Subgroups — D₃×#63367 #13137663

Order 1728.

426 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₃×C₉×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
2	D₃×C₄⋊Dic₉	C₂	1	1	yes	maximal, characteristic
3	C₂×D₃×C₉⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#11720526	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#63367	C₂	1	1	yes	maximal, characteristic, complemented
6	#11720265	C₂	1	1	yes	maximal, characteristic, complemented
7	#11720264	C₂	1	1	yes	maximal, characteristic, complemented
8	D₃×#808	—	3	1	no	maximal
9	C₂×#63367	—	3	1	no	maximal
10	C₂×C₄×D₃×C₉	—	2	1	no	—
11	C₂×C₄×D₃×C₉	C₂²	1	1	yes	characteristic
12	C₂×D₃×Dic₉	—	2	1	no	—
13	C₃×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₄⋊(C₃⋊Dic₉)	C₂²	1	1	yes	characteristic
15	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	1	yes	characteristic
16	C₉×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
17	C₉×Dic₃⋊C₄	—	2	1	no	—
18	C₃×C₄⋊Dic₉	C₂²	1	1	yes	characteristic
19	C₂×C₃×C₉⋊C₈	C₂²	1	1	yes	characteristic
20	Dic₃⋊Dic₉	—	2	1	no	—
21	D₃×C₉⋊C₈	C₄	1	4	yes	complemented
22	C₃×D₃×C₄⋊C₄	D₃	1	1	yes	characteristic
23	D₃×C₄⋊Dic₃	—	3	1	no	—
24	C₂×D₃×C₃⋊C₈	—	3	1	no	—
25	#63635	—	3	1	no	—
26	C₃×#808	—	3	1	no	—
27	#63574	—	3	1	no	—
28	#63573	—	3	1	no	—
29	C₂×C₉×C₄⋊C₄	—	3	1	no	—
30	C₂×C₄⋊Dic₉	—	3	1	no	—
31	C₂²×C₉⋊C₈	—	3	1	no	—
32	#63367	—	3	1	no	—
33	#63367	—	3	1	no	—
34	#63367	—	3	1	no	—
35	#63367	—	6	1	no	—
36	#63367	D₃	1	1	yes	characteristic, direct product factor
37	C₂²×D₃×C₉	D₄	1	1	yes	characteristic
38	C₂×C₃×C₄×C₉	—	2	1	no	—
39	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
40	C₂×C₃⋊Dic₉	—	2	1	no	—
41	C₂×C₉×Dic₃	—	2	1	no	—
42	C₂×C₉×Dic₃	D₄	1	1	yes	characteristic
43	C₂×C₃×Dic₉	—	2	1	no	—
44	C₄×D₃×C₉	—	4	2	no	—
45	C₄×D₃×C₉	C₂×C₄	1	2	yes	—
46	C₄×D₃×C₉	Q₈	1	2	yes	—
47	D₃×Dic₉	—	4	2	no	—
48	C₃⋊(C₉⋊C₈)	C₂×C₄	1	2	yes	complemented
49	C₃×C₉⋊C₈	C₂×C₄	1	2	yes	complemented
50	C₂×#808	—	9	1	no	—
51	D₃×#106	—	9	1	no	—
52	C₂×C₃×C₄×D₃	—	2	1	no	—
53	C₂×C₃×C₄×D₃	C₂×D₃	1	1	yes	characteristic
54	C₂×D₃×Dic₃	—	6	1	no	—
55	C₃²×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
56	C₄⋊(C₃⋊Dic₃)	—	3	1	no	—
57	C₂×C₃⋊(C₃⋊C₈)	—	3	1	no	—
