Subgroups — C₃²×#126 #64175

Order 288.

180 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₄⋊C₄	C₂	1	3	yes	maximal, complemented
2	C₃²×C₂²⋊C₄	C₂	1	3	yes	maximal
3	C₃²×C₄²	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#126	C₃	1	4	yes	maximal, direct product factor
5	C₂³×C₃²	C₂²	1	1	yes	characteristic
6	C₂×C₃²×C₄	C₂²	1	3	yes	—
7	C₂×C₃²×C₄	C₂²	1	3	yes	—
8	C₃×C₄⋊C₄	C₂×C₃	1	12	yes	complemented
9	C₃×C₂²⋊C₄	C₂×C₃	1	12	yes	—
10	C₃×C₄²	C₂×C₃	1	4	yes	complemented
11	C₂²×C₃²	—	2	3	no	—
12	C₂²×C₃²	C₂³	1	1	yes	characteristic
13	C₃²×C₄	—	2	3	no	—
14	C₃²×C₄	—	2	3	no	—
15	#126	C₃²	1	1	yes	characteristic, direct product factor, Sylow
16	C₂³×C₃	C₂²×C₃	1	4	yes	—
17	C₂×C₃×C₄	C₂²×C₃	1	12	yes	—
18	C₂×C₃×C₄	C₂²×C₃	1	12	yes	—
19	C₂×C₃²	—	4	1	no	—
20	C₂×C₃²	#41	1	3	yes	—
21	C₄⋊C₄	C₂×C₃²	1	3	yes	complemented
22	C₂²⋊C₄	C₂×C₃²	1	3	yes	—
23	C₄²	C₂×C₃²	1	1	yes	characteristic, complemented
24	C₂²×C₃	—	2	12	no	—
25	C₂²×C₃	C₂³×C₃	1	4	yes	—
26	C₃×C₄	—	2	12	no	—
27	C₃×C₄	—	2	12	no	—
28	C₃²	#126	1	1	yes	characteristic, direct product factor, Sylow
29	C₂³	C₂²×C₃²	1	1	yes	characteristic
30	C₂×C₄	C₂²×C₃²	1	3	yes	—
31	C₂×C₄	C₂²×C₃²	1	3	yes	—
32	C₂×C₃	—	4	4	no	—
33	C₂×C₃	C₃×#41	1	12	yes	—
34	C₂²	—	2	3	no	—
35	C₂²	C₂³×C₃²	1	1	yes	characteristic
36	C₄	—	2	3	no	—
37	C₄	—	2	3	no	—
38	C₃	C₃×#126	1	4	yes	direct product factor
39	C₂	—	4	1	no	—
40	C₂	C₃²×#41	1	3	yes	—

	subgroup	G/H
center, socle	C₂²×C₃²	C₂³
commutator, Frattini, cosocle	C₂²	C₂³×C₃²
prime radical	C₂³×C₃²	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃² ◁ G

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.