Subgroups — C₂×C₃²×D₃×C₁₆ #13162335

Order 1728.

700 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×D₃×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃³×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃²×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃²×D₃×C₁₆	C₂	1	4	yes	maximal, direct product factor
5	C₂²×C₃²×C₁₆	—	3	1	no	maximal
6	C₂×C₃×D₃×C₁₆	C₃	1	4	yes	maximal, direct product factor
7	C₂×C₃²×C₄×D₃	C₄	1	1	yes	characteristic
8	C₂×C₃³×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₃²×C₃⋊C₈	C₄	1	1	yes	characteristic
10	C₃²×D₃×C₈	C₄	1	2	yes	—
11	C₃²×D₃×C₈	C₂²	1	2	yes	—
12	C₃³×C₁₆	C₂²	1	2	yes	complemented
13	C₃²×C₃⋊C₁₆	C₂²	1	2	yes	complemented
14	C₂²×C₃²×C₈	—	3	1	no	—
15	C₂×C₃×D₃×C₈	C₂×C₃	1	4	yes	—
16	C₂×C₃²×C₁₆	—	2	4	no	—
17	C₂×C₃²×C₁₆	—	3	1	no	—
18	C₂×C₃²×C₁₆	—	3	4	no	—
19	C₂×C₃²×C₁₆	D₃	1	1	yes	characteristic, direct product factor
20	C₂×C₃²×C₁₆	C₂×C₃	1	4	yes	complemented
21	C₂×C₃×C₃⋊C₁₆	C₂×C₃	1	4	yes	complemented
22	C₃×D₃×C₁₆	C₂×C₃	1	16	yes	direct product factor
23	C₂²×C₃²×D₃	C₈	1	1	yes	characteristic
24	C₂×C₃³×C₄	C₂×C₄	1	1	yes	characteristic
25	C₂×C₃²×Dic₃	C₈	1	1	yes	characteristic
26	C₃²×C₄×D₃	C₈	1	2	yes	—
27	C₃²×C₄×D₃	C₂×C₄	1	2	yes	—
28	C₃³×C₈	C₂×C₄	1	1	yes	characteristic
29	C₃³×C₈	C₂³	1	1	yes	characteristic
30	C₃²×C₃⋊C₈	C₂×C₄	1	2	yes	—
31	C₂²×C₃×C₁₆	—	3	4	no	—
32	C₂×D₃×C₁₆	C₃²	1	1	yes	characteristic, direct product factor
33	C₂²×C₃²×C₄	—	3	1	no	—
34	C₂×C₃×C₄×D₃	C₃×C₄	1	4	yes	—
35	C₂×C₃²×C₈	—	2	4	no	—
36	C₂×C₃²×C₈	—	3	1	no	—
37	C₂×C₃²×C₈	—	3	2	no	—
38	C₂×C₃²×C₈	—	3	2	no	—
39	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
40	C₂×C₃²×C₈	C₂²×C₃	1	4	yes	—
41	C₂×C₃×C₃⋊C₈	C₃×C₄	1	4	yes	—
42	C₃×D₃×C₈	C₃×C₄	1	8	yes	—
43	C₃×D₃×C₈	C₂²×C₃	1	8	yes	—
44	C₃²×C₁₆	—	2	8	no	—
45	C₃²×C₁₆	—	3	2	no	—
46	C₃²×C₁₆	C₂×D₃	1	2	yes	direct product factor
47	C₃²×C₁₆	C₂²×C₃	1	8	yes	complemented
48	C₃×C₃⋊C₁₆	C₂²×C₃	1	8	yes	complemented
49	C₂²×C₃³	C₂×C₈	1	1	yes	characteristic
50	C₂×C₃²×D₃	C₁₆	1	4	yes	direct product factor
51	C₂×C₃²×D₃	C₂×C₈	1	2	yes	—
52	C₃³×C₄	C₂×C₈	1	1	yes	characteristic
53	C₃³×C₄	C₂²×C₄	1	1	yes	characteristic
54	C₃²×Dic₃	C₂×C₈	1	2	yes	—
55	C₂²×C₃×C₈	—	3	4	no	—
56	C₂×D₃×C₈	C₂×C₃²	1	1	yes	characteristic
