Subgroups — C₃⋊(C₃⋊GL(2,𝔽₃)) #88973

Order 432.

208 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×SL(2,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃⋊GL(2,𝔽₃)	—	3	12	no	maximal
3	#3755	—	3	1	no	maximal
4	C₂×C₃⋊(C₃⋊D₃)	—	4	1	no	maximal
5	C₃²×Q₈	D₃	1	1	yes	characteristic, complemented
6	C₄⋊(C₃⋊D₃)	—	3	1	no	—
7	C₃×SL(2,𝔽₃)	D₃	1	12	yes	complemented
8	C₃⋊(C₃⋊C₈)	—	3	1	no	—
9	C₂×C₃³	—	4	1	no	—
10	C₃⋊(C₃⋊D₃)	—	4	2	no	—
11	GL(2,𝔽₃)	—	9	9	no	—
12	#215	—	9	4	no	—
13	C₂×C₃⋊D₃	—	6	1	no	—
14	C₂×C₃⋊D₃	—	12	12	no	—
15	C₃²×C₄	—	3	1	no	—
16	C₃³	—	4	1	no	Sylow
17	C₃×Q₈	C₃⋊D₃	1	4	yes	complemented
18	D₁₂	—	9	4	no	—
19	SL(2,𝔽₃)	C₃⋊D₃	1	9	yes	complemented
20	C₃⋊C₈	—	9	4	no	—
21	C₂×C₃²	—	4	12	no	—
22	C₂×C₃²	S₄	1	1	yes	characteristic
23	C₃⋊D₃	—	12	24	no	—
24	C₃⋊D₃	—	12	1	no	—
25	SD₁₆	—	27	1	no	Sylow
26	C₂×D₃	—	18	4	no	—
27	C₂×D₃	—	36	9	no	—
28	C₃×C₄	—	3	4	no	—
29	C₃²	—	4	12	no	—
30	C₃²	GL(2,𝔽₃)	1	1	yes	characteristic, complemented
31	Q₈	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic, complemented
32	D₄	—	27	1	no	—
33	C₈	—	27	1	no	—
34	C₂×C₃	—	4	9	no	—
35	C₂×C₃	C₃⋊S₄	1	4	yes	—
36	D₃	—	36	18	no	—
37	D₃	—	36	4	no	—
38	C₂²	—	54	1	no	—
39	C₄	—	3	1	no	—
40	C₃	—	4	9	no	—
41	C₃	C₃⋊GL(2,𝔽₃)	1	4	yes	complemented
42	C₂	—	108	1	no	—
43	C₂	C₃⋊(C₃⋊S₄)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	C₃⋊(C₃⋊S₄)
commutator, cosocle, hypocenter	C₃²×SL(2,𝔽₃)	C₂
Fitting	C₃²×Q₈	D₃
socle	C₂×C₃²	S₄
LCS	G ▷ C₃²×SL(2,𝔽₃)
UCS	C₁ ◁ C₂

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.