Subgroups — C₃×#10592419 #13162759

Order 1728.

776 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#64135	C₂	1	1	yes	maximal, characteristic
2	C₃³×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₃×#63657	C₂	1	1	yes	maximal, characteristic
4	C₃×#63642	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#63641	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#63641	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×C₈×C₃⋊Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#10592419	C₃	1	1	yes	maximal, characteristic, direct product factor
9	C₃×#4720	—	3	4	no	maximal
10	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
11	C₂×C₃³×C₈	C₂²	1	1	yes	characteristic
12	C₂×C₃³×C₈	C₂²	1	1	yes	characteristic
13	C₃×#3757	C₄	1	1	yes	characteristic
14	C₃×#3757	C₄	1	1	yes	characteristic
15	C₃×C₄⋊(C₃⋊Dic₃)	C₄	1	1	yes	characteristic
16	C₃×C₄⋊(C₃⋊Dic₃)	C₄	1	1	yes	characteristic
17	C₃×C₄×C₃⋊Dic₃	C₂²	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	#64135	C₂×C₃	1	1	yes	characteristic
22	C₃×#926	—	3	4	no	—
23	C₃²×C₂²⋊C₈	—	2	4	no	—
24	C₃²×C₂²⋊C₈	D₃	1	4	yes	—
25	C₃²×C₂²⋊C₈	C₂×C₃	1	1	yes	characteristic
26	#63657	C₂×C₃	1	1	yes	characteristic
27	#63642	C₂×C₃	1	1	yes	characteristic, complemented
28	#63641	C₂×C₃	1	1	yes	characteristic, complemented
29	#63641	C₂×C₃	1	1	yes	characteristic, complemented
30	C₈×C₃⋊Dic₃	C₂×C₃	1	1	yes	characteristic, complemented
31	C₃×#830	—	3	4	no	—
32	C₃×#815	—	3	4	no	—
33	C₃×Dic₃⋊C₈	—	3	4	no	—
34	C₃×Dic₃⋊C₈	—	3	4	no	—
35	C₃×C₈×Dic₃	—	3	4	no	—
36	C₂³×C₃³	C₂×C₄	1	1	yes	characteristic
37	C₂×C₃³×C₄	—	2	1	no	—
38	C₂×C₃³×C₄	—	2	1	no	—
39	C₂×C₃³×C₄	C₂×C₄	1	1	yes	characteristic
40	C₂×C₃³×C₄	C₂³	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	C₂×C₃×C₃⋊Dic₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₃×C₃⋊Dic₃	C₂×C₄	1	1	yes	characteristic
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	#4720	—	3	4	no	—
51	C₂²×C₃²×C₄	—	2	4	no	—
52	C₂²×C₃²×C₄	C₂×D₃	1	4	yes	—
53	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
54	C₂×C₃²×C₈	—	2	4	no	—
55	C₂×C₃²×C₈	—	2	4	no	—
56	C₂×C₃²×C₈	C₂×D₃	1	4	yes	—
57	C₂×C₃²×C₈	C₂×D₃	1	4	yes	—
58	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
