Subgroups — C₃×D₃×#498 #11874874

Order 1152.

897 conjugacy classes of subgroups.

The subgroup lattice has 305 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×D₃×C₄⋊Q₈	C₂	1	2	yes	maximal
2	C₃²×#498	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#5080	C₂	1	2	yes	maximal, complemented
4	C₃×D₃×#106	C₂	1	4	yes	maximal
5	C₃×#4855	C₂	1	4	yes	maximal, complemented
6	C₃×C₄×D₃×C₈	C₂	1	1	yes	maximal, characteristic
7	C₃×#4680	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃×#498	—	3	1	no	maximal
9	D₃×#498	C₃	1	1	yes	maximal, characteristic, direct product factor
10	C₂×C₃×D₃×Q₈	—	2	2	no	—
11	C₃²×C₄⋊Q₈	C₂²	1	2	yes	—
12	C₃×#944	—	2	2	no	—
13	C₂×C₃×D₃×C₈	C₂²	1	2	yes	—
14	C₃×D₃×C₄⋊C₄	—	2	2	no	—
15	C₃×D₃×C₄⋊C₄	C₂²	1	4	yes	—
16	C₃×#888	—	2	2	no	—
17	C₃×#888	C₂²	1	4	yes	—
18	C₃×C₄²×D₃	C₂²	1	1	yes	characteristic
19	C₃×#869	C₂²	1	2	yes	—
20	C₃²×#106	C₂²	1	4	yes	—
21	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
22	C₃×#817	C₂²	1	4	yes	—
23	C₃×C₈×Dic₃	C₂²	1	2	yes	—
24	C₃×#808	C₂²	1	8	yes	—
25	C₃×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
26	C₂×C₃×C₄⋊Q₈	—	3	2	no	—
27	D₃×C₄⋊Q₈	C₂×C₃	1	2	yes	—
28	C₃×#498	—	2	1	no	—
29	C₃×#498	—	3	2	no	—
30	C₃×#498	—	3	4	no	—
31	C₃×#498	—	3	1	no	—
32	C₃×#498	D₃	1	1	yes	characteristic, direct product factor
33	C₃×#498	C₂×C₃	1	1	yes	characteristic, complemented
34	C₂×C₃×#106	—	3	4	no	—
35	C₂×C₃×C₄×C₈	—	3	1	no	—
36	#5080	C₂×C₃	1	2	yes	complemented
37	D₃×#106	C₂×C₃	1	4	yes	—
38	#4855	C₂×C₃	1	4	yes	complemented
39	C₄×D₃×C₈	C₂×C₃	1	1	yes	characteristic
40	#4680	C₂×C₃	1	1	yes	characteristic, complemented
41	C₂×C₃²×Q₈	—	2	2	no	—
42	C₃×D₃×Q₈	—	4	8	no	—
43	C₂×C₃×C₄×D₃	—	2	4	no	—
44	C₂×C₃×C₄×D₃	D₄	1	2	yes	—
45	C₂×C₃×C₄×D₃	C₂³	1	1	yes	characteristic
46	C₂×C₃×Dic₆	—	2	2	no	—
47	C₂×C₃×Dic₆	—	2	4	no	—
48	C₂×C₃²×C₈	C₂³	1	2	yes	—
49	C₃²×C₄⋊C₄	—	2	2	no	—
50	C₃²×C₄⋊C₄	C₂³	1	4	yes	—
51	C₃²×C₄²	C₂³	1	1	yes	characteristic
52	C₃×C₄⋊Dic₃	—	2	2	no	—
53	C₃×C₄⋊Dic₃	C₂³	1	4	yes	—
54	C₃×Dic₃⋊C₄	—	2	4	no	—
55	C₃×Dic₃⋊C₄	—	2	4	no	—
56	C₃×C₄×Dic₃	D₄	1	2	yes	—
57	C₃×C₄×Dic₃	C₂³	1	1	yes	characteristic
58	C₂×C₃×C₃⋊C₈	C₂³	1	2	yes	—
59	C₃×D₃×C₈	Q₈	1	8	yes	complemented
60	C₂×#498	—	3	1	no	Sylow
61	C₂²×C₃×Q₈	—	6	2	no	—
62	C₂×D₃×Q₈	—	2	2	no	—
63	C₂²×C₃×C₈	—	3	2	no	—
64	C₃×C₄⋊Q₈	—	2	2	no	—
65	C₃×C₄⋊Q₈	—	3	4	no	—
66	C₃×C₄⋊Q₈	—	3	2	no	—
67	C₃×C₄⋊Q₈	—	6	2	no	—
