Subgroups — C₂×#6204 #88688

Order 432.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#6204	C₂	1	1	yes	maximal, characteristic, direct product factor
2	C₂×C₃²×Q₈	C₃	1	1	yes	maximal, characteristic
3	C₂×Q₈⋊C₉	C₃	1	3	yes	maximal, complemented
4	C₂²×3₋¹⁺²	—	4	1	no	maximal
5	C₃²×Q₈	—	3	1	no	—
6	C₃²×Q₈	C₂×C₃	1	1	yes	characteristic
7	C₂×C₃²×C₄	—	3	1	no	—
8	Q₈⋊C₉	C₂×C₃	1	3	yes	complemented
9	C₂×3₋¹⁺²	—	4	2	no	—
10	C₂×3₋¹⁺²	—	4	1	no	—
11	C₂×C₃×Q₈	—	3	1	no	—
12	C₂×C₃×Q₈	C₃²	1	1	yes	characteristic
13	C₂²×C₃²	A₄	1	1	yes	characteristic
14	C₃²×C₄	—	3	1	no	—
15	C₃²×C₄	—	3	1	no	—
16	C₂²×C₉	—	4	3	no	—
17	3₋¹⁺²	—	4	1	no	Sylow
18	C₃×Q₈	—	3	1	no	—
19	C₃×Q₈	—	3	3	no	—
20	C₃×Q₈	—	3	1	no	—
21	C₃×Q₈	C₂×C₃²	1	1	yes	characteristic
22	C₂×C₃×C₄	—	3	3	no	—
23	C₂×C₃×C₄	—	3	1	no	—
24	C₂×C₃²	SL(2,𝔽₃)	1	2	yes	—
25	C₂×C₃²	C₂×A₄	1	1	yes	characteristic
26	C₂×C₉	—	4	6	no	—
27	C₂×C₉	—	4	3	no	—
28	C₂×Q₈	3₋¹⁺²	1	1	yes	characteristic, complemented, Sylow
29	C₂²×C₃	—	3	1	no	—
30	C₂²×C₃	C₃×A₄	1	1	yes	characteristic
31	C₃×C₄	—	3	3	no	—
32	C₃×C₄	—	3	3	no	—
33	C₃×C₄	—	3	1	no	—
34	C₃×C₄	—	3	1	no	—
35	C₃²	C₂×SL(2,𝔽₃)	1	1	yes	characteristic
36	C₉	—	4	3	no	—
37	Q₈	—	3	1	no	—
38	Q₈	C₂×3₋¹⁺²	1	1	yes	characteristic, complemented
39	C₂×C₄	—	3	1	no	—
40	C₂×C₃	—	3	2	no	—
41	C₂×C₃	—	3	1	no	—
42	C₂×C₃	C₃×SL(2,𝔽₃)	1	2	yes	—
43	C₂×C₃	C₂×C₃×A₄	1	1	yes	characteristic
44	C₂²	#1094	1	1	yes	characteristic
45	C₄	—	3	1	no	—
46	C₄	—	3	1	no	—
47	C₃	—	3	1	no	—
48	C₃	C₂×C₃×SL(2,𝔽₃)	1	1	yes	characteristic
49	C₂	#6204	1	2	yes	direct product factor
50	C₂	C₂×#1094	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	C₃×A₄
commutator, cosocle	C₃×Q₈	C₂×C₃²
Frattini	C₂×C₃	C₂×C₃×A₄
Fitting	C₂×C₃²×Q₈	C₃
prime radical, hypercenter	C₂²×C₃²	A₄
hypocenter	Q₈	C₂×3₋¹⁺²
LCS	G ▷ C₃×Q₈ ▷ Q₈
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃²

Complements

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

N	H ≅ G/N	type
1. #6204	C₂	direct (N × H)
3. C₂×Q₈⋊C₉	C₃	semidirect
8. Q₈⋊C₉	C₂×C₃	semidirect
28. C₂×Q₈	3₋¹⁺²	semidirect
38. Q₈	C₂×3₋¹⁺²	semidirect
49. C₂	#6204	direct (N × H)

Maximal subgroups

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

subgroup	index
#6204	2
C₂×C₃²×Q₈	3
C₂×Q₈⋊C₉	3
C₂²×3₋¹⁺²	4

Subgroups — C2×#6204 #88688

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#6204 #88688