Subgroups — C₃×#10588523 #13142019

Order 1728.

380 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#3663	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₈×D₉	C₂	1	1	yes	maximal, characteristic
3	C₃×#63434	C₂	1	1	yes	maximal, characteristic
4	C₃×C₉×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×Dic₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×Dic₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×C₄⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#4839	—	3	1	no	maximal
9	#10588523	C₃	1	1	yes	maximal, characteristic, direct product factor
10	C₂×C₃×C₄×D₉	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
13	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
14	C₃×(C₂×D₉)⋊C₄	C₄	1	1	yes	characteristic
15	C₃×(C₂×D₉)⋊C₄	C₄	1	1	yes	characteristic
16	C₃×Dic₉⋊C₄	C₄	1	1	yes	characteristic
17	C₃×Dic₉⋊C₄	C₄	1	1	yes	characteristic
18	C₃×C₄×Dic₉	C₂²	1	1	yes	characteristic
19	C₂×C₃×C₉⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₃×C₉⋊C₈	C₂²	1	1	yes	characteristic
21	C₃×#3663	—	2	2	no	—
22	C₃×C₈×D₉	—	2	2	no	—
23	C₂×C₃×ΓL(1,𝔽₂₅)	—	3	1	no	—
24	C₂×C₃×D₃×C₈	—	3	1	no	—
25	C₃×#872	—	3	1	no	—
26	C₃²×C₄⋊C₈	D₃	1	1	yes	characteristic
27	C₃×Dic₃⋊C₈	—	3	1	no	—
28	C₃×Dic₃⋊C₈	—	3	1	no	—
29	C₃×C₄⋊(C₃⋊C₈)	—	3	1	no	—
30	C₂×#3663	C₂×C₃	1	1	yes	characteristic
31	C₂×C₈×D₉	C₂×C₃	1	1	yes	characteristic
32	#63434	C₂×C₃	1	1	yes	characteristic
33	C₉×C₄⋊C₈	—	2	1	no	—
34	C₉×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
35	Dic₉⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
36	Dic₉⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
37	C₄⋊(C₉⋊C₈)	C₂×C₃	1	1	yes	characteristic, complemented
38	C₂²×C₃×D₉	C₂×C₄	1	1	yes	characteristic
39	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
40	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
41	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
42	C₂×C₃×Dic₉	C₂×C₄	1	1	yes	characteristic
43	C₂×C₃×Dic₉	C₂×C₄	1	1	yes	characteristic
44	C₂×C₃×Dic₉	C₂×C₄	1	1	yes	characteristic
45	C₃×C₄×D₉	D₄	1	2	yes	—
46	C₃×C₄×D₉	Q₈	1	2	yes	—
47	C₃×C₈×C₉	—	2	1	no	—
48	C₃×C₈×C₉	—	2	1	no	—
49	C₃×C₉⋊C₈	—	2	1	no	—
50	C₃×C₉⋊C₈	—	2	1	no	—
51	C₃×#424	—	9	1	no	—
52	#4839	—	3	1	no	—
53	C₂×C₃×C₄×D₃	—	3	1	no	—
54	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
55	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
