Subgroups — C₃³×#568 #13180023

Order 1728.

3612 conjugacy classes of subgroups.

The subgroup lattice has 120 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃³×#41	C₂	1	1	yes	maximal, characteristic
2	C₃³×#131	C₂	1	8	yes	maximal, complemented
3	C₂×C₃³×M₁₆	C₂	1	6	yes	maximal, complemented
4	C₃²×#568	C₃	1	13	yes	maximal, direct product factor
5	C₃³×#41	C₄	1	4	yes	—
6	C₃³×#41	C₂²	1	4	yes	—
7	C₂×C₃³×Q₈	C₄	1	1	yes	characteristic
8	C₂×C₃³×D₄	C₄	1	3	yes	—
9	C₂²×C₃³×C₄	C₂²	1	3	yes	—
10	C₃³×M₁₆	C₂²	1	4	yes	—
11	C₃³×M₁₆	C₂²	1	12	yes	complemented
12	C₂×C₃³×C₈	C₂²	1	12	yes	complemented
13	C₂×C₃²×#41	C₂×C₃	1	13	yes	—
14	C₃²×#131	C₂×C₃	1	104	yes	complemented
15	C₂×C₃²×M₁₆	C₂×C₃	1	78	yes	complemented
16	C₂³×C₃³	C₂×C₄	1	3	yes	—
17	C₃³×Q₈	C₂×C₄	1	4	yes	—
18	C₃³×D₄	C₂×C₄	1	12	yes	—
19	C₂×C₃³×C₄	C₂×C₄	1	3	yes	—
20	C₂×C₃³×C₄	C₂×C₄	1	6	yes	—
21	C₂×C₃³×C₄	C₂³	1	6	yes	—
22	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
23	C₃³×C₈	C₂³	1	8	yes	—
24	C₃×#568	C₃²	1	13	yes	direct product factor
25	C₃²×#41	C₃×C₄	1	52	yes	—
26	C₃²×#41	C₂²×C₃	1	52	yes	—
27	C₂×C₃²×Q₈	C₃×C₄	1	13	yes	—
28	C₂×C₃²×D₄	C₃×C₄	1	39	yes	—
29	C₂²×C₃²×C₄	C₂²×C₃	1	39	yes	—
30	C₃²×M₁₆	C₂²×C₃	1	52	yes	—
31	C₃²×M₁₆	C₂²×C₃	1	156	yes	complemented
32	C₂×C₃²×C₈	C₂²×C₃	1	156	yes	complemented
33	C₂²×C₃³	—	4	3	no	—
34	C₂²×C₃³	C₂²×C₄	1	1	yes	characteristic
35	C₂²×C₃³	C₂²×C₄	1	6	yes	—
36	C₃³×C₄	C₂²×C₄	1	1	yes	characteristic
37	C₃³×C₄	C₂²×C₄	1	6	yes	—
38	C₃³×C₄	C₂⁴	1	1	yes	characteristic
39	C₂×C₃×#41	C₂×C₃²	1	13	yes	—
40	C₃×#131	C₂×C₃²	1	104	yes	complemented
41	C₂×C₃×M₁₆	C₂×C₃²	1	78	yes	complemented
42	C₂³×C₃²	C₂×C₃×C₄	1	39	yes	—
43	C₃²×Q₈	C₂×C₃×C₄	1	52	yes	—
44	C₃²×D₄	C₂×C₃×C₄	1	156	yes	—
45	C₂×C₃²×C₄	C₂×C₃×C₄	1	39	yes	—
46	C₂×C₃²×C₄	C₂×C₃×C₄	1	78	yes	—
47	C₂×C₃²×C₄	C₂³×C₃	1	78	yes	—
48	C₂×C₃²×C₄	C₂³×C₃	1	13	yes	—
49	C₃²×C₈	C₂³×C₃	1	104	yes	—
50	#568	C₃³	1	1	yes	characteristic, direct product factor, Sylow
51	C₂×C₃³	—	2	1	no	—
52	C₂×C₃³	—	2	6	no	—
53	C₂×C₃³	C₂³×C₄	1	1	yes	characteristic
54	C₃×#41	C₃²×C₄	1	52	yes	—
55	C₃×#41	C₂²×C₃²	1	52	yes	—
56	C₂×C₃×Q₈	C₃²×C₄	1	13	yes	—
57	C₂×C₃×D₄	C₃²×C₄	1	39	yes	—
58	C₂²×C₃×C₄	C₂²×C₃²	1	39	yes	—
59	C₃×M₁₆	C₂²×C₃²	1	52	yes	—
60	C₃×M₁₆	C₂²×C₃²	1	156	yes	complemented
61	C₂×C₃×C₈	C₂²×C₃²	1	156	yes	complemented
62	C₂²×C₃²	—	4	39	no	—
63	C₂²×C₃²	C₂²×C₃×C₄	1	13	yes	—
64	C₂²×C₃²	C₂²×C₃×C₄	1	78	yes	—
65	C₃²×C₄	C₂²×C₃×C₄	1	13	yes	—
66	C₃²×C₄	C₂²×C₃×C₄	1	78	yes	—
67	C₃²×C₄	C₂⁴×C₃	1	13	yes	—
68	C₂×#41	C₂×C₃³	1	1	yes	characteristic
69	#131	C₂×C₃³	1	8	yes	complemented
70	C₂×M₁₆	C₂×C₃³	1	6	yes	complemented
