Subgroups — C₂⁴×H(3,𝔽₃) #88916

Order 432.

737 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×H(3,𝔽₃)	C₂	1	15	yes	maximal, direct product factor
2	C₂⁴×C₃²	C₃	1	4	yes	maximal, complemented
3	C₂²×H(3,𝔽₃)	C₂²	1	35	yes	direct product factor
4	C₂³×C₃²	C₂×C₃	1	60	yes	complemented
5	C₂×H(3,𝔽₃)	C₂³	1	15	yes	direct product factor
6	C₂⁴×C₃	—	3	4	no	—
7	C₂⁴×C₃	C₃²	1	1	yes	characteristic
8	C₂²×C₃²	C₂²×C₃	1	140	yes	complemented
9	H(3,𝔽₃)	C₂⁴	1	1	yes	characteristic, direct product factor, Sylow
10	C₂³×C₃	—	3	60	no	—
11	C₂³×C₃	C₂×C₃²	1	15	yes	—
12	C₂×C₃²	C₂³×C₃	1	60	yes	complemented
13	C₂⁴	H(3,𝔽₃)	1	1	yes	characteristic, direct product factor, Sylow
14	C₂²×C₃	—	3	140	no	—
15	C₂²×C₃	C₂²×C₃²	1	35	yes	—
16	C₃²	C₂⁴×C₃	1	4	yes	complemented
17	C₂³	C₂×H(3,𝔽₃)	1	15	yes	direct product factor
18	C₂×C₃	—	3	60	no	—
19	C₂×C₃	C₂³×C₃²	1	15	yes	—
20	C₂²	C₂²×H(3,𝔽₃)	1	35	yes	direct product factor
21	C₃	—	3	4	no	—
22	C₃	C₂⁴×C₃²	1	1	yes	characteristic
23	C₂	C₂³×H(3,𝔽₃)	1	15	yes	direct product factor

	subgroup	G/H
center, socle	C₂⁴×C₃	C₃²
commutator, Frattini, cosocle	C₃	C₂⁴×C₃²
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃ ◁ G

Splittings G = N ⋊ H with N normal; H ≅ G/N.

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂³×H(3,𝔽₃)	2
C₂⁴×C₃²	3