Subgroups — C₃×#491469 #72225816

Order 1536.

1692 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#33717	C₂	1	4	yes	maximal, complemented
2	C₃×#19603	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#1712	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#12519	C₂	1	4	yes	maximal, complemented
5	C₃×#12517	C₂	1	2	yes	maximal, complemented
6	C₃×#12515	C₂	1	2	yes	maximal, complemented
7	C₃×C₈×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
8	#491469	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
9	C₂×C₃×#509	—	2	8	no	—
10	C₃×#3169	—	2	2	no	—
11	C₃×#3147	C₂²	1	2	yes	—
12	C₂×C₃×#493	—	2	2	no	—
13	C₂×C₃×#493	C₂²	1	2	yes	—
14	C₃×#2250	—	2	8	no	—
15	C₃×#2248	C₂²	1	8	yes	complemented
16	C₃×#2246	C₂²	1	8	yes	complemented
17	C₃×C₄×M₃₂	C₂²	1	2	yes	—
18	C₃×#1713	C₂²	1	1	yes	characteristic, complemented
19	C₃×#1712	—	2	2	no	—
20	C₃×#1712	C₂²	1	2	yes	—
21	C₃×#1712	C₂²	1	2	yes	—
22	C₃×#1712	C₂²	1	1	yes	characteristic, complemented
23	C₃×#1709	C₂²	1	2	yes	complemented
24	C₃×#1706	—	2	2	no	—
25	C₂×C₃×C₈²	C₂²	1	1	yes	characteristic
26	C₃×#1311	C₂²	1	2	yes	complemented
27	C₃×C₈×C₁₆	C₂²	1	2	yes	complemented
28	#33717	C₂×C₃	1	4	yes	complemented
29	#19603	C₂×C₃	1	1	yes	characteristic, complemented
30	C₂×#1712	C₂×C₃	1	1	yes	characteristic, complemented
31	#12519	C₂×C₃	1	4	yes	complemented
32	#12517	C₂×C₃	1	2	yes	complemented
33	#12515	C₂×C₃	1	2	yes	complemented
34	C₈×M₃₂	C₂×C₃	1	1	yes	characteristic, complemented
35	C₂×C₃×#135	—	2	4	no	—
36	C₂²×C₃×D₈	—	2	4	no	—
37	C₂²×C₃×D₈	—	4	4	no	—
38	C₃×#532	—	2	2	no	—
39	C₂×C₃×#127	—	2	2	no	—
40	C₃×#509	—	4	32	no	—
41	C₂×C₃×SD₃₂	—	2	16	no	—
42	C₂×C₃×D₁₆	—	2	16	no	—
43	C₂×C₃×M₃₂	D₄	1	4	yes	—
44	C₃×#500	—	2	2	no	—
45	C₃×#495	—	2	4	no	—
46	C₃×#494	C₂³	1	2	yes	—
47	C₃×#493	—	2	4	no	—
48	C₃×#493	—	2	4	no	—
49	C₃×#493	—	2	2	no	—
50	C₃×#493	—	2	2	no	—
51	C₃×#493	—	2	4	no	—
52	C₃×#493	—	4	4	no	—
53	C₃×#493	C₂³	1	2	yes	—
54	C₃×#493	C₂³	1	4	yes	—
55	C₃×#492	—	2	2	no	—
56	C₃×#486	—	2	4	no	—
57	C₃×#466	—	4	4	no	—
58	C₃×C₄×D₈	—	4	4	no	—
59	C₂×C₃×C₄×C₈	D₄	1	2	yes	—
60	C₂×C₃×C₄×C₈	C₂³	1	1	yes	characteristic
61	C₃×#346	D₄	1	4	yes	—
62	C₃×C₄×C₁₆	C₂³	1	4	yes	—
63	C₃×C₈²	D₄	1	2	yes	—
64	C₃×C₈²	C₂³	1	1	yes	characteristic
