Subgroups — C₃×D₃×#494 #11874201

Order 1152.

1165 conjugacy classes of subgroups.

The subgroup lattice has 301 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×D₃×Q₁₆	C₂	1	4	yes	maximal
2	C₃×D₃×C₄⋊Q₈	C₂	1	2	yes	maximal
3	C₃²×#494	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#5183	C₂	1	4	yes	maximal, complemented
5	C₃×#5092	C₂	1	2	yes	maximal, complemented
6	C₃×#4699	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×C₄×D₃×C₈	C₂	1	1	yes	maximal, characteristic
8	C₂×C₃×#494	—	3	1	no	maximal
9	D₃×#494	C₃	1	1	yes	maximal, characteristic, direct product factor
10	C₂×C₃×D₃×Q₈	C₂²	1	4	yes	—
11	C₂×C₃²×Q₁₆	C₂²	1	4	yes	—
12	C₃²×C₄⋊Q₈	C₂²	1	2	yes	—
13	C₃×#944	C₂²	1	4	yes	—
14	C₂×C₃×#216	C₂²	1	8	yes	—
15	C₃×D₃×Q₁₆	—	2	16	no	—
16	C₂×C₃×Dic₁₂	C₂²	1	4	yes	—
17	C₂×C₃×D₃×C₈	C₂²	1	2	yes	—
18	C₃×D₃×C₄⋊C₄	—	2	4	no	—
19	C₃×#888	—	2	4	no	—
20	C₃×C₄²×D₃	C₂²	1	1	yes	characteristic
21	C₃×#869	C₂²	1	2	yes	—
22	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
23	C₃×C₈×Dic₃	C₂²	1	2	yes	—
24	C₃×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
25	C₂²×C₃×Q₁₆	—	3	4	no	—
26	C₂×C₃×C₄⋊Q₈	—	3	2	no	—
27	C₂×D₃×Q₁₆	C₂×C₃	1	4	yes	—
28	D₃×C₄⋊Q₈	C₂×C₃	1	2	yes	—
29	C₃×#494	—	2	1	no	—
30	C₃×#494	—	3	4	no	—
31	C₃×#494	—	3	2	no	—
32	C₃×#494	—	3	1	no	—
33	C₃×#494	D₃	1	1	yes	characteristic, direct product factor
34	C₃×#494	C₂×C₃	1	1	yes	characteristic, complemented
35	C₂×C₃×C₄×C₈	—	3	1	no	—
36	#5183	C₂×C₃	1	4	yes	complemented
37	#5092	C₂×C₃	1	2	yes	complemented
38	#4699	C₂×C₃	1	1	yes	characteristic, complemented
39	C₄×D₃×C₈	C₂×C₃	1	1	yes	characteristic
40	C₂×C₃²×Q₈	C₂³	1	4	yes	—
41	C₃×D₃×Q₈	—	2	16	no	—
42	C₂×C₃×C₄×D₃	—	2	4	no	—
43	C₂×C₃×C₄×D₃	D₄	1	2	yes	—
44	C₂×C₃×C₄×D₃	C₂³	1	1	yes	characteristic
45	C₂×C₃×Dic₆	—	2	4	no	—
46	C₂×C₃×Dic₆	C₂³	1	4	yes	—
47	C₃²×Q₁₆	—	2	8	no	—
48	C₂×C₃²×C₈	C₂³	1	2	yes	—
49	C₃²×C₄⋊C₄	—	2	4	no	—
50	C₃²×C₄²	C₂³	1	1	yes	characteristic
51	C₃×#216	—	2	16	no	—
52	C₃×C₄⋊Dic₃	—	2	4	no	—
53	C₃×Dic₃⋊C₄	—	2	8	no	—
54	C₃×C₄×Dic₃	D₄	1	2	yes	—
55	C₃×C₄×Dic₃	C₂³	1	1	yes	characteristic
56	C₂×C₃×C₃⋊C₈	C₂³	1	2	yes	—