68	C₃×C₄⋊Q₈	—	6	2	no	—
69	C₃×C₄⋊Q₈	C₂×D₃	1	2	yes	—
70	C₃×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
71	C₂×C₃×C₄⋊C₄	—	3	4	no	—
72	C₂×C₃×C₄⋊C₄	—	6	2	no	—
73	C₂×C₃×C₄²	—	3	1	no	—
74	#944	—	2	2	no	—
75	C₂×D₃×C₈	C₂²×C₃	1	2	yes	—
76	D₃×C₄⋊C₄	—	2	2	no	—
77	D₃×C₄⋊C₄	C₂²×C₃	1	4	yes	—
78	#888	—	2	2	no	—
79	#888	C₂²×C₃	1	4	yes	—
80	C₄²×D₃	C₂²×C₃	1	1	yes	characteristic
81	#869	C₂²×C₃	1	2	yes	—
82	C₃×#106	—	2	4	no	—
83	C₃×#106	—	3	4	no	—
84	C₃×#106	—	3	8	no	—
85	C₃×#106	C₂×D₃	1	4	yes	—
86	C₃×#106	C₂²×C₃	1	4	yes	—
87	C₃×C₄×C₈	—	2	1	no	—
88	C₃×C₄×C₈	—	3	2	no	—
89	C₃×C₄×C₈	—	3	1	no	—
90	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
91	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
92	#817	C₂²×C₃	1	4	yes	—
93	C₈×Dic₃	C₂²×C₃	1	2	yes	—
94	#808	C₂²×C₃	1	8	yes	—
95	C₄×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
96	C₂²×C₃×D₃	C₂×D₄	1	1	yes	characteristic
97	C₃²×Q₈	—	4	4	no	—
98	C₂×C₃²×C₄	—	2	4	no	—
99	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
100	C₂×C₃²×C₄	C₂⁴	1	1	yes	characteristic
101	C₂×C₃×Dic₃	—	2	4	no	—
102	C₂×C₃×Dic₃	C₂×D₄	1	1	yes	characteristic
103	C₂×C₃×Dic₃	C₂×D₄	1	2	yes	—
104	C₃×C₄×D₃	—	4	8	no	—
105	C₃×C₄×D₃	SD₁₆	1	8	yes	—
106	C₃×C₄×D₃	C₂×Q₈	1	4	yes	—
107	C₃×Dic₆	—	4	4	no	—
108	C₃×Dic₆	—	4	8	no	—
109	C₃²×C₈	C₂×Q₈	1	4	yes	complemented
110	C₃×C₃⋊C₈	C₂×Q₈	1	4	yes	complemented
111	C₂×C₄⋊Q₈	—	3	2	no	—
112	#498	—	3	2	no	—
113	#498	—	3	4	no	—
114	#498	—	3	1	no	—
115	#498	C₃×D₃	1	1	yes	characteristic, direct product factor
116	C₂×#106	—	3	4	no	—
117	C₂×C₄×C₈	—	3	1	no	—
118	C₂×C₃×Q₈	—	2	2	no	—
119	C₂×C₃×Q₈	—	2	2	no	—
120	C₂×C₃×Q₈	—	4	2	no	—
121	C₂×C₃×Q₈	—	6	4	no	—
122	C₂×C₃×Q₈	—	6	2	no	—
123	C₂×C₃×Q₈	—	12	8	no	—
124	C₂²×C₃×C₄	—	3	2	no	—
125	C₂²×C₃×C₄	—	3	1	no	—
126	C₂²×C₃×C₄	—	6	4	no	—
127	D₃×Q₈	—	4	8	no	—
128	C₂×C₄×D₃	—	2	4	no	—
129	C₂×C₄×D₃	C₃×D₄	1	2	yes	—
130	C₂×C₄×D₃	C₂³×C₃	1	1	yes	characteristic
131	C₂×Dic₆	—	2	2	no	—
132	C₂×Dic₆	—	2	4	no	—
133	C₂×C₃×C₈	—	2	2	no	—
134	C₂×C₃×C₈	—	3	2	no	—
135	C₂×C₃×C₈	—	3	8	no	—
136	C₂×C₃×C₈	C₂²×D₃	1	2	yes	—
137	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
138	C₃×C₄⋊C₄	—	2	2	no	—
139	C₃×C₄⋊C₄	—	2	2	no	—
140	C₃×C₄⋊C₄	—	2	4	no	—
141	C₃×C₄⋊C₄	—	3	4	no	—
142	C₃×C₄⋊C₄	—	4	2	no	—
143	C₃×C₄⋊C₄	—	6	4	no	—
144	C₃×C₄⋊C₄	—	6	4	no	—
145	C₃×C₄⋊C₄	—	6	2	no	—
146	C₃×C₄⋊C₄	C₂²×D₃	1	4	yes	—
