Subgroups — C₂²×C₃³×M₁₆ #13180021

Order 1728.

4172 conjugacy classes of subgroups.

The subgroup lattice has 124 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃³×C₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃³×M₁₆	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₃³×C₈	C₂	1	2	yes	maximal, complemented
4	C₂²×C₃²×M₁₆	C₃	1	13	yes	maximal, direct product factor
5	C₂⁴×C₃³	C₄	1	1	yes	characteristic
6	C₂²×C₃³×C₄	C₄	1	1	yes	characteristic
7	C₂²×C₃³×C₄	C₄	1	6	yes	—
8	C₂²×C₃³×C₄	C₂²	1	6	yes	—
9	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
10	C₃³×M₁₆	C₂²	1	16	yes	direct product factor
11	C₂×C₃³×C₈	C₂²	1	12	yes	complemented
12	C₂³×C₃²×C₄	C₂×C₃	1	13	yes	—
13	C₂×C₃²×M₁₆	C₂×C₃	1	156	yes	direct product factor
14	C₂²×C₃²×C₈	C₂×C₃	1	26	yes	complemented
15	C₂³×C₃³	—	2	4	no	—
16	C₂³×C₃³	C₂×C₄	1	6	yes	—
17	C₂³×C₃³	C₂×C₄	1	1	yes	characteristic
18	C₂×C₃³×C₄	C₂×C₄	1	12	yes	—
19	C₂×C₃³×C₄	C₂×C₄	1	6	yes	—
20	C₂×C₃³×C₄	C₂×C₄	1	3	yes	—
21	C₂×C₃³×C₄	C₂³	1	3	yes	—
22	C₂×C₃³×C₄	C₂³	1	4	yes	—
23	C₃³×C₈	C₂³	1	8	yes	complemented
24	C₂²×C₃×M₁₆	C₃²	1	13	yes	direct product factor
25	C₂⁴×C₃²	C₃×C₄	1	13	yes	—
26	C₂²×C₃²×C₄	C₃×C₄	1	78	yes	—
27	C₂²×C₃²×C₄	C₃×C₄	1	13	yes	—
28	C₂²×C₃²×C₄	C₂²×C₃	1	13	yes	—
29	C₂²×C₃²×C₄	C₂²×C₃	1	78	yes	—
30	C₃²×M₁₆	C₂²×C₃	1	208	yes	direct product factor
31	C₂×C₃²×C₈	C₂²×C₃	1	156	yes	complemented
32	C₂²×C₃³	—	2	12	no	—
33	C₂²×C₃³	M₁₆	1	4	yes	direct product factor
34	C₂²×C₃³	C₂²×C₄	1	4	yes	—
35	C₂²×C₃³	C₂²×C₄	1	3	yes	—
36	C₃³×C₄	C₂²×C₄	1	4	yes	—
37	C₃³×C₄	C₂²×C₄	1	3	yes	—
38	C₃³×C₄	C₂⁴	1	1	yes	characteristic
39	C₂³×C₃×C₄	C₂×C₃²	1	13	yes	—
40	C₂×C₃×M₁₆	C₂×C₃²	1	156	yes	direct product factor
41	C₂²×C₃×C₈	C₂×C₃²	1	26	yes	complemented
42	C₂³×C₃²	—	2	52	no	—
43	C₂³×C₃²	C₂×C₃×C₄	1	78	yes	—
44	C₂³×C₃²	C₂×C₃×C₄	1	13	yes	—
45	C₂×C₃²×C₄	C₂×C₃×C₄	1	156	yes	—
46	C₂×C₃²×C₄	C₂×C₃×C₄	1	78	yes	—
47	C₂×C₃²×C₄	C₂×C₃×C₄	1	39	yes	—
48	C₂×C₃²×C₄	C₂³×C₃	1	39	yes	—
49	C₂×C₃²×C₄	C₂³×C₃	1	52	yes	—
50	C₃²×C₈	C₂³×C₃	1	104	yes	complemented
51	C₂²×M₁₆	C₃³	1	1	yes	characteristic, direct product factor, Sylow
