Subgroups — C₂×Dic₃×C₃⋊C₁₆ #11795816

Order 1152.

481 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃⋊(C₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₈×Dic₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₃×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	Dic₃×C₃⋊C₁₆	C₂	1	4	yes	maximal, direct product factor
5	C₂×Dic₃×C₁₆	—	3	1	no	maximal
6	C₂×C₄×C₃⋊C₁₆	—	3	1	no	maximal
7	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₃×C₄×Dic₃	C₄	1	1	yes	characteristic
9	C₂²×C₃×C₃⋊C₈	C₄	1	1	yes	characteristic
10	C₂×C₃⋊(C₃⋊C₁₆)	C₄	1	4	yes	complemented
11	C₂×C₃⋊(C₃⋊C₁₆)	C₂²	1	2	yes	—
12	C₃×C₈×Dic₃	C₄	1	2	yes	—
13	C₃×C₈×Dic₃	C₂²	1	2	yes	—
14	C₂×C₃×C₃⋊C₁₆	C₄	1	4	yes	complemented
15	C₂×C₃×C₃⋊C₁₆	C₂²	1	2	yes	—
16	C₂²×C₃×C₁₆	—	3	1	no	—
17	C₂×C₃×C₄×C₈	—	3	1	no	—
18	C₂×C₈×Dic₃	D₃	1	1	yes	characteristic
19	C₂²×C₃⋊C₁₆	—	3	1	no	—
20	C₂²×C₃⋊C₁₆	—	3	1	no	—
21	C₂²×C₃⋊C₁₆	—	6	1	no	—
22	C₂²×C₃⋊C₁₆	D₃	1	1	yes	characteristic
23	Dic₃×C₁₆	—	3	4	no	—
24	C₄×C₃⋊C₁₆	—	3	4	no	—
25	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₃×Dic₃	C₈	1	2	yes	—
27	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
29	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
30	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
31	C₃×C₄×Dic₃	C₈	1	2	yes	—
32	C₃×C₄×Dic₃	C₂×C₄	1	2	yes	—
33	C₂×C₃×C₃⋊C₈	C₈	1	4	yes	—
34	C₂×C₃×C₃⋊C₈	C₂×C₄	1	2	yes	—
35	C₃⋊(C₃⋊C₁₆)	C₂×C₄	1	4	yes	complemented
36	C₃×C₃⋊C₁₆	C₂×C₄	1	4	yes	complemented
37	C₂×C₄×C₁₆	—	9	1	no	Sylow
38	C₂²×C₃×C₈	—	2	1	no	—
39	C₂²×C₃×C₈	—	3	1	no	—
40	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
41	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
42	C₂×C₃×C₄²	—	3	1	no	—
43	C₂×C₄×Dic₃	Dic₃	1	1	yes	characteristic
44	C₂²×C₃⋊C₈	Dic₃	1	1	yes	characteristic
45	C₂×C₃×C₁₆	—	3	4	no	—
46	C₂×C₃×C₁₆	—	3	2	no	—
47	C₃×C₄×C₈	—	3	2	no	—
48	C₃×C₄×C₈	—	3	2	no	—
49	C₈×Dic₃	Dic₃	1	2	yes	—
50	C₈×Dic₃	C₂×D₃	1	2	yes	—
51	C₂×C₃⋊C₁₆	—	3	4	no	—
52	C₂×C₃⋊C₁₆	—	3	4	no	—
53	C₂×C₃⋊C₁₆	—	3	2	no	—
54	C₂×C₃⋊C₁₆	—	3	2	no	—
55	C₂×C₃⋊C₁₆	—	6	4	no	—
56	C₂×C₃⋊C₁₆	—	6	2	no	—
57	C₂×C₃⋊C₁₆	Dic₃	1	4	yes	direct product factor
58	C₂×C₃⋊C₁₆	C₂×D₃	1	2	yes	—
