Subgroups — C₃×Dic₃×#106 #11784438

Order 1152.

625 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×#106	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#817	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₈×Dic₃	C₂	1	1	yes	maximal, characteristic
4	C₃×Dic₃×C₄⋊C₄	C₂	1	2	yes	maximal
5	C₃×#4529	C₂	1	2	yes	maximal
6	C₃×C₄×#106	—	3	1	no	maximal
7	Dic₃×#106	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₃²×C₄⋊C₄	C₂²	1	2	yes	—
10	C₂×C₃×C₄⋊Dic₃	C₂²	1	2	yes	—
11	C₂×C₃×C₄×Dic₃	—	2	2	no	—
12	C₂×C₃×C₄×Dic₃	C₂²	1	1	yes	characteristic
13	C₂²×C₃×C₃⋊C₈	C₂²	1	1	yes	characteristic
14	C₃²×#106	C₄	1	4	yes	complemented
15	C₃×C₃⋊#95	—	2	2	no	—
16	C₃×#817	C₄	1	4	yes	complemented
17	C₃×C₈×Dic₃	C₄	1	4	yes	complemented
18	C₂×C₃×#106	—	2	1	no	—
19	C₂×C₃×#106	—	3	1	no	—
20	C₂×C₃×#106	D₃	1	1	yes	characteristic
21	C₂×C₃×#106	C₂×C₃	1	1	yes	characteristic
22	C₂×C₃×C₄×C₈	—	3	1	no	—
23	C₃×C₄×C₄⋊C₄	—	3	2	no	—
24	C₂×#817	C₂×C₃	1	1	yes	characteristic
25	C₂×C₈×Dic₃	C₂×C₃	1	1	yes	characteristic
26	Dic₃×C₄⋊C₄	C₂×C₃	1	2	yes	—
27	C₃×#340	—	3	2	no	—
28	#4529	C₂×C₃	1	2	yes	—
29	C₂²×C₃²×C₄	—	2	2	no	—
30	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
31	C₂²×C₃×Dic₃	—	2	2	no	—
32	C₂²×C₃×Dic₃	D₄	1	1	yes	characteristic
33	C₂²×C₃×Dic₃	D₄	1	1	yes	characteristic
34	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
35	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
36	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
37	C₃²×C₄⋊C₄	—	2	2	no	—
38	C₃²×C₄⋊C₄	C₂×C₄	1	4	yes	—
39	C₃×C₄⋊Dic₃	—	2	2	no	—
40	C₃×C₄⋊Dic₃	C₂×C₄	1	4	yes	—
41	C₃×C₄×Dic₃	—	4	4	no	—
42	C₃×C₄×Dic₃	C₂×C₄	1	2	yes	—
43	C₃×C₄×Dic₃	Q₈	1	2	yes	—
44	C₂×C₃×C₃⋊C₈	—	2	1	no	—
45	C₂×C₃×C₃⋊C₈	—	2	1	no	—
46	C₂×C₃×C₃⋊C₈	C₂×C₄	1	2	yes	—
47	C₄×#106	—	3	1	no	Sylow
48	C₂²×C₃×C₈	—	2	1	no	—
49	C₂²×C₃×C₈	—	3	1	no	—
50	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
51	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
52	C₂×C₃×C₄⋊C₄	—	2	2	no	—
53	C₂×C₃×C₄⋊C₄	—	3	2	no	—
54	C₂×C₃×C₄⋊C₄	C₂×D₃	1	2	yes	—
55	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
56	C₂×C₃×C₄²	—	3	1	no	—
57	C₂×C₃×C₄²	—	6	2	no	—
58	C₂×C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
59	C₂×C₄×Dic₃	—	2	2	no	—
60	C₂×C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
61	C₂²×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
62	C₃×#106	—	2	4	no	—
63	C₃×#106	—	3	4	no	—
