Subgroups — C₃×Q₁₆×C₃⋊Dic₃ #13163216

Order 1728.

900 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃³×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	C₃×Q₈×C₃⋊Dic₃	C₂	1	2	yes	maximal
3	C₃×#63662	C₂	1	2	yes	maximal
4	C₃×#63645	C₂	1	1	yes	maximal, characteristic
5	C₃×C₈×C₃⋊Dic₃	C₂	1	1	yes	maximal, characteristic
6	Q₁₆×C₃⋊Dic₃	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₃×Dic₃×Q₁₆	—	3	4	no	maximal
8	C₂×C₃³×Q₈	C₂²	1	2	yes	—
9	C₃³×Q₁₆	C₄	1	4	yes	complemented
10	C₂×C₃³×C₈	C₂²	1	1	yes	characteristic
11	C₃×C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
12	C₃×C₄⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
13	C₃×C₄×C₃⋊Dic₃	—	2	2	no	—
14	C₃×C₄×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
16	C₂×C₃²×Q₁₆	—	2	4	no	—
17	C₂×C₃²×Q₁₆	D₃	1	4	yes	—
18	C₂×C₃²×Q₁₆	C₂×C₃	1	1	yes	characteristic
19	Q₈×C₃⋊Dic₃	C₂×C₃	1	2	yes	—
20	C₃×Q₈×Dic₃	—	3	8	no	—
21	#63662	C₂×C₃	1	2	yes	—
22	#63645	C₂×C₃	1	1	yes	characteristic
23	C₈×C₃⋊Dic₃	C₂×C₃	1	1	yes	characteristic
24	C₃×#835	—	3	8	no	—
25	C₃×#818	—	3	4	no	—
26	C₃×C₈×Dic₃	—	3	4	no	—
27	C₃³×Q₈	—	2	2	no	—
28	C₃³×Q₈	C₂×C₄	1	4	yes	—
29	C₂×C₃³×C₄	—	2	2	no	—
30	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
31	C₂×C₃×C₃⋊Dic₃	—	2	2	no	—
32	C₂×C₃×C₃⋊Dic₃	D₄	1	1	yes	characteristic
33	C₂×C₃×C₃⋊Dic₃	D₄	1	1	yes	characteristic
34	C₃³×C₈	C₂×C₄	1	2	yes	—
35	C₃×C₃⋊(C₃⋊C₈)	—	2	1	no	—
36	C₃×C₄×Q₁₆	—	9	1	no	—
37	Dic₃×Q₁₆	—	3	4	no	—
38	C₂×C₃²×Q₈	—	2	8	no	—
39	C₂×C₃²×Q₈	C₂×D₃	1	8	yes	—
40	C₂×C₃²×Q₈	C₂²×C₃	1	2	yes	—
41	C₃²×Q₁₆	—	2	16	no	—
42	C₃²×Q₁₆	Dic₃	1	16	yes	complemented
43	C₃²×Q₁₆	C₃×C₄	1	4	yes	complemented
44	C₂×C₃²×C₈	—	2	4	no	—
45	C₂×C₃²×C₈	C₂×D₃	1	4	yes	—
46	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
47	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
48	C₄⋊(C₃⋊Dic₃)	C₂²×C₃	1	2	yes	—
49	C₄×C₃⋊Dic₃	—	2	2	no	—
50	C₄×C₃⋊Dic₃	C₂²×C₃	1	1	yes	characteristic
51	C₂×C₃⋊(C₃⋊C₈)	C₂²×C₃	1	1	yes	characteristic
52	C₃×C₄⋊Dic₃	—	3	8	no	—
53	C₃×C₄⋊Dic₃	—	6	8	no	—
54	C₃×C₄×Dic₃	—	3	4	no	—
55	C₃×C₄×Dic₃	—	6	8	no	—
56	C₂×C₃×C₃⋊C₈	—	3	4	no	—
57	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
58	C₃³×C₄	—	2	4	no	—
59	C₃³×C₄	C₂²×C₄	1	1	yes	characteristic
60	C₃³×C₄	#41	1	1	yes	characteristic
61	C₃×C₃⋊Dic₃	—	2	1	no	—
62	C₃×C₃⋊Dic₃	—	4	2	no	—
63	C₃×C₃⋊Dic₃	Q₁₆	1	2	yes	direct product factor
64	C₂×C₃×Q₁₆	—	2	4	no	—
65	C₂×C₃×Q₁₆	C₃×D₃	1	4	yes	—
66	C₂×C₃×Q₁₆	C₃⋊D₃	1	1	yes	characteristic
67	C₃×C₄×Q₈	—	9	2	no	—
68	Q₈×Dic₃	—	3	8	no	—
69	C₃×#107	—	9	1	no	—
70	C₃×#103	—	9	2	no	—
71	C₃×C₄×C₈	—	9	1	no	—
72	#835	—	3	8	no	—
73	#818	—	3	4	no	—
74	C₈×Dic₃	—	3	4	no	—
75	C₃²×Q₈	—	2	2	no	—
