Subgroups — C₃×#10592539 #13162880

Order 1728.

728 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#64081	C₂	1	1	yes	maximal, characteristic
2	C₃³×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#63647	C₂	1	1	yes	maximal, characteristic
4	C₃×#63647	C₂	1	1	yes	maximal, characteristic
5	C₃×#63642	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×C₈×C₃⋊Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×C₄⋊(C₃⋊(C₃⋊C₈))	C₂	1	1	yes	maximal, characteristic, complemented
8	#10592539	C₃	1	1	yes	maximal, characteristic, direct product factor
9	C₃×#4840	—	3	4	no	maximal
10	C₂×C₃×C₄×C₃⋊D₃	C₂²	1	1	yes	characteristic
11	C₂×C₃³×C₈	C₂²	1	1	yes	characteristic
12	C₂×C₃³×C₈	C₂²	1	1	yes	characteristic
13	C₃³×C₄²	C₂²	1	1	yes	characteristic
14	C₃×#3752	C₄	1	1	yes	characteristic
15	C₃×#3752	C₄	1	1	yes	characteristic
16	C₃×#3750	C₄	1	1	yes	characteristic
17	C₃×#3750	C₄	1	1	yes	characteristic
18	C₃×C₄×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
19	C₂×C₃×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
20	C₂×C₃×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
21	#64081	C₂×C₃	1	1	yes	characteristic
22	C₃×#872	—	3	4	no	—
23	C₃²×C₄⋊C₈	—	2	4	no	—
24	C₃²×C₄⋊C₈	D₃	1	4	yes	complemented
25	C₃²×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
26	#63647	C₂×C₃	1	1	yes	characteristic
27	#63647	C₂×C₃	1	1	yes	characteristic
28	#63642	C₂×C₃	1	1	yes	characteristic, complemented
29	C₈×C₃⋊Dic₃	C₂×C₃	1	1	yes	characteristic, complemented
30	C₄⋊(C₃⋊(C₃⋊C₈))	C₂×C₃	1	1	yes	characteristic, complemented
31	C₃×(C₂×D₃)⋊C₈	—	3	4	no	—
32	C₃×(C₂×D₃)⋊C₈	—	3	4	no	—
33	C₃×#815	—	3	4	no	—
34	C₃×C₈×Dic₃	—	3	4	no	—
35	C₃×C₄⋊(C₃⋊C₈)	—	3	4	no	—
36	C₂²×C₃×C₃⋊D₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₃³×C₄	C₂×C₄	1	1	yes	characteristic
38	C₂×C₃³×C₄	C₂×C₄	1	1	yes	characteristic
39	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
40	C₂×C₃×C₃⋊Dic₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₃×C₃⋊Dic₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₃×C₃⋊Dic₃	C₂×C₄	1	1	yes	characteristic
43	C₃×C₄×C₃⋊D₃	—	2	1	no	—
44	C₃×C₄×C₃⋊D₃	—	2	1	no	—
45	C₃³×C₈	—	2	1	no	—
46	C₃³×C₈	—	2	1	no	—
47	C₃×C₃⋊(C₃⋊C₈)	—	2	1	no	—
48	C₃×C₃⋊(C₃⋊C₈)	—	2	1	no	—
49	C₃×#433	—	9	1	no	—
50	#4840	—	3	4	no	—
51	C₂×C₄×C₃⋊D₃	C₂²×C₃	1	1	yes	characteristic
