Subgroups — C₂×C₃×#64156 #13179565

Order 1728.

2604 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	C₂²×C₃³×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃×C₄⋊(C₃⋊D₃)	C₂	1	1	yes	maximal, characteristic
3	C₃×#64156	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₃×#3752	C₂	1	4	yes	maximal, complemented
5	C₂×C₃×C₄×C₃⋊Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#64156	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₂×C₃×#947	—	3	4	no	maximal
8	C₂³×C₃×C₃⋊D₃	C₂²	1	2	yes	—
9	C₂×C₃³×Q₈	—	2	4	no	—
10	C₂×C₃³×Q₈	C₂²	1	4	yes	complemented
11	C₂²×C₃³×C₄	C₂²	1	2	yes	—
12	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
13	C₂²×C₃×C₃⋊Dic₃	C₂²	1	2	yes	—
14	C₂×C₃×C₄⋊(C₃⋊D₃)	—	2	4	no	—
15	C₂×C₃×C₄⋊(C₃⋊D₃)	C₂²	1	4	yes	—
16	C₃×#3752	C₂²	1	16	yes	complemented
17	C₃×C₄×C₃⋊Dic₃	C₂²	1	4	yes	complemented
18	C₂²×C₃²×Q₈	—	2	4	no	—
19	C₂²×C₃²×Q₈	D₃	1	4	yes	complemented
20	C₂²×C₃²×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
21	C₂²×C₄⋊(C₃⋊D₃)	C₂×C₃	1	1	yes	characteristic
22	C₂²×C₃×D₁₂	—	3	4	no	—
23	#64156	C₂×C₃	1	8	yes	direct product factor
24	C₂×#3752	C₂×C₃	1	4	yes	complemented
25	C₂×C₄×C₃⋊Dic₃	C₂×C₃	1	1	yes	characteristic, complemented
26	C₃×#947	—	3	32	no	—
27	C₂×C₃×(C₂×D₃)⋊C₄	—	3	16	no	—
28	C₂×C₃×C₄×Dic₃	—	3	4	no	—
29	C₂³×C₃³	C₂³	1	1	yes	characteristic
30	C₂²×C₃×C₃⋊D₃	—	2	4	no	—
31	C₂²×C₃×C₃⋊D₃	—	2	8	no	—
32	C₂²×C₃×C₃⋊D₃	C₂³	1	4	yes	—
33	C₃³×Q₈	—	2	8	no	—
34	C₂×C₃³×C₄	—	2	4	no	—
35	C₂×C₃³×C₄	D₄	1	4	yes	—
36	C₂×C₃³×C₄	C₂³	1	4	yes	—
37	C₂×C₃³×C₄	C₂³	1	2	yes	—
38	C₂×C₃×C₃⋊Dic₃	—	2	4	no	—
39	C₂×C₃×C₃⋊Dic₃	C₂³	1	4	yes	—
40	C₃×C₄⋊(C₃⋊D₃)	—	2	8	no	—
41	C₂×C₃×#124	—	9	1	no	—
42	C₂×#947	—	3	4	no	—
43	C₂³×C₃⋊D₃	C₂²×C₃	1	2	yes	—
44	C₂³×C₃×D₃	—	3	8	no	—
45	C₂×C₃²×Q₈	—	2	32	no	—
46	C₂×C₃²×Q₈	—	2	16	no	—
47	C₂×C₃²×Q₈	—	2	16	no	—
48	C₂×C₃²×Q₈	—	2	4	no	—
49	C₂×C₃²×Q₈	C₂×D₃	1	16	yes	complemented
50	C₂×C₃²×Q₈	C₂²×C₃	1	4	yes	complemented
51	C₂²×C₃²×C₄	—	2	8	no	—
52	C₂²×C₃²×C₄	—	2	4	no	—
53	C₂²×C₃²×C₄	C₂×D₃	1	8	yes	—
54	C₂²×C₃²×C₄	C₂×D₃	1	4	yes	—
55	C₂²×C₃²×C₄	C₂²×C₃	1	2	yes	—
56	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
