Subgroups — C₃×#10592547 #13162887

Order 1728.

920 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#3744	C₂	1	2	yes	maximal, complemented
2	C₃×#64083	C₂	1	1	yes	maximal, characteristic
3	C₃×C₄×#634	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃³×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#63648	C₂	1	2	yes	maximal, complemented
6	#10592547	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₃×#4848	—	3	4	no	maximal
8	C₂×C₃×C₄⋊(C₃⋊D₃)	—	2	1	no	—
9	C₂×C₃×C₄⋊(C₃⋊D₃)	—	2	1	no	—
10	C₂×C₃×C₄⋊(C₃⋊D₃)	C₂²	1	2	yes	—
11	C₂×C₃×#634	C₂²	1	1	yes	characteristic
12	C₂×C₃³×C₈	C₂²	1	2	yes	—
13	C₃³×C₄²	C₂²	1	1	yes	characteristic
14	C₃×C₄⋊(C₃⋊Dic₃)	C₂²	1	1	yes	characteristic
15	C₃×#3750	—	2	1	no	—
16	C₃×C₄×C₃⋊Dic₃	—	2	1	no	—
17	C₃×#3744	—	2	4	no	—
18	C₂×#3744	C₂×C₃	1	2	yes	complemented
19	#64083	C₂×C₃	1	1	yes	characteristic
20	C₄×#634	C₂×C₃	1	1	yes	characteristic, complemented
21	C₂×C₃×#204	—	3	8	no	—
22	C₃×#874	—	3	4	no	—
23	C₃×C₄×Dic₆	—	3	4	no	—
24	C₃²×C₄⋊C₈	—	2	4	no	—
25	C₃²×C₄⋊C₈	D₃	1	4	yes	complemented
26	C₃²×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
27	#63648	C₂×C₃	1	2	yes	complemented
28	C₃×#821	—	3	8	no	—
29	C₂²×C₃×C₃⋊D₃	—	2	2	no	—
30	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
31	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
32	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
33	C₂×C₃×C₃⋊Dic₃	—	2	1	no	—
34	C₂×C₃×C₃⋊Dic₃	—	2	1	no	—
35	C₃×C₄⋊(C₃⋊D₃)	—	2	2	no	—
36	C₃×C₄⋊(C₃⋊D₃)	—	2	2	no	—
37	C₃×C₄⋊(C₃⋊D₃)	—	4	2	no	—
38	C₃×C₄⋊(C₃⋊D₃)	—	4	2	no	—
39	C₃×#634	—	2	1	no	—
40	C₃×#634	D₄	1	2	yes	complemented
41	C₃³×C₈	—	2	2	no	—
42	C₃×#460	—	9	1	no	—
43	#4848	—	3	4	no	—
44	C₂×C₄⋊(C₃⋊D₃)	—	2	1	no	—
45	C₂×C₄⋊(C₃⋊D₃)	—	2	1	no	—
46	C₂×C₄⋊(C₃⋊D₃)	C₂²×C₃	1	2	yes	—
47	C₂×#634	C₂²×C₃	1	1	yes	characteristic
48	C₂×C₃×D₁₂	—	3	8	no	—
49	C₂×C₃×D₁₂	—	6	4	no	—
50	C₂×C₃×D₁₂	—	6	4	no	—
51	C₂×C₃×Dic₆	—	3	4	no	—
52	C₂×C₃²×C₈	—	2	8	no	—
53	C₂×C₃²×C₈	C₂×D₃	1	8	yes	—
54	C₂×C₃²×C₈	C₂²×C₃	1	2	yes	—
55	C₃²×C₄²	—	2	4	no	—
56	C₃²×C₄²	C₂×D₃	1	4	yes	—
57	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
58	C₄⋊(C₃⋊Dic₃)	C₂²×C₃	1	1	yes	characteristic
59	#3750	—	2	1	no	—
60	C₄×C₃⋊Dic₃	—	2	1	no	—
61	#3744	—	2	4	no	—
62	C₃×C₄⋊Dic₃	—	3	4	no	—
63	C₃×Dic₃⋊C₄	—	6	4	no	—
64	C₃×C₄×Dic₃	—	6	4	no	—
65	C₃×#204	—	6	16	no	—
66	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
67	C₂×C₃×C₃⋊D₃	—	4	2	no	—
68	C₂×C₃×C₃⋊D₃	—	4	2	no	—
69	C₂×C₃×C₃⋊D₃	—	4	2	no	—
70	C₃³×C₄	—	2	1	no	—
71	C₃³×C₄	SD₁₆	1	2	yes	complemented
72	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
73	C₃³×C₄	#41	1	1	yes	characteristic
