Subgroups — C₃×#483961 #72226049

Order 1536.

950 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#33748	C₂	1	1	yes	maximal, characteristic
2	C₃×#17962	C₂	1	2	yes	maximal, complemented
3	C₃×#12089	C₂	1	4	yes	maximal, complemented
4	C₃×#12067	C₂	1	4	yes	maximal, complemented
5	C₃×#12020	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#11680	C₂	1	2	yes	maximal, complemented
7	C₃×#11404	C₂	1	1	yes	maximal, characteristic, complemented
8	#483961	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
9	C₃×#3169	C₂²	1	2	yes	—
10	C₃×#2931	—	2	2	no	—
11	C₃×#2254	C₄	1	1	yes	characteristic
12	C₃×#2246	C₄	1	1	yes	characteristic
13	C₃×#2240	C₄	1	2	yes	—
14	C₃×#2218	C₄	1	4	yes	—
15	C₃×#2177	C₂²	1	4	yes	—
16	C₃×C₄×M₃₂	C₂²	1	1	yes	characteristic
17	C₃×C₄×M₃₂	C₂²	1	1	yes	characteristic
18	C₃×#1594	—	2	2	no	—
19	C₃×#1594	C₂²	1	4	yes	—
20	C₃×#1580	—	2	2	no	—
21	C₃×#1580	C₂²	1	4	yes	—
22	C₃×#1566	C₂²	1	2	yes	—
23	C₃×#1492	—	2	2	no	—
24	C₂×C₃×#322	C₂²	1	1	yes	characteristic
25	C₃×#1368	C₂²	1	4	yes	—
26	C₃×#1333	C₂²	1	8	yes	—
27	C₃×#1313	C₂²	1	2	yes	—
28	C₃×#1313	C₂²	1	2	yes	—
29	#33748	C₂×C₃	1	1	yes	characteristic
30	#17962	C₂×C₃	1	2	yes	complemented
31	#12089	C₂×C₃	1	4	yes	complemented
32	#12067	C₂×C₃	1	4	yes	complemented
33	#12020	C₂×C₃	1	1	yes	characteristic, complemented
34	#11680	C₂×C₃	1	2	yes	complemented
35	#11404	C₂×C₃	1	1	yes	characteristic, complemented
36	C₂×C₃×#131	—	4	2	no	—
37	C₃×#532	—	2	2	no	—
38	C₂×C₃×D₁₆	C₂×C₄	1	4	yes	—
39	C₂×C₃×M₃₂	—	2	1	no	—
40	C₂×C₃×M₃₂	C₂×C₄	1	2	yes	—
41	C₃×#500	C₂×C₄	1	2	yes	—
42	C₃×#493	C₂×C₄	1	2	yes	—
43	C₃×#486	—	2	2	no	—
44	C₃×#466	—	2	2	no	—
45	C₃×#466	C₂×C₄	1	4	yes	—
46	C₃×C₄×D₈	—	2	2	no	—
47	C₃×C₄×D₈	C₂³	1	4	yes	—
48	C₃×#436	—	2	4	no	—
49	C₃×#436	—	4	2	no	—
50	C₃×C₈×D₄	—	2	4	no	—
51	C₃×C₈×D₄	—	4	2	no	—
52	C₃×#423	—	2	2	no	—
53	C₃×#408	—	4	2	no	—
54	C₂×C₃×C₄×C₈	D₄	1	2	yes	—
55	C₂×C₃×C₄×C₈	C₂³	1	1	yes	characteristic
56	C₃×#367	C₂×C₄	1	4	yes	—
57	C₃×#357	C₂×C₄	1	8	yes	—
58	C₃×#346	C₂³	1	2	yes	—
59	C₃×#346	C₂³	1	2	yes	—
60	C₃×C₄×C₁₆	—	2	1	no	—
61	C₃×C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
62	C₃×C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
63	C₃×#334	—	2	2	no	—
64	C₃×#334	C₂³	1	4	yes	—
