Subgroups — C₃×C₄×D₃×3₋¹⁺² #14518888

Order 1944.

926 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×D₃×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic
2	C₃²×C₄×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×Dic₃×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃³×C₄×D₃	C₃	1	1	yes	maximal, characteristic
5	C₂×C₃×C₄×3₋¹⁺²	—	3	1	no	maximal
6	C₄×D₃×3₋¹⁺²	C₃	1	9	yes	maximal, direct product factor
7	C₃×C₄×D₃×C₉	C₃	1	3	yes	maximal, complemented
8	C₂×C₃²×3₋¹⁺²	C₂²	1	1	yes	characteristic
9	C₃×D₃×3₋¹⁺²	C₄	1	2	yes	direct product factor
10	C₂×C₃³×D₃	C₂×C₃	1	1	yes	characteristic
11	C₃⁴×C₄	C₂×C₃	1	1	yes	characteristic
12	C₃³×Dic₃	C₂×C₃	1	1	yes	characteristic
13	C₂²×C₃×3₋¹⁺²	—	3	1	no	—
14	C₂×D₃×3₋¹⁺²	C₂×C₃	1	9	yes	—
15	C₂×C₃×D₃×C₉	C₂×C₃	1	3	yes	—
16	C₃×C₄×3₋¹⁺²	—	2	1	no	—
17	C₃×C₄×3₋¹⁺²	—	2	9	no	—
18	C₃×C₄×3₋¹⁺²	—	2	3	no	—
19	C₃×C₄×3₋¹⁺²	—	3	1	no	—
20	C₃×C₄×3₋¹⁺²	D₃	1	1	yes	characteristic, direct product factor
21	C₃×C₄×3₋¹⁺²	C₂×C₃	1	9	yes	complemented
22	C₃²×C₄×C₉	C₂×C₃	1	3	yes	complemented
23	Dic₃×3₋¹⁺²	C₂×C₃	1	9	yes	complemented
24	C₃×C₉×Dic₃	C₂×C₃	1	3	yes	complemented
25	C₃²×3₋¹⁺²	C₂×C₄	1	1	yes	characteristic, complemented, Sylow
26	C₂×C₃³×C₄	—	3	1	no	—
27	C₃²×C₄×D₃	—	3	3	no	—
28	C₃²×C₄×D₃	C₃²	1	3	yes	—
29	C₃²×C₄×D₃	C₃²	1	1	yes	characteristic
30	C₂×C₄×3₋¹⁺²	—	3	9	no	—
31	C₂×C₃×C₄×C₉	—	3	3	no	—
32	C₄×D₃×C₉	C₃²	1	9	yes	complemented
33	C₂×C₃⁴	C₂²×C₃	1	1	yes	characteristic
34	C₃³×D₃	C₃×C₄	1	2	yes	—
35	C₂×C₃×3₋¹⁺²	—	2	1	no	—
36	C₂×C₃×3₋¹⁺²	—	2	9	no	—
37	C₂×C₃×3₋¹⁺²	—	2	3	no	—
38	C₂×C₃×3₋¹⁺²	—	3	2	no	—
39	C₂×C₃×3₋¹⁺²	C₂×D₃	1	1	yes	characteristic
40	C₂×C₃×3₋¹⁺²	C₂²×C₃	1	9	yes	—
41	C₂×C₃²×C₉	C₂²×C₃	1	3	yes	—
42	D₃×3₋¹⁺²	C₃×C₄	1	18	yes	direct product factor
43	C₃×D₃×C₉	C₃×C₄	1	6	yes	complemented
44	C₂²×C₃³	—	3	1	no	—
45	C₂×C₃²×D₃	—	3	3	no	—
