Subgroups — C₂×C₃×#64287 #13181493

Order 1728.

968 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#3830	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#3792	C₂	1	2	yes	maximal, complemented
3	C₂²×C₃×#622	C₂	1	2	yes	maximal, complemented
4	C₃×#64287	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃×#3788	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃×#3787	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#64287	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂×C₃³×D₄	C₄	1	1	yes	characteristic
9	C₂×C₃×#638	C₄	1	2	yes	—
10	C₂²×C₃×C₃⋊Dic₃	C₂²	1	2	yes	—
11	C₃×#3830	C₄	1	4	yes	—
12	C₃×#3830	C₂²	1	4	yes	—
13	C₂×C₃×C₄×C₃⋊D₃	C₂²	1	1	yes	characteristic
14	C₂×C₃×#634	C₄	1	1	yes	characteristic
15	C₃×#3792	C₂²	1	8	yes	complemented
16	C₂×C₃×#622	C₂²	1	1	yes	characteristic
17	C₂×C₃×#622	C₂²	1	1	yes	characteristic
18	C₂×C₃×#622	C₂²	1	2	yes	—
19	C₂×C₃×#622	C₂²	1	8	yes	complemented
20	C₃×#3788	C₂²	1	4	yes	complemented
21	C₃×#3787	C₂²	1	4	yes	complemented
22	C₂×#3830	C₂×C₃	1	1	yes	characteristic
23	C₂×C₃×#237	—	6	2	no	—
24	C₂×#3792	C₂×C₃	1	2	yes	complemented
25	C₂²×#622	C₂×C₃	1	2	yes	complemented
26	#64287	C₂×C₃	1	8	yes	direct product factor
27	C₂×#3788	C₂×C₃	1	1	yes	characteristic, complemented
28	C₂×#3787	C₂×C₃	1	1	yes	characteristic, complemented
29	C₂³×C₃³	C₂×C₄	1	2	yes	—
30	C₂²×C₃×C₃⋊D₃	C₂×C₄	1	1	yes	characteristic
31	C₃³×D₄	C₂×C₄	1	4	yes	—
32	C₂×C₃³×C₄	C₂×C₄	1	1	yes	characteristic
33	C₃×#638	C₂×C₄	1	8	yes	—
34	C₂×C₃×C₃⋊Dic₃	C₂×C₄	1	2	yes	—
35	C₂×C₃×C₃⋊Dic₃	C₂×C₄	1	4	yes	—
36	C₂×C₃×C₃⋊Dic₃	C₂³	1	1	yes	characteristic
37	C₂×C₃×C₃⋊Dic₃	C₂³	1	4	yes	—
38	C₃×C₄×C₃⋊D₃	C₂×C₄	1	2	yes	—
39	C₃×C₄×C₃⋊D₃	C₂³	1	2	yes	—
40	C₃×#634	C₂×C₄	1	4	yes	—
41	C₃×#622	C₂³	1	2	yes	—
42	C₃×#622	C₂³	1	2	yes	—
43	C₃×#622	C₂³	1	4	yes	—
44	C₂×C₃×#131	—	9	1	no	maximal
45	C₂×C₃²×D₄	—	2	2	no	—
46	C₂×C₃²×D₄	—	4	2	no	—
47	C₂×C₃²×D₄	C₃×C₄	1	1	yes	characteristic
48	C₂×#638	C₃×C₄	1	2	yes	—
