Subgroups — C₃×#10589421 #13145979

Order 1728.

932 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃³×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₃×#64135	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₄×C₃⋊Dic₃	C₂	1	1	yes	maximal, characteristic
4	#10589421	C₃	1	1	yes	maximal, characteristic, direct product factor
5	C₃×#4555	—	3	4	no	maximal
6	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
7	C₂²×C₃×C₃⋊Dic₃	—	2	1	no	—
8	C₃³×M₁₆	—	2	1	no	—
9	C₃³×M₁₆	C₄	1	2	yes	complemented
10	C₂×C₃³×C₈	—	2	1	no	—
11	C₃×#3757	—	2	1	no	—
12	C₃×C₄⋊(C₃⋊Dic₃)	C₄	1	2	yes	complemented
13	C₃×C₄×C₃⋊Dic₃	—	2	1	no	—
14	C₃×C₄×C₃⋊Dic₃	—	2	1	no	—
15	C₃×C₄×C₃⋊Dic₃	C₄	1	2	yes	complemented
16	C₂×C₃²×M₁₆	—	2	4	no	—
17	C₂×C₃²×M₁₆	D₃	1	4	yes	—
18	C₂×C₃²×M₁₆	C₂×C₃	1	1	yes	characteristic
19	#64135	C₂×C₃	1	1	yes	characteristic
20	C₂×C₄×C₃⋊Dic₃	C₂×C₃	1	1	yes	characteristic
21	C₃×#926	—	3	4	no	—
22	C₂×C₃×C₄×Dic₃	—	3	4	no	—
23	C₂³×C₃³	D₄	1	1	yes	characteristic
24	C₂×C₃³×C₄	C₂×C₄	1	1	yes	characteristic
25	C₂×C₃³×C₄	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃³×C₄	C₂×C₄	1	1	yes	characteristic
27	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
28	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
29	C₂×C₃³×C₄	Q₈	1	1	yes	characteristic
30	C₂×C₃×C₃⋊Dic₃	—	2	4	no	—
31	C₂×C₃×C₃⋊Dic₃	—	2	2	no	—
32	C₂×C₃×C₃⋊Dic₃	—	2	2	no	—
33	C₃³×C₈	—	2	2	no	—
34	C₃×#339	—	9	1	no	—
35	#4555	—	3	4	no	—
36	C₂²×C₃²×C₄	—	2	4	no	—
37	C₂²×C₃²×C₄	C₂×D₃	1	4	yes	—
38	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
39	C₂²×C₃⋊Dic₃	—	2	1	no	—
40	C₂²×C₃×Dic₃	—	6	4	no	—
41	C₃²×M₁₆	—	2	8	no	—
42	C₃²×M₁₆	—	2	8	no	—
43	C₃²×M₁₆	—	2	4	no	—
44	C₃²×M₁₆	—	2	1	no	—
45	C₃²×M₁₆	Dic₃	1	8	yes	complemented
46	C₃²×M₁₆	C₃×C₄	1	2	yes	complemented
47	C₂×C₃²×C₈	—	2	8	no	—
48	C₂×C₃²×C₈	—	2	4	no	—
49	C₂×C₃²×C₈	—	2	1	no	—
50	#3757	—	2	1	no	—
51	C₄⋊(C₃⋊Dic₃)	C₃×C₄	1	2	yes	complemented
52	C₄×C₃⋊Dic₃	—	2	1	no	—
53	C₄×C₃⋊Dic₃	—	2	1	no	—
54	C₄×C₃⋊Dic₃	C₃×C₄	1	2	yes	complemented
55	C₃×#217	—	6	4	no	—
56	C₃×C₄⋊Dic₃	—	3	8	no	—
57	C₃×C₄×Dic₃	—	3	8	no	—