147	C₃×C₄⋊C₄	C₂³×C₃	1	4	yes	—
148	C₃×C₄²	—	2	1	no	—
149	C₃×C₄²	—	3	2	no	—
150	C₃×C₄²	—	3	1	no	—
151	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
152	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
153	C₄⋊Dic₃	—	2	2	no	—
154	C₄⋊Dic₃	C₂³×C₃	1	4	yes	—
155	Dic₃⋊C₄	—	2	4	no	—
156	Dic₃⋊C₄	—	2	4	no	—
157	C₄×Dic₃	C₃×D₄	1	2	yes	—
158	C₄×Dic₃	C₂³×C₃	1	1	yes	characteristic
159	C₂×C₃⋊C₈	C₂³×C₃	1	2	yes	—
160	D₃×C₈	C₃×Q₈	1	8	yes	complemented
161	C₂²×C₃²	C₂²×D₄	1	1	yes	characteristic
162	C₂×C₃×D₃	C₄⋊Q₈	1	2	yes	—
163	C₂×C₃×D₃	C₂×SD₁₆	1	4	yes	—
164	C₃²×C₄	—	4	4	no	—
165	C₃²×C₄	C₂×SD₁₆	1	4	yes	—
166	C₃²×C₄	C₂²×Q₈	1	2	yes	—
167	C₃×Dic₃	—	4	4	no	—
168	C₃×Dic₃	C₄⋊Q₈	1	2	yes	—
169	C₃×Dic₃	C₂×SD₁₆	1	4	yes	—
170	C₂²×Q₈	—	6	2	no	—
171	C₂²×C₈	—	3	2	no	—
172	C₄⋊Q₈	—	3	4	no	—
173	C₄⋊Q₈	—	3	2	no	—
174	C₄⋊Q₈	—	6	2	no	—
175	C₄⋊Q₈	—	6	2	no	—
176	C₄⋊Q₈	C₂×C₃×D₃	1	2	yes	—
177	C₂×C₄⋊C₄	—	3	4	no	—
178	C₂×C₄⋊C₄	—	6	2	no	—
179	C₂×C₄²	—	3	1	no	—
180	#106	—	3	4	no	—
181	#106	—	3	8	no	—
182	#106	C₂×C₃×D₃	1	4	yes	—
183	C₄×C₈	—	3	2	no	—
184	C₄×C₈	—	3	1	no	—
185	C₄×C₈	C₂×C₃×D₃	1	1	yes	characteristic
186	C₂³×C₃	—	3	1	no	—
187	C₂²×D₃	C₂×C₃×D₄	1	1	yes	characteristic
188	C₃×Q₈	—	4	4	no	—
189	C₃×Q₈	—	4	4	no	—
190	C₃×Q₈	—	8	4	no	—
191	C₃×Q₈	—	12	8	no	—
192	C₃×Q₈	—	12	4	no	—
193	C₂×C₃×C₄	—	2	4	no	—
194	C₂×C₃×C₄	—	2	4	no	—
195	C₂×C₃×C₄	—	2	1	no	—
196	C₂×C₃×C₄	—	2	2	no	—
197	C₂×C₃×C₄	—	3	1	no	—
198	C₂×C₃×C₄	—	3	2	no	—
199	C₂×C₃×C₄	—	3	8	no	—
200	C₂×C₃×C₄	—	3	4	no	—
201	C₂×C₃×C₄	—	4	4	no	—
202	C₂×C₃×C₄	—	6	4	no	—
203	C₂×C₃×C₄	—	12	8	no	—
204	C₂×C₃×C₄	D₃×D₄	1	2	yes	—
205	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
206	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
207	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
208	C₂×Dic₃	—	2	4	no	—
209	C₂×Dic₃	C₂×C₃×D₄	1	1	yes	characteristic
210	C₂×Dic₃	C₂×C₃×D₄	1	2	yes	—
211	C₄×D₃	—	4	8	no	—
212	C₄×D₃	C₃×SD₁₆	1	8	yes	—
213	C₄×D₃	C₂×C₃×Q₈	1	4	yes	—
214	Dic₆	—	4	4	no	—
215	Dic₆	—	4	8	no	—
216	C₃×C₈	—	2	4	no	—
217	C₃×C₈	—	3	4	no	—
218	C₃×C₈	D₃×Q₈	1	4	yes	complemented
219	C₃×C₈	C₂×C₃×Q₈	1	4	yes	complemented
220	C₃⋊C₈	C₂×C₃×Q₈	1	4	yes	complemented
221	C₂×C₃²	C₂×C₄⋊Q₈	1	1	yes	characteristic
222	C₂×C₃²	C₂²×SD₁₆	1	2	yes	—
223	C₃×D₃	#498	1	4	yes	direct product factor
224	C₂×Q₈	—	2	2	no	—
225	C₂×Q₈	—	6	4	no	—
226	C₂×Q₈	—	6	2	no	—
227	C₂×Q₈	—	12	8	no	—