64	C₃×#106	Dic₃	1	4	yes	direct product factor
65	C₃×#106	C₃×C₄	1	4	yes	complemented
66	C₃×C₄×C₈	—	3	4	no	—
67	C₃×#95	—	6	2	no	—
68	C₃⋊#95	—	2	2	no	—
69	#817	C₃×C₄	1	4	yes	complemented
70	C₈×Dic₃	C₃×C₄	1	4	yes	complemented
71	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
72	C₂×C₃²×C₄	—	2	4	no	—
73	C₂×C₃²×C₄	—	4	2	no	—
74	C₂×C₃²×C₄	—	4	2	no	—
75	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
79	C₂×C₃²×C₄	#41	1	1	yes	characteristic
80	C₂×C₃²×C₄	#41	1	1	yes	characteristic
81	C₂×C₃×Dic₃	—	2	2	no	—
82	C₂×C₃×Dic₃	—	2	4	no	—
83	C₂×C₃×Dic₃	—	4	2	no	—
84	C₂×C₃×Dic₃	—	4	2	no	—
85	C₂×C₃×Dic₃	C₄⋊C₄	1	2	yes	—
86	C₂×C₃×Dic₃	C₄⋊C₄	1	2	yes	—
87	C₂×C₃×Dic₃	SD₁₆	1	4	yes	—
88	C₃²×C₈	C₄²	1	4	yes	complemented
89	C₃×C₃⋊C₈	—	2	2	no	—
90	C₂×#106	—	3	1	no	—
91	C₂×#106	C₃×D₃	1	1	yes	characteristic
92	C₂×C₄×C₈	—	3	1	no	—
93	C₄×C₄⋊C₄	—	3	2	no	—
94	#340	—	3	2	no	—
95	C₂²×C₃×C₄	—	2	2	no	—
96	C₂²×C₃×C₄	—	2	2	no	—
97	C₂²×C₃×C₄	—	2	1	no	—
98	C₂²×C₃×C₄	—	3	1	no	—
99	C₂²×C₃×C₄	—	3	1	no	—
100	C₂²×C₃×C₄	—	4	2	no	—
101	C₂²×C₃×C₄	—	6	2	no	—
102	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
103	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
104	C₂²×Dic₃	—	2	2	no	—
105	C₂²×Dic₃	C₃×D₄	1	1	yes	characteristic
106	C₂²×Dic₃	C₃×D₄	1	1	yes	characteristic
107	C₂×C₃×C₈	—	2	2	no	—
108	C₂×C₃×C₈	—	2	2	no	—
109	C₂×C₃×C₈	—	2	2	no	—
110	C₂×C₃×C₈	—	3	2	no	—
111	C₂×C₃×C₈	—	6	1	no	—
112	C₂×C₃×C₈	—	6	1	no	—
113	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
114	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
115	C₂×C₃×C₈	C₂×Dic₃	1	2	yes	—
116	C₂×C₃×C₈	C₂×C₃×C₄	1	2	yes	—
117	C₂×C₃×C₈	C₂×C₃×C₄	1	2	yes	—
118	C₂×C₃×C₈	C₂×C₃×C₄	1	2	yes	—
119	C₃×C₄⋊C₄	—	2	2	no	—
120	C₃×C₄⋊C₄	—	2	2	no	—
121	C₃×C₄⋊C₄	—	2	4	no	—
122	C₃×C₄⋊C₄	—	3	4	no	—
123	C₃×C₄⋊C₄	—	4	2	no	—
124	C₃×C₄⋊C₄	—	6	2	no	—
125	C₃×C₄⋊C₄	C₂×Dic₃	1	4	yes	—
126	C₃×C₄⋊C₄	C₂×C₃×C₄	1	4	yes	—
127	C₃×C₄²	—	3	2	no	—
128	C₃×C₄²	—	3	2	no	—
129	C₃×C₄²	—	12	4	no	—
130	C₄⋊Dic₃	—	2	2	no	—
131	C₄⋊Dic₃	C₂×C₃×C₄	1	4	yes	—
132	C₄×Dic₃	—	4	4	no	—
133	C₄×Dic₃	C₂×C₃×C₄	1	2	yes	—
134	C₄×Dic₃	C₃×Q₈	1	2	yes	—
135	C₂×C₃⋊C₈	—	2	1	no	—
136	C₂×C₃⋊C₈	—	2	1	no	—
137	C₂×C₃⋊C₈	C₂×C₃×C₄	1	2	yes	—
138	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
139	C₂²×C₃²	C₄×D₄	1	1	yes	characteristic
140	C₂²×C₃²	C₄×D₄	1	1	yes	characteristic
141	C₂²×C₃²	C₂×SD₁₆	1	2	yes	—
142	C₂²×C₃²	#135	1	2	yes	—
143	C₃²×C₄	—	4	4	no	—