57	C₂²×C₃⋊Dic₃	C₂²×C₃	1	2	yes	—
58	C₂×C₄⋊(C₃⋊D₃)	—	2	4	no	—
59	C₂×C₄⋊(C₃⋊D₃)	C₂²×C₃	1	4	yes	—
60	C₂²×C₃×Dic₃	—	3	8	no	—
61	C₂×C₃×D₁₂	—	3	16	no	—
62	C₂×C₃×D₁₂	—	6	16	no	—
63	#3752	C₂²×C₃	1	16	yes	complemented
64	C₄×C₃⋊Dic₃	C₂²×C₃	1	4	yes	complemented
65	C₃×(C₂×D₃)⋊C₄	—	3	64	no	—
66	C₃×C₄×Dic₃	—	3	16	no	—
67	C₂²×C₃³	C₂×D₄	1	2	yes	—
68	C₂²×C₃³	#41	1	4	yes	—
69	C₂²×C₃³	C₂⁴	1	1	yes	characteristic
70	C₂×C₃×C₃⋊D₃	—	2	4	no	—
71	C₂×C₃×C₃⋊D₃	—	2	8	no	—
72	C₂×C₃×C₃⋊D₃	—	4	8	no	—
73	C₃³×C₄	—	2	4	no	—
74	C₃³×C₄	C₂×D₄	1	4	yes	—
75	C₃×C₃⋊Dic₃	—	2	4	no	—
76	C₂²×C₃×Q₈	—	2	4	no	—
77	C₂²×C₃×Q₈	C₃×D₃	1	4	yes	complemented
78	C₂²×C₃×Q₈	C₃⋊D₃	1	1	yes	characteristic, complemented
79	C₂²×C₃×D₄	—	9	1	no	—
80	C₂²×D₁₂	—	3	4	no	—
81	C₃×#124	—	9	8	no	—
82	C₂×C₃×C₂²⋊C₄	—	9	4	no	—
83	C₂×C₃×C₄²	—	9	1	no	—
84	#947	—	3	32	no	—
85	C₂×(C₂×D₃)⋊C₄	—	3	16	no	—
86	C₂×C₄×Dic₃	—	3	4	no	—
87	C₂³×C₃²	—	2	4	no	—
88	C₂³×C₃²	C₂²×D₃	1	4	yes	—
89	C₂³×C₃²	C₂³×C₃	1	1	yes	characteristic
90	C₂²×C₃⋊D₃	—	2	4	no	—
91	C₂²×C₃⋊D₃	—	2	8	no	—
92	C₂²×C₃⋊D₃	C₂³×C₃	1	4	yes	—
93	C₂²×C₃×D₃	—	3	16	no	—
94	C₂²×C₃×D₃	—	6	32	no	—
95	C₂²×C₃×D₃	—	6	16	no	—
96	C₃²×Q₈	—	2	64	no	—
97	C₃²×Q₈	—	2	32	no	—
98	C₃²×Q₈	—	2	8	no	—
99	C₂×C₃²×C₄	—	2	32	no	—
100	C₂×C₃²×C₄	—	2	16	no	—
101	C₂×C₃²×C₄	—	2	4	no	—
102	C₂×C₃²×C₄	—	2	16	no	—
103	C₂×C₃²×C₄	—	2	16	no	—
104	C₂×C₃²×C₄	—	2	8	no	—
105	C₂×C₃²×C₄	#67	1	16	yes	—
106	C₂×C₃²×C₄	C₃×D₄	1	4	yes	—
107	C₂×C₃²×C₄	C₂²×D₃	1	16	yes	—
108	C₂×C₃²×C₄	C₂²×D₃	1	8	yes	—
109	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
110	C₂×C₃²×C₄	C₂³×C₃	1	2	yes	—
111	C₂×C₃⋊Dic₃	—	2	4	no	—
112	C₂×C₃⋊Dic₃	C₂³×C₃	1	4	yes	—
113	C₄⋊(C₃⋊D₃)	—	2	8	no	—
114	C₂×C₃×Dic₃	—	3	16	no	—
115	C₂×C₃×Dic₃	—	6	16	no	—
116	C₃×D₁₂	—	6	32	no	—
117	C₂×#124	—	9	1	no	Sylow
118	C₂×C₃³	#124	1	4	yes	complemented
119	C₂×C₃³	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₃³	C₂×#41	1	2	yes	—
121	C₃×C₃⋊D₃	—	4	4	no	—
122	C₂⁴×C₃	—	9	2	no	—
123	C₂³×D₃	—	3	8	no	—
124	C₂×C₃×Q₈	—	2	32	no	—
125	C₂×C₃×Q₈	—	2	16	no	—
126	C₂×C₃×Q₈	—	2	4	no	—
127	C₂×C₃×Q₈	—	2	16	no	—
128	C₂×C₃×Q₈	C₂×C₃×D₃	1	16	yes	complemented
129	C₂×C₃×Q₈	C₂×C₃⋊D₃	1	4	yes	complemented