228	C₂²×C₄	—	3	1	no	—
229	C₂²×C₄	—	3	2	no	—
230	C₂²×C₄	—	6	4	no	—
231	C₂×C₈	—	3	2	no	—
232	C₂×C₈	—	3	8	no	—
233	C₂×C₈	C₂²×C₃×D₃	1	2	yes	—
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	3	4	no	—
236	C₄⋊C₄	—	6	4	no	—
237	C₄⋊C₄	—	6	4	no	—
238	C₄⋊C₄	—	6	2	no	—
239	C₄⋊C₄	C₂²×C₃×D₃	1	4	yes	—
240	C₄²	—	3	1	no	—
241	C₄²	—	3	2	no	—
242	C₄²	C₂²×C₃×D₃	1	1	yes	characteristic
243	C₂²×C₃	—	2	1	no	—
244	C₂²×C₃	—	3	2	no	—
245	C₂²×C₃	—	3	4	no	—
246	C₂²×C₃	C₂×D₃×D₄	1	1	yes	characteristic
247	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
248	C₂×D₃	C₃×C₄⋊Q₈	1	2	yes	—
249	C₂×D₃	C₂×C₃×SD₁₆	1	4	yes	—
250	C₃×C₄	—	2	4	no	—
251	C₃×C₄	—	2	2	no	—
252	C₃×C₄	—	3	2	no	—
253	C₃×C₄	—	3	4	no	—
254	C₃×C₄	—	4	4	no	—
255	C₃×C₄	—	4	4	no	—
256	C₃×C₄	—	8	4	no	—
257	C₃×C₄	—	12	4	no	—
258	C₃×C₄	D₃×SD₁₆	1	4	yes	—
259	C₃×C₄	C₂×C₃×SD₁₆	1	4	yes	—
260	C₃×C₄	C₂×D₃×Q₈	1	2	yes	—
261	C₃×C₄	C₂²×C₃×Q₈	1	2	yes	—
262	Dic₃	—	4	4	no	—
263	Dic₃	C₃×C₄⋊Q₈	1	2	yes	—
264	Dic₃	C₂×C₃×SD₁₆	1	4	yes	—
265	C₃²	C₂×#498	1	1	yes	characteristic, complemented, Sylow
266	C₂³	—	3	1	no	—
267	Q₈	—	4	4	no	—
268	Q₈	—	12	8	no	—
269	Q₈	—	12	4	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	3	1	no	—
272	C₂×C₄	—	3	2	no	—
273	C₂×C₄	—	3	8	no	—
274	C₂×C₄	—	3	4	no	—
275	C₂×C₄	—	6	4	no	—
276	C₂×C₄	—	12	8	no	—
277	C₂×C₄	C₃×D₃×D₄	1	2	yes	—
278	C₂×C₄	C₂³×C₃×D₃	1	1	yes	characteristic
279	C₈	—	3	4	no	—
280	C₈	C₃×D₃×Q₈	1	4	yes	complemented
281	C₂×C₃	—	2	2	no	—
282	C₂×C₃	—	2	1	no	—
283	C₂×C₃	—	3	4	no	—
284	C₂×C₃	D₃×C₄⋊Q₈	1	1	yes	characteristic
285	C₂×C₃	C₂×D₃×SD₁₆	1	2	yes	—
286	C₂×C₃	C₂×C₃×C₄⋊Q₈	1	1	yes	characteristic
287	C₂×C₃	C₂²×C₃×SD₁₆	1	2	yes	—
288	D₃	C₃×#498	1	4	yes	direct product factor
289	C₂²	—	3	4	no	—
290	C₂²	—	3	2	no	—
291	C₂²	C₂×C₃×D₃×D₄	1	1	yes	characteristic
292	C₄	—	3	4	no	—
293	C₄	—	3	2	no	—
294	C₄	—	4	4	no	—
295	C₄	—	12	4	no	—
296	C₄	C₃×D₃×SD₁₆	1	4	yes	—
297	C₄	C₂×C₃×D₃×Q₈	1	2	yes	—
298	C₃	—	2	1	no	—
299	C₃	D₃×#498	1	1	yes	characteristic, direct product factor
300	C₃	C₂×C₃×#498	1	1	yes	characteristic, complemented
301	C₂	—	3	4	no	—
302	C₂	C₃×D₃×C₄⋊Q₈	1	1	yes	characteristic
303	C₂	C₂×C₃×D₃×SD₁₆	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×#498	C₂
socle	C₂²×C₃²	C₂²×D₄
prime radical	C₂²×C₃×D₃	C₂×D₄
hypercenter	C₃×#498	D₃
hypocenter	C₃	C₂×C₃×#498
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×#498

