Subgroups — C₃×#10590812 #13157787

Order 1728.

590 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×Dic₁₂	C₂	1	1	yes	maximal, characteristic
2	C₃×Dic₃×Dic₆	C₂	1	2	yes	maximal
3	C₃×#63644	C₂	1	1	yes	maximal, characteristic
4	C₃²×#815	C₂	1	1	yes	maximal, characteristic
5	C₃×#63566	C₂	1	2	yes	maximal
6	C₃×#5184	—	3	1	no	maximal
7	C₃×#4716	—	3	1	no	maximal
8	#10590812	C₃	1	1	yes	maximal, characteristic, direct product factor
9	C₂×C₃²×Dic₆	C₂²	1	2	yes	—
10	C₂×C₃³×C₈	C₂²	1	1	yes	characteristic
11	C₃×C₄⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
12	C₃²×C₄×Dic₃	C₂²	1	1	yes	characteristic
13	C₂×C₃²×C₃⋊C₈	C₂²	1	1	yes	characteristic
14	C₃²×Dic₁₂	C₄	1	4	yes	complemented
15	C₃×Dic₃⋊Dic₃	—	2	2	no	—
16	C₃×Dic₃²	—	2	2	no	—
17	C₂×C₃²×Q₁₆	—	3	1	no	—
18	C₃×Q₈×Dic₃	—	3	2	no	—
19	C₂×C₃×Dic₁₂	—	2	1	no	—
20	C₂×C₃×Dic₁₂	D₃	1	1	yes	characteristic
21	C₂×C₃×Dic₁₂	C₂×C₃	1	1	yes	characteristic
22	C₃×C₄×Dic₆	—	3	2	no	—
23	Dic₃×Dic₆	C₂×C₃	1	2	yes	—
24	C₃²×#97	—	3	1	no	—
25	#63644	C₂×C₃	1	1	yes	characteristic
26	C₃×#835	—	3	2	no	—
27	C₃×#817	—	3	1	no	—
28	C₃×#817	—	3	1	no	—
29	C₃×#817	—	6	1	no	—
30	C₃×#816	—	3	2	no	—
31	C₃×#815	—	2	1	no	—
32	C₃×#815	D₃	1	1	yes	characteristic
33	C₃×#815	C₂×C₃	1	1	yes	characteristic
34	#63566	C₂×C₃	1	2	yes	—
35	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
36	C₂×C₃×C₃⋊Dic₃	—	2	2	no	—
37	C₂×C₃²×Dic₃	—	2	2	no	—
38	C₂×C₃²×Dic₃	D₄	1	2	yes	—
39	C₃²×Dic₆	—	2	2	no	—
40	C₃²×Dic₆	C₂×C₄	1	4	yes	—
41	C₃³×C₈	C₂×C₄	1	2	yes	—
42	C₃²×C₃⋊C₈	—	2	1	no	—
43	C₃×#441	—	9	1	no	—
44	#5184	—	3	1	no	—
45	#4716	—	3	1	no	—
46	C₂×C₃²×Q₈	—	3	2	no	—
47	C₂×C₃×Dic₆	—	2	2	no	—
48	C₂×C₃×Dic₆	C₂×D₃	1	2	yes	—
49	C₂×C₃×Dic₆	C₂²×C₃	1	2	yes	—
50	C₃²×Q₁₆	—	3	4	no	—
51	C₂×C₃²×C₈	—	2	1	no	—
52	C₂×C₃²×C₈	—	2	1	no	—
53	C₂×C₃²×C₈	—	2	1	no	—
54	C₂×C₃²×C₈	—	3	1	no	—
55	C₂×C₃²×C₈	—	4	1	no	—
56	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
57	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
58	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
59	C₃²×C₄²	—	3	1	no	—
60	C₄⋊(C₃⋊Dic₃)	C₂²×C₃	1	2	yes	—
61	C₃×C₄⋊Dic₃	—	3	2	no	—
62	C₃×C₄⋊Dic₃	—	3	2	no	—
63	C₃×C₄⋊Dic₃	—	6	2	no	—
64	C₃×C₄⋊Dic₃	—	6	2	no	—
65	C₃×Dic₃⋊C₄	—	6	2	no	—
66	C₃×C₄×Dic₃	—	2	1	no	—
67	C₃×C₄×Dic₃	—	6	2	no	—
68	C₃×C₄×Dic₃	—	6	2	no	—