130	C₂×C₃×D₄	—	9	4	no	—
131	C₂×C₃×D₄	—	18	4	no	—
132	C₂²×C₃×C₄	—	2	8	no	—
133	C₂²×C₃×C₄	—	2	4	no	—
134	C₂²×C₃×C₄	—	9	2	no	—
135	C₂²×C₃×C₄	C₂×C₃×D₃	1	8	yes	—
136	C₂²×C₃×C₄	C₂×C₃×D₃	1	4	yes	—
137	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	2	yes	—
138	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
139	C₂²×Dic₃	—	3	8	no	—
140	C₂×D₁₂	—	3	16	no	—
141	C₂×D₁₂	—	6	16	no	—
142	C₃×C₂²⋊C₄	—	9	16	no	—
143	C₃×C₄²	—	9	4	no	—
144	(C₂×D₃)⋊C₄	—	3	64	no	—
145	C₄×Dic₃	—	3	16	no	—
146	C₂²×C₃²	—	2	4	no	—
147	C₂²×C₃²	—	2	16	no	—
148	C₂²×C₃²	—	2	8	no	—
149	C₂²×C₃²	#239	1	16	yes	—
150	C₂²×C₃²	C₂×#67	1	8	yes	—
151	C₂²×C₃²	C₂×C₃×D₄	1	2	yes	—
152	C₂²×C₃²	C₃×#41	1	4	yes	—
153	C₂²×C₃²	C₂³×D₃	1	4	yes	—
154	C₂²×C₃²	C₂⁴×C₃	1	1	yes	characteristic
155	C₂×C₃⋊D₃	—	2	4	no	—
156	C₂×C₃⋊D₃	—	2	8	no	—
157	C₂×C₃⋊D₃	—	4	8	no	—
158	C₂×C₃×D₃	—	6	32	no	—
159	C₂×C₃×D₃	—	6	16	no	—
160	C₂×C₃×D₃	—	12	32	no	—
161	C₃²×C₄	—	2	32	no	—
162	C₃²×C₄	—	2	16	no	—
163	C₃²×C₄	—	2	4	no	—
164	C₃²×C₄	—	2	16	no	—
165	C₃²×C₄	C₂×#67	1	16	yes	—
166	C₃²×C₄	C₂×C₃×D₄	1	4	yes	—
167	C₃⋊Dic₃	—	2	4	no	—
168	C₃×Dic₃	—	6	16	no	—
169	C₂²×Q₈	C₃×C₃⋊D₃	1	1	yes	characteristic, complemented
170	C₂²×D₄	—	9	1	no	—
171	#124	—	9	8	no	—
172	C₂×C₂²⋊C₄	—	9	4	no	—
173	C₂×C₄²	—	9	1	no	—
174	C₃³	C₂×#124	1	1	yes	characteristic, complemented, Sylow
175	C₂³×C₃	—	2	4	no	—
176	C₂³×C₃	—	9	4	no	—
177	C₂³×C₃	—	18	8	no	—
178	C₂³×C₃	—	18	4	no	—
179	C₂³×C₃	C₂²×C₃×D₃	1	4	yes	—
180	C₂³×C₃	C₂²×C₃⋊D₃	1	1	yes	characteristic
181	C₂²×D₃	—	3	16	no	—
182	C₂²×D₃	—	6	32	no	—
183	C₂²×D₃	—	6	16	no	—
184	C₃×Q₈	—	2	64	no	—
185	C₃×Q₈	—	2	32	no	—
186	C₃×Q₈	—	2	8	no	—
187	C₃×D₄	—	18	8	no	—
188	C₂×C₃×C₄	—	2	32	no	—
189	C₂×C₃×C₄	—	2	16	no	—
190	C₂×C₃×C₄	—	2	4	no	—
191	C₂×C₃×C₄	—	2	16	no	—
192	C₂×C₃×C₄	—	2	16	no	—
193	C₂×C₃×C₄	—	2	8	no	—
194	C₂×C₃×C₄	—	9	4	no	—
195	C₂×C₃×C₄	—	18	4	no	—
196	C₂×C₃×C₄	C₃×#67	1	16	yes	—
197	C₂×C₃×C₄	#638	1	4	yes	—
198	C₂×C₃×C₄	C₂²×C₃×D₃	1	16	yes	—
199	C₂×C₃×C₄	C₂²×C₃×D₃	1	8	yes	—
200	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	4	yes	—
201	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	2	yes	—
