Subgroups — C₃³×#576 #13180031

Order 1728.

3752 conjugacy classes of subgroups.

The subgroup lattice has 152 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×H(5,𝔽₂)	C₂	1	2	yes	maximal, complemented
2	C₃³×AGL(1,ℤ/8)	C₂	1	6	yes	maximal, complemented
3	C₃³×#135	C₂	1	3	yes	maximal, complemented
4	C₂×C₃³×D₈	C₂	1	3	yes	maximal, complemented
5	C₃³×#131	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃²×#576	C₃	1	13	yes	maximal, direct product factor
7	C₃³×#41	—	2	2	no	—
8	C₃³×#41	C₂²	1	1	yes	characteristic
9	C₃³×#41	C₂²	1	6	yes	complemented
10	C₂×C₃³×D₄	—	2	6	no	—
11	C₂×C₃³×D₄	C₂²	1	6	yes	complemented
12	C₃³×Q₁₆	C₂²	1	1	yes	characteristic, complemented
13	C₃³×SD₁₆	C₂²	1	6	yes	complemented
14	C₃³×D₈	C₂²	1	3	yes	complemented
15	C₃³×D₈	C₂²	1	6	yes	complemented
16	C₃³×M₁₆	C₂²	1	3	yes	complemented
17	C₂×C₃³×C₈	C₂²	1	3	yes	complemented
18	C₃²×H(5,𝔽₂)	C₂×C₃	1	26	yes	complemented
19	C₃²×AGL(1,ℤ/8)	C₂×C₃	1	78	yes	complemented
20	C₃²×#135	C₂×C₃	1	39	yes	complemented
21	C₂×C₃²×D₈	C₂×C₃	1	39	yes	complemented
22	C₃²×#131	C₂×C₃	1	13	yes	complemented
23	C₂³×C₃³	—	2	6	no	—
24	C₃³×Q₈	D₄	1	1	yes	characteristic
25	C₃³×Q₈	C₂³	1	2	yes	—
26	C₃³×D₄	—	2	6	no	—
27	C₃³×D₄	—	2	6	no	—
28	C₃³×D₄	D₄	1	3	yes	—
29	C₃³×D₄	C₂³	1	6	yes	—
30	C₂×C₃³×C₄	—	2	6	no	—
31	C₂×C₃³×C₄	C₂³	1	3	yes	—
32	C₃³×C₈	C₂³	1	1	yes	characteristic
33	C₃³×C₈	C₂³	1	3	yes	—
34	C₃×#576	C₃²	1	13	yes	direct product factor
35	C₃²×#41	—	2	26	no	—
36	C₃²×#41	C₂²×C₃	1	13	yes	—
37	C₃²×#41	C₂²×C₃	1	78	yes	complemented
38	C₂×C₃²×D₄	—	2	78	no	—
39	C₂×C₃²×D₄	C₂²×C₃	1	78	yes	complemented
40	C₃²×Q₁₆	C₂²×C₃	1	13	yes	complemented
41	C₃²×SD₁₆	C₂²×C₃	1	78	yes	complemented
42	C₃²×D₈	C₂²×C₃	1	39	yes	complemented
43	C₃²×D₈	C₂²×C₃	1	78	yes	complemented
44	C₃²×M₁₆	C₂²×C₃	1	39	yes	complemented
45	C₂×C₃²×C₈	C₂²×C₃	1	39	yes	complemented
46	C₂²×C₃³	—	2	6	no	—
47	C₂²×C₃³	—	8	6	no	—
48	C₂²×C₃³	C₂×D₄	1	3	yes	—
49	C₃³×C₄	—	2	2	no	—
50	C₃³×C₄	C₂×D₄	1	3	yes	—
51	C₃³×C₄	C₂⁴	1	1	yes	characteristic
52	C₃×H(5,𝔽₂)	C₂×C₃²	1	26	yes	complemented
53	C₃×AGL(1,ℤ/8)	C₂×C₃²	1	78	yes	complemented
54	C₃×#135	C₂×C₃²	1	39	yes	complemented
55	C₂×C₃×D₈	C₂×C₃²	1	39	yes	complemented
56	C₃×#131	C₂×C₃²	1	13	yes	complemented
57	C₂³×C₃²	—	2	78	no	—
58	C₃²×Q₈	C₃×D₄	1	13	yes	—
59	C₃²×Q₈	C₂³×C₃	1	26	yes	—
60	C₃²×D₄	—	2	78	no	—
61	C₃²×D₄	—	2	78	no	—
62	C₃²×D₄	C₃×D₄	1	39	yes	—
63	C₃²×D₄	C₂³×C₃	1	78	yes	—
64	C₂×C₃²×C₄	—	2	78	no	—
65	C₂×C₃²×C₄	C₂³×C₃	1	39	yes	—
