Subgroups — C₃²×#125 #64174

Order 288.

168 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₄⋊C₄	C₂	1	4	yes	maximal
2	C₃²×C₄⋊C₄	C₂	1	2	yes	maximal
3	C₃²×C₄²	C₂	1	1	yes	maximal, characteristic
4	C₃×#125	C₃	1	4	yes	maximal, direct product factor
5	C₂×C₃²×C₄	C₂²	1	2	yes	—
6	C₂×C₃²×C₄	C₂²	1	4	yes	—
7	C₂×C₃²×C₄	C₂²	1	1	yes	characteristic
8	C₃×C₄⋊C₄	C₂×C₃	1	16	yes	—
9	C₃×C₄⋊C₄	C₂×C₃	1	8	yes	—
10	C₃×C₄²	C₂×C₃	1	4	yes	—
11	C₂²×C₃²	C₂³	1	1	yes	characteristic
12	C₃²×C₄	—	2	2	no	—
13	C₃²×C₄	—	2	4	no	—
14	C₃²×C₄	Q₈	1	2	yes	—
15	#125	C₃²	1	1	yes	characteristic, direct product factor, Sylow
16	C₂×C₃×C₄	C₂²×C₃	1	8	yes	—
17	C₂×C₃×C₄	C₂²×C₃	1	16	yes	—
18	C₂×C₃×C₄	C₂²×C₃	1	4	yes	—
19	C₂×C₃²	C₂×Q₈	1	1	yes	characteristic
20	C₂×C₃²	#41	1	2	yes	—
21	C₄⋊C₄	C₂×C₃²	1	4	yes	—
22	C₄⋊C₄	C₂×C₃²	1	2	yes	—
23	C₄²	C₂×C₃²	1	1	yes	characteristic
24	C₂²×C₃	C₂³×C₃	1	4	yes	—
25	C₃×C₄	—	2	8	no	—
26	C₃×C₄	—	2	16	no	—
27	C₃×C₄	C₃×Q₈	1	8	yes	—
28	C₃²	#125	1	1	yes	characteristic, direct product factor, Sylow
29	C₂×C₄	C₂²×C₃²	1	2	yes	—
30	C₂×C₄	C₂²×C₃²	1	4	yes	—
31	C₂×C₄	C₂²×C₃²	1	1	yes	characteristic
32	C₂×C₃	C₂×C₃×Q₈	1	4	yes	—
33	C₂×C₃	C₃×#41	1	8	yes	—
34	C₂²	C₂³×C₃²	1	1	yes	characteristic
35	C₄	—	2	2	no	—
36	C₄	—	2	4	no	—
37	C₄	C₃²×Q₈	1	2	yes	—
38	C₃	C₃×#125	1	4	yes	direct product factor
39	C₂	C₂×C₃²×Q₈	1	1	yes	characteristic
40	C₂	C₃²×#41	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂²×C₃²	C₂³
commutator, Frattini, cosocle	C₂²	C₂³×C₃²
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.