Subgroups — C₃×#10592542 #13162883

Order 1728.

992 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#3745	C₂	1	2	yes	maximal, complemented
2	C₃×#64083	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₄×C₄⋊(C₃⋊D₃)	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃³×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#63648	C₂	1	2	yes	maximal, complemented
6	#10592542	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₃×#4843	—	3	4	no	maximal
8	C₂×C₃×C₄⋊(C₃⋊D₃)	—	2	1	no	—
9	C₂×C₃×C₄⋊(C₃⋊D₃)	—	2	1	no	—
10	C₂×C₃×C₄⋊(C₃⋊D₃)	C₂²	1	2	yes	—
11	C₂×C₃×C₄⋊(C₃⋊D₃)	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₄×C₃⋊D₃	—	2	1	no	—
13	C₂×C₃³×C₈	C₂²	1	2	yes	—
14	C₃³×C₄²	C₂²	1	1	yes	characteristic
15	C₃×#3752	—	2	1	no	—
16	C₃×C₄⋊(C₃⋊Dic₃)	C₂²	1	1	yes	characteristic
17	C₃×#3745	—	2	4	no	—
18	C₂×#3745	C₂×C₃	1	2	yes	complemented
19	#64083	C₂×C₃	1	1	yes	characteristic, complemented
20	C₄×C₄⋊(C₃⋊D₃)	C₂×C₃	1	1	yes	characteristic, complemented
21	C₂×C₃×D₂₄	—	3	8	no	—
22	C₃×#874	—	3	4	no	—
23	C₃×C₄×D₁₂	—	3	4	no	—
24	C₃²×C₄⋊C₈	—	2	4	no	—
25	C₃²×C₄⋊C₈	D₃	1	4	yes	complemented
26	C₃²×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
27	#63648	C₂×C₃	1	2	yes	complemented
28	C₃×#821	—	3	8	no	—
29	C₂²×C₃×C₃⋊D₃	—	2	1	no	—
30	C₂²×C₃×C₃⋊D₃	—	2	2	no	—
31	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
32	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
33	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
34	C₂×C₃×C₃⋊Dic₃	—	2	1	no	—
35	C₃×C₄⋊(C₃⋊D₃)	—	2	2	no	—
36	C₃×C₄⋊(C₃⋊D₃)	—	2	1	no	—
37	C₃×C₄⋊(C₃⋊D₃)	—	2	2	no	—
38	C₃×C₄⋊(C₃⋊D₃)	—	4	2	no	—
39	C₃×C₄⋊(C₃⋊D₃)	—	4	2	no	—
40	C₃×C₄⋊(C₃⋊D₃)	D₄	1	2	yes	complemented
41	C₃×C₄×C₃⋊D₃	—	4	2	no	—
42	C₃³×C₈	—	2	2	no	—
43	C₃×#459	—	9	1	no	—
44	#4843	—	3	4	no	—
45	C₂×C₄⋊(C₃⋊D₃)	—	2	1	no	—
46	C₂×C₄⋊(C₃⋊D₃)	—	2	1	no	—
47	C₂×C₄⋊(C₃⋊D₃)	C₂²×C₃	1	2	yes	—
48	C₂×C₄⋊(C₃⋊D₃)	C₂²×C₃	1	1	yes	characteristic
49	C₂×C₄×C₃⋊D₃	—	2	1	no	—
50	C₂×C₃×D₁₂	—	3	4	no	—
51	C₂×C₃×D₁₂	—	3	8	no	—
52	C₂×C₃×D₁₂	—	6	4	no	—
53	C₂×C₃×D₁₂	—	6	4	no	—
54	C₂×C₃×C₄×D₃	—	6	4	no	—
55	C₂×C₃²×C₈	—	2	8	no	—
56	C₂×C₃²×C₈	C₂×D₃	1	8	yes	—
57	C₂×C₃²×C₈	C₂²×C₃	1	2	yes	—
58	C₃²×C₄²	—	2	4	no	—
59	C₃²×C₄²	C₂×D₃	1	4	yes	—