69	C₃×C₄×Dic₃	C₂×D₃	1	1	yes	characteristic
70	C₃×C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
71	C₂×C₃×C₃⋊C₈	—	2	1	no	—
72	C₂×C₃×C₃⋊C₈	C₂×D₃	1	1	yes	characteristic
73	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
74	C₃×Dic₁₂	—	2	4	no	—
75	C₃×Dic₁₂	Dic₃	1	4	yes	complemented
76	C₃×Dic₁₂	C₃×C₄	1	4	yes	complemented
77	Dic₃⋊Dic₃	—	2	2	no	—
78	Dic₃²	—	2	2	no	—
79	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
80	C₃³×C₄	C₂²×C₄	1	1	yes	characteristic
81	C₃³×C₄	#41	1	1	yes	characteristic
82	C₃×C₃⋊Dic₃	—	4	2	no	—
83	C₃²×Dic₃	—	2	4	no	—
84	C₃²×Dic₃	—	2	2	no	—
85	C₂×C₃×Q₁₆	—	3	1	no	—
86	C₂×C₃×Q₁₆	—	3	1	no	—
87	C₂×C₃×Q₁₆	—	6	1	no	—
88	C₃×C₄×Q₈	—	9	2	no	—
89	Q₈×Dic₃	—	3	2	no	—
90	C₂×Dic₁₂	C₃×D₃	1	1	yes	characteristic
91	C₄×Dic₆	—	3	2	no	—
92	C₃×#106	—	9	1	no	—
93	C₃×#103	—	9	2	no	—
94	C₃×#97	—	3	1	no	—
95	C₃×#97	—	3	1	no	—
96	C₃×#97	—	6	1	no	—
97	#835	—	3	2	no	—
98	#817	—	3	1	no	—
99	#817	—	3	1	no	—
100	#817	—	6	1	no	—
101	#816	—	3	2	no	—
102	#815	C₃×D₃	1	1	yes	characteristic
103	C₃²×Q₈	—	3	4	no	—
104	C₃²×Q₈	—	6	2	no	—
105	C₂×C₃²×C₄	—	2	1	no	—
106	C₂×C₃²×C₄	—	2	1	no	—
107	C₂×C₃²×C₄	—	2	1	no	—
108	C₂×C₃²×C₄	—	3	2	no	—
109	C₂×C₃²×C₄	—	4	1	no	—
110	C₂×C₃²×C₄	—	6	2	no	—
111	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
112	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
113	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
114	C₂×C₃⋊Dic₃	—	2	2	no	—
115	C₂×C₃×Dic₃	—	2	2	no	—
116	C₂×C₃×Dic₃	—	2	2	no	—
117	C₂×C₃×Dic₃	—	2	2	no	—
118	C₂×C₃×Dic₃	—	4	2	no	—
119	C₂×C₃×Dic₃	—	6	2	no	—
120	C₂×C₃×Dic₃	—	6	2	no	—
121	C₂×C₃×Dic₃	—	12	2	no	—
122	C₂×C₃×Dic₃	D₁₂	1	2	yes	—
123	C₂×C₃×Dic₃	C₃×D₄	1	2	yes	—
124	C₃×Dic₆	—	2	2	no	—
125	C₃×Dic₆	—	2	2	no	—
126	C₃×Dic₆	—	2	4	no	—
127	C₃×Dic₆	—	4	2	no	—
128	C₃×Dic₆	C₂×Dic₃	1	4	yes	—
129	C₃×Dic₆	C₂×C₃×C₄	1	4	yes	—
130	C₃²×C₈	—	2	2	no	—
131	C₃²×C₈	—	2	2	no	—
132	C₃²×C₈	—	2	2	no	—
133	C₃²×C₈	—	4	2	no	—
134	C₃²×C₈	—	6	1	no	—
135	C₃²×C₈	C₄×D₃	1	2	yes	—
136	C₃²×C₈	C₂×Dic₃	1	2	yes	—
137	C₃²×C₈	C₂×C₃×C₄	1	2	yes	—
138	C₃×C₃⋊C₈	—	2	2	no	—
139	C₃×C₃⋊C₈	—	2	1	no	—
140	C₃×C₃⋊C₈	—	2	1	no	—
141	#441	—	9	1	no	Sylow
142	C₂×C₃³	C₄×D₄	1	1	yes	characteristic
143	C₂×C₃³	#137	1	2	yes	—
144	C₂×C₃×Q₈	—	3	2	no	—
145	C₂×C₃×Q₈	—	3	2	no	—
146	C₂×C₃×Q₈	—	6	2	no	—