49	C₂²×C₃⋊Dic₃	C₂²×C₃	1	2	yes	—
50	#3830	C₃×C₄	1	4	yes	—
51	#3830	C₂²×C₃	1	4	yes	—
52	C₂×C₄×C₃⋊D₃	C₂²×C₃	1	1	yes	characteristic
53	C₂×#634	C₃×C₄	1	1	yes	characteristic
54	C₂×C₃×#67	—	6	4	no	—
55	C₂²×C₃×Dic₃	—	6	4	no	—
56	C₃×#237	—	6	8	no	—
57	C₃×#237	—	6	8	no	—
58	C₂×C₃×C₄×D₃	—	6	2	no	—
59	C₂×C₃×Dic₆	—	6	2	no	—
60	#3792	C₂²×C₃	1	8	yes	complemented
61	C₂×#622	C₂²×C₃	1	1	yes	characteristic
62	C₂×#622	C₂²×C₃	1	1	yes	characteristic
63	C₂×#622	C₂²×C₃	1	2	yes	—
64	C₂×#622	C₂²×C₃	1	8	yes	complemented
65	#3788	C₂²×C₃	1	4	yes	complemented
66	#3787	C₂²×C₃	1	4	yes	complemented
67	C₂²×C₃³	—	2	4	no	—
68	C₂²×C₃³	C₂²×C₄	1	4	yes	—
69	C₂²×C₃³	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₃×C₃⋊D₃	—	2	2	no	—
71	C₂×C₃×C₃⋊D₃	C₂²×C₄	1	2	yes	—
72	C₃³×C₄	C₂²×C₄	1	2	yes	—
73	C₃×C₃⋊Dic₃	C₂²×C₄	1	4	yes	—
74	C₃×C₃⋊Dic₃	C₂²×C₄	1	1	yes	characteristic
75	C₃×C₃⋊Dic₃	C₂⁴	1	1	yes	characteristic
76	C₂×C₃×#41	—	9	1	no	—
77	C₂×#237	—	6	2	no	—
78	C₃×#131	—	9	8	no	—
79	C₂×C₃×M₁₆	—	9	1	no	—
80	C₂×C₃×M₁₆	—	9	2	no	—
81	C₂²×C₃×C₈	—	9	1	no	—
82	C₂²×C₃×C₈	—	9	2	no	—
83	C₂³×C₃²	—	2	4	no	—
84	C₂³×C₃²	—	4	4	no	—
85	C₂³×C₃²	C₂×C₃×C₄	1	2	yes	—
86	C₂²×C₃⋊D₃	C₂×C₃×C₄	1	1	yes	characteristic
87	C₂²×C₃×D₃	—	6	2	no	—
88	C₃²×D₄	—	2	8	no	—
89	C₃²×D₄	—	4	8	no	—
90	C₃²×D₄	C₂×C₃×C₄	1	4	yes	—
91	C₂×C₃²×C₄	—	2	2	no	—
92	C₂×C₃²×C₄	—	4	2	no	—
93	C₂×C₃²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
94	#638	C₂×C₃×C₄	1	8	yes	—
95	C₂×C₃⋊Dic₃	C₂×C₃×C₄	1	2	yes	—
96	C₂×C₃⋊Dic₃	C₂×C₃×C₄	1	4	yes	—
97	C₂×C₃⋊Dic₃	C₂³×C₃	1	1	yes	characteristic
98	C₂×C₃⋊Dic₃	C₂³×C₃	1	4	yes	—
99	C₄×C₃⋊D₃	C₂×C₃×C₄	1	2	yes	—
100	C₄×C₃⋊D₃	C₂³×C₃	1	2	yes	—
101	#634	C₂×C₃×C₄	1	4	yes	—
102	C₃×#67	—	6	16	no	—
103	C₂×C₃×Dic₃	—	6	4	no	—
104	C₂×C₃×Dic₃	—	6	8	no	—
105	C₂×C₃×Dic₃	—	6	2	no	—
106	C₂×C₃×Dic₃	—	6	8	no	—
107	C₃×C₄×D₃	—	6	4	no	—
108	C₃×C₄×D₃	—	6	4	no	—
109	C₃×Dic₆	—	6	8	no	—
110	#622	C₂³×C₃	1	2	yes	—
111	#622	C₂³×C₃	1	2	yes	—
112	#622	C₂³×C₃	1	4	yes	—
113	C₂×#131	—	9	1	no	Sylow