58	C₃×C₄×Dic₃	—	6	4	no	—
59	C₃×C₄×Dic₃	—	6	4	no	—
60	C₂²×C₃³	—	2	2	no	—
61	C₂²×C₃³	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₃³	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₃³	C₄⋊C₄	1	1	yes	characteristic
64	C₃³×C₄	C₄²	1	1	yes	characteristic
65	C₃³×C₄	C₂²⋊C₄	1	1	yes	characteristic
66	C₃³×C₄	C₄⋊C₄	1	1	yes	characteristic
67	C₃³×C₄	C₄⋊C₄	1	1	yes	characteristic
68	C₃×C₃⋊Dic₃	—	2	4	no	—
69	C₃×C₃⋊Dic₃	—	4	2	no	—
70	C₂×C₃×M₁₆	—	2	4	no	—
71	C₂×C₃×M₁₆	C₃×D₃	1	4	yes	—
72	C₂×C₃×M₁₆	C₃⋊D₃	1	1	yes	characteristic
73	C₃×#117	—	9	1	no	—
74	C₂×C₃×C₄²	—	9	1	no	—
75	#926	—	3	4	no	—
76	C₂×C₄×Dic₃	—	3	4	no	—
77	C₂³×C₃²	—	2	4	no	—
78	C₂³×C₃²	D₁₂	1	4	yes	—
79	C₂³×C₃²	C₃×D₄	1	1	yes	characteristic
80	C₂×C₃²×C₄	—	2	4	no	—
81	C₂×C₃²×C₄	—	2	4	no	—
82	C₂×C₃²×C₄	—	2	4	no	—
83	C₂×C₃²×C₄	—	2	4	no	—
84	C₂×C₃²×C₄	—	2	4	no	—
85	C₂×C₃²×C₄	—	2	4	no	—
86	C₂×C₃²×C₄	Dic₆	1	4	yes	—
87	C₂×C₃²×C₄	C₄×D₃	1	4	yes	—
88	C₂×C₃²×C₄	C₄×D₃	1	4	yes	—
89	C₂×C₃²×C₄	C₂×Dic₃	1	4	yes	—
90	C₂×C₃²×C₄	#67	1	4	yes	—
91	C₂×C₃²×C₄	#67	1	4	yes	—
92	C₂×C₃²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
93	C₂×C₃²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
94	C₂×C₃²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
95	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
96	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
97	C₂×C₃²×C₄	C₃×Q₈	1	1	yes	characteristic
98	C₂×C₃⋊Dic₃	—	2	4	no	—
99	C₂×C₃⋊Dic₃	—	2	2	no	—
100	C₂×C₃⋊Dic₃	—	2	2	no	—
101	C₂×C₃×Dic₃	—	6	16	no	—
102	C₂×C₃×Dic₃	—	6	8	no	—
103	C₂×C₃×Dic₃	—	6	8	no	—
104	C₃²×C₈	—	2	16	no	—
105	C₃²×C₈	—	2	8	no	—
106	C₃²×C₈	—	2	2	no	—
107	#339	—	9	1	no	Sylow
108	C₂×C₃³	—	2	1	no	—
109	C₂×C₃³	—	2	1	no	—
110	C₂×C₃³	#95	1	1	yes	characteristic
111	C₂×C₃³	#104	1	2	yes	—
112	C₂²×C₃×C₄	—	2	4	no	—
113	C₂²×C₃×C₄	—	18	1	no	—
114	C₂²×C₃×C₄	C₂×C₃×D₃	1	4	yes	—
115	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
116	C₂²×Dic₃	—	6	4	no	—
117	C₃×M₁₆	—	2	8	no	—
118	C₃×M₁₆	—	2	4	no	—
119	C₃×M₁₆	—	2	1	no	—
120	C₃×M₁₆	—	2	8	no	—
121	C₃×M₁₆	C₃×Dic₃	1	8	yes	complemented
122	C₃×M₁₆	C₃⋊Dic₃	1	2	yes	complemented