66	C₃²×C₈	C₂³×C₃	1	13	yes	—
67	C₃²×C₈	C₂³×C₃	1	39	yes	—
68	#576	C₃³	1	1	yes	characteristic, direct product factor, Sylow
69	C₂×C₃³	—	2	3	no	—
70	C₂×C₃³	—	4	6	no	—
71	C₂×C₃³	C₂²×D₄	1	1	yes	characteristic
72	C₃×#41	—	2	26	no	—
73	C₃×#41	C₂²×C₃²	1	13	yes	—
74	C₃×#41	C₂²×C₃²	1	78	yes	complemented
75	C₂×C₃×D₄	—	2	78	no	—
76	C₂×C₃×D₄	C₂²×C₃²	1	78	yes	complemented
77	C₃×Q₁₆	C₂²×C₃²	1	13	yes	complemented
78	C₃×SD₁₆	C₂²×C₃²	1	78	yes	complemented
79	C₃×D₈	C₂²×C₃²	1	39	yes	complemented
80	C₃×D₈	C₂²×C₃²	1	78	yes	complemented
81	C₃×M₁₆	C₂²×C₃²	1	39	yes	complemented
82	C₂×C₃×C₈	C₂²×C₃²	1	39	yes	complemented
83	C₂²×C₃²	—	2	78	no	—
84	C₂²×C₃²	—	8	78	no	—
85	C₂²×C₃²	C₂×C₃×D₄	1	39	yes	—
86	C₃²×C₄	—	2	26	no	—
87	C₃²×C₄	C₂×C₃×D₄	1	39	yes	—
88	C₃²×C₄	C₂⁴×C₃	1	13	yes	—
89	H(5,𝔽₂)	C₂×C₃³	1	2	yes	complemented
90	AGL(1,ℤ/8)	C₂×C₃³	1	6	yes	complemented
91	#135	C₂×C₃³	1	3	yes	complemented
92	C₂×D₈	C₂×C₃³	1	3	yes	complemented
93	#131	C₂×C₃³	1	1	yes	characteristic, complemented
94	C₃³	#576	1	1	yes	characteristic, direct product factor, Sylow
95	C₂³×C₃	—	2	78	no	—
96	C₃×Q₈	C₃²×D₄	1	13	yes	—
97	C₃×Q₈	C₂³×C₃²	1	26	yes	—
98	C₃×D₄	—	2	78	no	—
99	C₃×D₄	—	2	78	no	—
100	C₃×D₄	C₃²×D₄	1	39	yes	—
101	C₃×D₄	C₂³×C₃²	1	78	yes	—
102	C₂×C₃×C₄	—	2	78	no	—
103	C₂×C₃×C₄	C₂³×C₃²	1	39	yes	—
104	C₃×C₈	C₂³×C₃²	1	13	yes	—
105	C₃×C₈	C₂³×C₃²	1	39	yes	—
106	C₂×C₃²	—	2	39	no	—
107	C₂×C₃²	—	4	78	no	—
108	C₂×C₃²	C₂²×C₃×D₄	1	13	yes	—
109	#41	—	2	2	no	—
110	#41	C₂²×C₃³	1	1	yes	characteristic
111	#41	C₂²×C₃³	1	6	yes	complemented
112	C₂×D₄	—	2	6	no	—
113	C₂×D₄	C₂²×C₃³	1	6	yes	complemented
114	Q₁₆	C₂²×C₃³	1	1	yes	characteristic, complemented
115	SD₁₆	C₂²×C₃³	1	6	yes	complemented
116	D₈	C₂²×C₃³	1	3	yes	complemented
117	D₈	C₂²×C₃³	1	6	yes	complemented
118	M₁₆	C₂²×C₃³	1	3	yes	complemented
119	C₂×C₈	C₂²×C₃³	1	3	yes	complemented
120	C₂²×C₃	—	2	78	no	—
121	C₂²×C₃	—	8	78	no	—
122	C₂²×C₃	C₂×C₃²×D₄	1	39	yes	—
123	C₃×C₄	—	2	26	no	—
124	C₃×C₄	C₂×C₃²×D₄	1	39	yes	—
125	C₃×C₄	C₂⁴×C₃²	1	13	yes	—
126	C₃²	C₃×#576	1	13	yes	direct product factor
127	C₂³	—	2	6	no	—
128	Q₈	C₃³×D₄	1	1	yes	characteristic
129	Q₈	C₂³×C₃³	1	2	yes	—
130	D₄	—	2	6	no	—
131	D₄	—	2	6	no	—
132	D₄	C₃³×D₄	1	3	yes	—
133	D₄	C₂³×C₃³	1	6	yes	—
134	C₂×C₄	—	2	6	no	—
135	C₂×C₄	C₂³×C₃³	1	3	yes	—
136	C₈	C₂³×C₃³	1	1	yes	characteristic
137	C₈	C₂³×C₃³	1	3	yes	—
138	C₂×C₃	—	2	39	no	—
139	C₂×C₃	—	4	78	no	—
140	C₂×C₃	C₂²×C₃²×D₄	1	13	yes	—
141	C₂²	—	2	6	no	—