59	C₂³×C₃²	C₂×C₈	1	1	yes	characteristic
60	C₂×C₃²×C₄	C₄²	1	2	yes	—
61	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
62	C₂×C₃²×C₄	C₂×C₈	1	2	yes	—
63	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₃×Dic₃	C₁₆	1	8	yes	complemented
65	C₂×C₃×Dic₃	C₂×C₈	1	4	yes	—
66	C₃²×C₈	C₄²	1	2	yes	—
67	C₃²×C₈	C₂²×C₄	1	1	yes	characteristic
68	C₃²×C₈	C₂²×C₄	1	1	yes	characteristic
69	C₃×C₃⋊C₈	C₂×C₈	1	4	yes	—
70	C₂²×C₁₆	—	3	1	no	—
71	C₂²×C₁₆	—	9	1	no	—
72	C₂×C₄×C₈	—	3	1	no	—
73	C₄×C₁₆	—	9	4	no	—
74	C₂²×C₃×C₄	—	2	1	no	—
75	C₂²×C₃×C₄	—	3	2	no	—
76	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
77	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
78	C₂²×Dic₃	C₃⋊C₈	1	2	yes	—
79	C₂×C₃×C₈	—	2	1	no	—
80	C₂×C₃×C₈	—	2	2	no	—
81	C₂×C₃×C₈	—	2	2	no	—
82	C₂×C₃×C₈	—	2	1	no	—
83	C₂×C₃×C₈	—	3	4	no	—
84	C₂×C₃×C₈	—	3	2	no	—
85	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
86	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
87	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
88	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
89	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
90	C₂×C₃×C₈	C₂×Dic₃	1	2	yes	—
91	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
92	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
93	C₃×C₄²	—	3	2	no	—
94	C₃×C₄²	—	3	2	no	—
95	C₄×Dic₃	C₃⋊C₈	1	2	yes	—
96	C₄×Dic₃	C₂×Dic₃	1	2	yes	—
97	C₂×C₃⋊C₈	C₃⋊C₈	1	4	yes	—
98	C₂×C₃⋊C₈	C₂×Dic₃	1	2	yes	—
99	C₃×C₁₆	—	3	4	no	—
100	C₃⋊C₁₆	—	3	4	no	—
101	C₃⋊C₁₆	—	3	4	no	—
102	C₃⋊C₁₆	—	6	4	no	—
103	C₃⋊C₁₆	C₂×Dic₃	1	4	yes	direct product factor
104	C₂²×C₃²	C₄×C₈	1	2	yes	—
105	C₂²×C₃²	C₂×C₁₆	1	4	yes	—
106	C₂²×C₃²	C₂²×C₈	1	1	yes	characteristic
107	C₃²×C₄	C₄×C₈	1	2	yes	—
108	C₃²×C₄	C₂×C₄²	1	1	yes	characteristic
109	C₃²×C₄	C₂²×C₈	1	1	yes	characteristic
110	C₃×Dic₃	C₂×C₁₆	1	8	yes	complemented
111	C₂²×C₈	—	3	1	no	—
112	C₂²×C₈	D₃²	1	1	yes	characteristic
113	C₂×C₄²	—	3	1	no	—
114	C₂×C₁₆	—	3	4	no	—
115	C₂×C₁₆	—	3	2	no	—
116	C₂×C₁₆	—	9	4	no	—
117	C₂×C₁₆	—	9	2	no	—
118	C₄×C₈	—	3	2	no	—
119	C₄×C₈	—	3	2	no	—
120	C₂³×C₃	—	2	1	no	—
121	C₂³×C₃	D₃×C₈	1	1	yes	characteristic
122	C₂³×C₃	C₂×C₃⋊C₈	1	1	yes	characteristic