59	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
60	#3757	C₃×C₄	1	1	yes	characteristic
61	#3757	C₃×C₄	1	1	yes	characteristic
62	C₄⋊(C₃⋊Dic₃)	C₃×C₄	1	1	yes	characteristic
63	C₄⋊(C₃⋊Dic₃)	C₃×C₄	1	1	yes	characteristic
64	C₄×C₃⋊Dic₃	C₂²×C₃	1	1	yes	characteristic
65	C₄×C₃⋊Dic₃	C₂²×C₃	1	1	yes	characteristic
66	C₂×C₃⋊(C₃⋊C₈)	C₂²×C₃	1	1	yes	characteristic
67	C₂×C₃⋊(C₃⋊C₈)	C₂²×C₃	1	1	yes	characteristic
68	C₃×#217	—	3	4	no	—
69	C₃×#217	—	3	4	no	—
70	C₃×C₄⋊Dic₃	—	3	4	no	—
71	C₃×C₄⋊Dic₃	—	3	4	no	—
72	C₃×C₄×Dic₃	—	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	—	2	1	no	—
77	C₂²×C₃³	—	2	1	no	—
78	C₂²×C₃³	—	2	1	no	—
79	C₂²×C₃³	C₂²×C₄	1	1	yes	characteristic
80	C₃³×C₄	—	2	1	no	—
81	C₃³×C₄	#41	1	1	yes	characteristic
82	C₃³×C₄	#41	1	1	yes	characteristic
83	C₃×C₃⋊Dic₃	—	2	1	no	—
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	#926	—	3	4	no	—
89	C₃×C₄⋊C₈	—	9	1	no	—
90	C₃×C₄⋊C₈	—	9	1	no	—
91	C₃×C₂²⋊C₈	—	2	4	no	—
92	C₃×C₂²⋊C₈	—	9	1	no	—
93	C₃×C₂²⋊C₈	C₃×D₃	1	4	yes	—
94	C₃×C₂²⋊C₈	C₃⋊D₃	1	1	yes	characteristic
95	C₃×#97	—	9	1	no	—
96	C₃×C₄×C₈	—	9	1	no	—
97	#830	—	3	4	no	—
98	#815	—	3	4	no	—
99	Dic₃⋊C₈	—	3	4	no	—
100	Dic₃⋊C₈	—	3	4	no	—
101	C₈×Dic₃	—	3	4	no	—
102	C₂³×C₃²	—	2	4	no	—
103	C₂³×C₃²	C₄×D₃	1	4	yes	—
104	C₂³×C₃²	C₂×C₃×C₄	1	1	yes	characteristic
105	C₂×C₃²×C₄	—	2	1	no	—
106	C₂×C₃²×C₄	—	2	1	no	—
107	C₂×C₃²×C₄	—	2	4	no	—
108	C₂×C₃²×C₄	—	2	4	no	—
109	C₂×C₃²×C₄	—	2	4	no	—
110	C₂×C₃²×C₄	—	2	4	no	—
111	C₂×C₃²×C₄	—	4	4	no	—
112	C₂×C₃²×C₄	—	4	4	no	—
113	C₂×C₃²×C₄	C₄×D₃	1	4	yes	—
114	C₂×C₃²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
115	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
116	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
117	C₂×C₃⋊Dic₃	C₂×C₃×C₄	1	1	yes	characteristic
118	C₂×C₃⋊Dic₃	C₂×C₃×C₄	1	1	yes	characteristic
119	C₂×C₃⋊Dic₃	C₂×C₃×C₄	1	1	yes	characteristic
120	C₂×C₃⋊Dic₃	C₂×C₃×C₄	1	1	yes	characteristic
121	C₂×C₃×Dic₃	—	3	4	no	—
122	C₂×C₃×Dic₃	—	3	4	no	—
123	C₂×C₃×Dic₃	—	3	4	no	—
124	C₂×C₃×Dic₃	—	3	4	no	—
125	C₃²×C₈	—	2	1	no	—
126	C₃²×C₈	—	2	1	no	—
127	C₃²×C₈	—	2	4	no	—
128	C₃²×C₈	—	2	4	no	—
129	C₃²×C₈	—	4	4	no	—
130	C₃²×C₈	—	4	4	no	—
131	C₃⋊(C₃⋊C₈)	—	2	1	no	—
132	C₃⋊(C₃⋊C₈)	—	2	1	no	—
