Subgroups — C₃×#10590369 #13157386

Order 1728.

818 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#64078	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃²×#874	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#3716	C₂	1	4	yes	maximal, complemented
4	C₃²×C₄×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#5076	—	3	1	no	maximal
6	C₃×#4693	—	3	1	no	maximal
7	#10590369	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂×C₃×#634	C₂²	1	2	yes	—
9	C₂×C₃²×D₁₂	—	2	2	no	—
10	C₂×C₃²×D₁₂	C₂²	1	2	yes	—
11	C₃³×C₄²	C₂²	1	1	yes	characteristic
12	C₃×C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
13	C₂×C₃²×C₃⋊C₈	C₂²	1	2	yes	—
14	C₃×#3716	—	2	8	no	—
15	C₃²×#127	—	3	1	no	—
16	#64078	C₂×C₃	1	1	yes	characteristic, complemented
17	C₂×C₃×#214	—	3	4	no	—
18	C₂×C₃×#204	—	3	4	no	—
19	C₃×#874	—	2	1	no	—
20	C₃×#874	D₃	1	1	yes	characteristic
21	C₃×#874	C₂×C₃	1	1	yes	characteristic
22	C₃×#869	—	3	1	no	—
23	C₃×#869	—	3	1	no	—
24	C₃×#869	—	6	1	no	—
25	C₂×#3716	C₂×C₃	1	4	yes	complemented
26	C₃²×C₄×C₈	—	3	1	no	—
27	C₃×C₄×C₃⋊C₈	—	2	1	no	—
28	C₃×C₄×C₃⋊C₈	D₃	1	1	yes	characteristic, complemented
29	C₃×C₄×C₃⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
30	C₂²×C₃²×D₃	—	2	2	no	—
31	C₂×C₃³×C₄	D₄	1	2	yes	—
32	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
33	C₂×C₃×C₃⋊Dic₃	—	2	2	no	—
34	C₃×#634	—	2	4	no	—
35	C₃²×D₁₂	—	2	4	no	—
36	C₃²×D₁₂	—	4	4	no	—
37	C₃²×C₃⋊C₈	D₄	1	4	yes	complemented
38	C₃×#492	—	9	1	no	—
39	#5076	—	3	1	no	—
40	#4693	—	3	1	no	—
41	C₂×C₃²×D₄	—	3	2	no	—
42	C₂×C₃²×D₄	—	6	2	no	—
43	C₂×#634	C₂²×C₃	1	2	yes	—
44	C₂×C₃×D₁₂	—	2	4	no	—
45	C₂×C₃×D₁₂	—	2	2	no	—
46	C₂×C₃×D₁₂	—	2	2	no	—
47	C₂×C₃×D₁₂	—	2	2	no	—
48	C₂×C₃×D₁₂	C₂×D₃	1	2	yes	—
49	C₂×C₃×D₁₂	C₂²×C₃	1	2	yes	—
50	C₂×C₃×Dic₆	—	3	2	no	—
51	C₂×C₃×Dic₆	—	3	2	no	—
52	C₂×C₃×Dic₆	—	6	2	no	—
53	C₂×C₃²×C₈	—	3	2	no	—
54	C₃²×C₄²	—	2	1	no	—
55	C₃²×C₄²	—	2	1	no	—
56	C₃²×C₄²	—	2	1	no	—
57	C₃²×C₄²	—	4	1	no	—
58	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
59	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
60	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
61	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
62	C₃×#214	—	6	8	no	—
63	C₃×C₄⋊Dic₃	—	6	4	no	—
64	C₃×C₄⋊Dic₃	—	6	2	no	—
65	C₃×C₄⋊Dic₃	—	6	2	no	—
66	C₂×C₃×C₃⋊C₈	—	2	2	no	—
67	C₂×C₃×C₃⋊C₈	C₂×D₃	1	2	yes	—
68	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	2	yes	—
69	C₃×#204	—	6	8	no	—
70	#3716	—	2	8	no	—
71	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
72	C₂×C₃²×D₃	—	4	2	no	—
73	C₂×C₃²×D₃	—	4	4	no	—
74	C₃³×C₄	SD₁₆	1	4	yes	complemented