57	C₃×Dic₁₂	—	2	8	no	—
58	C₃×D₃×C₈	D₄	1	8	yes	—
59	C₂×#494	—	3	1	no	Sylow
60	C₂²×C₃×Q₈	—	3	4	no	—
61	C₂×D₃×Q₈	C₂²×C₃	1	4	yes	—
62	C₂×C₃×Q₁₆	—	2	4	no	—
63	C₂×C₃×Q₁₆	—	3	8	no	—
64	C₂×C₃×Q₁₆	—	3	4	no	—
65	C₂×C₃×Q₁₆	—	6	16	no	—
66	C₂×C₃×Q₁₆	C₂×D₃	1	4	yes	—
67	C₂×C₃×Q₁₆	C₂²×C₃	1	4	yes	—
68	C₂²×C₃×C₈	—	3	2	no	—
69	C₃×C₄⋊Q₈	—	2	2	no	—
70	C₃×C₄⋊Q₈	—	3	4	no	—
71	C₃×C₄⋊Q₈	—	3	2	no	—
72	C₃×C₄⋊Q₈	—	6	4	no	—
73	C₃×C₄⋊Q₈	C₂×D₃	1	2	yes	—
74	C₃×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
75	C₂×C₃×C₄⋊C₄	—	6	4	no	—
76	C₂×C₃×C₄²	—	3	1	no	—
77	#944	C₂²×C₃	1	4	yes	—
78	C₂×#216	C₂²×C₃	1	8	yes	—
79	D₃×Q₁₆	—	2	16	no	—
80	C₂×Dic₁₂	C₂²×C₃	1	4	yes	—
81	C₂×D₃×C₈	C₂²×C₃	1	2	yes	—
82	D₃×C₄⋊C₄	—	2	4	no	—
83	#888	—	2	4	no	—
84	C₄²×D₃	C₂²×C₃	1	1	yes	characteristic
85	#869	C₂²×C₃	1	2	yes	—
86	C₃×C₄×C₈	—	2	1	no	—
87	C₃×C₄×C₈	—	3	2	no	—
88	C₃×C₄×C₈	—	3	1	no	—
89	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
90	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
91	C₈×Dic₃	C₂²×C₃	1	2	yes	—
92	C₄×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
93	C₂²×C₃×D₃	C₂×D₄	1	1	yes	characteristic
94	C₃²×Q₈	—	2	8	no	—
95	C₂×C₃²×C₄	—	2	4	no	—
96	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
97	C₂×C₃²×C₄	C₂⁴	1	1	yes	characteristic
98	C₂×C₃×Dic₃	—	2	4	no	—
99	C₂×C₃×Dic₃	C₂×D₄	1	1	yes	characteristic
100	C₂×C₃×Dic₃	C₂×D₄	1	2	yes	—
101	C₃×C₄×D₃	—	4	8	no	—
102	C₃×C₄×D₃	Q₁₆	1	8	yes	complemented
103	C₃×C₄×D₃	C₂×D₄	1	4	yes	—
104	C₃×Dic₆	—	2	8	no	—
105	C₃×Dic₆	—	4	8	no	—
106	C₃²×C₈	C₂×D₄	1	4	yes	—
107	C₃×C₃⋊C₈	C₂×D₄	1	4	yes	—
108	C₂²×Q₁₆	—	3	4	no	—
109	C₂×C₄⋊Q₈	—	3	2	no	—
110	#494	—	3	4	no	—
111	#494	—	3	2	no	—
112	#494	—	3	1	no	—
113	#494	C₃×D₃	1	1	yes	characteristic, direct product factor
114	C₂×C₄×C₈	—	3	1	no	—
115	C₂×C₃×Q₈	—	2	4	no	—
116	C₂×C₃×Q₈	—	3	4	no	—
117	C₂×C₃×Q₈	—	6	4	no	—
118	C₂×C₃×Q₈	—	6	16	no	—
119	C₂×C₃×Q₈	C₂²×D₃	1	4	yes	—
120	C₂×C₃×Q₈	C₂³×C₃	1	4	yes	—
121	C₂²×C₃×C₄	—	3	2	no	—
122	C₂²×C₃×C₄	—	3	1	no	—
123	C₂²×C₃×C₄	—	6	4	no	—
124	D₃×Q₈	—	2	16	no	—
125	C₂×C₄×D₃	—	2	4	no	—
126	C₂×C₄×D₃	C₃×D₄	1	2	yes	—
