | C2×C42×H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 173 | C2 | H(3,𝔽3)×#131 | 1 | 2 | yes | — |
Complements
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|
| 4. H(3,𝔽3)×#97 | C2 | semidirect |
| 5. C4×C8×H(3,𝔽3) | C2 | semidirect |
| 6. C32×#405 | C3 | semidirect |
| 8. M16×H(3,𝔽3) | C4 | semidirect |
| 9. C2×C8×H(3,𝔽3) | C4 | semidirect |
| 19. C32×#97 | C2×C3 | semidirect |
| 20. C32×C4×C8 | C2×C3 | semidirect |
| 33. C32×M16 | C3×C4 | semidirect |
| 34. C2×C32×C8 | C3×C4 | semidirect |
| 67. #405 | H(3,𝔽3) | direct (N × H) |
| 99. #97 | C2×H(3,𝔽3) | semidirect |
| 100. C4×C8 | C2×H(3,𝔽3) | semidirect |
| 101. H(3,𝔽3) | #405 | direct (N × H) |
| 124. M16 | C4×H(3,𝔽3) | semidirect |
| 125. C2×C8 | C4×H(3,𝔽3) | semidirect |
| 146. C32 | C3×#405 | semidirect |
Maximal subgroups
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
Finite Group Databasebeta