Subgroups — C₃×#10596196 #13179477

Order 1728.

2132 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#3832	C₂	1	2	yes	maximal, complemented
2	C₃³×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#64156	C₂	1	2	yes	maximal, complemented
4	C₃×#64104	C₂	1	4	yes	maximal, complemented
5	C₃×#64102	C₂	1	4	yes	maximal, complemented
6	C₃×#64083	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×C₄²×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#10596196	C₃	1	1	yes	maximal, characteristic, direct product factor
9	C₃×#5725	—	3	4	no	maximal
10	C₂×C₃³×Q₈	C₂²	1	2	yes	—
11	C₃×#3832	—	2	8	no	—
12	C₂×C₃×C₄⋊(C₃⋊D₃)	C₂²	1	4	yes	—
13	C₂×C₃×C₄⋊(C₃⋊D₃)	C₂²	1	2	yes	—
14	C₂×C₃×C₄⋊(C₃⋊D₃)	C₂²	1	4	yes	—
15	C₂×C₃×C₄×C₃⋊D₃	C₂²	1	2	yes	—
16	C₂×C₃×C₄×C₃⋊D₃	C₂²	1	4	yes	—
17	C₂×C₃×C₄×C₃⋊D₃	C₂²	1	1	yes	characteristic
18	C₃³×C₄⋊C₄	C₂²	1	4	yes	—
19	C₃³×C₄²	C₂²	1	1	yes	characteristic
20	C₃×#3752	C₂²	1	8	yes	—
21	C₃×C₄×C₃⋊Dic₃	C₂²	1	2	yes	—
22	C₃×C₄×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
23	C₂×#3832	C₂×C₃	1	2	yes	complemented
24	C₂×C₃×#239	—	3	8	no	—
25	C₃²×C₄⋊Q₈	—	2	4	no	—
26	C₃²×C₄⋊Q₈	D₃	1	4	yes	complemented
27	C₃²×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
28	#64156	C₂×C₃	1	2	yes	complemented
29	#64104	C₂×C₃	1	4	yes	complemented
30	#64102	C₂×C₃	1	4	yes	complemented
31	#64083	C₂×C₃	1	1	yes	characteristic, complemented
32	C₄²×C₃⋊D₃	C₂×C₃	1	1	yes	characteristic, complemented
33	C₃×#947	—	3	8	no	—
34	C₃×#895	—	3	16	no	—
35	C₃×#893	—	3	16	no	—
36	C₃×#874	—	3	4	no	—
37	C₃×C₄²×D₃	—	3	4	no	—
38	C₂²×C₃×C₃⋊D₃	C₂³	1	4	yes	—
39	C₂²×C₃×C₃⋊D₃	C₂³	1	1	yes	characteristic
40	C₃³×Q₈	—	2	4	no	—
41	C₂×C₃³×C₄	C₂³	1	4	yes	—
42	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
43	C₂×C₃³×C₄	C₂³	1	2	yes	—
44	C₂×C₃×C₃⋊Dic₃	C₂³	1	2	yes	—
45	C₂×C₃×C₃⋊Dic₃	C₂³	1	1	yes	characteristic
46	C₃×C₄⋊(C₃⋊D₃)	—	2	8	no	—
47	C₃×C₄⋊(C₃⋊D₃)	—	2	4	no	—
48	C₃×C₄⋊(C₃⋊D₃)	—	2	8	no	—
49	C₃×C₄×C₃⋊D₃	—	2	4	no	—
50	C₃×C₄×C₃⋊D₃	—	2	8	no	—
51	C₃×C₄×C₃⋊D₃	D₄	1	4	yes	complemented
52	C₃×#532	—	9	1	no	—
53	#5725	—	3	4	no	—
54	C₂×C₃²×Q₈	—	2	8	no	—
55	C₂×C₃²×Q₈	C₂×D₃	1	8	yes	—
56	C₂×C₃²×Q₈	C₂²×C₃	1	2	yes	—
57	#3832	—	2	8	no	—
58	C₂×C₄⋊(C₃⋊D₃)	C₂²×C₃	1	4	yes	—
59	C₂×C₄⋊(C₃⋊D₃)	C₂²×C₃	1	2	yes	—
60	C₂×C₄⋊(C₃⋊D₃)	C₂²×C₃	1	4	yes	—
61	C₂×C₄×C₃⋊D₃	C₂²×C₃	1	2	yes	—
62	C₂×C₄×C₃⋊D₃	C₂²×C₃	1	4	yes	—
63	C₂×C₄×C₃⋊D₃	C₂²×C₃	1	1	yes	characteristic
64	C₃×#239	—	6	32	no	—
65	C₂×C₃×D₁₂	—	3	16	no	—
66	C₂×C₃×D₁₂	—	3	16	no	—
