Subgroups — Dic₃×#817 #11784448

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×#63644	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	Dic₃×C₄⋊Dic₃	C₂	1	2	yes	maximal
5	#10589164	C₂	1	2	yes	maximal
6	Dic₃×#106	—	3	1	no	maximal
7	C₄×#817	—	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	#63644	C₄	1	4	yes	complemented
15	C₃×#817	C₄	1	4	yes	complemented
16	C₃×C₈×Dic₃	C₄	1	4	yes	complemented
17	#63579	—	2	2	no	—
18	C₂×C₃×#106	—	3	1	no	—
19	C₂×C₃×C₄×C₈	—	3	1	no	—
20	C₂×#817	—	3	1	no	—
21	C₂×#817	—	3	1	no	—
22	C₂×#817	—	6	1	no	—
23	C₂×#817	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×#106	—	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×#106	—	3	4	no	—
63	C₃×C₄×C₈	—	3	4	no	—
64	C₃⋊#95	—	6	2	no	—
65	C₃⋊#95	—	6	2	no	—
66	#817	—	3	4	no	—
67	#817	—	3	4	no	—
68	#817	—	6	4	no	—
69	#817	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	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	—	9	1	no	—
92	C₂×C₄×C₈	—	3	1	no	—
93	C₄×C₄⋊C₄	—	9	2	no	—
94	#340	—	9	2	no	—
95	C₂²×C₃×C₄	—	2	1	no	—
96	C₂²×C₃×C₄	—	3	1	no	—
97	C₂²×C₃×C₄	—	3	1	no	—
98	C₂²×C₃×C₄	—	6	2	no	—
99	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
100	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
101	C₂²×Dic₃	—	2	2	no	—
102	C₂²×Dic₃	—	6	2	no	—
103	C₂²×Dic₃	—	6	2	no	—
104	C₂²×Dic₃	—	12	2	no	—
105	C₂²×Dic₃	D₁₂	1	1	yes	characteristic
106	C₂²×Dic₃	D₁₂	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	C₄×D₃	1	2	yes	—
116	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
117	C₂×C₃×C₈	C₂×Dic₃	1	2	yes	—
118	C₂×C₃×C₈	C₂×Dic₃	1	2	yes	—
119	C₃×C₄⋊C₄	—	3	4	no	—
120	C₃×C₄⋊C₄	—	6	2	no	—
121	C₃×C₄²	—	3	2	no	—
122	C₃×C₄²	—	3	2	no	—
123	C₄⋊Dic₃	—	2	2	no	—
124	C₄⋊Dic₃	—	3	4	no	—
125	C₄⋊Dic₃	—	3	4	no	—
126	C₄⋊Dic₃	—	6	4	no	—
127	C₄⋊Dic₃	—	6	2	no	—
128	C₄⋊Dic₃	—	6	2	no	—
129	C₄⋊Dic₃	—	12	2	no	—
130	C₄⋊Dic₃	C₂×Dic₃	1	4	yes	—
131	C₄×Dic₃	—	12	4	no	—
132	C₄×Dic₃	—	12	4	no	—
133	C₄×Dic₃	Dic₆	1	2	yes	—
134	C₄×Dic₃	C₂×Dic₃	1	2	yes	—
135	C₂×C₃⋊C₈	—	2	1	no	—
136	C₂×C₃⋊C₈	—	2	1	no	—
137	C₂×C₃⋊C₈	C₂×Dic₃	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	C₂×C₄²	1	1	yes	characteristic
144	C₃²×C₄	#117	1	1	yes	characteristic
145	C₃²×C₄	C₄×Q₈	1	1	yes	characteristic
146	C₃²×C₄	C₄×Q₈	1	1	yes	characteristic
147	C₃⋊Dic₃	—	4	4	no	—
148	C₃×Dic₃	—	2	2	no	—
149	C₃×Dic₃	—	4	4	no	—
150	C₃×Dic₃	#106	1	4	yes	complemented
151	C₂²×C₈	—	3	1	no	—
152	C₂²×C₈	D₃²	1	1	yes	characteristic
153	C₂×C₄⋊C₄	—	3	2	no	—
154	C₂×C₄⋊C₄	—	9	2	no	—
155	C₂×C₄²	—	3	1	no	—
156	C₂×C₄²	—	18	2	no	—
157	#106	—	3	4	no	—
158	#106	—	9	4	no	—
159	C₄×C₈	—	3	4	no	—
160	#95	—	18	2	no	—
161	C₂³×C₃	—	2	1	no	—
162	C₂³×C₃	C₂×D₁₂	1	1	yes	characteristic
163	C₂³×C₃	D₃×D₄	1	1	yes	characteristic
