Subgroups — C₃³×#366 #13146159

Order 1728.

644 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃³×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₃³×#107	C₂	1	2	yes	maximal
3	C₃²×#366	C₃	1	13	yes	maximal, direct product factor
4	C₂×C₃³×C₈	C₂²	1	1	yes	characteristic
5	C₃³×C₄⋊C₄	—	2	2	no	—
6	C₃³×C₁₆	C₄	1	2	yes	complemented
7	C₂×C₃²×C₁₆	C₂×C₃	1	13	yes	—
8	C₃²×#107	C₂×C₃	1	26	yes	—
9	C₂×C₃³×C₄	—	4	2	no	—
10	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
11	C₃³×C₈	C₂×C₄	1	1	yes	characteristic
12	C₃³×C₈	Q₈	1	1	yes	characteristic
13	C₃×#366	C₃²	1	13	yes	direct product factor
14	C₂×C₃²×C₈	C₂²×C₃	1	13	yes	—
15	C₃²×C₄⋊C₄	—	2	26	no	—
16	C₃²×C₁₆	C₃×C₄	1	26	yes	complemented
17	C₂²×C₃³	D₈	1	1	yes	characteristic
18	C₃³×C₄	—	8	2	no	—
19	C₃³×C₄	C₄⋊C₄	1	1	yes	characteristic
20	C₃³×C₄	Q₁₆	1	1	yes	characteristic
21	C₂×C₃×C₁₆	C₂×C₃²	1	13	yes	—
22	C₃×#107	C₂×C₃²	1	26	yes	—
23	C₂×C₃²×C₄	—	4	26	no	—
24	C₂×C₃²×C₄	C₃×D₄	1	13	yes	—
25	C₃²×C₈	C₂×C₃×C₄	1	13	yes	—
26	C₃²×C₈	C₃×Q₈	1	13	yes	—
27	#366	C₃³	1	1	yes	characteristic, direct product factor, Sylow
28	C₂×C₃³	#107	1	1	yes	characteristic
29	C₂×C₃³	D₁₆	1	1	yes	characteristic
30	C₂×C₃³	Q₃₂	1	1	yes	characteristic
31	C₂×C₃×C₈	C₂²×C₃²	1	13	yes	—
32	C₃×C₄⋊C₄	—	2	26	no	—
33	C₃×C₁₆	C₃²×C₄	1	26	yes	complemented
34	C₂²×C₃²	C₃×D₈	1	13	yes	—
35	C₃²×C₄	—	8	26	no	—
36	C₃²×C₄	C₃×C₄⋊C₄	1	13	yes	—
37	C₃²×C₄	C₃×Q₁₆	1	13	yes	—
38	C₂×C₁₆	C₂×C₃³	1	1	yes	characteristic
39	#107	C₂×C₃³	1	2	yes	—
40	C₃³	#366	1	1	yes	characteristic, direct product factor, Sylow
41	C₂×C₃×C₄	—	4	26	no	—
42	C₂×C₃×C₄	C₃²×D₄	1	13	yes	—
43	C₃×C₈	C₂×C₃²×C₄	1	13	yes	—
44	C₃×C₈	C₃²×Q₈	1	13	yes	—
45	C₂×C₃²	C₃×#107	1	13	yes	—
46	C₂×C₃²	C₃×D₁₆	1	13	yes	—
47	C₂×C₃²	C₃×Q₃₂	1	13	yes	—
48	C₂×C₈	C₂²×C₃³	1	1	yes	characteristic
49	C₄⋊C₄	—	2	2	no	—
50	C₁₆	C₃³×C₄	1	2	yes	complemented
51	C₂²×C₃	C₃²×D₈	1	13	yes	—
52	C₃×C₄	—	8	26	no	—
53	C₃×C₄	C₃²×C₄⋊C₄	1	13	yes	—
54	C₃×C₄	C₃²×Q₁₆	1	13	yes	—
55	C₃²	C₃×#366	1	13	yes	direct product factor
56	C₂×C₄	—	4	2	no	—
57	C₂×C₄	C₃³×D₄	1	1	yes	characteristic
58	C₈	C₂×C₃³×C₄	1	1	yes	characteristic
59	C₈	C₃³×Q₈	1	1	yes	characteristic
60	C₂×C₃	C₃²×#107	1	13	yes	—
61	C₂×C₃	C₃²×D₁₆	1	13	yes	—
62	C₂×C₃	C₃²×Q₃₂	1	13	yes	—
63	C₂²	C₃³×D₈	1	1	yes	characteristic
64	C₄	—	8	2	no	—
65	C₄	C₃³×C₄⋊C₄	1	1	yes	characteristic
66	C₄	C₃³×Q₁₆	1	1	yes	characteristic
67	C₃	C₃²×#366	1	13	yes	direct product factor
68	C₂	C₃³×#107	1	1	yes	characteristic
69	C₂	C₃³×D₁₆	1	1	yes	characteristic
70	C₂	C₃³×Q₃₂	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₃³	D₈
commutator	C₈	C₂×C₃³×C₄
Frattini, cosocle	C₂×C₈	C₂²×C₃³
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃³ ◁ C₂×C₃³×C₄ ◁ C₂×C₃³×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃²×#366	C₃	direct (N × H)
6. C₃³×C₁₆	C₄	semidirect
13. C₃×#366	C₃²	direct (N × H)
16. C₃²×C₁₆	C₃×C₄	semidirect
27. #366	C₃³	direct (N × H)
33. C₃×C₁₆	C₃²×C₄	semidirect
40. C₃³	#366	direct (N × H)
50. C₁₆	C₃³×C₄	semidirect
55. C₃²	C₃×#366	direct (N × H)
67. C₃	C₃²×#366	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃³×C₁₆	2
C₃³×#107	2
C₃²×#366	3

Subgroups — C33×#366 #13146159

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃³×#366 #13146159