127	C₂×C₄×D₃	C₂³×C₃	1	1	yes	characteristic
128	C₂×Dic₆	—	2	4	no	—
129	C₂×Dic₆	C₂³×C₃	1	4	yes	—
130	C₃×Q₁₆	—	2	8	no	—
131	C₃×Q₁₆	—	2	8	no	—
132	C₃×Q₁₆	—	4	8	no	—
133	C₃×Q₁₆	—	6	16	no	—
134	C₃×Q₁₆	—	6	8	no	—
135	C₂×C₃×C₈	—	2	2	no	—
136	C₂×C₃×C₈	—	3	2	no	—
137	C₂×C₃×C₈	—	3	8	no	—
138	C₂×C₃×C₈	C₂²×D₃	1	2	yes	—
139	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
140	C₃×C₄⋊C₄	—	2	4	no	—
141	C₃×C₄⋊C₄	—	2	4	no	—
142	C₃×C₄⋊C₄	—	4	4	no	—
143	C₃×C₄⋊C₄	—	6	8	no	—
144	C₃×C₄⋊C₄	—	6	4	no	—
145	C₃×C₄²	—	2	1	no	—
146	C₃×C₄²	—	3	2	no	—
147	C₃×C₄²	—	3	1	no	—
148	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
149	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
150	#216	—	2	16	no	—
151	C₄⋊Dic₃	—	2	4	no	—
152	Dic₃⋊C₄	—	2	8	no	—
153	C₄×Dic₃	C₃×D₄	1	2	yes	—
154	C₄×Dic₃	C₂³×C₃	1	1	yes	characteristic
155	C₂×C₃⋊C₈	C₂³×C₃	1	2	yes	—
156	Dic₁₂	—	2	8	no	—
157	D₃×C₈	C₃×D₄	1	8	yes	—
158	C₂²×C₃²	C₂²×D₄	1	1	yes	characteristic
159	C₂×C₃×D₃	#127	1	2	yes	—
160	C₂×C₃×D₃	C₂×Q₁₆	1	4	yes	—
161	C₃²×C₄	—	4	4	no	—
162	C₃²×C₄	C₂×Q₁₆	1	4	yes	complemented
163	C₃²×C₄	C₂²×D₄	1	2	yes	—
164	C₃×Dic₃	—	4	4	no	—
165	C₃×Dic₃	#127	1	2	yes	—
166	C₃×Dic₃	C₂×Q₁₆	1	4	yes	complemented
167	C₂²×Q₈	—	3	4	no	—
168	C₂×Q₁₆	—	3	8	no	—
169	C₂×Q₁₆	—	3	4	no	—
170	C₂×Q₁₆	—	6	16	no	—
171	C₂×Q₁₆	C₂×C₃×D₃	1	4	yes	—
172	C₂²×C₈	—	3	2	no	—
173	C₄⋊Q₈	—	3	4	no	—
174	C₄⋊Q₈	—	3	2	no	—
175	C₄⋊Q₈	—	6	4	no	—
176	C₄⋊Q₈	C₂×C₃×D₃	1	2	yes	—
177	C₂×C₄⋊C₄	—	6	4	no	—
178	C₂×C₄²	—	3	1	no	—
179	C₄×C₈	—	3	2	no	—
180	C₄×C₈	—	3	1	no	—
181	C₄×C₈	C₂×C₃×D₃	1	1	yes	characteristic
182	C₂³×C₃	—	3	1	no	—
183	C₂²×D₃	C₂×C₃×D₄	1	1	yes	characteristic
184	C₃×Q₈	—	2	8	no	—
185	C₃×Q₈	—	2	8	no	—
186	C₃×Q₈	—	4	8	no	—
187	C₃×Q₈	—	6	8	no	—
188	C₃×Q₈	—	12	8	no	—
189	C₂×C₃×C₄	—	2	4	no	—
190	C₂×C₃×C₄	—	2	4	no	—
191	C₂×C₃×C₄	—	2	1	no	—
192	C₂×C₃×C₄	—	2	2	no	—
193	C₂×C₃×C₄	—	3	2	no	—
194	C₂×C₃×C₄	—	3	1	no	—
195	C₂×C₃×C₄	—	3	8	no	—
196	C₂×C₃×C₄	—	3	4	no	—
197	C₂×C₃×C₄	—	4	4	no	—
198	C₂×C₃×C₄	—	6	4	no	—
199	C₂×C₃×C₄	—	12	8	no	—
200	C₂×C₃×C₄	D₃×D₄	1	2	yes	—
201	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
