Subgroups — C₃³×#578 #13180033

Order 1728.

3472 conjugacy classes of subgroups.

The subgroup lattice has 148 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×2₋¹⁺⁴	C₂	1	2	yes	maximal, complemented
2	C₃³×#137	C₂	1	6	yes	maximal, complemented
3	C₃³×#135	C₂	1	3	yes	maximal, complemented
4	C₂×C₃³×Q₁₆	C₂	1	3	yes	maximal, complemented
5	C₃³×#131	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃²×#578	C₃	1	13	yes	maximal, direct product factor
7	C₃³×#41	—	2	6	no	—
8	C₃³×#41	C₂²	1	6	yes	—
9	C₃³×#41	C₂²	1	1	yes	characteristic
10	C₂×C₃³×Q₈	—	2	2	no	—
11	C₂×C₃³×Q₈	C₂²	1	6	yes	—
12	C₃³×Q₁₆	C₂²	1	3	yes	—
13	C₃³×Q₁₆	C₂²	1	6	yes	—
14	C₃³×SD₁₆	C₂²	1	6	yes	—
15	C₃³×D₈	C₂²	1	1	yes	characteristic
16	C₃³×M₁₆	C₂²	1	3	yes	—
17	C₂×C₃³×C₈	C₂²	1	3	yes	—
18	C₃²×2₋¹⁺⁴	C₂×C₃	1	26	yes	complemented
19	C₃²×#137	C₂×C₃	1	78	yes	complemented
20	C₃²×#135	C₂×C₃	1	39	yes	complemented
21	C₂×C₃²×Q₁₆	C₂×C₃	1	39	yes	complemented
22	C₃²×#131	C₂×C₃	1	13	yes	complemented
23	C₃³×Q₈	—	2	6	no	—
24	C₃³×Q₈	D₄	1	1	yes	characteristic
25	C₃³×Q₈	C₂³	1	6	yes	—
26	C₃³×D₄	—	2	6	no	—
27	C₃³×D₄	D₄	1	3	yes	—
28	C₃³×D₄	C₂³	1	2	yes	—
29	C₂×C₃³×C₄	—	2	6	no	—
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×#578	C₃²	1	13	yes	direct product factor
35	C₃²×#41	—	2	78	no	—
36	C₃²×#41	C₂²×C₃	1	78	yes	—
37	C₃²×#41	C₂²×C₃	1	13	yes	—
38	C₂×C₃²×Q₈	—	2	26	no	—
39	C₂×C₃²×Q₈	C₂²×C₃	1	78	yes	—
40	C₃²×Q₁₆	C₂²×C₃	1	39	yes	—
41	C₃²×Q₁₆	C₂²×C₃	1	78	yes	—
42	C₃²×SD₁₆	C₂²×C₃	1	78	yes	—
43	C₃²×D₈	C₂²×C₃	1	13	yes	—
44	C₃²×M₁₆	C₂²×C₃	1	39	yes	—
45	C₂×C₃²×C₈	C₂²×C₃	1	39	yes	—
46	C₂²×C₃³	—	2	2	no	—
47	C₂²×C₃³	C₂×D₄	1	3	yes	—
48	C₃³×C₄	—	2	6	no	—
49	C₃³×C₄	C₂×D₄	1	3	yes	—
50	C₃³×C₄	C₂⁴	1	1	yes	characteristic
51	C₃×2₋¹⁺⁴	C₂×C₃²	1	26	yes	complemented
52	C₃×#137	C₂×C₃²	1	78	yes	complemented
53	C₃×#135	C₂×C₃²	1	39	yes	complemented
54	C₂×C₃×Q₁₆	C₂×C₃²	1	39	yes	complemented
55	C₃×#131	C₂×C₃²	1	13	yes	complemented
56	C₃²×Q₈	—	2	78	no	—
57	C₃²×Q₈	C₃×D₄	1	13	yes	—
58	C₃²×Q₈	C₂³×C₃	1	78	yes	—
59	C₃²×D₄	—	2	78	no	—
60	C₃²×D₄	C₃×D₄	1	39	yes	—
61	C₃²×D₄	C₂³×C₃	1	26	yes	—
62	C₂×C₃²×C₄	—	2	78	no	—
63	C₂×C₃²×C₄	—	2	78	no	—
64	C₂×C₃²×C₄	C₂³×C₃	1	39	yes	—
65	C₃²×C₈	C₂³×C₃	1	13	yes	—
66	C₃²×C₈	C₂³×C₃	1	39	yes	—
67	#578	C₃³	1	1	yes	characteristic, direct product factor, Sylow
68	C₂×C₃³	—	2	3	no	—
69	C₂×C₃³	—	4	2	no	—
70	C₂×C₃³	C₂²×D₄	1	1	yes	characteristic
71	C₃×#41	—	2	78	no	—
72	C₃×#41	C₂²×C₃²	1	78	yes	—
73	C₃×#41	C₂²×C₃²	1	13	yes	—
74	C₂×C₃×Q₈	—	2	26	no	—
75	C₂×C₃×Q₈	C₂²×C₃²	1	78	yes	—
76	C₃×Q₁₆	C₂²×C₃²	1	39	yes	—
77	C₃×Q₁₆	C₂²×C₃²	1	78	yes	—
78	C₃×SD₁₆	C₂²×C₃²	1	78	yes	—
79	C₃×D₈	C₂²×C₃²	1	13	yes	—
80	C₃×M₁₆	C₂²×C₃²	1	39	yes	—
81	C₂×C₃×C₈	C₂²×C₃²	1	39	yes	—
82	C₂²×C₃²	—	2	26	no	—
83	C₂²×C₃²	C₂×C₃×D₄	1	39	yes	—
84	C₃²×C₄	—	2	78	no	—
85	C₃²×C₄	C₂×C₃×D₄	1	39	yes	—
