Subgroups — C₃²×#127 #64176

Order 288.

324 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×D₄	C₂	1	6	yes	maximal, complemented
2	C₃²×C₄²	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#127	C₃	1	4	yes	maximal, direct product factor
4	C₂³×C₃²	C₂²	1	4	yes	—
5	C₃²×D₄	—	2	12	no	—
6	C₂×C₃²×C₄	C₂²	1	3	yes	—
7	C₂×C₃×D₄	C₂×C₃	1	24	yes	complemented
8	C₃×C₄²	C₂×C₃	1	4	yes	complemented
9	C₂²×C₃²	—	2	12	no	—
10	C₂²×C₃²	C₂³	1	1	yes	characteristic
11	C₃²×C₄	D₄	1	6	yes	complemented
12	#127	C₃²	1	1	yes	characteristic, direct product factor, Sylow
13	C₂³×C₃	C₂²×C₃	1	16	yes	—
14	C₃×D₄	—	2	48	no	—
15	C₂×C₃×C₄	C₂²×C₃	1	12	yes	—
16	C₂×C₃²	—	4	4	no	—
17	C₂×C₃²	C₂×D₄	1	3	yes	—
18	C₂×D₄	C₂×C₃²	1	6	yes	complemented
19	C₄²	C₂×C₃²	1	1	yes	characteristic, complemented
20	C₂²×C₃	—	2	48	no	—
21	C₂²×C₃	C₂³×C₃	1	4	yes	—
22	C₃×C₄	C₃×D₄	1	24	yes	complemented
23	C₃²	#127	1	1	yes	characteristic, direct product factor, Sylow
24	C₂³	C₂²×C₃²	1	4	yes	—
25	D₄	—	2	12	no	—
26	C₂×C₄	C₂²×C₃²	1	3	yes	—
27	C₂×C₃	—	4	16	no	—
28	C₂×C₃	C₂×C₃×D₄	1	12	yes	—
29	C₂²	—	2	12	no	—
30	C₂²	C₂³×C₃²	1	1	yes	characteristic
31	C₄	C₃²×D₄	1	6	yes	complemented
32	C₃	C₃×#127	1	4	yes	direct product factor
33	C₂	—	4	4	no	—
34	C₂	C₂×C₃²×D₄	1	3	yes	—

	subgroup	G/H
center, socle	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.