202	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
203	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
204	C₂×Dic₃	—	2	4	no	—
205	C₂×Dic₃	C₂×C₃×D₄	1	1	yes	characteristic
206	C₂×Dic₃	C₂×C₃×D₄	1	2	yes	—
207	C₄×D₃	—	4	8	no	—
208	C₄×D₃	C₃×Q₁₆	1	8	yes	complemented
209	C₄×D₃	C₂×C₃×D₄	1	4	yes	—
210	Dic₆	—	2	8	no	—
211	Dic₆	—	4	8	no	—
212	C₃×C₈	—	2	4	no	—
213	C₃×C₈	—	3	4	no	—
214	C₃×C₈	D₃×D₄	1	4	yes	—
215	C₃×C₈	C₂×C₃×D₄	1	4	yes	—
216	C₃⋊C₈	C₂×C₃×D₄	1	4	yes	—
217	C₂×C₃²	C₂×#127	1	1	yes	characteristic
218	C₂×C₃²	C₂²×Q₁₆	1	2	yes	—
219	C₃×D₃	#494	1	4	yes	direct product factor
220	C₂×Q₈	—	3	4	no	—
221	C₂×Q₈	—	6	4	no	—
222	C₂×Q₈	—	6	16	no	—
223	C₂×Q₈	C₂²×C₃×D₃	1	4	yes	—
224	C₂²×C₄	—	3	1	no	—
225	C₂²×C₄	—	3	2	no	—
226	C₂²×C₄	—	6	4	no	—
227	Q₁₆	—	2	8	no	—
228	Q₁₆	—	6	16	no	—
229	Q₁₆	—	6	8	no	—
230	C₂×C₈	—	3	2	no	—
231	C₂×C₈	—	3	8	no	—
232	C₂×C₈	C₂²×C₃×D₃	1	2	yes	—
233	C₄⋊C₄	—	2	4	no	—
234	C₄⋊C₄	—	6	8	no	—
235	C₄⋊C₄	—	6	4	no	—
236	C₄²	—	3	1	no	—
237	C₄²	—	3	2	no	—
238	C₄²	C₂²×C₃×D₃	1	1	yes	characteristic
239	C₂²×C₃	—	2	1	no	—
240	C₂²×C₃	—	3	2	no	—
241	C₂²×C₃	—	3	4	no	—
242	C₂²×C₃	C₂×D₃×D₄	1	1	yes	characteristic
243	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
244	C₂×D₃	C₃×#127	1	2	yes	—
245	C₂×D₃	C₂×C₃×Q₁₆	1	4	yes	—
246	C₃×C₄	—	2	4	no	—
247	C₃×C₄	—	2	2	no	—
248	C₃×C₄	—	3	2	no	—
249	C₃×C₄	—	3	4	no	—
250	C₃×C₄	—	4	4	no	—
251	C₃×C₄	—	4	4	no	—
252	C₃×C₄	—	8	4	no	—
253	C₃×C₄	—	12	4	no	—
254	C₃×C₄	D₃×Q₁₆	1	4	yes	complemented
255	C₃×C₄	C₂×C₃×Q₁₆	1	4	yes	complemented
256	C₃×C₄	C₂×D₃×D₄	1	2	yes	—
257	C₃×C₄	C₂²×C₃×D₄	1	2	yes	—
258	Dic₃	—	4	4	no	—
259	Dic₃	C₃×#127	1	2	yes	—
260	Dic₃	C₂×C₃×Q₁₆	1	4	yes	complemented
261	C₃²	C₂×#494	1	1	yes	characteristic, complemented, Sylow
262	C₂³	—	3	1	no	—
263	Q₈	—	2	8	no	—
264	Q₈	—	6	8	no	—
265	Q₈	—	12	8	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	3	2	no	—
268	C₂×C₄	—	3	1	no	—
269	C₂×C₄	—	3	8	no	—
270	C₂×C₄	—	3	4	no	—
271	C₂×C₄	—	6	4	no	—
272	C₂×C₄	—	12	8	no	—
273	C₂×C₄	C₃×D₃×D₄	1	2	yes	—
274	C₂×C₄	C₂³×C₃×D₃	1	1	yes	characteristic
275	C₈	—	3	4	no	—