65	C₃×C₈²	C₂³	1	1	yes	characteristic
66	C₂×#509	—	2	8	no	—
67	#3169	—	2	2	no	—
68	#3147	C₂²×C₃	1	2	yes	—
69	C₂×#493	—	2	2	no	—
70	C₂×#493	C₂²×C₃	1	2	yes	—
71	#2250	—	2	8	no	—
72	#2248	C₂²×C₃	1	8	yes	complemented
73	#2246	C₂²×C₃	1	8	yes	complemented
74	C₄×M₃₂	C₂²×C₃	1	2	yes	—
75	#1713	C₂²×C₃	1	1	yes	characteristic, complemented
76	#1712	—	2	2	no	—
77	#1712	C₂²×C₃	1	2	yes	—
78	#1712	C₂²×C₃	1	2	yes	—
79	#1712	C₂²×C₃	1	1	yes	characteristic, complemented
80	#1709	C₂²×C₃	1	2	yes	complemented
81	#1706	—	2	2	no	—
82	C₂×C₈²	C₂²×C₃	1	1	yes	characteristic
83	#1311	C₂²×C₃	1	2	yes	complemented
84	C₈×C₁₆	C₂²×C₃	1	2	yes	complemented
85	C₂×C₃×#41	—	4	2	no	—
86	C₂²×C₃×D₄	—	4	2	no	—
87	C₂²×C₃×D₄	—	4	4	no	—
88	C₃×#135	—	4	16	no	—
89	C₂×C₃×Q₁₆	—	2	4	no	—
90	C₂×C₃×SD₁₆	—	4	4	no	—
91	C₂×C₃×D₈	—	2	4	no	—
92	C₂×C₃×D₈	—	2	8	no	—
93	C₂×C₃×D₈	—	4	4	no	—
94	C₂×C₃×D₈	—	4	16	no	—
95	C₂×C₃×D₈	—	4	8	no	—
96	C₂×C₃×D₈	—	4	8	no	—
97	C₂×C₃×D₈	—	4	4	no	—
98	C₂×C₃×D₈	—	8	16	no	—
99	C₂²×C₃×C₈	D₈	1	4	yes	—
100	C₂²×C₃×C₈	C₂×D₄	1	2	yes	—
101	C₃×C₄⋊Q₈	—	2	2	no	—
102	C₃×#127	—	2	4	no	—
103	C₃×#127	—	2	2	no	—
104	C₃×#127	—	4	2	no	—
105	C₃×#127	—	4	4	no	—
106	C₃×#124	—	4	2	no	—
107	C₃×#121	—	4	4	no	—
108	C₃×C₄×D₄	—	4	4	no	—
109	C₂×C₃×C₄²	C₂×D₄	1	1	yes	characteristic
110	C₃×SD₃₂	—	4	32	no	—
111	C₃×D₁₆	—	4	32	no	—
112	C₃×M₃₂	D₈	1	16	yes	complemented
113	C₂×C₃×C₁₆	C₂×D₄	1	8	yes	—
114	C₃×#107	—	4	4	no	—
115	C₃×#102	—	4	8	no	—
116	C₃×C₄×C₈	D₈	1	4	yes	—
117	C₃×C₄×C₈	C₂×D₄	1	2	yes	—
118	C₃×C₄×C₈	C₂×D₄	1	1	yes	characteristic
119	C₃×C₄×C₈	C₂×D₄	1	2	yes	—
120	C₃×C₄×C₈	C₂×D₄	1	2	yes	—
121	C₃×C₄×C₈	C₂⁴	1	1	yes	characteristic
122	C₂×#135	—	2	4	no	—
123	C₂²×D₈	—	2	4	no	—
124	C₂²×D₈	—	4	4	no	—
125	#532	—	2	2	no	—
126	C₂×#127	—	2	2	no	—
127	#509	—	4	32	no	—
128	C₂×SD₃₂	—	2	16	no	—
129	C₂×D₁₆	—	2	16	no	—
130	C₂×M₃₂	C₃×D₄	1	4	yes	—
131	#500	—	2	2	no	—
132	#495	—	2	4	no	—
133	#494	C₂³×C₃	1	2	yes	—
134	#493	—	2	4	no	—
135	#493	—	2	4	no	—
136	#493	—	2	2	no	—
137	#493	—	2	2	no	—
138	#493	—	2	4	no	—
139	#493	—	4	4	no	—
140	#493	C₂³×C₃	1	4	yes	—
141	#493	C₂³×C₃	1	2	yes	—
142	#492	—	2	2	no	—
143	#486	—	2	4	no	—
144	#466	—	4	4	no	—
145	C₄×D₈	—	4	4	no	—
146	C₂×C₄×C₈	C₃×D₄	1	2	yes	—
