Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
52 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #470 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #349 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | H(3,ℤ/4) | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | C2×Q16 | C4 | 1 | 1 | yes | characteristic, complemented |
| 5 | C2×M16 | C22 | 1 | 1 | yes | characteristic |
| 6 | C22⋊Q8 | — | 2 | 1 | no | — |
| 7 | #117 | — | 2 | 1 | no | — |
| 8 | M32 | — | 2 | 1 | no | — |
| 9 | #106 | C4 | 1 | 1 | yes | characteristic, complemented |
| 10 | #103 | — | 2 | 1 | no | — |
| 11 | C2×Q8 | — | 2 | 1 | no | — |
| 12 | C22×C4 | D4 | 1 | 1 | yes | characteristic |
| 13 | Q16 | — | 4 | 1 | no | — |
| 14 | M16 | — | 4 | 1 | no | — |
| 15 | C2×C8 | C2×C4 | 1 | 1 | yes | characteristic |
| 16 | C2×C8 | D4 | 1 | 1 | yes | characteristic |
| 17 | C4⋊C4 | — | 2 | 1 | no | — |
| 18 | C4⋊C4 | — | 4 | 1 | no | — |
| 19 | C4⋊C4 | — | 4 | 1 | no | — |
| 20 | C22⋊C4 | — | 4 | 1 | no | — |
| 21 | C22⋊C4 | — | 4 | 1 | no | — |
| 22 | C42 | — | 2 | 2 | no | — |
| 23 | C16 | — | 4 | 1 | no | — |
| 24 | C23 | D8 | 1 | 1 | yes | characteristic |
| 25 | Q8 | — | 4 | 1 | no | — |
| 26 | Q8 | — | 4 | 1 | no | — |
| 27 | C2×C4 | — | 2 | 1 | no | — |
| 28 | C2×C4 | — | 2 | 1 | no | — |
| 29 | C2×C4 | — | 4 | 2 | no | — |
| 30 | C2×C4 | — | 4 | 1 | no | — |
| 31 | C2×C4 | — | 4 | 1 | no | — |
| 32 | C2×C4 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 33 | C2×C4 | SD16 | 1 | 1 | yes | characteristic |
| 34 | C8 | — | 2 | 1 | no | — |
| 35 | C8 | — | 2 | 1 | no | — |
| 36 | C22 | — | 2 | 1 | no | — |
| 37 | C22 | — | 4 | 1 | no | — |
| 38 | C22 | #102 | 1 | 1 | yes | characteristic |
| 39 | C4 | — | 2 | 1 | no | — |
| 40 | C4 | — | 8 | 1 | no | — |
| 41 | C4 | — | 8 | 2 | no | — |
| 42 | C4 | — | 8 | 1 | no | — |
| 43 | C4 | #99 | 1 | 1 | yes | characteristic |
| 44 | C4 | #104 | 1 | 1 | yes | characteristic |
| 45 | C2 | — | 2 | 1 | no | — |
| 46 | C2 | — | 4 | 1 | no | — |
| 47 | C2 | #327 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #327 |
| commutator | C2×C8 | C2×C4 |
| Frattini, cosocle | C2×M16 | C22 |
| prime radical | C23 | D8 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. C2×Q16 | C4 | semidirect |
| 9. #106 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #470 | 2 |
| #349 | 2 |
| H(3,ℤ/4) | 2 |