65	C₃×D₄⋊C₈	—	2	4	no	—
66	C₃×D₄⋊C₈	—	2	4	no	—
67	C₃×#322	D₄	1	2	yes	—
68	C₃×#322	C₂³	1	1	yes	characteristic
69	C₃×#322	C₂³	1	1	yes	characteristic
70	#3169	C₂²×C₃	1	2	yes	—
71	#2931	—	2	2	no	—
72	#2254	C₃×C₄	1	1	yes	characteristic
73	#2246	C₃×C₄	1	1	yes	characteristic
74	#2240	C₃×C₄	1	2	yes	—
75	#2218	C₃×C₄	1	4	yes	—
76	#2177	C₂²×C₃	1	4	yes	—
77	C₄×M₃₂	C₂²×C₃	1	1	yes	characteristic
78	C₄×M₃₂	C₂²×C₃	1	1	yes	characteristic
79	#1594	—	2	2	no	—
80	#1594	C₂²×C₃	1	4	yes	—
81	#1580	—	2	2	no	—
82	#1580	C₂²×C₃	1	4	yes	—
83	#1566	C₂²×C₃	1	2	yes	—
84	#1492	—	2	2	no	—
85	C₂×#322	C₂²×C₃	1	1	yes	characteristic
86	#1368	C₂²×C₃	1	4	yes	—
87	#1333	C₂²×C₃	1	8	yes	—
88	#1313	C₂²×C₃	1	2	yes	—
89	#1313	C₂²×C₃	1	2	yes	—
90	C₂×C₃×#41	—	4	2	no	—
91	C₂×C₃×D₈	—	2	2	no	—
92	C₂×C₃×D₈	C₂²×C₄	1	4	yes	—
93	C₃×#131	—	8	8	no	—
94	C₂×C₃×M₁₆	—	4	4	no	—
95	C₂×C₃×M₁₆	—	4	2	no	—
96	C₂²×C₃×C₈	—	2	2	no	—
97	C₂²×C₃×C₈	—	4	4	no	—
98	C₂²×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
99	C₂²×C₃×C₈	#41	1	1	yes	characteristic
100	C₃×C₄⋊Q₈	—	2	2	no	—
101	C₃×#127	—	2	2	no	—
102	C₃×#124	—	4	2	no	—
103	C₃×#121	—	2	4	no	—
104	C₃×#121	—	4	2	no	—
105	C₃×C₄×D₄	—	2	4	no	—
106	C₃×C₄×D₄	—	4	2	no	—
107	C₂×C₃×C₄²	C₂×D₄	1	1	yes	characteristic
108	C₃×D₁₆	—	2	8	no	—
109	C₃×M₃₂	—	2	4	no	—
110	C₃×M₃₂	—	4	2	no	—
111	C₂×C₃×C₁₆	—	2	2	no	—
112	C₂×C₃×C₁₆	C₂²×C₄	1	2	yes	—
113	C₂×C₃×C₁₆	C₂²×C₄	1	2	yes	—
114	C₃×#107	—	2	2	no	—
115	C₃×#107	C₂²×C₄	1	4	yes	—
116	C₃×C₄⋊C₈	—	2	4	no	—
117	C₃×C₄⋊C₈	—	4	2	no	—
118	C₃×#102	—	2	4	no	—
119	C₃×#102	—	2	4	no	—
120	C₃×C₂²⋊C₈	—	4	4	no	—
121	C₃×C₂²⋊C₈	—	4	4	no	—
122	C₃×C₄×C₈	—	2	2	no	—
123	C₃×C₄×C₈	C₂²×C₄	1	1	yes	characteristic
124	C₃×C₄×C₈	C₂×D₄	1	2	yes	—
125	C₃×C₄×C₈	C₂×D₄	1	2	yes	—
126	C₃×C₄×C₈	C₂×D₄	1	1	yes	characteristic
127	C₃×C₄×C₈	#41	1	1	yes	characteristic
128	C₃×C₄×C₈	C₂⁴	1	1	yes	characteristic
129	C₂×#131	—	4	2	no	—
130	#532	—	2	2	no	—
131	C₂×D₁₆	C₂×C₃×C₄	1	4	yes	—
132	C₂×M₃₂	—	2	1	no	—
133	C₂×M₃₂	C₂×C₃×C₄	1	2	yes	—
134	#500	C₂×C₃×C₄	1	2	yes	—
135	#493	C₂×C₃×C₄	1	2	yes	—
136	#486	—	2	2	no	—
137	#466	—	2	2	no	—
138	#466	C₂×C₃×C₄	1	4	yes	—
139	C₄×D₈	—	2	2	no	—
140	C₄×D₈	C₂³×C₃	1	4	yes	—
141	#436	—	2	4	no	—
142	#436	—	4	2	no	—
143	C₈×D₄	—	2	4	no	—
144	C₈×D₄	—	4	2	no	—
145	#423	—	2	2	no	—
