Subgroups — Dic₃×#818 #11784475

Order 1152.

625 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#63645	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#818	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₈×Dic₃	C₂	1	1	yes	maximal, characteristic
4	Dic₃×C₄⋊Dic₃	C₂	1	2	yes	maximal
5	#10589164	C₂	1	2	yes	maximal
6	Dic₃×#107	—	3	1	no	maximal
7	C₄×#818	—	3	1	no	maximal
8	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
10	C₂×C₃×C₄⋊Dic₃	C₂²	1	2	yes	—
11	C₂×C₃×C₄×Dic₃	C₂²	1	1	yes	characteristic
12	C₂²×C₃×C₃⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×Dic₃²	—	2	2	no	—
14	#63645	C₄	1	4	yes	complemented
15	C₃×#818	C₄	1	4	yes	complemented
16	C₃×C₈×Dic₃	C₄	1	4	yes	complemented
17	#63579	—	2	2	no	—
18	C₂×C₃×#107	—	3	1	no	—
19	C₂×C₃×C₄×C₈	—	3	1	no	—
20	C₂×#818	—	3	1	no	—
21	C₂×#818	—	3	1	no	—
22	C₂×#818	—	6	1	no	—
23	C₂×#818	D₃	1	1	yes	characteristic
24	C₂×C₈×Dic₃	D₃	1	1	yes	characteristic
25	Dic₃×C₄⋊C₄	—	3	2	no	—
26	C₄×C₄⋊Dic₃	—	3	2	no	—
27	#4551	—	3	2	no	—
28	#4529	—	3	2	no	—
29	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
30	C₂²×C₃⋊Dic₃	—	2	2	no	—
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	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
38	C₄⋊(C₃⋊Dic₃)	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	C₂×C₄	1	2	yes	—
42	C₃×C₄×Dic₃	Q₈	1	2	yes	—
43	C₂×C₃×C₃⋊C₈	—	2	1	no	—
44	C₂×C₃×C₃⋊C₈	—	2	1	no	—
45	C₂×C₃×C₃⋊C₈	C₂×C₄	1	2	yes	—
46	Dic₃²	—	4	4	no	—
47	C₄×#107	—	9	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	C₂×D₃	1	1	yes	characteristic
52	C₂×C₃×C₄⋊C₄	—	3	2	no	—
53	C₂×C₃×C₄²	—	3	1	no	—
54	C₂×C₄⋊Dic₃	—	3	2	no	—
55	C₂×C₄⋊Dic₃	—	3	2	no	—
56	C₂×C₄⋊Dic₃	—	6	2	no	—
57	C₂×C₄⋊Dic₃	C₂×D₃	1	2	yes	—
58	C₂×C₄×Dic₃	—	6	2	no	—
59	C₂×C₄×Dic₃	—	6	2	no	—
60	C₂×C₄×Dic₃	C₂×D₃	1	1	yes	characteristic
61	C₂²×C₃⋊C₈	C₂×D₃	1	1	yes	characteristic
62	C₃×#107	—	3	4	no	—
63	C₃×C₄×C₈	—	3	4	no	—
64	C₃⋊#95	—	6	2	no	—
65	C₃⋊#95	—	6	2	no	—
66	#818	—	3	4	no	—
67	#818	—	3	4	no	—
68	#818	—	6	4	no	—
69	#818	Dic₃	1	4	yes	direct product factor
70	C₈×Dic₃	Dic₃	1	4	yes	complemented
71	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
72	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
76	C₂×C₃²×C₄	#41	1	1	yes	characteristic
77	C₂×C₃²×C₄	#41	1	1	yes	characteristic
78	C₂×C₃⋊Dic₃	—	2	4	no	—
79	C₂×C₃⋊Dic₃	—	4	2	no	—
80	C₂×C₃⋊Dic₃	—	4	2	no	—
81	C₂×C₃×Dic₃	—	2	1	no	—