58	C₃×C₄⋊Dic₃	—	3	1	no	—
59	C₃×C₄⋊Dic₃	C₂×D₃	1	1	yes	characteristic
60	C₃×Dic₃⋊C₄	—	2	1	no	—
61	C₂×C₃×C₃⋊C₈	—	3	1	no	—
62	Dic₃⋊Dic₃	—	6	1	no	—
63	D₃×C₃⋊C₈	—	3	4	no	—
64	C₂²×C₄×C₉	—	3	1	no	—
65	C₂²×C₄×C₉	—	6	1	no	—
66	C₂²×Dic₉	—	6	1	no	—
67	C₉×C₄⋊C₄	—	2	1	no	—
68	C₉×C₄⋊C₄	—	3	1	no	—
69	C₉×C₄⋊C₄	—	6	1	no	—
70	C₉×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
71	C₄⋊Dic₉	—	3	1	no	—
72	C₄⋊Dic₉	—	6	1	no	—
73	C₄⋊Dic₉	—	6	1	no	—
74	C₄⋊Dic₉	C₂×D₃	1	1	yes	characteristic
75	C₂×C₉⋊C₈	—	3	1	no	—
76	C₂×C₉⋊C₈	—	3	4	no	—
77	C₂×C₉⋊C₈	—	6	1	no	—
78	C₂×C₉⋊C₈	C₂×D₃	1	1	yes	characteristic
79	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
80	C₂×D₃×C₉	C₄⋊C₄	1	2	yes	—
81	C₂×D₃×C₉	SD₁₆	1	2	yes	—
82	C₂×D₃×C₉	SD₁₆	1	2	yes	—
83	C₃×C₄×C₉	—	4	1	no	—
84	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
85	C₃×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
86	C₃⋊Dic₉	—	4	1	no	—
87	C₉×Dic₃	—	4	1	no	—
88	C₉×Dic₃	C₄⋊C₄	1	2	yes	—
89	C₃×Dic₉	—	4	1	no	—
90	C₂×C₃×C₄⋊C₄	—	3	1	no	—
91	C₂×C₄⋊Dic₃	—	9	1	no	—
92	C₂²×C₃⋊C₈	—	9	1	no	—
93	C₂×D₃×C₈	—	9	1	no	—
94	D₃×C₄⋊C₄	—	9	1	no	—
95	D₃×C₄⋊C₄	D₉	1	1	yes	characteristic
96	C₃×#106	—	9	1	no	—
97	#817	—	9	1	no	—
98	#808	—	3	1	no	—
99	#808	—	9	1	no	—
100	#808	—	9	1	no	—
101	#808	—	9	1	no	—
102	#808	—	9	1	no	—
103	#808	—	9	1	no	—
104	#808	—	18	1	no	—
105	C₂²×C₃×D₃	#67	1	1	yes	characteristic
106	C₂×C₃²×C₄	—	2	1	no	—
107	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
108	C₂×C₃⋊Dic₃	—	6	1	no	—
109	C₂×C₃×Dic₃	—	2	1	no	—
110	C₂×C₃×Dic₃	—	6	1	no	—
111	C₂×C₃×Dic₃	#67	1	1	yes	characteristic
112	C₃×C₄×D₃	—	4	2	no	—
113	C₃×C₄×D₃	Dic₆	1	2	yes	—
114	C₃×C₄×D₃	C₄×D₃	1	2	yes	—
115	D₃×Dic₃	—	12	2	no	—
116	C₂³×C₉	—	3	1	no	—
117	C₃⋊(C₃⋊C₈)	—	3	2	no	—
118	C₃×C₃⋊C₈	—	3	2	no	—
119	C₂×C₄×C₉	—	2	1	no	—
120	C₂×C₄×C₉	—	2	1	no	—
121	C₂×C₄×C₉	—	3	1	no	—
122	C₂×C₄×C₉	—	3	2	no	—
123	C₂×C₄×C₉	—	3	2	no	—
124	C₂×C₄×C₉	—	4	1	no	—
125	C₂×C₄×C₉	—	6	1	no	—
126	C₂×C₄×C₉	—	12	2	no	—
127	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
128	C₂×Dic₉	—	2	1	no	—
129	C₂×Dic₉	—	6	1	no	—
130	C₂×Dic₉	—	12	1	no	—
131	C₂×Dic₉	—	12	2	no	—
132	C₉⋊C₈	—	3	2	no	—
133	C₉⋊C₈	—	6	2	no	—
134	C₉⋊C₈	C₄×D₃	1	2	yes	complemented
