Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
178 conjugacy classes of subgroups; only the 27 normal subgroups are shown.
This group has 178 conjugacy classes of subgroups — too many to draw. Only its 27 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 | 2−1+4⋊C27 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #11720652 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #11720109 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #10588926 | — | 3 | 1 | no | maximal |
| 5 | #41⋊C27 | C22 | 1 | 1 | yes | characteristic |
| 6 | C2×Q8⋊C27 | — | 2 | 1 | no | — |
| 7 | #88391 | — | 2 | 1 | no | — |
| 8 | #88369 | — | 4 | 1 | no | maximal |
| 9 | C9×2−1+4 | D3 | 1 | 1 | yes | characteristic |
| 10 | #63512 | — | 3 | 1 | no | — |
| 11 | #63512 | — | 3 | 1 | no | — |
| 12 | #63509 | — | 3 | 1 | no | — |
| 13 | #63396 | — | 3 | 1 | no | — |
| 14 | #63396 | — | 3 | 1 | no | — |
| 15 | #63394 | — | 3 | 1 | no | — |
| 16 | D4×C27 | — | 4 | 1 | no | — |
| 17 | D108 | — | 4 | 1 | no | — |
| 18 | Q8⋊C27 | D4 | 1 | 1 | yes | characteristic |
| 19 | C27⋊C8 | — | 4 | 1 | no | — |
| 20 | #5245 | — | 9 | 1 | no | — |
| 21 | C9×#41 | — | 3 | 1 | no | — |
| 22 | C9×#41 | — | 6 | 1 | no | — |
| 23 | C9×#41 | C2×D3 | 1 | 1 | yes | characteristic |
| 24 | C2×Q8×C9 | — | 2 | 1 | no | — |
| 25 | C2×Q8×C9 | — | 3 | 1 | no | — |
| 26 | C2×D36 | — | 3 | 1 | no | — |
| 27 | #3675 | — | 6 | 1 | no | — |
| 28 | #3675 | — | 6 | 1 | no | — |
| 29 | #3673 | — | 6 | 1 | no | — |
| 30 | #3673 | — | 6 | 1 | no | — |
| 31 | (C2×D9)⋊C4 | — | 6 | 2 | no | — |
| 32 | C4×Dic9 | — | 3 | 1 | no | — |
| 33 | #3667 | — | 3 | 1 | no | — |
| 34 | #3667 | — | 3 | 1 | no | — |
| 35 | C22×C27 | — | 8 | 1 | no | — |
| 36 | C2×D27 | — | 8 | 1 | no | — |
| 37 | C4×C27 | — | 4 | 1 | no | — |
| 38 | C3×2−1+4 | D9 | 1 | 1 | yes | characteristic |
| 39 | #949 | — | 9 | 1 | no | — |
| 40 | #949 | — | 9 | 1 | no | — |
| 41 | #947 | — | 9 | 1 | no | — |
| 42 | #837 | — | 9 | 1 | no | — |
| 43 | #837 | — | 9 | 1 | no | — |
| 44 | #836 | — | 9 | 1 | no | — |
| 45 | C22×D9 | — | 6 | 1 | no | — |
| 46 | Q8×C9 | — | 3 | 1 | no | — |
| 47 | Q8×C9 | — | 6 | 1 | no | — |
| 48 | Q8×C9 | #67 | 1 | 1 | yes | characteristic |
| 49 | D4×C9 | — | 3 | 1 | no | — |
| 50 | D4×C9 | — | 6 | 1 | no | — |
| 51 | D4×C9 | S4 | 1 | 1 | yes | characteristic |
| 52 | C2×C4×C9 | — | 3 | 1 | no | — |
| 53 | C2×C4×C9 | — | 3 | 2 | no | — |
| 54 | C2×C4×C9 | — | 6 | 1 | no | — |
| 55 | C2×Dic9 | — | 6 | 1 | no | — |