123	C₂×C₃×C₈	—	2	8	no	—
124	C₂×C₃×C₈	—	2	4	no	—
125	C₂×C₃×C₈	—	2	1	no	—
126	C₃×C₄⋊C₄	—	9	2	no	—
127	C₃×C₂²⋊C₄	—	18	1	no	—
128	C₃×C₄²	—	9	2	no	—
129	C₃×C₄²	—	18	1	no	—
130	C₃×C₄²	—	18	1	no	—
131	#217	—	6	4	no	—
132	C₄⋊Dic₃	—	3	8	no	—
133	C₄×Dic₃	—	3	8	no	—
134	C₄×Dic₃	—	6	4	no	—
135	C₄×Dic₃	—	6	4	no	—
136	C₂²×C₃²	—	2	2	no	—
137	C₂²×C₃²	—	2	4	no	—
138	C₂²×C₃²	—	2	4	no	—
139	C₂²×C₃²	—	2	4	no	—
140	C₂²×C₃²	—	2	8	no	—
141	C₂²×C₃²	—	4	8	no	—
142	C₂²×C₃²	C₄⋊Dic₃	1	4	yes	—
143	C₂²×C₃²	(C₂×D₃)⋊C₄	1	4	yes	—
144	C₂²×C₃²	(C₂×D₃)⋊C₄	1	4	yes	—
145	C₂²×C₃²	C₃×C₂²⋊C₄	1	1	yes	characteristic
146	C₂²×C₃²	C₃×C₂²⋊C₄	1	1	yes	characteristic
147	C₂²×C₃²	C₃×C₄⋊C₄	1	1	yes	characteristic
148	C₃²×C₄	—	2	4	no	—
149	C₃²×C₄	—	2	4	no	—
150	C₃²×C₄	—	2	4	no	—
151	C₃²×C₄	—	2	4	no	—
152	C₃²×C₄	C₄×Dic₃	1	4	yes	—
153	C₃²×C₄	Dic₃⋊C₄	1	4	yes	—
154	C₃²×C₄	Dic₃⋊C₄	1	4	yes	—
155	C₃²×C₄	#217	1	4	yes	—
156	C₃²×C₄	C₃×C₄²	1	1	yes	characteristic
157	C₃²×C₄	C₃×C₂²⋊C₄	1	1	yes	characteristic
158	C₃²×C₄	C₃×C₄⋊C₄	1	1	yes	characteristic
159	C₃²×C₄	C₃×C₄⋊C₄	1	1	yes	characteristic
160	C₃⋊Dic₃	—	2	4	no	—
161	C₃⋊Dic₃	—	4	2	no	—
162	C₃×Dic₃	—	6	16	no	—
163	C₃×Dic₃	—	12	8	no	—
164	C₂×M₁₆	C₃×C₃⋊D₃	1	1	yes	characteristic
165	#117	—	9	1	no	—
166	C₂×C₄²	—	9	1	no	—
167	C₃³	#339	1	1	yes	characteristic, complemented, Sylow
168	C₂³×C₃	—	2	4	no	—
169	C₂³×C₃	C₃×D₁₂	1	4	yes	—
170	C₂³×C₃	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
171	C₂×C₃×C₄	—	2	4	no	—
172	C₂×C₃×C₄	—	2	4	no	—
173	C₂×C₃×C₄	—	2	4	no	—
174	C₂×C₃×C₄	—	2	4	no	—
175	C₂×C₃×C₄	—	2	4	no	—
176	C₂×C₃×C₄	—	2	4	no	—
177	C₂×C₃×C₄	—	18	4	no	—
178	C₂×C₃×C₄	—	18	2	no	—
179	C₂×C₃×C₄	—	18	2	no	—
180	C₂×C₃×C₄	C₃×Dic₆	1	4	yes	—
181	C₂×C₃×C₄	C₃×C₄×D₃	1	4	yes	—
182	C₂×C₃×C₄	C₃×C₄×D₃	1	4	yes	—
183	C₂×C₃×C₄	C₂×C₃×Dic₃	1	4	yes	—
184	C₂×C₃×C₄	C₃×#67	1	4	yes	—
185	C₂×C₃×C₄	C₃×#67	1	4	yes	—
186	C₂×C₃×C₄	#634	1	1	yes	characteristic
187	C₂×C₃×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
188	C₂×C₃×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
189	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
