Subgroups — C₃×#10592825 #13163164

Order 1728.

1412 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₄×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃³×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#3748	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#63647	C₂	1	4	yes	maximal, complemented
5	#10592825	C₃	1	1	yes	maximal, characteristic, direct product factor
6	C₃×#5126	—	3	4	no	maximal
7	C₂³×C₃×C₃⋊D₃	C₄	1	1	yes	characteristic
8	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
9	C₂²×C₃×C₃⋊Dic₃	C₄	1	1	yes	characteristic
10	C₂×C₃×C₄×C₃⋊D₃	—	2	2	no	—
11	C₂×C₃×C₄×C₃⋊D₃	—	2	2	no	—
12	C₂×C₃×C₄×C₃⋊D₃	C₄	1	2	yes	—
13	C₂×C₃×C₄×C₃⋊D₃	C₂²	1	2	yes	—
14	C₃³×M₁₆	—	2	2	no	—
15	C₂×C₃³×C₈	C₂²	1	2	yes	complemented
16	C₃×#3748	—	2	2	no	—
17	C₂×C₃×C₃⋊(C₃⋊C₈)	C₂²	1	2	yes	complemented
18	C₂²×C₄×C₃⋊D₃	C₂×C₃	1	1	yes	characteristic
19	C₂²×C₃×C₄×D₃	—	3	4	no	—
20	C₂×C₃²×M₁₆	—	2	4	no	—
21	C₂×C₃²×M₁₆	D₃	1	4	yes	complemented
22	C₂×C₃²×M₁₆	C₂×C₃	1	1	yes	characteristic, complemented
23	C₂×#3748	C₂×C₃	1	1	yes	characteristic, complemented
24	C₂×C₃×#208	—	3	4	no	—
25	#63647	C₂×C₃	1	4	yes	complemented
26	C₃×(C₂×D₃)⋊C₈	—	3	16	no	—
27	C₂³×C₃³	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₃×C₃⋊D₃	—	2	2	no	—
29	C₂²×C₃×C₃⋊D₃	—	2	4	no	—
30	C₂²×C₃×C₃⋊D₃	C₂×C₄	1	2	yes	—
31	C₂×C₃³×C₄	C₂×C₄	1	1	yes	characteristic
32	C₂×C₃³×C₄	D₄	1	2	yes	—
33	C₂×C₃³×C₄	D₄	1	2	yes	—
34	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
35	C₂×C₃×C₃⋊Dic₃	—	2	2	no	—
36	C₂×C₃×C₃⋊Dic₃	C₂×C₄	1	2	yes	—
37	C₃×C₄×C₃⋊D₃	—	2	4	no	—
38	C₃×C₄×C₃⋊D₃	—	2	2	no	—
39	C₃×C₄×C₃⋊D₃	—	2	2	no	—
40	C₃³×C₈	—	2	2	no	—
41	C₃×C₃⋊(C₃⋊C₈)	—	2	2	no	—
42	C₃×#407	—	9	1	no	—
43	#5126	—	3	4	no	—
44	C₂³×C₃⋊D₃	C₃×C₄	1	1	yes	characteristic
45	C₂³×C₃×D₃	—	3	4	no	—
46	C₂²×C₃²×C₄	—	2	4	no	—
47	C₂²×C₃²×C₄	C₂×D₃	1	4	yes	—
48	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
49	C₂²×C₃⋊Dic₃	C₃×C₄	1	1	yes	characteristic
50	C₂×C₄×C₃⋊D₃	—	2	2	no	—
51	C₂×C₄×C₃⋊D₃	—	2	2	no	—
52	C₂×C₄×C₃⋊D₃	C₃×C₄	1	2	yes	—
53	C₂×C₄×C₃⋊D₃	C₂²×C₃	1	2	yes	—
54	C₂²×C₃×Dic₃	—	3	4	no	—
55	C₂×C₃×C₄×D₃	—	3	8	no	—
56	C₂×C₃×C₄×D₃	—	3	8	no	—
57	C₂×C₃×C₄×D₃	—	6	8	no	—
58	C₂×C₃×C₄×D₃	—	6	8	no	—
59	C₃²×M₁₆	—	2	16	no	—
60	C₃²×M₁₆	—	2	8	no	—
61	C₃²×M₁₆	—	2	2	no	—