164	C₂×C₃×C₄	—	2	1	no	—
165	C₂×C₃×C₄	—	2	1	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	—	3	2	no	—
171	C₂×C₃×C₄	—	3	4	no	—
172	C₂×C₃×C₄	—	3	2	no	—
173	C₂×C₃×C₄	—	6	2	no	—
174	C₂×C₃×C₄	—	6	4	no	—
175	C₂×C₃×C₄	—	12	2	no	—
176	C₂×C₃×C₄	—	12	2	no	—
177	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
178	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
179	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
180	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
181	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
182	C₂×C₃×C₄	#235	1	1	yes	characteristic
183	C₂×C₃×C₄	#235	1	1	yes	characteristic
184	C₂×C₃×C₄	#237	1	1	yes	characteristic
185	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
186	C₂×C₃×C₄	#239	1	1	yes	characteristic
187	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
188	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
189	C₂×Dic₃	—	2	2	no	—
190	C₂×Dic₃	—	2	4	no	—
191	C₂×Dic₃	—	4	2	no	—
192	C₂×Dic₃	—	4	2	no	—
193	C₂×Dic₃	—	6	4	no	—
194	C₂×Dic₃	—	6	4	no	—
195	C₂×Dic₃	—	12	4	no	—
196	C₂×Dic₃	—	12	2	no	—
197	C₂×Dic₃	—	12	2	no	—
198	C₂×Dic₃	—	12	2	no	—
199	C₂×Dic₃	—	12	2	no	—
200	C₂×Dic₃	—	24	2	no	—
201	C₂×Dic₃	—	24	2	no	—
202	C₂×Dic₃	#204	1	4	yes	—
203	C₂×Dic₃	C₄⋊Dic₃	1	2	yes	—
204	C₂×Dic₃	C₄⋊Dic₃	1	2	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	C₄×Dic₃	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	—	3	1	no	—
216	C₂²×C₄	—	3	1	no	—
217	C₂²×C₄	—	6	2	no	—
218	C₂²×C₄	—	18	2	no	—
219	C₂²×C₄	C₂×D₃²	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	D₃×Dic₃	1	2	yes	—
224	C₂×C₈	D₃×Dic₃	1	2	yes	—
225	C₂×C₈	#624	1	2	yes	—
226	C₄⋊C₄	—	3	4	no	—
227	C₄⋊C₄	—	6	2	no	—
228	C₄⋊C₄	—	9	4	no	—
229	C₄⋊C₄	—	18	2	no	—
230	C₄²	—	3	2	no	—
231	C₄²	—	3	2	no	—
232	C₄²	—	36	4	no	—
233	C₂²×C₃	—	2	2	no	—
234	C₂²×C₃	—	2	2	no	—
235	C₂²×C₃	—	2	1	no	—
236	C₂²×C₃	—	2	1	no	—
237	C₂²×C₃	—	2	1	no	—
238	C₂²×C₃	C₄×D₁₂	1	1	yes	characteristic
239	C₂²×C₃	C₄×D₁₂	1	1	yes	characteristic
240	C₂²×C₃	D₃×C₄⋊C₄	1	1	yes	characteristic
241	C₂²×C₃	#893	1	1	yes	characteristic
242	C₂²×C₃	C₂×#204	1	2	yes	—
243	C₂²×C₃	#904	1	2	yes	—
244	C₂²×C₃	D₃×SD₁₆	1	2	yes	—
245	C₂²×C₃	#916	1	2	yes	—
246	C₂²×C₃	C₂×C₄⋊Dic₃	1	1	yes	characteristic
247	C₂²×C₃	D₄×Dic₃	1	1	yes	characteristic
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	—	6	2	no	—
254	C₃×C₄	—	12	4	no	—
255	C₃×C₄	C₄×Dic₆	1	1	yes	characteristic
256	C₃×C₄	C₄×Dic₆	1	1	yes	characteristic
257	C₃×C₄	#887	1	1	yes	characteristic
258	C₃×C₄	#892	1	1	yes	characteristic
259	C₃×C₄	C₂×C₄×Dic₃	1	1	yes	characteristic
260	C₃×C₄	C₂×C₄×Dic₃	1	1	yes	characteristic
261	C₃×C₄	#926	1	1	yes	characteristic
262	C₃×C₄	Q₈×Dic₃	1	1	yes	characteristic
263	Dic₃	—	2	2	no	—