Complements

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

N	H ≅ G/N	type
2. C₃²×#498	C₂	semidirect
3. C₃×#5080	C₂	semidirect
5. C₃×#4855	C₂	semidirect
7. C₃×#4680	C₂	semidirect
9. D₃×#498	C₃	direct (N × H)
32. C₃×#498	D₃	direct (N × H)
33. C₃×#498	C₂×C₃	semidirect
36. #5080	C₂×C₃	semidirect
38. #4855	C₂×C₃	semidirect
40. #4680	C₂×C₃	semidirect
59. C₃×D₃×C₈	Q₈	semidirect
109. C₃²×C₈	C₂×Q₈	semidirect
110. C₃×C₃⋊C₈	C₂×Q₈	semidirect
115. #498	C₃×D₃	direct (N × H)
160. D₃×C₈	C₃×Q₈	semidirect
218. C₃×C₈	D₃×Q₈	semidirect
219. C₃×C₈	C₂×C₃×Q₈	semidirect
220. C₃⋊C₈	C₂×C₃×Q₈	semidirect
223. C₃×D₃	#498	direct (N × H)
265. C₃²	C₂×#498	semidirect
280. C₈	C₃×D₃×Q₈	semidirect
288. D₃	C₃×#498	direct (N × H)
299. C₃	D₃×#498	direct (N × H)
300. C₃	C₂×C₃×#498	semidirect

Maximal subgroups

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

subgroup	index
C₃×D₃×C₄⋊Q₈	2
C₃²×#498	2
C₃×#5080	2
C₃×D₃×#106	2
C₃×#4855	2
C₃×C₄×D₃×C₈	2
C₃×#4680	2
C₂×C₃×#498	3
D₃×#498	3

Subgroups — C3×D3×#498 #11874874

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×D₃×#498 #11874874