75	C₃³×C₄	C₂×D₄	1	2	yes	—
76	C₃×C₃⋊Dic₃	—	4	2	no	—
77	C₂×C₃×SD₁₆	—	9	4	no	—
78	C₃×C₄⋊Q₈	—	9	1	no	—
79	C₃×#127	—	3	1	no	—
80	C₃×#127	—	3	1	no	—
81	C₃×#127	—	6	1	no	—
82	C₂×#214	—	3	4	no	—
83	C₂×#204	—	3	4	no	—
84	#874	C₃×D₃	1	1	yes	characteristic
85	#869	—	3	1	no	—
86	#869	—	3	1	no	—
87	#869	—	6	1	no	—
88	C₃×C₄×C₈	—	3	1	no	—
89	C₃×C₄×C₈	—	3	1	no	—
90	C₃×C₄×C₈	—	6	1	no	—
91	C₄×C₃⋊C₈	C₃×D₃	1	1	yes	characteristic, complemented
92	C₂³×C₃²	—	6	2	no	—
93	C₂²×C₃×D₃	—	2	4	no	—
94	C₂²×C₃×D₃	—	2	2	no	—
95	C₂²×C₃×D₃	—	2	2	no	—
96	C₃²×D₄	—	6	4	no	—
97	C₃²×D₄	—	12	4	no	—
98	C₂×C₃²×C₄	—	2	2	no	—
99	C₂×C₃²×C₄	—	2	2	no	—
100	C₂×C₃²×C₄	—	2	2	no	—
101	C₂×C₃²×C₄	—	2	1	no	—
102	C₂×C₃²×C₄	—	2	1	no	—
103	C₂×C₃²×C₄	—	2	1	no	—
104	C₂×C₃²×C₄	—	4	2	no	—
105	C₂×C₃²×C₄	—	4	1	no	—
106	C₂×C₃²×C₄	D₁₂	1	2	yes	—
107	C₂×C₃²×C₄	#67	1	2	yes	—
108	C₂×C₃²×C₄	C₃×D₄	1	2	yes	—
109	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
110	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
111	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
112	C₂×C₃⋊Dic₃	—	2	2	no	—
113	#634	—	2	4	no	—
114	C₂×C₃×Dic₃	—	6	4	no	—
115	C₂×C₃×Dic₃	—	6	2	no	—
116	C₂×C₃×Dic₃	—	6	2	no	—
117	C₃×D₁₂	—	2	4	no	—
118	C₃×D₁₂	—	2	4	no	—
119	C₃×D₁₂	—	4	4	no	—
120	C₃×D₁₂	—	4	8	no	—
121	C₃×D₁₂	—	4	4	no	—
122	C₃×D₁₂	—	4	4	no	—
123	C₃×Dic₆	—	6	4	no	—
124	C₃×Dic₆	—	6	4	no	—
125	C₃×Dic₆	—	12	4	no	—
126	C₃²×C₈	—	3	4	no	—
127	C₃×C₃⋊C₈	—	2	4	no	—
128	C₃×C₃⋊C₈	D₁₂	1	4	yes	complemented
129	C₃×C₃⋊C₈	C₃×D₄	1	4	yes	complemented
130	#492	—	9	1	no	Sylow
131	C₂×C₃³	#127	1	1	yes	characteristic
132	C₂×C₃³	C₂×SD₁₆	1	2	yes	—
133	C₃²×D₃	—	8	2	no	—
134	C₂×C₃×Q₈	—	9	2	no	—
135	C₂×C₃×D₄	—	3	2	no	—
136	C₂×C₃×D₄	—	3	2	no	—
137	C₂×C₃×D₄	—	6	2	no	—
138	C₂×C₃×D₄	—	6	4	no	—
139	C₂×C₃×D₄	—	6	2	no	—
140	C₂×C₃×D₄	—	6	2	no	—
141	C₂×D₁₂	—	2	2	no	—
142	C₂×D₁₂	C₂×C₃×D₃	1	2	yes	—
143	C₂×Dic₆	—	3	2	no	—
144	C₂×Dic₆	—	3	2	no	—
145	C₂×Dic₆	—	6	2	no	—
146	C₃×SD₁₆	—	18	8	no	—
147	C₂×C₃×C₈	—	3	2	no	—
148	C₂×C₃×C₈	—	3	2	no	—
149	C₂×C₃×C₈	—	6	2	no	—
150	C₃×C₄⋊C₄	—	18	2	no	—
151	C₃×C₄²	—	2	1	no	—
152	C₃×C₄²	—	2	1	no	—
153	C₃×C₄²	—	2	1	no	—
154	C₃×C₄²	—	4	1	no	—
155	C₃×C₄²	D₃²	1	1	yes	characteristic
156	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
157	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
158	#214	—	6	8	no	—
159	C₄⋊Dic₃	—	6	4	no	—
160	C₄⋊Dic₃	—	6	2	no	—
161	C₄⋊Dic₃	—	6	2	no	—
162	C₂×C₃⋊C₈	C₂×C₃×D₃	1	2	yes	—
163	#204	—	6	8	no	—
164	C₂²×C₃²	—	2	1	no	—
165	C₂²×C₃²	—	2	1	no	—
166	C₂²×C₃²	—	2	1	no	—
167	C₂²×C₃²	—	4	1	no	—
168	C₂²×C₃²	—	12	2	no	—
169	C₂²×C₃²	—	12	4	no	—