62	C₂×C₃²×C₈	—	2	8	no	—
63	C₂×C₃²×C₈	C₂×D₃	1	8	yes	complemented
64	C₂×C₃²×C₈	C₂²×C₃	1	2	yes	complemented
65	#3748	—	2	2	no	—
66	C₂×C₃⋊(C₃⋊C₈)	C₂²×C₃	1	2	yes	complemented
67	C₃×#208	—	6	8	no	—
68	C₂×C₃×C₃⋊C₈	—	3	8	no	—
69	C₂²×C₃³	—	2	2	no	—
70	C₂²×C₃³	C₂²⋊C₄	1	2	yes	—
71	C₂²×C₃³	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₃×C₃⋊D₃	—	2	8	no	—
73	C₂×C₃×C₃⋊D₃	—	2	2	no	—
74	C₂×C₃×C₃⋊D₃	—	2	2	no	—
75	C₂×C₃×C₃⋊D₃	M₁₆	1	4	yes	complemented
76	C₃³×C₄	C₂²⋊C₄	1	2	yes	—
77	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
78	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
79	C₃×C₃⋊Dic₃	—	2	2	no	—
80	C₂³×C₃×C₄	—	9	1	no	—
81	C₂²×C₄×D₃	—	3	4	no	—
82	C₂×C₃×M₁₆	—	2	4	no	—
83	C₂×C₃×M₁₆	—	9	1	no	—
84	C₂×C₃×M₁₆	C₃×D₃	1	4	yes	complemented
85	C₂×C₃×M₁₆	C₃⋊D₃	1	1	yes	characteristic, complemented
86	C₂×#208	—	3	4	no	—
87	C₃×C₂²⋊C₈	—	9	4	no	—
88	(C₂×D₃)⋊C₈	—	3	16	no	—
89	C₂³×C₃²	—	2	4	no	—
90	C₂³×C₃²	C₄×D₃	1	4	yes	—
91	C₂³×C₃²	C₂×C₃×C₄	1	1	yes	characteristic
92	C₂²×C₃⋊D₃	—	2	2	no	—
93	C₂²×C₃⋊D₃	—	2	4	no	—
94	C₂²×C₃⋊D₃	C₂×C₃×C₄	1	2	yes	—
95	C₂²×C₃×D₃	—	3	8	no	—
96	C₂²×C₃×D₃	—	6	16	no	—
97	C₂²×C₃×D₃	—	6	8	no	—
98	C₂×C₃²×C₄	—	2	4	no	—
99	C₂×C₃²×C₄	—	2	8	no	—
100	C₂×C₃²×C₄	—	2	8	no	—
101	C₂×C₃²×C₄	—	2	4	no	—
102	C₂×C₃²×C₄	C₄×D₃	1	4	yes	—
103	C₂×C₃²×C₄	D₁₂	1	8	yes	—
104	C₂×C₃²×C₄	#67	1	8	yes	—
105	C₂×C₃²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
106	C₂×C₃²×C₄	C₃×D₄	1	2	yes	—
107	C₂×C₃²×C₄	C₃×D₄	1	2	yes	—
108	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
109	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
110	C₂×C₃⋊Dic₃	—	2	2	no	—
111	C₂×C₃⋊Dic₃	C₂×C₃×C₄	1	2	yes	—
112	C₄×C₃⋊D₃	—	2	4	no	—
113	C₄×C₃⋊D₃	—	2	2	no	—
114	C₄×C₃⋊D₃	—	2	2	no	—
115	C₂×C₃×Dic₃	—	3	8	no	—
116	C₂×C₃×Dic₃	—	6	8	no	—
117	C₃×C₄×D₃	—	6	16	no	—
118	C₃×C₄×D₃	—	6	8	no	—
119	C₃×C₄×D₃	—	6	8	no	—
120	C₃²×C₈	—	2	16	no	—
121	C₃²×C₈	—	2	8	no	—
122	C₃²×C₈	—	2	2	no	—
123	C₃⋊(C₃⋊C₈)	—	2	2	no	—
124	C₃×C₃⋊C₈	—	6	8	no	—
125	#407	—	9	1	no	Sylow
126	C₂×C₃³	—	2	2	no	—
127	C₂×C₃³	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	C₂×C₃³	C₂×M₁₆	1	2	yes	—
129	C₃×C₃⋊D₃	—	2	4	no	—
130	C₂⁴×C₃	—	9	1	no	—
131	C₂³×D₃	—	3	4	no	—
132	C₂²×C₃×C₄	—	2	4	no	—
133	C₂²×C₃×C₄	—	9	1	no	—
134	C₂²×C₃×C₄	—	9	2	no	—