123	C₂×C₃×C₄	—	2	1	no	—
124	C₂×C₃×C₄	—	2	2	no	—
125	C₂×C₃×C₄	—	2	2	no	—
126	C₂×C₃×C₄	—	2	1	no	—
127	C₂×C₃×C₄	—	3	8	no	—
128	C₂×C₃×C₄	—	3	4	no	—
129	C₂×C₃×C₄	D₃×C₈	1	1	yes	characteristic
130	C₂×C₃×C₄	D₃×C₈	1	2	yes	—
131	C₂×C₃×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
132	C₂×C₃×C₄	C₂×C₃⋊C₈	1	2	yes	—
133	C₂×C₃×C₄	C₄×Dic₃	1	2	yes	—
134	C₂×C₃×C₄	C₄×Dic₃	1	2	yes	—
135	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
136	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
137	C₂×Dic₃	C₃⋊C₁₆	1	8	yes	direct product factor
138	C₂×Dic₃	C₂×C₃⋊C₈	1	4	yes	—
139	C₃×C₈	—	2	2	no	—
140	C₃×C₈	—	2	1	no	—
141	C₃×C₈	—	2	1	no	—
142	C₃×C₈	—	3	4	no	—
143	C₃×C₈	C₄×Dic₃	1	2	yes	—
144	C₃×C₈	C₄×Dic₃	1	2	yes	—
145	C₃×C₈	C₂×C₄×D₃	1	1	yes	characteristic
146	C₃×C₈	C₂×C₄×D₃	1	1	yes	characteristic
147	C₃×C₈	C₂×C₄×D₃	1	1	yes	characteristic
148	C₃×C₈	C₂²×Dic₃	1	1	yes	characteristic
149	C₃⋊C₈	C₂×C₃⋊C₈	1	4	yes	—
150	C₂×C₃²	C₄×C₁₆	1	4	yes	complemented
151	C₂×C₃²	C₂×C₄×C₈	1	1	yes	characteristic
152	C₂×C₃²	C₂²×C₁₆	1	2	yes	—
153	C₂²×C₄	—	3	2	no	—
154	C₂²×C₄	D₃×Dic₃	1	1	yes	characteristic
155	C₂×C₈	—	3	4	no	—
156	C₂×C₈	—	3	2	no	—
157	C₂×C₈	D₃×Dic₃	1	1	yes	characteristic
158	C₂×C₈	D₃×Dic₃	1	2	yes	—
159	C₂×C₈	#624	1	2	yes	—
160	C₂×C₈	C₂×D₃²	1	1	yes	characteristic
161	C₄²	—	3	2	no	—
162	C₄²	—	3	2	no	—
163	C₁₆	—	3	4	no	—
164	C₁₆	—	9	4	no	—
165	C₂²×C₃	—	2	4	no	—
166	C₂²×C₃	—	2	2	no	—
167	C₂²×C₃	—	2	1	no	—
168	C₂²×C₃	D₃×C₁₆	1	4	yes	—
169	C₂²×C₃	C₄×C₃⋊C₈	1	2	yes	—
170	C₂²×C₃	C₂×C₃⋊C₁₆	1	4	yes	—
171	C₂²×C₃	C₈×Dic₃	1	2	yes	—
172	C₂²×C₃	C₂×D₃×C₈	1	1	yes	characteristic
173	C₂²×C₃	C₂²×C₃⋊C₈	1	1	yes	characteristic
174	C₃×C₄	—	2	2	no	—
175	C₃×C₄	—	2	1	no	—
176	C₃×C₄	—	2	1	no	—
177	C₃×C₄	—	3	8	no	—
178	C₃×C₄	C₄×C₃⋊C₈	1	2	yes	—
179	C₃×C₄	C₈×Dic₃	1	2	yes	—
180	C₃×C₄	C₂×D₃×C₈	1	1	yes	characteristic
181	C₃×C₄	C₂²×C₃⋊C₈	1	1	yes	characteristic
182	C₃×C₄	C₂×C₄×Dic₃	1	1	yes	characteristic
183	C₃×C₄	C₂×C₄×Dic₃	1	1	yes	characteristic
184	Dic₃	C₂×C₃⋊C₁₆	1	8	yes	direct product factor
185	C₃²	C₂×C₄×C₁₆	1	1	yes	characteristic, complemented, Sylow
186	C₂³	D₃×C₃⋊C₈	1	1	yes	characteristic
187	C₂×C₄	—	3	8	no	—
188	C₂×C₄	—	3	4	no	—