56	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
57	C₃×(C₂×D₃)⋊C₄	—	3	1	no	—
58	C₃×(C₂×D₃)⋊C₄	—	3	1	no	—
59	C₃×Dic₃⋊C₄	—	3	1	no	—
60	C₃×Dic₃⋊C₄	—	3	1	no	—
61	C₃×C₄×Dic₃	—	3	1	no	—
62	C₂×C₃×C₃⋊C₈	—	3	1	no	—
63	C₂×C₃×C₃⋊C₈	—	3	1	no	—
64	C₃×ΓL(1,𝔽₂₅)	—	6	2	no	—
65	C₃×D₃×C₈	—	6	2	no	—
66	C₂×C₄×D₉	C₂²×C₃	1	1	yes	characteristic
67	C₂×C₈×C₉	—	2	1	no	—
68	C₂×C₈×C₉	—	2	1	no	—
69	C₂×C₈×C₉	C₂²×C₃	1	1	yes	characteristic
70	C₂×C₈×C₉	C₂²×C₃	1	1	yes	characteristic
71	C₄²×C₉	—	2	1	no	—
72	C₄²×C₉	C₂²×C₃	1	1	yes	characteristic
73	(C₂×D₉)⋊C₄	C₃×C₄	1	1	yes	characteristic
74	(C₂×D₉)⋊C₄	C₃×C₄	1	1	yes	characteristic
75	Dic₉⋊C₄	C₃×C₄	1	1	yes	characteristic
76	Dic₉⋊C₄	C₃×C₄	1	1	yes	characteristic
77	C₄×Dic₉	C₂²×C₃	1	1	yes	characteristic
78	C₂×C₉⋊C₈	C₂²×C₃	1	1	yes	characteristic
79	C₂×C₉⋊C₈	C₂²×C₃	1	1	yes	characteristic
80	#3663	—	2	2	no	—
81	C₈×D₉	—	2	2	no	—
82	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₃×D₉	—	2	1	no	—
84	C₂×C₃×D₉	—	2	1	no	—
85	C₂×C₃×D₉	C₄⋊C₄	1	2	yes	—
86	C₃×C₄×C₉	—	2	1	no	—
87	C₃×C₄×C₉	—	2	1	no	—
88	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
89	C₃×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
90	C₃×Dic₉	—	2	1	no	—
91	C₃×Dic₉	—	2	1	no	—
92	C₃×Dic₉	C₄⋊C₄	1	2	yes	—
93	C₂×C₃×M₁₆	—	9	1	no	—
94	C₂²×C₃×C₈	—	9	1	no	—
95	C₃×#117	—	9	1	no	—
96	C₂×ΓL(1,𝔽₂₅)	—	3	1	no	—
97	C₂×D₃×C₈	—	3	1	no	—
98	#872	—	3	1	no	—
99	C₃×C₄⋊C₈	—	2	1	no	—
100	C₃×C₄⋊C₈	—	9	1	no	—
101	C₃×C₄⋊C₈	—	9	1	no	—
102	C₃×C₄⋊C₈	—	9	1	no	—
103	C₃×C₄⋊C₈	D₉	1	1	yes	characteristic, complemented
104	C₃×C₄⋊C₈	C₃×D₃	1	1	yes	characteristic
105	Dic₃⋊C₈	—	3	1	no	—
106	Dic₃⋊C₈	—	3	1	no	—
107	C₄⋊(C₃⋊C₈)	—	3	1	no	—
108	C₂²×C₃×D₃	—	3	1	no	—
109	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
110	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
111	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
112	C₂×C₃×Dic₃	—	3	1	no	—
113	C₂×C₃×Dic₃	—	3	1	no	—
114	C₂×C₃×Dic₃	—	3	1	no	—
115	C₃×C₄×D₃	—	3	2	no	—
116	C₃×C₄×D₃	—	3	2	no	—
117	C₂²×D₉	C₂×C₃×C₄	1	1	yes	characteristic
118	C₃²×C₈	—	2	1	no	—
119	C₃²×C₈	—	2	1	no	—
120	C₃×C₃⋊C₈	—	6	1	no	—
121	C₃×C₃⋊C₈	—	6	1	no	—
122	C₂×C₄×C₉	—	2	1	no	—
123	C₂×C₄×C₉	—	2	1	no	—
124	C₂×C₄×C₉	—	2	1	no	—
125	C₂×C₄×C₉	C₂×C₃×C₄	1	1	yes	characteristic
126	C₂×C₄×C₉	C₂×C₃×C₄	1	1	yes	characteristic
127	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