190	C₂×C₃×C₄	#638	1	1	yes	characteristic
191	C₂×C₃×C₄	#638	1	1	yes	characteristic
192	C₂×Dic₃	—	6	16	no	—
193	C₂×Dic₃	—	6	8	no	—
194	C₂×Dic₃	—	6	8	no	—
195	C₃×C₈	—	2	16	no	—
196	C₃×C₈	—	2	8	no	—
197	C₃×C₈	—	2	2	no	—
198	C₂×C₃²	—	2	1	no	—
199	C₂×C₃²	—	2	1	no	—
200	C₂×C₃²	—	2	8	no	—
201	C₂×C₃²	—	2	4	no	—
202	C₂×C₃²	—	2	4	no	—
203	C₂×C₃²	—	2	4	no	—
204	C₂×C₃²	—	4	4	no	—
205	C₂×C₃²	—	4	4	no	—
206	C₂×C₃²	#825	1	8	yes	—
207	C₂×C₃²	C₃⋊#95	1	4	yes	—
208	C₂×C₃²	C₃×#95	1	1	yes	characteristic
209	C₂×C₃²	C₃×#104	1	2	yes	—
210	C₂²×C₄	—	18	1	no	—
211	C₂²×C₄	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
212	M₁₆	—	2	1	no	—
213	M₁₆	C₃×C₃⋊Dic₃	1	2	yes	complemented
214	C₂×C₈	—	2	1	no	—
215	C₄⋊C₄	—	9	2	no	—
216	C₂²⋊C₄	—	18	1	no	—
217	C₄²	—	9	2	no	—
218	C₄²	—	18	1	no	—
219	C₄²	—	18	1	no	—
220	C₂²×C₃	—	2	8	no	—
221	C₂²×C₃	—	2	2	no	—
222	C₂²×C₃	—	2	4	no	—
223	C₂²×C₃	—	2	4	no	—
224	C₂²×C₃	—	2	4	no	—
225	C₂²×C₃	—	4	8	no	—
226	C₂²×C₃	C₃×C₄⋊Dic₃	1	4	yes	—
227	C₂²×C₃	C₃×(C₂×D₃)⋊C₄	1	4	yes	—
228	C₂²×C₃	C₃×(C₂×D₃)⋊C₄	1	4	yes	—
229	C₂²×C₃	C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
230	C₂²×C₃	#3752	1	1	yes	characteristic
231	C₂²×C₃	#3752	1	1	yes	characteristic
232	C₃×C₄	—	2	4	no	—
233	C₃×C₄	—	2	4	no	—
234	C₃×C₄	—	2	4	no	—
235	C₃×C₄	—	2	4	no	—
236	C₃×C₄	—	18	4	no	—
237	C₃×C₄	—	36	2	no	—
238	C₃×C₄	C₃×C₄×Dic₃	1	4	yes	—
239	C₃×C₄	C₃×Dic₃⋊C₄	1	4	yes	—
240	C₃×C₄	C₃×Dic₃⋊C₄	1	4	yes	—
241	C₃×C₄	C₃×#217	1	4	yes	—
242	C₃×C₄	C₄×C₃⋊Dic₃	1	1	yes	characteristic
243	C₃×C₄	#3750	1	1	yes	characteristic
244	C₃×C₄	#3750	1	1	yes	characteristic
245	C₃×C₄	#3757	1	1	yes	characteristic
246	Dic₃	—	6	16	no	—
247	Dic₃	—	12	8	no	—
248	C₃²	—	2	4	no	—
249	C₃²	#4555	1	4	yes	complemented
250	C₃²	C₃×#339	1	1	yes	characteristic, complemented
251	C₂³	C₃×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
252	C₂×C₄	—	18	4	no	—
253	C₂×C₄	—	18	2	no	—
254	C₂×C₄	—	18	2	no	—
255	C₂×C₄	C₃×#634	1	1	yes	characteristic
256	C₂×C₄	C₃×C₄×C₃⋊D₃	1	1	yes	characteristic
257	C₂×C₄	C₃×C₄×C₃⋊D₃	1	1	yes	characteristic
258	C₂×C₄	C₂×C₃×C₃⋊Dic₃	1	1	yes	characteristic