135	C₂×#106	—	27	1	no	Sylow
136	C₂×C₃×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
137	C₂×C₃×C₉	C₂×SD₁₆	1	1	yes	characteristic
138	C₂×C₃×C₉	C₂×SD₁₆	1	1	yes	characteristic
139	D₃×C₉	#106	1	4	yes	complemented
140	C₂²×C₃×C₄	—	3	1	no	—
141	C₂²×C₃×C₄	—	6	1	no	—
142	C₂²×Dic₃	—	18	1	no	—
143	C₂×C₄×D₃	—	2	1	no	—
144	C₂×C₄×D₃	—	18	1	no	—
145	C₂×C₄×D₃	C₂×D₉	1	1	yes	characteristic
146	C₂×C₃×C₈	—	9	1	no	—
147	C₃×C₄⋊C₄	—	2	1	no	—
148	C₃×C₄⋊C₄	—	3	1	no	—
149	C₃×C₄⋊C₄	—	6	1	no	—
150	C₃×C₄⋊C₄	—	9	1	no	—
151	C₃×C₄⋊C₄	C₂×D₉	1	1	yes	characteristic
152	C₃×C₄⋊C₄	D₃²	1	1	yes	characteristic
153	C₄⋊Dic₃	—	3	1	no	—
154	C₄⋊Dic₃	—	9	1	no	—
155	C₄⋊Dic₃	—	9	1	no	—
156	C₄⋊Dic₃	—	18	1	no	—
157	C₄⋊Dic₃	—	18	1	no	—
158	C₄⋊Dic₃	C₂×D₉	1	1	yes	characteristic
159	Dic₃⋊C₄	—	2	1	no	—
160	Dic₃⋊C₄	—	18	1	no	—
161	C₂×C₃⋊C₈	—	3	1	no	—
162	C₂×C₃⋊C₈	—	9	4	no	—
163	C₂×C₃⋊C₈	—	9	1	no	—
164	C₂×C₃⋊C₈	—	9	1	no	—
165	C₂×C₃⋊C₈	—	18	1	no	—
166	D₃×C₈	—	9	4	no	—
167	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
168	C₂×C₃×D₃	Dic₃⋊C₄	1	2	yes	—
169	C₂×C₃×D₃	#214	1	2	yes	—
170	C₂×C₃×D₃	#215	1	2	yes	—
171	C₃²×C₄	—	4	1	no	—
172	C₃²×C₄	C₂×Dic₆	1	1	yes	characteristic
173	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
174	C₃⋊Dic₃	—	12	1	no	—
175	C₃×Dic₃	—	4	1	no	—
176	C₃×Dic₃	—	12	1	no	—
177	C₃×Dic₃	Dic₃⋊C₄	1	2	yes	—
178	C₂²×C₉	—	2	1	no	—
179	C₂²×C₉	—	3	2	no	—
180	C₂²×C₉	—	3	2	no	—
181	C₂²×C₉	—	3	2	no	—
182	C₂²×C₉	D₃×D₄	1	1	yes	characteristic
183	C₄×C₉	—	2	1	no	—
184	C₄×C₉	—	2	1	no	—
185	C₄×C₉	—	3	2	no	—
186	C₄×C₉	—	4	1	no	—
187	C₄×C₉	—	8	1	no	—
188	C₄×C₉	—	12	1	no	—
189	C₄×C₉	C₂×C₄×D₃	1	1	yes	characteristic
190	C₄×C₉	D₃×Q₈	1	1	yes	characteristic
191	Dic₉	—	4	1	no	—
192	Dic₉	—	12	1	no	—
193	Dic₉	—	24	1	no	—
194	C₂²×C₈	—	27	1	no	—
195	C₂×C₄⋊C₄	—	3	1	no	—
196	C₂×C₄⋊C₄	—	27	1	no	—
197	#106	—	9	1	no	—
198	#106	—	27	1	no	—
199	#106	—	27	1	no	—
200	#106	—	27	1	no	—
201	C₃×C₉	C₂×#106	1	1	yes	characteristic, complemented, Sylow
202	C₂³×C₃	—	3	1	no	—
203	C₂²×D₃	#611	1	1	yes	characteristic
204	C₂×C₃×C₄	—	2	1	no	—
205	C₂×C₃×C₄	—	2	1	no	—
206	C₂×C₃×C₄	—	2	1	no	—
207	C₂×C₃×C₄	—	3	1	no	—
208	C₂×C₃×C₄	—	3	2	no	—
209	C₂×C₃×C₄	—	3	2	no	—
