Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
1216 conjugacy classes of subgroups; only the 202 normal subgroups are shown.
This group has 1216 conjugacy classes of subgroups — too many to draw. Only its 202 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 | #5945 | C2 | 1 | 6 | yes | maximal, complemented |
| 2 | C2×2−1+4⋊C3 | C2 | 1 | 9 | yes | maximal, complemented |
| 3 | 2−1+6 | C3 | 1 | 1 | yes | maximal, characteristic, complemented, Sylow |
| 4 | C3×H(5,𝔽2) | — | 4 | 1 | no | maximal |
| 5 | 2−1+4⋊C3 | C22 | 1 | 18 | yes | complemented |
| 6 | #994 | C22 | 1 | 2 | yes | complemented |
| 7 | C2×#41⋊C3 | C22 | 1 | 9 | yes | complemented |
| 8 | C22×SL(2,𝔽3) | C22 | 1 | 6 | yes | complemented |
| 9 | #585 | — | 3 | 1 | no | — |
| 10 | #585 | — | 3 | 9 | no | — |
| 11 | #585 | C2×C3 | 1 | 6 | yes | complemented |
| 12 | C2×2−1+4 | — | 3 | 6 | no | — |
| 13 | C2×2−1+4 | C2×C3 | 1 | 9 | yes | complemented |
| 14 | C3×#41 | — | 4 | 6 | no | — |
| 15 | C2×C3×D4 | — | 4 | 9 | no | — |
| 16 | #41⋊C3 | C23 | 1 | 6 | yes | — |
| 17 | C2×SL(2,𝔽3) | C23 | 1 | 9 | yes | — |
| 18 | 2−1+4 | — | 3 | 36 | no | — |
| 19 | 2−1+4 | — | 3 | 18 | no | — |
| 20 | 2−1+4 | — | 3 | 6 | no | — |
| 21 | 2−1+4 | — | 3 | 6 | no | — |
| 22 | 2−1+4 | C22×C3 | 1 | 18 | yes | complemented |
| 23 | H(5,𝔽2) | — | 3 | 12 | no | — |
| 24 | H(5,𝔽2) | — | 3 | 9 | no | — |
| 25 | H(5,𝔽2) | — | 3 | 18 | no | — |
| 26 | H(5,𝔽2) | A4 | 1 | 1 | yes | characteristic |
| 27 | H(5,𝔽2) | C22×C3 | 1 | 2 | yes | complemented |
| 28 | C2×#41 | — | 3 | 18 | no | — |
| 29 | C2×#41 | — | 3 | 36 | no | — |
| 30 | C2×#41 | — | 3 | 6 | no | — |
| 31 | C2×#41 | — | 3 | 9 | no | — |
| 32 | C2×#41 | — | 3 | 18 | no | — |
| 33 | C2×#41 | C22×C3 | 1 | 9 | yes | complemented |
| 34 | C22×Q8 | — | 3 | 9 | no | — |
| 35 | C22×Q8 | — | 3 | 4 | no | — |
| 36 | C22×Q8 | C22×C3 | 1 | 6 | yes | complemented |
| 37 | C23×C3 | — | 4 | 6 | no | — |
| 38 | C3×Q8 | — | 4 | 2 | no | — |
| 39 | C3×D4 | — | 4 | 18 | no | — |
| 40 | C2×C3×C4 | — | 4 | 9 | no | — |
| 41 | SL(2,𝔽3) | C24 | 1 | 1 | yes | characteristic |
| 42 | #41 | — | 3 | 72 | no | — |
| 43 | #41 | — | 3 | 36 | no | — |
| 44 | #41 | — | 3 | 2 | no | — |
| 45 | #41 | — | 3 | 36 | no | — |
| 46 | #41 | — | 3 | 18 | no | — |
| 47 | #41 | — | 3 | 36 | no | — |
| 48 | #41 | — | 3 | 18 | no | — |
| 49 | #41 | — | 3 | 18 | no | — |
| 50 | #41 | C2×A4 | 1 | 6 | yes | — |
| 51 | #41 | C23×C3 | 1 | 6 | yes | — |
| 52 | C2×Q8 | — | 3 | 18 | no | — |
| 53 | C2×Q8 | — | 3 | 36 | no | — |
| 54 | C2×Q8 | — | 3 | 6 | no | — |
| 55 | C2×Q8 | — | 3 | 9 | no | — |
