Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
1222 conjugacy classes of subgroups; only the 204 normal subgroups are shown.
This group has 1222 conjugacy classes of subgroups — too many to draw. Only its 204 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 | A4×#41 | C2 | 1 | 10 | yes | maximal, complemented |
| 2 | C2×Q8×A4 | C2 | 1 | 5 | yes | maximal, complemented |
| 3 | C22×2−1+4 | C3 | 1 | 1 | yes | maximal, characteristic, complemented, Sylow |
| 4 | C3×2−1+4 | — | 4 | 1 | no | maximal |
| 5 | Q8×A4 | C22 | 1 | 10 | yes | — |
| 6 | D4×A4 | C22 | 1 | 10 | yes | — |
| 7 | C2×C4×A4 | C22 | 1 | 15 | yes | — |
| 8 | C2×2−1+4 | — | 3 | 10 | no | — |
| 9 | C2×2−1+4 | — | 3 | 5 | no | — |
| 10 | C2×2−1+4 | — | 3 | 1 | no | — |
| 11 | C22×#41 | C2×C3 | 1 | 10 | yes | complemented |
| 12 | C23×Q8 | C2×C3 | 1 | 5 | yes | complemented |
| 13 | C22×A4 | C23 | 1 | 5 | yes | — |
| 14 | C3×#41 | — | 4 | 10 | no | — |
| 15 | C2×C3×Q8 | — | 4 | 5 | no | — |
| 16 | C4×A4 | C23 | 1 | 10 | yes | — |
| 17 | 2−1+4 | — | 3 | 20 | no | — |
| 18 | 2−1+4 | — | 3 | 30 | no | — |
| 19 | 2−1+4 | — | 3 | 20 | no | — |
| 20 | 2−1+4 | — | 3 | 10 | no | — |
| 21 | 2−1+4 | — | 3 | 5 | no | — |
| 22 | 2−1+4 | A4 | 1 | 1 | yes | characteristic, direct product factor |
| 23 | C2×#41 | — | 3 | 30 | no | — |
| 24 | C2×#41 | — | 3 | 30 | no | — |
| 25 | C2×#41 | — | 3 | 10 | no | — |
| 26 | C2×#41 | — | 3 | 10 | no | — |
| 27 | C22×Q8 | — | 3 | 15 | no | — |
| 28 | C22×Q8 | — | 3 | 20 | no | — |
| 29 | C22×Q8 | — | 3 | 5 | no | — |
| 30 | C22×Q8 | C22×C3 | 1 | 10 | yes | — |
| 31 | C22×D4 | C22×C3 | 1 | 10 | yes | — |
| 32 | C23×C4 | C22×C3 | 1 | 15 | yes | — |
| 33 | C2×A4 | — | 2 | 5 | no | — |
| 34 | C2×A4 | C24 | 1 | 1 | yes | characteristic |
| 35 | C3×Q8 | — | 4 | 10 | no | — |
| 36 | C3×D4 | — | 4 | 10 | no | — |
| 37 | C2×C3×C4 | — | 4 | 15 | no | — |
| 38 | C24 | C23×C3 | 1 | 5 | yes | — |
| 39 | #41 | — | 3 | 60 | no | — |
| 40 | #41 | — | 3 | 60 | no | — |
| 41 | #41 | — | 3 | 20 | no | — |
| 42 | #41 | — | 3 | 10 | no | — |
| 43 | #41 | — | 3 | 30 | no | — |
| 44 | #41 | — | 3 | 30 | no | — |
| 45 | #41 | C2×A4 | 1 | 10 | yes | complemented |
| 46 | C2×Q8 | — | 3 | 30 | no | — |
| 47 | C2×Q8 | — | 3 | 60 | no | — |
| 48 | C2×Q8 | — | 3 | 10 | no | — |
| 49 | C2×Q8 | — | 3 | 10 | no | — |
| 50 | C2×Q8 | — | 3 | 20 | no | — |
| 51 | C2×Q8 | — | 3 | 15 | no | — |
| 52 | C2×Q8 | C2×A4 | 1 | 5 | yes | complemented |
| 53 | C2×D4 | — | 3 | 10 | no | — |