67	C₂×C₃×D₁₂	—	3	8	no	—
68	C₂×C₃×C₄×D₃	—	3	4	no	—
69	C₂×C₃×C₄×D₃	—	3	16	no	—
70	C₂×C₃×C₄×D₃	—	3	8	no	—
71	C₃²×C₄⋊C₄	—	2	16	no	—
72	C₃²×C₄⋊C₄	C₂×D₃	1	16	yes	—
73	C₃²×C₄⋊C₄	C₂²×C₃	1	4	yes	—
74	C₃²×C₄²	—	2	4	no	—
75	C₃²×C₄²	C₂×D₃	1	4	yes	—
76	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
77	#3752	C₂²×C₃	1	8	yes	—
78	C₄×C₃⋊Dic₃	C₂²×C₃	1	2	yes	—
79	C₄×C₃⋊Dic₃	C₂²×C₃	1	1	yes	characteristic
80	C₃×(C₂×D₃)⋊C₄	—	3	32	no	—
81	C₃×C₄×Dic₃	—	3	8	no	—
82	C₃×C₄×Dic₃	—	3	4	no	—
83	C₂²×C₃³	C₂⁴	1	1	yes	characteristic
84	C₂×C₃×C₃⋊D₃	—	2	2	no	—
85	C₂×C₃×C₃⋊D₃	—	2	4	no	—
86	C₂×C₃×C₃⋊D₃	—	2	8	no	—
87	C₂×C₃×C₃⋊D₃	C₂×D₄	1	2	yes	—
88	C₃³×C₄	—	2	4	no	—
89	C₃³×C₄	C₂×D₄	1	2	yes	—
90	C₃³×C₄	#41	1	4	yes	complemented
91	C₃×C₃⋊Dic₃	—	2	2	no	—
92	C₃×C₃⋊Dic₃	C₂×D₄	1	2	yes	—
93	C₂×C₃×#41	—	9	2	no	—
94	C₂×#239	—	3	8	no	—
95	C₃×C₄⋊Q₈	—	2	4	no	—
96	C₃×C₄⋊Q₈	C₃×D₃	1	4	yes	complemented
97	C₃×C₄⋊Q₈	C₃⋊D₃	1	1	yes	characteristic, complemented
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	—	9	1	no	—
103	#947	—	3	8	no	—
104	#895	—	3	16	no	—
105	#893	—	3	16	no	—
106	#874	—	3	4	no	—
107	C₄²×D₃	—	3	4	no	—
108	C₂²×C₃⋊D₃	C₂³×C₃	1	4	yes	—
109	C₂²×C₃⋊D₃	C₂³×C₃	1	1	yes	characteristic
110	C₂²×C₃×D₃	—	3	4	no	—
111	C₂²×C₃×D₃	—	3	16	no	—
112	C₃²×Q₈	—	2	4	no	—
113	C₃²×Q₈	—	2	16	no	—
114	C₃²×Q₈	—	4	16	no	—
115	C₂×C₃²×C₄	—	2	8	no	—
116	C₂×C₃²×C₄	—	2	4	no	—
117	C₂×C₃²×C₄	—	2	16	no	—
118	C₂×C₃²×C₄	C₂²×D₃	1	16	yes	—
119	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
120	C₂×C₃²×C₄	C₂²×D₃	1	8	yes	—
121	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
122	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
123	C₂×C₃²×C₄	C₂³×C₃	1	2	yes	—
124	C₂×C₃⋊Dic₃	C₂³×C₃	1	2	yes	—
125	C₂×C₃⋊Dic₃	C₂³×C₃	1	1	yes	characteristic
126	C₄⋊(C₃⋊D₃)	—	2	8	no	—
127	C₄⋊(C₃⋊D₃)	—	2	4	no	—
128	C₄⋊(C₃⋊D₃)	—	2	8	no	—
129	C₄×C₃⋊D₃	—	2	4	no	—
130	C₄×C₃⋊D₃	—	2	8	no	—
131	C₄×C₃⋊D₃	C₃×D₄	1	4	yes	complemented
132	C₂×C₃×Dic₃	—	3	4	no	—
133	C₂×C₃×Dic₃	—	3	8	no	—
134	C₃×D₁₂	—	6	32	no	—
135	C₃×D₁₂	—	6	16	no	—
136	C₃×D₁₂	—	6	32	no	—
137	C₃×C₄×D₃	—	3	16	no	—
138	C₃×C₄×D₃	—	6	32	no	—
139	C₃×C₄×D₃	—	6	16	no	—
140	#532	—	9	1	no	Sylow
141	C₂×C₃³	C₂²×D₄	1	1	yes	characteristic
142	C₂×C₃³	C₂×#41	1	2	yes	—
143	C₃×C₃⋊D₃	—	2	2	no	—
144	C₃×C₃⋊D₃	—	4	4	no	—
145	C₃×#41	—	18	8	no	—
146	C₂×C₃×Q₈	—	2	8	no	—
147	C₂×C₃×Q₈	C₂×C₃×D₃	1	8	yes	—
148	C₂×C₃×Q₈	C₂×C₃⋊D₃	1	2	yes	—
149	C₂×C₃×D₄	—	9	4	no	—
150	C₂×C₃×D₄	—	9	4	no	—
151	C₂×C₃×D₄	—	9	2	no	—