76	C₃²×Q₈	—	2	16	no	—
77	C₃²×Q₈	—	2	8	no	—
78	C₃²×Q₈	—	4	8	no	—
79	C₃²×Q₈	C₂×Dic₃	1	16	yes	—
80	C₃²×Q₈	C₂×C₃×C₄	1	4	yes	—
81	C₂×C₃²×C₄	—	2	2	no	—
82	C₂×C₃²×C₄	—	2	4	no	—
83	C₂×C₃²×C₄	—	2	8	no	—
84	C₂×C₃²×C₄	—	4	8	no	—
85	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
86	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
87	C₂×C₃⋊Dic₃	—	2	2	no	—
88	C₂×C₃⋊Dic₃	C₃×D₄	1	1	yes	characteristic
89	C₂×C₃⋊Dic₃	C₃×D₄	1	1	yes	characteristic
90	C₂×C₃×Dic₃	—	3	4	no	—
91	C₂×C₃×Dic₃	—	3	4	no	—
92	C₂×C₃×Dic₃	—	6	8	no	—
93	C₃²×C₈	—	2	8	no	—
94	C₃²×C₈	C₂×Dic₃	1	8	yes	—
95	C₃²×C₈	C₂×C₃×C₄	1	2	yes	—
96	C₃⋊(C₃⋊C₈)	—	2	1	no	—
97	C₃×C₃⋊C₈	—	6	4	no	—
98	C₄×Q₁₆	—	9	1	no	Sylow
99	C₂×C₃³	C₄×D₄	1	1	yes	characteristic
100	C₂×C₃³	C₂×Q₁₆	1	1	yes	characteristic
101	C₂×C₃³	#135	1	1	yes	characteristic
102	C₂×C₃×Q₈	—	2	8	no	—
103	C₂×C₃×Q₈	C₂×C₃×D₃	1	8	yes	—
104	C₂×C₃×Q₈	C₂×C₃⋊D₃	1	2	yes	—
105	C₃×Q₁₆	—	2	16	no	—
106	C₃×Q₁₆	C₃×Dic₃	1	16	yes	complemented
107	C₃×Q₁₆	C₃⋊Dic₃	1	4	yes	direct product factor
108	C₂×C₃×C₈	—	2	4	no	—
109	C₂×C₃×C₈	—	9	1	no	—
110	C₂×C₃×C₈	C₂×C₃×D₃	1	4	yes	—
111	C₂×C₃×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
112	C₃×C₄⋊C₄	—	9	2	no	—
113	C₃×C₄⋊C₄	—	18	2	no	—
114	C₃×C₄²	—	9	1	no	—
115	C₃×C₄²	—	18	2	no	—
116	C₄⋊Dic₃	—	3	8	no	—
117	C₄⋊Dic₃	—	6	8	no	—
118	C₄×Dic₃	—	3	4	no	—
119	C₄×Dic₃	—	6	8	no	—
120	C₂×C₃⋊C₈	—	3	4	no	—
121	C₂²×C₃²	—	2	4	no	—
122	C₂²×C₃²	D₃×D₄	1	4	yes	—
123	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
124	C₃²×C₄	—	2	4	no	—
125	C₃²×C₄	—	2	4	no	—
126	C₃²×C₄	—	2	4	no	—
127	C₃²×C₄	—	2	16	no	—
128	C₃²×C₄	—	4	16	no	—
129	C₃²×C₄	#237	1	4	yes	—
130	C₃²×C₄	C₂²×Dic₃	1	4	yes	—
131	C₃²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
132	C₃²×C₄	C₃×#41	1	1	yes	characteristic
133	C₃⋊Dic₃	—	2	1	no	—
134	C₃⋊Dic₃	—	4	2	no	—
135	C₃⋊Dic₃	C₃×Q₁₆	1	2	yes	direct product factor
136	C₃×Dic₃	—	3	8	no	—
137	C₃×Dic₃	—	6	4	no	—
138	C₃×Dic₃	—	12	8	no	—
139	C₂×Q₁₆	C₃×C₃⋊D₃	1	1	yes	characteristic
140	C₄×Q₈	—	9	2	no	—
141	#107	—	9	1	no	—
142	#103	—	9	2	no	—
143	C₄×C₈	—	9	1	no	—
144	C₃³	C₄×Q₁₆	1	1	yes	characteristic, complemented, Sylow
145	C₃×Q₈	—	2	8	no	—
146	C₃×Q₈	—	2	2	no	—
147	C₃×Q₈	—	2	16	no	—
148	C₃×Q₈	—	4	8	no	—
149	C₃×Q₈	C₂×C₃×Dic₃	1	16	yes	—
150	C₃×Q₈	C₂×C₃⋊Dic₃	1	4	yes	—
151	C₂×C₃×C₄	—	2	8	no	—
152	C₂×C₃×C₄	—	2	2	no	—
153	C₂×C₃×C₄	—	2	4	no	—
154	C₂×C₃×C₄	—	4	8	no	—
155	C₂×C₃×C₄	—	9	1	no	—
156	C₂×C₃×C₄	—	9	1	no	—
157	C₂×C₃×C₄	—	18	2	no	—
158	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
159	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
160	C₂×Dic₃	—	3	4	no	—
161	C₂×Dic₃	—	3	4	no	—