74	C₃×C₃⋊Dic₃	—	2	2	no	—
75	C₃×C₃⋊Dic₃	—	4	1	no	—
76	C₂×C₃×SD₁₆	—	9	2	no	—
77	C₃×#127	—	9	1	no	—
78	C₃×C₄×Q₈	—	9	1	no	—
79	C₂×#204	—	3	8	no	—
80	#874	—	3	4	no	—
81	C₄×Dic₆	—	3	4	no	—
82	C₃×C₄⋊C₈	—	2	4	no	—
83	C₃×C₄⋊C₈	C₃×D₃	1	4	yes	complemented
84	C₃×C₄⋊C₈	C₃⋊D₃	1	1	yes	characteristic, complemented
85	C₃×#102	—	9	2	no	—
86	#821	—	3	8	no	—
87	C₂²×C₃⋊D₃	—	2	2	no	—
88	C₂²×C₃×D₃	—	6	8	no	—
89	C₂×C₃²×C₄	—	2	4	no	—
90	C₂×C₃²×C₄	—	2	4	no	—
91	C₂×C₃²×C₄	—	2	4	no	—
92	C₂×C₃²×C₄	D₁₂	1	4	yes	—
93	C₂×C₃²×C₄	D₁₂	1	4	yes	—
94	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
95	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
96	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
97	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
98	C₂×C₃⋊Dic₃	—	2	1	no	—
99	C₂×C₃⋊Dic₃	—	2	1	no	—
100	C₄⋊(C₃⋊D₃)	—	2	2	no	—
101	C₄⋊(C₃⋊D₃)	—	2	2	no	—
102	C₄⋊(C₃⋊D₃)	—	4	2	no	—
103	C₄⋊(C₃⋊D₃)	—	4	2	no	—
104	#634	—	2	1	no	—
105	#634	C₃×D₄	1	2	yes	complemented
106	C₂×C₃×Dic₃	—	6	4	no	—
107	C₂×C₃×Dic₃	—	6	4	no	—
108	C₃×D₁₂	—	6	8	no	—
109	C₃×D₁₂	—	6	8	no	—
110	C₃×D₁₂	—	12	8	no	—
111	C₃×D₁₂	—	12	8	no	—
112	C₃×Dic₆	—	3	8	no	—
113	C₃×Dic₆	—	6	4	no	—
114	C₃²×C₈	—	2	2	no	—
115	C₃²×C₈	—	2	8	no	—
116	C₃²×C₈	—	4	8	no	—
117	#460	—	9	1	no	Sylow
118	C₂×C₃³	#121	1	1	yes	characteristic
119	C₂×C₃³	C₂×SD₁₆	1	1	yes	characteristic
120	C₂×C₃³	AGL(1,ℤ/8)	1	1	yes	characteristic
121	C₃×C₃⋊D₃	—	8	2	no	—
122	C₂×C₃×Q₈	—	9	1	no	—
123	C₂×C₃×D₄	—	9	2	no	—
124	C₂×C₃×D₄	—	18	1	no	—
125	C₂×C₃×D₄	—	18	1	no	—
126	C₂×D₁₂	—	3	8	no	—
127	C₂×D₁₂	—	6	4	no	—
128	C₂×D₁₂	—	6	4	no	—
129	C₂×Dic₆	—	3	4	no	—
130	C₃×SD₁₆	—	18	4	no	—
131	C₂×C₃×C₈	—	2	8	no	—
132	C₂×C₃×C₈	C₂×C₃×D₃	1	8	yes	—
133	C₂×C₃×C₈	C₂×C₃⋊D₃	1	2	yes	—
134	C₃×C₄⋊C₄	—	9	1	no	—
135	C₃×C₄⋊C₄	—	18	1	no	—
136	C₃×C₄²	—	2	4	no	—
137	C₃×C₄²	—	18	1	no	—
138	C₃×C₄²	C₂×C₃×D₃	1	4	yes	—
139	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
140	C₄⋊Dic₃	—	3	4	no	—
141	Dic₃⋊C₄	—	6	4	no	—
142	C₄×Dic₃	—	6	4	no	—
143	#204	—	6	16	no	—
144	C₂²×C₃²	—	2	4	no	—
145	C₂²×C₃²	C₂×D₁₂	1	4	yes	—
146	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
147	C₂×C₃⋊D₃	—	4	2	no	—
148	C₂×C₃⋊D₃	—	4	2	no	—
149	C₂×C₃⋊D₃	—	4	2	no	—
150	C₂×C₃×D₃	—	12	8	no	—
151	C₂×C₃×D₃	—	12	8	no	—
152	C₂×C₃×D₃	—	12	8	no	—
153	C₃²×C₄	—	2	1	no	—
154	C₃²×C₄	—	2	8	no	—
155	C₃²×C₄	—	2	4	no	—
156	C₃²×C₄	—	2	4	no	—
157	C₃²×C₄	—	2	4	no	—
158	C₃²×C₄	—	4	4	no	—
159	C₃²×C₄	#204	1	8	yes	complemented
160	C₃²×C₄	C₃×SD₁₆	1	2	yes	complemented
161	C₃²×C₄	D₃×D₄	1	4	yes	—
162	C₃²×C₄	#239	1	4	yes	—
163	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