170	C₂²×C₃²	C₂×D₁₂	1	1	yes	characteristic
171	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
172	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
173	C₂×C₃×D₃	—	4	8	no	—
174	C₂×C₃×D₃	—	4	4	no	—
175	C₂×C₃×D₃	—	4	4	no	—
176	C₂×C₃×D₃	—	4	2	no	—
177	C₂×C₃×D₃	—	4	2	no	—
178	C₂×C₃×D₃	—	4	4	no	—
179	C₃²×C₄	—	2	4	no	—
180	C₃²×C₄	—	2	4	no	—
181	C₃²×C₄	—	2	4	no	—
182	C₃²×C₄	—	2	2	no	—
183	C₃²×C₄	—	2	2	no	—
184	C₃²×C₄	—	2	2	no	—
185	C₃²×C₄	—	4	4	no	—
186	C₃²×C₄	—	4	2	no	—
187	C₃²×C₄	#204	1	4	yes	complemented
188	C₃²×C₄	#214	1	4	yes	complemented
189	C₃²×C₄	C₃×SD₁₆	1	4	yes	complemented
190	C₃²×C₄	C₂×D₁₂	1	2	yes	—
191	C₃²×C₄	D₃×D₄	1	2	yes	—
192	C₃²×C₄	C₂×C₃×D₄	1	2	yes	—
193	C₃⋊Dic₃	—	4	2	no	—
194	C₃×Dic₃	—	12	4	no	—
195	C₃×Dic₃	—	12	2	no	—
196	C₃×Dic₃	—	12	2	no	—
197	C₂×SD₁₆	—	9	4	no	—
198	C₄⋊Q₈	—	9	1	no	—
199	#127	—	3	1	no	—
200	C₄×C₈	—	3	1	no	—
201	C₃³	#492	1	1	yes	characteristic, complemented, Sylow
202	C₂³×C₃	—	6	4	no	—
203	C₂³×C₃	—	6	2	no	—
204	C₂³×C₃	—	6	2	no	—
205	C₂²×D₃	—	2	2	no	—
206	C₃×Q₈	—	18	4	no	—
207	C₃×D₄	—	6	4	no	—
208	C₃×D₄	—	6	4	no	—
209	C₃×D₄	—	12	4	no	—
210	C₃×D₄	—	12	8	no	—
211	C₃×D₄	—	12	4	no	—
212	C₃×D₄	—	12	4	no	—
213	C₂×C₃×C₄	—	2	2	no	—
214	C₂×C₃×C₄	—	2	2	no	—
215	C₂×C₃×C₄	—	2	2	no	—
216	C₂×C₃×C₄	—	2	1	no	—
217	C₂×C₃×C₄	—	2	1	no	—
218	C₂×C₃×C₄	—	2	1	no	—
219	C₂×C₃×C₄	—	4	2	no	—
220	C₂×C₃×C₄	—	4	1	no	—
221	C₂×C₃×C₄	—	18	2	no	—
222	C₂×C₃×C₄	#626	1	2	yes	—
223	C₂×C₃×C₄	C₃×D₁₂	1	2	yes	—
224	C₂×C₃×C₄	C₃×#67	1	2	yes	—
225	C₂×C₃×C₄	C₂×D₃²	1	1	yes	characteristic
226	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
227	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
228	C₂×Dic₃	—	6	4	no	—
229	C₂×Dic₃	—	6	2	no	—
230	C₂×Dic₃	—	6	2	no	—
231	D₁₂	—	2	4	no	—
232	D₁₂	—	4	4	no	—
233	Dic₆	—	6	4	no	—
234	Dic₆	—	6	4	no	—
235	Dic₆	—	12	4	no	—
236	C₃×C₈	—	3	4	no	—
237	C₃×C₈	—	3	4	no	—
238	C₃×C₈	—	6	4	no	—
239	C₃⋊C₈	C₃×D₁₂	1	4	yes	complemented
240	C₂×C₃²	—	2	2	no	—
241	C₂×C₃²	—	2	2	no	—
242	C₂×C₃²	—	2	1	no	—
243	C₂×C₃²	—	2	2	no	—
244	C₂×C₃²	—	2	1	no	—
245	C₂×C₃²	—	2	1	no	—
246	C₂×C₃²	—	4	1	no	—
247	C₂×C₃²	—	4	2	no	—
248	C₂×C₃²	—	24	2	no	—
249	C₂×C₃²	#874	1	1	yes	characteristic
250	C₂×C₃²	C₂×#204	1	2	yes	—
251	C₂×C₃²	C₂×#214	1	2	yes	—
252	C₂×C₃²	#940	1	1	yes	characteristic
253	C₂×C₃²	C₃×#127	1	1	yes	characteristic
254	C₂×C₃²	C₂×C₃×SD₁₆	1	2	yes	—
255	C₃×D₃	—	8	4	no	—
256	C₃×D₃	—	8	2	no	—
257	C₃×D₃	—	8	2	no	—
258	C₂×Q₈	—	9	2	no	—
259	C₂×D₄	—	3	2	no	—
260	C₂×D₄	—	6	2	no	—
261	SD₁₆	—	18	8	no	—
262	C₂×C₈	—	3	2	no	—
263	C₄⋊C₄	—	18	2	no	—
