Order 3 has only one type of magic
cube. It is simple but has the extra feature of being associated.
The next larger order is 4 and has cubes with a
variety of features. Of most importance, probably, is the introduction of
a major class of magic cubes, the pantriagonal cube.
Simple,
associated or not 
Schubert's simple associated order 4 cube was
published in 1898.
Weidemann's simple cube, published in 1922, is not associated.

Pantriagonal 
John Hendricks introduced the name pan3agonal in
several JRM articles in 1972, and it's mathematical relationship to
pandiagonal magic squares in 1980.

More Order 4
cubes 
Royal Heath published a pantriagonal cube in 1930.
Kanji Setsuda has produced many pantriagonal cubes with different
features.

Magic
Squares to cubes 
Adler and Shuoyen demonstrate conversion between
magic squares and cubes.

Simple, associated or not

This cube was published in
Germany in 1898 [1] along with
an order 5 magic cube. These are the first published cubes I have
been able to locate that are magic by the now accepted rules (except
fot A.H. Frost and F. A. P. Barnard). These rules, first published
by W. S. Andrews [2][3] page
69 state
A magic cube consists of a series of
numbers so arranged in cubical form that each row of numbers running
parallel with any of its edges, and also the four great diagonals
shall sum the same amount.
This is an associated magic cube because
number pairs on opposite sides of the center of the cube sum to 65,
which is the sum of the first and last numbers in the series used.
Examples are : 1 + 64, 51 + 14, 42 + 23, etc.
It contains no magic squares. However, it is unique because both
diagonals sum to the same value for each of the 12 planar arrays.
Different aspects of this cube were also
published by Adler and Shuoyen (1977) and Benson & Jacoby (1981). 
[1]Hermann Schubert, Mathematical Essays
and Recreations. Translated from German to English by Thomas J. McCormack,
Open Court, 1899.
[2] W. S. Andrews, Magic Squares & Cubes, Open Court, 1908.
[3] W. S. Andrews, Magic Squares & Cubes, 2nd edition, Dover Publ. 1960.

Alternate horizontal planes are
shown in these illustrations in alternate colors as an aid to
clarity. This cube by Ingenieur
Weidemann was first published in 1922.
[4]
A semipantriagonal, bent triagonal (see next
cube) magic cube. Not associated, but diametrically opposed cells
sum to either 64 or 66.
The 4 horizontal squares only, are pandiagonal magic. Also, all 2x2
squares in these 4 planes sum to the magic constant.
[4] Weidemann, Ingenieur, Zauberquadrate und
andere magische Zahlenfiguren der Ebene und des Raumes, Oscar Leiner,
1922 (German) Translated title is Magic squares and other plane and
solid magic figures. 

This semipantriagonal magic cube
by Hendricks appears in his 1999 book
[5]. It is magic because all rows,
columns and the 4 main triagonals sum correctly. It also is not
associated.
It has the unique feature of containing bent
triagonals. Examples of these are 56 + 9 + 24 + 41, 56 + 9 + 3 + 62,
56 + 9 + 54 + 11, 56 + 9 + 64 + 1, etc.
It is semipantriagonal because opposite short
triagonals such as 13 + 52 + 39 + 26 = 130.
All associated (center symmetric) cubes are also semipantriagonal.
[5] John R. Hendricks, Inlaid Magic Squares and
Cubes, selfpublished, 1999, 0968470017 
Pantriagonal magic is the third order of complexity
after simple and associated. (Actually, it is the second of six classes.
Associated cubes are found in all of these classes.)
John R. Hendricks published this cube in 1972 when he introduced the
concept [1][2].
Of course, the pantriagonal magic cube was not a new
invention. In fact, a rotated version of this same cube was published in
1939 [3]. However, the
characteristics were not recognized (and a name for this class not
assigned).
Required for the cube to be pantriagonal magic:
 Rows, columns pillars and 4 main triagonals must
sum correctly (the basic requirement for a magic cube).
 All broken triagonals must also sum correctly.
These will each consist of 2 or 3 segments. There are a total of 4m^{2}
triagonals in a cube.