210	C₂×C₃×C₄	—	4	1	no	—
211	C₂×C₃×C₄	—	6	1	no	—
212	C₂×C₃×C₄	—	12	2	no	—
213	C₂×C₃×C₄	—	18	1	no	—
214	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
215	C₂×C₃×C₄	C₂×D₃²	1	1	yes	characteristic
216	C₂×Dic₃	—	2	1	no	—
217	C₂×Dic₃	—	6	1	no	—
218	C₂×Dic₃	—	18	1	no	—
219	C₂×Dic₃	—	18	1	no	—
220	C₂×Dic₃	—	36	1	no	—
221	C₂×Dic₃	—	36	2	no	—
222	C₂×Dic₃	#611	1	1	yes	characteristic
223	C₄×D₃	—	4	2	no	—
224	C₄×D₃	—	36	2	no	—
225	C₄×D₃	Dic₁₈	1	2	yes	—
226	C₄×D₃	C₄×D₉	1	2	yes	—
227	C₃×C₈	—	9	2	no	—
228	C₃⋊C₈	—	3	2	no	—
229	C₃⋊C₈	—	9	2	no	—
230	C₃⋊C₈	—	9	2	no	—
231	C₃⋊C₈	—	18	2	no	—
232	C₂×C₃²	C₂×Dic₃⋊C₄	1	1	yes	characteristic
233	C₂×C₃²	C₂×#214	1	1	yes	characteristic
234	C₂×C₃²	C₂×#215	1	1	yes	characteristic
235	C₃×D₃	#808	1	4	yes	—
236	C₂×C₉	—	2	1	no	—
237	C₂×C₉	—	2	1	no	—
238	C₂×C₉	—	2	1	no	—
239	C₂×C₉	—	3	4	no	—
240	C₂×C₉	D₃×C₄⋊C₄	1	1	yes	characteristic
241	C₂×C₉	D₃×SD₁₆	1	1	yes	characteristic
242	C₂×C₉	D₃×SD₁₆	1	1	yes	characteristic
243	C₂²×C₄	—	3	1	no	—
244	C₂²×C₄	—	6	1	no	—
245	C₂²×C₄	—	54	1	no	—
246	C₂×C₈	—	9	1	no	—
247	C₂×C₈	—	27	4	no	—
248	C₂×C₈	—	27	1	no	—
249	C₄⋊C₄	—	3	1	no	—
250	C₄⋊C₄	—	6	1	no	—
251	C₄⋊C₄	—	9	1	no	—
252	C₄⋊C₄	—	27	1	no	—
253	C₄⋊C₄	—	54	1	no	—
254	C₄⋊C₄	D₃×D₉	1	1	yes	characteristic
255	C₂²×C₃	—	2	1	no	—
256	C₂²×C₃	—	3	2	no	—
257	C₂²×C₃	—	3	2	no	—
258	C₂²×C₃	—	3	2	no	—
259	C₂²×C₃	C₂×#611	1	1	yes	characteristic
260	C₂²×C₃	D₃×#67	1	1	yes	characteristic
261	C₂×D₃	Dic₉⋊C₄	1	2	yes	—
262	C₂×D₃	#3672	1	2	yes	—
263	C₂×D₃	#3675	1	2	yes	—
264	C₃×C₄	—	2	1	no	—
265	C₃×C₄	—	2	1	no	—
266	C₃×C₄	—	3	2	no	—
267	C₃×C₄	—	4	1	no	—
268	C₃×C₄	—	4	1	no	—
269	C₃×C₄	—	8	1	no	—
270	C₃×C₄	—	12	1	no	—
271	C₃×C₄	—	36	1	no	—
272	C₃×C₄	C₂×Dic₁₈	1	1	yes	characteristic
273	C₃×C₄	C₂×C₄×D₉	1	1	yes	characteristic
274	C₃×C₄	D₃×Dic₆	1	1	yes	characteristic
275	C₃×C₄	C₄×D₃²	1	1	yes	characteristic
276	Dic₃	—	4	1	no	—
277	Dic₃	—	12	1	no	—
278	Dic₃	—	36	1	no	—
279	Dic₃	—	36	1	no	—
280	Dic₃	—	72	1	no	—
281	Dic₃	Dic₉⋊C₄	1	2	yes	—
282	C₃²	C₂×#808	1	1	yes	characteristic
283	C₉	—	2	1	no	—
284	C₉	D₃×#106	1	1	yes	characteristic, complemented
285	C₂³	—	3	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	3	1	no	—