264	C₄²	C₃×D₃²	1	1	yes	characteristic
265	C₂²×C₃	—	2	1	no	—
266	C₂²×C₃	—	2	1	no	—
267	C₂²×C₃	—	2	1	no	—
268	C₂²×C₃	—	4	1	no	—
269	C₂²×C₃	—	12	8	no	—
270	C₂²×C₃	—	12	4	no	—
271	C₂²×C₃	—	12	4	no	—
272	C₂²×C₃	—	12	4	no	—
273	C₂²×C₃	—	12	2	no	—
274	C₂²×C₃	—	12	2	no	—
275	C₂²×C₃	C₂×#626	1	1	yes	characteristic
276	C₂²×C₃	C₂×C₃×D₁₂	1	1	yes	characteristic
277	C₂²×C₃	C₂×C₃×#67	1	1	yes	characteristic
278	C₂×D₃	—	4	4	no	—
279	C₂×D₃	—	4	2	no	—
280	C₃×C₄	—	2	2	no	—
281	C₃×C₄	—	2	2	no	—
282	C₃×C₄	—	2	2	no	—
283	C₃×C₄	—	2	4	no	—
284	C₃×C₄	—	2	4	no	—
285	C₃×C₄	—	2	4	no	—
286	C₃×C₄	—	4	2	no	—
287	C₃×C₄	—	4	4	no	—
288	C₃×C₄	—	36	2	no	—
289	C₃×C₄	#3716	1	4	yes	complemented
290	C₃×C₄	C₃×#204	1	4	yes	complemented
291	C₃×C₄	C₃×#214	1	4	yes	complemented
292	C₃×C₄	D₃×D₁₂	1	2	yes	—
293	C₃×C₄	C₂×C₃×D₁₂	1	2	yes	—
294	C₃×C₄	C₃×D₃×D₄	1	2	yes	—
295	Dic₃	—	12	4	no	—
296	Dic₃	—	12	2	no	—
297	Dic₃	—	12	2	no	—
298	C₃²	—	2	1	no	—
299	C₃²	—	2	1	no	—
300	C₃²	—	2	1	no	—
301	C₃²	—	4	1	no	—
302	C₃²	#4693	1	1	yes	characteristic, complemented
303	C₃²	#5076	1	1	yes	characteristic, complemented
304	C₃²	C₃×#492	1	1	yes	characteristic, complemented
305	C₂³	—	6	2	no	—
306	Q₈	—	18	4	no	—
307	D₄	—	6	4	no	—
308	D₄	—	12	4	no	—
309	C₂×C₄	—	18	2	no	—
310	C₂×C₄	C₃×#626	1	2	yes	—
311	C₂×C₄	C₂×C₃×D₃²	1	1	yes	characteristic
312	C₈	—	3	4	no	—
313	C₂×C₃	—	2	2	no	—
314	C₂×C₃	—	2	2	no	—
315	C₂×C₃	—	2	2	no	—
316	C₂×C₃	—	2	1	no	—
317	C₂×C₃	—	2	1	no	—
318	C₂×C₃	—	2	1	no	—
319	C₂×C₃	—	4	2	no	—
320	C₂×C₃	—	4	1	no	—
321	C₂×C₃	—	24	4	no	—
322	C₂×C₃	—	24	2	no	—
323	C₂×C₃	—	24	2	no	—
324	C₂×C₃	C₂×#3716	1	2	yes	—
325	C₂×C₃	#63911	1	1	yes	characteristic
326	C₂×C₃	C₃×#874	1	1	yes	characteristic
327	C₂×C₃	C₂×C₃×#204	1	2	yes	—
328	C₂×C₃	C₂×C₃×#214	1	2	yes	—
329	C₂×C₃	C₃×#940	1	1	yes	characteristic
330	D₃	—	8	2	no	—
331	C₂²	—	12	4	no	—
332	C₂²	—	12	2	no	—
333	C₂²	C₂×C₃×#626	1	1	yes	characteristic
334	C₄	—	36	2	no	—
335	C₄	C₃×#3716	1	4	yes	complemented
336	C₄	C₃×D₃×D₁₂	1	2	yes	—
337	C₃	—	2	1	no	—
338	C₃	—	2	1	no	—
339	C₃	—	2	1	no	—
340	C₃	—	4	1	no	—
341	C₃	#10590369	1	1	yes	characteristic, direct product factor
342	C₃	C₃×#4693	1	1	yes	characteristic, complemented
343	C₃	C₃×#5076	1	1	yes	characteristic, complemented
344	C₂	—	24	2	no	—
345	C₂	C₂×C₃×#3716	1	2	yes	—
346	C₂	C₃×#63911	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂×#626
commutator, cosocle	C₂×C₃²×C₄	C₂³×C₃
Frattini	C₂×C₄	C₂×C₃×D₃²
Fitting	C₃³×C₄²	C₂²
socle	C₂²×C₃³	C₂×D₄
prime radical	C₃²×#874	C₂
hypercenter	C₃×C₄²	D₃²
hypocenter	C₃²	C₃×#492
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄²