57	C₂×C₃×C₁₆	—	2	4	no	—
58	C₂×C₃×C₁₆	—	3	4	no	—
59	C₂×C₃×C₁₆	—	3	16	no	—
60	C₂×C₃×C₁₆	C₃×D₃	1	4	yes	direct product factor
61	C₂×C₃×C₁₆	C₂×C₃²	1	1	yes	characteristic, complemented
62	C₂×C₃⋊C₁₆	C₂×C₃²	1	1	yes	characteristic, complemented
63	D₃×C₁₆	C₂×C₃²	1	4	yes	direct product factor
64	C₂³×C₃²	—	3	1	no	—
65	C₂²×C₃×D₃	C₃×C₈	1	4	yes	—
66	C₂×C₃²×C₄	—	2	4	no	—
67	C₂×C₃²×C₄	—	3	1	no	—
68	C₂×C₃²×C₄	—	3	2	no	—
69	C₂×C₃²×C₄	—	3	2	no	—
70	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
71	C₂×C₃²×C₄	C₂×C₃×C₄	1	4	yes	—
72	C₂×C₃×Dic₃	C₃×C₈	1	4	yes	—
73	C₃×C₄×D₃	C₃×C₈	1	8	yes	—
74	C₃×C₄×D₃	C₂×C₃×C₄	1	8	yes	—
75	C₃²×C₈	—	2	4	no	—
76	C₃²×C₈	—	2	4	no	—
77	C₃²×C₈	—	3	2	no	—
78	C₃²×C₈	C₄×D₃	1	1	yes	characteristic
79	C₃²×C₈	C₂×C₃×C₄	1	4	yes	—
80	C₃²×C₈	C₂²×D₃	1	1	yes	characteristic
81	C₃²×C₈	C₂³×C₃	1	4	yes	—
82	C₃×C₃⋊C₈	C₂×C₃×C₄	1	8	yes	—
83	C₂²×C₁₆	—	3	1	no	Sylow
84	C₂×C₃³	C₂×C₁₆	1	2	yes	complemented
85	C₂×C₃³	C₂²×C₈	1	1	yes	characteristic
86	C₃²×D₃	C₂×C₁₆	1	4	yes	direct product factor
87	C₂²×C₃×C₄	—	3	4	no	—
88	C₂×C₄×D₃	C₃²×C₄	1	1	yes	characteristic
89	C₂×C₃×C₈	—	2	4	no	—
90	C₂×C₃×C₈	—	3	4	no	—
91	C₂×C₃×C₈	—	3	8	no	—
92	C₂×C₃×C₈	—	3	8	no	—
93	C₂×C₃×C₈	C₂×C₃×D₃	1	4	yes	—
94	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
95	C₂×C₃⋊C₈	C₃²×C₄	1	1	yes	characteristic
96	D₃×C₈	C₃²×C₄	1	2	yes	—
97	D₃×C₈	C₂²×C₃²	1	2	yes	—
98	C₃×C₁₆	—	2	8	no	—
99	C₃×C₁₆	—	3	8	no	—
100	C₃×C₁₆	C₂×C₃×D₃	1	8	yes	direct product factor
101	C₃×C₁₆	C₂²×C₃²	1	2	yes	complemented
102	C₃⋊C₁₆	C₂²×C₃²	1	2	yes	complemented
103	C₂²×C₃²	—	2	4	no	—
104	C₂²×C₃²	—	3	4	no	—
105	C₂²×C₃²	—	3	2	no	—
106	C₂²×C₃²	D₃×C₈	1	1	yes	characteristic
107	C₂²×C₃²	C₂×C₃×C₈	1	4	yes	—
108	C₂×C₃×D₃	C₃×C₁₆	1	16	yes	direct product factor
109	C₂×C₃×D₃	C₂×C₃×C₈	1	8	yes	—
110	C₃²×C₄	—	2	4	no	—
111	C₃²×C₄	—	2	4	no	—
112	C₃²×C₄	—	3	2	no	—
113	C₃²×C₄	D₃×C₈	1	1	yes	characteristic
114	C₃²×C₄	C₂×C₃×C₈	1	4	yes	—
115	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
116	C₃²×C₄	C₂²×C₃×C₄	1	4	yes	—
117	C₃×Dic₃	C₂×C₃×C₈	1	8	yes	—
118	C₂²×C₈	—	3	1	no	—
119	C₂×C₁₆	—	3	1	no	—
120	C₂×C₁₆	—	3	4	no	—
121	C₂×C₁₆	C₃²×D₃	1	1	yes	characteristic, direct product factor
122	C₃³	C₂²×C₁₆	1	1	yes	characteristic, complemented, Sylow
