Subgroups — C₃×#10590386 #13157399

Order 1728.

698 conjugacy classes of subgroups.

The subgroup lattice has 318 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
2	C₃²×#869	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#3719	C₂	1	4	yes	maximal
4	C₃²×C₄×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₃×#5092	—	3	1	no	maximal
6	C₃×#4699	—	3	1	no	maximal
7	#10590386	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂×C₃×#634	C₂²	1	2	yes	—
9	C₂×C₃²×Dic₆	C₂²	1	2	yes	—
10	C₃³×C₄²	C₂²	1	1	yes	characteristic
11	C₃×C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
12	C₃²×C₄⋊Dic₃	—	2	2	no	—
13	C₂×C₃²×C₃⋊C₈	C₂²	1	2	yes	—
14	C₃×#3719	—	2	8	no	—
15	C₃²×C₄⋊Q₈	—	3	1	no	—
16	#64078	C₂×C₃	1	1	yes	characteristic
17	C₂×C₃×#216	—	3	4	no	—
18	C₂×C₃×Dic₁₂	—	3	4	no	—
19	C₃×#869	—	2	1	no	—
20	C₃×#869	—	3	1	no	—
21	C₃×#869	—	3	1	no	—
22	C₃×#869	—	6	1	no	—
23	C₃×#869	D₃	1	1	yes	characteristic
24	C₃×#869	C₂×C₃	1	1	yes	characteristic
25	C₂×#3719	C₂×C₃	1	4	yes	—
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
29	C₃×C₄×C₃⋊C₈	C₂×C₃	1	1	yes	characteristic
30	C₂×C₃³×C₄	D₄	1	2	yes	—
31	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
32	C₂×C₃×C₃⋊Dic₃	—	2	2	no	—
33	C₃×#634	—	2	4	no	—
34	C₂×C₃²×Dic₃	—	2	2	no	—
35	C₃²×Dic₆	—	2	4	no	—
36	C₃²×C₃⋊C₈	D₄	1	4	yes	—
37	C₃×#494	—	9	1	no	—
38	#5092	—	3	1	no	—
39	#4699	—	3	1	no	—
40	C₂×C₃²×Q₈	—	3	2	no	—
41	C₂×#634	C₂²×C₃	1	2	yes	—
42	C₂×C₃×Dic₆	—	2	2	no	—
43	C₂×C₃×Dic₆	—	3	2	no	—
44	C₂×C₃×Dic₆	—	3	2	no	—
45	C₂×C₃×Dic₆	—	6	2	no	—
46	C₂×C₃×Dic₆	C₂×D₃	1	2	yes	—
47	C₂×C₃×Dic₆	C₂²×C₃	1	2	yes	—
48	C₂×C₃²×C₈	—	3	2	no	—
49	C₃²×C₄⋊C₄	—	6	2	no	—
50	C₃²×C₄²	—	2	1	no	—
51	C₃²×C₄²	—	2	1	no	—
52	C₃²×C₄²	—	2	1	no	—
53	C₃²×C₄²	—	4	1	no	—
54	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
55	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
56	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
57	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
58	C₃×#216	—	6	8	no	—
59	C₃×C₄⋊Dic₃	—	2	4	no	—
60	C₃×C₄⋊Dic₃	—	2	2	no	—
61	C₃×C₄⋊Dic₃	—	2	2	no	—
62	C₃×C₄⋊Dic₃	—	6	4	no	—
63	C₃×C₄⋊Dic₃	—	6	2	no	—
64	C₃×C₄⋊Dic₃	—	6	2	no	—
65	C₂×C₃×C₃⋊C₈	—	2	2	no	—
66	C₂×C₃×C₃⋊C₈	C₂×D₃	1	2	yes	—
67	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	2	yes	—
68	C₃×Dic₁₂	—	6	8	no	—
69	#3719	—	2	8	no	—
70	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
71	C₃³×C₄	Q₁₆	1	4	yes	complemented
72	C₃³×C₄	C₂×D₄	1	2	yes	—
73	C₃×C₃⋊Dic₃	—	4	2	no	—
74	C₃²×Dic₃	—	4	2	no	—
75	C₂×C₃×Q₁₆	—	9	4	no	—
