Subgroups — C₃×Dic₃×M₁₆ #10592214

Order 576.

318 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄×Dic₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#208	C₂	1	1	yes	maximal, characteristic
4	C₃×#815	C₂	1	2	yes	maximal, complemented
5	C₃×C₈×Dic₃	C₂	1	2	yes	maximal, complemented
6	C₃×C₄×M₁₆	—	3	1	no	maximal
7	Dic₃×M₁₆	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂²×C₃²×C₄	C₂²	1	1	yes	characteristic
9	C₂²×C₃×Dic₃	C₄	1	2	yes	—
10	C₃²×M₁₆	C₄	1	4	yes	complemented
11	C₂×C₃²×C₈	C₂²	1	2	yes	—
12	C₃×C₄×Dic₃	C₄	1	2	yes	—
13	C₃×C₄×Dic₃	C₂²	1	1	yes	characteristic
14	C₃×C₄×Dic₃	C₂²	1	1	yes	characteristic
15	C₃×#208	C₄	1	4	yes	complemented
16	C₂×C₃×C₃⋊C₈	C₂²	1	2	yes	—
17	C₂×C₃×M₁₆	—	2	1	no	—
18	C₂×C₃×M₁₆	—	3	1	no	—
19	C₂×C₃×M₁₆	D₃	1	1	yes	characteristic
20	C₂×C₃×M₁₆	C₂×C₃	1	1	yes	characteristic
21	C₂×C₃×C₄²	—	3	1	no	—
22	C₂×C₄×Dic₃	C₂×C₃	1	1	yes	characteristic
23	C₂×#208	C₂×C₃	1	1	yes	characteristic
24	C₃×#97	—	3	2	no	—
25	C₃×C₄×C₈	—	3	2	no	—
26	#815	C₂×C₃	1	2	yes	complemented
27	C₈×Dic₃	C₂×C₃	1	2	yes	complemented
28	C₂³×C₃²	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃²×C₄	C₂×C₄	1	2	yes	—
31	C₂×C₃²×C₄	C₂×C₄	1	2	yes	—
32	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
33	C₂×C₃×Dic₃	—	2	4	no	—
34	C₂×C₃×Dic₃	C₂×C₄	1	2	yes	—
35	C₂×C₃×Dic₃	C₂×C₄	1	2	yes	—
36	C₃²×C₈	C₂×C₄	1	4	yes	complemented
37	C₃×C₃⋊C₈	C₂×C₄	1	4	yes	complemented
38	C₄×M₁₆	—	3	1	no	Sylow
39	C₂²×C₃×C₄	—	2	1	no	—
40	C₂²×C₃×C₄	—	3	2	no	—
41	C₂²×C₃×C₄	C₂×D₃	1	1	yes	characteristic
42	C₂²×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
43	C₂²×Dic₃	C₃×C₄	1	2	yes	—
44	C₃×M₁₆	—	2	4	no	—
45	C₃×M₁₆	—	3	4	no	—
46	C₃×M₁₆	Dic₃	1	4	yes	direct product factor
47	C₃×M₁₆	C₃×C₄	1	4	yes	complemented
48	C₂×C₃×C₈	—	2	2	no	—
49	C₂×C₃×C₈	—	3	2	no	—
50	C₂×C₃×C₈	C₂×D₃	1	2	yes	—
51	C₂×C₃×C₈	C₂²×C₃	1	2	yes	—
52	C₃×C₄²	—	3	2	no	—
53	C₃×C₄²	—	3	1	no	—
54	C₃×C₄²	—	3	1	no	—
55	C₄×Dic₃	C₃×C₄	1	2	yes	—
56	C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
57	C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
58	#208	C₃×C₄	1	4	yes	complemented
59	C₂×C₃⋊C₈	C₂²×C₃	1	2	yes	—
60	C₂²×C₃²	—	2	2	no	—
61	C₂²×C₃²	C₄²	1	2	yes	—
62	C₂²×C₃²	C₂²×C₄	1	1	yes	characteristic
63	C₃²×C₄	C₄²	1	2	yes	—
64	C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
65	C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
66	C₃×Dic₃	—	2	2	no	—
67	C₃×Dic₃	M₁₆	1	4	yes	direct product factor
68	C₂×M₁₆	—	3	1	no	—
69	C₂×M₁₆	C₃×D₃	1	1	yes	characteristic
70	C₂×C₄²	—	3	1	no	—
71	#97	—	3	2	no	—
72	C₄×C₈	—	3	2	no	—
73	C₂³×C₃	—	2	1	no	—
74	C₂³×C₃	C₄×D₃	1	1	yes	characteristic
75	C₂³×C₃	C₂×C₃×C₄	1	1	yes	characteristic
76	C₂×C₃×C₄	—	2	1	no	—
77	C₂×C₃×C₄	—	2	2	no	—
78	C₂×C₃×C₄	—	2	2	no	—
79	C₂×C₃×C₄	—	2	1	no	—
80	C₂×C₃×C₄	—	3	2	no	—
81	C₂×C₃×C₄	—	3	2	no	—
82	C₂×C₃×C₄	—	6	4	no	—
83	C₂×C₃×C₄	C₄×D₃	1	1	yes	characteristic
84	C₂×C₃×C₄	C₄×D₃	1	2	yes	—
85	C₂×C₃×C₄	C₂×Dic₃	1	2	yes	—
86	C₂×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