60	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
61	#3752	—	2	1	no	—
62	C₄⋊(C₃⋊Dic₃)	C₂²×C₃	1	1	yes	characteristic
63	#3745	—	2	4	no	—
64	C₃×(C₂×D₃)⋊C₄	—	6	4	no	—
65	C₃×C₄⋊Dic₃	—	3	4	no	—
66	C₃×D₂₄	—	6	16	no	—
67	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
68	C₂×C₃×C₃⋊D₃	—	2	2	no	—
69	C₂×C₃×C₃⋊D₃	—	4	1	no	—
70	C₂×C₃×C₃⋊D₃	—	4	1	no	—
71	C₂×C₃×C₃⋊D₃	—	4	2	no	—
72	C₂×C₃×C₃⋊D₃	—	4	2	no	—
73	C₂×C₃×C₃⋊D₃	—	4	2	no	—
74	C₃³×C₄	—	2	1	no	—
75	C₃³×C₄	D₈	1	2	yes	complemented
76	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
77	C₃³×C₄	#41	1	1	yes	characteristic
78	C₃×C₃⋊Dic₃	—	4	1	no	—
79	C₂×C₃×D₈	—	9	2	no	—
80	C₃×#127	—	9	1	no	—
81	C₃×C₄×D₄	—	9	1	no	—
82	C₂×D₂₄	—	3	8	no	—
83	#874	—	3	4	no	—
84	C₄×D₁₂	—	3	4	no	—
85	C₃×C₄⋊C₈	—	2	4	no	—
86	C₃×C₄⋊C₈	C₃×D₃	1	4	yes	complemented
87	C₃×C₄⋊C₈	C₃⋊D₃	1	1	yes	characteristic, complemented
88	C₃×#102	—	9	2	no	—
89	#821	—	3	8	no	—
90	C₂²×C₃⋊D₃	—	2	1	no	—
91	C₂²×C₃⋊D₃	—	2	2	no	—
92	C₂²×C₃×D₃	—	6	4	no	—
93	C₂²×C₃×D₃	—	6	8	no	—
94	C₂×C₃²×C₄	—	2	4	no	—
95	C₂×C₃²×C₄	—	2	4	no	—
96	C₂×C₃²×C₄	—	2	4	no	—
97	C₂×C₃²×C₄	D₁₂	1	4	yes	—
98	C₂×C₃²×C₄	D₁₂	1	4	yes	—
99	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
100	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
101	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
102	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
103	C₂×C₃⋊Dic₃	—	2	1	no	—
104	C₄⋊(C₃⋊D₃)	—	2	2	no	—
105	C₄⋊(C₃⋊D₃)	—	2	1	no	—
106	C₄⋊(C₃⋊D₃)	—	2	2	no	—
107	C₄⋊(C₃⋊D₃)	—	4	2	no	—
108	C₄⋊(C₃⋊D₃)	—	4	2	no	—
109	C₄⋊(C₃⋊D₃)	C₃×D₄	1	2	yes	complemented
110	C₄×C₃⋊D₃	—	4	2	no	—
111	C₂×C₃×Dic₃	—	6	4	no	—
112	C₃×D₁₂	—	3	8	no	—
113	C₃×D₁₂	—	6	8	no	—
114	C₃×D₁₂	—	6	4	no	—
115	C₃×D₁₂	—	6	8	no	—
116	C₃×D₁₂	—	12	8	no	—
117	C₃×D₁₂	—	12	8	no	—
118	C₃×C₄×D₃	—	12	8	no	—
119	C₃²×C₈	—	2	2	no	—
120	C₃²×C₈	—	2	8	no	—
121	C₃²×C₈	—	4	8	no	—
122	#459	—	9	1	no	Sylow
123	C₂×C₃³	#121	1	1	yes	characteristic
124	C₂×C₃³	C₂×D₈	1	1	yes	characteristic
125	C₂×C₃³	AGL(1,ℤ/8)	1	1	yes	characteristic
126	C₃×C₃⋊D₃	—	4	2	no	—
127	C₃×C₃⋊D₃	—	8	2	no	—
128	C₂×C₃×D₄	—	9	1	no	—
129	C₂×C₃×D₄	—	9	2	no	—
130	C₂×C₃×D₄	—	18	1	no	—
131	C₂×C₃×D₄	—	18	1	no	—
132	C₂²×C₃×C₄	—	18	1	no	—
133	C₂×D₁₂	—	3	4	no	—
134	C₂×D₁₂	—	3	8	no	—
