Subgroups — C₃⋊(#120⋊C₃) #63763

Order 288.

77 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₂²⋊A₄	C₂	1	1	yes	maximal, characteristic, complemented
2	#953	C₃	1	1	yes	maximal, characteristic, complemented
3	#120⋊C₃	—	3	1	no	maximal
4	D₃×A₄	—	4	1	no	maximal
5	C₂⁴×C₃	C₂×C₃	1	1	yes	characteristic, complemented
6	C₂²⋊A₄	—	2	1	no	—
7	C₂²⋊A₄	D₃	1	1	yes	characteristic, complemented
8	C₂×#67	—	3	1	no	—
9	#217	—	3	1	no	—
10	C₃×A₄	—	4	1	no	—
11	C₃×A₄	—	8	2	no	—
12	#120	—	3	1	no	Sylow
13	C₂³×C₃	—	3	1	no	—
14	C₂³×C₃	—	6	2	no	—
15	C₂²×D₃	A₄	1	1	yes	characteristic, complemented
16	C₂×A₄	—	12	1	no	—
17	#67	—	6	2	no	—
18	C₂×Dic₃	—	3	1	no	—
19	C₃×D₃	—	16	1	no	—
20	C₂⁴	C₃×D₃	1	1	yes	characteristic, complemented
21	C₂×D₄	—	9	1	no	—
22	C₂²⋊C₄	—	9	1	no	—
23	C₂²×C₃	—	2	2	no	—
24	C₂²×C₃	—	3	2	no	—
25	C₂²×C₃	—	6	2	no	—
26	C₂²×C₃	—	6	2	no	—
27	C₂²×C₃	C₂×A₄	1	1	yes	characteristic
28	C₂×D₃	—	6	1	no	—
29	A₄	—	4	1	no	—
30	A₄	—	8	2	no	—
31	A₄	—	8	4	no	—
32	A₄	—	8	1	no	—
33	Dic₃	—	6	1	no	—
34	C₃²	—	16	1	no	Sylow
35	C₂³	—	3	1	no	—
36	C₂³	—	3	1	no	—
37	C₂³	—	6	2	no	—
38	D₄	—	18	2	no	—
39	C₂×C₄	—	9	1	no	—
40	C₂×C₃	—	3	1	no	—
41	C₂×C₃	—	6	2	no	—
42	C₂×C₃	—	48	1	no	—
43	D₃	—	4	1	no	—
44	C₂²	—	2	2	no	—
45	C₂²	—	3	2	no	—
46	C₂²	—	6	2	no	—
47	C₂²	—	6	2	no	—
48	C₂²	—	18	1	no	—
49	C₂²	D₃×A₄	1	1	yes	characteristic
50	C₄	—	18	1	no	—
51	C₃	—	16	1	no	—
52	C₃	—	32	1	no	—
53	C₃	#120⋊C₃	1	1	yes	characteristic, complemented
54	C₂	—	3	1	no	—
55	C₂	—	6	2	no	—
56	C₂	—	12	1	no	—

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₂⁴×C₃	C₂×C₃
Frattini	C₂²	D₃×A₄
socle	C₂²×C₃	C₂×A₄
LCS	G ▷ C₂⁴×C₃
UCS	C₁

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

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