This cube is not associated. Associated
pantriagonal magic cubes start at order 5. One triagonal is shown in red
for clarity.
Because 3 segment triagonals are rather difficult to envision, here is one
that parallels the main triagonal shown in red: 13, 38, 52, 27.
This cube has two other features only found in order 4
cubes (or 4x). All the pantriagonals cubes I have looked at, have either
one or both of these.
Is this true for all pantriagonals?
The two additional features are:
 Compact: every 2 x 2 square, horizontal or
vertical, in the cube sums to 130, the magic constant. Abe
[4] defined the term in reference
to magic squares. Kanji Setsuda extended it to cubes
[5] but used the term ‘composite’.
I have reverted back to ‘compact’ because composite has a different
meaning when used with magic squares (and by extension, with magic
cubes).
 Complete: Every pantriagonal contains m/2
complement pairs spaced m/2 apart. This is a characteristic of
mostperfect magic squares.
The pandiagonal magic square can be transformed to a
different magic square by moving a row or column from 1 side of the square
to the other. In exactly the same way, a magic cube may be transformed
into another magic cube. However, this time we move an orthogonal plane
from one side to the other. Here the bottom plane of the above cube has
been moved to the top.
John Hendricks published another paper on
Pantriagonal magic cubes in 1980 [6].
In it, he explored the mathematics, and various transformations. 
[1] John R. Hendricks, The Pan3Agonal Magic Cube, JRM
5:1:1972, pp 5154
[2] John R. Hendricks, The Pan3Agonal Magic Cube of Order 5, JRM
5:3:1972, pp 205206
[3] RouseBall & Coxeter, Mathematical Recreations & Essays, 11th edition,
1939.
[4] Gakuho Abe, Fifty Problems of Magic Squares, Self published 1950.
Later republished in Discrete Math, 127, 1994, pp 313.
[5] . http://homepage2.nifty.com/KanjiSetsuda/pages/EnglishP1.html
[6] John R. Hendricks, the Pan3agonal Magic Cube of Order 4, JRM 13:4,
198081, pp274281
More Order 4 cubes
Below I show some more order 4 cubes.
Encyclopedia Britannica  1911 edition
This is simple associated magic. No special features.
Heath 1938 pantriagonal
[1]
Because the four horizontal planes of this cube are simple magic, they may
form the quadrants of an order 8 magic square. This square has the unique
feature that alternating numbers in each row, column and the two main
diagonals sum to 130. Two oblique squares are pandiagonal magic. It is
‘complete’ because all triagonals consist of 2 complement pairs.
Kanji Setsuda lists a great many order 4 magic cubes
on his site [2]. Here I show 4
pantriagonal cubes. None of Setsuda's cubes shown here contain magic
squares.
[1] RouseBall & Coxeter, Mathematical Recreations &
Essays, 11 edition, 1939. Chapter VII. Also
Editions 12, University of Toronto Press, 1974, 0802061899 and
editions 13, Dover Publ., 1989, 0486253570
[2] Kanji Setsuda’s
http://homepage2.nifty.com/KanjiSetsuda/pages/EnglishP1.html
Magic Squares to cubes
A. Adler, & R. Shuoyen, in a 1977 paper
[1], demonstrate a method for
transforming an order m magic cube into an order 2m magic square. Diagram
(a) shows the originating cube. Diagram (b) the resulting magic square
when every 2 rows of the cube are put into 1 row of the square. (c) shows
transposing 2 columns into 1 column, and (d) shows transposing 2 pillars
into 1 row.
Each of the resulting magic squares can then be
transformed to another magic cube by changing rows or columns into 2 rows,
columns or pillars of a new magic cube.
I tried this method on six different types of magic
cubes, including a cube with prime numbers and another cube using only
every sixth number. It worked for all of them except for the Setsuda
complete (the last cube shown above). so it seems this method works for
most, but not all, order 4 magic cubes. The method can probably be
extended to other orders.
By the way. The Adler cube shown here is
semipantriagonal, associated. It is the same as the cube published in
1911 in the Encyclopedia Britannica. The other cubes I tried were simple,
not associated; semipantriagonal, associated; pantriagonal, not
associated (normal and not normal); and pantriagonal associated (not
normal, prime).
Top 2 3 Bottom
01 63 62 04 48 18 19 45 32 34 35 29 49 15 14 52
60 06 07 57 21 43 42 24 37 27 26 40 12 54 55 09
56 10 11 53 25 39 38 28 41 23 22 44 08 58 59 05
13 51 50 16 36 30 31 33 20 46 47 17 61 03 02 64
(a) originating magic cube
01 63 62 04 60 06 07 57 01 62 48 19 32 35 49 14 01 48 32 49 63 18 34 15
56 10 11 53 13 51 50 16 60 07 21 42 37 26 12 55 62 19 35 14 04 45 29 52
48 18 19 45 21 43 42 24 56 11 25 38 41 22 08 59 60 21 37 12 06 48 27 54
25 39 38 28 36 30 31 33 13 30 36 31 20 47 61 02 07 42 26 55 57 24 40 09
32 34 35 29 37 27 26 40 63 04 18 45 34 29 15 52 56 25 41 08 10 39 23 58
41 23 22 44 20 46 47 17 06 57 43 24 27 40 54 09 11 38 22 59 53 28 44 05
49 15 14 52 12 54 55 09 10 53 39 28 23 44 58 05 13 36 20 61 51 30 46 03
08 58 59 05 61 03 02 64 51 16 30 33 46 17 03 64 50 31 47 02 16 33 17 64
(b) rowbyrow (c) columnbycolumn (d) pillarbypillar
[1] A. Adler, & R. Shuoyen, Magic Cubes and Prouhet
Sequences, American Mathematical Monthly, vol. 84(8), 1977, p. 618627.