76	C₃×C₄⋊Q₈	—	3	1	no	—
77	C₃×C₄⋊Q₈	—	3	1	no	—
78	C₃×C₄⋊Q₈	—	6	1	no	—
79	C₃×C₄⋊Q₈	—	9	1	no	—
80	C₂×#216	—	3	4	no	—
81	C₂×Dic₁₂	—	3	4	no	—
82	#869	—	3	1	no	—
83	#869	—	3	1	no	—
84	#869	—	6	1	no	—
85	#869	C₃×D₃	1	1	yes	characteristic
86	C₃×C₄×C₈	—	3	1	no	—
87	C₃×C₄×C₈	—	3	1	no	—
88	C₃×C₄×C₈	—	6	1	no	—
89	C₄×C₃⋊C₈	C₃×D₃	1	1	yes	characteristic
90	C₃²×Q₈	—	6	4	no	—
91	C₂×C₃²×C₄	—	2	2	no	—
92	C₂×C₃²×C₄	—	2	2	no	—
93	C₂×C₃²×C₄	—	2	2	no	—
94	C₂×C₃²×C₄	—	2	1	no	—
95	C₂×C₃²×C₄	—	2	1	no	—
96	C₂×C₃²×C₄	—	2	1	no	—
97	C₂×C₃²×C₄	—	4	2	no	—
98	C₂×C₃²×C₄	—	4	1	no	—
99	C₂×C₃²×C₄	—	6	2	no	—
100	C₂×C₃²×C₄	D₁₂	1	2	yes	—
101	C₂×C₃²×C₄	#67	1	2	yes	—
102	C₂×C₃²×C₄	C₃×D₄	1	2	yes	—
103	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
104	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
105	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
106	C₂×C₃⋊Dic₃	—	2	2	no	—
107	#634	—	2	4	no	—
108	C₂×C₃×Dic₃	—	2	4	no	—
109	C₂×C₃×Dic₃	—	2	2	no	—
110	C₂×C₃×Dic₃	—	2	2	no	—
111	C₂×C₃×Dic₃	—	6	4	no	—
112	C₂×C₃×Dic₃	—	6	2	no	—
113	C₂×C₃×Dic₃	—	6	2	no	—
114	C₃×Dic₆	—	2	4	no	—
115	C₃×Dic₆	—	2	4	no	—
116	C₃×Dic₆	—	4	4	no	—
117	C₃×Dic₆	—	6	4	no	—
118	C₃×Dic₆	—	6	4	no	—
119	C₃×Dic₆	—	12	4	no	—
120	C₃²×C₈	—	3	4	no	—
121	C₃×C₃⋊C₈	—	2	4	no	—
122	C₃×C₃⋊C₈	D₁₂	1	4	yes	—
123	C₃×C₃⋊C₈	C₃×D₄	1	4	yes	—
124	#494	—	9	1	no	Sylow
125	C₂×C₃³	#127	1	1	yes	characteristic
126	C₂×C₃³	C₂×Q₁₆	1	2	yes	—
127	C₂×C₃×Q₈	—	3	2	no	—
128	C₂×C₃×Q₈	—	3	2	no	—
129	C₂×C₃×Q₈	—	6	2	no	—
130	C₂×C₃×Q₈	—	9	2	no	—
131	C₂×Dic₆	—	3	2	no	—
132	C₂×Dic₆	—	3	2	no	—
133	C₂×Dic₆	—	6	2	no	—
134	C₂×Dic₆	C₂×C₃×D₃	1	2	yes	—
135	C₃×Q₁₆	—	18	8	no	—
136	C₂×C₃×C₈	—	3	2	no	—
137	C₂×C₃×C₈	—	3	2	no	—
138	C₂×C₃×C₈	—	6	2	no	—
139	C₃×C₄⋊C₄	—	6	4	no	—
140	C₃×C₄⋊C₄	—	6	2	no	—
141	C₃×C₄⋊C₄	—	6	2	no	—
142	C₃×C₄⋊C₄	—	18	2	no	—
143	C₃×C₄²	—	2	1	no	—
144	C₃×C₄²	—	2	1	no	—
145	C₃×C₄²	—	2	1	no	—
146	C₃×C₄²	—	4	1	no	—
147	C₃×C₄²	D₃²	1	1	yes	characteristic
148	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
149	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
150	#216	—	6	8	no	—
151	C₄⋊Dic₃	—	2	2	no	—
152	C₄⋊Dic₃	—	6	4	no	—
153	C₄⋊Dic₃	—	6	2	no	—
154	C₄⋊Dic₃	—	6	2	no	—
155	C₂×C₃⋊C₈	C₂×C₃×D₃	1	2	yes	—
156	Dic₁₂	—	6	8	no	—
157	C₂²×C₃²	—	2	1	no	—
158	C₂²×C₃²	—	2	1	no	—
159	C₂²×C₃²	—	2	1	no	—
160	C₂²×C₃²	—	4	1	no	—
161	C₂²×C₃²	C₂×D₁₂	1	1	yes	characteristic
162	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
163	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
164	C₃²×C₄	—	2	4	no	—
165	C₃²×C₄	—	2	4	no	—
