Subgroups — C₃×#10589579 #13152285

Order 1728.

1038 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#3697	C₂	1	2	yes	maximal, complemented
2	C₂×C₃×C₄²×C₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#63492	C₂	1	4	yes	maximal, complemented
4	C₃×#63437	C₂	1	2	yes	maximal, complemented
5	C₃×#63436	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×C₄×D₃₆	C₂	1	4	yes	maximal, complemented
7	C₃×#63431	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#5477	—	3	1	no	maximal
9	#10589579	C₃	1	1	yes	maximal, characteristic, direct product factor
10	C₂²×C₃×C₄×C₉	C₂²	1	2	yes	—
11	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
12	C₂×C₃×#611	C₂²	1	4	yes	—
13	C₃×#3697	—	2	8	no	—
14	C₂×C₃×D₃₆	C₂²	1	2	yes	—
15	C₂×C₃×D₃₆	C₂²	1	4	yes	—
16	C₂×C₃×C₄×D₉	C₂²	1	4	yes	—
17	C₂×C₃×Dic₁₈	C₂²	1	2	yes	—
18	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
19	C₃×C₄²×C₉	C₂²	1	2	yes	—
20	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
21	C₃×(C₂×D₉)⋊C₄	C₂²	1	8	yes	—
22	C₃×C₄⋊Dic₉	C₂²	1	4	yes	—
23	C₂×C₃×#235	—	3	2	no	—
24	C₂×C₃²×C₄²	D₃	1	1	yes	characteristic
25	C₃×#930	—	3	4	no	—
26	C₃×#875	—	3	2	no	—
27	C₃×#874	—	3	1	no	—
28	C₃×C₄×D₁₂	—	3	4	no	—
29	C₃×#869	—	3	1	no	—
30	C₂×#3697	C₂×C₃	1	2	yes	complemented
31	C₂×C₄²×C₉	—	2	1	no	—
32	C₂×C₄²×C₉	C₂×C₃	1	1	yes	characteristic, complemented
33	#63492	C₂×C₃	1	4	yes	complemented
34	#63437	C₂×C₃	1	2	yes	complemented
35	#63436	C₂×C₃	1	1	yes	characteristic, complemented
36	C₄×D₃₆	C₂×C₃	1	4	yes	complemented
37	#63431	C₂×C₃	1	1	yes	characteristic, complemented
38	C₂³×C₃×C₉	C₂³	1	1	yes	characteristic
39	C₂²×C₃×D₉	C₂³	1	4	yes	—
40	C₂×C₃×C₄×C₉	—	2	4	no	—
41	C₂×C₃×C₄×C₉	D₄	1	4	yes	complemented
42	C₂×C₃×C₄×C₉	C₂³	1	2	yes	—
43	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
44	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
45	C₂×C₃×C₄×C₉	C₂³	1	2	yes	—
46	C₃×#611	—	2	8	no	—
47	C₂×C₃×Dic₉	C₂³	1	4	yes	—
48	C₃×D₃₆	—	2	4	no	—
49	C₃×D₃₆	—	2	8	no	—
50	C₃×C₄×D₉	—	2	8	no	—
51	C₃×Dic₁₈	—	2	4	no	—
52	C₃×#532	—	9	1	no	—
53	#5477	—	3	1	no	—
54	C₂²×C₃²×C₄	C₂×D₃	1	2	yes	—
55	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
56	C₂×C₃×#67	—	3	4	no	—
57	C₃×#235	—	6	8	no	—
58	C₂×C₃×D₁₂	—	3	4	no	—
59	C₂×C₃×D₁₂	—	3	2	no	—
60	C₂×C₃×C₄×D₃	—	3	4	no	—
61	C₂×C₃×Dic₆	—	3	2	no	—
62	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