146	#408	—	4	2	no	—
147	C₂×C₄×C₈	C₃×D₄	1	2	yes	—
148	C₂×C₄×C₈	C₂³×C₃	1	1	yes	characteristic
149	#367	C₂×C₃×C₄	1	4	yes	—
150	#357	C₂×C₃×C₄	1	8	yes	—
151	#346	C₂³×C₃	1	2	yes	—
152	#346	C₂³×C₃	1	2	yes	—
153	C₄×C₁₆	—	2	1	no	—
154	C₄×C₁₆	C₂×C₃×C₄	1	1	yes	characteristic
155	C₄×C₁₆	C₂×C₃×C₄	1	1	yes	characteristic
156	#334	—	2	2	no	—
157	#334	C₂³×C₃	1	4	yes	—
158	D₄⋊C₈	—	2	4	no	—
159	D₄⋊C₈	—	2	4	no	—
160	#322	C₃×D₄	1	2	yes	—
161	#322	C₂³×C₃	1	1	yes	characteristic
162	#322	C₂³×C₃	1	1	yes	characteristic
163	C₃×#41	—	8	4	no	—
164	C₃×#41	—	8	4	no	—
165	C₂×C₃×Q₈	—	4	2	no	—
166	C₂×C₃×D₄	—	2	4	no	—
167	C₂×C₃×D₄	—	4	4	no	—
168	C₂×C₃×D₄	—	4	2	no	—
169	C₂×C₃×D₄	—	4	2	no	—
170	C₂²×C₃×C₄	—	4	4	no	—
171	C₂²×C₃×C₄	C₄×D₄	1	1	yes	characteristic
172	C₂²×C₃×C₄	AGL(1,ℤ/8)	1	2	yes	—
173	C₃×D₈	—	2	8	no	—
174	C₃×D₈	—	4	4	no	—
175	C₃×M₁₆	—	8	8	no	—
176	C₃×M₁₆	—	8	4	no	—
177	C₂×C₃×C₈	—	2	2	no	—
178	C₂×C₃×C₈	—	2	2	no	—
179	C₂×C₃×C₈	—	2	2	no	—
180	C₂×C₃×C₈	—	2	2	no	—
181	C₂×C₃×C₈	—	4	4	no	—
182	C₂×C₃×C₈	—	4	4	no	—
183	C₂×C₃×C₈	—	8	8	no	—
184	C₂×C₃×C₈	C₄×D₄	1	1	yes	characteristic
185	C₂×C₃×C₈	C₄×D₄	1	1	yes	characteristic
186	C₂×C₃×C₈	C₂³×C₄	1	1	yes	characteristic
187	C₂×C₃×C₈	C₂×#41	1	1	yes	characteristic
188	C₃×C₄⋊C₄	—	2	4	no	—
189	C₃×C₄⋊C₄	—	4	2	no	—
190	C₃×C₂²⋊C₄	—	4	4	no	—
191	C₃×C₂²⋊C₄	—	4	4	no	—
192	C₃×C₄²	C₄×D₄	1	1	yes	characteristic
193	C₃×C₄²	AGL(1,ℤ/8)	1	2	yes	—
194	C₃×C₄²	C₂²×D₄	1	1	yes	characteristic
195	C₃×C₁₆	—	2	2	no	—
196	C₃×C₁₆	—	4	2	no	—
197	C₃×C₁₆	#131	1	4	yes	complemented
198	C₂×#41	—	4	2	no	—
199	C₂×D₈	—	2	2	no	—
200	C₂×D₈	C₂²×C₃×C₄	1	4	yes	—
201	#131	—	8	8	no	—
202	C₂×M₁₆	—	4	4	no	—
203	C₂×M₁₆	—	4	2	no	—
204	C₂²×C₈	—	2	2	no	—
205	C₂²×C₈	—	4	4	no	—
206	C₂²×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
207	C₂²×C₈	C₃×#41	1	1	yes	characteristic
208	C₄⋊Q₈	—	2	2	no	—
209	#127	—	2	2	no	—
210	#124	—	4	2	no	—
211	#121	—	2	4	no	—
212	#121	—	4	2	no	—
213	C₄×D₄	—	2	4	no	—
214	C₄×D₄	—	4	2	no	—
215	C₂×C₄²	C₂×C₃×D₄	1	1	yes	characteristic
216	D₁₆	—	2	8	no	—
217	M₃₂	—	2	4	no	—
218	M₃₂	—	4	2	no	—
219	C₂×C₁₆	—	2	2	no	—
220	C₂×C₁₆	C₂²×C₃×C₄	1	2	yes	—
221	C₂×C₁₆	C₂²×C₃×C₄	1	2	yes	—
222	#107	—	2	2	no	—
223	#107	C₂²×C₃×C₄	1	4	yes	—
224	C₄⋊C₈	—	2	4	no	—
225	C₄⋊C₈	—	4	2	no	—