133	C₃×C₃⋊C₈	—	6	4	no	—
134	C₃×C₃⋊C₈	—	6	4	no	—
135	#433	—	9	1	no	Sylow
136	C₂×C₃³	—	4	1	no	—
137	C₂×C₃³	#117	1	1	yes	characteristic
138	C₂×C₃³	#131	1	1	yes	characteristic
139	C₂×C₃³	#131	1	1	yes	characteristic
140	C₂²×C₃×C₄	—	2	4	no	—
141	C₂²×C₃×C₄	C₂×C₃×D₃	1	4	yes	—
142	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
143	C₂×C₃×C₈	—	2	4	no	—
144	C₂×C₃×C₈	—	2	4	no	—
145	C₂×C₃×C₈	—	9	1	no	—
146	C₂×C₃×C₈	—	9	1	no	—
147	C₂×C₃×C₈	C₂×C₃×D₃	1	4	yes	—
148	C₂×C₃×C₈	C₂×C₃×D₃	1	4	yes	—
149	C₂×C₃×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
150	C₂×C₃×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
151	C₃×C₄⋊C₄	—	9	1	no	—
152	C₃×C₄⋊C₄	—	9	1	no	—
153	C₃×C₂²⋊C₄	—	9	1	no	—
154	C₃×C₂²⋊C₄	—	9	1	no	—
155	C₃×C₄²	—	9	1	no	—
156	C₃×C₄²	—	9	1	no	—
157	#217	—	3	4	no	—
158	#217	—	3	4	no	—
159	C₄⋊Dic₃	—	3	4	no	—
160	C₄⋊Dic₃	—	3	4	no	—
161	C₄×Dic₃	—	3	4	no	—
162	C₄×Dic₃	—	3	4	no	—
163	C₂×C₃⋊C₈	—	3	4	no	—
164	C₂×C₃⋊C₈	—	3	4	no	—
165	C₂²×C₃²	—	2	8	no	—
166	C₂²×C₃²	—	2	4	no	—
167	C₂²×C₃²	—	2	1	no	—
168	C₂²×C₃²	—	2	1	no	—
169	C₂²×C₃²	—	2	1	no	—
170	C₂²×C₃²	—	2	4	no	—
171	C₂²×C₃²	—	2	4	no	—
172	C₂²×C₃²	—	2	4	no	—
173	C₂²×C₃²	—	4	4	no	—
174	C₂²×C₃²	—	4	4	no	—
175	C₂²×C₃²	C₂×C₄×D₃	1	4	yes	—
176	C₂²×C₃²	C₂²×C₃×C₄	1	1	yes	characteristic
177	C₃²×C₄	—	2	1	no	—
178	C₃²×C₄	—	2	4	no	—
179	C₃²×C₄	—	2	4	no	—
180	C₃²×C₄	—	2	4	no	—
181	C₃²×C₄	—	4	4	no	—
182	C₃²×C₄	#237	1	4	yes	—
183	C₃²×C₄	#237	1	4	yes	—
184	C₃²×C₄	C₃×#41	1	1	yes	characteristic
185	C₃²×C₄	C₃×#41	1	1	yes	characteristic
186	C₃⋊Dic₃	—	2	1	no	—
187	C₃⋊Dic₃	—	2	1	no	—
188	C₃⋊Dic₃	—	2	1	no	—
189	C₃⋊Dic₃	—	2	1	no	—
190	C₃×Dic₃	—	6	4	no	—
191	C₃×Dic₃	—	6	4	no	—
192	C₃×Dic₃	—	6	4	no	—
193	C₃×Dic₃	—	6	4	no	—
194	#117	—	9	1	no	—
195	C₄⋊C₈	—	9	1	no	—
196	C₄⋊C₈	—	9	1	no	—
197	C₂²⋊C₈	—	9	1	no	—
198	C₂²⋊C₈	C₃×C₃⋊D₃	1	1	yes	characteristic
199	#97	—	9	1	no	—
200	C₄×C₈	—	9	1	no	—
201	C₃³	#433	1	1	yes	characteristic, complemented, Sylow
202	C₂³×C₃	—	2	4	no	—
203	C₂³×C₃	C₃×C₄×D₃	1	4	yes	—
204	C₂³×C₃	C₄×C₃⋊D₃	1	1	yes	characteristic
205	C₂×C₃×C₄	—	2	4	no	—
206	C₂×C₃×C₄	—	2	4	no	—
207	C₂×C₃×C₄	—	2	1	no	—
208	C₂×C₃×C₄	—	2	1	no	—
209	C₂×C₃×C₄	—	2	4	no	—
210	C₂×C₃×C₄	—	2	4	no	—
