Subgroups — #88940

Order 432.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#157	C₂	1	1	yes	maximal, characteristic
2	#6294	C₂	1	2	yes	maximal
3	#3773	—	3	1	no	maximal
4	C₂×C₃×C₃⋊D₃	C₂²	1	1	yes	characteristic
5	C₃×#157	C₄	1	2	yes	complemented
6	C₃×C₃⋊Dic₃	—	2	2	no	—
7	C₂×#157	D₃	1	1	yes	characteristic
8	#624	—	3	2	no	—
9	D₃×Dic₃	—	6	2	no	—
10	C₂×C₃³	D₄	1	1	yes	characteristic
11	C₃×C₃⋊D₃	C₂×C₄	1	1	yes	characteristic
12	C₃×C₃⋊D₃	Q₈	1	1	yes	characteristic
13	C₄⋊Dic₃	—	9	1	no	maximal
14	C₂×C₃⋊D₃	C₂×D₃	1	1	yes	characteristic
15	C₂×C₃×D₃	—	6	2	no	—
16	#157	Dic₃	1	2	yes	complemented
17	C₃⋊Dic₃	—	2	2	no	—
18	C₃×Dic₃	—	6	2	no	—
19	C₃×Dic₃	—	6	2	no	—
20	C₃×Dic₃	—	12	2	no	—
21	C₃³	C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
22	C₂×C₃×C₄	—	9	1	no	—
23	C₂×Dic₃	—	9	2	no	—
24	C₄×D₃	—	18	2	no	—
25	C₂×C₃²	—	2	2	no	—
26	C₂×C₃²	—	4	2	no	—
27	C₂×C₃²	D₁₂	1	1	yes	characteristic
28	C₃⋊D₃	Dic₆	1	1	yes	characteristic
29	C₃⋊D₃	C₂×Dic₃	1	1	yes	characteristic
30	C₃×D₃	—	6	2	no	—
31	C₃×D₃	—	6	2	no	—
32	C₄⋊C₄	—	27	1	no	Sylow
33	C₂²×C₃	—	9	1	no	—
34	C₂×D₃	—	6	2	no	—
35	C₃×C₄	—	9	2	no	—
36	C₃×C₄	—	18	2	no	—
37	Dic₃	—	6	2	no	—
38	Dic₃	—	6	2	no	—
39	Dic₃	—	12	2	no	—
40	C₃²	—	2	2	no	—
41	C₃²	—	4	2	no	—
42	C₃²	C₄⋊Dic₃	1	1	yes	characteristic, complemented
43	C₂×C₄	—	9	1	no	—
44	C₂×C₄	—	27	2	no	—
45	C₂×C₃	—	2	2	no	—
46	C₂×C₃	—	4	2	no	—
47	C₂×C₃	—	9	1	no	—
48	C₂×C₃	—	9	1	no	—
49	C₂×C₃	GO⁺(4,𝔽₂)	1	1	yes	characteristic
50	D₃	—	6	2	no	—
51	D₃	—	6	2	no	—
52	C₂²	—	9	1	no	—
53	C₄	—	9	2	no	—
54	C₄	—	18	2	no	—
55	C₃	—	2	2	no	—
56	C₃	—	4	2	no	—
57	C₃	#3773	1	1	yes	characteristic, complemented
58	C₂	—	9	1	no	—
59	C₂	—	9	1	no	—
60	C₂	#157⋊D₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	#157⋊D₃
commutator	C₃×C₃⋊D₃	C₂×C₄
Fitting, socle	C₂×C₃³	D₄
prime radical, cosocle	C₂×C₃×C₃⋊D₃	C₂²
hypocenter	C₃³	C₄⋊C₄
LCS	G ▷ C₃×C₃⋊D₃ ▷ C₃³
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
5. C₃×#157	C₄	semidirect
16. #157	Dic₃	semidirect
21. C₃³	C₄⋊C₄	semidirect
42. C₃²	C₄⋊Dic₃	semidirect
57. C₃	#3773	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#157	2
#6294	2
#3773	3
C₄⋊Dic₃	9