147	C₂×Dic₆	C₂×C₃×D₃	1	2	yes	—
148	C₃×Q₁₆	—	3	4	no	—
149	C₃×Q₁₆	—	3	4	no	—
150	C₃×Q₁₆	—	6	4	no	—
151	C₂×C₃×C₈	—	2	1	no	—
152	C₂×C₃×C₈	—	2	1	no	—
153	C₂×C₃×C₈	—	2	1	no	—
154	C₂×C₃×C₈	—	3	1	no	—
155	C₂×C₃×C₈	—	3	1	no	—
156	C₂×C₃×C₈	—	4	1	no	—
157	C₂×C₃×C₈	—	6	1	no	—
158	C₂×C₃×C₈	D₃²	1	1	yes	characteristic
159	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
160	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
161	C₃×C₄⋊C₄	—	9	2	no	—
162	C₃×C₄⋊C₄	—	18	2	no	—
163	C₃×C₄²	—	3	1	no	—
164	C₃×C₄²	—	3	1	no	—
165	C₃×C₄²	—	6	1	no	—
166	C₃×C₄²	—	18	2	no	—
167	C₄⋊Dic₃	—	3	2	no	—
168	C₄⋊Dic₃	—	3	2	no	—
169	C₄⋊Dic₃	—	6	2	no	—
170	C₄⋊Dic₃	—	6	2	no	—
171	Dic₃⋊C₄	—	6	2	no	—
172	C₄×Dic₃	—	6	2	no	—
173	C₄×Dic₃	—	6	2	no	—
174	C₄×Dic₃	C₂×C₃×D₃	1	1	yes	characteristic
175	C₂×C₃⋊C₈	C₂×C₃×D₃	1	1	yes	characteristic
176	Dic₁₂	C₃×Dic₃	1	4	yes	complemented
177	C₂²×C₃²	—	2	1	no	—
178	C₂²×C₃²	—	2	1	no	—
179	C₂²×C₃²	—	2	1	no	—
180	C₂²×C₃²	—	4	1	no	—
181	C₂²×C₃²	C₂×D₁₂	1	1	yes	characteristic
182	C₂²×C₃²	D₃×D₄	1	1	yes	characteristic
183	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
184	C₃²×C₄	—	2	1	no	—
185	C₃²×C₄	—	2	1	no	—
186	C₃²×C₄	—	2	1	no	—
187	C₃²×C₄	—	2	1	no	—
188	C₃²×C₄	—	2	1	no	—
189	C₃²×C₄	—	2	1	no	—
190	C₃²×C₄	—	4	1	no	—
191	C₃²×C₄	—	4	1	no	—
192	C₃²×C₄	—	6	4	no	—
193	C₃²×C₄	—	6	2	no	—
194	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
195	C₃²×C₄	#235	1	1	yes	characteristic
196	C₃²×C₄	#237	1	1	yes	characteristic
197	C₃²×C₄	C₂²×Dic₃	1	1	yes	characteristic
198	C₃²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
199	C₃²×C₄	C₃×#41	1	1	yes	characteristic
200	C₃⋊Dic₃	—	4	2	no	—
201	C₃×Dic₃	—	2	2	no	—
202	C₃×Dic₃	—	2	4	no	—
203	C₃×Dic₃	—	2	4	no	—
204	C₃×Dic₃	—	2	2	no	—
205	C₃×Dic₃	—	4	4	no	—
206	C₃×Dic₃	—	4	2	no	—
207	C₃×Dic₃	—	12	2	no	—
208	C₃×Dic₃	—	12	2	no	—
209	C₃×Dic₃	—	24	2	no	—
210	C₂×Q₁₆	—	3	1	no	—
211	C₄×Q₈	—	9	2	no	—
212	#106	—	9	1	no	—
213	#103	—	9	2	no	—
214	#97	—	3	1	no	—
215	C₃³	#441	1	1	yes	characteristic, complemented, Sylow
216	C₃×Q₈	—	3	4	no	—
217	C₃×Q₈	—	3	4	no	—
218	C₃×Q₈	—	6	4	no	—
219	C₃×Q₈	—	6	2	no	—
220	C₃×Q₈	—	6	2	no	—
221	C₃×Q₈	—	12	2	no	—
222	C₂×C₃×C₄	—	2	1	no	—
223	C₂×C₃×C₄	—	2	1	no	—
224	C₂×C₃×C₄	—	2	1	no	—
225	C₂×C₃×C₄	—	3	2	no	—
226	C₂×C₃×C₄	—	3	2	no	—