| 56 | D36 | — | 6 | 1 | no | — |
| 57 | D36 | — | 6 | 1 | no | — |
| 58 | C9⋊C8 | — | 6 | 1 | no | — |
| 59 | C9⋊C8 | — | 6 | 1 | no | — |
| 60 | #454 | — | 27 | 1 | no | Sylow |
| 61 | C2×C27 | — | 4 | 1 | no | — |
| 62 | C2×C27 | — | 16 | 1 | no | — |
| 63 | D27 | — | 16 | 1 | no | — |
| 64 | C3×#41 | — | 3 | 1 | no | — |
| 65 | C3×#41 | — | 6 | 1 | no | — |
| 66 | C3×#41 | C2×D9 | 1 | 1 | yes | characteristic |
| 67 | C2×C3×Q8 | — | 2 | 1 | no | — |
| 68 | C2×C3×Q8 | — | 3 | 1 | no | — |
| 69 | C2×D12 | — | 9 | 1 | no | — |
| 70 | #215 | — | 18 | 1 | no | — |
| 71 | #215 | — | 18 | 1 | no | — |
| 72 | #213 | — | 18 | 1 | no | — |
| 73 | #213 | — | 18 | 1 | no | — |
| 74 | (C2×D3)⋊C4 | — | 18 | 2 | no | — |
| 75 | C4×Dic3 | — | 9 | 1 | no | — |
| 76 | #208 | — | 9 | 1 | no | — |
| 77 | #208 | — | 9 | 1 | no | — |
| 78 | C22×C9 | — | 2 | 1 | no | — |
| 79 | C22×C9 | — | 3 | 1 | no | — |
| 80 | C2×D9 | — | 12 | 2 | no | — |
| 81 | C2×D9 | — | 12 | 1 | no | — |
| 82 | C4×C9 | — | 3 | 1 | no | — |
| 83 | C4×C9 | — | 6 | 1 | no | — |
| 84 | C4×C9 | C2×S4 | 1 | 1 | yes | characteristic |
| 85 | Dic9 | — | 12 | 1 | no | — |
| 86 | 2−1+4 | D27 | 1 | 1 | yes | characteristic, complemented |
| 87 | AGL(1,ℤ/8) | — | 27 | 1 | no | — |
| 88 | AGL(1,ℤ/8) | — | 27 | 1 | no | — |
| 89 | #124 | — | 27 | 1 | no | — |
| 90 | #104 | — | 27 | 1 | no | — |
| 91 | #104 | — | 27 | 1 | no | — |
| 92 | #101 | — | 27 | 1 | no | — |
| 93 | C27 | — | 4 | 1 | no | Sylow |
| 94 | C22×D3 | — | 18 | 1 | no | — |
| 95 | C3×Q8 | — | 3 | 1 | no | — |
| 96 | C3×Q8 | — | 6 | 1 | no | — |
| 97 | C3×Q8 | #611 | 1 | 1 | yes | characteristic |
| 98 | C3×D4 | — | 3 | 1 | no | — |
| 99 | C3×D4 | — | 6 | 1 | no | — |
| 100 | C3×D4 | (C22⋊C9)⋊C2 | 1 | 1 | yes | characteristic |
| 101 | C2×C3×C4 | — | 3 | 1 | no | — |
| 102 | C2×C3×C4 | — | 3 | 2 | no | — |
| 103 | C2×C3×C4 | — | 6 | 1 | no | — |
| 104 | C2×Dic3 | — | 18 | 1 | no | — |
| 105 | D12 | — | 18 | 1 | no | — |
| 106 | D12 | — | 18 | 1 | no | — |
| 107 | C3⋊C8 | — | 18 | 1 | no | — |
| 108 | C3⋊C8 | — | 18 | 1 | no | — |
| 109 | C2×C9 | — | 4 | 1 | no | — |
| 110 | C2×C9 | — | 6 | 1 | no | — |
| 111 | C2×C9 | GL(2,ℤ/4) | 1 | 1 | yes | characteristic |
| 112 | D9 | — | 24 | 1 | no | — |
| 113 | #41 | — | 3 | 1 | no | — |
| 114 | #41 | — | 6 | 1 | no | — |
| 115 | #41 | C2×D27 | 1 | 1 | yes | characteristic |
| 116 | C2×Q8 | — | 2 | 1 | no | — |
| 117 | C2×Q8 | — | 3 | 1 | no | — |
| 118 | C2×D4 | — | 27 | 1 | no | — |