144	C₃²×C₄	C₂×C₄²	1	1	yes	characteristic
145	C₃²×C₄	#117	1	1	yes	characteristic
146	C₃²×C₄	C₄×Q₈	1	1	yes	characteristic
147	C₃²×C₄	C₄×Q₈	1	1	yes	characteristic
148	C₃×Dic₃	—	2	2	no	—
149	C₃×Dic₃	—	4	4	no	—
150	C₃×Dic₃	#106	1	4	yes	direct product factor
151	C₂²×C₈	—	3	1	no	—
152	C₂²×C₈	C₂×C₃×D₃	1	1	yes	characteristic
153	C₂×C₄⋊C₄	—	3	2	no	—
154	C₂×C₄⋊C₄	C₂×C₃×D₃	1	2	yes	—
155	C₂×C₄²	—	3	1	no	—
156	C₂×C₄²	—	6	2	no	—
157	#106	—	3	4	no	—
158	#106	C₃×Dic₃	1	4	yes	direct product factor
159	C₄×C₈	—	3	4	no	—
160	#95	—	6	2	no	—
161	C₂³×C₃	—	2	1	no	—
162	C₂³×C₃	D₃×D₄	1	1	yes	characteristic
163	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
164	C₂×C₃×C₄	—	2	4	no	—
165	C₂×C₃×C₄	—	2	4	no	—
166	C₂×C₃×C₄	—	2	1	no	—
167	C₂×C₃×C₄	—	2	1	no	—
168	C₂×C₃×C₄	—	2	1	no	—
169	C₂×C₃×C₄	—	2	1	no	—
170	C₂×C₃×C₄	—	2	1	no	—
171	C₂×C₃×C₄	—	2	1	no	—
172	C₂×C₃×C₄	—	3	2	no	—
173	C₂×C₃×C₄	—	3	4	no	—
174	C₂×C₃×C₄	—	3	2	no	—
175	C₂×C₃×C₄	—	4	4	no	—
176	C₂×C₃×C₄	—	4	2	no	—
177	C₂×C₃×C₄	—	4	2	no	—
178	C₂×C₃×C₄	—	4	2	no	—
179	C₂×C₃×C₄	—	4	2	no	—
180	C₂×C₃×C₄	—	6	2	no	—
181	C₂×C₃×C₄	—	6	4	no	—
182	C₂×C₃×C₄	—	8	2	no	—
183	C₂×C₃×C₄	—	8	2	no	—
184	C₂×C₃×C₄	—	12	2	no	—
185	C₂×C₃×C₄	—	12	2	no	—
186	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
187	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
188	C₂×C₃×C₄	#237	1	1	yes	characteristic
189	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
190	C₂×C₃×C₄	#239	1	1	yes	characteristic
191	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
192	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
193	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
194	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
195	C₂×C₃×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
196	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
197	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
198	C₂×Dic₃	—	2	2	no	—
199	C₂×Dic₃	—	2	4	no	—
200	C₂×Dic₃	—	4	2	no	—
201	C₂×Dic₃	—	4	2	no	—
202	C₂×Dic₃	C₃×C₄⋊C₄	1	2	yes	—
203	C₂×Dic₃	C₃×C₄⋊C₄	1	2	yes	—
204	C₂×Dic₃	C₃×SD₁₆	1	4	yes	—
205	C₃×C₈	—	2	4	no	—
206	C₃×C₈	—	6	2	no	—
207	C₃×C₈	C₄×Dic₃	1	4	yes	complemented
208	C₃×C₈	C₃×C₄²	1	4	yes	complemented
209	C₃⋊C₈	—	2	2	no	—
210	C₂×C₃²	C₄×C₄⋊C₄	1	1	yes	characteristic
211	C₂×C₃²	C₂×#106	1	1	yes	characteristic
212	C₂×C₃²	#427	1	1	yes	characteristic
213	C₂×C₃²	C₄×SD₁₆	1	2	yes	—
