Subgroups — D₄×3₋¹⁺² #6241

Order 216.

64 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×3₋¹⁺²	C₂	1	2	yes	maximal, complemented
2	C₄×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×D₄	C₃	1	1	yes	maximal, characteristic
4	D₄×C₉	C₃	1	3	yes	maximal, complemented
5	C₂×3₋¹⁺²	—	2	2	no	—
6	C₂×3₋¹⁺²	C₂²	1	1	yes	characteristic
7	C₂²×C₃²	C₂×C₃	1	2	yes	—
8	C₃²×C₄	C₂×C₃	1	1	yes	characteristic
9	C₂²×C₉	C₂×C₃	1	6	yes	complemented
10	C₄×C₉	C₂×C₃	1	3	yes	complemented
11	3₋¹⁺²	D₄	1	1	yes	characteristic, direct product factor, Sylow
12	C₃×D₄	—	3	1	no	—
13	C₃×D₄	C₃²	1	1	yes	characteristic
14	C₂×C₃²	—	2	2	no	—
15	C₂×C₃²	C₂²×C₃	1	1	yes	characteristic
16	C₂×C₉	—	2	6	no	—
17	C₂×C₉	C₂²×C₃	1	3	yes	—
18	C₂²×C₃	—	3	2	no	—
19	C₂²×C₃	C₂×C₃²	1	2	yes	—
20	C₃×C₄	—	3	1	no	—
21	C₃×C₄	C₂×C₃²	1	1	yes	characteristic
22	C₃²	C₃×D₄	1	1	yes	characteristic
23	C₉	C₃×D₄	1	3	yes	complemented
24	D₄	3₋¹⁺²	1	1	yes	characteristic, direct product factor, Sylow
25	C₂×C₃	—	2	2	no	—
26	C₂×C₃	—	3	1	no	—
27	C₂×C₃	—	6	2	no	—
28	C₂×C₃	C₂²×C₃²	1	1	yes	characteristic
29	C₂²	C₂×3₋¹⁺²	1	2	yes	complemented
30	C₄	C₂×3₋¹⁺²	1	1	yes	characteristic, complemented
31	C₃	—	3	1	no	—
32	C₃	C₃²×D₄	1	1	yes	characteristic
33	C₂	—	2	2	no	—
34	C₂	C₂²×3₋¹⁺²	1	1	yes	characteristic

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂×C₃	C₂²×C₃²
prime radical	C₃²×D₄	C₃
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.