202	C₂×Dic₃	—	3	16	no	—
203	C₂×Dic₃	—	6	16	no	—
204	D₁₂	—	6	32	no	—
205	C₂×C₃²	—	2	16	no	—
206	C₂×C₃²	—	2	8	no	—
207	C₂×C₃²	—	2	4	no	—
208	C₂×C₃²	#947	1	16	yes	complemented
209	C₂×C₃²	C₃×#124	1	4	yes	complemented
210	C₂×C₃²	C₂×#239	1	8	yes	—
211	C₂×C₃²	C₂²×#67	1	4	yes	—
212	C₂×C₃²	C₂²×C₃×D₄	1	1	yes	characteristic
213	C₂×C₃²	C₂×C₃×#41	1	2	yes	—
214	C₃⋊D₃	—	4	4	no	—
215	C₃×D₃	—	12	16	no	—
216	C₂⁴	—	9	2	no	—
217	C₂×Q₈	—	2	4	no	—
218	C₂×Q₈	C₂×C₃×C₃⋊D₃	1	4	yes	complemented
219	C₂×D₄	—	9	4	no	—
220	C₂×D₄	—	18	4	no	—
221	C₂²×C₄	—	9	2	no	—
222	C₂²×C₄	C₂×C₃×C₃⋊D₃	1	2	yes	—
223	C₂²×C₄	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
224	C₂²⋊C₄	—	9	16	no	—
225	C₄²	—	9	4	no	—
226	C₂²×C₃	—	2	4	no	—
227	C₂²×C₃	—	2	16	no	—
228	C₂²×C₃	—	2	8	no	—
229	C₂²×C₃	—	18	8	no	—
230	C₂²×C₃	—	18	4	no	—
231	C₂²×C₃	—	36	8	no	—
232	C₂²×C₃	C₃×#239	1	16	yes	—
233	C₂²×C₃	C₂×C₃×#67	1	8	yes	—
234	C₂²×C₃	#3832	1	4	yes	—
235	C₂²×C₃	C₂×#638	1	2	yes	—
236	C₂²×C₃	C₂³×C₃×D₃	1	4	yes	—
237	C₂²×C₃	C₂³×C₃⋊D₃	1	1	yes	characteristic
238	C₂×D₃	—	6	32	no	—
239	C₂×D₃	—	6	16	no	—
240	C₂×D₃	—	12	32	no	—
241	C₃×C₄	—	2	32	no	—
242	C₃×C₄	—	2	16	no	—
243	C₃×C₄	—	2	4	no	—
244	C₃×C₄	—	2	16	no	—
245	C₃×C₄	—	18	4	no	—
246	C₃×C₄	C₂×C₃×#67	1	16	yes	—
247	C₃×C₄	C₂×#638	1	4	yes	—
248	Dic₃	—	6	16	no	—
249	C₃²	—	2	4	no	—
250	C₃²	C₂×#947	1	4	yes	complemented
251	C₃²	C₂×C₃×#124	1	1	yes	characteristic, complemented
252	C₂³	—	9	4	no	—
253	C₂³	—	18	8	no	—
254	C₂³	—	18	4	no	—
255	C₂³	C₂²×C₃×C₃⋊D₃	1	1	yes	characteristic
256	Q₈	—	2	8	no	—
257	D₄	—	18	8	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	9	4	no	—
260	C₂×C₄	—	18	4	no	—
261	C₂×C₄	C₃×#638	1	4	yes	—
262	C₂×C₄	C₂²×C₃×C₃⋊D₃	1	4	yes	—
263	C₂×C₄	C₂²×C₃×C₃⋊D₃	1	2	yes	—
264	C₂×C₃	—	2	8	no	—
265	C₂×C₃	—	2	16	no	—
266	C₂×C₃	—	2	4	no	—
267	C₂×C₃	—	36	4	no	—
268	C₂×C₃	C₃×#947	1	16	yes	complemented
269	C₂×C₃	#64156	1	4	yes	direct product factor
270	C₂×C₃	C₂×C₃×#239	1	8	yes	—
271	C₂×C₃	C₂²×C₃×#67	1	4	yes	—
272	C₂×C₃	C₂×#3832	1	2	yes	—
273	C₂×C₃	C₂²×#638	1	1	yes	characteristic
274	D₃	—	12	16	no	—
275	C₂²	—	18	8	no	—
276	C₂²	—	18	4	no	—
