Finite Group DatabasebetaFinite Group DatabasebetaOrder 3024.
219 conjugacy classes of subgroups; only the 12 normal subgroups are shown.
This group has 219 conjugacy classes of subgroups — too many to draw. Only its 12 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 | C32×GL(3,𝔽2) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×C3×GL(3,𝔽2) | — | 3 | 1 | no | maximal |
| 3 | D3×GL(3,𝔽2) | C3 | 1 | 1 | yes | maximal, characteristic, direct product factor |
| 4 | C3×GL(3,𝔽2) | — | 2 | 1 | no | — |
| 5 | C3×GL(3,𝔽2) | D3 | 1 | 1 | yes | characteristic, direct product factor |
| 6 | C3×GL(3,𝔽2) | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 7 | C3×D3×S4 | — | 7 | 2 | no | maximal, Hall |
| 8 | C3×D3×ΓL(1,𝔽8) | — | 8 | 1 | no | maximal |
| 9 | C2×GL(3,𝔽2) | — | 3 | 1 | no | — |
| 10 | C3×D3×A4 | — | 7 | 2 | no | — |
| 11 | C3×C3⋊S4 | — | 7 | 2 | no | — |
| 12 | C32×S4 | — | 7 | 2 | no | — |
| 13 | C32×ΓL(1,𝔽8) | — | 8 | 1 | no | Hall |
| 14 | GL(3,𝔽2) | C3×D3 | 1 | 1 | yes | characteristic, direct product factor |
| 15 | C2×C3×S4 | — | 21 | 2 | no | — |
| 16 | D3×S4 | — | 7 | 2 | no | — |
| 17 | C3×D3×D4 | — | 21 | 1 | no | — |
| 18 | C3×D3×C7 | — | 8 | 1 | no | — |
| 19 | C2×C3×ΓL(1,𝔽8) | — | 24 | 1 | no | — |
| 20 | D3×ΓL(1,𝔽8) | — | 8 | 1 | no | — |
| 21 | D3×ΓL(1,𝔽8) | — | 8 | 2 | no | — |
| 22 | C32×A4 | — | 7 | 2 | no | — |
| 23 | C3×D32 | — | 28 | 1 | no | — |
| 24 | C22×C3×D3 | — | 7 | 2 | no | — |
| 25 | C2×C3×A4 | — | 21 | 2 | no | — |
| 26 | D3×A4 | — | 7 | 2 | no | — |
| 27 | D3×A4 | — | 14 | 2 | no | — |
| 28 | C3⋊S4 | — | 7 | 2 | no | — |
| 29 | C3×S4 | — | 7 | 2 | no | — |
| 30 | C3×S4 | — | 7 | 2 | no | — |
| 31 | C3×S4 | — | 14 | 2 | no | — |
| 32 | C3×S4 | — | 21 | 2 | no | — |
| 33 | C3×S4 | — | 42 | 2 | no | — |
| 34 | C32×D4 | — | 21 | 1 | no | — |
| 35 | C3×#67 | — | 21 | 2 | no | — |
| 36 | C3×D12 | — | 21 | 1 | no | — |
| 37 | C3×C4×D3 | — | 21 | 1 | no | — |
| 38 | C32×C7 | — | 8 | 1 | no | — |
| 39 | C3×ΓL(1,𝔽8) | — | 8 | 1 | no | — |
| 40 | C3×ΓL(1,𝔽8) | — | 8 | 1 | no | — |
| 41 | C3×ΓL(1,𝔽8) | — | 8 | 2 | no | — |
| 42 | C3×ΓL(1,𝔽8) | — | 16 | 1 | no | — |
| 43 | C3×ΓL(1,𝔽8) | — | 16 | 1 | no | — |
| 44 | C3×ΓL(1,𝔽8) | — | 16 | 2 | no | — |
| 45 | C3×C3⋊D3 | — | 28 | 1 | no | — |
| 46 | C32×D3 | — | 28 | 1 | no | — |
| 47 | C32×D3 | — | 28 | 1 | no | — |
| 48 | C2×S4 | — | 21 | 2 | no | — |
| 49 | C2×C3×D4 | — | 63 | 1 | no | — |
| 50 | D3×D4 | — | 21 | 1 | no | — |
| 51 | C2×C3×C7 | — | 24 | 1 | no | — |
| 52 | D3×C7 | — | 8 | 1 | no | — |
| 53 | C2×ΓL(1,𝔽8) | — | 24 | 1 | no | — |