189	C₂×C₄	D₃×C₃⋊C₈	1	1	yes	characteristic
190	C₂×C₄	D₃×C₃⋊C₈	1	2	yes	—
191	C₂×C₄	Dic₃²	1	2	yes	—
192	C₂×C₄	C₂×D₃×Dic₃	1	1	yes	characteristic
193	C₈	—	3	4	no	—
194	C₈	Dic₃²	1	2	yes	—
195	C₈	C₂×D₃×Dic₃	1	1	yes	characteristic
196	C₈	C₂×#624	1	1	yes	characteristic
197	C₂×C₃	—	2	1	no	—
198	C₂×C₃	—	2	2	no	—
199	C₂×C₃	—	2	4	no	—
200	C₂×C₃	C₄×C₃⋊C₁₆	1	4	yes	complemented
201	C₂×C₃	Dic₃×C₁₆	1	4	yes	complemented
202	C₂×C₃	C₂×D₃×C₁₆	1	2	yes	—
203	C₂×C₃	C₂×C₄×C₃⋊C₈	1	1	yes	characteristic
204	C₂×C₃	C₂²×C₃⋊C₁₆	1	2	yes	—
205	C₂×C₃	C₂×C₈×Dic₃	1	1	yes	characteristic
206	C₂²	D₃×C₃⋊C₁₆	1	4	yes	—
207	C₂²	Dic₃×C₃⋊C₈	1	2	yes	—
208	C₂²	C₂×D₃×C₃⋊C₈	1	1	yes	characteristic
209	C₄	—	3	8	no	—
210	C₄	Dic₃×C₃⋊C₈	1	2	yes	—
211	C₄	C₂×D₃×C₃⋊C₈	1	1	yes	characteristic
212	C₄	C₂×Dic₃²	1	1	yes	characteristic
213	C₃	—	2	1	no	—
214	C₃	C₂×C₄×C₃⋊C₁₆	1	1	yes	characteristic, complemented
215	C₃	C₂×Dic₃×C₁₆	1	1	yes	characteristic, complemented
216	C₂	Dic₃×C₃⋊C₁₆	1	4	yes	direct product factor
217	C₂	C₂×D₃×C₃⋊C₁₆	1	2	yes	—
218	C₂	C₂×Dic₃×C₃⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²×C₈	D₃²
commutator, hypocenter	C₃²	C₂×C₄×C₁₆
Frattini	C₂×C₈	C₂×D₃²
Fitting	C₂²×C₃²×C₈	C₂²
socle, prime radical	C₂³×C₃²	C₂×C₈
cosocle	C₂×C₃²×C₈	C₂³
LCS	G ▷ C₃²
UCS	C₁ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
4. Dic₃×C₃⋊C₁₆	C₂	direct (N × H)
10. C₂×C₃⋊(C₃⋊C₁₆)	C₄	semidirect
14. C₂×C₃×C₃⋊C₁₆	C₄	semidirect
35. C₃⋊(C₃⋊C₁₆)	C₂×C₄	semidirect
36. C₃×C₃⋊C₁₆	C₂×C₄	semidirect
57. C₂×C₃⋊C₁₆	Dic₃	direct (N × H)
64. C₂×C₃×Dic₃	C₁₆	semidirect
103. C₃⋊C₁₆	C₂×Dic₃	direct (N × H)
110. C₃×Dic₃	C₂×C₁₆	semidirect
137. C₂×Dic₃	C₃⋊C₁₆	direct (N × H)
150. C₂×C₃²	C₄×C₁₆	semidirect
184. Dic₃	C₂×C₃⋊C₁₆	direct (N × H)
185. C₃²	C₂×C₄×C₁₆	semidirect
200. C₂×C₃	C₄×C₃⋊C₁₆	semidirect
201. C₂×C₃	Dic₃×C₁₆	semidirect
214. C₃	C₂×C₄×C₃⋊C₁₆	semidirect
215. C₃	C₂×Dic₃×C₁₆	semidirect
216. C₂	Dic₃×C₃⋊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
C₂²×C₃⋊(C₃⋊C₁₆)	2
C₂×C₃×C₈×Dic₃	2
C₂²×C₃×C₃⋊C₁₆	2
Dic₃×C₃⋊C₁₆	2
C₂×Dic₃×C₁₆	3
C₂×C₄×C₃⋊C₁₆	3

Subgroups — C2×Dic3×C3⋊C16 #11795816

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Dic₃×C₃⋊C₁₆ #11795816