288	C₂×C₄	—	3	2	no	—
289	C₂×C₄	—	3	2	no	—
290	C₂×C₄	—	6	1	no	—
291	C₂×C₄	—	12	2	no	—
292	C₂×C₄	—	18	1	no	—
293	C₂×C₄	—	54	1	no	—
294	C₂×C₄	—	108	2	no	—
295	C₂×C₄	C₂×D₃×D₉	1	1	yes	characteristic
296	C₈	—	9	2	no	—
297	C₈	—	27	2	no	—
298	C₂×C₃	—	2	1	no	—
299	C₂×C₃	—	2	1	no	—
300	C₂×C₃	—	2	1	no	—
301	C₂×C₃	—	3	4	no	—
302	C₂×C₃	C₂×Dic₉⋊C₄	1	1	yes	characteristic
303	C₂×C₃	C₂×#3672	1	1	yes	characteristic
304	C₂×C₃	C₂×#3675	1	1	yes	characteristic
305	C₂×C₃	D₃×Dic₃⋊C₄	1	1	yes	characteristic
306	C₂×C₃	D₃×#214	1	1	yes	characteristic
307	C₂×C₃	D₃×#215	1	1	yes	characteristic
308	D₃	#63367	1	4	yes	direct product factor
309	C₂²	—	3	2	no	—
310	C₂²	—	3	2	no	—
311	C₂²	—	3	2	no	—
312	C₂²	D₃×#611	1	1	yes	characteristic
313	C₄	—	3	2	no	—
314	C₄	—	4	1	no	—
315	C₄	—	12	1	no	—
316	C₄	—	36	1	no	—
317	C₄	—	108	1	no	—
318	C₄	D₃×Dic₁₈	1	1	yes	characteristic
319	C₄	C₄×D₃×D₉	1	1	yes	characteristic
320	C₃	—	2	1	no	—
321	C₃	C₂×#63367	1	1	yes	characteristic, complemented
322	C₃	D₃×#808	1	1	yes	characteristic
323	C₂	—	3	4	no	—
324	C₂	D₃×Dic₉⋊C₄	1	1	yes	characteristic
325	C₂	D₃×#3672	1	1	yes	characteristic
326	C₂	D₃×#3675	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₃×#611
commutator	C₃×C₄×C₉	C₂²×C₄
Frattini	C₂×C₃×C₄	C₂×D₃²
Fitting	C₃×C₉×C₄⋊C₄	C₂²
socle	C₂²×C₃²	C₂×#67
prime radical	C₂²×C₃×D₃	#67
cosocle	C₂×C₃×C₄×C₉	C₂³
hypercenter	C₄⋊C₄	D₃×D₉
hypocenter	C₃×C₉	C₂×#106
LCS	G ▷ C₃×C₄×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
4. #11720526	C₂	semidirect
5. C₃×#63367	C₂	semidirect
6. #11720265	C₂	semidirect
7. #11720264	C₂	semidirect
21. D₃×C₉⋊C₈	C₄	semidirect
36. #63367	D₃	direct (N × H)
48. C₃⋊(C₉⋊C₈)	C₂×C₄	semidirect
49. C₃×C₉⋊C₈	C₂×C₄	semidirect
134. C₉⋊C₈	C₄×D₃	semidirect
139. D₃×C₉	#106	semidirect
201. C₃×C₉	C₂×#106	semidirect
284. C₉	D₃×#106	semidirect
308. D₃	#63367	direct (N × H)
321. C₃	C₂×#63367	semidirect

Maximal subgroups

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

subgroup	index
D₃×C₉×C₄⋊C₄	2
D₃×C₄⋊Dic₉	2
C₂×D₃×C₉⋊C₈	2
#11720526	2
C₃×#63367	2
#11720265	2
#11720264	2
D₃×#808	3
C₂×#63367	3

Subgroups — D3×#63367 #13137663

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₃×#63367 #13137663