114	C₂×C₃³	—	2	4	no	—
115	C₂×C₃³	#131	1	2	yes	complemented
116	C₂×C₃³	C₂³×C₄	1	1	yes	characteristic
117	C₃×C₃⋊D₃	—	2	2	no	—
118	C₃×#41	—	9	4	no	—
119	C₃×#41	—	9	4	no	—
120	C₂×C₃×Q₈	—	9	1	no	—
121	C₂×C₃×D₄	—	2	2	no	—
122	C₂×C₃×D₄	—	4	2	no	—
123	C₂×C₃×D₄	—	9	2	no	—
124	C₂×C₃×D₄	#157	1	1	yes	characteristic
125	C₂²×C₃×C₄	—	9	2	no	—
126	C₂²×C₃×C₄	—	9	1	no	—
127	C₂×#67	—	6	4	no	—
128	C₂²×Dic₃	—	6	4	no	—
129	#237	—	6	8	no	—
130	#237	—	6	8	no	—
131	C₂×C₄×D₃	—	6	2	no	—
132	C₂×Dic₆	—	6	2	no	—
133	C₃×M₁₆	—	9	4	no	—
134	C₃×M₁₆	—	9	8	no	—
135	C₂×C₃×C₈	—	9	4	no	—
136	C₂×C₃×C₈	—	9	1	no	—
137	C₂×C₃×C₈	—	9	1	no	—
138	C₂×C₃×C₈	—	9	2	no	—
139	C₂×C₃×C₈	—	9	8	no	—
140	C₂²×C₃²	—	2	8	no	—
141	C₂²×C₃²	—	2	2	no	—
142	C₂²×C₃²	—	2	4	no	—
143	C₂²×C₃²	—	4	2	no	—
144	C₂²×C₃²	—	4	8	no	—
145	C₂²×C₃²	—	4	8	no	—
146	C₂²×C₃²	—	8	8	no	—
147	C₂²×C₃²	C₂²×C₃×C₄	1	4	yes	—
148	C₂²×C₃²	C₂²×C₃×C₄	1	1	yes	characteristic
149	C₂×C₃⋊D₃	—	2	2	no	—
150	C₂×C₃⋊D₃	C₂²×C₃×C₄	1	2	yes	—
151	C₂×C₃×D₃	—	6	4	no	—
152	C₂×C₃×D₃	—	12	4	no	—
153	C₃²×C₄	—	2	4	no	—
154	C₃²×C₄	—	4	4	no	—
155	C₃²×C₄	C₂²×C₃×C₄	1	2	yes	—
156	C₃⋊Dic₃	C₂²×C₃×C₄	1	4	yes	—
157	C₃⋊Dic₃	C₂²×C₃×C₄	1	1	yes	characteristic
158	C₃⋊Dic₃	C₂⁴×C₃	1	1	yes	characteristic
159	C₃×Dic₃	—	6	8	no	—
160	C₃×Dic₃	—	6	2	no	—
161	C₃×Dic₃	—	6	2	no	—
162	C₂×#41	—	9	1	no	—
163	#131	—	9	8	no	—
164	C₂×M₁₆	—	9	1	no	—
165	C₂×M₁₆	—	9	2	no	—
166	C₂²×C₈	—	9	1	no	—
167	C₂²×C₈	—	9	2	no	—
168	C₃³	C₂×#131	1	1	yes	characteristic, complemented, Sylow
169	C₂³×C₃	—	2	4	no	—
170	C₂³×C₃	—	4	4	no	—
171	C₂³×C₃	—	9	1	no	—
172	C₂³×C₃	C₂×#157	1	2	yes	—
173	C₂²×D₃	—	6	2	no	—
174	C₃×Q₈	—	9	4	no	—
175	C₃×D₄	—	2	8	no	—
176	C₃×D₄	—	4	8	no	—
177	C₃×D₄	—	9	8	no	—
178	C₃×D₄	C₂×#157	1	4	yes	—
179	C₂×C₃×C₄	—	2	2	no	—
180	C₂×C₃×C₄	—	4	2	no	—
181	C₂×C₃×C₄	—	9	4	no	—
182	C₂×C₃×C₄	—	9	4	no	—
183	C₂×C₃×C₄	—	9	1	no	—
184	C₂×C₃×C₄	—	9	2	no	—
185	C₂×C₃×C₄	—	9	2	no	—
186	C₂×C₃×C₄	—	9	2	no	—
187	C₂×C₃×C₄	C₂×#157	1	1	yes	characteristic
