Subgroups — C₃×#123403 #72226103

Order 1536.

644 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#12515	C₂	1	6	yes	maximal, complemented
2	C₃×C₁₆²	C₂	1	1	yes	maximal, characteristic, complemented
3	#123403	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
4	C₃×#2246	—	2	12	no	—
5	C₃×#1712	C₂²	1	4	yes	—
6	C₃×C₈×C₁₆	C₂²	1	3	yes	—
7	#12515	C₂×C₃	1	6	yes	complemented
8	C₁₆²	C₂×C₃	1	1	yes	characteristic, complemented
9	C₂×C₃×D₁₆	—	4	24	no	—
10	C₃×#493	—	2	12	no	—
11	C₃×C₄×C₁₆	D₄	1	6	yes	—
12	C₃×C₈²	C₂³	1	1	yes	characteristic
13	#2246	—	2	12	no	—
14	#1712	C₂²×C₃	1	4	yes	—
15	C₈×C₁₆	C₂²×C₃	1	3	yes	—
16	C₂×C₃×D₈	—	4	24	no	—
17	C₃×#127	—	4	4	no	—
18	C₃×D₁₆	—	8	48	no	—
19	C₂×C₃×C₁₆	D₈	1	12	yes	—
20	C₃×C₄×C₈	C₂×D₄	1	3	yes	—
21	C₂×D₁₆	—	4	24	no	—
22	#493	—	2	12	no	—
23	C₄×C₁₆	C₃×D₄	1	6	yes	—
24	C₈²	C₂³×C₃	1	1	yes	characteristic
25	C₂×C₃×D₄	—	8	12	no	—
26	C₃×D₈	—	8	48	no	—
27	C₂×C₃×C₈	C₂×D₈	1	6	yes	—
28	C₃×C₄²	#127	1	1	yes	characteristic
29	C₃×C₁₆	D₁₆	1	24	yes	complemented
30	C₂×D₈	—	4	24	no	—
31	#127	—	4	4	no	—
32	D₁₆	—	8	48	no	—
33	C₂×C₁₆	C₃×D₈	1	12	yes	—
34	C₄×C₈	C₂×C₃×D₄	1	3	yes	—
35	C₂³×C₃	—	16	4	no	—
36	C₃×D₄	—	16	24	no	—
37	C₂×C₃×C₄	#493	1	3	yes	—
38	C₃×C₈	C₂×D₁₆	1	12	yes	—
39	C₂×D₄	—	8	12	no	—
40	D₈	—	8	48	no	—
41	C₂×C₈	C₂×C₃×D₈	1	6	yes	—
42	C₄²	C₃×#127	1	1	yes	characteristic
43	C₁₆	C₃×D₁₆	1	24	yes	complemented
44	C₂²×C₃	—	32	12	no	—
45	C₂²×C₃	#1712	1	1	yes	characteristic
46	C₃×C₄	#2246	1	6	yes	—
47	C₂³	—	16	4	no	—
48	D₄	—	16	24	no	—
49	C₂×C₄	C₃×#493	1	3	yes	—
50	C₈	C₂×C₃×D₁₆	1	12	yes	—
51	C₂×C₃	—	64	4	no	—
52	C₂×C₃	#12515	1	3	yes	—
53	C₂²	—	32	12	no	—
54	C₂²	C₃×#1712	1	1	yes	characteristic
55	C₄	C₃×#2246	1	6	yes	—
56	C₃	#123403	1	1	yes	characteristic, direct product factor, Sylow
57	C₂	—	64	4	no	—
58	C₂	C₃×#12515	1	3	yes	—

	subgroup	G/H
center, socle	C₂²×C₃	#1712
commutator, Frattini, cosocle	C₈²	C₂³×C₃
LCS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×C₈² ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×#12515	C₂	semidirect
2. C₃×C₁₆²	C₂	semidirect
3. #123403	C₃	direct (N × H)
7. #12515	C₂×C₃	semidirect
8. C₁₆²	C₂×C₃	semidirect
29. C₃×C₁₆	D₁₆	semidirect
43. C₁₆	C₃×D₁₆	semidirect
56. C₃	#123403	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×#12515	2
C₃×C₁₆²	2
#123403	3

Subgroups — C3×#123403 #72226103

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#123403 #72226103