164	C₃²×C₄	C₃×#41	1	1	yes	characteristic
165	C₃⋊Dic₃	—	2	2	no	—
166	C₃⋊Dic₃	—	4	1	no	—
167	C₃×Dic₃	—	6	8	no	—
168	C₃×Dic₃	—	12	4	no	—
169	C₂×SD₁₆	—	9	2	no	—
170	#127	—	9	1	no	—
171	C₄×Q₈	—	9	1	no	—
172	C₄⋊C₈	C₃×C₃⋊D₃	1	1	yes	characteristic, complemented
173	#102	—	9	2	no	—
174	C₃³	#460	1	1	yes	characteristic, complemented, Sylow
175	C₂³×C₃	—	18	2	no	—
176	C₂²×D₃	—	6	8	no	—
177	C₃×Q₈	—	9	2	no	—
178	C₃×Q₈	—	18	1	no	—
179	C₃×D₄	—	18	2	no	—
180	C₃×D₄	—	18	2	no	—
181	C₃×D₄	—	36	2	no	—
182	C₃×D₄	—	36	2	no	—
183	C₂×C₃×C₄	—	2	4	no	—
184	C₂×C₃×C₄	—	2	4	no	—
185	C₂×C₃×C₄	—	2	4	no	—
186	C₂×C₃×C₄	—	18	1	no	—
187	C₂×C₃×C₄	—	18	1	no	—
188	C₂×C₃×C₄	C₃×D₁₂	1	4	yes	—
189	C₂×C₃×C₄	C₃×D₁₂	1	4	yes	—
190	C₂×C₃×C₄	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
191	C₂×C₃×C₄	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
192	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
193	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
194	C₂×Dic₃	—	6	4	no	—
195	C₂×Dic₃	—	6	4	no	—
196	D₁₂	—	6	8	no	—
197	D₁₂	—	6	8	no	—
198	D₁₂	—	12	8	no	—
199	D₁₂	—	12	8	no	—
200	Dic₆	—	3	8	no	—
201	Dic₆	—	6	4	no	—
202	C₃×C₈	—	2	8	no	—
203	C₃×C₈	—	2	2	no	—
204	C₃×C₈	—	4	8	no	—
205	C₂×C₃²	—	2	4	no	—
206	C₂×C₃²	—	2	4	no	—
207	C₂×C₃²	—	2	4	no	—
208	C₂×C₃²	#895	1	4	yes	—
209	C₂×C₃²	C₂×#204	1	4	yes	—
210	C₂×C₃²	#908	1	4	yes	—
211	C₂×C₃²	C₃×#121	1	1	yes	characteristic
212	C₂×C₃²	C₂×C₃×SD₁₆	1	1	yes	characteristic
213	C₂×C₃²	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
214	C₃⋊D₃	—	8	2	no	—
215	C₃×D₃	—	24	8	no	—
216	C₂×Q₈	—	9	1	no	—
217	C₂×D₄	—	9	2	no	—
218	C₂×D₄	—	18	1	no	—
219	C₂×D₄	—	18	1	no	—
220	SD₁₆	—	18	4	no	—
221	C₂×C₈	C₂×C₃×C₃⋊D₃	1	2	yes	—
222	C₄⋊C₄	—	9	1	no	—
223	C₄⋊C₄	—	18	1	no	—
224	C₄²	—	18	1	no	—
225	C₄²	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
226	C₂²×C₃	—	2	4	no	—
227	C₂²×C₃	—	36	2	no	—
228	C₂²×C₃	—	36	2	no	—
229	C₂²×C₃	—	36	2	no	—
230	C₂²×C₃	C₂×C₃×D₁₂	1	4	yes	—
231	C₂²×C₃	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
232	C₂×D₃	—	12	8	no	—
233	C₂×D₃	—	12	8	no	—
234	C₂×D₃	—	12	8	no	—
235	C₃×C₄	—	2	4	no	—
236	C₃×C₄	—	2	1	no	—
237	C₃×C₄	—	2	8	no	—
238	C₃×C₄	—	2	4	no	—
239	C₃×C₄	—	2	4	no	—
240	C₃×C₄	—	4	4	no	—
241	C₃×C₄	—	18	2	no	—
242	C₃×C₄	—	36	1	no	—
243	C₃×C₄	C₃×#204	1	8	yes	complemented
244	C₃×C₄	#3744	1	2	yes	complemented
245	C₃×C₄	C₃×D₃×D₄	1	4	yes	—
246	C₃×C₄	C₃×#239	1	4	yes	—
247	C₃×C₄	D₄×C₃⋊D₃	1	1	yes	characteristic
248	C₃×C₄	#3832	1	1	yes	characteristic
249	Dic₃	—	6	8	no	—
250	Dic₃	—	12	4	no	—
251	C₃²	—	2	4	no	—
252	C₃²	#4848	1	4	yes	complemented
