Finite Group DatabasebetaFinite Group DatabasebetaOrder 1152.
288 conjugacy classes of subgroups; only the 22 normal subgroups are shown.
This group has 288 conjugacy classes of subgroups — too many to draw. Only its 22 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×#5465 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #85560 | C3 | 1 | 1 | yes | maximal, characteristic, direct product factor |
| 3 | C3×#1655 | — | 3 | 1 | no | maximal |
| 4 | C2×C3×GL(2,𝔽3) | — | 4 | 3 | no | maximal |
| 5 | C3×H(3,𝔽4) | D3 | 1 | 1 | yes | characteristic, complemented |
| 6 | #5465 | — | 2 | 1 | no | — |
| 7 | #5465 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 8 | C3×#496 | — | 3 | 3 | no | — |
| 9 | C3×#329 | — | 3 | 3 | no | — |
| 10 | C22×C3×A4 | — | 8 | 1 | no | — |
| 11 | C2×C3×SL(2,𝔽3) | — | 4 | 3 | no | — |
| 12 | C3×GL(2,𝔽3) | — | 4 | 6 | no | — |
| 13 | #1655 | — | 3 | 1 | no | Sylow |
| 14 | C2×GL(2,𝔽3) | — | 4 | 3 | no | — |
| 15 | C2×C3×SD16 | — | 6 | 3 | no | — |
| 16 | C2×C3×D8 | — | 6 | 3 | no | — |
| 17 | C3×#127 | — | 3 | 1 | no | — |
| 18 | C3×#124 | — | 3 | 3 | no | — |
| 19 | C3×#120 | — | 6 | 1 | no | — |
| 20 | C3×#97 | — | 3 | 3 | no | — |
| 21 | C22×C3×D3 | — | 16 | 1 | no | — |
| 22 | C2×C3×A4 | — | 8 | 3 | no | — |
| 23 | C3×SL(2,𝔽3) | — | 4 | 3 | no | — |
| 24 | H(3,𝔽4) | C3×D3 | 1 | 1 | yes | characteristic, complemented |
| 25 | #496 | — | 3 | 3 | no | — |
| 26 | #329 | — | 3 | 3 | no | — |
| 27 | C24×C3 | — | 2 | 1 | no | — |
| 28 | C22×A4 | — | 8 | 1 | no | — |
| 29 | C22×A4 | — | 8 | 2 | no | — |
| 30 | C2×C3×Q8 | S4 | 1 | 3 | yes | — |
| 31 | C2×C3×D4 | — | 3 | 3 | no | — |
| 32 | C2×C3×D4 | — | 6 | 3 | no | — |
| 33 | C2×SL(2,𝔽3) | — | 4 | 3 | no | — |
| 34 | C2×SL(2,𝔽3) | — | 8 | 3 | no | — |
| 35 | GL(2,𝔽3) | — | 4 | 6 | no | — |
| 36 | C3×SD16 | — | 12 | 6 | no | — |
| 37 | C3×D8 | — | 12 | 6 | no | — |
| 38 | C2×C3×C8 | — | 6 | 3 | no | — |
| 39 | C3×C22⋊C4 | — | 6 | 3 | no | — |
| 40 | C3×C42 | — | 3 | 1 | no | — |
| 41 | C22×C32 | — | 16 | 1 | no | — |
| 42 | C2×C3×D3 | — | 16 | 6 | no | — |
| 43 | C3×A4 | — | 8 | 1 | no | — |
| 44 | C2×SD16 | — | 6 | 3 | no | — |
| 45 | C2×D8 | — | 6 | 3 | no | — |
| 46 | #127 | — | 3 | 1 | no | — |
| 47 | #124 | — | 3 | 3 | no | — |
| 48 | #120 | — | 6 | 1 | no | — |
| 49 | #97 | — | 3 | 3 | no | — |
| 50 | C23×C3 | — | 6 | 1 | no | — |
| 51 | C23×C3 | — | 8 | 3 | no | — |
| 52 | C23×C3 | — | 12 | 1 | no | — |
| 53 | C22×D3 | — | 16 | 1 | no | — |
| 54 | C2×A4 | — | 8 | 3 | no | — |
| 55 | C2×A4 | — | 8 | 6 | no | — |
| 56 | C3×Q8 | — | 4 | 3 | no | — |
| 57 | C3×D4 | — | 12 | 3 | no | — |
| 58 | C3×D4 | — | 12 | 3 | no | — |