211	C₂×C₃×C₄	—	4	4	no	—
212	C₂×C₃×C₄	—	4	4	no	—
213	C₂×C₃×C₄	—	9	1	no	—
214	C₂×C₃×C₄	—	9	1	no	—
215	C₂×C₃×C₄	—	9	1	no	—
216	C₂×C₃×C₄	—	9	1	no	—
217	C₂×C₃×C₄	C₃×C₄×D₃	1	4	yes	—
218	C₂×C₃×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
219	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
220	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
221	C₂×Dic₃	—	3	4	no	—
222	C₂×Dic₃	—	3	4	no	—
223	C₂×Dic₃	—	3	4	no	—
224	C₂×Dic₃	—	3	4	no	—
225	C₃×C₈	—	2	4	no	—
226	C₃×C₈	—	2	4	no	—
227	C₃×C₈	—	2	1	no	—
228	C₃×C₈	—	2	1	no	—
229	C₃×C₈	—	4	4	no	—
230	C₃×C₈	—	4	4	no	—
231	C₃×C₈	—	18	1	no	—
232	C₃×C₈	—	18	1	no	—
233	C₃⋊C₈	—	6	4	no	—
234	C₃⋊C₈	—	6	4	no	—
235	C₂×C₃²	—	2	4	no	—
236	C₂×C₃²	—	2	4	no	—
237	C₂×C₃²	—	2	4	no	—
238	C₂×C₃²	—	4	8	no	—
239	C₂×C₃²	—	4	4	no	—
240	C₂×C₃²	—	4	1	no	—
241	C₂×C₃²	#877	1	4	yes	—
242	C₂×C₃²	#901	1	4	yes	—
243	C₂×C₃²	#907	1	4	yes	—
244	C₂×C₃²	C₃×#117	1	1	yes	characteristic
245	C₂×C₃²	C₃×#131	1	1	yes	characteristic
246	C₂×C₃²	C₃×#131	1	1	yes	characteristic
247	C₂²×C₄	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
248	C₂×C₈	—	9	1	no	—
249	C₂×C₈	—	9	1	no	—
250	C₂×C₈	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
251	C₂×C₈	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
252	C₄⋊C₄	—	9	1	no	—
253	C₄⋊C₄	—	9	1	no	—
254	C₂²⋊C₄	—	9	1	no	—
255	C₂²⋊C₄	—	9	1	no	—
256	C₄²	—	9	1	no	—
257	C₄²	—	9	1	no	—
258	C₂²×C₃	—	2	8	no	—
259	C₂²×C₃	—	2	4	no	—
260	C₂²×C₃	—	2	4	no	—
261	C₂²×C₃	—	2	4	no	—
262	C₂²×C₃	—	2	1	no	—
263	C₂²×C₃	—	2	1	no	—
264	C₂²×C₃	—	2	1	no	—
265	C₂²×C₃	—	2	4	no	—
266	C₂²×C₃	—	4	4	no	—
267	C₂²×C₃	—	4	4	no	—
268	C₂²×C₃	C₂×C₃×C₄×D₃	1	4	yes	—
269	C₂²×C₃	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
270	C₃×C₄	—	2	4	no	—
271	C₃×C₄	—	2	1	no	—
272	C₃×C₄	—	2	4	no	—
273	C₃×C₄	—	2	4	no	—
274	C₃×C₄	—	4	4	no	—
275	C₃×C₄	—	18	1	no	—
276	C₃×C₄	—	18	1	no	—
277	C₃×C₄	—	18	1	no	—
278	C₃×C₄	—	18	1	no	—
279	C₃×C₄	C₃×#237	1	4	yes	—
280	C₃×C₄	C₃×#237	1	4	yes	—
281	C₃×C₄	#3830	1	1	yes	characteristic
282	C₃×C₄	#3830	1	1	yes	characteristic
283	Dic₃	—	6	4	no	—
284	Dic₃	—	6	4	no	—
285	Dic₃	—	6	4	no	—
286	Dic₃	—	6	4	no	—
287	C₃²	—	2	4	no	—
288	C₃²	#4720	1	4	yes	complemented