135	C₂²×C₃×C₄	—	9	2	no	—
136	C₂²×C₃×C₄	—	18	2	no	—
137	C₂²×C₃×C₄	—	18	2	no	—
138	C₂²×C₃×C₄	C₂×C₃×D₃	1	4	yes	—
139	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
140	C₂²×Dic₃	—	3	4	no	—
141	C₂×C₄×D₃	—	3	8	no	—
142	C₂×C₄×D₃	—	3	8	no	—
143	C₂×C₄×D₃	—	6	8	no	—
144	C₂×C₄×D₃	—	6	8	no	—
145	C₃×M₁₆	—	2	16	no	—
146	C₃×M₁₆	—	2	8	no	—
147	C₃×M₁₆	—	2	2	no	—
148	C₃×M₁₆	—	18	2	no	—
149	C₂×C₃×C₈	—	2	8	no	—
150	C₂×C₃×C₈	—	9	2	no	—
151	C₂×C₃×C₈	C₂×C₃×D₃	1	8	yes	complemented
152	C₂×C₃×C₈	C₂×C₃⋊D₃	1	2	yes	complemented
153	#208	—	6	8	no	—
154	C₂×C₃⋊C₈	—	3	8	no	—
155	C₂²×C₃²	—	2	2	no	—
156	C₂²×C₃²	—	2	8	no	—
157	C₂²×C₃²	—	2	4	no	—
158	C₂²×C₃²	—	2	8	no	—
159	C₂²×C₃²	—	4	8	no	—
160	C₂²×C₃²	(C₂×D₃)⋊C₄	1	8	yes	—
161	C₂²×C₃²	C₃×C₂²⋊C₄	1	2	yes	—
162	C₂²×C₃²	C₂×C₄×D₃	1	4	yes	—
163	C₂²×C₃²	C₂²×C₃×C₄	1	1	yes	characteristic
164	C₂×C₃⋊D₃	—	2	8	no	—
165	C₂×C₃⋊D₃	—	2	2	no	—
166	C₂×C₃⋊D₃	—	2	2	no	—
167	C₂×C₃⋊D₃	C₃×M₁₆	1	4	yes	complemented
168	C₂×C₃×D₃	—	3	16	no	—
169	C₂×C₃×D₃	—	6	32	no	—
170	C₂×C₃×D₃	—	6	8	no	—
171	C₂×C₃×D₃	—	6	8	no	—
172	C₃²×C₄	—	2	8	no	—
173	C₃²×C₄	—	2	4	no	—
174	C₃²×C₄	—	2	4	no	—
175	C₃²×C₄	(C₂×D₃)⋊C₄	1	8	yes	—
176	C₃²×C₄	C₃×C₂²⋊C₄	1	2	yes	—
177	C₃²×C₄	C₂×D₁₂	1	4	yes	—
178	C₃²×C₄	C₂×#67	1	4	yes	—
179	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
180	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
181	C₃⋊Dic₃	—	2	2	no	—
182	C₃×Dic₃	—	6	8	no	—
183	C₂³×C₄	—	9	1	no	—
184	C₂×M₁₆	—	9	1	no	—
185	C₂×M₁₆	C₃×C₃⋊D₃	1	1	yes	characteristic, complemented
186	C₂²⋊C₈	—	9	4	no	—
187	C₃³	#407	1	1	yes	characteristic, complemented, Sylow
188	C₂³×C₃	—	2	4	no	—
189	C₂³×C₃	—	9	2	no	—
190	C₂³×C₃	—	18	4	no	—
191	C₂³×C₃	—	18	2	no	—
192	C₂³×C₃	C₃×C₄×D₃	1	4	yes	—
193	C₂³×C₃	C₄×C₃⋊D₃	1	1	yes	characteristic
194	C₂²×D₃	—	3	8	no	—
195	C₂²×D₃	—	6	16	no	—
196	C₂²×D₃	—	6	8	no	—
197	C₂×C₃×C₄	—	2	4	no	—
198	C₂×C₃×C₄	—	2	8	no	—
199	C₂×C₃×C₄	—	2	8	no	—
200	C₂×C₃×C₄	—	2	4	no	—
201	C₂×C₃×C₄	—	9	2	no	—
202	C₂×C₃×C₄	—	18	2	no	—
203	C₂×C₃×C₄	—	18	4	no	—
204	C₂×C₃×C₄	—	18	2	no	—
205	C₂×C₃×C₄	—	18	2	no	—
206	C₂×C₃×C₄	C₃×C₄×D₃	1	4	yes	—
207	C₂×C₃×C₄	C₃×D₁₂	1	8	yes	—
208	C₂×C₃×C₄	C₃×#67	1	8	yes	—
209	C₂×C₃×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
210	C₂×C₃×C₄	C₄⋊(C₃⋊D₃)	1	2	yes	—
211	C₂×C₃×C₄	#638	1	2	yes	—