128	C₂×Dic₉	C₂×C₃×C₄	1	1	yes	characteristic
129	C₂×Dic₉	C₂×C₃×C₄	1	1	yes	characteristic
130	C₂×Dic₉	C₂×C₃×C₄	1	1	yes	characteristic
131	C₄×D₉	C₃×D₄	1	2	yes	—
132	C₄×D₉	C₃×Q₈	1	2	yes	—
133	C₈×C₉	—	2	1	no	—
134	C₈×C₉	—	2	1	no	—
135	C₈×C₉	—	4	1	no	—
136	C₈×C₉	—	4	1	no	—
137	C₉⋊C₈	—	2	1	no	—
138	C₉⋊C₈	—	2	1	no	—
139	#424	—	9	1	no	Sylow
140	C₂×C₃×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
141	C₂×C₃×C₉	#131	1	1	yes	characteristic
142	C₂×C₃×C₉	#131	1	1	yes	characteristic
143	C₃×D₉	—	2	2	no	—
144	C₂²×C₃×C₄	—	9	1	no	—
145	C₂×C₄×D₃	—	3	1	no	—
146	C₃×M₁₆	—	18	2	no	—
147	C₂×C₃×C₈	—	2	1	no	—
148	C₂×C₃×C₈	—	2	1	no	—
149	C₂×C₃×C₈	—	9	1	no	—
150	C₂×C₃×C₈	—	9	1	no	—
151	C₂×C₃×C₈	—	18	2	no	—
152	C₂×C₃×C₈	C₂×D₉	1	1	yes	characteristic
153	C₂×C₃×C₈	C₂×D₉	1	1	yes	characteristic
154	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
155	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
156	C₃×C₄⋊C₄	—	9	1	no	—
157	C₃×C₄⋊C₄	—	9	1	no	—
158	C₃×C₂²⋊C₄	—	9	1	no	—
159	C₃×C₂²⋊C₄	—	9	1	no	—
160	C₃×C₄²	—	2	1	no	—
161	C₃×C₄²	—	9	1	no	—
162	C₃×C₄²	C₂×D₉	1	1	yes	characteristic
163	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
164	(C₂×D₃)⋊C₄	—	3	1	no	—
165	(C₂×D₃)⋊C₄	—	3	1	no	—
166	Dic₃⋊C₄	—	3	1	no	—
167	Dic₃⋊C₄	—	3	1	no	—
168	C₄×Dic₃	—	3	1	no	—
169	C₂×C₃⋊C₈	—	3	1	no	—
170	C₂×C₃⋊C₈	—	3	1	no	—
171	ΓL(1,𝔽₂₅)	—	6	2	no	—
172	D₃×C₈	—	6	2	no	—
173	C₂²×C₃²	C₂×C₄×D₃	1	1	yes	characteristic
174	C₂×C₃×D₃	—	3	2	no	—
175	C₂×C₃×D₃	—	6	1	no	—
176	C₂×C₃×D₃	—	6	1	no	—
177	C₃²×C₄	—	2	1	no	—
178	C₃²×C₄	—	2	1	no	—
179	C₃²×C₄	D₃×D₄	1	1	yes	characteristic
180	C₃²×C₄	D₃×Q₈	1	1	yes	characteristic
181	C₃×Dic₃	—	3	2	no	—
182	C₃×Dic₃	—	6	1	no	—
183	C₃×Dic₃	—	6	1	no	—
184	C₂²×C₉	—	2	1	no	—
185	C₂²×C₉	C₂²×C₃×C₄	1	1	yes	characteristic
186	C₂×D₉	—	2	1	no	—
187	C₂×D₉	—	2	1	no	—
188	C₂×D₉	C₃×C₄⋊C₄	1	2	yes	—
189	C₄×C₉	—	2	2	no	—
190	C₄×C₉	—	2	1	no	—
191	C₄×C₉	—	2	1	no	—
192	C₄×C₉	—	2	1	no	—
193	C₄×C₉	—	2	1	no	—
194	C₄×C₉	—	4	1	no	—
195	C₄×C₉	C₂×C₃×D₄	1	1	yes	characteristic
196	C₄×C₉	C₂×C₃×Q₈	1	1	yes	characteristic
197	Dic₉	—	2	1	no	—
198	Dic₉	—	2	1	no	—
199	Dic₉	C₃×C₄⋊C₄	1	2	yes	—
200	C₂×M₁₆	—	9	1	no	—
201	C₂²×C₈	—	9	1	no	—
202	#117	—	9	1	no	—
203	C₄⋊C₈	—	9	1	no	—
204	C₄⋊C₈	—	9	1	no	—
205	C₄⋊C₈	—	9	1	no	—
206	C₄⋊C₈	C₃×D₉	1	1	yes	characteristic, complemented
207	C₃×C₉	#424	1	1	yes	characteristic, complemented, Sylow