227	C₂×C₃×C₄	—	4	1	no	—
228	C₂×C₃×C₄	—	6	2	no	—
229	C₂×C₃×C₄	—	6	2	no	—
230	C₂×C₃×C₄	—	6	2	no	—
231	C₂×C₃×C₄	—	12	2	no	—
232	C₂×C₃×C₄	—	18	2	no	—
233	C₂×C₃×C₄	C₂×D₃²	1	1	yes	characteristic
234	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
235	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
236	C₂×Dic₃	—	2	2	no	—
237	C₂×Dic₃	—	6	2	no	—
238	C₂×Dic₃	—	6	2	no	—
239	C₂×Dic₃	—	12	2	no	—
240	C₂×Dic₃	C₃×D₁₂	1	2	yes	—
241	Dic₆	—	2	2	no	—
242	Dic₆	C₂×C₃×Dic₃	1	4	yes	—
243	C₃×C₈	—	2	2	no	—
244	C₃×C₈	—	2	2	no	—
245	C₃×C₈	—	2	2	no	—
246	C₃×C₈	—	4	2	no	—
247	C₃×C₈	—	6	2	no	—
248	C₃×C₈	—	6	1	no	—
249	C₃×C₈	—	6	1	no	—
250	C₃×C₈	D₃×Dic₃	1	2	yes	—
251	C₃×C₈	C₃×C₄×D₃	1	2	yes	—
252	C₃×C₈	C₂×C₃×Dic₃	1	2	yes	—
253	C₃⋊C₈	—	2	1	no	—
254	C₂×C₃²	—	2	2	no	—
255	C₂×C₃²	—	2	2	no	—
256	C₂×C₃²	—	2	2	no	—
257	C₂×C₃²	—	2	1	no	—
258	C₂×C₃²	—	2	1	no	—
259	C₂×C₃²	—	2	1	no	—
260	C₂×C₃²	—	4	1	no	—
261	C₂×C₃²	—	4	2	no	—
262	C₂×C₃²	C₄×D₁₂	1	1	yes	characteristic
263	C₂×C₃²	#909	1	2	yes	—
264	C₂×C₃²	#918	1	2	yes	—
265	C₂×C₃²	D₄×Dic₃	1	1	yes	characteristic
266	C₂×C₃²	C₃×C₄×D₄	1	1	yes	characteristic
267	C₂×C₃²	C₃×#137	1	2	yes	—
268	C₂×Q₈	—	3	2	no	—
269	Q₁₆	—	3	4	no	—
270	C₂×C₈	—	3	1	no	—
271	C₂×C₈	C₃×D₃²	1	1	yes	characteristic
272	C₄⋊C₄	—	9	2	no	—
273	C₄⋊C₄	—	18	2	no	—
274	C₄²	—	3	1	no	—
275	C₄²	—	18	2	no	—
276	C₂²×C₃	—	2	1	no	—
277	C₂²×C₃	—	2	1	no	—
278	C₂²×C₃	—	2	1	no	—
279	C₂²×C₃	—	4	1	no	—
280	C₂²×C₃	D₃×D₁₂	1	1	yes	characteristic
281	C₂²×C₃	C₂×C₃×D₁₂	1	1	yes	characteristic
282	C₂²×C₃	C₃×D₃×D₄	1	1	yes	characteristic
283	C₃×C₄	—	2	1	no	—
284	C₃×C₄	—	2	1	no	—
285	C₃×C₄	—	2	1	no	—
286	C₃×C₄	—	2	1	no	—
287	C₃×C₄	—	2	1	no	—
288	C₃×C₄	—	2	1	no	—
289	C₃×C₄	—	4	1	no	—
290	C₃×C₄	—	4	1	no	—
291	C₃×C₄	—	6	2	no	—
292	C₃×C₄	—	6	2	no	—
293	C₃×C₄	—	6	4	no	—
294	C₃×C₄	—	6	4	no	—
295	C₃×C₄	—	12	2	no	—
296	C₃×C₄	—	12	4	no	—
297	C₃×C₄	—	36	2	no	—
298	C₃×C₄	#3795	1	1	yes	characteristic
299	C₃×C₄	C₂×D₃×Dic₃	1	1	yes	characteristic
300	C₃×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
301	C₃×C₄	C₃×#235	1	1	yes	characteristic
302	C₃×C₄	C₃×#237	1	1	yes	characteristic
303	C₃×C₄	C₂²×C₃×Dic₃	1	1	yes	characteristic
304	Dic₃	—	2	2	no	—
305	Dic₃	—	2	4	no	—
306	Dic₃	—	12	2	no	—
307	Dic₃	—	12	2	no	—
308	Dic₃	—	24	2	no	—
309	C₃²	—	2	1	no	—