214	C₂×C₃²	C₄×SD₁₆	1	2	yes	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	3	1	no	—
217	C₂²×C₄	—	3	1	no	—
218	C₂²×C₄	—	6	2	no	—
219	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
220	C₂×C₈	—	3	2	no	—
221	C₂×C₈	—	6	1	no	—
222	C₂×C₈	—	6	1	no	—
223	C₂×C₈	C₃×C₄×D₃	1	2	yes	—
224	C₂×C₈	C₃×C₄×D₃	1	2	yes	—
225	C₂×C₈	C₂×C₃×Dic₃	1	2	yes	—
226	C₄⋊C₄	—	2	2	no	—
227	C₄⋊C₄	—	3	4	no	—
228	C₄⋊C₄	—	6	2	no	—
229	C₄⋊C₄	C₂×C₃×Dic₃	1	4	yes	—
230	C₄²	—	3	2	no	—
231	C₄²	—	3	2	no	—
232	C₄²	—	12	4	no	—
233	C₂²×C₃	—	2	1	no	—
234	C₂²×C₃	—	2	1	no	—
235	C₂²×C₃	—	2	2	no	—
236	C₂²×C₃	—	2	2	no	—
237	C₂²×C₃	—	2	1	no	—
238	C₂²×C₃	D₃×C₄⋊C₄	1	1	yes	characteristic
239	C₂²×C₃	#893	1	1	yes	characteristic
240	C₂²×C₃	D₃×SD₁₆	1	2	yes	—
241	C₂²×C₃	#916	1	2	yes	—
242	C₂²×C₃	D₄×Dic₃	1	1	yes	characteristic
243	C₂²×C₃	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
244	C₂²×C₃	C₃×C₄×D₄	1	1	yes	characteristic
245	C₂²×C₃	C₃×C₄×D₄	1	1	yes	characteristic
246	C₂²×C₃	C₂×C₃×SD₁₆	1	2	yes	—
247	C₂²×C₃	C₃×#135	1	2	yes	—
248	C₃×C₄	—	2	1	no	—
249	C₃×C₄	—	2	1	no	—
250	C₃×C₄	—	2	1	no	—
251	C₃×C₄	—	2	1	no	—
252	C₃×C₄	—	3	4	no	—
253	C₃×C₄	—	4	4	no	—
254	C₃×C₄	—	4	4	no	—
255	C₃×C₄	—	6	2	no	—
256	C₃×C₄	—	8	4	no	—
257	C₃×C₄	—	12	4	no	—
258	C₃×C₄	#887	1	1	yes	characteristic
259	C₃×C₄	#892	1	1	yes	characteristic
260	C₃×C₄	C₂×C₄×Dic₃	1	1	yes	characteristic
261	C₃×C₄	Q₈×Dic₃	1	1	yes	characteristic
262	C₃×C₄	C₂×C₃×C₄²	1	1	yes	characteristic
263	C₃×C₄	C₃×#117	1	1	yes	characteristic
264	C₃×C₄	C₃×C₄×Q₈	1	1	yes	characteristic
265	C₃×C₄	C₃×C₄×Q₈	1	1	yes	characteristic
266	Dic₃	—	2	2	no	—
267	Dic₃	—	4	4	no	—
268	Dic₃	C₃×#106	1	4	yes	direct product factor
269	C₃²	C₄×#106	1	1	yes	characteristic, complemented, Sylow
270	C₂³	C₃×D₃×D₄	1	1	yes	characteristic
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	3	2	no	—
273	C₂×C₄	—	3	4	no	—
274	C₂×C₄	—	3	2	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	6	2	no	—
278	C₂×C₄	—	6	4	no	—
279	C₂×C₄	—	12	2	no	—
280	C₂×C₄	—	12	2	no	—
281	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
282	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
283	C₂×C₄	C₃×#237	1	1	yes	characteristic
284	C₂×C₄	C₃×D₃×Q₈	1	1	yes	characteristic
285	C₂×C₄	C₃×#239	1	1	yes	characteristic
286	C₂×C₄	C₂²×C₃×Dic₃	1	1	yes	characteristic
287	C₈	—	6	2	no	—
288	C₈	C₃×C₄×Dic₃	1	4	yes	complemented
289	C₂×C₃	—	2	2	no	—
290	C₂×C₃	—	2	1	no	—
291	C₂×C₃	—	2	2	no	—
