Subgroups — C₃×#5464 #10595561

Order 576.

172 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×H(3,𝔽₄)	C₃	1	1	yes	maximal, characteristic, complemented
2	#5464	C₃	1	3	yes	maximal, direct product factor
3	C₃×C₂²⋊A₄	—	4	2	no	maximal
4	C₃×C₄²⋊C₃	—	4	3	no	maximal
5	C₃×#124	—	3	3	no	—
6	C₃×#120	—	3	2	no	—
7	H(3,𝔽₄)	C₃²	1	1	yes	characteristic, complemented, Sylow
8	C₂⁴×C₃	A₄	1	2	yes	complemented
9	C₂²⋊A₄	—	4	6	no	—
10	C₂×C₃×Q₈	—	3	1	no	—
11	C₂×C₃×D₄	—	3	3	no	—
12	C₃×C₂²⋊C₄	—	3	6	no	—
13	C₃×C₄²	A₄	1	3	yes	complemented
14	C₄²⋊C₃	—	4	9	no	—
15	C₃×A₄	—	16	8	no	—
16	C₃×A₄	—	16	1	no	—
17	#124	—	3	3	no	—
18	#120	—	3	2	no	—
19	C₂³×C₃	—	3	2	no	—
20	C₂³×C₃	—	12	2	no	—
21	C₃×Q₈	—	12	1	no	—
22	C₃×D₄	—	12	3	no	—
23	C₂×C₃×C₄	—	3	3	no	—
24	C₂⁴	C₃×A₄	1	2	yes	complemented
25	C₂×Q₈	—	3	1	no	—
26	C₂×D₄	—	3	3	no	—
27	C₂²⋊C₄	—	3	6	no	—
28	C₄²	C₃×A₄	1	3	yes	complemented
29	C₂²×C₃	—	4	8	no	—
30	C₂²×C₃	—	6	6	no	—
31	C₂²×C₃	C₂²⋊A₄	1	1	yes	characteristic
32	A₄	—	16	24	no	—
33	A₄	—	16	3	no	—
34	C₃×C₄	—	6	3	no	—
35	C₃²	—	64	1	no	Sylow
36	C₂³	—	3	2	no	—
37	C₂³	—	12	2	no	—
38	Q₈	—	12	1	no	—
39	D₄	—	12	3	no	—
40	C₂×C₄	—	3	3	no	—
41	C₂×C₃	—	3	1	no	—
42	C₂×C₃	—	12	2	no	—
43	C₂²	—	4	8	no	—
44	C₂²	—	6	6	no	—
45	C₂²	C₃×C₂²⋊A₄	1	1	yes	characteristic
46	C₄	—	6	3	no	—
47	C₃	—	64	3	no	—
48	C₃	#5464	1	1	yes	characteristic, direct product factor
49	C₂	—	3	1	no	—
50	C₂	—	12	2	no	—

	subgroup	G/H
center, hypercenter	C₃	#5464
commutator, cosocle, hypocenter	H(3,𝔽₄)	C₃²
Frattini	C₂²	C₃×C₂²⋊A₄
Fitting	C₃×H(3,𝔽₄)	C₃
socle	C₂²×C₃	C₂²⋊A₄
LCS	G ▷ H(3,𝔽₄)
UCS	C₁ ◁ C₃

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.