147	C₂×C₄×C₈	C₂³×C₃	1	1	yes	characteristic
148	#346	C₃×D₄	1	4	yes	—
149	C₄×C₁₆	C₂³×C₃	1	4	yes	—
150	C₈²	C₃×D₄	1	2	yes	—
151	C₈²	C₂³×C₃	1	1	yes	characteristic
152	C₈²	C₂³×C₃	1	1	yes	characteristic
153	C₂⁴×C₃	—	8	2	no	—
154	C₃×#41	—	8	8	no	—
155	C₂×C₃×Q₈	—	4	2	no	—
156	C₂×C₃×D₄	—	4	2	no	—
157	C₂×C₃×D₄	—	4	4	no	—
158	C₂×C₃×D₄	—	4	4	no	—
159	C₂×C₃×D₄	—	4	8	no	—
160	C₂×C₃×D₄	—	8	16	no	—
161	C₂×C₃×D₄	—	8	2	no	—
162	C₂×C₃×D₄	—	8	8	no	—
163	C₂×C₃×D₄	—	8	4	no	—
164	C₂×C₃×D₄	—	8	2	no	—
165	C₂²×C₃×C₄	—	8	2	no	—
166	C₂²×C₃×C₄	#127	1	1	yes	characteristic
167	C₂²×C₃×C₄	C₂×D₈	1	2	yes	—
168	C₃×Q₁₆	—	4	8	no	—
169	C₃×SD₁₆	—	8	8	no	—
170	C₃×D₈	—	4	8	no	—
171	C₃×D₈	—	4	16	no	—
172	C₃×D₈	—	8	16	no	—
173	C₃×D₈	—	8	8	no	—
174	C₃×D₈	—	8	8	no	—
175	C₃×D₈	—	8	16	no	—
176	C₂×C₃×C₈	—	2	8	no	—
177	C₂×C₃×C₈	#127	1	2	yes	—
178	C₂×C₃×C₈	C₂×D₈	1	4	yes	—
179	C₂×C₃×C₈	C₂×D₈	1	8	yes	—
180	C₂×C₃×C₈	C₂×D₈	1	4	yes	—
181	C₂×C₃×C₈	C₂²×D₄	1	2	yes	—
182	C₃×C₄⋊C₄	—	4	4	no	—
183	C₃×C₂²⋊C₄	—	8	4	no	—
184	C₃×C₄²	#127	1	1	yes	characteristic
185	C₃×C₄²	C₂×D₈	1	2	yes	—
186	C₃×C₄²	C₂²×D₄	1	1	yes	characteristic
187	C₃×C₁₆	C₂×D₈	1	16	yes	complemented
188	C₂×#41	—	4	2	no	—
189	C₂²×D₄	—	4	2	no	—
190	C₂²×D₄	—	4	4	no	—
191	#135	—	4	16	no	—
192	C₂×Q₁₆	—	2	4	no	—
193	C₂×SD₁₆	—	4	4	no	—
194	C₂×D₈	—	2	8	no	—
195	C₂×D₈	—	2	4	no	—
196	C₂×D₈	—	4	4	no	—
197	C₂×D₈	—	4	16	no	—
198	C₂×D₈	—	4	8	no	—
199	C₂×D₈	—	4	8	no	—
200	C₂×D₈	—	4	4	no	—
201	C₂×D₈	—	8	16	no	—
202	C₂²×C₈	C₃×D₈	1	4	yes	—
203	C₂²×C₈	C₂×C₃×D₄	1	2	yes	—
204	C₄⋊Q₈	—	2	2	no	—
205	#127	—	2	4	no	—
206	#127	—	2	2	no	—
207	#127	—	4	2	no	—
208	#127	—	4	4	no	—
209	#124	—	4	2	no	—
210	#121	—	4	4	no	—
211	C₄×D₄	—	4	4	no	—
212	C₂×C₄²	C₂×C₃×D₄	1	1	yes	characteristic
213	SD₃₂	—	4	32	no	—
214	D₁₆	—	4	32	no	—
215	M₃₂	C₃×D₈	1	16	yes	complemented
216	C₂×C₁₆	C₂×C₃×D₄	1	8	yes	—
217	#107	—	4	4	no	—
218	#102	—	4	8	no	—
219	C₄×C₈	C₃×D₈	1	4	yes	—
220	C₄×C₈	C₂×C₃×D₄	1	2	yes	—
221	C₄×C₈	C₂×C₃×D₄	1	1	yes	characteristic
222	C₄×C₈	C₂×C₃×D₄	1	2	yes	—
223	C₄×C₈	C₂×C₃×D₄	1	2	yes	—
224	C₄×C₈	C₂⁴×C₃	1	1	yes	characteristic
225	C₂³×C₃	—	8	2	no	—
226	C₂³×C₃	—	8	4	no	—
227	C₂³×C₃	—	16	8	no	—
228	C₂³×C₃	—	16	4	no	—
229	C₂³×C₃	#493	1	1	yes	characteristic
230	C₃×Q₈	—	8	4	no	—
231	C₃×D₄	—	8	8	no	—
