Subgroups — #10620175

Order 648.

1272 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃⁴×C₄	C₂	1	1	yes	maximal, characteristic
2	C₃⋊(C₃⋊(C₃⋊Dic₃))	C₂	1	2	yes	maximal
3	#6308	—	3	40	no	maximal
4	C₂×C₃⁴	C₂²	1	1	yes	characteristic
5	C₃³×C₄	D₃	1	40	yes	—
6	C₃⋊(C₃⋊Dic₃)	—	3	80	no	—
7	C₃⁴	Q₈	1	1	yes	characteristic, complemented, Sylow
8	#634	—	9	130	no	—
9	C₂×C₃³	C₂×D₃	1	40	yes	—
10	C₃²×C₄	C₃⋊D₃	1	130	yes	—
11	C₃⋊Dic₃	—	9	260	no	—
12	C₃³	Dic₆	1	40	yes	complemented
13	Dic₆	—	27	40	no	—
14	C₂×C₃²	C₂×C₃⋊D₃	1	130	yes	—
15	C₃×C₄	C₃⋊(C₃⋊D₃)	1	40	yes	—
16	Dic₃	—	27	80	no	—
17	C₃²	#634	1	130	yes	complemented
18	Q₈	—	81	1	no	Sylow
19	C₂×C₃	C₂×C₃⋊(C₃⋊D₃)	1	40	yes	—
20	C₄	—	81	2	no	—
21	C₄	C₃⋊(C₃⋊(C₃⋊D₃))	1	1	yes	characteristic
22	C₃	#6308	1	40	yes	complemented
23	C₂	C₂×C₃⋊(C₃⋊(C₃⋊D₃))	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂×C₃⋊(C₃⋊(C₃⋊D₃))
commutator, socle, prime radical, cosocle	C₂×C₃⁴	C₂²
Fitting	C₃⁴×C₄	C₂
hypercenter	C₄	C₃⋊(C₃⋊(C₃⋊D₃))
hypocenter	C₃⁴	Q₈
LCS	G ▷ C₂×C₃⁴ ▷ C₃⁴
UCS	C₁ ◁ C₂ ◁ C₄

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.