46	C₂×C₃²×D₃	C₂×C₃²	1	3	yes	—
47	C₂×C₃²×D₃	C₂×C₃²	1	1	yes	characteristic
48	C₃³×C₄	—	2	3	no	—
49	C₃³×C₄	—	2	1	no	—
50	C₃³×C₄	—	3	3	no	—
51	C₃³×C₄	—	3	1	no	—
52	C₃³×C₄	—	6	3	no	—
53	C₃³×C₄	C₃×D₃	1	1	yes	characteristic
54	C₃³×C₄	C₂×C₃²	1	1	yes	characteristic
55	C₃³×C₄	C₂×C₃²	1	3	yes	—
56	C₃²×Dic₃	—	3	3	no	—
57	C₃²×Dic₃	C₂×C₃²	1	3	yes	—
58	C₃²×Dic₃	C₂×C₃²	1	1	yes	characteristic
59	C₂²×3₋¹⁺²	—	3	9	no	—
60	C₂²×C₃×C₉	—	3	3	no	—
61	C₂×D₃×C₉	C₂×C₃²	1	9	yes	—
62	C₄×3₋¹⁺²	—	2	9	no	—
63	C₄×3₋¹⁺²	—	2	27	no	—
64	C₄×3₋¹⁺²	—	3	9	no	—
65	C₄×3₋¹⁺²	C₃×D₃	1	9	yes	direct product factor
66	C₃×C₄×C₉	—	2	3	no	—
67	C₃×C₄×C₉	—	2	9	no	—
68	C₃×C₄×C₉	—	3	3	no	—
69	C₃×C₄×C₉	C₃×D₃	1	3	yes	complemented
70	C₃×C₄×C₉	C₂×C₃²	1	9	yes	complemented
71	C₉×Dic₃	C₂×C₃²	1	9	yes	complemented
72	C₃⁴	C₂×C₃×C₄	1	1	yes	characteristic
73	C₃×3₋¹⁺²	—	2	1	no	—
74	C₃×3₋¹⁺²	—	2	9	no	—
75	C₃×3₋¹⁺²	—	2	3	no	—
76	C₃×3₋¹⁺²	C₄×D₃	1	1	yes	characteristic, direct product factor
77	C₃×3₋¹⁺²	C₂×C₃×C₄	1	9	yes	complemented
78	C₃²×C₉	C₂×C₃×C₄	1	3	yes	complemented
79	C₂×C₃²×C₄	—	3	1	no	—
80	C₂×C₃²×C₄	—	3	3	no	—
81	C₂×C₃²×C₄	—	9	3	no	—
82	C₃×C₄×D₃	—	3	3	no	—
83	C₃×C₄×D₃	3₋¹⁺²	1	3	yes	direct product factor
84	C₃×C₄×D₃	C₃³	1	1	yes	characteristic
85	C₂×C₄×C₉	—	3	9	no	—
86	C₂×C₃³	—	2	3	no	—
87	C₂×C₃³	—	2	1	no	—
88	C₂×C₃³	—	3	3	no	—
89	C₂×C₃³	—	3	2	no	—
90	C₂×C₃³	—	6	3	no	—
91	C₂×C₃³	C₂×C₃×D₃	1	1	yes	characteristic
92	C₂×C₃³	C₂²×C₃²	1	1	yes	characteristic
93	C₂×C₃³	C₂²×C₃²	1	3	yes	—
94	C₃²×D₃	—	3	6	no	—
95	C₃²×D₃	C₃²×C₄	1	6	yes	—
96	C₃²×D₃	C₃²×C₄	1	2	yes	—
97	C₂×3₋¹⁺²	—	2	9	no	—
98	C₂×3₋¹⁺²	—	2	27	no	—
99	C₂×3₋¹⁺²	—	3	18	no	—
100	C₂×3₋¹⁺²	C₂×C₃×D₃	1	9	yes	—
101	C₂×C₃×C₉	—	2	3	no	—
102	C₂×C₃×C₉	—	2	9	no	—
103	C₂×C₃×C₉	—	3	6	no	—
104	C₂×C₃×C₉	C₂×C₃×D₃	1	3	yes	—
105	C₂×C₃×C₉	C₂²×C₃²	1	9	yes	—
106	D₃×C₉	C₃²×C₄	1	18	yes	complemented