232	C₃×D₄	—	8	4	no	—
233	C₃×D₄	—	8	16	no	—
234	C₃×D₄	—	8	8	no	—
235	C₃×D₄	—	16	8	no	—
236	C₃×D₄	—	16	4	no	—
237	C₃×D₄	—	16	4	no	—
238	C₂×C₃×C₄	—	2	4	no	—
239	C₂×C₃×C₄	—	8	2	no	—
240	C₂×C₃×C₄	—	16	4	no	—
241	C₂×C₃×C₄	#493	1	2	yes	—
242	C₂×C₃×C₄	#493	1	4	yes	—
243	C₂×C₃×C₄	#493	1	1	yes	characteristic
244	C₂×C₃×C₄	C₂×#127	1	1	yes	characteristic
245	C₂×C₃×C₄	C₂²×D₈	1	2	yes	—
246	C₃×C₈	—	2	4	no	—
247	C₃×C₈	#493	1	4	yes	—
248	C₃×C₈	#509	1	8	yes	complemented
249	C₃×C₈	C₂²×D₈	1	4	yes	—
250	C₂⁴	—	8	2	no	—
251	#41	—	8	8	no	—
252	C₂×Q₈	—	4	2	no	—
253	C₂×D₄	—	4	4	no	—
254	C₂×D₄	—	4	8	no	—
255	C₂×D₄	—	4	2	no	—
256	C₂×D₄	—	4	4	no	—
257	C₂×D₄	—	8	16	no	—
258	C₂×D₄	—	8	2	no	—
259	C₂×D₄	—	8	8	no	—
260	C₂×D₄	—	8	4	no	—
261	C₂×D₄	—	8	2	no	—
262	C₂²×C₄	—	8	2	no	—
263	C₂²×C₄	C₃×#127	1	1	yes	characteristic
264	C₂²×C₄	C₂×C₃×D₈	1	2	yes	—
265	Q₁₆	—	4	8	no	—
266	SD₁₆	—	8	8	no	—
267	D₈	—	4	8	no	—
268	D₈	—	4	16	no	—
269	D₈	—	8	16	no	—
270	D₈	—	8	16	no	—
271	D₈	—	8	8	no	—
272	D₈	—	8	8	no	—
273	C₂×C₈	—	2	8	no	—
274	C₂×C₈	C₃×#127	1	2	yes	—
275	C₂×C₈	C₂×C₃×D₈	1	4	yes	—
276	C₂×C₈	C₂×C₃×D₈	1	4	yes	—
277	C₂×C₈	C₂×C₃×D₈	1	8	yes	—
278	C₂×C₈	C₂²×C₃×D₄	1	2	yes	—
279	C₄⋊C₄	—	4	4	no	—
280	C₂²⋊C₄	—	8	4	no	—
281	C₄²	C₃×#127	1	1	yes	characteristic
282	C₄²	C₂×C₃×D₈	1	2	yes	—
283	C₄²	C₂²×C₃×D₄	1	1	yes	characteristic
284	C₁₆	C₂×C₃×D₈	1	16	yes	complemented
285	C₂²×C₃	—	2	2	no	—
286	C₂²×C₃	—	16	2	no	—
287	C₂²×C₃	—	16	4	no	—
288	C₂²×C₃	—	16	8	no	—
289	C₂²×C₃	—	16	4	no	—
290	C₂²×C₃	—	32	8	no	—
291	C₂²×C₃	#1712	1	2	yes	—
292	C₂²×C₃	C₂×#493	1	1	yes	characteristic
293	C₃×C₄	—	2	2	no	—
294	C₃×C₄	—	16	2	no	—
295	C₃×C₄	#1712	1	2	yes	—
296	C₃×C₄	C₂×#493	1	2	yes	—
297	C₃×C₄	C₂×#509	1	4	yes	—
298	C₂³	—	8	2	no	—
299	C₂³	—	8	4	no	—
300	C₂³	—	16	8	no	—
301	C₂³	—	16	4	no	—
302	C₂³	C₃×#493	1	1	yes	characteristic
303	Q₈	—	8	4	no	—
304	D₄	—	8	8	no	—
305	D₄	—	8	4	no	—
306	D₄	—	8	16	no	—
307	D₄	—	8	8	no	—
308	D₄	—	16	8	no	—
309	D₄	—	16	4	no	—
310	D₄	—	16	4	no	—
311	C₂×C₄	—	2	4	no	—
312	C₂×C₄	—	8	2	no	—
313	C₂×C₄	—	16	4	no	—
314	C₂×C₄	C₃×#493	1	2	yes	—
315	C₂×C₄	C₃×#493	1	4	yes	—
316	C₂×C₄	C₃×#493	1	1	yes	characteristic
317	C₂×C₄	C₂×C₃×#127	1	1	yes	characteristic
318	C₂×C₄	C₂²×C₃×D₈	1	2	yes	—
319	C₈	—	2	4	no	—
