Subgroups — C₂²×GL(2,𝔽₃) #5916

Order 192.

213 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×SL(2,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×GL(2,𝔽₃)	C₂	1	6	yes	maximal, direct product factor
3	C₂²×SD₁₆	—	3	1	no	maximal, Sylow
4	C₂³×D₃	—	4	1	no	maximal
5	C₂×SL(2,𝔽₃)	C₂²	1	3	yes	complemented
6	GL(2,𝔽₃)	C₂²	1	4	yes	direct product factor
7	C₂²×Q₈	D₃	1	1	yes	characteristic, complemented
8	C₂²×D₄	—	3	1	no	—
9	C₂×SD₁₆	—	3	6	no	—
10	C₂×SD₁₆	—	3	6	no	—
11	C₂²×C₈	—	3	1	no	—
12	C₂³×C₃	—	4	1	no	—
13	C₂²×D₃	—	4	6	no	—
14	C₂²×D₃	—	4	8	no	—
15	SL(2,𝔽₃)	C₂³	1	1	yes	characteristic, complemented
16	C₂⁴	—	6	1	no	—
17	C₂×Q₈	—	3	3	no	—
18	C₂×Q₈	C₂×D₃	1	3	yes	complemented
19	C₂×D₄	—	3	6	no	—
20	C₂×D₄	—	6	3	no	—
21	C₂²×C₄	—	3	1	no	—
22	SD₁₆	—	3	12	no	—
23	SD₁₆	—	3	4	no	—
24	C₂×C₈	—	3	6	no	—
25	C₂²×C₃	—	4	4	no	—
26	C₂²×C₃	—	4	3	no	—
27	C₂×D₃	—	4	4	no	—
28	C₂×D₃	—	4	24	no	—
29	C₂³	—	6	6	no	—
30	C₂³	—	12	4	no	—
31	C₂³	S₄	1	1	yes	characteristic
32	Q₈	—	3	3	no	—
33	Q₈	—	6	1	no	—
34	Q₈	C₂²×D₃	1	1	yes	characteristic, complemented
35	D₄	—	3	4	no	—
36	D₄	—	6	6	no	—
37	C₂×C₄	—	3	3	no	—
38	C₂×C₄	—	3	3	no	—
39	C₈	—	3	4	no	—
40	C₂×C₃	—	4	6	no	—
41	C₂×C₃	—	4	1	no	—
42	D₃	—	4	8	no	—
43	C₂²	—	6	4	no	—
44	C₂²	—	12	12	no	—
45	C₂²	GL(2,𝔽₃)	1	4	yes	direct product factor
46	C₂²	C₂×S₄	1	3	yes	—
47	C₄	—	3	1	no	—
48	C₄	—	3	3	no	—
49	C₃	—	4	1	no	Sylow
50	C₂	—	12	4	no	—
51	C₂	C₂×GL(2,𝔽₃)	1	6	yes	direct product factor
52	C₂	C₂²×S₄	1	1	yes	characteristic

	subgroup	G/H
center, socle, hypercenter	C₂³	S₄
commutator, cosocle, hypocenter	SL(2,𝔽₃)	C₂³
Frattini	C₂	C₂²×S₄
Fitting	C₂²×Q₈	D₃
LCS	G ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂³

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.