276	C₈	C₃×D₃×D₄	1	4	yes	—
277	C₂×C₃	—	2	2	no	—
278	C₂×C₃	—	2	1	no	—
279	C₂×C₃	—	3	4	no	—
280	C₂×C₃	D₃×#127	1	1	yes	characteristic
281	C₂×C₃	C₂×D₃×Q₁₆	1	2	yes	—
282	C₂×C₃	C₂×C₃×#127	1	1	yes	characteristic
283	C₂×C₃	C₂²×C₃×Q₁₆	1	2	yes	—
284	D₃	C₃×#494	1	4	yes	direct product factor
285	C₂²	—	3	4	no	—
286	C₂²	—	3	2	no	—
287	C₂²	C₂×C₃×D₃×D₄	1	1	yes	characteristic
288	C₄	—	3	4	no	—
289	C₄	—	3	2	no	—
290	C₄	—	4	4	no	—
291	C₄	—	12	4	no	—
292	C₄	C₃×D₃×Q₁₆	1	4	yes	complemented
293	C₄	C₂×C₃×D₃×D₄	1	2	yes	—
294	C₃	—	2	1	no	—
295	C₃	D₃×#494	1	1	yes	characteristic, direct product factor
296	C₃	C₂×C₃×#494	1	1	yes	characteristic, complemented
297	C₂	—	3	4	no	—
298	C₂	C₃×D₃×#127	1	1	yes	characteristic
299	C₂	C₂×C₃×D₃×Q₁₆	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃	C₂×D₃×D₄
commutator, cosocle	C₂×C₃×C₄	C₂⁴×C₃
Frattini	C₂×C₄	C₂³×C₃×D₃
Fitting	C₃²×#494	C₂
socle	C₂²×C₃²	C₂²×D₄
prime radical	C₂²×C₃×D₃	C₂×D₄
hypercenter	C₃×#494	D₃
hypocenter	C₃	C₂×C₃×#494
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×#494

Complements

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

N	H ≅ G/N	type
3. C₃²×#494	C₂	semidirect
4. C₃×#5183	C₂	semidirect
5. C₃×#5092	C₂	semidirect
6. C₃×#4699	C₂	semidirect
9. D₃×#494	C₃	direct (N × H)
33. C₃×#494	D₃	direct (N × H)
34. C₃×#494	C₂×C₃	semidirect
36. #5183	C₂×C₃	semidirect
37. #5092	C₂×C₃	semidirect
38. #4699	C₂×C₃	semidirect
102. C₃×C₄×D₃	Q₁₆	semidirect
113. #494	C₃×D₃	direct (N × H)
162. C₃²×C₄	C₂×Q₁₆	semidirect
166. C₃×Dic₃	C₂×Q₁₆	semidirect
208. C₄×D₃	C₃×Q₁₆	semidirect
219. C₃×D₃	#494	direct (N × H)
254. C₃×C₄	D₃×Q₁₆	semidirect
255. C₃×C₄	C₂×C₃×Q₁₆	semidirect
260. Dic₃	C₂×C₃×Q₁₆	semidirect
261. C₃²	C₂×#494	semidirect
284. D₃	C₃×#494	direct (N × H)
292. C₄	C₃×D₃×Q₁₆	semidirect
295. C₃	D₃×#494	direct (N × H)
296. C₃	C₂×C₃×#494	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×D₃×Q₁₆	2
C₃×D₃×C₄⋊Q₈	2
C₃²×#494	2
C₃×#5183	2
C₃×#5092	2
C₃×#4699	2
C₃×C₄×D₃×C₈	2
C₂×C₃×#494	3
D₃×#494	3

Subgroups — C3×D3×#494 #11874201

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×D₃×#494 #11874201