Subgroups — C₄²×H(3,𝔽₃) #88554

Order 432.

165 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×H(3,𝔽₃)	C₂	1	3	yes	maximal
2	C₃²×C₄²	C₃	1	4	yes	maximal, complemented
3	C₂²×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
4	C₄×H(3,𝔽₃)	C₄	1	6	yes	direct product factor
5	C₂×C₃²×C₄	C₂×C₃	1	12	yes	—
6	C₂×H(3,𝔽₃)	C₂×C₄	1	3	yes	—
7	C₃×C₄²	—	3	4	no	—
8	C₃×C₄²	C₃²	1	1	yes	characteristic
9	C₂²×C₃²	C₂²×C₃	1	4	yes	—
10	C₃²×C₄	C₃×C₄	1	24	yes	complemented
11	H(3,𝔽₃)	C₄²	1	1	yes	characteristic, direct product factor, Sylow
12	C₂×C₃×C₄	—	3	12	no	—
13	C₂×C₃×C₄	C₂×C₃²	1	3	yes	—
14	C₂×C₃²	C₂×C₃×C₄	1	12	yes	—
15	C₄²	H(3,𝔽₃)	1	1	yes	characteristic, direct product factor, Sylow
16	C₂²×C₃	—	3	4	no	—
17	C₂²×C₃	C₂²×C₃²	1	1	yes	characteristic
18	C₃×C₄	—	3	24	no	—
19	C₃×C₄	C₃²×C₄	1	6	yes	—
20	C₃²	C₃×C₄²	1	4	yes	complemented
21	C₂×C₄	C₂×H(3,𝔽₃)	1	3	yes	—
22	C₂×C₃	—	3	12	no	—
23	C₂×C₃	C₂×C₃²×C₄	1	3	yes	—
24	C₂²	C₂²×H(3,𝔽₃)	1	1	yes	characteristic
25	C₄	C₄×H(3,𝔽₃)	1	6	yes	direct product factor
26	C₃	—	3	4	no	—
27	C₃	C₃²×C₄²	1	1	yes	characteristic
28	C₂	C₂×C₄×H(3,𝔽₃)	1	3	yes	—

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
C₂×C₄×H(3,𝔽₃)	2
C₃²×C₄²	3