264	Dic₃	—	4	4	no	—
265	Dic₃	—	12	4	no	—
266	Dic₃	—	12	4	no	—
267	Dic₃	—	24	4	no	—
268	Dic₃	#817	1	4	yes	direct product factor
269	C₃²	C₄×#106	1	1	yes	characteristic, complemented, Sylow
270	C₂³	D₃×D₁₂	1	1	yes	characteristic
271	C₂×C₄	—	3	4	no	—
272	C₂×C₄	—	3	2	no	—
273	C₂×C₄	—	3	2	no	—
274	C₂×C₄	—	6	2	no	—
275	C₂×C₄	—	6	4	no	—
276	C₂×C₄	—	12	2	no	—
277	C₂×C₄	—	12	2	no	—
278	C₂×C₄	—	18	4	no	—
279	C₂×C₄	—	36	2	no	—
280	C₂×C₄	—	36	2	no	—
281	C₂×C₄	D₃×Dic₆	1	1	yes	characteristic
282	C₂×C₄	#3795	1	1	yes	characteristic
283	C₂×C₄	#3799	1	1	yes	characteristic
284	C₂×C₄	C₂×D₃×Dic₃	1	1	yes	characteristic
285	C₂×C₄	C₂×D₃×Dic₃	1	1	yes	characteristic
286	C₂×C₄	C₂×#624	1	1	yes	characteristic
287	C₈	—	6	2	no	—
288	C₈	Dic₃²	1	4	yes	complemented
289	C₂×C₃	—	2	1	no	—
290	C₂×C₃	—	2	2	no	—
291	C₂×C₃	—	2	2	no	—
292	C₂×C₃	—	2	1	no	—
293	C₂×C₃	—	2	1	no	—
294	C₂×C₃	C₄×#204	1	2	yes	—
295	C₂×C₃	C₄×#204	1	2	yes	—
296	C₂×C₃	#4852	1	2	yes	—
297	C₂×C₃	D₃×#106	1	1	yes	characteristic
298	C₂×C₃	#4860	1	1	yes	characteristic
299	C₂×C₃	C₄×C₄⋊Dic₃	1	1	yes	characteristic
300	C₂×C₃	Dic₃×C₄⋊C₄	1	1	yes	characteristic
301	C₂×C₃	C₂×#817	1	1	yes	characteristic
302	C₂×C₃	#5106	1	1	yes	characteristic
303	C₂×C₃	Dic₃×SD₁₆	1	2	yes	—
304	C₂²	D₃×#204	1	2	yes	—
305	C₂²	#63806	1	2	yes	—
306	C₂²	D₃×C₄⋊Dic₃	1	1	yes	characteristic
307	C₂²	Dic₃×D₁₂	1	1	yes	characteristic
308	C₂²	#63894	1	1	yes	characteristic
309	C₄	—	3	4	no	—
310	C₄	—	6	2	no	—
311	C₄	—	12	4	no	—
312	C₄	—	36	4	no	—
313	C₄	#63841	1	1	yes	characteristic
314	C₄	Dic₃×Dic₆	1	1	yes	characteristic
315	C₄	#63844	1	1	yes	characteristic
316	C₄	C₂×Dic₃²	1	1	yes	characteristic
317	C₃	—	2	1	no	—
318	C₃	C₄×#817	1	1	yes	characteristic, complemented
319	C₃	Dic₃×#106	1	1	yes	characteristic, complemented
320	C₂	D₃×#817	1	1	yes	characteristic
321	C₂	#10590706	1	1	yes	characteristic
322	C₂	Dic₃×#204	1	2	yes	—
323	C₂	#10590758	1	2	yes	—
324	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×#106
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. #63644	C₄	semidirect
15. C₃×#817	C₄	semidirect
16. C₃×C₈×Dic₃	C₄	semidirect
69. #817	Dic₃	direct (N × H)
70. C₈×Dic₃	Dic₃	semidirect
88. C₃²×C₈	C₄²	semidirect
150. C₃×Dic₃	#106	semidirect
207. C₃×C₈	C₄×Dic₃	semidirect
208. C₃×C₈	C₄×Dic₃	semidirect
268. Dic₃	#817	direct (N × H)
269. C₃²	C₄×#106	semidirect
288. C₈	Dic₃²	semidirect
318. C₃	C₄×#817	semidirect
319. C₃	Dic₃×#106	semidirect

Maximal subgroups

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

subgroup	index
C₂×#63644	2
C₂×C₃×#817	2
C₂×C₃×C₈×Dic₃	2
Dic₃×C₄⋊Dic₃	2
#10589164	2
Dic₃×#106	3
C₄×#817	3

Subgroups — Dic3×#817 #11784448

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Dic₃×#817 #11784448