Subgroups — C₃×#10592404 #13162745

Order 1728.

820 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄²×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic
2	C₃³×#97	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#63647	C₂	1	2	yes	maximal
4	C₃×#63641	C₂	1	2	yes	maximal, complemented
5	C₃×#63630	C₂	1	1	yes	maximal, characteristic, complemented
6	#10592404	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₃×#4705	—	3	4	no	maximal
8	C₂×C₃×C₄×C₃⋊D₃	C₄	1	1	yes	characteristic
9	C₂×C₃×C₄×C₃⋊D₃	C₄	1	1	yes	characteristic
10	C₂×C₃×C₄×C₃⋊D₃	C₂²	1	1	yes	characteristic
11	C₂×C₃³×C₈	C₂²	1	2	yes	—
12	C₃³×C₄²	C₂²	1	1	yes	characteristic
13	C₃×C₄×C₃⋊Dic₃	C₄	1	1	yes	characteristic
14	C₃×C₄×C₃⋊Dic₃	C₄	1	1	yes	characteristic
15	C₃×C₄×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
16	C₂×C₃×C₃⋊(C₃⋊C₈)	C₂²	1	2	yes	—
17	C₄²×C₃⋊D₃	C₂×C₃	1	1	yes	characteristic
18	C₃×C₄²×D₃	—	3	4	no	—
19	C₃²×#97	—	2	4	no	—
20	C₃²×#97	D₃	1	4	yes	complemented
21	C₃²×#97	C₂×C₃	1	1	yes	characteristic, complemented
22	#63647	C₂×C₃	1	2	yes	—
23	#63641	C₂×C₃	1	2	yes	complemented
24	#63630	C₂×C₃	1	1	yes	characteristic, complemented
25	C₃×(C₂×D₃)⋊C₈	—	3	8	no	—
26	C₃×Dic₃⋊C₈	—	3	8	no	—
27	C₃×#803	—	3	4	no	—
28	C₂²×C₃×C₃⋊D₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃³×C₄	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃³×C₄	C₂×C₄	1	1	yes	characteristic
31	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
32	C₂×C₃×C₃⋊Dic₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃×C₃⋊Dic₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₃×C₃⋊Dic₃	C₂×C₄	1	1	yes	characteristic
35	C₃×C₄×C₃⋊D₃	—	2	2	no	—
36	C₃×C₄×C₃⋊D₃	—	2	2	no	—
37	C₃×C₄×C₃⋊D₃	—	2	2	no	—
38	C₃³×C₈	—	2	2	no	—
39	C₃×C₃⋊(C₃⋊C₈)	—	2	2	no	—
40	C₃×#432	—	9	1	no	—
41	#4705	—	3	4	no	—
42	C₂×C₄×C₃⋊D₃	C₃×C₄	1	1	yes	characteristic
43	C₂×C₄×C₃⋊D₃	C₃×C₄	1	1	yes	characteristic
44	C₂×C₄×C₃⋊D₃	C₂²×C₃	1	1	yes	characteristic
45	C₂×C₃×C₄×D₃	—	3	4	no	—
46	C₂×C₃×C₄×D₃	—	3	4	no	—
47	C₂×C₃×C₄×D₃	—	3	4	no	—
48	C₂×C₃²×C₈	—	2	8	no	—
49	C₂×C₃²×C₈	C₂×D₃	1	8	yes	—
50	C₂×C₃²×C₈	C₂²×C₃	1	2	yes	—
51	C₃²×C₄²	—	2	4	no	—
52	C₃²×C₄²	C₂×D₃	1	4	yes	—
53	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
54	C₄×C₃⋊Dic₃	C₃×C₄	1	1	yes	characteristic
55	C₄×C₃⋊Dic₃	C₃×C₄	1	1	yes	characteristic
56	C₄×C₃⋊Dic₃	C₂²×C₃	1	1	yes	characteristic
57	C₂×C₃⋊(C₃⋊C₈)	C₂²×C₃	1	2	yes	—
58	C₃×C₄×Dic₃	—	3	4	no	—
59	C₃×C₄×Dic₃	—	3	4	no	—
60	C₃×C₄×Dic₃	—	3	4	no	—
61	C₂×C₃×C₃⋊C₈	—	3	8	no	—