63	C₃²×C₄²	C₂×D₃	1	2	yes	—
64	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
65	C₃×(C₂×D₃)⋊C₄	—	3	8	no	—
66	C₃×C₄⋊Dic₃	—	3	4	no	—
67	C₂²×C₄×C₉	—	2	2	no	—
68	C₂²×C₄×C₉	—	2	1	no	—
69	C₂²×C₄×C₉	C₂²×C₃	1	2	yes	—
70	C₂²×C₄×C₉	C₂²×C₃	1	1	yes	characteristic
71	C₂×#611	C₂²×C₃	1	4	yes	—
72	#3697	—	2	8	no	—
73	C₂×D₃₆	C₂²×C₃	1	4	yes	—
74	C₂×D₃₆	C₂²×C₃	1	2	yes	—
75	C₂×C₄×D₉	C₂²×C₃	1	4	yes	—
76	C₂×Dic₁₈	C₂²×C₃	1	2	yes	—
77	C₄²×C₉	—	2	1	no	—
78	C₄²×C₉	—	2	2	no	—
79	C₄²×C₉	—	2	1	no	—
80	C₄²×C₉	C₂²×C₃	1	1	yes	characteristic
81	C₄²×C₉	C₂²×C₃	1	2	yes	—
82	C₄²×C₉	C₂²×C₃	1	1	yes	characteristic
83	(C₂×D₉)⋊C₄	C₂²×C₃	1	8	yes	—
84	C₄⋊Dic₉	C₂²×C₃	1	4	yes	—
85	C₂²×C₃×C₉	—	2	2	no	—
86	C₂²×C₃×C₉	C₂×D₄	1	2	yes	—
87	C₂²×C₃×C₉	C₂⁴	1	1	yes	characteristic
88	C₂×C₃×D₉	—	2	4	no	—
89	C₂×C₃×D₉	—	2	8	no	—
90	C₃×C₄×C₉	—	2	2	no	—
91	C₃×C₄×C₉	C₂×D₄	1	2	yes	—
92	C₃×C₄×C₉	C₂×D₄	1	2	yes	—
93	C₃×C₄×C₉	#41	1	4	yes	complemented
94	C₃×Dic₉	—	2	4	no	—
95	C₂×C₃×#41	—	9	2	no	—
96	C₂×#235	—	3	2	no	—
97	C₃×C₄⋊Q₈	—	9	1	no	—
98	C₃×#127	—	9	1	no	—
99	C₃×#124	—	9	2	no	—
100	C₃×#121	—	9	4	no	—
101	C₃×C₄×D₄	—	9	4	no	—
102	C₂×C₃×C₄²	—	2	1	no	—
103	C₂×C₃×C₄²	D₉	1	1	yes	characteristic, complemented
104	C₂×C₃×C₄²	C₃×D₃	1	1	yes	characteristic
105	#930	—	3	4	no	—
106	#875	—	3	2	no	—
107	#874	—	3	1	no	—
108	C₄×D₁₂	—	3	4	no	—
109	#869	—	3	1	no	—
110	C₂³×C₃²	C₂²×D₃	1	1	yes	characteristic
111	C₂²×C₃×D₃	—	3	4	no	—
112	C₂×C₃²×C₄	—	2	4	no	—
113	C₂×C₃²×C₄	D₁₂	1	4	yes	—
114	C₂×C₃²×C₄	C₂²×D₃	1	2	yes	—
115	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
116	C₂×C₃²×C₄	C₂²×D₃	1	2	yes	—
117	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
118	C₃×#67	—	6	8	no	—
119	C₂×C₃×Dic₃	—	3	4	no	—
120	C₃×D₁₂	—	6	4	no	—
121	C₃×D₁₂	—	6	8	no	—
122	C₃×C₄×D₃	—	6	8	no	—
123	C₃×Dic₆	—	6	4	no	—
124	C₂³×C₉	—	2	1	no	—
125	C₂³×C₉	C₂³×C₃	1	1	yes	characteristic
126	C₂²×D₉	C₂³×C₃	1	4	yes	—
127	C₂×C₄×C₉	—	2	8	no	—
128	C₂×C₄×C₉	—	2	4	no	—
129	C₂×C₄×C₉	—	2	4	no	—
130	C₂×C₄×C₉	—	2	2	no	—
131	C₂×C₄×C₉	—	2	1	no	—
132	C₂×C₄×C₉	—	2	1	no	—
133	C₂×C₄×C₉	—	2	2	no	—
134	C₂×C₄×C₉	C₃×D₄	1	4	yes	complemented
135	C₂×C₄×C₉	C₂³×C₃	1	2	yes	—
136	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
137	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
138	C₂×C₄×C₉	C₂³×C₃	1	2	yes	—
139	#611	—	2	8	no	—
140	C₂×Dic₉	C₂³×C₃	1	4	yes	—
141	D₃₆	—	2	4	no	—