135	C₂×D₁₂	—	6	4	no	—
136	C₂×D₁₂	—	6	4	no	—
137	C₂×C₄×D₃	—	6	4	no	—
138	C₃×D₈	—	18	4	no	—
139	C₂×C₃×C₈	—	2	8	no	—
140	C₂×C₃×C₈	C₂×C₃×D₃	1	8	yes	—
141	C₂×C₃×C₈	C₂×C₃⋊D₃	1	2	yes	—
142	C₃×C₄⋊C₄	—	9	1	no	—
143	C₃×C₂²⋊C₄	—	18	1	no	—
144	C₃×C₄²	—	2	4	no	—
145	C₃×C₄²	C₂×C₃×D₃	1	4	yes	—
146	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
147	(C₂×D₃)⋊C₄	—	6	4	no	—
148	C₄⋊Dic₃	—	3	4	no	—
149	D₂₄	—	6	16	no	—
150	C₂²×C₃²	—	2	4	no	—
151	C₂²×C₃²	C₂×D₁₂	1	4	yes	—
152	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
153	C₂×C₃⋊D₃	—	2	2	no	—
154	C₂×C₃⋊D₃	—	4	1	no	—
155	C₂×C₃⋊D₃	—	4	1	no	—
156	C₂×C₃⋊D₃	—	4	2	no	—
157	C₂×C₃⋊D₃	—	4	2	no	—
158	C₂×C₃⋊D₃	—	4	2	no	—
159	C₂×C₃×D₃	—	6	8	no	—
160	C₂×C₃×D₃	—	12	4	no	—
161	C₂×C₃×D₃	—	12	4	no	—
162	C₂×C₃×D₃	—	12	8	no	—
163	C₂×C₃×D₃	—	12	8	no	—
164	C₂×C₃×D₃	—	12	8	no	—
165	C₃²×C₄	—	2	1	no	—
166	C₃²×C₄	—	2	8	no	—
167	C₃²×C₄	—	2	4	no	—
168	C₃²×C₄	—	2	4	no	—
169	C₃²×C₄	—	2	4	no	—
170	C₃²×C₄	—	4	4	no	—
171	C₃²×C₄	D₂₄	1	8	yes	complemented
172	C₃²×C₄	C₃×D₈	1	2	yes	complemented
173	C₃²×C₄	D₃×D₄	1	4	yes	—
174	C₃²×C₄	#239	1	4	yes	—
175	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
176	C₃²×C₄	C₃×#41	1	1	yes	characteristic
177	C₃⋊Dic₃	—	4	1	no	—
178	C₃×Dic₃	—	12	4	no	—
179	C₂×D₈	—	9	2	no	—
180	#127	—	9	1	no	—
181	C₄×D₄	—	9	1	no	—
182	C₄⋊C₈	C₃×C₃⋊D₃	1	1	yes	characteristic, complemented
183	#102	—	9	2	no	—
184	C₃³	#459	1	1	yes	characteristic, complemented, Sylow
185	C₂³×C₃	—	18	1	no	—
186	C₂³×C₃	—	18	2	no	—
187	C₂²×D₃	—	6	4	no	—
188	C₂²×D₃	—	6	8	no	—
189	C₃×D₄	—	9	2	no	—
190	C₃×D₄	—	18	2	no	—
191	C₃×D₄	—	18	1	no	—
192	C₃×D₄	—	18	2	no	—
193	C₃×D₄	—	36	2	no	—
194	C₃×D₄	—	36	2	no	—
195	C₂×C₃×C₄	—	2	4	no	—
196	C₂×C₃×C₄	—	2	4	no	—
197	C₂×C₃×C₄	—	2	4	no	—
198	C₂×C₃×C₄	—	18	1	no	—
199	C₂×C₃×C₄	—	36	2	no	—
200	C₂×C₃×C₄	C₃×D₁₂	1	4	yes	—
201	C₂×C₃×C₄	C₃×D₁₂	1	4	yes	—
202	C₂×C₃×C₄	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
203	C₂×C₃×C₄	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
204	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
205	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
206	C₂×Dic₃	—	6	4	no	—
207	D₁₂	—	3	8	no	—
208	D₁₂	—	6	8	no	—
209	D₁₂	—	6	4	no	—
210	D₁₂	—	6	8	no	—
