Subgroups — C₂³×#4208 #13049356

Order 1296.

1267 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#737	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#4208	C₂	1	14	yes	maximal, direct product factor
3	C₂³×#272	C₃	1	1	yes	maximal, characteristic
4	C₂³×D₃×C₉	—	3	4	no	maximal
5	C₂²×#737	C₂²	1	7	yes	complemented
6	C₂×#4208	C₂²	1	28	yes	direct product factor
7	C₂³×3₋¹⁺²	—	2	4	no	—
8	C₂³×H(3,𝔽₃)	C₂×C₃	1	1	yes	characteristic
9	C₂³×C₃×C₉	D₃	1	4	yes	complemented
10	C₂²×#272	C₂×C₃	1	14	yes	—
11	C₂²×D₃×C₉	—	3	56	no	—
12	C₂×#737	C₂³	1	7	yes	complemented
13	#4208	C₂³	1	8	yes	direct product factor
14	C₂³×C₃×D₃	—	3	4	no	—
15	C₂⁴×C₉	—	9	1	no	—
16	C₂²×3₋¹⁺²	—	2	28	no	—
17	C₂²×H(3,𝔽₃)	C₂²×C₃	1	7	yes	—
18	C₂²×C₃×C₉	C₂×D₃	1	28	yes	complemented
19	C₂×#272	C₂²×C₃	1	28	yes	—
20	C₂×D₃×C₉	—	3	112	no	—
21	#737	C₂⁴	1	1	yes	characteristic, complemented, Sylow
22	C₂³×C₃²	C₃×D₃	1	4	yes	—
23	C₂²×C₃×D₃	—	3	56	no	—
24	C₂³×C₉	—	2	4	no	—
25	C₂³×C₉	—	9	14	no	—
26	C₂³×C₉	C₃⋊D₃	1	1	yes	characteristic
27	C₂×3₋¹⁺²	—	2	28	no	—
28	C₂×H(3,𝔽₃)	C₂³×C₃	1	7	yes	—
29	C₂×C₃×C₉	C₂²×D₃	1	28	yes	complemented
30	#272	C₂³×C₃	1	8	yes	—
31	D₃×C₉	—	3	32	no	—
32	C₂⁴×C₃	—	9	1	no	—
33	C₂³×D₃	—	3	4	no	—
34	C₂²×C₃²	C₂×C₃×D₃	1	28	yes	—
35	C₂×C₃×D₃	—	3	112	no	—
36	C₂²×C₉	—	2	28	no	—
37	C₂²×C₉	—	9	28	no	—
38	C₂²×C₉	C₂×C₃⋊D₃	1	7	yes	—
39	3₋¹⁺²	—	2	4	no	—
40	H(3,𝔽₃)	C₂⁴×C₃	1	1	yes	characteristic
41	C₃×C₉	C₂³×D₃	1	4	yes	complemented
42	C₂³×C₃	—	3	4	no	—
43	C₂³×C₃	—	9	14	no	—
44	C₂³×C₃	C₃×C₃⋊D₃	1	1	yes	characteristic
45	C₂²×D₃	—	3	56	no	—
46	C₂×C₃²	C₂²×C₃×D₃	1	28	yes	—
47	C₃×D₃	—	3	32	no	—
48	C₂×C₉	—	2	28	no	—
49	C₂×C₉	—	9	8	no	—
50	C₂×C₉	C₂²×C₃⋊D₃	1	7	yes	—
51	C₂⁴	—	9	1	no	Sylow
52	C₂²×C₃	—	3	28	no	—
53	C₂²×C₃	—	9	28	no	—
54	C₂²×C₃	C₂×C₃×C₃⋊D₃	1	7	yes	—
55	C₂×D₃	—	3	112	no	—
56	C₃²	C₂³×C₃×D₃	1	4	yes	—
57	C₉	—	2	4	no	—
58	C₉	C₂³×C₃⋊D₃	1	1	yes	characteristic
59	C₂³	—	9	14	no	—
60	C₂³	#4208	1	1	yes	characteristic, direct product factor
61	C₂×C₃	—	3	28	no	—
62	C₂×C₃	—	9	8	no	—
63	C₂×C₃	C₂²×C₃×C₃⋊D₃	1	7	yes	—
64	D₃	—	3	32	no	—
65	C₂²	—	9	28	no	—
66	C₂²	C₂×#4208	1	7	yes	direct product factor
67	C₃	—	3	4	no	—
68	C₃	C₂³×C₃×C₃⋊D₃	1	1	yes	characteristic
69	C₂	—	9	8	no	—
70	C₂	C₂²×#4208	1	7	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂³×C₉	C₃⋊D₃
commutator, cosocle, hypocenter	H(3,𝔽₃)	C₂⁴×C₃
Frattini	C₃	C₂³×C₃×C₃⋊D₃
Fitting	C₂³×#737	C₂
socle	C₂³×C₃	C₃×C₃⋊D₃
prime radical	C₂³×#272	C₃
LCS	G ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂³×C₉

Complements

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

N	H ≅ G/N	type
1. C₂³×#737	C₂	semidirect
2. C₂²×#4208	C₂	direct (N × H)
5. C₂²×#737	C₂²	semidirect
6. C₂×#4208	C₂²	direct (N × H)
9. C₂³×C₃×C₉	D₃	semidirect
12. C₂×#737	C₂³	semidirect
13. #4208	C₂³	direct (N × H)
18. C₂²×C₃×C₉	C₂×D₃	semidirect
21. #737	C₂⁴	semidirect
29. C₂×C₃×C₉	C₂²×D₃	semidirect
41. C₃×C₉	C₂³×D₃	semidirect
60. C₂³	#4208	direct (N × H)
66. C₂²	C₂×#4208	direct (N × H)
70. C₂	C₂²×#4208	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×#737	2
C₂²×#4208	2
C₂³×#272	3
C₂³×D₃×C₉	3

Subgroups — C23×#4208 #13049356

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#4208 #13049356