188	#67	—	6	16	no	—
189	C₂×Dic₃	—	6	4	no	—
190	C₂×Dic₃	—	6	8	no	—
191	C₂×Dic₃	—	6	2	no	—
192	C₂×Dic₃	—	6	8	no	—
193	C₄×D₃	—	6	4	no	—
194	C₄×D₃	—	6	4	no	—
195	Dic₆	—	6	8	no	—
196	C₃×C₈	—	9	2	no	—
197	C₃×C₈	—	9	2	no	—
198	C₃×C₈	—	9	4	no	—
199	C₂×C₃²	—	2	4	no	—
200	C₂×C₃²	—	2	2	no	—
201	C₂×C₃²	—	2	4	no	—
202	C₂×C₃²	—	4	4	no	—
203	C₂×C₃²	—	4	2	no	—
204	C₂×C₃²	—	4	8	no	—
205	C₂×C₃²	—	8	8	no	—
206	C₂×C₃²	C₃×#131	1	2	yes	complemented
207	C₂×C₃²	C₂³×C₃×C₄	1	1	yes	characteristic
208	C₃⋊D₃	—	2	2	no	—
209	C₃×D₃	—	12	4	no	—
210	#41	—	9	4	no	—
211	#41	—	9	4	no	—
212	C₂×Q₈	—	9	1	no	—
213	C₂×D₄	—	9	2	no	—
214	C₂×D₄	C₃×#157	1	1	yes	characteristic
215	C₂²×C₄	—	9	2	no	—
216	C₂²×C₄	—	9	1	no	—
217	M₁₆	—	9	4	no	—
218	M₁₆	—	9	8	no	—
219	C₂×C₈	—	9	1	no	—
220	C₂×C₈	—	9	1	no	—
221	C₂×C₈	—	9	2	no	—
222	C₂×C₈	—	9	4	no	—
223	C₂×C₈	—	9	8	no	—
224	C₂²×C₃	—	2	4	no	—
225	C₂²×C₃	—	2	2	no	—
226	C₂²×C₃	—	2	8	no	—
227	C₂²×C₃	—	4	8	no	—
228	C₂²×C₃	—	4	2	no	—
229	C₂²×C₃	—	4	8	no	—
230	C₂²×C₃	—	8	8	no	—
231	C₂²×C₃	—	9	2	no	—
232	C₂²×C₃	—	18	2	no	—
233	C₂²×C₃	C₂²×#157	1	4	yes	—
234	C₂²×C₃	C₂²×#157	1	1	yes	characteristic
235	C₂×D₃	—	6	4	no	—
236	C₂×D₃	—	12	4	no	—
237	C₃×C₄	—	2	4	no	—
238	C₃×C₄	—	4	4	no	—
239	C₃×C₄	—	9	1	no	—
240	C₃×C₄	—	9	1	no	—
241	C₃×C₄	—	9	4	no	—
242	C₃×C₄	C₂²×#157	1	2	yes	—
243	Dic₃	—	6	8	no	—
244	Dic₃	—	6	2	no	—
245	Dic₃	—	6	2	no	—
246	C₃²	—	2	2	no	—
247	C₃²	—	4	2	no	—
248	C₃²	C₂×C₃×#131	1	1	yes	characteristic, complemented
249	C₂³	—	9	1	no	—
250	C₂³	C₂×C₃×#157	1	2	yes	—
251	Q₈	—	9	4	no	—
252	D₄	—	9	8	no	—
253	D₄	C₂×C₃×#157	1	4	yes	—
254	C₂×C₄	—	9	1	no	—
255	C₂×C₄	—	9	4	no	—
256	C₂×C₄	—	9	2	no	—
257	C₂×C₄	—	9	4	no	—
258	C₂×C₄	—	9	2	no	—
259	C₂×C₄	—	9	2	no	—
260	C₂×C₄	C₂×C₃×#157	1	1	yes	characteristic
261	C₈	—	9	2	no	—
262	C₈	—	9	2	no	—
263	C₈	—	9	4	no	—
264	C₂×C₃	—	2	4	no	—
265	C₂×C₃	—	2	2	no	—
266	C₂×C₃	—	2	4	no	—
267	C₂×C₃	—	4	8	no	—
268	C₂×C₃	—	4	2	no	—
269	C₂×C₃	—	4	4	no	—
270	C₂×C₃	—	8	8	no	—