212	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
213	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
214	C₂×Dic₃	—	3	8	no	—
215	C₂×Dic₃	—	6	8	no	—
216	C₄×D₃	—	6	16	no	—
217	C₄×D₃	—	6	8	no	—
218	C₄×D₃	—	6	8	no	—
219	C₃×C₈	—	2	16	no	—
220	C₃×C₈	—	2	8	no	—
221	C₃×C₈	—	2	2	no	—
222	C₃×C₈	—	18	2	no	—
223	C₃⋊C₈	—	6	8	no	—
224	C₂×C₃²	—	2	2	no	—
225	C₂×C₃²	—	2	8	no	—
226	C₂×C₃²	—	2	4	no	—
227	C₂×C₃²	—	2	8	no	—
228	C₂×C₃²	—	4	8	no	—
229	C₂×C₃²	D₃×M₁₆	1	8	yes	—
230	C₂×C₃²	C₂×(C₂×D₃)⋊C₄	1	4	yes	—
231	C₂×C₃²	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
232	C₂×C₃²	C₂×C₃×M₁₆	1	2	yes	—
233	C₃⋊D₃	—	2	4	no	—
234	C₃×D₃	—	6	16	no	—
235	C₂⁴	—	9	1	no	—
236	C₂²×C₄	—	9	1	no	—
237	C₂²×C₄	—	9	2	no	—
238	C₂²×C₄	—	9	2	no	—
239	C₂²×C₄	—	18	2	no	—
240	C₂²×C₄	—	18	2	no	—
241	C₂²×C₄	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
242	M₁₆	—	2	2	no	—
243	M₁₆	—	18	2	no	—
244	C₂×C₈	—	9	2	no	—
245	C₂×C₈	C₂×C₃×C₃⋊D₃	1	2	yes	complemented
246	C₂²×C₃	—	2	8	no	—
247	C₂²×C₃	—	2	2	no	—
248	C₂²×C₃	—	2	8	no	—
249	C₂²×C₃	—	2	4	no	—
250	C₂²×C₃	—	4	8	no	—
251	C₂²×C₃	—	9	4	no	—
252	C₂²×C₃	—	18	8	no	—
253	C₂²×C₃	—	18	2	no	—
254	C₂²×C₃	—	18	2	no	—
255	C₂²×C₃	C₃×(C₂×D₃)⋊C₄	1	8	yes	—
256	C₂²×C₃	#3752	1	2	yes	—
257	C₂²×C₃	C₂×C₃×C₄×D₃	1	4	yes	—
258	C₂²×C₃	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
259	C₂×D₃	—	3	16	no	—
260	C₂×D₃	—	6	32	no	—
261	C₂×D₃	—	6	8	no	—
262	C₂×D₃	—	6	8	no	—
263	C₃×C₄	—	2	8	no	—
264	C₃×C₄	—	2	4	no	—
265	C₃×C₄	—	2	4	no	—
266	C₃×C₄	—	18	2	no	—
267	C₃×C₄	C₃×(C₂×D₃)⋊C₄	1	8	yes	—
268	C₃×C₄	#3752	1	2	yes	—
269	C₃×C₄	C₂×C₃×D₁₂	1	4	yes	—
270	C₃×C₄	C₂×C₃×#67	1	4	yes	—
271	C₃×C₄	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
272	C₃×C₄	C₂×#638	1	1	yes	characteristic
273	Dic₃	—	6	8	no	—
274	C₃²	—	2	4	no	—
275	C₃²	#5126	1	4	yes	complemented
276	C₃²	C₃×#407	1	1	yes	characteristic, complemented
277	C₂³	—	9	2	no	—
278	C₂³	—	18	4	no	—
279	C₂³	—	18	2	no	—
280	C₂³	C₃×C₄×C₃⋊D₃	1	1	yes	characteristic
281	C₂×C₄	—	9	2	no	—
282	C₂×C₄	—	18	2	no	—
283	C₂×C₄	—	18	4	no	—
284	C₂×C₄	—	18	2	no	—
285	C₂×C₄	—	18	2	no	—
286	C₂×C₄	C₃×C₄×C₃⋊D₃	1	1	yes	characteristic
287	C₂×C₄	C₃×C₄⋊(C₃⋊D₃)	1	2	yes	—
288	C₂×C₄	C₃×#638	1	2	yes	—
289	C₂×C₄	C₂²×C₃×C₃⋊D₃	1	1	yes	characteristic