142	D₃₆	—	2	8	no	—
143	C₄×D₉	—	2	8	no	—
144	Dic₁₈	—	2	4	no	—
145	#532	—	9	1	no	Sylow
146	C₂×C₃×C₉	—	2	2	no	—
147	C₂×C₃×C₉	C₂²×D₄	1	1	yes	characteristic
148	C₂×C₃×C₉	C₂×#41	1	2	yes	—
149	C₃×D₉	—	4	4	no	—
150	C₃×#41	—	18	8	no	—
151	C₂×C₃×Q₈	—	9	2	no	—
152	C₂×C₃×D₄	—	9	4	no	—
153	C₂×C₃×D₄	—	9	4	no	—
154	C₂×C₃×D₄	—	9	2	no	—
155	C₂²×C₃×C₄	—	2	2	no	—
156	C₂²×C₃×C₄	—	2	1	no	—
157	C₂²×C₃×C₄	—	9	4	no	—
158	C₂²×C₃×C₄	C₂×D₉	1	2	yes	—
159	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
160	C₂²×C₃×C₄	C₂×C₃×D₃	1	2	yes	—
161	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
162	C₂×#67	—	3	4	no	—
163	#235	—	6	8	no	—
164	C₂×D₁₂	—	3	4	no	—
165	C₂×D₁₂	—	3	2	no	—
166	C₂×C₄×D₃	—	3	4	no	—
167	C₂×Dic₆	—	3	2	no	—
168	C₃×C₄⋊C₄	—	9	4	no	—
169	C₃×C₂²⋊C₄	—	9	8	no	—
170	C₃×C₄²	—	2	2	no	—
171	C₃×C₄²	—	2	1	no	—
172	C₃×C₄²	—	2	1	no	—
173	C₃×C₄²	C₂×D₉	1	1	yes	characteristic
174	C₃×C₄²	C₂×D₉	1	1	yes	characteristic
175	C₃×C₄²	C₂×D₉	1	2	yes	—
176	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
177	C₃×C₄²	C₂×C₃×D₃	1	2	yes	—
178	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
179	(C₂×D₃)⋊C₄	—	3	8	no	—
180	C₄⋊Dic₃	—	3	4	no	—
181	C₂²×C₃²	—	2	2	no	—
182	C₂²×C₃²	C₂×D₁₂	1	2	yes	—
183	C₂²×C₃²	C₂³×D₃	1	1	yes	characteristic
184	C₂×C₃×D₃	—	6	4	no	—
185	C₂×C₃×D₃	—	6	8	no	—
186	C₃²×C₄	—	2	2	no	—
187	C₃²×C₄	C₂×D₁₂	1	2	yes	—
188	C₃²×C₄	C₂×D₁₂	1	2	yes	—
189	C₃²×C₄	#235	1	4	yes	—
190	C₃×Dic₃	—	6	4	no	—
191	C₂²×C₉	—	2	4	no	—
192	C₂²×C₉	—	2	2	no	—
193	C₂²×C₉	—	2	1	no	—
194	C₂²×C₉	—	2	2	no	—
195	C₂²×C₉	C₂×C₃×D₄	1	2	yes	—
196	C₂²×C₉	C₂⁴×C₃	1	1	yes	characteristic
197	C₂×D₉	—	2	4	no	—
198	C₂×D₉	—	2	8	no	—
199	C₄×C₉	—	2	4	no	—
200	C₄×C₉	—	2	2	no	—
201	C₄×C₉	—	2	2	no	—
202	C₄×C₉	—	2	2	no	—
203	C₄×C₉	—	2	4	no	—
204	C₄×C₉	C₂×C₃×D₄	1	2	yes	—
205	C₄×C₉	C₂×C₃×D₄	1	2	yes	—
206	C₄×C₉	C₃×#41	1	4	yes	complemented
207	Dic₉	—	2	4	no	—
208	C₂×#41	—	9	2	no	—
209	C₄⋊Q₈	—	9	1	no	—
210	#127	—	9	1	no	—
211	#124	—	9	2	no	—
212	#121	—	9	4	no	—
213	C₄×D₄	—	9	4	no	—
214	C₂×C₄²	C₃×D₉	1	1	yes	characteristic, complemented
215	C₃×C₉	#532	1	1	yes	characteristic, complemented, Sylow
216	C₂³×C₃	—	2	1	no	—
217	C₂³×C₃	—	9	4	no	—
218	C₂³×C₃	C₂²×D₉	1	1	yes	characteristic
219	C₂³×C₃	C₂²×C₃×D₃	1	1	yes	characteristic
220	C₂²×D₃	—	3	4	no	—
221	C₃×Q₈	—	18	4	no	—
222	C₃×D₄	—	18	8	no	—
223	C₃×D₄	—	18	4	no	—
224	C₃×D₄	—	18	8	no	—