226	#102	—	2	4	no	—
227	#102	—	2	4	no	—
228	C₂²⋊C₈	—	4	4	no	—
229	C₂²⋊C₈	—	4	4	no	—
230	C₄×C₈	—	2	2	no	—
231	C₄×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
232	C₄×C₈	C₂×C₃×D₄	1	2	yes	—
233	C₄×C₈	C₂×C₃×D₄	1	2	yes	—
234	C₄×C₈	C₂×C₃×D₄	1	1	yes	characteristic
235	C₄×C₈	C₃×#41	1	1	yes	characteristic
236	C₄×C₈	C₂⁴×C₃	1	1	yes	characteristic
237	C₂³×C₃	—	4	4	no	—
238	C₂³×C₃	#442	1	1	yes	characteristic
239	C₃×Q₈	—	8	4	no	—
240	C₃×D₄	—	4	4	no	—
241	C₃×D₄	—	4	4	no	—
242	C₃×D₄	—	8	8	no	—
243	C₃×D₄	—	8	4	no	—
244	C₃×D₄	—	8	4	no	—
245	C₂×C₃×C₄	—	2	1	no	—
246	C₂×C₃×C₄	—	2	4	no	—
247	C₂×C₃×C₄	—	2	1	no	—
248	C₂×C₃×C₄	—	4	4	no	—
249	C₂×C₃×C₄	—	8	4	no	—
250	C₂×C₃×C₄	—	8	4	no	—
251	C₂×C₃×C₄	#442	1	1	yes	characteristic
252	C₂×C₃×C₄	#442	1	2	yes	—
253	C₂×C₃×C₄	C₂×C₄×D₄	1	1	yes	characteristic
254	C₂×C₃×C₄	C₂×AGL(1,ℤ/8)	1	2	yes	—
255	C₃×C₈	—	2	1	no	—
256	C₃×C₈	—	2	1	no	—
257	C₃×C₈	—	4	2	no	—
258	C₃×C₈	—	4	2	no	—
259	C₃×C₈	—	8	4	no	—
260	C₃×C₈	#444	1	2	yes	—
261	C₃×C₈	C₂×#131	1	2	yes	—
262	#41	—	8	4	no	—
263	#41	—	8	4	no	—
264	C₂×Q₈	—	4	2	no	—
265	C₂×D₄	—	2	4	no	—
266	C₂×D₄	—	4	4	no	—
267	C₂×D₄	—	4	2	no	—
268	C₂×D₄	—	4	2	no	—
269	C₂²×C₄	—	4	4	no	—
270	C₂²×C₄	C₃×C₄×D₄	1	1	yes	characteristic
271	C₂²×C₄	C₃×AGL(1,ℤ/8)	1	2	yes	—
272	D₈	—	2	8	no	—
273	D₈	—	4	4	no	—
274	M₁₆	—	8	8	no	—
275	M₁₆	—	8	4	no	—
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	—	2	2	no	—
278	C₂×C₈	—	2	2	no	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	—	4	4	no	—
281	C₂×C₈	—	4	4	no	—
282	C₂×C₈	—	8	8	no	—
283	C₂×C₈	C₃×C₄×D₄	1	1	yes	characteristic
284	C₂×C₈	C₃×C₄×D₄	1	1	yes	characteristic
285	C₂×C₈	C₂³×C₃×C₄	1	1	yes	characteristic
286	C₂×C₈	C₂×C₃×#41	1	1	yes	characteristic
287	C₄⋊C₄	—	2	4	no	—
288	C₄⋊C₄	—	4	2	no	—
289	C₂²⋊C₄	—	4	4	no	—
290	C₂²⋊C₄	—	4	4	no	—
291	C₄²	C₃×C₄×D₄	1	1	yes	characteristic
292	C₄²	C₃×AGL(1,ℤ/8)	1	2	yes	—
293	C₄²	C₂²×C₃×D₄	1	1	yes	characteristic
294	C₁₆	—	2	2	no	—
295	C₁₆	—	4	2	no	—
296	C₁₆	C₃×#131	1	4	yes	complemented
297	C₂²×C₃	—	2	1	no	—
298	C₂²×C₃	—	2	2	no	—
299	C₂²×C₃	—	8	4	no	—
300	C₂²×C₃	—	8	4	no	—
301	C₂²×C₃	—	8	4	no	—
302	C₂²×C₃	C₂×#442	1	1	yes	characteristic
303	C₃×C₄	—	2	1	no	—
304	C₃×C₄	—	2	2	no	—
305	C₃×C₄	—	8	4	no	—
306	C₃×C₄	AGL(1,ℤ/16)	1	4	yes	—