71	C₃³	#568	1	1	yes	characteristic, direct product factor, Sylow
72	C₂³×C₃	C₂×C₃²×C₄	1	39	yes	—
73	C₃×Q₈	C₂×C₃²×C₄	1	52	yes	—
74	C₃×D₄	C₂×C₃²×C₄	1	156	yes	—
75	C₂×C₃×C₄	C₂×C₃²×C₄	1	39	yes	—
76	C₂×C₃×C₄	C₂×C₃²×C₄	1	78	yes	—
77	C₂×C₃×C₄	C₂³×C₃²	1	78	yes	—
78	C₂×C₃×C₄	C₂³×C₃²	1	13	yes	—
79	C₃×C₈	C₂³×C₃²	1	104	yes	—
80	C₂×C₃²	—	2	13	no	—
81	C₂×C₃²	—	2	78	no	—
82	C₂×C₃²	C₂³×C₃×C₄	1	13	yes	—
83	#41	C₃³×C₄	1	4	yes	—
84	#41	C₂²×C₃³	1	4	yes	—
85	C₂×Q₈	C₃³×C₄	1	1	yes	characteristic
86	C₂×D₄	C₃³×C₄	1	3	yes	—
87	C₂²×C₄	C₂²×C₃³	1	3	yes	—
88	M₁₆	C₂²×C₃³	1	4	yes	—
89	M₁₆	C₂²×C₃³	1	12	yes	complemented
90	C₂×C₈	C₂²×C₃³	1	12	yes	complemented
91	C₂²×C₃	—	4	39	no	—
92	C₂²×C₃	C₂²×C₃²×C₄	1	13	yes	—
93	C₂²×C₃	C₂²×C₃²×C₄	1	78	yes	—
94	C₃×C₄	C₂²×C₃²×C₄	1	13	yes	—
95	C₃×C₄	C₂²×C₃²×C₄	1	78	yes	—
96	C₃×C₄	C₂⁴×C₃²	1	13	yes	—
97	C₃²	C₃×#568	1	13	yes	direct product factor
98	C₂³	C₂×C₃³×C₄	1	3	yes	—
99	Q₈	C₂×C₃³×C₄	1	4	yes	—
100	D₄	C₂×C₃³×C₄	1	12	yes	—
101	C₂×C₄	C₂×C₃³×C₄	1	3	yes	—
102	C₂×C₄	C₂×C₃³×C₄	1	6	yes	—
103	C₂×C₄	C₂³×C₃³	1	6	yes	—
104	C₂×C₄	C₂³×C₃³	1	1	yes	characteristic
105	C₈	C₂³×C₃³	1	8	yes	—
106	C₂×C₃	—	2	13	no	—
107	C₂×C₃	—	2	78	no	—
108	C₂×C₃	C₂³×C₃²×C₄	1	13	yes	—
109	C₂²	—	4	3	no	—
110	C₂²	C₂²×C₃³×C₄	1	1	yes	characteristic
111	C₂²	C₂²×C₃³×C₄	1	6	yes	—
112	C₄	C₂²×C₃³×C₄	1	1	yes	characteristic
113	C₄	C₂²×C₃³×C₄	1	6	yes	—
114	C₄	C₂⁴×C₃³	1	1	yes	characteristic
115	C₃	C₃²×#568	1	13	yes	direct product factor
116	C₂	—	2	1	no	—
117	C₂	—	2	6	no	—
118	C₂	C₂³×C₃³×C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₃³×C₄	C₂⁴
commutator	C₂	C₂³×C₃³×C₄
Frattini, cosocle	C₄	C₂⁴×C₃³
socle	C₂×C₃³	C₂³×C₄
prime radical	C₂×C₃³×#41	C₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃³×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃³×#131	C₂	semidirect
3. C₂×C₃³×M₁₆	C₂	semidirect
4. C₃²×#568	C₃	direct (N × H)
11. C₃³×M₁₆	C₂²	semidirect
12. C₂×C₃³×C₈	C₂²	semidirect
14. C₃²×#131	C₂×C₃	semidirect
15. C₂×C₃²×M₁₆	C₂×C₃	semidirect
24. C₃×#568	C₃²	direct (N × H)
31. C₃²×M₁₆	C₂²×C₃	semidirect
32. C₂×C₃²×C₈	C₂²×C₃	semidirect
40. C₃×#131	C₂×C₃²	semidirect
41. C₂×C₃×M₁₆	C₂×C₃²	semidirect
50. #568	C₃³	direct (N × H)
60. C₃×M₁₆	C₂²×C₃²	semidirect
61. C₂×C₃×C₈	C₂²×C₃²	semidirect
69. #131	C₂×C₃³	semidirect
70. C₂×M₁₆	C₂×C₃³	semidirect
71. C₃³	#568	direct (N × H)
89. M₁₆	C₂²×C₃³	semidirect
90. C₂×C₈	C₂²×C₃³	semidirect
97. C₃²	C₃×#568	direct (N × H)
115. C₃	C₃²×#568	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×#41	2
C₃³×#131	2
C₂×C₃³×M₁₆	2
C₃²×#568	3

Subgroups — C33×#568 #13180023

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃³×#568 #13180023