162	C₂×Dic₃	—	6	8	no	—
163	C₃×C₈	—	2	8	no	—
164	C₃×C₈	—	18	1	no	—
165	C₃×C₈	C₂×C₃×Dic₃	1	8	yes	—
166	C₃×C₈	C₂×C₃⋊Dic₃	1	2	yes	—
167	C₃⋊C₈	—	6	4	no	—
168	C₂×C₃²	—	2	4	no	—
169	C₂×C₃²	—	2	4	no	—
170	C₂×C₃²	—	2	4	no	—
171	C₂×C₃²	D₃×Q₁₆	1	4	yes	—
172	C₂×C₃²	#919	1	4	yes	—
173	C₂×C₃²	D₄×Dic₃	1	4	yes	—
174	C₂×C₃²	C₃×C₄×D₄	1	1	yes	characteristic
175	C₂×C₃²	C₂×C₃×Q₁₆	1	1	yes	characteristic
176	C₂×C₃²	C₃×#135	1	1	yes	characteristic
177	C₂×Q₈	C₂×C₃×C₃⋊D₃	1	2	yes	—
178	Q₁₆	C₃×C₃⋊Dic₃	1	4	yes	direct product factor
179	C₂×C₈	—	9	1	no	—
180	C₂×C₈	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
181	C₄⋊C₄	—	9	2	no	—
182	C₄⋊C₄	—	18	2	no	—
183	C₄²	—	9	1	no	—
184	C₄²	—	18	2	no	—
185	C₂²×C₃	—	2	4	no	—
186	C₂²×C₃	C₃×D₃×D₄	1	4	yes	—
187	C₂²×C₃	D₄×C₃⋊D₃	1	1	yes	characteristic
188	C₃×C₄	—	2	16	no	—
189	C₃×C₄	—	2	4	no	—
190	C₃×C₄	—	2	4	no	—
191	C₃×C₄	—	2	4	no	—
192	C₃×C₄	—	4	16	no	—
193	C₃×C₄	—	9	2	no	—
194	C₃×C₄	—	18	1	no	—
195	C₃×C₄	—	36	2	no	—
196	C₃×C₄	C₃×#237	1	4	yes	—
197	C₃×C₄	C₂²×C₃×Dic₃	1	4	yes	—
198	C₃×C₄	#3830	1	1	yes	characteristic
199	C₃×C₄	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
200	Dic₃	—	3	8	no	—
201	Dic₃	—	6	4	no	—
202	Dic₃	—	12	8	no	—
203	C₃²	—	2	4	no	—
204	C₃²	Dic₃×Q₁₆	1	4	yes	complemented
205	C₃²	C₃×C₄×Q₁₆	1	1	yes	characteristic, complemented
206	Q₈	—	2	2	no	—
207	Q₈	C₂×C₃×C₃⋊Dic₃	1	4	yes	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	9	1	no	—
210	C₂×C₄	—	9	1	no	—
211	C₂×C₄	—	18	2	no	—
212	C₂×C₄	C₂²×C₃×C₃⋊D₃	1	1	yes	characteristic
213	C₈	—	18	1	no	—
214	C₈	C₂×C₃×C₃⋊Dic₃	1	2	yes	—
215	C₂×C₃	—	2	4	no	—
216	C₂×C₃	—	2	4	no	—
217	C₂×C₃	—	2	4	no	—
218	C₂×C₃	C₃×D₃×Q₁₆	1	4	yes	—
219	C₂×C₃	C₃×#919	1	4	yes	—
220	C₂×C₃	C₃×D₄×Dic₃	1	4	yes	—
221	C₂×C₃	Q₁₆×C₃⋊D₃	1	1	yes	characteristic
222	C₂×C₃	#64128	1	1	yes	characteristic
223	C₂×C₃	D₄×C₃⋊Dic₃	1	1	yes	characteristic
224	C₂²	C₃×D₄×C₃⋊D₃	1	1	yes	characteristic
225	C₄	—	2	4	no	—
226	C₄	—	9	2	no	—
227	C₄	—	18	1	no	—
228	C₄	—	36	2	no	—
229	C₄	C₃×#3830	1	1	yes	characteristic
230	C₄	C₂²×C₃×C₃⋊Dic₃	1	1	yes	characteristic
231	C₃	—	2	4	no	—
232	C₃	C₃×Dic₃×Q₁₆	1	4	yes	complemented
233	C₃	Q₁₆×C₃⋊Dic₃	1	1	yes	characteristic, direct product factor
234	C₂	C₃×Q₁₆×C₃⋊D₃	1	1	yes	characteristic
235	C₂	C₃×#64128	1	1	yes	characteristic
236	C₂	C₃×D₄×C₃⋊Dic₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	D₄×C₃⋊D₃
commutator	C₃²×C₄	C₂²×C₃×C₄
Frattini	C₂×C₄	C₂²×C₃×C₃⋊D₃
Fitting	C₂×C₃³×Q₁₆	C₂
socle, prime radical	C₂²×C₃³	C₂×D₄
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₂×C₃×Q₁₆	C₃⋊D₃
hypocenter	C₃²	C₃×C₄×Q₁₆
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄ ◁ C₂×C₃×Q₁₆