253	C₃²	C₃×#460	1	1	yes	characteristic, complemented
254	C₂³	—	18	2	no	—
255	Q₈	—	9	2	no	—
256	Q₈	—	18	1	no	—
257	D₄	—	18	2	no	—
258	D₄	—	18	2	no	—
259	D₄	—	36	2	no	—
260	D₄	—	36	2	no	—
261	C₂×C₄	—	18	1	no	—
262	C₂×C₄	—	18	1	no	—
263	C₂×C₄	C₃×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
264	C₂×C₄	C₃×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
265	C₂×C₄	C₂²×C₃×C₃⋊D₃	1	1	yes	characteristic
266	C₈	—	2	2	no	—
267	C₂×C₃	—	2	4	no	—
268	C₂×C₃	—	2	4	no	—
269	C₂×C₃	—	2	4	no	—
270	C₂×C₃	—	72	2	no	—
271	C₂×C₃	C₃×#895	1	4	yes	—
272	C₂×C₃	C₂×C₃×#204	1	4	yes	—
273	C₂×C₃	C₃×#908	1	4	yes	—
274	C₂×C₃	#64104	1	1	yes	characteristic
275	C₂×C₃	C₂×#3744	1	1	yes	characteristic
276	C₂×C₃	#64117	1	1	yes	characteristic
277	D₃	—	24	8	no	—
278	C₂²	—	36	2	no	—
279	C₂²	—	36	2	no	—
280	C₂²	—	36	2	no	—
281	C₂²	C₂×C₃×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
282	C₄	—	2	1	no	—
283	C₄	—	18	2	no	—
284	C₄	—	36	1	no	—
285	C₄	C₃×#3744	1	2	yes	complemented
286	C₄	C₃×D₄×C₃⋊D₃	1	1	yes	characteristic
287	C₄	C₃×#3832	1	1	yes	characteristic
288	C₃	—	2	4	no	—
289	C₃	C₃×#4848	1	4	yes	complemented
290	C₃	#10592547	1	1	yes	characteristic, direct product factor
291	C₂	—	72	2	no	—
292	C₂	C₃×#64104	1	1	yes	characteristic
293	C₂	C₂×C₃×#3744	1	1	yes	characteristic
294	C₂	C₃×#64117	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂×C₄⋊(C₃⋊D₃)
commutator, cosocle	C₂×C₃²×C₄	C₂³×C₃
Frattini	C₂×C₄	C₂²×C₃×C₃⋊D₃
Fitting	C₃³×C₄⋊C₈	C₂
socle	C₂²×C₃³	C₂×D₄
prime radical	C₃×#64083	C₂
hypercenter	C₃×C₄⋊C₈	C₃⋊D₃
hypocenter	C₃²	C₃×#460
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#3744	C₂	semidirect
3. C₃×C₄×#634	C₂	semidirect
4. C₃³×C₄⋊C₈	C₂	semidirect
5. C₃×#63648	C₂	semidirect
6. #10592547	C₃	direct (N × H)
18. C₂×#3744	C₂×C₃	semidirect
20. C₄×#634	C₂×C₃	semidirect
25. C₃²×C₄⋊C₈	D₃	semidirect
26. C₃²×C₄⋊C₈	C₂×C₃	semidirect
27. #63648	C₂×C₃	semidirect
40. C₃×#634	D₄	semidirect
71. C₃³×C₄	SD₁₆	semidirect
83. C₃×C₄⋊C₈	C₃×D₃	semidirect
84. C₃×C₄⋊C₈	C₃⋊D₃	semidirect
105. #634	C₃×D₄	semidirect
159. C₃²×C₄	#204	semidirect
160. C₃²×C₄	C₃×SD₁₆	semidirect
172. C₄⋊C₈	C₃×C₃⋊D₃	semidirect
174. C₃³	#460	semidirect
243. C₃×C₄	C₃×#204	semidirect
244. C₃×C₄	#3744	semidirect
252. C₃²	#4848	semidirect
253. C₃²	C₃×#460	semidirect
285. C₄	C₃×#3744	semidirect
289. C₃	C₃×#4848	semidirect
290. C₃	#10592547	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×#3744	2
C₃×#64083	2
C₃×C₄×#634	2
C₃³×C₄⋊C₈	2
C₃×#63648	2
#10592547	3
C₃×#4848	3

Subgroups — C3×#10592547 #13162887

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10592547 #13162887