211	D₁₂	—	12	8	no	—
212	D₁₂	—	12	8	no	—
213	C₄×D₃	—	12	8	no	—
214	C₃×C₈	—	2	8	no	—
215	C₃×C₈	—	2	2	no	—
216	C₃×C₈	—	4	8	no	—
217	C₂×C₃²	—	2	4	no	—
218	C₂×C₃²	—	2	4	no	—
219	C₂×C₃²	—	2	4	no	—
220	C₂×C₃²	#895	1	4	yes	—
221	C₂×C₃²	C₂×D₂₄	1	4	yes	—
222	C₂×C₃²	#908	1	4	yes	—
223	C₂×C₃²	C₃×#121	1	1	yes	characteristic
224	C₂×C₃²	C₂×C₃×D₈	1	1	yes	characteristic
225	C₂×C₃²	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
226	C₃⋊D₃	—	4	2	no	—
227	C₃⋊D₃	—	8	2	no	—
228	C₃×D₃	—	12	8	no	—
229	C₃×D₃	—	24	8	no	—
230	C₂×D₄	—	9	1	no	—
231	C₂×D₄	—	9	2	no	—
232	C₂×D₄	—	18	1	no	—
233	C₂×D₄	—	18	1	no	—
234	C₂²×C₄	—	18	1	no	—
235	D₈	—	18	4	no	—
236	C₂×C₈	C₂×C₃×C₃⋊D₃	1	2	yes	—
237	C₄⋊C₄	—	9	1	no	—
238	C₂²⋊C₄	—	18	1	no	—
239	C₄²	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
240	C₂²×C₃	—	2	4	no	—
241	C₂²×C₃	—	18	2	no	—
242	C₂²×C₃	—	36	1	no	—
243	C₂²×C₃	—	36	1	no	—
244	C₂²×C₃	—	36	2	no	—
245	C₂²×C₃	—	36	2	no	—
246	C₂²×C₃	—	36	2	no	—
247	C₂²×C₃	C₂×C₃×D₁₂	1	4	yes	—
248	C₂²×C₃	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
249	C₂×D₃	—	6	8	no	—
250	C₂×D₃	—	12	4	no	—
251	C₂×D₃	—	12	4	no	—
252	C₂×D₃	—	12	8	no	—
253	C₂×D₃	—	12	8	no	—
254	C₂×D₃	—	12	8	no	—
255	C₃×C₄	—	2	4	no	—
256	C₃×C₄	—	2	1	no	—
257	C₃×C₄	—	2	8	no	—
258	C₃×C₄	—	2	4	no	—
259	C₃×C₄	—	2	4	no	—
260	C₃×C₄	—	4	4	no	—
261	C₃×C₄	—	36	1	no	—
262	C₃×C₄	C₃×D₂₄	1	8	yes	complemented
263	C₃×C₄	#3745	1	2	yes	complemented
264	C₃×C₄	C₃×D₃×D₄	1	4	yes	—
265	C₃×C₄	C₃×#239	1	4	yes	—
266	C₃×C₄	D₄×C₃⋊D₃	1	1	yes	characteristic
267	C₃×C₄	#3832	1	1	yes	characteristic
268	Dic₃	—	12	4	no	—
269	C₃²	—	2	4	no	—
270	C₃²	#4843	1	4	yes	complemented
271	C₃²	C₃×#459	1	1	yes	characteristic, complemented
272	C₂³	—	18	1	no	—
273	C₂³	—	18	2	no	—
274	D₄	—	9	2	no	—
275	D₄	—	18	2	no	—
276	D₄	—	18	1	no	—
277	D₄	—	18	2	no	—
278	D₄	—	36	2	no	—
279	D₄	—	36	2	no	—
280	C₂×C₄	—	18	1	no	—
281	C₂×C₄	—	36	2	no	—
282	C₂×C₄	C₃×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
283	C₂×C₄	C₃×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
284	C₂×C₄	C₂²×C₃×C₃⋊D₃	1	1	yes	characteristic
285	C₈	—	2	2	no	—
286	C₂×C₃	—	2	4	no	—
287	C₂×C₃	—	2	4	no	—
288	C₂×C₃	—	2	4	no	—
289	C₂×C₃	—	36	2	no	—
290	C₂×C₃	—	72	2	no	—
