Finite Group DatabasebetaFinite Group DatabasebetaOrder 1152.
256 conjugacy classes of subgroups; only the 16 normal subgroups are shown.
This group has 256 conjugacy classes of subgroups — too many to draw. Only its 16 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 | C3×#5464 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #85559 | C3 | 1 | 1 | yes | maximal, characteristic, direct product factor |
| 3 | C3×#2199 | — | 3 | 1 | no | maximal |
| 4 | C3×#1020 | — | 4 | 2 | no | maximal |
| 5 | C3×#857 | — | 4 | 1 | no | maximal |
| 6 | C3×H(3,𝔽4) | D3 | 1 | 1 | yes | characteristic, complemented |
| 7 | #5464 | — | 2 | 1 | no | — |
| 8 | #5464 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 9 | C3×B(4,𝔽2) | — | 3 | 2 | no | — |
| 10 | C3×#455 | — | 3 | 1 | no | — |
| 11 | C3×#354 | — | 3 | 1 | no | — |
| 12 | C3×AΣL(1,𝔽16) | — | 3 | 2 | no | — |
| 13 | C3×C22⋊A4 | — | 4 | 2 | no | — |
| 14 | C3×C42⋊C3 | — | 4 | 1 | no | — |
| 15 | C3×C42⋊C3 | — | 8 | 1 | no | — |
| 16 | #2199 | — | 3 | 1 | no | Sylow |
| 17 | #1020 | — | 4 | 2 | no | — |
| 18 | C3×H(5,𝔽2) | — | 3 | 1 | no | — |
| 19 | C3×#137 | — | 6 | 1 | no | — |
| 20 | C3×#124 | — | 3 | 1 | no | — |
| 21 | C3×#124 | — | 6 | 1 | no | — |
| 22 | C3×#120 | — | 3 | 2 | no | — |
| 23 | C3×#120 | — | 6 | 2 | no | — |
| 24 | #857 | — | 4 | 1 | no | — |
| 25 | C3×#104 | — | 6 | 1 | no | — |
| 26 | C3×#100 | — | 3 | 1 | no | — |
| 27 | C3×#99 | — | 3 | 2 | no | — |
| 28 | C3×#99 | — | 6 | 2 | no | — |
| 29 | C3×S4 | — | 16 | 1 | no | — |
| 30 | C3×S4 | — | 16 | 4 | no | — |
| 31 | H(3,𝔽4) | C3×D3 | 1 | 1 | yes | characteristic, complemented |
| 32 | B(4,𝔽2) | — | 3 | 2 | no | — |
| 33 | #455 | — | 3 | 1 | no | — |
| 34 | #354 | — | 3 | 1 | no | — |
| 35 | AΣL(1,𝔽16) | — | 3 | 2 | no | — |
| 36 | C24×C3 | S4 | 1 | 2 | yes | complemented |
| 37 | C22⋊A4 | — | 4 | 2 | no | — |
| 38 | C22⋊A4 | — | 8 | 2 | no | — |
| 39 | C3×#41 | — | 6 | 1 | no | — |
| 40 | C3×#41 | — | 12 | 1 | no | — |
| 41 | C2×C3×Q8 | — | 3 | 1 | no | — |
| 42 | C2×C3×D4 | — | 3 | 1 | no | — |
| 43 | C2×C3×D4 | — | 6 | 2 | no | — |
| 44 | C2×C3×D4 | — | 6 | 1 | no | — |
| 45 | C2×C3×D4 | — | 12 | 1 | no | — |
| 46 | C2×C3×D4 | — | 12 | 2 | no | — |
| 47 | C3×Q16 | — | 12 | 1 | no | — |
| 48 | C3×SD16 | — | 12 | 1 | no | — |
| 49 | C3×M16 | — | 6 | 1 | no | — |
| 50 | C3×C22⋊C4 | — | 3 | 2 | no | — |
| 51 | C3×C22⋊C4 | — | 6 | 2 | no | — |
| 52 | C3×C22⋊C4 | — | 6 | 2 | no | — |
| 53 | C3×C22⋊C4 | — | 12 | 2 | no | — |