277	C₂²	—	36	8	no	—
278	C₂²	C₃×#3832	1	4	yes	—
279	C₂²	C₂×C₃×#638	1	2	yes	—
280	C₂²	C₂³×C₃×C₃⋊D₃	1	1	yes	characteristic
281	C₄	—	2	4	no	—
282	C₄	—	18	4	no	—
283	C₄	C₂×C₃×#638	1	4	yes	—
284	C₃	—	2	4	no	—
285	C₃	C₂×C₃×#947	1	4	yes	complemented
286	C₃	C₂×#64156	1	1	yes	characteristic, direct product factor
287	C₂	—	36	4	no	—
288	C₂	C₃×#64156	1	4	yes	direct product factor
289	C₂	C₂×C₃×#3832	1	2	yes	—
290	C₂	C₂²×C₃×#638	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₃	C₂²×C₃⋊D₃
commutator, cosocle	C₂²×C₃²	C₂⁴×C₃
Frattini	C₂²	C₂³×C₃×C₃⋊D₃
Fitting	C₂²×C₃³×Q₈	C₂
socle	C₂³×C₃³	C₂³
prime radical	C₂²×C₃×C₄⋊(C₃⋊D₃)	C₂
hypercenter	C₂²×C₃×Q₈	C₃⋊D₃
hypocenter	C₃²	C₂×C₃×#124
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂³×C₃ ◁ C₂²×C₃×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃³×Q₈	C₂	semidirect
3. C₃×#64156	C₂	direct (N × H)
4. C₂×C₃×#3752	C₂	semidirect
5. C₂×C₃×C₄×C₃⋊Dic₃	C₂	semidirect
6. C₂×#64156	C₃	direct (N × H)
10. C₂×C₃³×Q₈	C₂²	semidirect
16. C₃×#3752	C₂²	semidirect
17. C₃×C₄×C₃⋊Dic₃	C₂²	semidirect
19. C₂²×C₃²×Q₈	D₃	semidirect
20. C₂²×C₃²×Q₈	C₂×C₃	semidirect
23. #64156	C₂×C₃	direct (N × H)
24. C₂×#3752	C₂×C₃	semidirect
25. C₂×C₄×C₃⋊Dic₃	C₂×C₃	semidirect
49. C₂×C₃²×Q₈	C₂×D₃	semidirect
50. C₂×C₃²×Q₈	C₂²×C₃	semidirect
63. #3752	C₂²×C₃	semidirect
64. C₄×C₃⋊Dic₃	C₂²×C₃	semidirect
77. C₂²×C₃×Q₈	C₃×D₃	semidirect
78. C₂²×C₃×Q₈	C₃⋊D₃	semidirect
118. C₂×C₃³	#124	semidirect
128. C₂×C₃×Q₈	C₂×C₃×D₃	semidirect
129. C₂×C₃×Q₈	C₂×C₃⋊D₃	semidirect
169. C₂²×Q₈	C₃×C₃⋊D₃	semidirect
174. C₃³	C₂×#124	semidirect
208. C₂×C₃²	#947	semidirect
209. C₂×C₃²	C₃×#124	semidirect
218. C₂×Q₈	C₂×C₃×C₃⋊D₃	semidirect
250. C₃²	C₂×#947	semidirect
251. C₃²	C₂×C₃×#124	semidirect
268. C₂×C₃	C₃×#947	semidirect
269. C₂×C₃	#64156	direct (N × H)
285. C₃	C₂×C₃×#947	semidirect
286. C₃	C₂×#64156	direct (N × H)
288. C₂	C₃×#64156	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃³×Q₈	2
C₂²×C₃×C₄⋊(C₃⋊D₃)	2
C₃×#64156	2
C₂×C₃×#3752	2
C₂×C₃×C₄×C₃⋊Dic₃	2
C₂×#64156	3
C₂×C₃×#947	3

Subgroups — C2×C3×#64156 #13179565

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#64156 #13179565