Subgroups — C₃×#10589669 #13152668

Order 1728.

670 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×Q₈×C₉	C₂	1	1	yes	maximal, characteristic
2	C₃×#63506	C₂	1	2	yes	maximal
3	C₃×#63451	C₂	1	4	yes	maximal
4	C₃×#63449	C₂	1	2	yes	maximal
5	C₃×#63432	C₂	1	2	yes	maximal
6	C₃×#63431	C₂	1	1	yes	maximal, characteristic
7	C₃×C₄×Dic₁₈	C₂	1	1	yes	maximal, characteristic
8	C₃×C₄×Dic₁₈	C₂	1	2	yes	maximal
9	C₃×#5567	—	3	1	no	maximal
10	#10589669	C₃	1	1	yes	maximal, characteristic, direct product factor
11	C₂×C₃×Q₈×C₉	C₂²	1	1	yes	characteristic
12	C₂×C₃×Dic₁₈	C₂²	1	2	yes	—
13	C₂×C₃×Dic₁₈	C₂²	1	1	yes	characteristic
14	C₃×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₃×C₉×C₄⋊C₄	C₂²	1	2	yes	—
16	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
17	C₃×C₄²×C₉	C₂²	1	2	yes	—
18	C₃×C₄⋊Dic₉	C₂²	1	2	yes	—
19	C₃×C₄⋊Dic₉	C₂²	1	1	yes	characteristic
20	C₃×C₄⋊Dic₉	C₂²	1	2	yes	—
21	C₃×Dic₉⋊C₄	C₂²	1	4	yes	—
22	C₃×Dic₉⋊C₄	C₂²	1	2	yes	—
23	C₃×Dic₉⋊C₄	C₂²	1	4	yes	—
24	C₃×Dic₉⋊C₄	C₂²	1	4	yes	—
25	C₃×C₄×Dic₉	C₂²	1	2	yes	—
26	C₃×C₄×Dic₉	C₂²	1	4	yes	—
27	C₃²×C₄×Q₈	D₃	1	1	yes	characteristic
28	C₃×#944	—	3	2	no	—
29	C₃×#889	—	3	4	no	—
30	C₃×#887	—	3	2	no	—
31	C₃×#870	—	3	2	no	—
32	C₃×#869	—	3	1	no	—
33	C₃×C₄×Dic₆	—	3	2	no	—
34	C₃×C₄×Dic₆	—	3	1	no	—
35	C₄×Q₈×C₉	—	2	1	no	—
36	C₄×Q₈×C₉	C₂×C₃	1	1	yes	characteristic
37	#63506	C₂×C₃	1	2	yes	—
38	#63451	C₂×C₃	1	4	yes	—
39	#63449	C₂×C₃	1	2	yes	—
40	#63432	C₂×C₃	1	2	yes	—
41	#63431	C₂×C₃	1	1	yes	characteristic
42	C₄×Dic₁₈	C₂×C₃	1	1	yes	characteristic
43	C₄×Dic₁₈	C₂×C₃	1	2	yes	—
44	C₃×Q₈×C₉	—	2	2	no	—
45	C₂×C₃×C₄×C₉	C₂³	1	2	yes	—
46	C₂×C₃×C₄×C₉	C₂³	1	2	yes	—
47	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
48	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
49	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
50	C₂×C₃×Dic₉	C₂³	1	4	yes	—
51	C₂×C₃×Dic₉	C₂³	1	2	yes	—
52	C₂×C₃×Dic₉	C₂³	1	2	yes	—
53	C₃×Dic₁₈	—	2	4	no	—
54	C₃×Dic₁₈	Q₈	1	4	yes	complemented
55	C₃×#557	—	9	1	no	—
56	#5567	—	3	1	no	—
57	C₂×C₃²×Q₈	C₂×D₃	1	1	yes	characteristic
58	C₂×C₃×Dic₆	—	3	1	no	—
59	C₂×C₃×Dic₆	—	3	2	no	—
60	C₃²×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
61	C₃²×C₄⋊C₄	C₂×D₃	1	2	yes	—