166	C₃²×C₄	—	2	4	no	—
167	C₃²×C₄	—	2	2	no	—
168	C₃²×C₄	—	2	2	no	—
169	C₃²×C₄	—	2	2	no	—
170	C₃²×C₄	—	4	4	no	—
171	C₃²×C₄	—	4	2	no	—
172	C₃²×C₄	—	12	2	no	—
173	C₃²×C₄	Dic₁₂	1	4	yes	complemented
174	C₃²×C₄	#216	1	4	yes	complemented
175	C₃²×C₄	C₃×Q₁₆	1	4	yes	complemented
176	C₃²×C₄	C₂×D₁₂	1	2	yes	—
177	C₃²×C₄	D₃×D₄	1	2	yes	—
178	C₃²×C₄	C₂×C₃×D₄	1	2	yes	—
179	C₃⋊Dic₃	—	4	2	no	—
180	C₃×Dic₃	—	4	4	no	—
181	C₃×Dic₃	—	4	2	no	—
182	C₃×Dic₃	—	4	2	no	—
183	C₃×Dic₃	—	12	4	no	—
184	C₃×Dic₃	—	12	2	no	—
185	C₃×Dic₃	—	12	2	no	—
186	C₂×Q₁₆	—	9	4	no	—
187	C₄⋊Q₈	—	3	1	no	—
188	C₄⋊Q₈	—	9	1	no	—
189	C₄×C₈	—	3	1	no	—
190	C₃³	#494	1	1	yes	characteristic, complemented, Sylow
191	C₃×Q₈	—	6	4	no	—
192	C₃×Q₈	—	6	4	no	—
193	C₃×Q₈	—	12	4	no	—
194	C₃×Q₈	—	18	4	no	—
195	C₂×C₃×C₄	—	2	2	no	—
196	C₂×C₃×C₄	—	2	2	no	—
197	C₂×C₃×C₄	—	2	2	no	—
198	C₂×C₃×C₄	—	2	1	no	—
199	C₂×C₃×C₄	—	2	1	no	—
200	C₂×C₃×C₄	—	2	1	no	—
201	C₂×C₃×C₄	—	4	2	no	—
202	C₂×C₃×C₄	—	4	1	no	—
203	C₂×C₃×C₄	—	6	4	no	—
204	C₂×C₃×C₄	—	6	2	no	—
205	C₂×C₃×C₄	—	6	2	no	—
206	C₂×C₃×C₄	—	18	2	no	—
207	C₂×C₃×C₄	#626	1	2	yes	—
208	C₂×C₃×C₄	C₃×D₁₂	1	2	yes	—
209	C₂×C₃×C₄	C₃×#67	1	2	yes	—
210	C₂×C₃×C₄	C₂×D₃²	1	1	yes	characteristic
211	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
212	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
213	C₂×Dic₃	—	2	2	no	—
214	C₂×Dic₃	—	6	4	no	—
215	C₂×Dic₃	—	6	2	no	—
216	C₂×Dic₃	—	6	2	no	—
217	Dic₆	—	2	4	no	—
218	Dic₆	—	6	4	no	—
219	Dic₆	—	6	4	no	—
220	Dic₆	—	12	4	no	—
221	C₃×C₈	—	3	4	no	—
222	C₃×C₈	—	3	4	no	—
223	C₃×C₈	—	6	4	no	—
224	C₃⋊C₈	C₃×D₁₂	1	4	yes	—
225	C₂×C₃²	—	2	2	no	—
226	C₂×C₃²	—	2	1	no	—
227	C₂×C₃²	—	2	2	no	—
228	C₂×C₃²	—	2	2	no	—
229	C₂×C₃²	—	2	1	no	—
230	C₂×C₃²	—	2	1	no	—
231	C₂×C₃²	—	4	1	no	—
232	C₂×C₃²	—	4	2	no	—
233	C₂×C₃²	#874	1	1	yes	characteristic
234	C₂×C₃²	C₂×Dic₁₂	1	2	yes	—
235	C₂×C₃²	#940	1	1	yes	characteristic
236	C₂×C₃²	C₂×#216	1	2	yes	—
237	C₂×C₃²	C₃×#127	1	1	yes	characteristic
238	C₂×C₃²	C₂×C₃×Q₁₆	1	2	yes	—
239	C₂×Q₈	—	3	2	no	—
240	C₂×Q₈	—	9	2	no	—
241	Q₁₆	—	18	8	no	—
242	C₂×C₈	—	3	2	no	—
243	C₄⋊C₄	—	6	2	no	—
244	C₄⋊C₄	—	18	2	no	—
245	C₄²	C₃×D₃²	1	1	yes	characteristic
246	C₂²×C₃	—	2	1	no	—
247	C₂²×C₃	—	2	1	no	—
248	C₂²×C₃	—	2	1	no	—
249	C₂²×C₃	—	4	1	no	—
250	C₂²×C₃	C₂×#626	1	1	yes	characteristic
251	C₂²×C₃	C₂×C₃×D₁₂	1	1	yes	characteristic
252	C₂²×C₃	C₂×C₃×#67	1	1	yes	characteristic
253	C₃×C₄	—	2	2	no	—
254	C₃×C₄	—	2	2	no	—
255	C₃×C₄	—	2	2	no	—
256	C₃×C₄	—	2	4	no	—
257	C₃×C₄	—	2	4	no	—
258	C₃×C₄	—	2	4	no	—
259	C₃×C₄	—	4	2	no	—