52	C₂×C₃×C₄×D₃	—	3	4	no	—
53	C₂×C₃²×C₈	—	2	4	no	—
54	C₂×C₃²×C₈	—	2	4	no	—
55	C₂×C₃²×C₈	C₂×D₃	1	4	yes	—
56	C₂×C₃²×C₈	C₂×D₃	1	4	yes	—
57	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
58	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
59	C₃²×C₄²	—	2	4	no	—
60	C₃²×C₄²	C₂×D₃	1	4	yes	—
61	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
62	#3752	C₃×C₄	1	1	yes	characteristic
63	#3752	C₃×C₄	1	1	yes	characteristic
64	#3750	C₃×C₄	1	1	yes	characteristic
65	#3750	C₃×C₄	1	1	yes	characteristic
66	C₄×C₃⋊Dic₃	C₂²×C₃	1	1	yes	characteristic
67	C₂×C₃⋊(C₃⋊C₈)	C₂²×C₃	1	1	yes	characteristic
68	C₂×C₃⋊(C₃⋊C₈)	C₂²×C₃	1	1	yes	characteristic
69	C₃×(C₂×D₃)⋊C₄	—	3	4	no	—
70	C₃×(C₂×D₃)⋊C₄	—	3	4	no	—
71	C₃×Dic₃⋊C₄	—	3	4	no	—
72	C₃×Dic₃⋊C₄	—	3	4	no	—
73	C₃×C₄×Dic₃	—	3	4	no	—
74	C₂×C₃×C₃⋊C₈	—	3	4	no	—
75	C₂×C₃×C₃⋊C₈	—	3	4	no	—
76	C₂²×C₃³	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₃×C₃⋊D₃	—	2	1	no	—
78	C₂×C₃×C₃⋊D₃	—	2	1	no	—
79	C₂×C₃×C₃⋊D₃	—	2	1	no	—
80	C₃³×C₄	—	2	1	no	—
81	C₃³×C₄	—	2	1	no	—
82	C₃³×C₄	#41	1	1	yes	characteristic
83	C₃³×C₄	#41	1	1	yes	characteristic
84	C₃×C₃⋊Dic₃	—	2	1	no	—
85	C₃×C₃⋊Dic₃	—	2	1	no	—
86	C₃×C₃⋊Dic₃	—	2	1	no	—
87	C₃×#117	—	9	1	no	—
88	#872	—	3	4	no	—
89	C₃×C₄⋊C₈	—	2	4	no	—
90	C₃×C₄⋊C₈	—	9	1	no	—
91	C₃×C₄⋊C₈	C₃×D₃	1	4	yes	complemented
92	C₃×C₄⋊C₈	C₃⋊D₃	1	1	yes	characteristic, complemented
93	C₃×C₂²⋊C₈	—	9	1	no	—
94	C₃×C₂²⋊C₈	—	9	1	no	—
95	C₃×#97	—	9	1	no	—
96	C₃×C₄×C₈	—	9	1	no	—
97	(C₂×D₃)⋊C₈	—	3	4	no	—
98	(C₂×D₃)⋊C₈	—	3	4	no	—
99	#815	—	3	4	no	—
100	C₈×Dic₃	—	3	4	no	—
101	C₄⋊(C₃⋊C₈)	—	3	4	no	—
102	C₂²×C₃⋊D₃	C₂×C₃×C₄	1	1	yes	characteristic
103	C₂²×C₃×D₃	—	3	4	no	—
104	C₂×C₃²×C₄	—	2	4	no	—
105	C₂×C₃²×C₄	—	2	4	no	—
106	C₂×C₃²×C₄	—	2	4	no	—
107	C₂×C₃²×C₄	C₄×D₃	1	4	yes	—
108	C₂×C₃²×C₄	C₄×D₃	1	4	yes	—
109	C₂×C₃²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
110	C₂×C₃²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
111	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
112	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
113	C₂×C₃⋊Dic₃	C₂×C₃×C₄	1	1	yes	characteristic
114	C₂×C₃⋊Dic₃	C₂×C₃×C₄	1	1	yes	characteristic
115	C₂×C₃⋊Dic₃	C₂×C₃×C₄	1	1	yes	characteristic
116	C₄×C₃⋊D₃	—	2	1	no	—
117	C₄×C₃⋊D₃	—	2	1	no	—
118	C₂×C₃×Dic₃	—	3	4	no	—
119	C₂×C₃×Dic₃	—	3	4	no	—