62	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
63	C₃²×C₄²	C₂×D₃	1	2	yes	—
64	C₃×C₄⋊Dic₃	—	3	2	no	—
65	C₃×C₄⋊Dic₃	—	3	2	no	—
66	C₃×C₄⋊Dic₃	—	3	1	no	—
67	C₃×Dic₃⋊C₄	—	3	4	no	—
68	C₃×Dic₃⋊C₄	—	3	2	no	—
69	C₃×Dic₃⋊C₄	—	3	4	no	—
70	C₃×Dic₃⋊C₄	—	3	4	no	—
71	C₃×C₄×Dic₃	—	3	4	no	—
72	C₃×C₄×Dic₃	—	3	2	no	—
73	C₂×Q₈×C₉	—	2	1	no	—
74	C₂×Q₈×C₉	C₂²×C₃	1	1	yes	characteristic
75	C₂×Dic₁₈	C₂²×C₃	1	2	yes	—
76	C₂×Dic₁₈	C₂²×C₃	1	1	yes	characteristic
77	C₉×C₄⋊C₄	—	2	1	no	—
78	C₉×C₄⋊C₄	—	2	2	no	—
79	C₉×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
80	C₉×C₄⋊C₄	C₂²×C₃	1	2	yes	—
81	C₄²×C₉	—	2	1	no	—
82	C₄²×C₉	—	2	2	no	—
83	C₄²×C₉	C₂²×C₃	1	1	yes	characteristic
84	C₄²×C₉	C₂²×C₃	1	2	yes	—
85	C₄⋊Dic₉	C₂²×C₃	1	2	yes	—
86	C₄⋊Dic₉	C₂²×C₃	1	1	yes	characteristic
87	C₄⋊Dic₉	C₂²×C₃	1	2	yes	—
88	Dic₉⋊C₄	C₂²×C₃	1	4	yes	—
89	Dic₉⋊C₄	C₂²×C₃	1	4	yes	—
90	Dic₉⋊C₄	C₂²×C₃	1	2	yes	—
91	Dic₉⋊C₄	C₂²×C₃	1	4	yes	—
92	C₄×Dic₉	C₂²×C₃	1	2	yes	—
93	C₄×Dic₉	C₂²×C₃	1	4	yes	—
94	C₂²×C₃×C₉	C₂⁴	1	1	yes	characteristic
95	C₃×C₄×C₉	—	2	2	no	—
96	C₃×C₄×C₉	—	2	2	no	—
97	C₃×C₄×C₉	—	2	1	no	—
98	C₃×C₄×C₉	C₂×Q₈	1	2	yes	—
99	C₃×C₄×C₉	#41	1	2	yes	—
100	C₃×Dic₉	—	2	4	no	—
101	C₃×Dic₉	—	2	2	no	—
102	C₃×Dic₉	C₂×Q₈	1	4	yes	—
103	C₃×C₄⋊Q₈	—	9	2	no	—
104	C₃×C₄⋊Q₈	—	9	1	no	—
105	C₃×#125	—	9	4	no	—
106	C₃×#125	—	9	2	no	—
107	C₃×C₄×Q₈	—	2	1	no	—
108	C₃×C₄×Q₈	—	9	2	no	—
109	C₃×C₄×Q₈	—	9	2	no	—
110	C₃×C₄×Q₈	—	9	1	no	—
111	C₃×C₄×Q₈	D₉	1	1	yes	characteristic
112	C₃×C₄×Q₈	C₃×D₃	1	1	yes	characteristic
113	#944	—	3	2	no	—
114	#889	—	3	4	no	—
115	#887	—	3	2	no	—
116	#870	—	3	2	no	—
117	#869	—	3	1	no	—
118	C₄×Dic₆	—	3	2	no	—
119	C₄×Dic₆	—	3	1	no	—
120	C₃²×Q₈	—	2	2	no	—
121	C₂×C₃²×C₄	C₂²×D₃	1	2	yes	—
122	C₂×C₃²×C₄	C₂²×D₃	1	2	yes	—
123	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
124	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
125	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
126	C₂×C₃×Dic₃	—	3	2	no	—
127	C₂×C₃×Dic₃	—	3	2	no	—
128	C₂×C₃×Dic₃	—	3	4	no	—
129	C₃×Dic₆	—	3	4	no	—
130	C₃×Dic₆	—	6	4	no	—
131	Q₈×C₉	—	2	4	no	—
132	Q₈×C₉	—	2	2	no	—
133	C₂×C₄×C₉	—	2	1	no	—
134	C₂×C₄×C₉	—	2	2	no	—
135	C₂×C₄×C₉	—	2	1	no	—
136	C₂×C₄×C₉	—	2	2	no	—
137	C₂×C₄×C₉	—	2	1	no	—
138	C₂×C₄×C₉	C₂³×C₃	1	2	yes	—