87	C₂×C₃×C₄	C₂×C₃×C₄	1	2	yes	—
88	C₂×C₃×C₄	C₂×C₃×C₄	1	2	yes	—
89	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
90	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
91	C₂×Dic₃	—	2	4	no	—
92	C₂×Dic₃	C₂×C₃×C₄	1	2	yes	—
93	C₂×Dic₃	C₂×C₃×C₄	1	2	yes	—
94	C₃×C₈	—	2	4	no	—
95	C₃×C₈	—	3	4	no	—
96	C₃×C₈	C₂×Dic₃	1	4	yes	complemented
97	C₃×C₈	C₂×C₃×C₄	1	4	yes	complemented
98	C₃⋊C₈	C₂×C₃×C₄	1	4	yes	complemented
99	C₂×C₃²	—	2	2	no	—
100	C₂×C₃²	C₂×C₄²	1	1	yes	characteristic
101	C₂×C₃²	C₂×M₁₆	1	2	yes	—
102	C₂²×C₄	—	3	2	no	—
103	C₂²×C₄	C₂×C₃×D₃	1	1	yes	characteristic
104	M₁₆	—	3	4	no	—
105	M₁₆	C₃×Dic₃	1	4	yes	direct product factor
106	C₂×C₈	—	3	2	no	—
107	C₂×C₈	C₂×C₃×D₃	1	2	yes	—
108	C₄²	—	3	2	no	—
109	C₄²	—	3	1	no	—
110	C₄²	—	3	1	no	—
111	C₂²×C₃	—	2	2	no	—
112	C₂²×C₃	—	2	2	no	—
113	C₂²×C₃	—	2	2	no	—
114	C₂²×C₃	—	2	1	no	—
115	C₂²×C₃	—	4	2	no	—
116	C₂²×C₃	C₄×Dic₃	1	2	yes	—
117	C₂²×C₃	C₃×C₄²	1	2	yes	—
118	C₂²×C₃	C₂×C₄×D₃	1	1	yes	characteristic
119	C₂²×C₃	C₂²×C₃×C₄	1	1	yes	characteristic
120	C₃×C₄	—	2	2	no	—
121	C₃×C₄	—	2	1	no	—
122	C₃×C₄	—	2	1	no	—
123	C₃×C₄	—	3	4	no	—
124	C₃×C₄	—	6	2	no	—
125	C₃×C₄	C₄×Dic₃	1	2	yes	—
126	C₃×C₄	C₃×C₄²	1	2	yes	—
127	C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
128	C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
129	C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
130	C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
131	Dic₃	—	2	2	no	—
132	Dic₃	C₃×M₁₆	1	4	yes	direct product factor
133	C₃²	C₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
134	C₂³	C₃×C₄×D₃	1	1	yes	characteristic
135	C₂×C₄	—	3	2	no	—
136	C₂×C₄	—	3	2	no	—
137	C₂×C₄	—	6	4	no	—
138	C₂×C₄	C₃×C₄×D₃	1	1	yes	characteristic
139	C₂×C₄	C₃×C₄×D₃	1	2	yes	—
140	C₂×C₄	C₂×C₃×Dic₃	1	2	yes	—
141	C₂×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
142	C₈	—	3	4	no	—
143	C₈	C₂×C₃×Dic₃	1	4	yes	complemented
144	C₂×C₃	—	2	2	no	—
145	C₂×C₃	—	2	2	no	—
146	C₂×C₃	—	2	2	no	—
147	C₂×C₃	—	2	1	no	—
148	C₂×C₃	—	4	2	no	—
149	C₂×C₃	D₃×M₁₆	1	2	yes	—
150	C₂×C₃	C₂×C₄×Dic₃	1	1	yes	characteristic
151	C₂×C₃	C₂×C₃×C₄²	1	1	yes	characteristic
152	C₂×C₃	C₂×C₃×M₁₆	1	2	yes	—
153	C₂²	—	2	2	no	—
154	C₂²	C₃×C₄×Dic₃	1	2	yes	—
155	C₂²	C₂×C₃×C₄×D₃	1	1	yes	characteristic
156	C₄	—	3	4	no	—
157	C₄	—	6	2	no	—
158	C₄	C₃×C₄×Dic₃	1	2	yes	—
159	C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
160	C₄	C₂²×C₃×Dic₃	1	1	yes	characteristic
161	C₃	—	2	1	no	—
162	C₃	Dic₃×M₁₆	1	1	yes	characteristic, direct product factor
163	C₃	C₃×C₄×M₁₆	1	1	yes	characteristic, complemented
164	C₂	—	2	2	no	—
165	C₂	C₃×D₃×M₁₆	1	2	yes	—
166	C₂	C₂×C₃×C₄×Dic₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂²×D₃
commutator	C₂×C₃	C₂×C₃×C₄²
Frattini	C₂×C₄	C₂²×C₃×D₃
Fitting	C₂×C₃²×M₁₆	C₂
socle	C₂²×C₃²	C₂²×C₄
prime radical	C₂³×C₃²	C₂×C₄
cosocle	C₂×C₃×C₄	C₂³×C₃
hypercenter	C₂×C₃×M₁₆	D₃
hypocenter	C₃	C₃×C₄×M₁₆
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×M₁₆