307	C₃×C₄	#2931	1	1	yes	characteristic
308	C₃×C₄	C₂×#444	1	1	yes	characteristic
309	C₂³	—	4	4	no	—
310	C₂³	C₃×#442	1	1	yes	characteristic
311	Q₈	—	8	4	no	—
312	D₄	—	4	4	no	—
313	D₄	—	4	4	no	—
314	D₄	—	8	8	no	—
315	D₄	—	8	4	no	—
316	D₄	—	8	4	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	4	no	—
320	C₂×C₄	—	4	4	no	—
321	C₂×C₄	—	8	4	no	—
322	C₂×C₄	—	8	4	no	—
323	C₂×C₄	C₃×#442	1	1	yes	characteristic
324	C₂×C₄	C₃×#442	1	2	yes	—
325	C₂×C₄	C₂×C₃×C₄×D₄	1	1	yes	characteristic
326	C₂×C₄	C₂×C₃×AGL(1,ℤ/8)	1	2	yes	—
327	C₈	—	2	1	no	—
328	C₈	—	2	1	no	—
329	C₈	—	4	2	no	—
330	C₈	—	4	2	no	—
331	C₈	—	8	4	no	—
332	C₈	C₃×#444	1	2	yes	—
333	C₈	C₂×C₃×#131	1	2	yes	—
334	C₂×C₃	—	4	1	no	—
335	C₂×C₃	—	16	4	no	—
336	C₂×C₃	#17962	1	1	yes	characteristic
337	C₂×C₃	C₂×AGL(1,ℤ/16)	1	2	yes	—
338	C₂²	—	2	1	no	—
339	C₂²	—	2	2	no	—
340	C₂²	—	8	4	no	—
341	C₂²	—	8	4	no	—
342	C₂²	—	8	4	no	—
343	C₂²	C₂×C₃×#442	1	1	yes	characteristic
344	C₄	—	2	1	no	—
345	C₄	—	2	2	no	—
346	C₄	—	8	4	no	—
347	C₄	C₃×AGL(1,ℤ/16)	1	4	yes	—
348	C₄	C₃×#2931	1	1	yes	characteristic
349	C₄	C₂×C₃×#444	1	1	yes	characteristic
350	C₃	#483961	1	1	yes	characteristic, direct product factor, Sylow
351	C₂	—	4	1	no	—
352	C₂	—	16	4	no	—
353	C₂	C₃×#17962	1	1	yes	characteristic
354	C₂	C₂×C₃×AGL(1,ℤ/16)	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃	C₂×#442
commutator	C₂×C₈	C₂³×C₃×C₄
Frattini, cosocle	C₄×C₈	C₂⁴×C₃
prime radical	C₃×#33748	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ C₂×C₃×#322 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×#17962	C₂	semidirect
3. C₃×#12089	C₂	semidirect
4. C₃×#12067	C₂	semidirect
5. C₃×#12020	C₂	semidirect
6. C₃×#11680	C₂	semidirect
7. C₃×#11404	C₂	semidirect
8. #483961	C₃	direct (N × H)
30. #17962	C₂×C₃	semidirect
31. #12089	C₂×C₃	semidirect
32. #12067	C₂×C₃	semidirect
33. #12020	C₂×C₃	semidirect
34. #11680	C₂×C₃	semidirect
35. #11404	C₂×C₃	semidirect
197. C₃×C₁₆	#131	semidirect
296. C₁₆	C₃×#131	semidirect
350. C₃	#483961	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×#33748	2
C₃×#17962	2
C₃×#12089	2
C₃×#12067	2
C₃×#12020	2
C₃×#11680	2
C₃×#11404	2
#483961	3

Subgroups — C3×#483961 #72226049

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#483961 #72226049