320	C₈	C₃×#493	1	4	yes	—
321	C₈	C₃×#509	1	8	yes	complemented
322	C₈	C₂²×C₃×D₈	1	4	yes	—
323	C₂×C₃	—	2	2	no	—
324	C₂×C₃	—	32	2	no	—
325	C₂×C₃	—	32	4	no	—
326	C₂×C₃	C₂×#1712	1	1	yes	characteristic
327	C₂×C₃	#33717	1	2	yes	—
328	C₂²	—	2	2	no	—
329	C₂²	—	16	2	no	—
330	C₂²	—	16	4	no	—
331	C₂²	—	16	8	no	—
332	C₂²	—	16	4	no	—
333	C₂²	—	32	8	no	—
334	C₂²	C₃×#1712	1	2	yes	—
335	C₂²	C₂×C₃×#493	1	1	yes	characteristic
336	C₄	—	2	2	no	—
337	C₄	—	16	2	no	—
338	C₄	C₃×#1712	1	2	yes	—
339	C₄	C₂×C₃×#493	1	2	yes	—
340	C₄	C₂×C₃×#509	1	4	yes	—
341	C₃	#491469	1	1	yes	characteristic, direct product factor, Sylow
342	C₂	—	2	2	no	—
343	C₂	—	32	4	no	—
344	C₂	—	32	2	no	—
345	C₂	C₂×C₃×#1712	1	1	yes	characteristic
346	C₂	C₃×#33717	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃	C₂×#493
commutator, Frattini, cosocle	C₄×C₈	C₂⁴×C₃
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ C₂×C₃×C₈² ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×#33717	C₂	semidirect
2. C₃×#19603	C₂	semidirect
3. C₂×C₃×#1712	C₂	semidirect
4. C₃×#12519	C₂	semidirect
5. C₃×#12517	C₂	semidirect
6. C₃×#12515	C₂	semidirect
7. C₃×C₈×M₃₂	C₂	semidirect
8. #491469	C₃	direct (N × H)
15. C₃×#2248	C₂²	semidirect
16. C₃×#2246	C₂²	semidirect
18. C₃×#1713	C₂²	semidirect
22. C₃×#1712	C₂²	semidirect
23. C₃×#1709	C₂²	semidirect
26. C₃×#1311	C₂²	semidirect
27. C₃×C₈×C₁₆	C₂²	semidirect
28. #33717	C₂×C₃	semidirect
29. #19603	C₂×C₃	semidirect
30. C₂×#1712	C₂×C₃	semidirect
31. #12519	C₂×C₃	semidirect
32. #12517	C₂×C₃	semidirect
33. #12515	C₂×C₃	semidirect
34. C₈×M₃₂	C₂×C₃	semidirect
72. #2248	C₂²×C₃	semidirect
73. #2246	C₂²×C₃	semidirect
75. #1713	C₂²×C₃	semidirect
79. #1712	C₂²×C₃	semidirect
80. #1709	C₂²×C₃	semidirect
83. #1311	C₂²×C₃	semidirect
84. C₈×C₁₆	C₂²×C₃	semidirect
112. C₃×M₃₂	D₈	semidirect
187. C₃×C₁₆	C₂×D₈	semidirect
215. M₃₂	C₃×D₈	semidirect
248. C₃×C₈	#509	semidirect
284. C₁₆	C₂×C₃×D₈	semidirect
321. C₈	C₃×#509	semidirect
341. C₃	#491469	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×#33717	2
C₃×#19603	2
C₂×C₃×#1712	2
C₃×#12519	2
C₃×#12517	2
C₃×#12515	2
C₃×C₈×M₃₂	2
#491469	3

Subgroups — C3×#491469 #72225816

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#491469 #72225816