120	C₂×C₃×Dic₃	—	3	4	no	—
121	C₃×C₄×D₃	—	6	4	no	—
122	C₃×C₄×D₃	—	6	4	no	—
123	C₃²×C₈	—	2	1	no	—
124	C₃²×C₈	—	2	1	no	—
125	C₃²×C₈	—	2	4	no	—
126	C₃²×C₈	—	2	4	no	—
127	C₃²×C₈	—	4	4	no	—
128	C₃²×C₈	—	4	4	no	—
129	C₃⋊(C₃⋊C₈)	—	2	1	no	—
130	C₃⋊(C₃⋊C₈)	—	2	1	no	—
131	C₃×C₃⋊C₈	—	6	4	no	—
132	C₃×C₃⋊C₈	—	6	4	no	—
133	#433	—	9	1	no	Sylow
134	C₂×C₃³	#117	1	1	yes	characteristic
135	C₂×C₃³	#131	1	1	yes	characteristic
136	C₂×C₃³	#131	1	1	yes	characteristic
137	C₃×C₃⋊D₃	—	4	1	no	—
138	C₂²×C₃×C₄	—	9	1	no	—
139	C₂×C₄×D₃	—	3	4	no	—
140	C₂×C₃×C₈	—	2	4	no	—
141	C₂×C₃×C₈	—	2	4	no	—
142	C₂×C₃×C₈	—	9	1	no	—
143	C₂×C₃×C₈	—	9	1	no	—
144	C₂×C₃×C₈	C₂×C₃×D₃	1	4	yes	—
145	C₂×C₃×C₈	C₂×C₃×D₃	1	4	yes	—
146	C₂×C₃×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
147	C₂×C₃×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
148	C₃×C₄⋊C₄	—	9	1	no	—
149	C₃×C₄⋊C₄	—	9	1	no	—
150	C₃×C₂²⋊C₄	—	9	1	no	—
151	C₃×C₂²⋊C₄	—	9	1	no	—
152	C₃×C₄²	—	2	4	no	—
153	C₃×C₄²	—	9	1	no	—
154	C₃×C₄²	C₂×C₃×D₃	1	4	yes	—
155	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
156	(C₂×D₃)⋊C₄	—	3	4	no	—
157	(C₂×D₃)⋊C₄	—	3	4	no	—
158	Dic₃⋊C₄	—	3	4	no	—
159	Dic₃⋊C₄	—	3	4	no	—
160	C₄×Dic₃	—	3	4	no	—
161	C₂×C₃⋊C₈	—	3	4	no	—
162	C₂×C₃⋊C₈	—	3	4	no	—
163	C₂²×C₃²	—	2	4	no	—
164	C₂²×C₃²	C₂×C₄×D₃	1	4	yes	—
165	C₂²×C₃²	C₂²×C₃×C₄	1	1	yes	characteristic
166	C₂×C₃⋊D₃	—	2	1	no	—
167	C₂×C₃⋊D₃	—	2	1	no	—
168	C₂×C₃⋊D₃	—	2	1	no	—
169	C₂×C₃×D₃	—	6	4	no	—
170	C₂×C₃×D₃	—	6	4	no	—
171	C₂×C₃×D₃	—	6	4	no	—
172	C₃²×C₄	—	2	8	no	—
173	C₃²×C₄	—	2	4	no	—
174	C₃²×C₄	—	2	1	no	—
175	C₃²×C₄	—	2	1	no	—
176	C₃²×C₄	—	2	4	no	—
177	C₃²×C₄	—	2	4	no	—
178	C₃²×C₄	—	2	4	no	—
179	C₃²×C₄	—	4	4	no	—
180	C₃²×C₄	#237	1	4	yes	—
181	C₃²×C₄	#239	1	4	yes	—
182	C₃²×C₄	C₃×#41	1	1	yes	characteristic
183	C₃²×C₄	C₃×#41	1	1	yes	characteristic
184	C₃⋊Dic₃	—	2	1	no	—
185	C₃⋊Dic₃	—	2	1	no	—
186	C₃⋊Dic₃	—	2	1	no	—
187	C₃×Dic₃	—	6	4	no	—
188	C₃×Dic₃	—	6	4	no	—
189	C₃×Dic₃	—	6	4	no	—
190	#117	—	9	1	no	—
191	C₄⋊C₈	—	9	1	no	—
192	C₄⋊C₈	C₃×C₃⋊D₃	1	1	yes	characteristic, complemented
193	C₂²⋊C₈	—	9	1	no	—
194	C₂²⋊C₈	—	9	1	no	—
195	#97	—	9	1	no	—
196	C₄×C₈	—	9	1	no	—
197	C₃³	#433	1	1	yes	characteristic, complemented, Sylow
198	C₂³×C₃	—	9	1	no	—
199	C₂²×D₃	—	3	4	no	—
200	C₂×C₃×C₄	—	2	4	no	—
201	C₂×C₃×C₄	—	2	4	no	—