123	C₂³×C₃	—	3	4	no	—
124	C₂²×D₃	C₃²×C₈	1	1	yes	characteristic
125	C₂×C₃×C₄	—	2	4	no	—
126	C₂×C₃×C₄	—	3	4	no	—
127	C₂×C₃×C₄	—	3	8	no	—
128	C₂×C₃×C₄	—	3	8	no	—
129	C₂×C₃×C₄	C₃×C₄×D₃	1	4	yes	—
130	C₂×C₃×C₄	C₂×C₃²×C₄	1	1	yes	characteristic
131	C₂×Dic₃	C₃²×C₈	1	1	yes	characteristic
132	C₄×D₃	C₃²×C₈	1	2	yes	—
133	C₄×D₃	C₂×C₃²×C₄	1	2	yes	—
134	C₃×C₈	—	2	4	no	—
135	C₃×C₈	—	2	4	no	—
136	C₃×C₈	—	3	8	no	—
137	C₃×C₈	C₃×C₄×D₃	1	4	yes	—
138	C₃×C₈	C₂×C₃²×C₄	1	1	yes	characteristic
139	C₃×C₈	C₂²×C₃×D₃	1	4	yes	—
140	C₃×C₈	C₂³×C₃²	1	1	yes	characteristic
141	C₃⋊C₈	C₂×C₃²×C₄	1	2	yes	—
142	C₂×C₃²	—	2	4	no	—
143	C₂×C₃²	—	2	8	no	—
144	C₂×C₃²	—	3	4	no	—
145	C₂×C₃²	D₃×C₁₆	1	2	yes	direct product factor
146	C₂×C₃²	C₂×C₃×C₁₆	1	8	yes	complemented
147	C₂×C₃²	C₂×D₃×C₈	1	1	yes	characteristic
148	C₂×C₃²	C₂²×C₃×C₈	1	4	yes	—
149	C₃×D₃	C₂×C₃×C₁₆	1	16	yes	direct product factor
150	C₂²×C₄	—	3	1	no	—
151	C₂×C₈	—	3	1	no	—
152	C₂×C₈	—	3	2	no	—
153	C₂×C₈	—	3	2	no	—
154	C₂×C₈	C₂×C₃²×D₃	1	1	yes	characteristic
155	C₁₆	—	3	2	no	—
156	C₁₆	C₂×C₃²×D₃	1	2	yes	direct product factor
157	C₂²×C₃	—	2	4	no	—
158	C₂²×C₃	—	3	8	no	—
159	C₂²×C₃	—	3	16	no	—
160	C₂²×C₃	C₃×D₃×C₈	1	4	yes	—
161	C₂²×C₃	C₂×C₃²×C₈	1	1	yes	characteristic
162	C₂×D₃	C₃²×C₁₆	1	4	yes	direct product factor
163	C₂×D₃	C₂×C₃²×C₈	1	2	yes	—
164	C₃×C₄	—	2	4	no	—
165	C₃×C₄	—	2	4	no	—
166	C₃×C₄	—	3	8	no	—
167	C₃×C₄	C₃×D₃×C₈	1	4	yes	—
168	C₃×C₄	C₂×C₃²×C₈	1	1	yes	characteristic
169	C₃×C₄	C₂×C₃×C₄×D₃	1	4	yes	—
170	C₃×C₄	C₂²×C₃²×C₄	1	1	yes	characteristic
171	Dic₃	C₂×C₃²×C₈	1	2	yes	—
172	C₃²	—	2	4	no	—
173	C₃²	C₂×D₃×C₁₆	1	1	yes	characteristic, direct product factor
174	C₃²	C₂²×C₃×C₁₆	1	4	yes	complemented
175	C₂³	—	3	1	no	—
176	C₂×C₄	—	3	1	no	—
177	C₂×C₄	—	3	2	no	—
178	C₂×C₄	—	3	2	no	—
179	C₂×C₄	C₃²×C₄×D₃	1	1	yes	characteristic
180	C₈	—	3	2	no	—
181	C₈	C₃²×C₄×D₃	1	1	yes	characteristic
182	C₈	C₂²×C₃²×D₃	1	1	yes	characteristic
183	C₂×C₃	—	2	8	no	—
184	C₂×C₃	—	2	4	no	—
185	C₂×C₃	—	3	16	no	—
186	C₂×C₃	C₃×D₃×C₁₆	1	8	yes	direct product factor
187	C₂×C₃	C₂×C₃²×C₁₆	1	2	yes	complemented
188	C₂×C₃	C₂×C₃×D₃×C₈	1	4	yes	—
189	C₂×C₃	C₂²×C₃²×C₈	1	1	yes	characteristic
190	D₃	C₂×C₃²×C₁₆	1	4	yes	direct product factor
191	C₂²	—	3	4	no	—
192	C₂²	—	3	2	no	—