292	C₂×C₃	—	2	1	no	—
293	C₂×C₃	—	2	1	no	—
294	C₂×C₃	#4852	1	2	yes	—
295	C₂×C₃	D₃×#106	1	1	yes	characteristic
296	C₂×C₃	#4860	1	1	yes	characteristic
297	C₂×C₃	Dic₃×C₄⋊C₄	1	1	yes	characteristic
298	C₂×C₃	Dic₃×SD₁₆	1	2	yes	—
299	C₂×C₃	C₃×C₄×C₄⋊C₄	1	1	yes	characteristic
300	C₂×C₃	C₂×C₃×#106	1	1	yes	characteristic
301	C₂×C₃	C₃×#427	1	1	yes	characteristic
302	C₂×C₃	C₃×C₄×SD₁₆	1	2	yes	—
303	C₂×C₃	C₃×C₄×SD₁₆	1	2	yes	—
304	C₂²	C₃×D₃×C₄⋊C₄	1	1	yes	characteristic
305	C₂²	C₃×#893	1	1	yes	characteristic
306	C₂²	C₃×D₃×SD₁₆	1	2	yes	—
307	C₂²	C₃×#916	1	2	yes	—
308	C₂²	C₃×D₄×Dic₃	1	1	yes	characteristic
309	C₄	—	3	4	no	—
310	C₄	—	4	4	no	—
311	C₄	—	6	2	no	—
312	C₄	—	12	4	no	—
313	C₄	C₃×#887	1	1	yes	characteristic
314	C₄	C₃×#892	1	1	yes	characteristic
315	C₄	C₂×C₃×C₄×Dic₃	1	1	yes	characteristic
316	C₄	C₃×Q₈×Dic₃	1	1	yes	characteristic
317	C₃	—	2	1	no	—
318	C₃	Dic₃×#106	1	1	yes	characteristic, direct product factor
319	C₃	C₃×C₄×#106	1	1	yes	characteristic, complemented
320	C₂	C₃×#4852	1	2	yes	—
321	C₂	C₃×D₃×#106	1	1	yes	characteristic
322	C₂	C₃×#4860	1	1	yes	characteristic
323	C₂	C₃×Dic₃×C₄⋊C₄	1	1	yes	characteristic
324	C₂	C₃×Dic₃×SD₁₆	1	2	yes	—

	subgroup	G/H
center	C₂³×C₃	D₃×D₄
commutator	C₃×C₄	C₂×C₃×C₄²
Frattini	C₂²×C₄	C₂²×C₃×D₃
Fitting	C₂×C₃²×#106	C₂
socle, prime radical	C₂³×C₃²	C₂×D₄
cosocle	C₂²×C₃×C₄	C₂³×C₃
hypercenter	C₂×C₃×#106	D₃
hypocenter	C₃	C₃×C₄×#106
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₃ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×#106

Complements

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

N	H ≅ G/N	type
7. Dic₃×#106	C₃	direct (N × H)
14. C₃²×#106	C₄	semidirect
16. C₃×#817	C₄	semidirect
17. C₃×C₈×Dic₃	C₄	semidirect
64. C₃×#106	Dic₃	direct (N × H)
65. C₃×#106	C₃×C₄	semidirect
69. #817	C₃×C₄	semidirect
70. C₈×Dic₃	C₃×C₄	semidirect
88. C₃²×C₈	C₄²	semidirect
150. C₃×Dic₃	#106	direct (N × H)
158. #106	C₃×Dic₃	direct (N × H)
207. C₃×C₈	C₄×Dic₃	semidirect
208. C₃×C₈	C₃×C₄²	semidirect
268. Dic₃	C₃×#106	direct (N × H)
269. C₃²	C₄×#106	semidirect
288. C₈	C₃×C₄×Dic₃	semidirect
318. C₃	Dic₃×#106	direct (N × H)
319. C₃	C₃×C₄×#106	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×#106	2
C₂×C₃×#817	2
C₂×C₃×C₈×Dic₃	2
C₃×Dic₃×C₄⋊C₄	2
C₃×#4529	2
C₃×C₄×#106	3
Dic₃×#106	3

Subgroups — C3×Dic3×#106 #11784438

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×Dic₃×#106 #11784438