Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
676 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 676 conjugacy classes of subgroups — too many to draw. Only its 48 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×#997 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C3×#5934 | C3 | 1 | 4 | yes | maximal, direct product factor |
| 3 | C32×B(4,𝔽2) | — | 3 | 1 | no | maximal |
| 4 | C2×C32×S4 | — | 4 | 3 | no | maximal |
| 5 | C32×H(5,𝔽2) | D3 | 1 | 1 | yes | characteristic, complemented |
| 6 | C3×#997 | — | 2 | 4 | no | — |
| 7 | C3×#997 | C2×C3 | 1 | 4 | yes | complemented |
| 8 | C32×#120 | — | 3 | 3 | no | — |
| 9 | C32×#99 | — | 3 | 3 | no | — |
| 10 | C2×C32×A4 | — | 4 | 3 | no | — |
| 11 | C32×S4 | — | 4 | 6 | no | — |
| 12 | C32×SL(2,𝔽3) | — | 8 | 1 | no | — |
| 13 | #5934 | C32 | 1 | 1 | yes | characteristic, direct product factor |
| 14 | C3×B(4,𝔽2) | — | 3 | 4 | no | — |
| 15 | C24×C32 | — | 3 | 1 | no | — |
| 16 | C2×C3×S4 | — | 4 | 12 | no | — |
| 17 | C32×#41 | — | 6 | 1 | no | — |
| 18 | C2×C32×D4 | — | 3 | 3 | no | — |
| 19 | C2×C32×D4 | — | 6 | 3 | no | — |
| 20 | C32×C22⋊C4 | — | 3 | 3 | no | — |
| 21 | C32×C22⋊C4 | — | 6 | 3 | no | — |
| 22 | C2×C32×D3 | — | 16 | 1 | no | — |
| 23 | C32×A4 | — | 4 | 3 | no | — |
| 24 | C3×H(5,𝔽2) | C3×D3 | 1 | 4 | yes | complemented |
| 25 | #997 | — | 2 | 4 | no | — |
| 26 | #997 | C2×C32 | 1 | 1 | yes | characteristic, complemented |
| 27 | C3×#120 | — | 3 | 12 | no | — |
| 28 | C3×#99 | — | 3 | 12 | no | — |
| 29 | C23×C32 | — | 3 | 1 | no | — |
| 30 | C23×C32 | — | 6 | 3 | no | — |
| 31 | C23×C32 | — | 12 | 2 | no | — |
| 32 | C23×C32 | S4 | 1 | 3 | yes | complemented |
| 33 | C2×C3×A4 | — | 4 | 12 | no | — |
| 34 | C2×C3×A4 | — | 8 | 12 | no | — |
| 35 | C3×S4 | — | 4 | 24 | no | — |
| 36 | C32×Q8 | — | 2 | 1 | no | — |
| 37 | C32×D4 | — | 6 | 3 | no | — |
| 38 | C32×D4 | — | 12 | 6 | no | — |
| 39 | C2×C32×C4 | — | 3 | 3 | no | — |
| 40 | C2×C32×C4 | — | 6 | 3 | no | — |
| 41 | C3×SL(2,𝔽3) | — | 8 | 4 | no | — |
| 42 | C3×SL(2,𝔽3) | — | 8 | 8 | no | — |
| 43 | B(4,𝔽2) | — | 3 | 1 | no | Sylow |
| 44 | C2×C33 | — | 16 | 1 | no | — |
| 45 | C32×D3 | — | 16 | 2 | no | — |
| 46 | C24×C3 | — | 3 | 4 | no | — |
| 47 | C2×S4 | — | 4 | 3 | no | — |
| 48 | C3×#41 | — | 6 | 4 | no | — |
| 49 | C2×C3×D4 | — | 3 | 12 | no | — |
| 50 | C2×C3×D4 | — | 6 | 12 | no | — |
| 51 | C3×C22⋊C4 | — | 3 | 12 | no | — |
| 52 | C3×C22⋊C4 | — | 6 | 12 | no | — |
| 53 | C22×C32 | — | 3 | 3 | no | — |
| 54 | C22×C32 | — | 4 | 3 | no | — |
| 55 | C22×C32 | — | 6 | 6 | no | — |
| 56 | C22×C32 | — | 12 | 1 | no | — |
| 57 | C22×C32 | — | 12 | 3 | no | — |
| 58 | C22×C32 | — | 12 | 1 | no | — |
| 59 | C2×C3×D3 | — | 16 | 4 | no | — |
| 60 | C3×A4 | — | 4 | 12 | no | — |
| 61 | C3×A4 | — | 8 | 12 | no | — |
| 62 | C32×C4 | — | 6 | 1 | no | — |
| 63 | C32×C4 | — | 12 | 3 | no | — |
| 64 | H(5,𝔽2) | C32×D3 | 1 | 1 | yes | characteristic, complemented |
| 65 | #120 | — | 3 | 3 | no | — |
| 66 | #99 | — | 3 | 3 | no | — |
| 67 | C33 | — | 16 | 1 | no | Sylow |
| 68 | C23×C3 | — | 3 | 4 | no | — |
| 69 | C23×C3 | — | 6 | 12 | no | — |
| 70 | C23×C3 | — | 12 | 8 | no | — |
| 71 | C23×C3 | C3×S4 | 1 | 12 | yes | complemented |
| 72 | C2×A4 | — | 4 | 3 | no | — |
