Finite Group DatabasebetaFinite Group DatabasebetaOrder 1152.
393 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 393 conjugacy classes of subgroups — too many to draw. Only its 68 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 | C9×H(3,𝔽4) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #10588739 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | #10588217 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | #71948 | — | 3 | 1 | no | maximal |
| 5 | C9×#124 | — | 2 | 3 | no | — |
| 6 | C9×#124 | C22 | 1 | 3 | yes | — |
| 7 | C9×#120 | — | 2 | 3 | no | — |
| 8 | C2×#3675 | — | 2 | 3 | no | — |
| 9 | C2×#3673 | — | 2 | 3 | no | — |
| 10 | #63436 | C22 | 1 | 1 | yes | characteristic, complemented |
| 11 | #63362 | C22 | 1 | 3 | yes | complemented |
| 12 | C3×H(3,𝔽4) | D3 | 1 | 1 | yes | characteristic |
| 13 | #5058 | — | 3 | 3 | no | — |
| 14 | #4540 | — | 3 | 3 | no | — |
| 15 | C24×C9 | — | 2 | 1 | no | — |
| 16 | C2×Q8×C9 | D4 | 1 | 3 | yes | — |
| 17 | C2×D4×C9 | — | 2 | 3 | no | — |
| 18 | C2×D4×C9 | D4 | 1 | 3 | yes | — |
| 19 | C2×D36 | — | 2 | 3 | no | — |
| 20 | C9×C22⋊C4 | — | 2 | 3 | no | — |
| 21 | C9×C22⋊C4 | — | 2 | 6 | no | — |
| 22 | C42×C9 | — | 2 | 1 | no | — |
| 23 | C42×C9 | C23 | 1 | 1 | yes | characteristic |
| 24 | #3675 | — | 4 | 6 | no | — |
| 25 | #3673 | — | 4 | 6 | no | — |
| 26 | C2×C9⋊C8 | — | 2 | 3 | no | — |
| 27 | #1655 | — | 9 | 1 | no | Sylow |
| 28 | C3×#124 | — | 2 | 3 | no | — |
| 29 | C3×#124 | C2×D3 | 1 | 3 | yes | — |
| 30 | C3×#120 | — | 2 | 3 | no | — |
| 31 | C2×#215 | — | 6 | 3 | no | — |
| 32 | C2×#213 | — | 6 | 3 | no | — |
| 33 | #874 | — | 3 | 1 | no | — |
| 34 | #803 | — | 3 | 3 | no | — |
| 35 | C23×C9 | — | 2 | 3 | no | — |
| 36 | C23×C9 | — | 8 | 3 | no | — |
| 37 | C22×D9 | — | 4 | 1 | no | — |
| 38 | Q8×C9 | — | 4 | 3 | no | — |
| 39 | D4×C9 | — | 4 | 3 | no | — |
| 40 | D4×C9 | — | 8 | 3 | no | — |
| 41 | C2×C4×C9 | — | 2 | 3 | no | — |
| 42 | C2×C4×C9 | C2×D4 | 1 | 3 | yes | — |
| 43 | D36 | — | 4 | 3 | no | — |
| 44 | D36 | — | 4 | 3 | no | — |
| 45 | C9⋊C8 | — | 2 | 6 | no | — |
| 46 | H(3,𝔽4) | D9 | 1 | 1 | yes | characteristic, complemented |
| 47 | #496 | — | 9 | 3 | no | — |
| 48 | #329 | — | 9 | 3 | no | — |
| 49 | C24×C3 | — | 2 | 1 | no | — |
| 50 | C2×C3×Q8 | #67 | 1 | 3 | yes | — |
| 51 | C2×C3×D4 | — | 2 | 3 | no | — |
| 52 | C2×C3×D4 | #67 | 1 | 3 | yes | — |
| 53 | C2×D12 | — | 6 | 3 | no | — |
| 54 | C3×C22⋊C4 | — | 2 | 3 | no | — |
| 55 | C3×C22⋊C4 | — | 2 | 6 | no | — |
| 56 | C3×C42 | — | 2 | 1 | no | — |
| 57 | C3×C42 | C22×D3 | 1 | 1 | yes | characteristic |
| 58 | #215 | — | 12 | 6 | no | — |
| 59 | #213 | — | 12 | 6 | no | — |
| 60 | C2×C3⋊C8 | — | 6 | 3 | no | — |
| 61 | C22×C9 | — | 4 | 3 | no | — |
| 62 | C22×C9 | — | 4 | 6 | no | — |
| 63 | C22×C9 | — | 8 | 4 | no | — |
| 64 | C22×C9 | #120 | 1 | 1 | yes | characteristic |
| 65 | C2×D9 | — | 8 | 3 | no | — |
| 66 | C4×C9 | — | 2 | 3 | no | — |
| 67 | C4×C9 | — | 4 | 3 | no | — |
| 68 | C2×SD16 | — | 18 | 3 | no | — |
| 69 | C2×D8 | — | 18 | 3 | no | — |
| 70 | #127 | — | 9 | 1 | no | — |
| 71 | #124 | — | 2 | 3 | no | — |