225	C₂×C₃×C₄	—	2	8	no	—
226	C₂×C₃×C₄	—	2	4	no	—
227	C₂×C₃×C₄	—	2	4	no	—
228	C₂×C₃×C₄	—	2	4	no	—
229	C₂×C₃×C₄	—	2	2	no	—
230	C₂×C₃×C₄	—	2	1	no	—
231	C₂×C₃×C₄	—	2	1	no	—
232	C₂×C₃×C₄	—	2	2	no	—
233	C₂×C₃×C₄	—	9	4	no	—
234	C₂×C₃×C₄	—	18	8	no	—
235	C₂×C₃×C₄	D₃₆	1	4	yes	complemented
236	C₂×C₃×C₄	C₂²×D₉	1	2	yes	—
237	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
238	C₂×C₃×C₄	C₂²×D₉	1	2	yes	—
239	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
240	C₂×C₃×C₄	C₃×D₁₂	1	4	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	2	yes	—
245	#67	—	6	8	no	—
246	C₂×Dic₃	—	3	4	no	—
247	D₁₂	—	6	4	no	—
248	D₁₂	—	6	8	no	—
249	C₄×D₃	—	6	8	no	—
250	Dic₆	—	6	4	no	—
251	C₂×C₃²	—	2	2	no	—
252	C₂×C₃²	C₂²×D₁₂	1	1	yes	characteristic
253	C₂×C₃²	C₂×#235	1	2	yes	—
254	C₃×D₃	—	12	4	no	—
255	C₂×C₉	—	2	4	no	—
256	C₂×C₉	—	2	2	no	—
257	C₂×C₉	—	2	1	no	—
258	C₂×C₉	—	2	2	no	—
259	C₂×C₉	C₂²×C₃×D₄	1	1	yes	characteristic
260	C₂×C₉	C₂×C₃×#41	1	2	yes	—
261	D₉	—	4	4	no	—
262	#41	—	18	8	no	—
263	C₂×Q₈	—	9	2	no	—
264	C₂×D₄	—	9	4	no	—
265	C₂×D₄	—	9	4	no	—
266	C₂×D₄	—	9	2	no	—
267	C₂²×C₄	—	9	4	no	—
268	C₂²×C₄	C₂×C₃×D₉	1	2	yes	—
269	C₂²×C₄	C₂×C₃×D₉	1	1	yes	characteristic
270	C₄⋊C₄	—	9	4	no	—
271	C₂²⋊C₄	—	9	8	no	—
272	C₄²	C₂×C₃×D₉	1	2	yes	—
273	C₄²	C₂×C₃×D₉	1	1	yes	characteristic
274	C₄²	C₂×C₃×D₉	1	1	yes	characteristic
275	C₂²×C₃	—	2	4	no	—
276	C₂²×C₃	—	2	2	no	—
277	C₂²×C₃	—	2	2	no	—
278	C₂²×C₃	—	2	1	no	—
279	C₂²×C₃	—	2	2	no	—
280	C₂²×C₃	—	18	4	no	—
281	C₂²×C₃	—	18	8	no	—
282	C₂²×C₃	C₂×D₃₆	1	2	yes	—
283	C₂²×C₃	C₂³×D₉	1	1	yes	characteristic
284	C₂²×C₃	C₂×C₃×D₁₂	1	2	yes	—
285	C₂²×C₃	C₂³×C₃×D₃	1	1	yes	characteristic
286	C₂×D₃	—	6	4	no	—
287	C₂×D₃	—	6	8	no	—
288	C₃×C₄	—	2	4	no	—
289	C₃×C₄	—	2	2	no	—
290	C₃×C₄	—	2	2	no	—
291	C₃×C₄	—	2	2	no	—
292	C₃×C₄	—	2	2	no	—
293	C₃×C₄	—	2	4	no	—
294	C₃×C₄	—	18	4	no	—
295	C₃×C₄	C₂×D₃₆	1	2	yes	—
296	C₃×C₄	C₂×D₃₆	1	2	yes	—
297	C₃×C₄	#3697	1	4	yes	complemented
298	C₃×C₄	C₂×C₃×D₁₂	1	2	yes	—
299	C₃×C₄	C₂×C₃×D₁₂	1	2	yes	—
300	C₃×C₄	C₃×#235	1	4	yes	—
301	Dic₃	—	6	4	no	—
302	C₃²	#5477	1	1	yes	characteristic
303	C₉	—	2	1	no	—
304	C₉	C₃×#532	1	1	yes	characteristic, complemented
305	C₂³	—	9	4	no	—
306	C₂³	C₂²×C₃×D₉	1	1	yes	characteristic
307	Q₈	—	18	4	no	—
308	D₄	—	18	8	no	—
309	D₄	—	18	4	no	—
310	D₄	—	18	8	no	—
311	C₂×C₄	—	2	4	no	—
312	C₂×C₄	—	9	4	no	—
313	C₂×C₄	—	18	8	no	—
