Subgroups — C₂×Q₈×H(3,𝔽₃) #88760

Order 432.

209 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×H(3,𝔽₃)	C₂	1	4	yes	maximal, direct product factor
2	C₂×C₄×H(3,𝔽₃)	C₂	1	3	yes	maximal
3	C₂×C₃²×Q₈	C₃	1	4	yes	maximal, complemented
4	C₂²×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
5	C₄×H(3,𝔽₃)	C₂²	1	6	yes	—
6	C₃²×Q₈	C₂×C₃	1	16	yes	complemented
7	C₂×C₃²×C₄	C₂×C₃	1	12	yes	—
8	C₂×H(3,𝔽₃)	Q₈	1	2	yes	direct product factor
9	C₂×H(3,𝔽₃)	C₂³	1	1	yes	characteristic
10	C₂×C₃×Q₈	—	3	4	no	—
11	C₂×C₃×Q₈	C₃²	1	1	yes	characteristic
12	C₂²×C₃²	C₂²×C₃	1	4	yes	—
13	C₃²×C₄	C₂²×C₃	1	24	yes	—
14	H(3,𝔽₃)	C₂×Q₈	1	1	yes	characteristic, direct product factor, Sylow
15	C₃×Q₈	—	3	16	no	—
16	C₃×Q₈	C₂×C₃²	1	4	yes	—
17	C₂×C₃×C₄	—	3	12	no	—
18	C₂×C₃×C₄	C₂×C₃²	1	3	yes	—
19	C₂×C₃²	C₃×Q₈	1	8	yes	complemented
20	C₂×C₃²	C₂³×C₃	1	4	yes	—
21	C₂×Q₈	H(3,𝔽₃)	1	1	yes	characteristic, direct product factor, Sylow
22	C₂²×C₃	—	3	4	no	—
23	C₂²×C₃	C₂²×C₃²	1	1	yes	characteristic
24	C₃×C₄	—	3	24	no	—
25	C₃×C₄	C₂²×C₃²	1	6	yes	—
26	C₃²	C₂×C₃×Q₈	1	4	yes	complemented
27	Q₈	C₂×H(3,𝔽₃)	1	4	yes	direct product factor
28	C₂×C₄	C₂×H(3,𝔽₃)	1	3	yes	—
29	C₂×C₃	—	3	8	no	—
30	C₂×C₃	—	3	4	no	—
31	C₂×C₃	C₃²×Q₈	1	2	yes	—
32	C₂×C₃	C₂³×C₃²	1	1	yes	characteristic
33	C₂²	C₂²×H(3,𝔽₃)	1	1	yes	characteristic
34	C₄	C₂²×H(3,𝔽₃)	1	6	yes	—
35	C₃	—	3	4	no	—
36	C₃	C₂×C₃²×Q₈	1	1	yes	characteristic
37	C₂	Q₈×H(3,𝔽₃)	1	2	yes	direct product factor
38	C₂	C₂³×H(3,𝔽₃)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	C₂²×C₃²
commutator, Frattini, cosocle	C₂×C₃	C₂³×C₃²
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
1. Q₈×H(3,𝔽₃)	C₂	direct (N × H)
3. C₂×C₃²×Q₈	C₃	semidirect
6. C₃²×Q₈	C₂×C₃	semidirect
8. C₂×H(3,𝔽₃)	Q₈	direct (N × H)
14. H(3,𝔽₃)	C₂×Q₈	direct (N × H)
19. C₂×C₃²	C₃×Q₈	semidirect
21. C₂×Q₈	H(3,𝔽₃)	direct (N × H)
26. C₃²	C₂×C₃×Q₈	semidirect
27. Q₈	C₂×H(3,𝔽₃)	direct (N × H)
37. C₂	Q₈×H(3,𝔽₃)	direct (N × H)

Maximal subgroups

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

subgroup	index
Q₈×H(3,𝔽₃)	2
C₂×C₄×H(3,𝔽₃)	2
C₂×C₃²×Q₈	3

Subgroups — C2×Q8×H(3,𝔽3) #88760

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Q₈×H(3,𝔽₃) #88760