193	C₂²	C₃²×D₃×C₈	1	1	yes	characteristic
194	C₄	—	3	2	no	—
195	C₄	C₃²×D₃×C₈	1	1	yes	characteristic
196	C₄	C₂×C₃²×C₄×D₃	1	1	yes	characteristic
197	C₃	—	2	4	no	—
198	C₃	C₂×C₃×D₃×C₁₆	1	4	yes	direct product factor
199	C₃	C₂²×C₃²×C₁₆	1	1	yes	characteristic, complemented
200	C₂	—	3	4	no	—
201	C₂	C₃²×D₃×C₁₆	1	2	yes	direct product factor
202	C₂	C₂×C₃²×D₃×C₈	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₃²×C₁₆	D₃
commutator, hypocenter	C₃	C₂²×C₃²×C₁₆
Frattini	C₈	C₂²×C₃²×D₃
Fitting	C₂×C₃³×C₁₆	C₂
socle	C₂²×C₃³	C₂×C₈
prime radical	C₂²×C₃²×D₃	C₈
cosocle	C₃×C₈	C₂³×C₃²
LCS	G ▷ C₃
UCS	C₁ ◁ C₂×C₃²×C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×C₃³×C₁₆	C₂	semidirect
3. C₂×C₃²×C₃⋊C₁₆	C₂	semidirect
4. C₃²×D₃×C₁₆	C₂	direct (N × H)
6. C₂×C₃×D₃×C₁₆	C₃	direct (N × H)
12. C₃³×C₁₆	C₂²	semidirect
13. C₃²×C₃⋊C₁₆	C₂²	semidirect
19. C₂×C₃²×C₁₆	D₃	direct (N × H)
20. C₂×C₃²×C₁₆	C₂×C₃	semidirect
21. C₂×C₃×C₃⋊C₁₆	C₂×C₃	semidirect
22. C₃×D₃×C₁₆	C₂×C₃	direct (N × H)
32. C₂×D₃×C₁₆	C₃²	direct (N × H)
46. C₃²×C₁₆	C₂×D₃	direct (N × H)
47. C₃²×C₁₆	C₂²×C₃	semidirect
48. C₃×C₃⋊C₁₆	C₂²×C₃	semidirect
50. C₂×C₃²×D₃	C₁₆	direct (N × H)
60. C₂×C₃×C₁₆	C₃×D₃	direct (N × H)
61. C₂×C₃×C₁₆	C₂×C₃²	semidirect
62. C₂×C₃⋊C₁₆	C₂×C₃²	semidirect
63. D₃×C₁₆	C₂×C₃²	direct (N × H)
84. C₂×C₃³	C₂×C₁₆	semidirect
86. C₃²×D₃	C₂×C₁₆	direct (N × H)
100. C₃×C₁₆	C₂×C₃×D₃	direct (N × H)
101. C₃×C₁₆	C₂²×C₃²	semidirect
102. C₃⋊C₁₆	C₂²×C₃²	semidirect
108. C₂×C₃×D₃	C₃×C₁₆	direct (N × H)
121. C₂×C₁₆	C₃²×D₃	direct (N × H)
122. C₃³	C₂²×C₁₆	semidirect
145. C₂×C₃²	D₃×C₁₆	direct (N × H)
146. C₂×C₃²	C₂×C₃×C₁₆	semidirect
149. C₃×D₃	C₂×C₃×C₁₆	direct (N × H)
156. C₁₆	C₂×C₃²×D₃	direct (N × H)
162. C₂×D₃	C₃²×C₁₆	direct (N × H)
173. C₃²	C₂×D₃×C₁₆	direct (N × H)
174. C₃²	C₂²×C₃×C₁₆	semidirect
186. C₂×C₃	C₃×D₃×C₁₆	direct (N × H)
187. C₂×C₃	C₂×C₃²×C₁₆	semidirect
190. D₃	C₂×C₃²×C₁₆	direct (N × H)
198. C₃	C₂×C₃×D₃×C₁₆	direct (N × H)
199. C₃	C₂²×C₃²×C₁₆	semidirect
201. C₂	C₃²×D₃×C₁₆	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×C₃²×D₃×C₈	2
C₂×C₃³×C₁₆	2
C₂×C₃²×C₃⋊C₁₆	2
C₃²×D₃×C₁₆	2
C₂²×C₃²×C₁₆	3
C₂×C₃×D₃×C₁₆	3

Subgroups — C2×C32×D3×C16 #13162335

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃²×D₃×C₁₆ #13162335