Complements

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

N	H ≅ G/N	type
6. Q₁₆×C₃⋊Dic₃	C₃	direct (N × H)
9. C₃³×Q₁₆	C₄	semidirect
42. C₃²×Q₁₆	Dic₃	semidirect
43. C₃²×Q₁₆	C₃×C₄	semidirect
63. C₃×C₃⋊Dic₃	Q₁₆	direct (N × H)
106. C₃×Q₁₆	C₃×Dic₃	semidirect
107. C₃×Q₁₆	C₃⋊Dic₃	direct (N × H)
135. C₃⋊Dic₃	C₃×Q₁₆	direct (N × H)
144. C₃³	C₄×Q₁₆	semidirect
178. Q₁₆	C₃×C₃⋊Dic₃	direct (N × H)
204. C₃²	Dic₃×Q₁₆	semidirect
205. C₃²	C₃×C₄×Q₁₆	semidirect
232. C₃	C₃×Dic₃×Q₁₆	semidirect
233. C₃	Q₁₆×C₃⋊Dic₃	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×C₃³×Q₁₆	2
C₃×Q₈×C₃⋊Dic₃	2
C₃×#63662	2
C₃×#63645	2
C₃×C₈×C₃⋊Dic₃	2
Q₁₆×C₃⋊Dic₃	3
C₃×Dic₃×Q₁₆	3

Subgroups — C3×Q16×C3⋊Dic3 #13163216

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×Q₁₆×C₃⋊Dic₃ #13163216