52	C₂×C₃³	—	2	4	no	—
53	C₂×C₃³	C₂×M₁₆	1	6	yes	direct product factor
54	C₂×C₃³	C₂³×C₄	1	1	yes	characteristic
55	C₂⁴×C₃	C₃²×C₄	1	13	yes	—
56	C₂²×C₃×C₄	C₃²×C₄	1	78	yes	—
57	C₂²×C₃×C₄	C₃²×C₄	1	13	yes	—
58	C₂²×C₃×C₄	C₂²×C₃²	1	13	yes	—
59	C₂²×C₃×C₄	C₂²×C₃²	1	78	yes	—
60	C₃×M₁₆	C₂²×C₃²	1	208	yes	direct product factor
61	C₂×C₃×C₈	C₂²×C₃²	1	156	yes	complemented
62	C₂²×C₃²	—	2	156	no	—
63	C₂²×C₃²	C₃×M₁₆	1	52	yes	direct product factor
64	C₂²×C₃²	C₂²×C₃×C₄	1	52	yes	—
65	C₂²×C₃²	C₂²×C₃×C₄	1	39	yes	—
66	C₃²×C₄	C₂²×C₃×C₄	1	39	yes	—
67	C₃²×C₄	C₂²×C₃×C₄	1	52	yes	—
68	C₃²×C₄	C₂⁴×C₃	1	13	yes	—
69	C₂³×C₄	C₂×C₃³	1	1	yes	characteristic
70	C₂×M₁₆	C₂×C₃³	1	12	yes	direct product factor
71	C₂²×C₈	C₂×C₃³	1	2	yes	complemented
72	C₃³	C₂²×M₁₆	1	1	yes	characteristic, direct product factor, Sylow
73	C₂³×C₃	—	2	52	no	—
74	C₂³×C₃	C₂×C₃²×C₄	1	78	yes	—
75	C₂³×C₃	C₂×C₃²×C₄	1	13	yes	—
76	C₂×C₃×C₄	C₂×C₃²×C₄	1	156	yes	—
77	C₂×C₃×C₄	C₂×C₃²×C₄	1	39	yes	—
78	C₂×C₃×C₄	C₂×C₃²×C₄	1	78	yes	—
79	C₂×C₃×C₄	C₂³×C₃²	1	39	yes	—
80	C₂×C₃×C₄	C₂³×C₃²	1	52	yes	—
81	C₃×C₈	C₂³×C₃²	1	104	yes	complemented
82	C₂×C₃²	—	2	52	no	—
83	C₂×C₃²	C₂×C₃×M₁₆	1	78	yes	direct product factor
84	C₂×C₃²	C₂³×C₃×C₄	1	13	yes	—
85	C₂⁴	C₃³×C₄	1	1	yes	characteristic
86	C₂²×C₄	C₃³×C₄	1	6	yes	—
87	C₂²×C₄	C₃³×C₄	1	1	yes	characteristic
88	C₂²×C₄	C₂²×C₃³	1	1	yes	characteristic
89	C₂²×C₄	C₂²×C₃³	1	6	yes	—
90	M₁₆	C₂²×C₃³	1	16	yes	direct product factor
91	C₂×C₈	C₂²×C₃³	1	12	yes	complemented
92	C₂²×C₃	—	2	156	no	—
93	C₂²×C₃	C₃²×M₁₆	1	52	yes	direct product factor
94	C₂²×C₃	C₂²×C₃²×C₄	1	52	yes	—
95	C₂²×C₃	C₂²×C₃²×C₄	1	39	yes	—
96	C₃×C₄	C₂²×C₃²×C₄	1	39	yes	—
97	C₃×C₄	C₂²×C₃²×C₄	1	52	yes	—
98	C₃×C₄	C₂⁴×C₃²	1	13	yes	—
99	C₃²	C₂²×C₃×M₁₆	1	13	yes	direct product factor
100	C₂³	—	2	4	no	—
101	C₂³	C₂×C₃³×C₄	1	6	yes	—
102	C₂³	C₂×C₃³×C₄	1	1	yes	characteristic
103	C₂×C₄	C₂×C₃³×C₄	1	12	yes	—
104	C₂×C₄	C₂×C₃³×C₄	1	3	yes	—
105	C₂×C₄	C₂×C₃³×C₄	1	6	yes	—
106	C₂×C₄	C₂³×C₃³	1	3	yes	—
107	C₂×C₄	C₂³×C₃³	1	4	yes	—
108	C₈	C₂³×C₃³	1	8	yes	complemented
