Subgroups — C₂³×#226 #87478

Order 384.

803 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×SL(2,𝔽₃)	C₂	1	1	yes	maximal, characteristic
2	C₂²×#226	C₂	1	14	yes	maximal, direct product factor
3	C₂³×Q₁₆	—	3	1	no	maximal, Sylow
4	C₂³×Dic₃	—	4	1	no	maximal
5	C₂²×SL(2,𝔽₃)	C₂²	1	7	yes	—
6	C₂×#226	C₂²	1	28	yes	direct product factor
7	C₂³×Q₈	—	3	1	no	—
8	C₂³×Q₈	D₃	1	1	yes	characteristic
9	C₂²×Q₁₆	—	3	14	no	—
10	C₂²×Q₁₆	—	3	14	no	—
11	C₂³×C₈	—	3	1	no	—
12	C₂⁴×C₃	—	4	1	no	—
13	C₂²×Dic₃	—	4	14	no	—
14	C₂×SL(2,𝔽₃)	C₂³	1	7	yes	—
15	#226	C₂³	1	8	yes	direct product factor
16	C₂²×Q₈	—	3	7	no	—
17	C₂²×Q₈	—	3	14	no	—
18	C₂²×Q₈	—	6	7	no	—
19	C₂²×Q₈	C₂×D₃	1	7	yes	—
20	C₂³×C₄	—	3	1	no	—
21	C₂³×C₄	—	6	1	no	—
22	C₂×Q₁₆	—	3	84	no	—
23	C₂×Q₁₆	—	3	28	no	—
24	C₂²×C₈	—	3	14	no	—
25	C₂³×C₃	—	4	8	no	—
26	C₂³×C₃	—	4	7	no	—
27	C₂×Dic₃	—	4	28	no	—
28	SL(2,𝔽₃)	C₂⁴	1	1	yes	characteristic
29	C₂⁴	S₄	1	1	yes	characteristic
30	C₂×Q₈	—	3	21	no	—
31	C₂×Q₈	—	3	28	no	—
32	C₂×Q₈	—	6	7	no	—
33	C₂×Q₈	—	6	42	no	—
34	C₂×Q₈	C₂²×D₃	1	7	yes	—
35	C₂²×C₄	—	3	7	no	—
36	C₂²×C₄	—	3	7	no	—
37	C₂²×C₄	—	6	14	no	—
38	Q₁₆	—	3	56	no	—
39	Q₁₆	—	3	8	no	—
40	C₂×C₈	—	3	28	no	—
41	C₂²×C₃	—	4	28	no	—
42	C₂²×C₃	—	4	7	no	—
43	Dic₃	—	4	8	no	—
44	C₂³	#226	1	8	yes	direct product factor
45	C₂³	C₂×S₄	1	7	yes	—
46	Q₈	—	3	7	no	—
47	Q₈	—	3	8	no	—
48	Q₈	—	6	7	no	—
49	Q₈	—	6	28	no	—
50	Q₈	C₂³×D₃	1	1	yes	characteristic
51	C₂×C₄	—	3	7	no	—
52	C₂×C₄	—	3	21	no	—
53	C₂×C₄	—	6	28	no	—
54	C₈	—	3	8	no	—
55	C₂×C₃	—	4	14	no	—
56	C₂×C₃	—	4	1	no	—
57	C₂²	C₂×#226	1	28	yes	direct product factor
58	C₂²	C₂²×S₄	1	7	yes	—
59	C₄	—	3	1	no	—
60	C₄	—	3	7	no	—
61	C₄	—	6	8	no	—
62	C₃	—	4	1	no	Sylow
63	C₂	C₂²×#226	1	14	yes	direct product factor
64	C₂	C₂³×S₄	1	1	yes	characteristic

	subgroup	G/H
center, socle, hypercenter	C₂⁴	S₄
commutator, cosocle, hypocenter	SL(2,𝔽₃)	C₂⁴
Frattini	C₂	C₂³×S₄
Fitting	C₂³×Q₈	D₃
prime radical	C₂³×SL(2,𝔽₃)	C₂
LCS	G ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂⁴

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₂²×#226	C₂	direct (N × H)
6. C₂×#226	C₂²	direct (N × H)
15. #226	C₂³	direct (N × H)
44. C₂³	#226	direct (N × H)
57. C₂²	C₂×#226	direct (N × H)
63. C₂	C₂²×#226	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂³×SL(2,𝔽₃)	2
C₂²×#226	2
C₂³×Q₁₆	3
C₂³×Dic₃	4

Subgroups — C23×#226 #87478

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#226 #87478