107	C₂²×C₃²	—	3	1	no	—
108	C₂²×C₃²	—	3	3	no	—
109	C₂²×C₃²	—	9	3	no	—
110	C₂×C₃×D₃	—	3	3	no	—
111	C₂×C₃×D₃	C₂×3₋¹⁺²	1	3	yes	—
112	C₂×C₃×D₃	C₂×C₃³	1	1	yes	characteristic
113	C₃²×C₄	—	2	3	no	—
114	C₃²×C₄	—	2	1	no	—
115	C₃²×C₄	—	2	3	no	—
116	C₃²×C₄	—	3	1	no	—
117	C₃²×C₄	—	3	3	no	—
118	C₃²×C₄	—	3	3	no	—
119	C₃²×C₄	—	3	3	no	—
120	C₃²×C₄	—	6	9	no	—
121	C₃²×C₄	—	6	3	no	—
122	C₃²×C₄	—	6	3	no	—
123	C₃²×C₄	—	9	3	no	—
124	C₃²×C₄	C₂×3₋¹⁺²	1	3	yes	complemented
125	C₃²×C₄	C₃²×D₃	1	3	yes	—
126	C₃²×C₄	C₃²×D₃	1	1	yes	characteristic
127	C₃²×C₄	C₂×C₃³	1	1	yes	characteristic
128	C₃×Dic₃	—	3	3	no	—
129	C₃×Dic₃	C₂×3₋¹⁺²	1	3	yes	complemented
130	C₃×Dic₃	C₂×C₃³	1	1	yes	characteristic
131	C₂²×C₉	—	3	9	no	—
132	C₄×C₉	—	2	9	no	—
133	C₄×C₉	—	3	9	no	—
134	C₄×C₉	C₃²×D₃	1	9	yes	complemented
135	C₃³	—	2	3	no	—
136	C₃³	—	2	1	no	—
137	C₃³	—	3	3	no	—
138	C₃³	—	6	3	no	—
139	C₃³	C₃×C₄×D₃	1	1	yes	characteristic
140	C₃³	C₂×C₃²×C₄	1	1	yes	characteristic
141	C₃³	C₂×C₃²×C₄	1	3	yes	—
142	3₋¹⁺²	—	2	9	no	—
143	3₋¹⁺²	—	2	27	no	—
144	3₋¹⁺²	C₃×C₄×D₃	1	9	yes	direct product factor
145	C₃×C₉	—	2	3	no	—
146	C₃×C₉	—	2	9	no	—
147	C₃×C₉	C₃×C₄×D₃	1	3	yes	complemented
148	C₃×C₉	C₂×C₃²×C₄	1	9	yes	complemented
149	C₂×C₃×C₄	—	3	3	no	—
150	C₂×C₃×C₄	—	3	1	no	—
151	C₂×C₃×C₄	—	9	3	no	—
152	C₄×D₃	C₃×3₋¹⁺²	1	1	yes	characteristic, direct product factor
153	C₂×C₃²	—	2	3	no	—
154	C₂×C₃²	—	2	1	no	—
155	C₂×C₃²	—	2	3	no	—
156	C₂×C₃²	—	3	2	no	—
157	C₂×C₃²	—	3	3	no	—
158	C₂×C₃²	—	3	3	no	—
159	C₂×C₃²	—	3	6	no	—
160	C₂×C₃²	—	6	9	no	—
161	C₂×C₃²	—	6	3	no	—
162	C₂×C₃²	—	6	3	no	—
163	C₂×C₃²	—	9	6	no	—
164	C₂×C₃²	C₂²×3₋¹⁺²	1	3	yes	—
165	C₂×C₃²	C₂×C₃²×D₃	1	3	yes	—
166	C₂×C₃²	C₂×C₃²×D₃	1	1	yes	characteristic
167	C₂×C₃²	C₂²×C₃³	1	1	yes	characteristic
168	C₃×D₃	—	3	6	no	—
169	C₃×D₃	C₄×3₋¹⁺²	1	6	yes	direct product factor