314	C₂×C₄	C₃×D₃₆	1	4	yes	complemented
315	C₂×C₄	C₂²×C₃×D₉	1	2	yes	—
316	C₂×C₄	C₂²×C₃×D₉	1	1	yes	characteristic
317	C₂×C₄	C₂²×C₃×D₉	1	1	yes	characteristic
318	C₂×C₄	C₂²×C₃×D₉	1	2	yes	—
319	C₂×C₃	—	2	4	no	—
320	C₂×C₃	—	2	2	no	—
321	C₂×C₃	—	2	2	no	—
322	C₂×C₃	—	2	1	no	—
323	C₂×C₃	—	2	2	no	—
324	C₂×C₃	—	36	4	no	—
325	C₂×C₃	C₂²×D₃₆	1	1	yes	characteristic
326	C₂×C₃	C₂×#3697	1	2	yes	—
327	C₂×C₃	C₂²×C₃×D₁₂	1	1	yes	characteristic
328	C₂×C₃	C₂×C₃×#235	1	2	yes	—
329	D₃	—	12	4	no	—
330	C₂²	—	2	2	no	—
331	C₂²	—	18	4	no	—
332	C₂²	—	18	8	no	—
333	C₂²	C₂×C₃×D₃₆	1	2	yes	—
334	C₂²	C₂³×C₃×D₉	1	1	yes	characteristic
335	C₄	—	2	2	no	—
336	C₄	—	18	4	no	—
337	C₄	C₂×C₃×D₃₆	1	2	yes	—
338	C₄	C₂×C₃×D₃₆	1	2	yes	—
339	C₄	C₃×#3697	1	4	yes	complemented
340	C₃	—	2	1	no	—
341	C₃	#10589579	1	1	yes	characteristic, direct product factor
342	C₃	C₃×#5477	1	1	yes	characteristic
343	C₂	—	2	2	no	—
344	C₂	—	36	4	no	—
345	C₂	C₂²×C₃×D₃₆	1	1	yes	characteristic
346	C₂	C₂×C₃×#3697	1	2	yes	—

	subgroup	G/H
center	C₂×C₃×C₄	C₂²×D₉
commutator, cosocle	C₂²×C₉	C₂⁴×C₃
Frattini	C₂²×C₃	C₂³×C₃×D₃
Fitting	C₂×C₃×C₄²×C₉	C₂
socle	C₂²×C₃²	C₂³×D₃
hypercenter	C₂×C₃×C₄²	D₉
hypocenter	C₉	C₃×#532
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₄²

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#3697	C₂	semidirect
2. C₂×C₃×C₄²×C₉	C₂	semidirect
3. C₃×#63492	C₂	semidirect
4. C₃×#63437	C₂	semidirect
5. C₃×#63436	C₂	semidirect
6. C₃×C₄×D₃₆	C₂	semidirect
7. C₃×#63431	C₂	semidirect
9. #10589579	C₃	direct (N × H)
30. C₂×#3697	C₂×C₃	semidirect
32. C₂×C₄²×C₉	C₂×C₃	semidirect
33. #63492	C₂×C₃	semidirect
34. #63437	C₂×C₃	semidirect
35. #63436	C₂×C₃	semidirect
36. C₄×D₃₆	C₂×C₃	semidirect
37. #63431	C₂×C₃	semidirect
41. C₂×C₃×C₄×C₉	D₄	semidirect
93. C₃×C₄×C₉	#41	semidirect
103. C₂×C₃×C₄²	D₉	semidirect
134. C₂×C₄×C₉	C₃×D₄	semidirect
206. C₄×C₉	C₃×#41	semidirect
214. C₂×C₄²	C₃×D₉	semidirect
215. C₃×C₉	#532	semidirect
235. C₂×C₃×C₄	D₃₆	semidirect
297. C₃×C₄	#3697	semidirect
304. C₉	C₃×#532	semidirect
314. C₂×C₄	C₃×D₃₆	semidirect
339. C₄	C₃×#3697	semidirect
341. C₃	#10589579	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×#3697	2
C₂×C₃×C₄²×C₉	2
C₃×#63492	2
C₃×#63437	2
C₃×#63436	2
C₃×C₄×D₃₆	2
C₃×#63431	2
C₃×#5477	3
#10589579	3

Subgroups — C3×#10589579 #13152285

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10589579 #13152285