202	C₂×C₃×C₄	—	2	4	no	—
203	C₂×C₃×C₄	—	9	1	no	—
204	C₂×C₃×C₄	—	9	1	no	—
205	C₂×C₃×C₄	—	9	1	no	—
206	C₂×C₃×C₄	—	18	1	no	—
207	C₂×C₃×C₄	—	18	1	no	—
208	C₂×C₃×C₄	C₃×C₄×D₃	1	4	yes	—
209	C₂×C₃×C₄	C₃×C₄×D₃	1	4	yes	—
210	C₂×C₃×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
211	C₂×C₃×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
212	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
213	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
214	C₂×Dic₃	—	3	4	no	—
215	C₂×Dic₃	—	3	4	no	—
216	C₂×Dic₃	—	3	4	no	—
217	C₄×D₃	—	6	4	no	—
218	C₄×D₃	—	6	4	no	—
219	C₃×C₈	—	2	4	no	—
220	C₃×C₈	—	2	4	no	—
221	C₃×C₈	—	2	1	no	—
222	C₃×C₈	—	2	1	no	—
223	C₃×C₈	—	4	4	no	—
224	C₃×C₈	—	4	4	no	—
225	C₃×C₈	—	18	1	no	—
226	C₃×C₈	—	18	1	no	—
227	C₃⋊C₈	—	6	4	no	—
228	C₃⋊C₈	—	6	4	no	—
229	C₂×C₃²	—	2	4	no	—
230	C₂×C₃²	—	2	4	no	—
231	C₂×C₃²	—	2	4	no	—
232	C₂×C₃²	#892	1	4	yes	—
233	C₂×C₃²	#901	1	4	yes	—
234	C₂×C₃²	#907	1	4	yes	—
235	C₂×C₃²	C₃×#117	1	1	yes	characteristic
236	C₂×C₃²	C₃×#131	1	1	yes	characteristic
237	C₂×C₃²	C₃×#131	1	1	yes	characteristic
238	C₃⋊D₃	—	4	1	no	—
239	C₃×D₃	—	12	4	no	—
240	C₂²×C₄	—	9	1	no	—
241	C₂×C₈	—	9	1	no	—
242	C₂×C₈	—	9	1	no	—
243	C₂×C₈	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
244	C₂×C₈	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
245	C₄⋊C₄	—	9	1	no	—
246	C₄⋊C₄	—	9	1	no	—
247	C₂²⋊C₄	—	9	1	no	—
248	C₂²⋊C₄	—	9	1	no	—
249	C₄²	—	9	1	no	—
250	C₄²	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
251	C₂²×C₃	—	2	4	no	—
252	C₂²×C₃	—	18	1	no	—
253	C₂²×C₃	—	18	1	no	—
254	C₂²×C₃	—	18	1	no	—
255	C₂²×C₃	C₂×C₃×C₄×D₃	1	4	yes	—
256	C₂²×C₃	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
257	C₂×D₃	—	6	4	no	—
258	C₂×D₃	—	6	4	no	—
259	C₂×D₃	—	6	4	no	—
260	C₃×C₄	—	2	8	no	—
261	C₃×C₄	—	2	4	no	—
262	C₃×C₄	—	2	4	no	—
263	C₃×C₄	—	2	1	no	—
264	C₃×C₄	—	2	1	no	—
265	C₃×C₄	—	2	4	no	—
266	C₃×C₄	—	2	4	no	—
267	C₃×C₄	—	4	4	no	—
268	C₃×C₄	—	18	1	no	—
269	C₃×C₄	—	18	1	no	—
270	C₃×C₄	—	18	1	no	—
271	C₃×C₄	C₃×#237	1	4	yes	—
272	C₃×C₄	C₃×#239	1	4	yes	—
273	C₃×C₄	#3830	1	1	yes	characteristic
274	C₃×C₄	#3832	1	1	yes	characteristic
275	Dic₃	—	6	4	no	—
276	Dic₃	—	6	4	no	—
277	Dic₃	—	6	4	no	—
278	C₃²	—	2	4	no	—
279	C₃²	#4840	1	4	yes	complemented
280	C₃²	C₃×#433	1	1	yes	characteristic, complemented
281	C₂³	—	9	1	no	—