| 73 | C2×A4 | — | 8 | 12 | no | — |
| 74 | S4 | — | 4 | 6 | no | — |
| 75 | C3×Q8 | — | 2 | 4 | no | — |
| 76 | C3×D4 | — | 6 | 12 | no | — |
| 77 | C3×D4 | — | 12 | 24 | no | — |
| 78 | C2×C3×C4 | — | 3 | 12 | no | — |
| 79 | C2×C3×C4 | — | 6 | 12 | no | — |
| 80 | SL(2,𝔽3) | — | 8 | 1 | no | — |
| 81 | SL(2,𝔽3) | — | 8 | 8 | no | — |
| 82 | C2×C32 | — | 6 | 3 | no | — |
| 83 | C2×C32 | — | 12 | 2 | no | — |
| 84 | C2×C32 | — | 16 | 4 | no | — |
| 85 | C2×C32 | — | 32 | 4 | no | — |
| 86 | C2×C32 | #1020 | 1 | 1 | yes | characteristic |
| 87 | C3×D3 | — | 16 | 8 | no | — |
| 88 | C24 | — | 3 | 1 | no | — |
| 89 | #41 | — | 6 | 1 | no | — |
| 90 | C2×D4 | — | 3 | 3 | no | — |
| 91 | C2×D4 | — | 6 | 3 | no | — |
| 92 | C22⋊C4 | — | 3 | 3 | no | — |
| 93 | C22⋊C4 | — | 6 | 3 | no | — |
| 94 | C22×C3 | — | 3 | 12 | no | — |
| 95 | C22×C3 | — | 4 | 12 | no | — |
| 96 | C22×C3 | — | 6 | 24 | no | — |
| 97 | C22×C3 | — | 12 | 4 | no | — |
| 98 | C22×C3 | — | 12 | 12 | no | — |
| 99 | C22×C3 | — | 12 | 4 | no | — |
| 100 | C2×D3 | — | 16 | 1 | no | — |
| 101 | A4 | — | 4 | 3 | no | — |
| 102 | A4 | — | 8 | 12 | no | — |
| 103 | C3×C4 | — | 6 | 4 | no | — |
| 104 | C3×C4 | — | 12 | 12 | no | — |
| 105 | C32 | — | 16 | 4 | no | — |
| 106 | C32 | — | 32 | 4 | no | — |
| 107 | C32 | #5934 | 1 | 1 | yes | characteristic, direct product factor |
| 108 | C23 | — | 3 | 1 | no | — |
| 109 | C23 | — | 6 | 3 | no | — |
| 110 | C23 | — | 12 | 2 | no | — |
| 111 | C23 | C32×S4 | 1 | 3 | yes | complemented |
| 112 | Q8 | — | 2 | 1 | no | — |
| 113 | D4 | — | 6 | 3 | no | — |
| 114 | D4 | — | 12 | 6 | no | — |
| 115 | C2×C4 | — | 3 | 3 | no | — |
| 116 | C2×C4 | — | 6 | 3 | no | — |
| 117 | C2×C3 | — | 6 | 12 | no | — |
| 118 | C2×C3 | — | 12 | 8 | no | — |
| 119 | C2×C3 | — | 16 | 1 | no | — |
| 120 | C2×C3 | — | 32 | 4 | no | — |
| 121 | C2×C3 | C3×#1020 | 1 | 4 | yes | — |
| 122 | D3 | — | 16 | 2 | no | — |
| 123 | C22 | — | 3 | 3 | no | — |
| 124 | C22 | — | 4 | 3 | no | — |
| 125 | C22 | — | 6 | 6 | no | — |
| 126 | C22 | — | 12 | 1 | no | — |
| 127 | C22 | — | 12 | 3 | no | — |
| 128 | C22 | — | 12 | 1 | no | — |
| 129 | C4 | — | 6 | 1 | no | — |
| 130 | C4 | — | 12 | 3 | no | — |
| 131 | C3 | — | 16 | 1 | no | — |
| 132 | C3 | — | 32 | 4 | no | — |
| 133 | C3 | C3×#5934 | 1 | 4 | yes | direct product factor |
| 134 | C2 | — | 6 | 3 | no | — |
| 135 | C2 | — | 12 | 2 | no | — |
| 136 | C2 | C32×#1020 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C2×C32 | #1020 |
| commutator, cosocle, hypocenter | #997 | C2×C32 |
| Frattini | C2 | C32×#1020 |
| Fitting | C32×H(5,𝔽2) | D3 |
| LCS | G ▷ #997 | |
| UCS | C1 ◁ C2×C32 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C32×#997 | C2 | semidirect |
| 2. C3×#5934 | C3 | direct (N × H) |
| 5. C32×H(5,𝔽2) | D3 | semidirect |
| 7. C3×#997 | C2×C3 | semidirect |
| 13. #5934 | C32 | direct (N × H) |
| 24. C3×H(5,𝔽2) | C3×D3 | semidirect |
| 26. #997 | C2×C32 | semidirect |
| 32. C23×C32 | S4 | semidirect |
| 64. H(5,𝔽2) | C32×D3 | semidirect |
| 71. C23×C3 | C3×S4 | semidirect |
| 107. C32 | #5934 | direct (N × H) |
| 111. C23 | C32×S4 | semidirect |
| 133. C3 | C3×#5934 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32×#997 | 2 |
| C3×#5934 | 3 |
| C32×B(4,𝔽2) | 3 |
| C2×C32×S4 | 4 |