Complements

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

N	H ≅ G/N	type
4. C₃×#815	C₂	semidirect
5. C₃×C₈×Dic₃	C₂	semidirect
7. Dic₃×M₁₆	C₃	direct (N × H)
10. C₃²×M₁₆	C₄	semidirect
15. C₃×#208	C₄	semidirect
26. #815	C₂×C₃	semidirect
27. C₈×Dic₃	C₂×C₃	semidirect
36. C₃²×C₈	C₂×C₄	semidirect
37. C₃×C₃⋊C₈	C₂×C₄	semidirect
46. C₃×M₁₆	Dic₃	direct (N × H)
47. C₃×M₁₆	C₃×C₄	semidirect
58. #208	C₃×C₄	semidirect
67. C₃×Dic₃	M₁₆	direct (N × H)
96. C₃×C₈	C₂×Dic₃	semidirect
97. C₃×C₈	C₂×C₃×C₄	semidirect
98. C₃⋊C₈	C₂×C₃×C₄	semidirect
105. M₁₆	C₃×Dic₃	direct (N × H)
132. Dic₃	C₃×M₁₆	direct (N × H)
133. C₃²	C₄×M₁₆	semidirect
143. C₈	C₂×C₃×Dic₃	semidirect
162. C₃	Dic₃×M₁₆	direct (N × H)
163. C₃	C₃×C₄×M₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×M₁₆	2
C₂×C₃×C₄×Dic₃	2
C₂×C₃×#208	2
C₃×#815	2
C₃×C₈×Dic₃	2
C₃×C₄×M₁₆	3
Dic₃×M₁₆	3

Subgroups — C3×Dic3×M16 #10592214

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×Dic₃×M₁₆ #10592214