139	C₂×C₄×C₉	C₂³×C₃	1	2	yes	—
140	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
141	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
142	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
143	C₂×Dic₉	C₂³×C₃	1	2	yes	—
144	C₂×Dic₉	C₂³×C₃	1	4	yes	—
145	C₂×Dic₉	C₂³×C₃	1	2	yes	—
146	Dic₁₈	—	2	4	no	—
147	Dic₁₈	C₃×Q₈	1	4	yes	complemented
148	#557	—	9	1	no	Sylow
149	C₂×C₃×C₉	C₂²×Q₈	1	1	yes	characteristic
150	C₂×C₃×C₉	C₂×#41	1	1	yes	characteristic
151	C₂×C₃×C₉	2₋¹⁺⁴	1	1	yes	characteristic
152	C₂×C₃×Q₈	—	2	1	no	—
153	C₂×C₃×Q₈	—	9	1	no	—
154	C₂×C₃×Q₈	—	9	2	no	—
155	C₂×C₃×Q₈	C₂×D₉	1	1	yes	characteristic
156	C₂×C₃×Q₈	C₂×C₃×D₃	1	1	yes	characteristic
157	C₂×Dic₆	—	3	1	no	—
158	C₂×Dic₆	—	3	2	no	—
159	C₃×C₄⋊C₄	—	2	1	no	—
160	C₃×C₄⋊C₄	—	2	2	no	—
161	C₃×C₄⋊C₄	—	9	2	no	—
162	C₃×C₄⋊C₄	—	9	4	no	—
163	C₃×C₄⋊C₄	—	9	4	no	—
164	C₃×C₄⋊C₄	—	9	4	no	—
165	C₃×C₄⋊C₄	—	9	1	no	—
166	C₃×C₄⋊C₄	—	9	2	no	—
167	C₃×C₄⋊C₄	—	9	2	no	—
168	C₃×C₄⋊C₄	C₂×D₉	1	1	yes	characteristic
169	C₃×C₄⋊C₄	C₂×D₉	1	2	yes	—
170	C₃×C₄⋊C₄	C₂×C₃×D₃	1	1	yes	characteristic
171	C₃×C₄⋊C₄	C₂×C₃×D₃	1	2	yes	—
172	C₃×C₄²	—	2	2	no	—
173	C₃×C₄²	—	2	1	no	—
174	C₃×C₄²	—	9	4	no	—
175	C₃×C₄²	—	9	2	no	—
176	C₃×C₄²	C₂×D₉	1	2	yes	—
177	C₃×C₄²	C₂×D₉	1	1	yes	characteristic
178	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
179	C₃×C₄²	C₂×C₃×D₃	1	2	yes	—
180	C₄⋊Dic₃	—	3	2	no	—
181	C₄⋊Dic₃	—	3	1	no	—
182	C₄⋊Dic₃	—	3	2	no	—
183	Dic₃⋊C₄	—	3	2	no	—
184	Dic₃⋊C₄	—	3	4	no	—
185	Dic₃⋊C₄	—	3	4	no	—
186	Dic₃⋊C₄	—	3	4	no	—
187	C₄×Dic₃	—	3	4	no	—
188	C₄×Dic₃	—	3	2	no	—
189	C₂²×C₃²	C₂³×D₃	1	1	yes	characteristic
190	C₃²×C₄	—	2	2	no	—
191	C₃²×C₄	—	2	2	no	—
192	C₃²×C₄	—	2	1	no	—
193	C₃²×C₄	#235	1	2	yes	—
194	C₃²×C₄	D₃×Q₈	1	2	yes	—
195	C₃×Dic₃	—	3	4	no	—
196	C₃×Dic₃	—	6	4	no	—
197	C₃×Dic₃	—	6	2	no	—
198	C₂²×C₉	—	2	1	no	—
199	C₂²×C₉	C₂⁴×C₃	1	1	yes	characteristic
200	C₄×C₉	—	2	4	no	—
201	C₄×C₉	—	2	2	no	—
202	C₄×C₉	—	2	2	no	—
203	C₄×C₉	—	2	1	no	—
204	C₄×C₉	—	2	2	no	—
205	C₄×C₉	—	2	2	no	—
206	C₄×C₉	—	4	2	no	—
207	C₄×C₉	—	4	1	no	—
208	C₄×C₉	C₂×C₃×Q₈	1	2	yes	—
209	C₄×C₉	C₃×#41	1	2	yes	—
210	Dic₉	—	2	4	no	—
211	Dic₉	—	2	2	no	—
212	Dic₉	C₂×C₃×Q₈	1	4	yes	—
213	C₄⋊Q₈	—	9	2	no	—
214	C₄⋊Q₈	—	9	1	no	—