| 119 | SD16 | — | 54 | 1 | no | — |
| 120 | SD16 | — | 54 | 1 | no | — |
| 121 | D8 | — | 54 | 1 | no | — |
| 122 | D8 | — | 54 | 1 | no | — |
| 123 | M16 | — | 27 | 1 | no | — |
| 124 | M16 | — | 27 | 1 | no | — |
| 125 | C22⋊C4 | — | 54 | 2 | no | — |
| 126 | C42 | — | 27 | 1 | no | — |
| 127 | C22×C3 | — | 2 | 1 | no | — |
| 128 | C22×C3 | — | 3 | 1 | no | — |
| 129 | C2×D3 | — | 36 | 2 | no | — |
| 130 | C2×D3 | — | 36 | 1 | no | — |
| 131 | C3×C4 | — | 3 | 1 | no | — |
| 132 | C3×C4 | — | 6 | 1 | no | — |
| 133 | C3×C4 | C2×(C22⋊C9)⋊C2 | 1 | 1 | yes | characteristic |
| 134 | Dic3 | — | 36 | 1 | no | — |
| 135 | C9 | #5431 | 1 | 1 | yes | characteristic |
| 136 | C23 | — | 54 | 1 | no | — |
| 137 | Q8 | — | 3 | 1 | no | — |
| 138 | Q8 | — | 6 | 1 | no | — |
| 139 | Q8 | #6171 | 1 | 1 | yes | characteristic |
| 140 | D4 | — | 3 | 1 | no | — |
| 141 | D4 | — | 6 | 1 | no | — |
| 142 | D4 | — | 54 | 1 | no | — |
| 143 | D4 | — | 54 | 1 | no | — |
| 144 | D4 | (C22⋊C27)⋊C2 | 1 | 1 | yes | characteristic |
| 145 | C2×C4 | — | 3 | 1 | no | — |
| 146 | C2×C4 | — | 3 | 2 | no | — |
| 147 | C2×C4 | — | 6 | 1 | no | — |
| 148 | C2×C4 | — | 54 | 1 | no | — |
| 149 | C8 | — | 54 | 1 | no | — |
| 150 | C8 | — | 54 | 1 | no | — |
| 151 | C2×C3 | — | 4 | 1 | no | — |
| 152 | C2×C3 | — | 6 | 1 | no | — |
| 153 | C2×C3 | #63694 | 1 | 1 | yes | characteristic |
| 154 | D3 | — | 72 | 1 | no | — |
| 155 | C22 | — | 2 | 1 | no | — |
| 156 | C22 | — | 3 | 1 | no | — |
| 157 | C22 | — | 108 | 2 | no | — |
| 158 | C22 | — | 108 | 1 | no | — |
| 159 | C4 | — | 3 | 1 | no | — |
| 160 | C4 | — | 6 | 1 | no | — |
| 161 | C4 | — | 108 | 1 | no | — |
| 162 | C4 | C2×(C22⋊C27)⋊C2 | 1 | 1 | yes | characteristic |
| 163 | C3 | #10589533 | 1 | 1 | yes | characteristic |
| 164 | C2 | — | 4 | 1 | no | — |
| 165 | C2 | — | 6 | 1 | no | — |
| 166 | C2 | — | 216 | 1 | no | — |
| 167 | C2 | #11720654 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #11720654 |
| commutator, cosocle | #41⋊C27 | C22 |
| Frattini | C4×C9 | C2×S4 |
| Fitting | C9×2−1+4 | D3 |
| socle | C2×C3 | #63694 |
| hypercenter | D4 | (C22⋊C27)⋊C2 |
| hypocenter | Q8⋊C27 | D4 |
| LCS | G ▷ #41⋊C27 ▷ Q8⋊C27 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. 2−1+4⋊C27 | C2 | semidirect |
| 2. #11720652 | C2 | semidirect |
| 3. #11720109 | C2 | semidirect |
| 86. 2−1+4 | D27 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| 2−1+4⋊C27 | 2 |
| #11720652 | 2 |
| #11720109 | 2 |
| #10588926 | 3 |
| #88369 | 4 |