282	C₂×C₄	—	9	1	no	—
283	C₂×C₄	—	9	1	no	—
284	C₂×C₄	—	9	1	no	—
285	C₂×C₄	—	18	1	no	—
286	C₂×C₄	—	18	1	no	—
287	C₂×C₄	C₃×C₄×C₃⋊D₃	1	1	yes	characteristic
288	C₂×C₄	C₃×C₄×C₃⋊D₃	1	1	yes	characteristic
289	C₂×C₄	C₂²×C₃×C₃⋊D₃	1	1	yes	characteristic
290	C₈	—	2	1	no	—
291	C₈	—	2	1	no	—
292	C₈	—	18	1	no	—
293	C₈	—	18	1	no	—
294	C₂×C₃	—	2	4	no	—
295	C₂×C₃	—	2	4	no	—
296	C₂×C₃	—	2	4	no	—
297	C₂×C₃	—	36	1	no	—
298	C₂×C₃	C₃×#892	1	4	yes	—
299	C₂×C₃	C₃×#901	1	4	yes	—
300	C₂×C₃	C₃×#907	1	4	yes	—
301	C₂×C₃	#64101	1	1	yes	characteristic
302	C₂×C₃	#64110	1	1	yes	characteristic
303	C₂×C₃	#64116	1	1	yes	characteristic
304	D₃	—	12	4	no	—
305	C₂²	—	18	1	no	—
306	C₂²	—	18	1	no	—
307	C₂²	—	18	1	no	—
308	C₂²	C₂×C₃×C₄×C₃⋊D₃	1	1	yes	characteristic
309	C₄	—	2	1	no	—
310	C₄	—	2	1	no	—
311	C₄	—	18	1	no	—
312	C₄	—	18	1	no	—
313	C₄	—	18	1	no	—
314	C₄	C₃×#3830	1	1	yes	characteristic
315	C₄	C₃×#3832	1	1	yes	characteristic
316	C₃	—	2	4	no	—
317	C₃	C₃×#4840	1	4	yes	complemented
318	C₃	#10592539	1	1	yes	characteristic, direct product factor
319	C₂	—	36	1	no	—
320	C₂	C₃×#64101	1	1	yes	characteristic
321	C₂	C₃×#64110	1	1	yes	characteristic
322	C₂	C₃×#64116	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×C₄⋊C₈	C₂
socle	C₂²×C₃³	C₂²×C₄
prime radical	C₂²×C₃×C₃⋊D₃	C₂×C₄
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₃×C₄⋊C₈	C₃⋊D₃
hypocenter	C₃²	C₃×#433
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₃×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₃³×C₄⋊C₈	C₂	semidirect
5. C₃×#63642	C₂	semidirect
6. C₃×C₈×C₃⋊Dic₃	C₂	semidirect
7. C₃×C₄⋊(C₃⋊(C₃⋊C₈))	C₂	semidirect
8. #10592539	C₃	direct (N × H)
24. C₃²×C₄⋊C₈	D₃	semidirect
25. C₃²×C₄⋊C₈	C₂×C₃	semidirect
28. #63642	C₂×C₃	semidirect
29. C₈×C₃⋊Dic₃	C₂×C₃	semidirect
30. C₄⋊(C₃⋊(C₃⋊C₈))	C₂×C₃	semidirect
91. C₃×C₄⋊C₈	C₃×D₃	semidirect
92. C₃×C₄⋊C₈	C₃⋊D₃	semidirect
192. C₄⋊C₈	C₃×C₃⋊D₃	semidirect
197. C₃³	#433	semidirect
279. C₃²	#4840	semidirect
280. C₃²	C₃×#433	semidirect
317. C₃	C₃×#4840	semidirect
318. C₃	#10592539	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×#64081	2
C₃³×C₄⋊C₈	2
C₃×#63647	2
C₃×#63642	2
C₃×C₈×C₃⋊Dic₃	2
C₃×C₄⋊(C₃⋊(C₃⋊C₈))	2
#10592539	3
C₃×#4840	3

Subgroups — C3×#10592539 #13162880

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10592539 #13162880