289	C₃²	C₃×#433	1	1	yes	characteristic, complemented
290	C₂³	C₃×C₄×C₃⋊D₃	1	1	yes	characteristic
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	9	1	no	—
294	C₂×C₄	—	9	1	no	—
295	C₂×C₄	—	9	1	no	—
296	C₂×C₄	—	9	1	no	—
297	C₂×C₄	C₃×C₄×C₃⋊D₃	1	1	yes	characteristic
298	C₂×C₄	C₂²×C₃×C₃⋊D₃	1	1	yes	characteristic
299	C₈	—	2	1	no	—
300	C₈	—	2	1	no	—
301	C₈	—	18	1	no	—
302	C₈	—	18	1	no	—
303	C₂×C₃	—	2	4	no	—
304	C₂×C₃	—	2	4	no	—
305	C₂×C₃	—	2	4	no	—
306	C₂×C₃	—	4	8	no	—
307	C₂×C₃	—	4	4	no	—
308	C₂×C₃	—	4	1	no	—
309	C₂×C₃	C₃×#877	1	4	yes	—
310	C₂×C₃	C₃×#901	1	4	yes	—
311	C₂×C₃	C₃×#907	1	4	yes	—
312	C₂×C₃	#64086	1	1	yes	characteristic
313	C₂×C₃	#64110	1	1	yes	characteristic
314	C₂×C₃	#64116	1	1	yes	characteristic
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	C₂×C₃×C₄×C₃⋊D₃	1	1	yes	characteristic
319	C₄	—	2	1	no	—
320	C₄	—	18	1	no	—
321	C₄	—	18	1	no	—
322	C₄	—	18	1	no	—
323	C₄	—	18	1	no	—
324	C₄	C₃×#3830	1	1	yes	characteristic
325	C₄	C₃×#3830	1	1	yes	characteristic
326	C₃	—	2	4	no	—
327	C₃	C₃×#4720	1	4	yes	complemented
328	C₃	#10592419	1	1	yes	characteristic, direct product factor
329	C₂	—	4	1	no	—
330	C₂	C₃×#64086	1	1	yes	characteristic
331	C₂	C₃×#64110	1	1	yes	characteristic
332	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×C₂²⋊C₈	C₂
socle	C₂²×C₃³	C₂²×C₄
prime radical	C₂³×C₃³	C₂×C₄
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₃×C₂²⋊C₈	C₃⋊D₃
hypocenter	C₃²	C₃×#433
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₃×C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
4. C₃×#63642	C₂	semidirect
5. C₃×#63641	C₂	semidirect
6. C₃×#63641	C₂	semidirect
7. C₃×C₈×C₃⋊Dic₃	C₂	semidirect
8. #10592419	C₃	direct (N × H)
27. #63642	C₂×C₃	semidirect
28. #63641	C₂×C₃	semidirect
29. #63641	C₂×C₃	semidirect
30. C₈×C₃⋊Dic₃	C₂×C₃	semidirect
201. C₃³	#433	semidirect
288. C₃²	#4720	semidirect
289. C₃²	C₃×#433	semidirect
327. C₃	C₃×#4720	semidirect
328. C₃	#10592419	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×#64135	2
C₃³×C₂²⋊C₈	2
C₃×#63657	2
C₃×#63642	2
C₃×#63641	2
C₃×C₈×C₃⋊Dic₃	2
#10592419	3
C₃×#4720	3

Subgroups — C3×#10592419 #13162759

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10592419 #13162759