82	C₂×C₃×Dic₃	—	2	1	no	—
83	C₂×C₃×Dic₃	—	2	4	no	—
84	C₂×C₃×Dic₃	—	4	2	no	—
85	C₂×C₃×Dic₃	—	4	2	no	—
86	C₂×C₃×Dic₃	C₄⋊C₄	1	2	yes	—
87	C₂×C₃×Dic₃	C₄⋊C₄	1	2	yes	—
88	C₂×C₃×Dic₃	D₈	1	2	yes	—
89	C₂×C₃×Dic₃	Q₁₆	1	2	yes	—
90	C₃²×C₈	C₄²	1	4	yes	complemented
91	C₃×C₃⋊C₈	—	2	2	no	—
92	C₂×#107	—	3	1	no	—
93	C₂×#107	—	9	1	no	—
94	C₂×C₄×C₈	—	3	1	no	—
95	C₄×C₄⋊C₄	—	9	2	no	—
96	#340	—	9	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	—	6	2	no	—
101	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
102	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
103	C₂²×Dic₃	—	2	2	no	—
104	C₂²×Dic₃	—	6	2	no	—
105	C₂²×Dic₃	—	6	2	no	—
106	C₂²×Dic₃	—	12	2	no	—
107	C₂²×Dic₃	D₁₂	1	1	yes	characteristic
108	C₂²×Dic₃	D₁₂	1	1	yes	characteristic
109	C₂×C₃×C₈	—	2	2	no	—
110	C₂×C₃×C₈	—	2	2	no	—
111	C₂×C₃×C₈	—	2	2	no	—
112	C₂×C₃×C₈	—	3	2	no	—
113	C₂×C₃×C₈	—	6	1	no	—
114	C₂×C₃×C₈	—	6	1	no	—
115	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
116	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
117	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
118	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
119	C₂×C₃×C₈	C₂×Dic₃	1	2	yes	—
120	C₂×C₃×C₈	C₂×Dic₃	1	2	yes	—
121	C₃×C₄⋊C₄	—	3	4	no	—
122	C₃×C₄⋊C₄	—	6	2	no	—
123	C₃×C₄²	—	3	2	no	—
124	C₃×C₄²	—	3	2	no	—
125	C₄⋊Dic₃	—	2	2	no	—
126	C₄⋊Dic₃	—	3	4	no	—
127	C₄⋊Dic₃	—	3	4	no	—
128	C₄⋊Dic₃	—	6	4	no	—
129	C₄⋊Dic₃	—	6	2	no	—
130	C₄⋊Dic₃	—	6	2	no	—
131	C₄⋊Dic₃	—	12	2	no	—
132	C₄⋊Dic₃	C₂×Dic₃	1	4	yes	—
133	C₄×Dic₃	—	12	4	no	—
134	C₄×Dic₃	—	12	4	no	—
135	C₄×Dic₃	Dic₆	1	2	yes	—
136	C₄×Dic₃	C₂×Dic₃	1	2	yes	—
137	C₂×C₃⋊C₈	—	2	1	no	—
138	C₂×C₃⋊C₈	—	2	1	no	—
139	C₂×C₃⋊C₈	C₂×Dic₃	1	2	yes	—
140	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
141	C₂²×C₃²	C₄×D₄	1	1	yes	characteristic
142	C₂²×C₃²	C₄×D₄	1	1	yes	characteristic
143	C₂²×C₃²	C₂×D₈	1	1	yes	characteristic
144	C₂²×C₃²	C₂×Q₁₆	1	1	yes	characteristic
145	C₂²×C₃²	#135	1	1	yes	characteristic
146	C₂²×C₃²	#135	1	1	yes	characteristic
147	C₃²×C₄	C₂×C₄²	1	1	yes	characteristic
148	C₃²×C₄	#117	1	1	yes	characteristic
149	C₃²×C₄	C₄×Q₈	1	1	yes	characteristic
150	C₃²×C₄	C₄×Q₈	1	1	yes	characteristic
151	C₃⋊Dic₃	—	4	4	no	—
152	C₃×Dic₃	—	2	2	no	—
153	C₃×Dic₃	—	4	4	no	—
154	C₃×Dic₃	#107	1	4	yes	complemented
155	C₂²×C₈	—	3	1	no	—
156	C₂²×C₈	D₃²	1	1	yes	characteristic
157	C₂×C₄⋊C₄	—	3	2	no	—
158	C₂×C₄⋊C₄	—	9	2	no	—