86	C₃²×C₄	C₂⁴×C₃	1	13	yes	—
87	2₋¹⁺⁴	C₂×C₃³	1	2	yes	complemented
88	#137	C₂×C₃³	1	6	yes	complemented
89	#135	C₂×C₃³	1	3	yes	complemented
90	C₂×Q₁₆	C₂×C₃³	1	3	yes	complemented
91	#131	C₂×C₃³	1	1	yes	characteristic, complemented
92	C₃³	#578	1	1	yes	characteristic, direct product factor, Sylow
93	C₃×Q₈	—	2	78	no	—
94	C₃×Q₈	C₃²×D₄	1	13	yes	—
95	C₃×Q₈	C₂³×C₃²	1	78	yes	—
96	C₃×D₄	—	2	78	no	—
97	C₃×D₄	C₃²×D₄	1	39	yes	—
98	C₃×D₄	C₂³×C₃²	1	26	yes	—
99	C₂×C₃×C₄	—	2	78	no	—
100	C₂×C₃×C₄	—	2	78	no	—
101	C₂×C₃×C₄	C₂³×C₃²	1	39	yes	—
102	C₃×C₈	C₂³×C₃²	1	13	yes	—
103	C₃×C₈	C₂³×C₃²	1	39	yes	—
104	C₂×C₃²	—	2	39	no	—
105	C₂×C₃²	—	4	26	no	—
106	C₂×C₃²	C₂²×C₃×D₄	1	13	yes	—
107	#41	—	2	6	no	—
108	#41	C₂²×C₃³	1	6	yes	—
109	#41	C₂²×C₃³	1	1	yes	characteristic
110	C₂×Q₈	—	2	2	no	—
111	C₂×Q₈	C₂²×C₃³	1	6	yes	—
112	Q₁₆	C₂²×C₃³	1	3	yes	—
113	Q₁₆	C₂²×C₃³	1	6	yes	—
114	SD₁₆	C₂²×C₃³	1	6	yes	—
115	D₈	C₂²×C₃³	1	1	yes	characteristic
116	M₁₆	C₂²×C₃³	1	3	yes	—
117	C₂×C₈	C₂²×C₃³	1	3	yes	—
118	C₂²×C₃	—	2	26	no	—
119	C₂²×C₃	C₂×C₃²×D₄	1	39	yes	—
120	C₃×C₄	—	2	78	no	—
121	C₃×C₄	C₂×C₃²×D₄	1	39	yes	—
122	C₃×C₄	C₂⁴×C₃²	1	13	yes	—
123	C₃²	C₃×#578	1	13	yes	direct product factor
124	Q₈	—	2	6	no	—
125	Q₈	C₃³×D₄	1	1	yes	characteristic
126	Q₈	C₂³×C₃³	1	6	yes	—
127	D₄	—	2	6	no	—
128	D₄	C₃³×D₄	1	3	yes	—
129	D₄	C₂³×C₃³	1	2	yes	—
130	C₂×C₄	—	2	6	no	—
131	C₂×C₄	—	2	6	no	—
132	C₂×C₄	C₂³×C₃³	1	3	yes	—
133	C₈	C₂³×C₃³	1	1	yes	characteristic
134	C₈	C₂³×C₃³	1	3	yes	—
135	C₂×C₃	—	2	39	no	—
136	C₂×C₃	—	4	26	no	—
137	C₂×C₃	C₂²×C₃²×D₄	1	13	yes	—
138	C₂²	—	2	2	no	—
139	C₂²	C₂×C₃³×D₄	1	3	yes	—
140	C₄	—	2	6	no	—
141	C₄	C₂×C₃³×D₄	1	3	yes	—
142	C₄	C₂⁴×C₃³	1	1	yes	characteristic
143	C₃	C₃²×#578	1	13	yes	direct product factor
144	C₂	—	2	3	no	—
145	C₂	—	4	2	no	—
146	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. C₃³×2₋¹⁺⁴	C₂	semidirect
2. C₃³×#137	C₂	semidirect
3. C₃³×#135	C₂	semidirect
4. C₂×C₃³×Q₁₆	C₂	semidirect
5. C₃³×#131	C₂	semidirect
6. C₃²×#578	C₃	direct (N × H)
18. C₃²×2₋¹⁺⁴	C₂×C₃	semidirect
19. C₃²×#137	C₂×C₃	semidirect
20. C₃²×#135	C₂×C₃	semidirect
21. C₂×C₃²×Q₁₆	C₂×C₃	semidirect
22. C₃²×#131	C₂×C₃	semidirect
34. C₃×#578	C₃²	direct (N × H)
51. C₃×2₋¹⁺⁴	C₂×C₃²	semidirect
52. C₃×#137	C₂×C₃²	semidirect
53. C₃×#135	C₂×C₃²	semidirect
54. C₂×C₃×Q₁₆	C₂×C₃²	semidirect
55. C₃×#131	C₂×C₃²	semidirect
67. #578	C₃³	direct (N × H)
87. 2₋¹⁺⁴	C₂×C₃³	semidirect
88. #137	C₂×C₃³	semidirect
89. #135	C₂×C₃³	semidirect
90. C₂×Q₁₆	C₂×C₃³	semidirect
91. #131	C₂×C₃³	semidirect
92. C₃³	#578	direct (N × H)
123. C₃²	C₃×#578	direct (N × H)
143. C₃	C₃²×#578	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃³×2₋¹⁺⁴	2
C₃³×#137	2
C₃³×#135	2
C₂×C₃³×Q₁₆	2
C₃³×#131	2
C₃²×#578	3

Subgroups — C33×#578 #13180033

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃³×#578 #13180033