| 54 | C2×ΓL(1,𝔽8) | — | 24 | 2 | no | — |
| 55 | C22×C32 | — | 7 | 2 | no | — |
| 56 | C2×C3×D3 | — | 21 | 1 | no | — |
| 57 | C2×C3×D3 | — | 21 | 2 | no | — |
| 58 | C2×C3×D3 | — | 84 | 1 | no | — |
| 59 | C3×A4 | — | 7 | 2 | no | — |
| 60 | C3×A4 | — | 7 | 2 | no | — |
| 61 | C3×A4 | — | 14 | 2 | no | — |
| 62 | C3×A4 | — | 14 | 2 | no | — |
| 63 | C3×A4 | — | 14 | 2 | no | — |
| 64 | C3×A4 | — | 28 | 2 | no | — |
| 65 | D32 | — | 28 | 1 | no | — |
| 66 | C32×C4 | — | 21 | 1 | no | — |
| 67 | C3×Dic3 | — | 21 | 1 | no | — |
| 68 | C33 | — | 28 | 1 | no | Sylow |
| 69 | C23×C3 | — | 21 | 2 | no | — |
| 70 | C22×D3 | — | 7 | 2 | no | — |
| 71 | C2×A4 | — | 21 | 2 | no | — |
| 72 | C2×A4 | — | 42 | 2 | no | — |
| 73 | S4 | — | 7 | 2 | no | — |
| 74 | S4 | — | 21 | 2 | no | — |
| 75 | S4 | — | 42 | 2 | no | — |
| 76 | C3×D4 | — | 21 | 1 | no | — |
| 77 | C3×D4 | — | 21 | 1 | no | — |
| 78 | C3×D4 | — | 42 | 1 | no | — |
| 79 | C3×D4 | — | 63 | 1 | no | — |
| 80 | C3×D4 | — | 63 | 2 | no | — |
| 81 | C2×C3×C4 | — | 63 | 1 | no | — |
| 82 | #67 | — | 21 | 2 | no | — |
| 83 | D12 | — | 21 | 1 | no | — |
| 84 | C4×D3 | — | 21 | 1 | no | — |
| 85 | C3×C7 | — | 8 | 1 | no | — |
| 86 | C3×C7 | — | 8 | 1 | no | — |
| 87 | C3×C7 | — | 16 | 1 | no | — |
| 88 | ΓL(1,𝔽8) | — | 8 | 1 | no | — |
| 89 | ΓL(1,𝔽8) | — | 8 | 2 | no | — |
| 90 | ΓL(1,𝔽8) | — | 16 | 1 | no | — |
| 91 | ΓL(1,𝔽8) | — | 16 | 2 | no | — |
| 92 | C2×C32 | — | 21 | 1 | no | — |
| 93 | C2×C32 | — | 84 | 1 | no | — |
| 94 | C3⋊D3 | — | 28 | 1 | no | — |
| 95 | C3×D3 | — | 21 | 1 | no | — |
| 96 | C3×D3 | — | 28 | 1 | no | — |
| 97 | C3×D3 | — | 28 | 1 | no | — |
| 98 | C3×D3 | — | 28 | 1 | no | — |
| 99 | C3×D3 | — | 56 | 1 | no | — |
| 100 | C3×D3 | — | 56 | 1 | no | — |
| 101 | C3×D3 | — | 84 | 1 | no | — |
| 102 | C3×D3 | — | 168 | 1 | no | — |
| 103 | C3×D3 | GL(3,𝔽2) | 1 | 1 | yes | maximal normal, characteristic, direct product factor |
| 104 | C2×D4 | — | 63 | 1 | no | Sylow |
| 105 | C2×C7 | — | 24 | 1 | no | — |
| 106 | C22×C3 | — | 7 | 2 | no | — |
| 107 | C22×C3 | — | 7 | 2 | no | — |
| 108 | C22×C3 | — | 14 | 2 | no | — |
| 109 | C22×C3 | — | 63 | 1 | no | — |
| 110 | C22×C3 | — | 63 | 2 | no | — |
| 111 | C2×D3 | — | 21 | 1 | no | — |
| 112 | C2×D3 | — | 21 | 2 | no | — |
| 113 | C2×D3 | — | 84 | 1 | no | — |
| 114 | A4 | — | 7 | 2 | no | — |
| 115 | A4 | — | 14 | 2 | no | — |
| 116 | A4 | — | 14 | 2 | no | — |
| 117 | A4 | — | 28 | 2 | no | — |
| 118 | C3×C4 | — | 21 | 1 | no | — |
| 119 | C3×C4 | — | 21 | 1 | no | — |
| 120 | C3×C4 | — | 42 | 1 | no | — |
