Subgroups — C₄⋊C₄×H(3,𝔽₃) #88560

Order 432.

143 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×H(3,𝔽₃)	C₂	1	2	yes	maximal
2	C₂×C₄×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic
3	C₃²×C₄⋊C₄	C₃	1	4	yes	maximal, complemented
4	C₂²×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
5	C₄×H(3,𝔽₃)	—	2	2	no	—
6	C₄×H(3,𝔽₃)	C₄	1	2	yes	complemented
7	C₂×C₃²×C₄	C₂×C₃	1	8	yes	—
8	C₂×C₃²×C₄	C₂×C₃	1	4	yes	—
9	C₂×H(3,𝔽₃)	C₂×C₄	1	1	yes	characteristic
10	C₂×H(3,𝔽₃)	D₄	1	1	yes	characteristic
11	C₂×H(3,𝔽₃)	Q₈	1	1	yes	characteristic
12	C₃×C₄⋊C₄	—	3	4	no	—
13	C₃×C₄⋊C₄	C₃²	1	1	yes	characteristic
14	C₂²×C₃²	C₂²×C₃	1	4	yes	—
15	C₃²×C₄	—	2	8	no	—
16	C₃²×C₄	C₃×C₄	1	8	yes	complemented
17	H(3,𝔽₃)	C₄⋊C₄	1	1	yes	characteristic, direct product factor, Sylow
18	C₂×C₃×C₄	—	3	4	no	—
19	C₂×C₃×C₄	—	3	8	no	—
20	C₂×C₃×C₄	C₂×C₃²	1	2	yes	—
21	C₂×C₃×C₄	C₂×C₃²	1	1	yes	characteristic
22	C₂×C₃²	C₂×C₃×C₄	1	4	yes	—
23	C₂×C₃²	C₃×D₄	1	4	yes	—
24	C₂×C₃²	C₃×Q₈	1	4	yes	—
25	C₄⋊C₄	H(3,𝔽₃)	1	1	yes	characteristic, direct product factor, Sylow
26	C₂²×C₃	—	3	4	no	—
27	C₂²×C₃	C₂²×C₃²	1	1	yes	characteristic
28	C₃×C₄	—	2	2	no	—
29	C₃×C₄	—	3	8	no	—
30	C₃×C₄	—	6	8	no	—
31	C₃×C₄	C₃²×C₄	1	2	yes	—
32	C₃²	C₃×C₄⋊C₄	1	4	yes	complemented
33	C₂×C₄	C₂×H(3,𝔽₃)	1	2	yes	—
34	C₂×C₄	C₂×H(3,𝔽₃)	1	1	yes	characteristic
35	C₂×C₃	—	3	4	no	—
36	C₂×C₃	—	3	4	no	—
37	C₂×C₃	—	3	4	no	—
38	C₂×C₃	C₂×C₃²×C₄	1	1	yes	characteristic
39	C₂×C₃	C₃²×D₄	1	1	yes	characteristic
40	C₂×C₃	C₃²×Q₈	1	1	yes	characteristic
41	C₂²	C₂²×H(3,𝔽₃)	1	1	yes	characteristic
42	C₄	—	2	2	no	—
43	C₄	C₄×H(3,𝔽₃)	1	2	yes	complemented
44	C₃	—	3	4	no	—
45	C₃	C₃²×C₄⋊C₄	1	1	yes	characteristic
46	C₂	C₂×C₄×H(3,𝔽₃)	1	1	yes	characteristic
47	C₂	D₄×H(3,𝔽₃)	1	1	yes	characteristic
48	C₂	Q₈×H(3,𝔽₃)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, socle, cosocle	C₂²×C₃	C₂²×C₃²
commutator	C₂×C₃	C₂×C₃²×C₄
prime radical	C₂²×H(3,𝔽₃)	C₂²
LCS	G ▷ C₂×C₃ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃²×C₄⋊C₄	C₃	semidirect
6. C₄×H(3,𝔽₃)	C₄	semidirect
16. C₃²×C₄	C₃×C₄	semidirect
17. H(3,𝔽₃)	C₄⋊C₄	direct (N × H)
25. C₄⋊C₄	H(3,𝔽₃)	direct (N × H)
32. C₃²	C₃×C₄⋊C₄	semidirect
43. C₄	C₄×H(3,𝔽₃)	semidirect

Maximal subgroups

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×C₄⋊C₄	3

Subgroups — C4⋊C4×H(3,𝔽3) #88560

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄⋊C₄×H(3,𝔽₃) #88560