62	C₂²×C₃³	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₃×C₃⋊D₃	—	2	1	no	—
64	C₂×C₃×C₃⋊D₃	—	2	1	no	—
65	C₂×C₃×C₃⋊D₃	M₁₆	1	2	yes	—
66	C₃³×C₄	—	2	1	no	—
67	C₃³×C₄	—	2	1	no	—
68	C₃³×C₄	#41	1	2	yes	—
69	C₃×C₃⋊Dic₃	—	2	1	no	—
70	C₃×C₃⋊Dic₃	—	2	1	no	—
71	C₃×C₃⋊Dic₃	M₁₆	1	2	yes	—
72	C₂×C₃×C₄²	—	9	1	no	—
73	C₄²×D₃	—	3	4	no	—
74	C₃×C₄⋊C₈	—	9	2	no	—
75	C₃×C₂²⋊C₈	—	9	2	no	—
76	C₃×#97	—	2	4	no	—
77	C₃×#97	—	9	1	no	—
78	C₃×#97	C₃×D₃	1	4	yes	complemented
79	C₃×#97	C₃⋊D₃	1	1	yes	characteristic, complemented
80	(C₂×D₃)⋊C₈	—	3	8	no	—
81	Dic₃⋊C₈	—	3	8	no	—
82	#803	—	3	4	no	—
83	C₂²×C₃⋊D₃	C₂×C₃×C₄	1	1	yes	characteristic
84	C₂²×C₃×D₃	—	3	4	no	—
85	C₂×C₃²×C₄	—	2	4	no	—
86	C₂×C₃²×C₄	—	2	4	no	—
87	C₂×C₃²×C₄	—	2	4	no	—
88	C₂×C₃²×C₄	C₄×D₃	1	4	yes	—
89	C₂×C₃²×C₄	C₄×D₃	1	4	yes	—
90	C₂×C₃²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
91	C₂×C₃²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
92	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
93	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
94	C₂×C₃⋊Dic₃	C₂×C₃×C₄	1	1	yes	characteristic
95	C₂×C₃⋊Dic₃	C₂×C₃×C₄	1	1	yes	characteristic
96	C₂×C₃⋊Dic₃	C₂×C₃×C₄	1	1	yes	characteristic
97	C₄×C₃⋊D₃	—	2	2	no	—
98	C₄×C₃⋊D₃	—	2	2	no	—
99	C₄×C₃⋊D₃	—	2	2	no	—
100	C₂×C₃×Dic₃	—	3	4	no	—
101	C₂×C₃×Dic₃	—	3	4	no	—
102	C₂×C₃×Dic₃	—	3	4	no	—
103	C₃×C₄×D₃	—	6	8	no	—
104	C₃×C₄×D₃	—	6	8	no	—
105	C₃×C₄×D₃	—	6	8	no	—
106	C₃²×C₈	—	2	16	no	—
107	C₃²×C₈	—	2	8	no	—
108	C₃²×C₈	—	2	2	no	—
109	C₃⋊(C₃⋊C₈)	—	2	2	no	—
110	C₃×C₃⋊C₈	—	6	8	no	—
111	#432	—	9	1	no	Sylow
112	C₂×C₃³	#117	1	1	yes	characteristic
113	C₂×C₃³	C₂×M₁₆	1	1	yes	characteristic
114	C₂×C₃³	C₂×M₁₆	1	1	yes	characteristic
115	C₃×C₃⋊D₃	—	2	2	no	—
116	C₂²×C₃×C₄	—	9	1	no	—
117	C₂²×C₃×C₄	—	9	1	no	—
118	C₂²×C₃×C₄	—	9	1	no	—
119	C₂×C₄×D₃	—	3	4	no	—
120	C₂×C₄×D₃	—	3	4	no	—
121	C₂×C₄×D₃	—	3	4	no	—
122	C₂×C₃×C₈	—	2	8	no	—
123	C₂×C₃×C₈	—	9	2	no	—
124	C₂×C₃×C₈	C₂×C₃×D₃	1	8	yes	—
125	C₂×C₃×C₈	C₂×C₃⋊D₃	1	2	yes	—
126	C₃×C₄²	—	2	4	no	—
127	C₃×C₄²	—	9	1	no	—
128	C₃×C₄²	—	9	1	no	—
129	C₃×C₄²	—	9	1	no	—
130	C₃×C₄²	C₂×C₃×D₃	1	4	yes	—
131	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
132	C₄×Dic₃	—	3	4	no	—
133	C₄×Dic₃	—	3	4	no	—
134	C₄×Dic₃	—	3	4	no	—
135	C₂×C₃⋊C₈	—	3	8	no	—
136	C₂²×C₃²	—	2	4	no	—
137	C₂²×C₃²	C₂×C₄×D₃	1	4	yes	—
138	C₂²×C₃²	C₂²×C₃×C₄	1	1	yes	characteristic
139	C₂×C₃⋊D₃	—	2	1	no	—
140	C₂×C₃⋊D₃	—	2	1	no	—
141	C₂×C₃⋊D₃	C₃×M₁₆	1	2	yes	—
142	C₂×C₃×D₃	—	3	8	no	—
143	C₂×C₃×D₃	—	6	4	no	—
