Subgroups — C₃²×H(5,𝔽₂) #64374

Order 288.

498 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#41	C₂	1	6	yes	maximal, complemented
2	C₂×C₃²×D₄	C₂	1	9	yes	maximal, complemented
3	C₃×H(5,𝔽₂)	C₃	1	4	yes	maximal, direct product factor
4	C₂³×C₃²	C₂²	1	6	yes	complemented
5	C₃²×Q₈	C₂²	1	2	yes	complemented
6	C₃²×D₄	C₂²	1	18	yes	complemented
7	C₂×C₃²×C₄	C₂²	1	9	yes	complemented
8	C₃×#41	C₂×C₃	1	24	yes	complemented
9	C₂×C₃×D₄	C₂×C₃	1	36	yes	complemented
10	C₂²×C₃²	—	4	6	no	—
11	C₂²×C₃²	C₂³	1	9	yes	—
12	C₃²×C₄	C₂³	1	6	yes	—
13	H(5,𝔽₂)	C₃²	1	1	yes	characteristic, direct product factor, Sylow
14	C₂³×C₃	C₂²×C₃	1	24	yes	complemented
15	C₃×Q₈	C₂²×C₃	1	8	yes	complemented
16	C₃×D₄	C₂²×C₃	1	72	yes	complemented
17	C₂×C₃×C₄	C₂²×C₃	1	36	yes	complemented
18	C₂×C₃²	—	2	9	no	—
19	C₂×C₃²	C₂⁴	1	1	yes	characteristic
20	#41	C₂×C₃²	1	6	yes	complemented
21	C₂×D₄	C₂×C₃²	1	9	yes	complemented
22	C₂²×C₃	—	4	24	no	—
23	C₂²×C₃	C₂³×C₃	1	36	yes	—
24	C₃×C₄	C₂³×C₃	1	24	yes	—
25	C₃²	H(5,𝔽₂)	1	1	yes	characteristic, direct product factor, Sylow
26	C₂³	C₂²×C₃²	1	6	yes	complemented
27	Q₈	C₂²×C₃²	1	2	yes	complemented
28	D₄	C₂²×C₃²	1	18	yes	complemented
29	C₂×C₄	C₂²×C₃²	1	9	yes	complemented
30	C₂×C₃	—	2	36	no	—
31	C₂×C₃	C₂⁴×C₃	1	4	yes	—
32	C₂²	—	4	6	no	—
33	C₂²	C₂³×C₃²	1	9	yes	—
34	C₄	C₂³×C₃²	1	6	yes	—
35	C₃	C₃×H(5,𝔽₂)	1	4	yes	direct product factor
36	C₂	—	2	9	no	—
37	C₂	C₂⁴×C₃²	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃²	C₂⁴
commutator, Frattini, cosocle	C₂	C₂⁴×C₃²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×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.