215	#125	—	9	4	no	—
216	#125	—	9	2	no	—
217	C₄×Q₈	—	9	2	no	—
218	C₄×Q₈	—	9	2	no	—
219	C₄×Q₈	—	9	1	no	—
220	C₄×Q₈	C₃×D₉	1	1	yes	characteristic
221	C₃×C₉	#557	1	1	yes	characteristic, complemented, Sylow
222	C₃×Q₈	—	2	4	no	—
223	C₃×Q₈	—	2	2	no	—
224	C₃×Q₈	—	2	2	no	—
225	C₃×Q₈	—	9	4	no	—
226	C₃×Q₈	—	18	4	no	—
227	C₂×C₃×C₄	—	2	1	no	—
228	C₂×C₃×C₄	—	2	2	no	—
229	C₂×C₃×C₄	—	2	1	no	—
230	C₂×C₃×C₄	—	2	2	no	—
231	C₂×C₃×C₄	—	2	1	no	—
232	C₂×C₃×C₄	—	9	2	no	—
233	C₂×C₃×C₄	—	9	2	no	—
234	C₂×C₃×C₄	—	9	4	no	—
235	C₂×C₃×C₄	C₂²×D₉	1	2	yes	—
236	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
237	C₂×C₃×C₄	C₂²×D₉	1	2	yes	—
238	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
239	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
240	C₂×C₃×C₄	C₂²×C₃×D₃	1	2	yes	—
241	C₂×C₃×C₄	C₂²×C₃×D₃	1	2	yes	—
242	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
243	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
244	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
245	C₂×Dic₃	—	3	2	no	—
246	C₂×Dic₃	—	3	2	no	—
247	C₂×Dic₃	—	3	4	no	—
248	Dic₆	—	3	4	no	—
249	Dic₆	—	6	4	no	—
250	C₂×C₃²	C₂×#235	1	1	yes	characteristic
251	C₂×C₃²	C₂×D₃×Q₈	1	1	yes	characteristic
252	C₂×C₃²	#1010	1	1	yes	characteristic
253	C₂×C₉	—	2	1	no	—
254	C₂×C₉	—	2	1	no	—
255	C₂×C₉	—	2	1	no	—
256	C₂×C₉	C₂²×C₃×Q₈	1	1	yes	characteristic
257	C₂×C₉	C₂×C₃×#41	1	1	yes	characteristic
258	C₂×C₉	C₃×2₋¹⁺⁴	1	1	yes	characteristic
259	C₂×Q₈	—	9	1	no	—
260	C₂×Q₈	—	9	2	no	—
261	C₂×Q₈	C₂×C₃×D₉	1	1	yes	characteristic
262	C₄⋊C₄	—	9	2	no	—
263	C₄⋊C₄	—	9	4	no	—
264	C₄⋊C₄	—	9	1	no	—
265	C₄⋊C₄	—	9	4	no	—
266	C₄⋊C₄	—	9	2	no	—
267	C₄⋊C₄	—	9	4	no	—
268	C₄⋊C₄	—	9	2	no	—
269	C₄⋊C₄	C₂×C₃×D₉	1	1	yes	characteristic
270	C₄⋊C₄	C₂×C₃×D₉	1	2	yes	—
271	C₄²	—	9	4	no	—
272	C₄²	—	9	2	no	—
273	C₄²	C₂×C₃×D₉	1	2	yes	—
274	C₄²	C₂×C₃×D₉	1	1	yes	characteristic
275	C₂²×C₃	—	2	1	no	—
276	C₂²×C₃	C₂³×D₉	1	1	yes	characteristic
277	C₂²×C₃	C₂³×C₃×D₃	1	1	yes	characteristic
278	C₃×C₄	—	2	4	no	—
279	C₃×C₄	—	2	2	no	—
280	C₃×C₄	—	2	2	no	—
281	C₃×C₄	—	2	1	no	—
282	C₃×C₄	—	2	2	no	—
283	C₃×C₄	—	2	2	no	—
284	C₃×C₄	—	2	1	no	—
285	C₃×C₄	—	2	2	no	—
286	C₃×C₄	—	2	2	no	—
287	C₃×C₄	—	4	2	no	—
288	C₃×C₄	—	4	1	no	—
289	C₃×C₄	—	9	4	no	—