208	C₂³×C₃	—	9	1	no	—
209	C₂²×D₃	—	3	1	no	—
210	C₂×C₃×C₄	—	2	1	no	—
211	C₂×C₃×C₄	—	2	1	no	—
212	C₂×C₃×C₄	—	2	1	no	—
213	C₂×C₃×C₄	—	9	1	no	—
214	C₂×C₃×C₄	—	9	2	no	—
215	C₂×C₃×C₄	—	9	2	no	—
216	C₂×C₃×C₄	—	9	1	no	—
217	C₂×C₃×C₄	—	9	1	no	—
218	C₂×C₃×C₄	C₄×D₉	1	1	yes	characteristic
219	C₂×C₃×C₄	C₄×D₉	1	1	yes	characteristic
220	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
221	C₂×C₃×C₄	C₃×C₄×D₃	1	1	yes	characteristic
222	C₂×C₃×C₄	C₃×C₄×D₃	1	1	yes	characteristic
223	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
224	C₂×Dic₃	—	3	1	no	—
225	C₂×Dic₃	—	3	1	no	—
226	C₂×Dic₃	—	3	1	no	—
227	C₄×D₃	—	3	2	no	—
228	C₄×D₃	—	3	2	no	—
229	C₃×C₈	—	2	1	no	—
230	C₃×C₈	—	2	1	no	—
231	C₃×C₈	—	2	1	no	—
232	C₃×C₈	—	2	1	no	—
233	C₃×C₈	—	4	1	no	—
234	C₃×C₈	—	4	1	no	—
235	C₃×C₈	—	18	1	no	—
236	C₃×C₈	—	18	1	no	—
237	C₃⋊C₈	—	6	1	no	—
238	C₃⋊C₈	—	6	1	no	—
239	C₂×C₃²	D₃×C₄⋊C₄	1	1	yes	characteristic
240	C₂×C₃²	#901	1	1	yes	characteristic
241	C₂×C₃²	#907	1	1	yes	characteristic
242	C₃×D₃	—	6	2	no	—
243	C₂×C₉	—	2	1	no	—
244	C₂×C₉	—	2	1	no	—
245	C₂×C₉	—	2	1	no	—
246	C₂×C₉	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
247	C₂×C₉	C₃×#131	1	1	yes	characteristic
248	C₂×C₉	C₃×#131	1	1	yes	characteristic
249	D₉	—	2	2	no	—
250	C₂²×C₄	—	9	1	no	—
251	M₁₆	—	18	2	no	—
252	C₂×C₈	—	9	1	no	—
253	C₂×C₈	—	9	1	no	—
254	C₂×C₈	—	18	2	no	—
255	C₂×C₈	C₂×C₃×D₉	1	1	yes	characteristic
256	C₂×C₈	C₂×C₃×D₉	1	1	yes	characteristic
257	C₄⋊C₄	—	9	1	no	—
258	C₄⋊C₄	—	9	1	no	—
259	C₂²⋊C₄	—	9	1	no	—
260	C₂²⋊C₄	—	9	1	no	—
261	C₄²	—	9	1	no	—
262	C₄²	C₂×C₃×D₉	1	1	yes	characteristic
263	C₂²×C₃	—	2	1	no	—
264	C₂²×C₃	—	9	2	no	—
265	C₂²×C₃	—	18	1	no	—
266	C₂²×C₃	—	18	1	no	—
267	C₂²×C₃	C₂×C₄×D₉	1	1	yes	characteristic
268	C₂²×C₃	C₂×C₃×C₄×D₃	1	1	yes	characteristic
269	C₂×D₃	—	3	2	no	—
270	C₂×D₃	—	6	1	no	—
271	C₂×D₃	—	6	1	no	—
272	C₃×C₄	—	2	2	no	—
273	C₃×C₄	—	2	1	no	—
274	C₃×C₄	—	2	1	no	—
275	C₃×C₄	—	2	1	no	—
276	C₃×C₄	—	2	1	no	—
277	C₃×C₄	—	2	1	no	—
278	C₃×C₄	—	2	1	no	—
279	C₃×C₄	—	4	1	no	—
280	C₃×C₄	—	9	2	no	—
281	C₃×C₄	—	18	1	no	—
282	C₃×C₄	—	18	1	no	—
283	C₃×C₄	D₄×D₉	1	1	yes	characteristic
284	C₃×C₄	Q₈×D₉	1	1	yes	characteristic
285	C₃×C₄	C₃×D₃×D₄	1	1	yes	characteristic
286	C₃×C₄	C₃×D₃×Q₈	1	1	yes	characteristic
287	Dic₃	—	3	2	no	—
288	Dic₃	—	6	1	no	—
289	Dic₃	—	6	1	no	—
290	C₃²	#4839	1	1	yes	characteristic