144	C₂×C₃×D₃	—	6	4	no	—
145	C₃²×C₄	—	2	1	no	—
146	C₃²×C₄	—	2	1	no	—
147	C₃²×C₄	—	2	8	no	—
148	C₃²×C₄	—	2	4	no	—
149	C₃²×C₄	—	2	4	no	—
150	C₃²×C₄	—	4	4	no	—
151	C₃²×C₄	—	4	4	no	—
152	C₃²×C₄	#235	1	8	yes	—
153	C₃²×C₄	C₃×#41	1	2	yes	—
154	C₃⋊Dic₃	—	2	1	no	—
155	C₃⋊Dic₃	—	2	1	no	—
156	C₃⋊Dic₃	C₃×M₁₆	1	2	yes	—
157	C₃×Dic₃	—	3	8	no	—
158	C₃×Dic₃	—	6	4	no	—
159	C₃×Dic₃	—	6	4	no	—
160	C₂×C₄²	—	9	1	no	—
161	C₄⋊C₈	—	9	2	no	—
162	C₂²⋊C₈	—	9	2	no	—
163	#97	—	9	1	no	—
164	#97	C₃×C₃⋊D₃	1	1	yes	characteristic, complemented
165	C₃³	#432	1	1	yes	characteristic, complemented, Sylow
166	C₂³×C₃	—	9	1	no	—
167	C₂²×D₃	—	3	4	no	—
168	C₂×C₃×C₄	—	2	4	no	—
169	C₂×C₃×C₄	—	2	4	no	—
170	C₂×C₃×C₄	—	2	4	no	—
171	C₂×C₃×C₄	—	9	1	no	—
172	C₂×C₃×C₄	—	9	1	no	—
173	C₂×C₃×C₄	—	9	1	no	—
174	C₂×C₃×C₄	—	18	2	no	—
175	C₂×C₃×C₄	—	18	2	no	—
176	C₂×C₃×C₄	—	18	2	no	—
177	C₂×C₃×C₄	C₃×C₄×D₃	1	4	yes	—
178	C₂×C₃×C₄	C₃×C₄×D₃	1	4	yes	—
179	C₂×C₃×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
180	C₂×C₃×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
181	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
182	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
183	C₂×Dic₃	—	3	4	no	—
184	C₂×Dic₃	—	3	4	no	—
185	C₂×Dic₃	—	3	4	no	—
186	C₄×D₃	—	6	8	no	—
187	C₄×D₃	—	6	8	no	—
188	C₄×D₃	—	6	8	no	—
189	C₃×C₈	—	2	16	no	—
190	C₃×C₈	—	2	8	no	—
191	C₃×C₈	—	2	2	no	—
192	C₃×C₈	—	18	2	no	—
193	C₃⋊C₈	—	6	8	no	—
194	C₂×C₃²	—	2	4	no	—
195	C₂×C₃²	—	2	4	no	—
196	C₂×C₃²	—	2	4	no	—
197	C₂×C₃²	#872	1	4	yes	—
198	C₂×C₃²	D₃×M₁₆	1	4	yes	—
199	C₂×C₃²	D₃×M₁₆	1	4	yes	—
200	C₂×C₃²	C₃×#117	1	1	yes	characteristic
201	C₂×C₃²	C₂×C₃×M₁₆	1	1	yes	characteristic
202	C₂×C₃²	C₂×C₃×M₁₆	1	1	yes	characteristic
203	C₃⋊D₃	—	2	2	no	—
204	C₃×D₃	—	6	8	no	—
205	C₂²×C₄	—	9	1	no	—
206	C₂²×C₄	—	9	1	no	—
207	C₂²×C₄	—	9	1	no	—
208	C₂×C₈	—	9	2	no	—
209	C₂×C₈	C₂×C₃×C₃⋊D₃	1	2	yes	—
210	C₄²	—	9	1	no	—
211	C₄²	—	9	1	no	—
212	C₄²	—	9	1	no	—
213	C₄²	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
214	C₂²×C₃	—	2	4	no	—
215	C₂²×C₃	—	9	2	no	—
216	C₂²×C₃	—	18	1	no	—
217	C₂²×C₃	—	18	1	no	—
218	C₂²×C₃	C₂×C₃×C₄×D₃	1	4	yes	—
219	C₂²×C₃	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
220	C₂×D₃	—	3	8	no	—
221	C₂×D₃	—	6	4	no	—
222	C₂×D₃	—	6	4	no	—
223	C₃×C₄	—	2	4	no	—
224	C₃×C₄	—	2	4	no	—
225	C₃×C₄	—	2	1	no	—
226	C₃×C₄	—	2	1	no	—
227	C₃×C₄	—	2	8	no	—
228	C₃×C₄	—	4	4	no	—
229	C₃×C₄	—	4	4	no	—
230	C₃×C₄	—	9	2	no	—
231	C₃×C₄	—	18	1	no	—
232	C₃×C₄	—	18	1	no	—