| 72 | #124 | C2×D9 | 1 | 3 | yes | — |
| 73 | #120 | — | 2 | 3 | no | — |
| 74 | #97 | — | 9 | 3 | no | — |
| 75 | C23×C3 | — | 2 | 3 | no | — |
| 76 | C23×C3 | — | 8 | 3 | no | — |
| 77 | C22×D3 | — | 12 | 1 | no | — |
| 78 | C3×Q8 | — | 4 | 3 | no | — |
| 79 | C3×D4 | — | 4 | 3 | no | — |
| 80 | C3×D4 | — | 8 | 3 | no | — |
| 81 | C2×C3×C4 | — | 2 | 3 | no | — |
| 82 | C2×C3×C4 | C2×#67 | 1 | 3 | yes | — |
| 83 | D12 | — | 12 | 3 | no | — |
| 84 | D12 | — | 12 | 3 | no | — |
| 85 | C3⋊C8 | — | 6 | 6 | no | — |
| 86 | C2×C9 | — | 8 | 3 | no | — |
| 87 | C2×C9 | #453 | 1 | 3 | yes | — |
| 88 | D9 | — | 16 | 1 | no | — |
| 89 | C24 | — | 2 | 1 | no | — |
| 90 | C2×Q8 | #611 | 1 | 3 | yes | — |
| 91 | C2×D4 | — | 2 | 3 | no | — |
| 92 | C2×D4 | — | 18 | 3 | no | — |
| 93 | C2×D4 | #611 | 1 | 3 | yes | — |
| 94 | SD16 | — | 36 | 6 | no | — |
| 95 | D8 | — | 36 | 6 | no | — |
| 96 | C2×C8 | — | 18 | 3 | no | — |
| 97 | C22⋊C4 | — | 2 | 3 | no | — |
| 98 | C22⋊C4 | — | 2 | 6 | no | — |
| 99 | C42 | — | 2 | 1 | no | — |
| 100 | C42 | C22×D9 | 1 | 1 | yes | characteristic |
| 101 | C22×C3 | — | 4 | 3 | no | — |
| 102 | C22×C3 | — | 4 | 6 | no | — |
| 103 | C22×C3 | — | 8 | 4 | no | — |
| 104 | C22×C3 | #953 | 1 | 1 | yes | characteristic |
| 105 | C2×D3 | — | 24 | 3 | no | — |
| 106 | C3×C4 | — | 2 | 3 | no | — |
| 107 | C3×C4 | — | 4 | 3 | no | — |
| 108 | C9 | #1655 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 109 | C23 | — | 2 | 3 | no | — |
| 110 | C23 | — | 8 | 3 | no | — |
| 111 | C23 | — | 36 | 1 | no | — |
| 112 | Q8 | — | 4 | 3 | no | — |
| 113 | D4 | — | 4 | 3 | no | — |
| 114 | D4 | — | 8 | 3 | no | — |
| 115 | D4 | — | 36 | 3 | no | — |
| 116 | D4 | — | 36 | 3 | no | — |
| 117 | C2×C4 | — | 2 | 3 | no | — |
| 118 | C2×C4 | C2×#611 | 1 | 3 | yes | — |
| 119 | C8 | — | 18 | 6 | no | — |
| 120 | C2×C3 | — | 8 | 3 | no | — |
| 121 | C2×C3 | #5241 | 1 | 3 | yes | — |
| 122 | D3 | — | 48 | 1 | no | — |
| 123 | C22 | — | 4 | 3 | no | — |
| 124 | C22 | — | 4 | 6 | no | — |
| 125 | C22 | — | 8 | 4 | no | — |
| 126 | C22 | — | 72 | 3 | no | — |
| 127 | C22 | #63515 | 1 | 1 | yes | characteristic |
| 128 | C4 | — | 2 | 3 | no | — |
| 129 | C4 | — | 4 | 3 | no | — |
| 130 | C3 | #71948 | 1 | 1 | yes | characteristic |
| 131 | C2 | — | 8 | 3 | no | — |
| 132 | C2 | — | 144 | 1 | no | — |
| 133 | C2 | #10588922 | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center | C22 | #63515 |
| commutator, cosocle | C42×C9 | C23 |
| Frattini | C3×C42 | C22×D3 |
| Fitting | C9×H(3,𝔽4) | C2 |
| socle | C22×C3 | #953 |
| hypercenter | H(3,𝔽4) | D9 |
| hypocenter | C9 | #1655 |
| LCS | G ▷ C42×C9 ▷ C22×C9 ▷ C9 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ H(3,𝔽4) | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C9×H(3,𝔽4) | C2 | semidirect |
| 2. #10588739 | C2 | semidirect |
| 3. #10588217 | C2 | semidirect |
| 10. #63436 | C22 | semidirect |
| 11. #63362 | C22 | semidirect |
| 46. H(3,𝔽4) | D9 | semidirect |
| 108. C9 | #1655 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C9×H(3,𝔽4) | 2 |
| #10588739 | 2 |
| #10588217 | 2 |
| #71948 | 3 |