Subgroups — D₄×H(3,𝔽₃) #6240

Order 216.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×H(3,𝔽₃)	C₂	1	2	yes	maximal, complemented
2	C₄×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×D₄	C₃	1	4	yes	maximal, complemented
4	C₂×H(3,𝔽₃)	—	2	2	no	—
5	C₂×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
6	C₂²×C₃²	C₂×C₃	1	8	yes	complemented
7	C₃²×C₄	C₂×C₃	1	4	yes	complemented
8	H(3,𝔽₃)	D₄	1	1	yes	characteristic, direct product factor, Sylow
9	C₃×D₄	—	3	4	no	—
10	C₃×D₄	C₃²	1	1	yes	characteristic
11	C₂×C₃²	—	2	8	no	—
12	C₂×C₃²	C₂²×C₃	1	4	yes	—
13	C₂²×C₃	—	3	8	no	—
14	C₂²×C₃	C₂×C₃²	1	2	yes	—
15	C₃×C₄	—	3	4	no	—
16	C₃×C₄	C₂×C₃²	1	1	yes	characteristic
17	C₃²	C₃×D₄	1	4	yes	complemented
18	D₄	H(3,𝔽₃)	1	1	yes	characteristic, direct product factor, Sylow
19	C₂×C₃	—	2	2	no	—
20	C₂×C₃	—	3	4	no	—
21	C₂×C₃	—	6	8	no	—
22	C₂×C₃	C₂²×C₃²	1	1	yes	characteristic
23	C₂²	C₂×H(3,𝔽₃)	1	2	yes	complemented
24	C₄	C₂×H(3,𝔽₃)	1	1	yes	characteristic, complemented
25	C₃	—	3	4	no	—
26	C₃	C₃²×D₄	1	1	yes	characteristic
27	C₂	—	2	2	no	—
28	C₂	C₂²×H(3,𝔽₃)	1	1	yes	characteristic

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂×C₃	C₂²×C₃²
LCS	G ▷ C₂×C₃ ▷ C₁
UCS	C₁ ◁ C₂×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.