Subgroups — C₃×#2840 #84704

Order 384.

580 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#393	C₂	1	12	yes	maximal, complemented
2	C₃×#379	C₂	1	3	yes	maximal
3	#2840	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
4	C₂⁵×C₃	C₂²	1	1	yes	characteristic
5	C₂²×C₃×Q₈	C₂²	1	4	yes	complemented
6	C₂×C₃×C₂²⋊C₄	C₂²	1	18	yes	—
7	C₃×#95	C₂²	1	12	yes	complemented
8	#393	C₂×C₃	1	12	yes	complemented
9	#379	C₂×C₃	1	3	yes	—
10	C₂⁴×C₃	—	4	6	no	—
11	C₂⁴×C₃	Q₈	1	4	yes	complemented
12	C₂⁴×C₃	C₂³	1	3	yes	—
13	C₂×C₃×Q₈	—	4	12	no	—
14	C₂²×C₃×C₄	C₂³	1	12	yes	—
15	C₃×C₂²⋊C₄	—	4	18	no	—
16	C₂⁵	C₂²×C₃	1	1	yes	characteristic
17	C₂²×Q₈	C₂²×C₃	1	4	yes	complemented
18	C₂×C₂²⋊C₄	C₂²×C₃	1	18	yes	—
19	#95	C₂²×C₃	1	12	yes	complemented
20	C₂³×C₃	—	2	18	no	—
21	C₂³×C₃	—	4	12	no	—
22	C₂³×C₃	—	4	16	no	—
23	C₂³×C₃	C₂×Q₈	1	6	yes	—
24	C₂³×C₃	C₂⁴	1	1	yes	characteristic
25	C₃×Q₈	—	4	16	no	—
26	C₂×C₃×C₄	—	2	36	no	—
27	C₂⁴	—	4	6	no	—
28	C₂⁴	C₃×Q₈	1	4	yes	complemented
29	C₂⁴	C₂³×C₃	1	3	yes	—
30	C₂×Q₈	—	4	12	no	—
31	C₂²×C₄	C₂³×C₃	1	12	yes	—
32	C₂²⋊C₄	—	4	18	no	—
33	C₂²×C₃	—	2	18	no	—
34	C₂²×C₃	—	4	16	no	—
35	C₂²×C₃	—	4	12	no	—
36	C₂²×C₃	C₂²×Q₈	1	1	yes	characteristic
37	C₂²×C₃	H(5,𝔽₂)	1	6	yes	—
38	C₃×C₄	—	4	12	no	—
39	C₂³	—	2	18	no	—
40	C₂³	—	4	12	no	—
41	C₂³	—	4	16	no	—
42	C₂³	C₂×C₃×Q₈	1	6	yes	—
43	C₂³	C₂⁴×C₃	1	1	yes	characteristic
44	Q₈	—	4	16	no	—
45	C₂×C₄	—	2	36	no	—
46	C₂×C₃	—	4	6	no	—
47	C₂×C₃	#543	1	3	yes	—
48	C₂×C₃	H(3,𝔽₄)	1	4	yes	—
49	C₂²	—	2	18	no	—
50	C₂²	—	4	16	no	—
51	C₂²	—	4	12	no	—
52	C₂²	C₂²×C₃×Q₈	1	1	yes	characteristic
53	C₂²	C₃×H(5,𝔽₂)	1	6	yes	—
54	C₄	—	4	12	no	—
55	C₃	#2840	1	1	yes	characteristic, direct product factor, Sylow
56	C₂	—	4	6	no	—
57	C₂	C₃×#543	1	3	yes	—
58	C₂	C₃×H(3,𝔽₄)	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃
prime radical	C₂⁵×C₃	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×#393	C₂	semidirect
3. #2840	C₃	direct (N × H)
5. C₂²×C₃×Q₈	C₂²	semidirect
7. C₃×#95	C₂²	semidirect
8. #393	C₂×C₃	semidirect
11. C₂⁴×C₃	Q₈	semidirect
17. C₂²×Q₈	C₂²×C₃	semidirect
19. #95	C₂²×C₃	semidirect
28. C₂⁴	C₃×Q₈	semidirect
55. C₃	#2840	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×#393	2
C₃×#379	2
#2840	3

Subgroups — C3×#2840 #84704

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2840 #84704