260	C₃×C₄	—	4	4	no	—
261	C₃×C₄	—	12	4	no	—
262	C₃×C₄	—	12	2	no	—
263	C₃×C₄	—	12	2	no	—
264	C₃×C₄	—	36	2	no	—
265	C₃×C₄	#3719	1	4	yes	complemented
266	C₃×C₄	C₃×Dic₁₂	1	4	yes	complemented
267	C₃×C₄	C₃×#216	1	4	yes	complemented
268	C₃×C₄	D₃×D₁₂	1	2	yes	—
269	C₃×C₄	C₂×C₃×D₁₂	1	2	yes	—
270	C₃×C₄	C₃×D₃×D₄	1	2	yes	—
271	Dic₃	—	4	2	no	—
272	Dic₃	—	12	4	no	—
273	Dic₃	—	12	2	no	—
274	Dic₃	—	12	2	no	—
275	C₃²	—	2	1	no	—
276	C₃²	—	2	1	no	—
277	C₃²	—	2	1	no	—
278	C₃²	—	4	1	no	—
279	C₃²	#4699	1	1	yes	characteristic, complemented
280	C₃²	#5092	1	1	yes	characteristic, complemented
281	C₃²	C₃×#494	1	1	yes	characteristic, complemented
282	Q₈	—	6	4	no	—
283	Q₈	—	18	4	no	—
284	C₂×C₄	—	6	2	no	—
285	C₂×C₄	—	18	2	no	—
286	C₂×C₄	C₃×#626	1	2	yes	—
287	C₂×C₄	C₂×C₃×D₃²	1	1	yes	characteristic
288	C₈	—	3	4	no	—
289	C₂×C₃	—	2	2	no	—
290	C₂×C₃	—	2	2	no	—
291	C₂×C₃	—	2	2	no	—
292	C₂×C₃	—	2	1	no	—
293	C₂×C₃	—	2	1	no	—
294	C₂×C₃	—	2	1	no	—
295	C₂×C₃	—	4	2	no	—
296	C₂×C₃	—	4	1	no	—
297	C₂×C₃	C₂×#3719	1	2	yes	—
298	C₂×C₃	#63911	1	1	yes	characteristic
299	C₂×C₃	C₃×#874	1	1	yes	characteristic
300	C₂×C₃	C₂×C₃×Dic₁₂	1	2	yes	—
301	C₂×C₃	C₃×#940	1	1	yes	characteristic
302	C₂×C₃	C₂×C₃×#216	1	2	yes	—
303	C₂²	C₂×C₃×#626	1	1	yes	characteristic
304	C₄	—	12	2	no	—
305	C₄	—	36	2	no	—
306	C₄	C₃×#3719	1	4	yes	complemented
307	C₄	C₃×D₃×D₁₂	1	2	yes	—
308	C₃	—	2	1	no	—
309	C₃	—	2	1	no	—
310	C₃	—	2	1	no	—
311	C₃	—	4	1	no	—
312	C₃	#10590386	1	1	yes	characteristic, direct product factor
313	C₃	C₃×#4699	1	1	yes	characteristic, complemented
314	C₃	C₃×#5092	1	1	yes	characteristic, complemented
315	C₂	C₂×C₃×#3719	1	2	yes	—
316	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, prime radical	C₂²×C₃³	C₂×D₄
hypercenter	C₃×C₄²	D₃²
hypocenter	C₃²	C₃×#494
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
7. #10590386	C₃	direct (N × H)
71. C₃³×C₄	Q₁₆	semidirect
173. C₃²×C₄	Dic₁₂	semidirect
174. C₃²×C₄	#216	semidirect
175. C₃²×C₄	C₃×Q₁₆	semidirect
190. C₃³	#494	semidirect
265. C₃×C₄	#3719	semidirect
266. C₃×C₄	C₃×Dic₁₂	semidirect
267. C₃×C₄	C₃×#216	semidirect
279. C₃²	#4699	semidirect
280. C₃²	#5092	semidirect
281. C₃²	C₃×#494	semidirect
306. C₄	C₃×#3719	semidirect
312. C₃	#10590386	direct (N × H)
313. C₃	C₃×#4699	semidirect
314. C₃	C₃×#5092	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×#869	2
C₂×C₃×#3719	2
C₃²×C₄×C₃⋊C₈	2
C₃×#5092	3
C₃×#4699	3
#10590386	3

Subgroups — C3×#10590386 #13157399

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10590386 #13157399