290	C₈	—	2	2	no	—
291	C₈	—	18	2	no	—
292	C₂×C₃	—	2	2	no	—
293	C₂×C₃	—	2	8	no	—
294	C₂×C₃	—	2	8	no	—
295	C₂×C₃	—	2	4	no	—
296	C₂×C₃	—	4	8	no	—
297	C₂×C₃	—	18	4	no	—
298	C₂×C₃	C₃×D₃×M₁₆	1	8	yes	—
299	C₂×C₃	C₂×C₃×(C₂×D₃)⋊C₄	1	4	yes	—
300	C₂×C₃	M₁₆×C₃⋊D₃	1	2	yes	—
301	C₂×C₃	C₂×#3752	1	1	yes	characteristic
302	D₃	—	6	16	no	—
303	C₂²	—	2	2	no	—
304	C₂²	—	9	4	no	—
305	C₂²	—	18	8	no	—
306	C₂²	—	18	2	no	—
307	C₂²	—	18	2	no	—
308	C₂²	C₃×#3752	1	2	yes	—
309	C₂²	C₂×C₃×C₄×C₃⋊D₃	1	1	yes	characteristic
310	C₄	—	18	2	no	—
311	C₄	C₃×#3752	1	2	yes	—
312	C₄	C₂×C₃×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
313	C₄	C₂×C₃×#638	1	1	yes	characteristic
314	C₃	—	2	4	no	—
315	C₃	C₃×#5126	1	4	yes	complemented
316	C₃	#10592825	1	1	yes	characteristic, direct product factor
317	C₂	—	2	2	no	—
318	C₂	—	18	4	no	—
319	C₂	C₃×M₁₆×C₃⋊D₃	1	2	yes	—
320	C₂	C₂×C₃×#3752	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂²×C₃⋊D₃
commutator	C₂²×C₃²	C₂²×C₃×C₄
Frattini	C₂×C₄	C₂²×C₃×C₃⋊D₃
Fitting	C₂×C₃³×M₁₆	C₂
socle	C₂²×C₃³	C₂²×C₄
prime radical	C₂³×C₃×C₃⋊D₃	C₄
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₂×C₃×M₁₆	C₃⋊D₃
hypocenter	C₃²	C₃×#407
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×C₃³×M₁₆	C₂	semidirect
3. C₂×C₃×#3748	C₂	semidirect
4. C₃×#63647	C₂	semidirect
5. #10592825	C₃	direct (N × H)
15. C₂×C₃³×C₈	C₂²	semidirect
17. C₂×C₃×C₃⋊(C₃⋊C₈)	C₂²	semidirect
21. C₂×C₃²×M₁₆	D₃	semidirect
22. C₂×C₃²×M₁₆	C₂×C₃	semidirect
23. C₂×#3748	C₂×C₃	semidirect
25. #63647	C₂×C₃	semidirect
63. C₂×C₃²×C₈	C₂×D₃	semidirect
64. C₂×C₃²×C₈	C₂²×C₃	semidirect
66. C₂×C₃⋊(C₃⋊C₈)	C₂²×C₃	semidirect
75. C₂×C₃×C₃⋊D₃	M₁₆	semidirect
84. C₂×C₃×M₁₆	C₃×D₃	semidirect
85. C₂×C₃×M₁₆	C₃⋊D₃	semidirect
151. C₂×C₃×C₈	C₂×C₃×D₃	semidirect
152. C₂×C₃×C₈	C₂×C₃⋊D₃	semidirect
167. C₂×C₃⋊D₃	C₃×M₁₆	semidirect
185. C₂×M₁₆	C₃×C₃⋊D₃	semidirect
187. C₃³	#407	semidirect
245. C₂×C₈	C₂×C₃×C₃⋊D₃	semidirect
275. C₃²	#5126	semidirect
276. C₃²	C₃×#407	semidirect
315. C₃	C₃×#5126	semidirect
316. C₃	#10592825	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₄×C₃⋊D₃	2
C₂×C₃³×M₁₆	2
C₂×C₃×#3748	2
C₃×#63647	2
#10592825	3
C₃×#5126	3

Subgroups — C3×#10592825 #13163164

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10592825 #13163164