259	C₂×C₄	C₃×#638	1	1	yes	characteristic
260	C₂×C₄	C₃×#638	1	1	yes	characteristic
261	C₈	—	2	2	no	—
262	C₂×C₃	—	2	4	no	—
263	C₂×C₃	—	2	4	no	—
264	C₂×C₃	—	2	1	no	—
265	C₂×C₃	—	2	1	no	—
266	C₂×C₃	—	2	8	no	—
267	C₂×C₃	—	2	4	no	—
268	C₂×C₃	—	4	4	no	—
269	C₂×C₃	—	4	4	no	—
270	C₂×C₃	C₃×#825	1	8	yes	—
271	C₂×C₃	C₃×C₃⋊#95	1	4	yes	—
272	C₂×C₃	#63652	1	2	yes	—
273	C₂×C₃	#63658	1	1	yes	characteristic
274	C₂²	—	2	2	no	—
275	C₂²	C₃×C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
276	C₂²	C₃×#3752	1	1	yes	characteristic
277	C₂²	C₃×#3752	1	1	yes	characteristic
278	C₄	—	18	4	no	—
279	C₄	—	36	2	no	—
280	C₄	C₃×C₄×C₃⋊Dic₃	1	1	yes	characteristic
281	C₄	C₃×#3750	1	1	yes	characteristic
282	C₄	C₃×#3750	1	1	yes	characteristic
283	C₄	C₃×#3757	1	1	yes	characteristic
284	C₃	—	2	4	no	—
285	C₃	C₃×#4555	1	4	yes	complemented
286	C₃	#10589421	1	1	yes	characteristic, direct product factor
287	C₂	—	2	1	no	—
288	C₂	—	2	1	no	—
289	C₂	C₃×#63652	1	2	yes	—
290	C₂	C₃×#63658	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	#638
commutator	C₃²×C₄	C₃×C₄²
Frattini	C₂²×C₄	C₂×C₃×C₃⋊D₃
Fitting	C₂×C₃³×M₁₆	C₂
socle	C₂²×C₃³	C₂²⋊C₄
prime radical	C₂³×C₃³	D₄
cosocle	C₂²×C₃²×C₄	C₂²×C₃
hypercenter	C₂×C₃×M₁₆	C₃⋊D₃
hypocenter	C₃²	C₃×#339
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×M₁₆

Complements

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

N	H ≅ G/N	type
4. #10589421	C₃	direct (N × H)
9. C₃³×M₁₆	C₄	semidirect
12. C₃×C₄⋊(C₃⋊Dic₃)	C₄	semidirect
15. C₃×C₄×C₃⋊Dic₃	C₄	semidirect
45. C₃²×M₁₆	Dic₃	semidirect
46. C₃²×M₁₆	C₃×C₄	semidirect
51. C₄⋊(C₃⋊Dic₃)	C₃×C₄	semidirect
54. C₄×C₃⋊Dic₃	C₃×C₄	semidirect
121. C₃×M₁₆	C₃×Dic₃	semidirect
122. C₃×M₁₆	C₃⋊Dic₃	semidirect
167. C₃³	#339	semidirect
213. M₁₆	C₃×C₃⋊Dic₃	semidirect
249. C₃²	#4555	semidirect
250. C₃²	C₃×#339	semidirect
285. C₃	C₃×#4555	semidirect
286. C₃	#10589421	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×M₁₆	2
C₃×#64135	2
C₂×C₃×C₄×C₃⋊Dic₃	2
#10589421	3
C₃×#4555	3

Subgroups — C3×#10589421 #13145979

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10589421 #13145979