159	C₂×C₄²	—	3	1	no	—
160	C₂×C₄²	—	18	2	no	—
161	#107	—	3	4	no	—
162	#107	—	9	4	no	—
163	C₄×C₈	—	3	4	no	—
164	#95	—	18	2	no	—
165	C₂³×C₃	—	2	1	no	—
166	C₂³×C₃	C₂×D₁₂	1	1	yes	characteristic
167	C₂³×C₃	D₃×D₄	1	1	yes	characteristic
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	—	2	1	no	—
173	C₂×C₃×C₄	—	2	1	no	—
174	C₂×C₃×C₄	—	3	2	no	—
175	C₂×C₃×C₄	—	3	2	no	—
176	C₂×C₃×C₄	—	3	2	no	—
177	C₂×C₃×C₄	—	3	2	no	—
178	C₂×C₃×C₄	—	6	1	no	—
179	C₂×C₃×C₄	—	6	1	no	—
180	C₂×C₃×C₄	—	6	4	no	—
181	C₂×C₃×C₄	—	12	2	no	—
182	C₂×C₃×C₄	—	12	2	no	—
183	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
184	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
185	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
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	#235	1	1	yes	characteristic
189	C₂×C₃×C₄	#235	1	1	yes	characteristic
190	C₂×C₃×C₄	#237	1	1	yes	characteristic
191	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
192	C₂×C₃×C₄	#239	1	1	yes	characteristic
193	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
194	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
195	C₂×Dic₃	—	2	1	no	—
196	C₂×Dic₃	—	2	1	no	—
197	C₂×Dic₃	—	2	4	no	—
198	C₂×Dic₃	—	4	2	no	—
199	C₂×Dic₃	—	4	2	no	—
200	C₂×Dic₃	—	6	4	no	—
201	C₂×Dic₃	—	6	4	no	—
202	C₂×Dic₃	—	12	4	no	—
203	C₂×Dic₃	—	12	2	no	—
204	C₂×Dic₃	—	12	2	no	—
205	C₂×Dic₃	—	12	2	no	—
206	C₂×Dic₃	—	12	2	no	—
207	C₂×Dic₃	—	24	2	no	—
208	C₂×Dic₃	—	24	2	no	—
209	C₂×Dic₃	D₂₄	1	2	yes	—
210	C₂×Dic₃	Dic₁₂	1	2	yes	—
211	C₂×Dic₃	C₄⋊Dic₃	1	2	yes	—
212	C₂×Dic₃	C₄⋊Dic₃	1	2	yes	—
213	C₃×C₈	—	2	4	no	—
214	C₃×C₈	—	6	2	no	—
215	C₃×C₈	C₄×Dic₃	1	4	yes	complemented
216	C₃×C₈	C₄×Dic₃	1	4	yes	complemented
217	C₃⋊C₈	—	2	2	no	—
218	C₂×C₃²	C₄×C₄⋊C₄	1	1	yes	characteristic
219	C₂×C₃²	C₂×#107	1	1	yes	characteristic
220	C₂×C₃²	#427	1	1	yes	characteristic
221	C₂×C₃²	C₄×D₈	1	1	yes	characteristic
222	C₂×C₃²	C₄×D₈	1	1	yes	characteristic
223	C₂×C₃²	C₄×Q₁₆	1	1	yes	characteristic
224	C₂×C₃²	C₄×Q₁₆	1	1	yes	characteristic
225	C₂²×C₄	—	3	1	no	—
226	C₂²×C₄	—	3	1	no	—
227	C₂²×C₄	—	6	2	no	—
228	C₂²×C₄	—	18	2	no	—
229	C₂²×C₄	C₂×D₃²	1	1	yes	characteristic
230	C₂×C₈	—	3	2	no	—
231	C₂×C₈	—	6	1	no	—
232	C₂×C₈	—	6	1	no	—
233	C₂×C₈	D₃×Dic₃	1	2	yes	—
234	C₂×C₈	D₃×Dic₃	1	2	yes	—
235	C₂×C₈	#624	1	2	yes	—
236	C₄⋊C₄	—	3	4	no	—
237	C₄⋊C₄	—	6	2	no	—
238	C₄⋊C₄	—	9	4	no	—
239	C₄⋊C₄	—	18	2	no	—