233	C₃×C₄	C₃×#235	1	8	yes	—
234	C₃×C₄	#3828	1	2	yes	—
235	Dic₃	—	3	8	no	—
236	Dic₃	—	6	4	no	—
237	Dic₃	—	6	4	no	—
238	C₃²	—	2	4	no	—
239	C₃²	#4705	1	4	yes	complemented
240	C₃²	C₃×#432	1	1	yes	characteristic, complemented
241	C₂³	—	9	1	no	—
242	C₂×C₄	—	9	1	no	—
243	C₂×C₄	—	9	1	no	—
244	C₂×C₄	—	9	1	no	—
245	C₂×C₄	—	18	2	no	—
246	C₂×C₄	—	18	2	no	—
247	C₂×C₄	—	18	2	no	—
248	C₂×C₄	C₃×C₄×C₃⋊D₃	1	1	yes	characteristic
249	C₂×C₄	C₃×C₄×C₃⋊D₃	1	1	yes	characteristic
250	C₂×C₄	C₂²×C₃×C₃⋊D₃	1	1	yes	characteristic
251	C₈	—	2	2	no	—
252	C₈	—	18	2	no	—
253	C₂×C₃	—	2	4	no	—
254	C₂×C₃	—	2	4	no	—
255	C₂×C₃	—	2	4	no	—
256	C₂×C₃	—	18	2	no	—
257	C₂×C₃	C₃×#872	1	4	yes	—
258	C₂×C₃	C₃×D₃×M₁₆	1	4	yes	—
259	C₂×C₃	C₃×D₃×M₁₆	1	4	yes	—
260	C₂×C₃	#64081	1	1	yes	characteristic
261	C₂×C₃	M₁₆×C₃⋊D₃	1	1	yes	characteristic
262	C₂×C₃	M₁₆×C₃⋊D₃	1	1	yes	characteristic
263	D₃	—	6	8	no	—
264	C₂²	—	9	2	no	—
265	C₂²	—	18	1	no	—
266	C₂²	—	18	1	no	—
267	C₂²	C₂×C₃×C₄×C₃⋊D₃	1	1	yes	characteristic
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	—	9	2	no	—
271	C₄	—	18	1	no	—
272	C₄	—	18	1	no	—
273	C₄	C₃×#3828	1	2	yes	—
274	C₃	—	2	4	no	—
275	C₃	C₃×#4705	1	4	yes	complemented
276	C₃	#10592404	1	1	yes	characteristic, direct product factor
277	C₂	—	18	2	no	—
278	C₂	C₃×#64081	1	1	yes	characteristic
279	C₂	C₃×M₁₆×C₃⋊D₃	1	1	yes	characteristic
280	C₂	C₃×M₁₆×C₃⋊D₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂²×C₃⋊D₃
commutator	C₂²×C₃²	C₂²×C₃×C₄
Frattini	C₂×C₄	C₂²×C₃×C₃⋊D₃
Fitting	C₃³×#97	C₂
socle	C₂²×C₃³	C₂²×C₄
prime radical	C₂²×C₃×C₃⋊D₃	C₂×C₄
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₃×#97	C₃⋊D₃
hypocenter	C₃²	C₃×#432
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₃×#97

Complements

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

N	H ≅ G/N	type
2. C₃³×#97	C₂	semidirect
4. C₃×#63641	C₂	semidirect
5. C₃×#63630	C₂	semidirect
6. #10592404	C₃	direct (N × H)
20. C₃²×#97	D₃	semidirect
21. C₃²×#97	C₂×C₃	semidirect
23. #63641	C₂×C₃	semidirect
24. #63630	C₂×C₃	semidirect
78. C₃×#97	C₃×D₃	semidirect
79. C₃×#97	C₃⋊D₃	semidirect
164. #97	C₃×C₃⋊D₃	semidirect
165. C₃³	#432	semidirect
239. C₃²	#4705	semidirect
240. C₃²	C₃×#432	semidirect
275. C₃	C₃×#4705	semidirect
276. C₃	#10592404	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₄²×C₃⋊D₃	2
C₃³×#97	2
C₃×#63647	2
C₃×#63641	2
C₃×#63630	2
#10592404	3
C₃×#4705	3

Subgroups — C3×#10592404 #13162745

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10592404 #13162745