| 59 | C3×D4 | — | 12 | 3 | no | — |
| 60 | C2×C3×C4 | — | 3 | 3 | no | — |
| 61 | SL(2,𝔽3) | — | 4 | 3 | no | — |
| 62 | SL(2,𝔽3) | — | 8 | 3 | no | — |
| 63 | C3×C8 | — | 6 | 6 | no | — |
| 64 | C2×C32 | — | 16 | 3 | no | — |
| 65 | C3×D3 | — | 16 | 4 | no | — |
| 66 | C24 | — | 2 | 1 | no | — |
| 67 | C2×Q8 | C3×S4 | 1 | 3 | yes | — |
| 68 | C2×D4 | — | 3 | 3 | no | — |
| 69 | C2×D4 | — | 6 | 3 | no | — |
| 70 | SD16 | — | 12 | 6 | no | — |
| 71 | D8 | — | 12 | 6 | no | — |
| 72 | C2×C8 | — | 6 | 3 | no | — |
| 73 | C22⋊C4 | — | 6 | 3 | no | — |
| 74 | C42 | — | 3 | 1 | no | — |
| 75 | C22×C3 | — | 8 | 1 | no | — |
| 76 | C22×C3 | — | 12 | 3 | no | — |
| 77 | C22×C3 | — | 16 | 1 | no | — |
| 78 | C22×C3 | — | 24 | 1 | no | — |
| 79 | C22×C3 | — | 24 | 3 | no | — |
| 80 | C22×C3 | — | 32 | 1 | no | — |
| 81 | C22×C3 | #1020 | 1 | 1 | yes | characteristic |
| 82 | C2×D3 | — | 16 | 6 | no | — |
| 83 | A4 | — | 8 | 1 | no | — |
| 84 | A4 | — | 8 | 2 | no | — |
| 85 | C3×C4 | — | 6 | 3 | no | — |
| 86 | C32 | — | 16 | 1 | no | Sylow |
| 87 | C23 | — | 6 | 1 | no | — |
| 88 | C23 | — | 8 | 3 | no | — |
| 89 | C23 | — | 12 | 1 | no | — |
| 90 | Q8 | — | 4 | 3 | no | — |
| 91 | D4 | — | 12 | 3 | no | — |
| 92 | D4 | — | 12 | 3 | no | — |
| 93 | D4 | — | 12 | 3 | no | — |
| 94 | C2×C4 | — | 3 | 3 | no | — |
| 95 | C8 | — | 6 | 6 | no | — |
| 96 | C2×C3 | — | 16 | 3 | no | — |
| 97 | C2×C3 | — | 24 | 1 | no | — |
| 98 | C2×C3 | — | 32 | 3 | no | — |
| 99 | C2×C3 | — | 48 | 1 | no | — |
| 100 | C2×C3 | #5935 | 1 | 3 | yes | — |
| 101 | D3 | — | 16 | 4 | no | — |
| 102 | C22 | — | 8 | 1 | no | — |
| 103 | C22 | — | 12 | 3 | no | — |
| 104 | C22 | — | 24 | 1 | no | — |
| 105 | C22 | — | 24 | 3 | no | — |
| 106 | C22 | C3×#1020 | 1 | 1 | yes | characteristic |
| 107 | C4 | — | 6 | 3 | no | — |
| 108 | C3 | — | 16 | 1 | no | — |
| 109 | C3 | — | 32 | 1 | no | — |
| 110 | C3 | #85560 | 1 | 1 | yes | characteristic, direct product factor |
| 111 | C2 | — | 24 | 1 | no | — |
| 112 | C2 | — | 48 | 1 | no | — |
| 113 | C2 | C3×#5935 | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C22×C3 | #1020 |
| commutator, cosocle, hypocenter | #5465 | C2×C3 |
| Frattini | C22 | C3×#1020 |
| Fitting | C3×H(3,𝔽4) | D3 |
| LCS | G ▷ #5465 | |
| UCS | C1 ◁ C22×C3 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#5465 | C2 | semidirect |
| 2. #85560 | C3 | direct (N × H) |
| 5. C3×H(3,𝔽4) | D3 | semidirect |
| 7. #5465 | C2×C3 | semidirect |
| 24. H(3,𝔽4) | C3×D3 | semidirect |
| 110. C3 | #85560 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#5465 | 2 |
| #85560 | 3 |
| C3×#1655 | 3 |
| C2×C3×GL(2,𝔽3) | 4 |