| 56 | C2×Q8 | — | 3 | 18 | no | — |
| 57 | C2×Q8 | C23×C3 | 1 | 9 | yes | — |
| 58 | C2×D4 | — | 3 | 9 | no | — |
| 59 | C2×D4 | — | 3 | 36 | no | — |
| 60 | C2×D4 | — | 3 | 18 | no | — |
| 61 | C2×D4 | — | 3 | 18 | no | — |
| 62 | C2×D4 | — | 3 | 6 | no | — |
| 63 | C2×D4 | C2×A4 | 1 | 9 | yes | — |
| 64 | C22×C4 | — | 3 | 18 | no | — |
| 65 | C22×C4 | — | 3 | 12 | no | — |
| 66 | C22×C4 | — | 3 | 6 | no | — |
| 67 | C22×C4 | — | 3 | 9 | no | — |
| 68 | C22×C3 | — | 4 | 9 | no | — |
| 69 | C22×C3 | — | 16 | 6 | no | — |
| 70 | C3×C4 | — | 4 | 6 | no | — |
| 71 | C23 | — | 3 | 4 | no | — |
| 72 | C23 | — | 3 | 9 | no | — |
| 73 | C23 | C22×A4 | 1 | 6 | yes | — |
| 74 | Q8 | — | 3 | 9 | no | — |
| 75 | Q8 | — | 3 | 12 | no | — |
| 76 | Q8 | — | 3 | 18 | no | — |
| 77 | Q8 | C22×A4 | 1 | 2 | yes | — |
| 78 | Q8 | C24×C3 | 1 | 1 | yes | characteristic |
| 79 | D4 | — | 3 | 6 | no | — |
| 80 | D4 | — | 3 | 18 | no | — |
| 81 | D4 | — | 3 | 36 | no | — |
| 82 | D4 | — | 3 | 6 | no | — |
| 83 | D4 | C22×A4 | 1 | 18 | yes | — |
| 84 | C2×C4 | — | 3 | 36 | no | — |
| 85 | C2×C4 | — | 3 | 9 | no | — |
| 86 | C2×C4 | — | 3 | 18 | no | — |
| 87 | C2×C4 | — | 3 | 18 | no | — |
| 88 | C2×C4 | — | 3 | 6 | no | — |
| 89 | C2×C4 | C22×A4 | 1 | 9 | yes | — |
| 90 | C2×C3 | — | 4 | 1 | no | — |
| 91 | C2×C3 | — | 8 | 9 | no | — |
| 92 | C22 | — | 3 | 6 | no | — |
| 93 | C22 | — | 4 | 6 | no | — |
| 94 | C22 | — | 12 | 9 | no | — |
| 95 | C22 | — | 12 | 4 | no | — |
| 96 | C22 | C23×A4 | 1 | 9 | yes | — |
| 97 | C4 | — | 3 | 9 | no | — |
| 98 | C4 | — | 3 | 1 | no | — |
| 99 | C4 | C23×A4 | 1 | 6 | yes | — |
| 100 | C3 | — | 4 | 1 | no | Sylow |
| 101 | C2 | — | 2 | 9 | no | — |
| 102 | C2 | — | 6 | 6 | no | — |
| 103 | C2 | C24×A4 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C2 | C24×A4 |
| commutator, cosocle, hypocenter | Q8 | C24×C3 |
| Fitting | 2−1+6 | C3 |
| hypercenter | H(5,𝔽2) | A4 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C2 ◁ H(5,𝔽2) | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #5945 | C2 | semidirect |
| 2. C2×2−1+4⋊C3 | C2 | semidirect |
| 3. 2−1+6 | C3 | semidirect |
| 5. 2−1+4⋊C3 | C22 | semidirect |
| 6. #994 | C22 | semidirect |
| 7. C2×#41⋊C3 | C22 | semidirect |
| 8. C22×SL(2,𝔽3) | C22 | semidirect |
| 11. #585 | C2×C3 | semidirect |
| 13. C2×2−1+4 | C2×C3 | semidirect |
| 22. 2−1+4 | C22×C3 | semidirect |
| 27. H(5,𝔽2) | C22×C3 | semidirect |
| 33. C2×#41 | C22×C3 | semidirect |
| 36. C22×Q8 | C22×C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #5945 | 2 |
| C2×2−1+4⋊C3 | 2 |
| 2−1+6 | 3 |
| C3×H(5,𝔽2) | 4 |