| 54 | C2×D4 | — | 3 | 10 | no | — |
| 55 | C2×D4 | — | 3 | 20 | no | — |
| 56 | C22×C4 | — | 3 | 15 | no | — |
| 57 | C22×C4 | — | 3 | 15 | no | — |
| 58 | C22×C4 | — | 3 | 30 | no | — |
| 59 | C22×C4 | C23×C3 | 1 | 10 | yes | — |
| 60 | C22×C3 | — | 4 | 5 | no | — |
| 61 | A4 | 2−1+4 | 1 | 1 | yes | characteristic, direct product factor |
| 62 | C3×C4 | — | 4 | 10 | no | — |
| 63 | C23 | — | 2 | 5 | no | — |
| 64 | C23 | — | 3 | 5 | no | — |
| 65 | C23 | — | 3 | 5 | no | — |
| 66 | C23 | — | 6 | 5 | no | — |
| 67 | C23 | C24×C3 | 1 | 1 | yes | characteristic |
| 68 | Q8 | — | 3 | 20 | no | — |
| 69 | Q8 | — | 3 | 30 | no | — |
| 70 | Q8 | C22×A4 | 1 | 10 | yes | — |
| 71 | D4 | — | 3 | 20 | no | — |
| 72 | D4 | — | 3 | 20 | no | — |
| 73 | D4 | — | 3 | 10 | no | — |
| 74 | D4 | C22×A4 | 1 | 10 | yes | — |
| 75 | C2×C4 | — | 3 | 30 | no | — |
| 76 | C2×C4 | — | 3 | 15 | no | — |
| 77 | C2×C4 | — | 3 | 10 | no | — |
| 78 | C2×C4 | — | 3 | 30 | no | — |
| 79 | C2×C4 | — | 3 | 10 | no | — |
| 80 | C2×C4 | C22×A4 | 1 | 15 | yes | — |
| 81 | C2×C3 | — | 4 | 1 | no | — |
| 82 | C2×C3 | — | 8 | 5 | no | — |
| 83 | C22 | — | 3 | 1 | no | — |
| 84 | C22 | — | 3 | 1 | no | — |
| 85 | C22 | — | 3 | 5 | no | — |
| 86 | C22 | — | 6 | 5 | no | — |
| 87 | C22 | — | 6 | 5 | no | — |
| 88 | C22 | — | 6 | 5 | no | — |
| 89 | C22 | — | 6 | 5 | no | — |
| 90 | C22 | C3×2−1+4 | 1 | 1 | yes | characteristic, complemented |
| 91 | C22 | C23×A4 | 1 | 5 | yes | — |
| 92 | C4 | — | 3 | 10 | no | — |
| 93 | C4 | C23×A4 | 1 | 10 | yes | — |
| 94 | C3 | — | 4 | 1 | no | Sylow |
| 95 | C2 | — | 2 | 5 | no | — |
| 96 | C2 | — | 3 | 1 | no | — |
| 97 | C2 | — | 3 | 1 | no | — |
| 98 | C2 | — | 6 | 5 | no | — |
| 99 | C2 | C24×A4 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C24×A4 |
| commutator, socle, cosocle | C23 | C24×C3 |
| Fitting | C22×2−1+4 | C3 |
| hypercenter | 2−1+4 | A4 |
| hypocenter | C22 | C3×2−1+4 |
| LCS | G ▷ C23 ▷ C22 | |
| UCS | C1 ◁ C2 ◁ 2−1+4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. A4×#41 | C2 | semidirect |
| 2. C2×Q8×A4 | C2 | semidirect |
| 3. C22×2−1+4 | C3 | semidirect |
| 11. C22×#41 | C2×C3 | semidirect |
| 12. C23×Q8 | C2×C3 | semidirect |
| 22. 2−1+4 | A4 | direct (N × H) |
| 45. #41 | C2×A4 | semidirect |
| 52. C2×Q8 | C2×A4 | semidirect |
| 61. A4 | 2−1+4 | direct (N × H) |
| 90. C22 | C3×2−1+4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| A4×#41 | 2 |
| C2×Q8×A4 | 2 |
| C22×2−1+4 | 3 |
| C3×2−1+4 | 4 |