271	C₂×C₃	—	18	2	no	—
272	C₂×C₃	#64287	1	2	yes	direct product factor
273	C₂×C₃	C₂³×#157	1	1	yes	characteristic
274	D₃	—	12	4	no	—
275	C₂²	—	2	4	no	—
276	C₂²	—	9	2	no	—
277	C₂²	—	18	2	no	—
278	C₂²	C₂²×C₃×#157	1	4	yes	—
279	C₂²	C₂²×C₃×#157	1	1	yes	characteristic
280	C₄	—	9	1	no	—
281	C₄	—	9	1	no	—
282	C₄	—	9	4	no	—
283	C₄	C₂²×C₃×#157	1	2	yes	—
284	C₃	—	2	2	no	—
285	C₃	—	4	2	no	—
286	C₃	C₂×#64287	1	1	yes	characteristic, direct product factor
287	C₂	—	2	4	no	—
288	C₂	—	18	2	no	—
289	C₂	C₃×#64287	1	2	yes	direct product factor
290	C₂	C₂³×C₃×#157	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂²×#157
commutator	C₂×C₃²	C₂³×C₃×C₄
Frattini	C₂	C₂³×C₃×#157
Fitting	C₂×C₃³×D₄	C₄
socle	C₂²×C₃³	C₂²×C₄
prime radical	C₂×C₃×#3830	C₂
cosocle	C₃⋊Dic₃	C₂⁴×C₃
hypercenter	C₂×C₃×D₄	#157
hypocenter	C₃²	C₂×C₃×#131
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×D₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×#3792	C₂	semidirect
3. C₂²×C₃×#622	C₂	semidirect
4. C₃×#64287	C₂	direct (N × H)
5. C₂×C₃×#3788	C₂	semidirect
6. C₂×C₃×#3787	C₂	semidirect
7. C₂×#64287	C₃	direct (N × H)
15. C₃×#3792	C₂²	semidirect
19. C₂×C₃×#622	C₂²	semidirect
20. C₃×#3788	C₂²	semidirect
21. C₃×#3787	C₂²	semidirect
24. C₂×#3792	C₂×C₃	semidirect
25. C₂²×#622	C₂×C₃	semidirect
26. #64287	C₂×C₃	direct (N × H)
27. C₂×#3788	C₂×C₃	semidirect
28. C₂×#3787	C₂×C₃	semidirect
60. #3792	C₂²×C₃	semidirect
64. C₂×#622	C₂²×C₃	semidirect
65. #3788	C₂²×C₃	semidirect
66. #3787	C₂²×C₃	semidirect
115. C₂×C₃³	#131	semidirect
168. C₃³	C₂×#131	semidirect
206. C₂×C₃²	C₃×#131	semidirect
248. C₃²	C₂×C₃×#131	semidirect
272. C₂×C₃	#64287	direct (N × H)
286. C₃	C₂×#64287	direct (N × H)
289. C₂	C₃×#64287	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×#3830	2
C₂×C₃×#3792	2
C₂²×C₃×#622	2
C₃×#64287	2
C₂×C₃×#3788	2
C₂×C₃×#3787	2
C₂×#64287	3
C₂×C₃×#131	9

Subgroups — C2×C3×#64287 #13181493

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#64287 #13181493