310	C₃²	—	2	1	no	—
311	C₃²	—	2	1	no	—
312	C₃²	—	4	1	no	—
313	C₃²	#4716	1	1	yes	characteristic, complemented
314	C₃²	#5184	1	1	yes	characteristic, complemented
315	C₃²	C₃×#441	1	1	yes	characteristic, complemented
316	Q₈	—	3	4	no	—
317	Q₈	—	6	2	no	—
318	C₂×C₄	—	3	2	no	—
319	C₂×C₄	—	6	2	no	—
320	C₂×C₄	—	18	2	no	—
321	C₂×C₄	C₂×C₃×D₃²	1	1	yes	characteristic
322	C₈	—	6	1	no	—
323	C₈	C₃×D₃×Dic₃	1	2	yes	—
324	C₂×C₃	—	2	2	no	—
325	C₂×C₃	—	2	2	no	—
326	C₂×C₃	—	2	2	no	—
327	C₂×C₃	—	2	1	no	—
328	C₂×C₃	—	2	1	no	—
329	C₂×C₃	—	2	1	no	—
330	C₂×C₃	—	4	2	no	—
331	C₂×C₃	—	4	1	no	—
332	C₂×C₃	#63801	1	2	yes	—
333	C₂×C₃	Dic₃×D₁₂	1	1	yes	characteristic
334	C₂×C₃	C₃×C₄×D₁₂	1	1	yes	characteristic
335	C₂×C₃	C₃×#909	1	2	yes	—
336	C₂×C₃	C₃×#918	1	2	yes	—
337	C₂×C₃	C₃×D₄×Dic₃	1	1	yes	characteristic
338	C₂²	C₃×D₃×D₁₂	1	1	yes	characteristic
339	C₄	—	6	2	no	—
340	C₄	—	6	4	no	—
341	C₄	—	36	2	no	—
342	C₄	C₃×#3795	1	1	yes	characteristic
343	C₄	C₂×C₃×D₃×Dic₃	1	1	yes	characteristic
344	C₃	—	2	1	no	—
345	C₃	—	2	1	no	—
346	C₃	—	2	1	no	—
347	C₃	—	4	1	no	—
348	C₃	#10590812	1	1	yes	characteristic, direct product factor
349	C₃	C₃×#4716	1	1	yes	characteristic, complemented
350	C₃	C₃×#5184	1	1	yes	characteristic, complemented
351	C₂	C₃×#63801	1	2	yes	—
352	C₂	C₃×Dic₃×D₁₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	D₃×D₁₂
commutator	C₃²×C₄	C₂²×C₃×C₄
Frattini	C₂×C₄	C₂×C₃×D₃²
Fitting	C₂×C₃³×C₈	C₂²
socle, prime radical	C₂²×C₃³	C₂×D₄
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₂×C₃×C₈	D₃²
hypocenter	C₃²	C₃×#441
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₈

Complements

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

N	H ≅ G/N	type
8. #10590812	C₃	direct (N × H)
14. C₃²×Dic₁₂	C₄	semidirect
75. C₃×Dic₁₂	Dic₃	semidirect
76. C₃×Dic₁₂	C₃×C₄	semidirect
176. Dic₁₂	C₃×Dic₃	semidirect
215. C₃³	#441	semidirect
313. C₃²	#4716	semidirect
314. C₃²	#5184	semidirect
315. C₃²	C₃×#441	semidirect
348. C₃	#10590812	direct (N × H)
349. C₃	C₃×#4716	semidirect
350. C₃	C₃×#5184	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×Dic₁₂	2
C₃×Dic₃×Dic₆	2
C₃×#63644	2
C₃²×#815	2
C₃×#63566	2
C₃×#5184	3
C₃×#4716	3
#10590812	3

Subgroups — C3×#10590812 #13157787

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10590812 #13157787