142	C₂²	—	8	6	no	—
143	C₂²	C₂×C₃³×D₄	1	3	yes	—
144	C₄	—	2	2	no	—
145	C₄	C₂×C₃³×D₄	1	3	yes	—
146	C₄	C₂⁴×C₃³	1	1	yes	characteristic
147	C₃	C₃²×#576	1	13	yes	direct product factor
148	C₂	—	2	3	no	—
149	C₂	—	4	6	no	—
150	C₂	C₂²×C₃³×D₄	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃³	C₂²×D₄
commutator, Frattini, cosocle	C₄	C₂⁴×C₃³
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃³ ◁ C₃³×#41 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃³×H(5,𝔽₂)	C₂	semidirect
2. C₃³×AGL(1,ℤ/8)	C₂	semidirect
3. C₃³×#135	C₂	semidirect
4. C₂×C₃³×D₈	C₂	semidirect
5. C₃³×#131	C₂	semidirect
6. C₃²×#576	C₃	direct (N × H)
9. C₃³×#41	C₂²	semidirect
11. C₂×C₃³×D₄	C₂²	semidirect
12. C₃³×Q₁₆	C₂²	semidirect
13. C₃³×SD₁₆	C₂²	semidirect
14. C₃³×D₈	C₂²	semidirect
15. C₃³×D₈	C₂²	semidirect
16. C₃³×M₁₆	C₂²	semidirect
17. C₂×C₃³×C₈	C₂²	semidirect
18. C₃²×H(5,𝔽₂)	C₂×C₃	semidirect
19. C₃²×AGL(1,ℤ/8)	C₂×C₃	semidirect
20. C₃²×#135	C₂×C₃	semidirect
21. C₂×C₃²×D₈	C₂×C₃	semidirect
22. C₃²×#131	C₂×C₃	semidirect
34. C₃×#576	C₃²	direct (N × H)
37. C₃²×#41	C₂²×C₃	semidirect
39. C₂×C₃²×D₄	C₂²×C₃	semidirect
40. C₃²×Q₁₆	C₂²×C₃	semidirect
41. C₃²×SD₁₆	C₂²×C₃	semidirect
42. C₃²×D₈	C₂²×C₃	semidirect
43. C₃²×D₈	C₂²×C₃	semidirect
44. C₃²×M₁₆	C₂²×C₃	semidirect
45. C₂×C₃²×C₈	C₂²×C₃	semidirect
52. C₃×H(5,𝔽₂)	C₂×C₃²	semidirect
53. C₃×AGL(1,ℤ/8)	C₂×C₃²	semidirect
54. C₃×#135	C₂×C₃²	semidirect
55. C₂×C₃×D₈	C₂×C₃²	semidirect
56. C₃×#131	C₂×C₃²	semidirect
68. #576	C₃³	direct (N × H)
74. C₃×#41	C₂²×C₃²	semidirect
76. C₂×C₃×D₄	C₂²×C₃²	semidirect
77. C₃×Q₁₆	C₂²×C₃²	semidirect
78. C₃×SD₁₆	C₂²×C₃²	semidirect
79. C₃×D₈	C₂²×C₃²	semidirect
80. C₃×D₈	C₂²×C₃²	semidirect
81. C₃×M₁₆	C₂²×C₃²	semidirect
82. C₂×C₃×C₈	C₂²×C₃²	semidirect
89. H(5,𝔽₂)	C₂×C₃³	semidirect
90. AGL(1,ℤ/8)	C₂×C₃³	semidirect
91. #135	C₂×C₃³	semidirect
92. C₂×D₈	C₂×C₃³	semidirect
93. #131	C₂×C₃³	semidirect
94. C₃³	#576	direct (N × H)
111. #41	C₂²×C₃³	semidirect
113. C₂×D₄	C₂²×C₃³	semidirect
114. Q₁₆	C₂²×C₃³	semidirect
115. SD₁₆	C₂²×C₃³	semidirect
116. D₈	C₂²×C₃³	semidirect
117. D₈	C₂²×C₃³	semidirect
118. M₁₆	C₂²×C₃³	semidirect
119. C₂×C₈	C₂²×C₃³	semidirect
126. C₃²	C₃×#576	direct (N × H)
147. C₃	C₃²×#576	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃³×H(5,𝔽₂)	2
C₃³×AGL(1,ℤ/8)	2
C₃³×#135	2
C₂×C₃³×D₈	2
C₃³×#131	2
C₃²×#576	3

Subgroups — C33×#576 #13180031

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃³×#576 #13180031