| 54 | C3×C42 | — | 2 | 1 | no | — |
| 55 | C3×C42 | S4 | 1 | 1 | yes | characteristic, complemented |
| 56 | C42⋊C3 | — | 4 | 1 | no | — |
| 57 | C42⋊C3 | — | 8 | 1 | no | — |
| 58 | C42⋊C3 | — | 8 | 2 | no | — |
| 59 | C42⋊C3 | — | 8 | 1 | no | — |
| 60 | C3×A4 | — | 16 | 4 | no | — |
| 61 | C3×A4 | — | 16 | 1 | no | — |
| 62 | C3×A4 | — | 32 | 2 | no | — |
| 63 | H(5,𝔽2) | — | 3 | 1 | no | — |
| 64 | #137 | — | 6 | 1 | no | — |
| 65 | #124 | — | 3 | 1 | no | — |
| 66 | #124 | — | 6 | 1 | no | — |
| 67 | #120 | — | 3 | 2 | no | — |
| 68 | #120 | — | 6 | 2 | no | — |
| 69 | #104 | — | 6 | 1 | no | — |
| 70 | #100 | — | 3 | 1 | no | — |
| 71 | #99 | — | 3 | 2 | no | — |
| 72 | #99 | — | 6 | 2 | no | — |
| 73 | C23×C3 | — | 3 | 2 | no | — |
| 74 | C23×C3 | — | 6 | 2 | no | — |
| 75 | C23×C3 | — | 12 | 2 | no | — |
| 76 | S4 | — | 16 | 1 | no | — |
| 77 | S4 | — | 16 | 4 | no | — |
| 78 | C3×Q8 | — | 6 | 1 | no | — |
| 79 | C3×Q8 | — | 12 | 1 | no | — |
| 80 | C3×D4 | — | 6 | 1 | no | — |
| 81 | C3×D4 | — | 12 | 1 | no | — |
| 82 | C3×D4 | — | 12 | 2 | no | — |
| 83 | C3×D4 | — | 12 | 1 | no | — |
| 84 | C3×D4 | — | 24 | 1 | no | — |
| 85 | C3×D4 | — | 24 | 4 | no | — |
| 86 | C2×C3×C4 | — | 3 | 1 | no | — |
| 87 | C2×C3×C4 | — | 6 | 2 | no | — |
| 88 | C2×C3×C4 | — | 6 | 1 | no | — |
| 89 | C2×C3×C4 | — | 12 | 1 | no | — |
| 90 | C2×C3×C4 | — | 12 | 2 | no | — |
| 91 | C3×C8 | — | 12 | 1 | no | — |
| 92 | C3×D3 | — | 64 | 1 | no | — |
| 93 | C24 | C3×S4 | 1 | 2 | yes | complemented |
| 94 | #41 | — | 6 | 1 | no | — |
| 95 | #41 | — | 12 | 1 | no | — |
| 96 | C2×Q8 | — | 3 | 1 | no | — |
| 97 | C2×D4 | — | 3 | 1 | no | — |
| 98 | C2×D4 | — | 6 | 2 | no | — |
| 99 | C2×D4 | — | 6 | 1 | no | — |
| 100 | C2×D4 | — | 12 | 1 | no | — |
| 101 | C2×D4 | — | 12 | 2 | no | — |
| 102 | Q16 | — | 12 | 1 | no | — |
| 103 | SD16 | — | 12 | 1 | no | — |
| 104 | M16 | — | 6 | 1 | no | — |
| 105 | C22⋊C4 | — | 3 | 2 | no | — |
| 106 | C22⋊C4 | — | 6 | 2 | no | — |
| 107 | C22⋊C4 | — | 6 | 2 | no | — |
| 108 | C22⋊C4 | — | 12 | 2 | no | — |
| 109 | C42 | — | 2 | 1 | no | — |
| 110 | C42 | C3×S4 | 1 | 1 | yes | characteristic, complemented |
| 111 | C22×C3 | — | 4 | 4 | no | — |
| 112 | C22×C3 | — | 6 | 2 | no | — |
| 113 | C22×C3 | — | 8 | 2 | no | — |
| 114 | C22×C3 | — | 12 | 2 | no | — |
| 115 | C22×C3 | — | 12 | 1 | no | — |
| 116 | C22×C3 | — | 24 | 2 | no | — |
| 117 | C22×C3 | #1020 | 1 | 1 | yes | characteristic |
| 118 | A4 | — | 16 | 4 | no | — |