291	C₂×C₃	C₃×#895	1	4	yes	—
292	C₂×C₃	C₂×C₃×D₂₄	1	4	yes	—
293	C₂×C₃	C₃×#908	1	4	yes	—
294	C₂×C₃	#64104	1	1	yes	characteristic
295	C₂×C₃	C₂×#3745	1	1	yes	characteristic
296	C₂×C₃	#64117	1	1	yes	characteristic
297	D₃	—	12	8	no	—
298	D₃	—	24	8	no	—
299	C₂²	—	18	2	no	—
300	C₂²	—	36	1	no	—
301	C₂²	—	36	1	no	—
302	C₂²	—	36	2	no	—
303	C₂²	—	36	2	no	—
304	C₂²	—	36	2	no	—
305	C₂²	C₂×C₃×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
306	C₄	—	2	1	no	—
307	C₄	—	36	1	no	—
308	C₄	C₃×#3745	1	2	yes	complemented
309	C₄	C₃×D₄×C₃⋊D₃	1	1	yes	characteristic
310	C₄	C₃×#3832	1	1	yes	characteristic
311	C₃	—	2	4	no	—
312	C₃	C₃×#4843	1	4	yes	complemented
313	C₃	#10592542	1	1	yes	characteristic, direct product factor
314	C₂	—	36	2	no	—
315	C₂	—	72	2	no	—
316	C₂	C₃×#64104	1	1	yes	characteristic
317	C₂	C₂×C₃×#3745	1	1	yes	characteristic
318	C₂	C₃×#64117	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂×C₄⋊(C₃⋊D₃)
commutator, cosocle	C₂×C₃²×C₄	C₂³×C₃
Frattini	C₂×C₄	C₂²×C₃×C₃⋊D₃
Fitting	C₃³×C₄⋊C₈	C₂
socle	C₂²×C₃³	C₂×D₄
hypercenter	C₃×C₄⋊C₈	C₃⋊D₃
hypocenter	C₃²	C₃×#459
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#3745	C₂	semidirect
2. C₃×#64083	C₂	semidirect
3. C₃×C₄×C₄⋊(C₃⋊D₃)	C₂	semidirect
4. C₃³×C₄⋊C₈	C₂	semidirect
5. C₃×#63648	C₂	semidirect
6. #10592542	C₃	direct (N × H)
18. C₂×#3745	C₂×C₃	semidirect
19. #64083	C₂×C₃	semidirect
20. C₄×C₄⋊(C₃⋊D₃)	C₂×C₃	semidirect
25. C₃²×C₄⋊C₈	D₃	semidirect
26. C₃²×C₄⋊C₈	C₂×C₃	semidirect
27. #63648	C₂×C₃	semidirect
40. C₃×C₄⋊(C₃⋊D₃)	D₄	semidirect
75. C₃³×C₄	D₈	semidirect
86. C₃×C₄⋊C₈	C₃×D₃	semidirect
87. C₃×C₄⋊C₈	C₃⋊D₃	semidirect
109. C₄⋊(C₃⋊D₃)	C₃×D₄	semidirect
171. C₃²×C₄	D₂₄	semidirect
172. C₃²×C₄	C₃×D₈	semidirect
182. C₄⋊C₈	C₃×C₃⋊D₃	semidirect
184. C₃³	#459	semidirect
262. C₃×C₄	C₃×D₂₄	semidirect
263. C₃×C₄	#3745	semidirect
270. C₃²	#4843	semidirect
271. C₃²	C₃×#459	semidirect
308. C₄	C₃×#3745	semidirect
312. C₃	C₃×#4843	semidirect
313. C₃	#10592542	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×C₃×#3745	2
C₃×#64083	2
C₃×C₄×C₄⋊(C₃⋊D₃)	2
C₃³×C₄⋊C₈	2
C₃×#63648	2
#10592542	3
C₃×#4843	3

Subgroups — C3×#10592542 #13162883

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10592542 #13162883