240	C₄²	—	3	2	no	—
241	C₄²	—	3	2	no	—
242	C₄²	—	36	4	no	—
243	C₂²×C₃	—	2	1	no	—
244	C₂²×C₃	—	2	1	no	—
245	C₂²×C₃	—	2	1	no	—
246	C₂²×C₃	—	2	1	no	—
247	C₂²×C₃	—	2	1	no	—
248	C₂²×C₃	—	2	1	no	—
249	C₂²×C₃	—	2	1	no	—
250	C₂²×C₃	C₄×D₁₂	1	1	yes	characteristic
251	C₂²×C₃	C₄×D₁₂	1	1	yes	characteristic
252	C₂²×C₃	D₃×C₄⋊C₄	1	1	yes	characteristic
253	C₂²×C₃	#893	1	1	yes	characteristic
254	C₂²×C₃	C₂×D₂₄	1	1	yes	characteristic
255	C₂²×C₃	#904	1	1	yes	characteristic
256	C₂²×C₃	#904	1	1	yes	characteristic
257	C₂²×C₃	C₂×Dic₁₂	1	1	yes	characteristic
258	C₂²×C₃	D₃×D₈	1	1	yes	characteristic
259	C₂²×C₃	#912	1	1	yes	characteristic
260	C₂²×C₃	D₃×Q₁₆	1	1	yes	characteristic
261	C₂²×C₃	#919	1	1	yes	characteristic
262	C₂²×C₃	C₂×C₄⋊Dic₃	1	1	yes	characteristic
263	C₂²×C₃	D₄×Dic₃	1	1	yes	characteristic
264	C₃×C₄	—	2	1	no	—
265	C₃×C₄	—	2	1	no	—
266	C₃×C₄	—	2	1	no	—
267	C₃×C₄	—	2	1	no	—
268	C₃×C₄	—	3	4	no	—
269	C₃×C₄	—	6	2	no	—
270	C₃×C₄	—	12	4	no	—
271	C₃×C₄	C₄×Dic₆	1	1	yes	characteristic
272	C₃×C₄	C₄×Dic₆	1	1	yes	characteristic
273	C₃×C₄	#887	1	1	yes	characteristic
274	C₃×C₄	#892	1	1	yes	characteristic
275	C₃×C₄	C₂×C₄×Dic₃	1	1	yes	characteristic
276	C₃×C₄	C₂×C₄×Dic₃	1	1	yes	characteristic
277	C₃×C₄	#926	1	1	yes	characteristic
278	C₃×C₄	Q₈×Dic₃	1	1	yes	characteristic
279	Dic₃	—	2	2	no	—
280	Dic₃	—	4	4	no	—
281	Dic₃	—	12	4	no	—
282	Dic₃	—	12	4	no	—
283	Dic₃	—	24	4	no	—
284	Dic₃	#818	1	4	yes	direct product factor
285	C₃²	C₄×#107	1	1	yes	characteristic, complemented, Sylow
286	C₂³	D₃×D₁₂	1	1	yes	characteristic
287	C₂×C₄	—	3	2	no	—
288	C₂×C₄	—	3	2	no	—
289	C₂×C₄	—	3	2	no	—
290	C₂×C₄	—	3	2	no	—
291	C₂×C₄	—	6	1	no	—
292	C₂×C₄	—	6	1	no	—
293	C₂×C₄	—	6	4	no	—
294	C₂×C₄	—	12	2	no	—
295	C₂×C₄	—	12	2	no	—
296	C₂×C₄	—	18	4	no	—
297	C₂×C₄	—	36	2	no	—
298	C₂×C₄	—	36	2	no	—
299	C₂×C₄	D₃×Dic₆	1	1	yes	characteristic
300	C₂×C₄	#3795	1	1	yes	characteristic
301	C₂×C₄	#3799	1	1	yes	characteristic
302	C₂×C₄	C₂×D₃×Dic₃	1	1	yes	characteristic
303	C₂×C₄	C₂×D₃×Dic₃	1	1	yes	characteristic
304	C₂×C₄	C₂×#624	1	1	yes	characteristic
305	C₈	—	6	2	no	—
306	C₈	Dic₃²	1	4	yes	complemented
307	C₂×C₃	—	2	1	no	—
308	C₂×C₃	—	2	1	no	—
309	C₂×C₃	—	2	1	no	—
310	C₂×C₃	—	2	1	no	—
311	C₂×C₃	—	2	1	no	—
312	C₂×C₃	—	2	1	no	—
313	C₂×C₃	—	2	1	no	—
314	C₂×C₃	C₄×D₂₄	1	1	yes	characteristic
315	C₂×C₃	C₄×D₂₄	1	1	yes	characteristic
316	C₂×C₃	C₄×Dic₁₂	1	1	yes	characteristic