291	C₉	—	2	1	no	—
292	C₉	C₃×#424	1	1	yes	characteristic, complemented
293	C₂³	—	9	1	no	—
294	C₂×C₄	—	9	1	no	—
295	C₂×C₄	—	9	2	no	—
296	C₂×C₄	—	9	2	no	—
297	C₂×C₄	—	9	1	no	—
298	C₂×C₄	—	9	1	no	—
299	C₂×C₄	C₃×C₄×D₉	1	1	yes	characteristic
300	C₂×C₄	C₃×C₄×D₉	1	1	yes	characteristic
301	C₂×C₄	C₂²×C₃×D₉	1	1	yes	characteristic
302	C₈	—	2	1	no	—
303	C₈	—	2	1	no	—
304	C₈	—	18	1	no	—
305	C₈	—	18	1	no	—
306	C₂×C₃	—	2	1	no	—
307	C₂×C₃	—	2	1	no	—
308	C₂×C₃	—	2	1	no	—
309	C₂×C₃	—	18	2	no	—
310	C₂×C₃	C₄⋊C₄×D₉	1	1	yes	characteristic
311	C₂×C₃	#63464	1	1	yes	characteristic
312	C₂×C₃	#63469	1	1	yes	characteristic
313	C₂×C₃	C₃×D₃×C₄⋊C₄	1	1	yes	characteristic
314	C₂×C₃	C₃×#901	1	1	yes	characteristic
315	C₂×C₃	C₃×#907	1	1	yes	characteristic
316	D₃	—	6	2	no	—
317	C₂²	—	9	2	no	—
318	C₂²	—	18	1	no	—
319	C₂²	—	18	1	no	—
320	C₂²	C₂×C₃×C₄×D₉	1	1	yes	characteristic
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	9	2	no	—
324	C₄	—	18	1	no	—
325	C₄	—	18	1	no	—
326	C₄	C₃×D₄×D₉	1	1	yes	characteristic
327	C₄	C₃×Q₈×D₉	1	1	yes	characteristic
328	C₃	—	2	1	no	—
329	C₃	#10588523	1	1	yes	characteristic, direct product factor
330	C₃	C₃×#4839	1	1	yes	characteristic
331	C₂	—	18	2	no	—
332	C₂	C₃×C₄⋊C₄×D₉	1	1	yes	characteristic
333	C₂	C₃×#63464	1	1	yes	characteristic
334	C₂	C₃×#63469	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂²×D₉
commutator	C₂²×C₉	C₂²×C₃×C₄
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₃×C₉×C₄⋊C₈	C₂
socle	C₂²×C₃²	C₂×C₄×D₃
prime radical	C₂²×C₃×D₉	C₂×C₄
cosocle	C₂×C₄×C₉	C₂³×C₃
hypercenter	C₃×C₄⋊C₈	D₉
hypocenter	C₉	C₃×#424
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₃×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
4. C₃×C₉×C₄⋊C₈	C₂	semidirect
5. C₃×Dic₉⋊C₈	C₂	semidirect
6. C₃×Dic₉⋊C₈	C₂	semidirect
7. C₃×C₄⋊(C₉⋊C₈)	C₂	semidirect
9. #10588523	C₃	direct (N × H)
34. C₉×C₄⋊C₈	C₂×C₃	semidirect
35. Dic₉⋊C₈	C₂×C₃	semidirect
36. Dic₉⋊C₈	C₂×C₃	semidirect
37. C₄⋊(C₉⋊C₈)	C₂×C₃	semidirect
103. C₃×C₄⋊C₈	D₉	semidirect
206. C₄⋊C₈	C₃×D₉	semidirect
207. C₃×C₉	#424	semidirect
292. C₉	C₃×#424	semidirect
329. C₃	#10588523	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×#3663	2
C₂×C₃×C₈×D₉	2
C₃×#63434	2
C₃×C₉×C₄⋊C₈	2
C₃×Dic₉⋊C₈	2
C₃×C₄⋊(C₉⋊C₈)	2
C₃×#4839	3
#10588523	3

Subgroups — C3×#10588523 #13142019

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10588523 #13142019