170	C₃×D₃	C₃³×C₄	1	2	yes	—
171	C₂×C₉	—	2	9	no	—
172	C₂×C₉	—	3	18	no	—
173	C₂×C₉	C₂×C₃²×D₃	1	9	yes	—
174	C₂²×C₃	—	3	3	no	—
175	C₂²×C₃	—	3	1	no	—
176	C₂²×C₃	—	9	3	no	—
177	C₂×D₃	C₂×C₃×3₋¹⁺²	1	1	yes	characteristic
178	C₃×C₄	—	2	3	no	—
179	C₃×C₄	—	2	1	no	—
180	C₃×C₄	—	3	3	no	—
181	C₃×C₄	—	3	1	no	—
182	C₃×C₄	—	3	3	no	—
183	C₃×C₄	—	6	3	no	—
184	C₃×C₄	—	9	3	no	—
185	C₃×C₄	D₃×3₋¹⁺²	1	3	yes	direct product factor
186	C₃×C₄	C₂×C₃×3₋¹⁺²	1	1	yes	characteristic, complemented
187	C₃×C₄	C₃³×D₃	1	1	yes	characteristic
188	Dic₃	C₂×C₃×3₋¹⁺²	1	1	yes	characteristic, complemented
189	C₃²	—	2	1	no	—
190	C₃²	—	2	3	no	—
191	C₃²	—	2	3	no	—
192	C₃²	—	3	3	no	—
193	C₃²	—	3	3	no	—
194	C₃²	—	6	3	no	—
195	C₃²	—	6	9	no	—
196	C₃²	—	6	3	no	—
197	C₃²	C₂×C₄×3₋¹⁺²	1	3	yes	complemented
198	C₃²	C₃²×C₄×D₃	1	3	yes	—
199	C₃²	C₃²×C₄×D₃	1	1	yes	characteristic
200	C₃²	C₂×C₃³×C₄	1	1	yes	characteristic
201	C₉	—	2	9	no	—
202	C₉	C₃²×C₄×D₃	1	9	yes	complemented
203	C₂×C₄	—	3	1	no	Sylow
204	C₂×C₃	—	2	3	no	—
205	C₂×C₃	—	2	1	no	—
206	C₂×C₃	—	3	2	no	—
207	C₂×C₃	—	3	6	no	—
208	C₂×C₃	—	3	3	no	—
209	C₂×C₃	—	6	3	no	—
210	C₂×C₃	—	9	6	no	—
211	C₂×C₃	C₂×D₃×3₋¹⁺²	1	3	yes	—
212	C₂×C₃	C₂²×C₃×3₋¹⁺²	1	1	yes	characteristic
213	C₂×C₃	C₂×C₃³×D₃	1	1	yes	characteristic
214	D₃	C₃×C₄×3₋¹⁺²	1	2	yes	direct product factor
215	C₂²	—	3	1	no	—
216	C₄	—	3	1	no	—
217	C₄	C₃×D₃×3₋¹⁺²	1	1	yes	characteristic, direct product factor
218	C₃	—	2	3	no	—
219	C₃	—	2	1	no	—
220	C₃	—	3	3	no	—
221	C₃	—	6	3	no	—
222	C₃	C₄×D₃×3₋¹⁺²	1	3	yes	direct product factor
223	C₃	C₂×C₃×C₄×3₋¹⁺²	1	1	yes	characteristic, complemented
224	C₃	C₃³×C₄×D₃	1	1	yes	characteristic
225	C₂	—	3	2	no	—
226	C₂	C₂×C₃×D₃×3₋¹⁺²	1	1	yes	characteristic

	subgroup	G/H
center	C₃²×C₄	C₃²×D₃
commutator	C₃²	C₂×C₃³×C₄
Frattini	C₂×C₃	C₂×C₃³×D₃
Fitting	C₃²×C₄×3₋¹⁺²	C₂
socle	C₂×C₃³	C₂²×C₃²