| 121 | C3×C4 | — | 63 | 1 | no | — |
| 122 | Dic3 | — | 21 | 1 | no | — |
| 123 | C32 | — | 28 | 1 | no | — |
| 124 | C32 | — | 28 | 1 | no | — |
| 125 | C32 | — | 56 | 1 | no | — |
| 126 | C32 | — | 56 | 1 | no | — |
| 127 | C32 | — | 56 | 1 | no | — |
| 128 | C32 | — | 112 | 1 | no | — |
| 129 | C32 | C2×GL(3,𝔽2) | 1 | 1 | yes | characteristic, complemented |
| 130 | C23 | — | 21 | 2 | no | — |
| 131 | D4 | — | 21 | 1 | no | — |
| 132 | D4 | — | 63 | 1 | no | — |
| 133 | D4 | — | 63 | 2 | no | — |
| 134 | C2×C4 | — | 63 | 1 | no | — |
| 135 | C7 | — | 8 | 1 | no | Sylow |
| 136 | C2×C3 | — | 3 | 1 | no | — |
| 137 | C2×C3 | — | 21 | 1 | no | — |
| 138 | C2×C3 | — | 21 | 1 | no | — |
| 139 | C2×C3 | — | 42 | 1 | no | — |
| 140 | C2×C3 | — | 63 | 1 | no | — |
| 141 | C2×C3 | — | 84 | 1 | no | — |
| 142 | C2×C3 | — | 168 | 1 | no | — |
| 143 | D3 | — | 21 | 1 | no | — |
| 144 | D3 | — | 28 | 1 | no | — |
| 145 | D3 | — | 84 | 1 | no | — |
| 146 | D3 | — | 168 | 1 | no | — |
| 147 | D3 | C3×GL(3,𝔽2) | 1 | 1 | yes | characteristic, direct product factor |
| 148 | C22 | — | 7 | 2 | no | — |
| 149 | C22 | — | 63 | 1 | no | — |
| 150 | C22 | — | 63 | 2 | no | — |
| 151 | C4 | — | 21 | 1 | no | — |
| 152 | C4 | — | 63 | 1 | no | — |
| 153 | C3 | — | 2 | 1 | no | — |
| 154 | C3 | — | 28 | 1 | no | — |
| 155 | C3 | — | 56 | 1 | no | — |
| 156 | C3 | — | 56 | 1 | no | — |
| 157 | C3 | — | 112 | 1 | no | — |
| 158 | C3 | D3×GL(3,𝔽2) | 1 | 1 | yes | characteristic, direct product factor |
| 159 | C3 | C2×C3×GL(3,𝔽2) | 1 | 1 | yes | characteristic, complemented |
| 160 | C2 | — | 3 | 1 | no | — |
| 161 | C2 | — | 21 | 1 | no | — |
| 162 | C2 | — | 63 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | D3×GL(3,𝔽2) |
| commutator, hypocenter | C3×GL(3,𝔽2) | C2×C3 |
| Frattini | C1 | G |
| Fitting | C32 | C2×GL(3,𝔽2) |
| socle | C32×GL(3,𝔽2) | C2 |
| cosocle | C3 | C2×C3×GL(3,𝔽2) |
| LCS | G ▷ C3×GL(3,𝔽2) | |
| UCS | C1 ◁ C3 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C32×GL(3,𝔽2) | C2 | semidirect |
| 3. D3×GL(3,𝔽2) | C3 | direct (N × H) |
| 5. C3×GL(3,𝔽2) | D3 | direct (N × H) |
| 6. C3×GL(3,𝔽2) | C2×C3 | semidirect |
| 14. GL(3,𝔽2) | C3×D3 | direct (N × H) |
| 103. C3×D3 | GL(3,𝔽2) | direct (N × H) |
| 129. C32 | C2×GL(3,𝔽2) | semidirect |
| 147. D3 | C3×GL(3,𝔽2) | direct (N × H) |
| 158. C3 | D3×GL(3,𝔽2) | direct (N × H) |
| 159. C3 | C2×C3×GL(3,𝔽2) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32×GL(3,𝔽2) | 2 |
| C2×C3×GL(3,𝔽2) | 3 |
| D3×GL(3,𝔽2) | 3 |
| C3×D3×S4 | 7 |
| C3×D3×ΓL(1,𝔽8) | 8 |