| 119 | A4 | — | 16 | 1 | no | — |
| 120 | A4 | — | 32 | 2 | no | — |
| 121 | A4 | — | 32 | 4 | no | — |
| 122 | A4 | — | 32 | 4 | no | — |
| 123 | A4 | — | 32 | 1 | no | — |
| 124 | C3×C4 | — | 6 | 1 | no | — |
| 125 | C3×C4 | — | 12 | 1 | no | — |
| 126 | C3×C4 | — | 12 | 1 | no | — |
| 127 | C3×C4 | — | 24 | 2 | no | — |
| 128 | C32 | — | 64 | 1 | no | Sylow |
| 129 | C23 | — | 3 | 2 | no | — |
| 130 | C23 | — | 6 | 2 | no | — |
| 131 | C23 | — | 12 | 2 | no | — |
| 132 | Q8 | — | 6 | 1 | no | — |
| 133 | Q8 | — | 12 | 1 | no | — |
| 134 | D4 | — | 6 | 1 | no | — |
| 135 | D4 | — | 12 | 1 | no | — |
| 136 | D4 | — | 12 | 2 | no | — |
| 137 | D4 | — | 12 | 1 | no | — |
| 138 | D4 | — | 24 | 1 | no | — |
| 139 | D4 | — | 24 | 4 | no | — |
| 140 | C2×C4 | — | 3 | 1 | no | — |
| 141 | C2×C4 | — | 6 | 2 | no | — |
| 142 | C2×C4 | — | 6 | 1 | no | — |
| 143 | C2×C4 | — | 12 | 1 | no | — |
| 144 | C2×C4 | — | 12 | 2 | no | — |
| 145 | C8 | — | 12 | 1 | no | — |
| 146 | C2×C3 | — | 3 | 1 | no | — |
| 147 | C2×C3 | — | 12 | 2 | no | — |
| 148 | C2×C3 | — | 24 | 1 | no | — |
| 149 | D3 | — | 64 | 1 | no | — |
| 150 | C22 | — | 4 | 4 | no | — |
| 151 | C22 | — | 6 | 2 | no | — |
| 152 | C22 | — | 8 | 2 | no | — |
| 153 | C22 | — | 12 | 2 | no | — |
| 154 | C22 | — | 12 | 1 | no | — |
| 155 | C22 | — | 24 | 2 | no | — |
| 156 | C22 | C3×#1020 | 1 | 1 | yes | characteristic |
| 157 | C4 | — | 6 | 1 | no | — |
| 158 | C4 | — | 12 | 1 | no | — |
| 159 | C4 | — | 12 | 1 | no | — |
| 160 | C4 | — | 24 | 2 | no | — |
| 161 | C3 | — | 64 | 1 | no | — |
| 162 | C3 | — | 128 | 1 | no | — |
| 163 | C3 | #85559 | 1 | 1 | yes | characteristic, direct product factor |
| 164 | C2 | — | 3 | 1 | no | — |
| 165 | C2 | — | 12 | 2 | no | — |
| 166 | C2 | — | 24 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | #85559 |
| commutator, cosocle, hypocenter | #5464 | C2×C3 |
| Frattini | C22 | C3×#1020 |
| Fitting | C3×H(3,𝔽4) | D3 |
| socle | C22×C3 | #1020 |
| LCS | G ▷ #5464 | |
| UCS | C1 ◁ C3 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#5464 | C2 | semidirect |
| 2. #85559 | C3 | direct (N × H) |
| 6. C3×H(3,𝔽4) | D3 | semidirect |
| 8. #5464 | C2×C3 | semidirect |
| 31. H(3,𝔽4) | C3×D3 | semidirect |
| 36. C24×C3 | S4 | semidirect |
| 55. C3×C42 | S4 | semidirect |
| 93. C24 | C3×S4 | semidirect |
| 110. C42 | C3×S4 | semidirect |
| 163. C3 | #85559 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#5464 | 2 |
| #85559 | 3 |
| C3×#2199 | 3 |
| C3×#1020 | 4 |
| C3×#857 | 4 |