prime radical	C₂×C₃³×D₃	C₂×C₃
cosocle	C₂×C₃²	C₂²×C₃³
hypercenter	C₃×C₄×3₋¹⁺²	D₃
hypocenter	C₃	C₂×C₃×C₄×3₋¹⁺²
LCS	G ▷ C₃² ▷ C₃
UCS	C₁ ◁ C₃²×C₄ ◁ C₃×C₄×3₋¹⁺²

Complements

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

N	H ≅ G/N	type
2. C₃²×C₄×3₋¹⁺²	C₂	semidirect
3. C₃×Dic₃×3₋¹⁺²	C₂	semidirect
6. C₄×D₃×3₋¹⁺²	C₃	direct (N × H)
7. C₃×C₄×D₃×C₉	C₃	semidirect
9. C₃×D₃×3₋¹⁺²	C₄	direct (N × H)
20. C₃×C₄×3₋¹⁺²	D₃	direct (N × H)
21. C₃×C₄×3₋¹⁺²	C₂×C₃	semidirect
22. C₃²×C₄×C₉	C₂×C₃	semidirect
23. Dic₃×3₋¹⁺²	C₂×C₃	semidirect
24. C₃×C₉×Dic₃	C₂×C₃	semidirect
25. C₃²×3₋¹⁺²	C₂×C₄	semidirect
32. C₄×D₃×C₉	C₃²	semidirect
42. D₃×3₋¹⁺²	C₃×C₄	direct (N × H)
43. C₃×D₃×C₉	C₃×C₄	semidirect
65. C₄×3₋¹⁺²	C₃×D₃	direct (N × H)
69. C₃×C₄×C₉	C₃×D₃	semidirect
70. C₃×C₄×C₉	C₂×C₃²	semidirect
71. C₉×Dic₃	C₂×C₃²	semidirect
76. C₃×3₋¹⁺²	C₄×D₃	direct (N × H)
77. C₃×3₋¹⁺²	C₂×C₃×C₄	semidirect
78. C₃²×C₉	C₂×C₃×C₄	semidirect
83. C₃×C₄×D₃	3₋¹⁺²	direct (N × H)
106. D₃×C₉	C₃²×C₄	semidirect
124. C₃²×C₄	C₂×3₋¹⁺²	semidirect
129. C₃×Dic₃	C₂×3₋¹⁺²	semidirect
134. C₄×C₉	C₃²×D₃	semidirect
144. 3₋¹⁺²	C₃×C₄×D₃	direct (N × H)
147. C₃×C₉	C₃×C₄×D₃	semidirect
148. C₃×C₉	C₂×C₃²×C₄	semidirect
152. C₄×D₃	C₃×3₋¹⁺²	direct (N × H)
169. C₃×D₃	C₄×3₋¹⁺²	direct (N × H)
185. C₃×C₄	D₃×3₋¹⁺²	direct (N × H)
186. C₃×C₄	C₂×C₃×3₋¹⁺²	semidirect
188. Dic₃	C₂×C₃×3₋¹⁺²	semidirect
197. C₃²	C₂×C₄×3₋¹⁺²	semidirect
202. C₉	C₃²×C₄×D₃	semidirect
214. D₃	C₃×C₄×3₋¹⁺²	direct (N × H)
217. C₄	C₃×D₃×3₋¹⁺²	direct (N × H)
222. C₃	C₄×D₃×3₋¹⁺²	direct (N × H)
223. C₃	C₂×C₃×C₄×3₋¹⁺²	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×D₃×3₋¹⁺²	2
C₃²×C₄×3₋¹⁺²	2
C₃×Dic₃×3₋¹⁺²	2
C₃³×C₄×D₃	3
C₂×C₃×C₄×3₋¹⁺²	3
C₄×D₃×3₋¹⁺²	3
C₃×C₄×D₃×C₉	3

Subgroups — C3×C4×D3×3−1+2 #14518888

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₄×D₃×3₋¹⁺² #14518888