109	C₂×C₃	—	2	52	no	—
110	C₂×C₃	C₂×C₃²×M₁₆	1	78	yes	direct product factor
111	C₂×C₃	C₂³×C₃²×C₄	1	13	yes	—
112	C₂²	—	2	12	no	—
113	C₂²	C₃³×M₁₆	1	4	yes	direct product factor
114	C₂²	C₂²×C₃³×C₄	1	4	yes	—
115	C₂²	C₂²×C₃³×C₄	1	3	yes	—
116	C₄	C₂²×C₃³×C₄	1	3	yes	—
117	C₄	C₂²×C₃³×C₄	1	4	yes	—
118	C₄	C₂⁴×C₃³	1	1	yes	characteristic
119	C₃	C₂²×C₃²×M₁₆	1	13	yes	direct product factor
120	C₂	—	2	4	no	—
121	C₂	C₂×C₃³×M₁₆	1	6	yes	direct product factor
122	C₂	C₂³×C₃³×C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃³×C₄	C₂²
commutator	C₂	C₂³×C₃³×C₄
Frattini, cosocle	C₄	C₂⁴×C₃³
socle	C₂³×C₃³	C₂×C₄
prime radical	C₂⁴×C₃³	C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃³×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃³×M₁₆	C₂	direct (N × H)
3. C₂²×C₃³×C₈	C₂	semidirect
4. C₂²×C₃²×M₁₆	C₃	direct (N × H)
10. C₃³×M₁₆	C₂²	direct (N × H)
11. C₂×C₃³×C₈	C₂²	semidirect
13. C₂×C₃²×M₁₆	C₂×C₃	direct (N × H)
14. C₂²×C₃²×C₈	C₂×C₃	semidirect
23. C₃³×C₈	C₂³	semidirect
24. C₂²×C₃×M₁₆	C₃²	direct (N × H)
30. C₃²×M₁₆	C₂²×C₃	direct (N × H)
31. C₂×C₃²×C₈	C₂²×C₃	semidirect
33. C₂²×C₃³	M₁₆	direct (N × H)
40. C₂×C₃×M₁₆	C₂×C₃²	direct (N × H)
41. C₂²×C₃×C₈	C₂×C₃²	semidirect
50. C₃²×C₈	C₂³×C₃	semidirect
51. C₂²×M₁₆	C₃³	direct (N × H)
53. C₂×C₃³	C₂×M₁₆	direct (N × H)
60. C₃×M₁₆	C₂²×C₃²	direct (N × H)
61. C₂×C₃×C₈	C₂²×C₃²	semidirect
63. C₂²×C₃²	C₃×M₁₆	direct (N × H)
70. C₂×M₁₆	C₂×C₃³	direct (N × H)
71. C₂²×C₈	C₂×C₃³	semidirect
72. C₃³	C₂²×M₁₆	direct (N × H)
81. C₃×C₈	C₂³×C₃²	semidirect
83. C₂×C₃²	C₂×C₃×M₁₆	direct (N × H)
90. M₁₆	C₂²×C₃³	direct (N × H)
91. C₂×C₈	C₂²×C₃³	semidirect
93. C₂²×C₃	C₃²×M₁₆	direct (N × H)
99. C₃²	C₂²×C₃×M₁₆	direct (N × H)
108. C₈	C₂³×C₃³	semidirect
110. C₂×C₃	C₂×C₃²×M₁₆	direct (N × H)
113. C₂²	C₃³×M₁₆	direct (N × H)
119. C₃	C₂²×C₃²×M₁₆	direct (N × H)
121. C₂	C₂×C₃³×M₁₆	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×C₃³×C₄	2
C₂×C₃³×M₁₆	2
C₂²×C₃³×C₈	2
C₂²×C₃²×M₁₆	3

Subgroups — C22×C33×M16 #13180021

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃³×M₁₆ #13180021