Complements

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

N	H ≅ G/N	type
1. C₃×#64078	C₂	semidirect
3. C₂×C₃×#3716	C₂	semidirect
4. C₃²×C₄×C₃⋊C₈	C₂	semidirect
7. #10590369	C₃	direct (N × H)
16. #64078	C₂×C₃	semidirect
25. C₂×#3716	C₂×C₃	semidirect
28. C₃×C₄×C₃⋊C₈	D₃	semidirect
29. C₃×C₄×C₃⋊C₈	C₂×C₃	semidirect
37. C₃²×C₃⋊C₈	D₄	semidirect
74. C₃³×C₄	SD₁₆	semidirect
91. C₄×C₃⋊C₈	C₃×D₃	semidirect
128. C₃×C₃⋊C₈	D₁₂	semidirect
129. C₃×C₃⋊C₈	C₃×D₄	semidirect
187. C₃²×C₄	#204	semidirect
188. C₃²×C₄	#214	semidirect
189. C₃²×C₄	C₃×SD₁₆	semidirect
201. C₃³	#492	semidirect
239. C₃⋊C₈	C₃×D₁₂	semidirect
289. C₃×C₄	#3716	semidirect
290. C₃×C₄	C₃×#204	semidirect
291. C₃×C₄	C₃×#214	semidirect
302. C₃²	#4693	semidirect
303. C₃²	#5076	semidirect
304. C₃²	C₃×#492	semidirect
335. C₄	C₃×#3716	semidirect
341. C₃	#10590369	direct (N × H)
342. C₃	C₃×#4693	semidirect
343. C₃	C₃×#5076	semidirect

Maximal subgroups

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

subgroup	index
C₃×#64078	2
C₃²×#874	2
C₂×C₃×#3716	2
C₃²×C₄×C₃⋊C₈	2
C₃×#5076	3
C₃×#4693	3
#10590369	3

Subgroups — C3×#10590369 #13157386

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10590369 #13157386