317	C₂×C₃	C₄×Dic₁₂	1	1	yes	characteristic
318	C₂×C₃	#4872	1	1	yes	characteristic
319	C₂×C₃	#4873	1	1	yes	characteristic
320	C₂×C₃	D₃×#107	1	1	yes	characteristic
321	C₂×C₃	#4880	1	1	yes	characteristic
322	C₂×C₃	C₄×C₄⋊Dic₃	1	1	yes	characteristic
323	C₂×C₃	Dic₃×C₄⋊C₄	1	1	yes	characteristic
324	C₂×C₃	C₂×#818	1	1	yes	characteristic
325	C₂×C₃	#5106	1	1	yes	characteristic
326	C₂×C₃	Dic₃×D₈	1	1	yes	characteristic
327	C₂×C₃	Dic₃×Q₁₆	1	1	yes	characteristic
328	C₂²	D₃×D₂₄	1	1	yes	characteristic
329	C₂²	D₃×Dic₁₂	1	1	yes	characteristic
330	C₂²	#63807	1	1	yes	characteristic
331	C₂²	#63808	1	1	yes	characteristic
332	C₂²	D₃×C₄⋊Dic₃	1	1	yes	characteristic
333	C₂²	Dic₃×D₁₂	1	1	yes	characteristic
334	C₂²	#63894	1	1	yes	characteristic
335	C₄	—	3	4	no	—
336	C₄	—	6	2	no	—
337	C₄	—	12	4	no	—
338	C₄	—	36	4	no	—
339	C₄	#63841	1	1	yes	characteristic
340	C₄	Dic₃×Dic₆	1	1	yes	characteristic
341	C₄	#63844	1	1	yes	characteristic
342	C₄	C₂×Dic₃²	1	1	yes	characteristic
343	C₃	—	2	1	no	—
344	C₃	C₄×#818	1	1	yes	characteristic, complemented
345	C₃	Dic₃×#107	1	1	yes	characteristic, complemented
346	C₂	D₃×#818	1	1	yes	characteristic
347	C₂	#10590707	1	1	yes	characteristic
348	C₂	Dic₃×D₂₄	1	1	yes	characteristic
349	C₂	#10590763	1	1	yes	characteristic
350	C₂	Dic₃×Dic₁₂	1	1	yes	characteristic
351	C₂	#10590807	1	1	yes	characteristic
352	C₂	Dic₃×C₄⋊Dic₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₃×D₁₂
commutator	C₃²×C₄	C₂×C₄²
Frattini	C₂²×C₄	C₂×D₃²
Fitting	C₂²×C₃²×C₈	C₂²
socle, prime radical	C₂³×C₃²	C₂×D₄
cosocle	C₂²×C₃²×C₄	C₂³
hypercenter	C₂²×C₈	D₃²
hypocenter	C₃²	C₄×#107
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
14. #63645	C₄	semidirect
15. C₃×#818	C₄	semidirect
16. C₃×C₈×Dic₃	C₄	semidirect
69. #818	Dic₃	direct (N × H)
70. C₈×Dic₃	Dic₃	semidirect
90. C₃²×C₈	C₄²	semidirect
154. C₃×Dic₃	#107	semidirect
215. C₃×C₈	C₄×Dic₃	semidirect
216. C₃×C₈	C₄×Dic₃	semidirect
284. Dic₃	#818	direct (N × H)
285. C₃²	C₄×#107	semidirect
306. C₈	Dic₃²	semidirect
344. C₃	C₄×#818	semidirect
345. C₃	Dic₃×#107	semidirect

Maximal subgroups

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

subgroup	index
C₂×#63645	2
C₂×C₃×#818	2
C₂×C₃×C₈×Dic₃	2
Dic₃×C₄⋊Dic₃	2
#10589164	2
Dic₃×#107	3
C₄×#818	3

Subgroups — Dic3×#818 #11784475

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Dic₃×#818 #11784475