290	C₃×C₄	—	18	4	no	—
291	C₃×C₄	—	18	2	no	—
292	C₃×C₄	#3697	1	2	yes	—
293	C₃×C₄	Q₈×D₉	1	2	yes	—
294	C₃×C₄	C₃×#235	1	2	yes	—
295	C₃×C₄	C₃×D₃×Q₈	1	2	yes	—
296	Dic₃	—	3	4	no	—
297	Dic₃	—	6	4	no	—
298	Dic₃	—	6	2	no	—
299	C₃²	#5567	1	1	yes	characteristic
300	C₉	—	2	1	no	—
301	C₉	C₃×#557	1	1	yes	characteristic, complemented
302	Q₈	—	2	2	no	—
303	Q₈	—	9	4	no	—
304	Q₈	—	18	4	no	—
305	C₂×C₄	—	9	2	no	—
306	C₂×C₄	—	9	2	no	—
307	C₂×C₄	—	9	4	no	—
308	C₂×C₄	C₂²×C₃×D₉	1	2	yes	—
309	C₂×C₄	C₂²×C₃×D₉	1	1	yes	characteristic
310	C₂×C₄	C₂²×C₃×D₉	1	2	yes	—
311	C₂×C₄	C₂²×C₃×D₉	1	1	yes	characteristic
312	C₂×C₄	C₂²×C₃×D₉	1	1	yes	characteristic
313	C₂×C₃	—	2	1	no	—
314	C₂×C₃	—	2	1	no	—
315	C₂×C₃	—	2	1	no	—
316	C₂×C₃	C₂×#3697	1	1	yes	characteristic
317	C₂×C₃	C₂×Q₈×D₉	1	1	yes	characteristic
318	C₂×C₃	#63716	1	1	yes	characteristic
319	C₂×C₃	C₂×C₃×#235	1	1	yes	characteristic
320	C₂×C₃	C₂×C₃×D₃×Q₈	1	1	yes	characteristic
321	C₂×C₃	C₃×#1010	1	1	yes	characteristic
322	C₂²	C₂³×C₃×D₉	1	1	yes	characteristic
323	C₄	—	2	2	no	—
324	C₄	—	2	2	no	—
325	C₄	—	2	1	no	—
326	C₄	—	9	4	no	—
327	C₄	—	18	4	no	—
328	C₄	—	18	2	no	—
329	C₄	C₃×#3697	1	2	yes	—
330	C₄	C₃×Q₈×D₉	1	2	yes	—
331	C₃	—	2	1	no	—
332	C₃	#10589669	1	1	yes	characteristic, direct product factor
333	C₃	C₃×#5567	1	1	yes	characteristic
334	C₂	C₂×C₃×#3697	1	1	yes	characteristic
335	C₂	C₂×C₃×Q₈×D₉	1	1	yes	characteristic
336	C₂	C₃×#63716	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂³×D₉
commutator, cosocle	C₂²×C₉	C₂⁴×C₃
Frattini	C₂²×C₃	C₂³×C₃×D₃
Fitting	C₃×C₄×Q₈×C₉	C₂
socle, prime radical	C₂²×C₃²	C₂³×D₃
hypercenter	C₃×C₄×Q₈	D₉
hypocenter	C₉	C₃×#557
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄×Q₈

Complements

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

N	H ≅ G/N	type
10. #10589669	C₃	direct (N × H)
54. C₃×Dic₁₈	Q₈	semidirect
147. Dic₁₈	C₃×Q₈	semidirect
221. C₃×C₉	#557	semidirect
301. C₉	C₃×#557	semidirect
332. C₃	#10589669	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₄×Q₈×C₉	2
C₃×#63506	2
C₃×#63451	2
C₃×#63449	2
C₃×#63432	2
C₃×#63431	2
C₃×C₄×Dic₁₈	2
C₃×#5567	3
#10589669	3

Subgroups — C3×#10589669 #13152668

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10589669 #13152668