152	C₂²×C₃×C₄	—	9	1	no	—
153	C₂²×C₃×C₄	—	9	2	no	—
154	C₂²×C₃×C₄	—	9	4	no	—
155	#239	—	6	32	no	—
156	C₂×D₁₂	—	3	16	no	—
157	C₂×D₁₂	—	3	8	no	—
158	C₂×D₁₂	—	3	16	no	—
159	C₂×C₄×D₃	—	3	4	no	—
160	C₂×C₄×D₃	—	3	8	no	—
161	C₂×C₄×D₃	—	3	16	no	—
162	C₃×C₄⋊C₄	—	2	16	no	—
163	C₃×C₄⋊C₄	C₂×C₃×D₃	1	16	yes	—
164	C₃×C₄⋊C₄	C₂×C₃⋊D₃	1	4	yes	—
165	C₃×C₂²⋊C₄	—	9	8	no	—
166	C₃×C₄²	—	2	4	no	—
167	C₃×C₄²	—	9	2	no	—
168	C₃×C₄²	—	9	1	no	—
169	C₃×C₄²	C₂×C₃×D₃	1	4	yes	—
170	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
171	(C₂×D₃)⋊C₄	—	3	32	no	—
172	C₄×Dic₃	—	3	8	no	—
173	C₄×Dic₃	—	3	4	no	—
174	C₂²×C₃²	—	2	4	no	—
175	C₂²×C₃²	C₂³×D₃	1	4	yes	—
176	C₂²×C₃²	C₂⁴×C₃	1	1	yes	characteristic
177	C₂×C₃⋊D₃	—	2	2	no	—
178	C₂×C₃⋊D₃	—	2	4	no	—
179	C₂×C₃⋊D₃	—	2	8	no	—
180	C₂×C₃⋊D₃	C₂×C₃×D₄	1	2	yes	—
181	C₂×C₃×D₃	—	3	8	no	—
182	C₂×C₃×D₃	—	6	8	no	—
183	C₂×C₃×D₃	—	6	16	no	—
184	C₂×C₃×D₃	—	6	32	no	—
185	C₃²×C₄	—	2	4	no	—
186	C₃²×C₄	—	2	8	no	—
187	C₃²×C₄	—	2	16	no	—
188	C₃²×C₄	—	2	16	no	—
189	C₃²×C₄	—	4	16	no	—
190	C₃²×C₄	D₃×D₄	1	8	yes	—
191	C₃²×C₄	#239	1	16	yes	complemented
192	C₃²×C₄	C₂×C₃×D₄	1	2	yes	—
193	C₃²×C₄	C₃×#41	1	4	yes	complemented
194	C₃⋊Dic₃	—	2	2	no	—
195	C₃⋊Dic₃	C₂×C₃×D₄	1	2	yes	—
196	C₃×Dic₃	—	3	8	no	—
197	C₃×Dic₃	—	6	8	no	—
198	C₂×#41	—	9	2	no	—
199	C₄⋊Q₈	C₃×C₃⋊D₃	1	1	yes	characteristic, complemented
200	#127	—	9	1	no	—
201	#124	—	9	2	no	—
202	#121	—	9	4	no	—
203	C₄×D₄	—	9	4	no	—
204	C₂×C₄²	—	9	1	no	—
205	C₃³	#532	1	1	yes	characteristic, complemented, Sylow
206	C₂³×C₃	—	9	1	no	—
207	C₂³×C₃	—	9	4	no	—
208	C₂²×D₃	—	3	4	no	—
209	C₂²×D₃	—	3	16	no	—
210	C₃×Q₈	—	2	16	no	—
211	C₃×Q₈	—	2	4	no	—
212	C₃×Q₈	—	4	16	no	—
213	C₃×D₄	—	18	8	no	—
214	C₃×D₄	—	18	4	no	—
215	C₃×D₄	—	18	8	no	—
216	C₂×C₃×C₄	—	2	8	no	—
217	C₂×C₃×C₄	—	2	4	no	—
218	C₂×C₃×C₄	—	2	16	no	—
219	C₂×C₃×C₄	—	9	1	no	—
220	C₂×C₃×C₄	—	9	4	no	—
221	C₂×C₃×C₄	—	9	2	no	—
222	C₂×C₃×C₄	—	18	8	no	—
223	C₂×C₃×C₄	—	18	4	no	—
224	C₂×C₃×C₄	C₂²×C₃×D₃	1	16	yes	—
225	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
226	C₂×C₃×C₄	C₂²×C₃×D₃	1	8	yes	—
227	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	4	yes	—
228	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
229	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	2	yes	—
230	C₂×Dic₃	—	3	4	no	—
231	C₂×Dic₃	—	3	8	no	—
232	D₁₂	—	6	32	no	—
233	D₁₂	—	6	16	no	—
234	D₁₂	—	6	32	no	—
235	C₄×D₃	—	3	16	no	—
236	C₄×D₃	—	6	32	no	—
237	C₄×D₃	—	6	16	no	—
238	C₂×C₃²	—	2	4	no	—
239	C₂×C₃²	—	2	8	no	—
240	C₂×C₃²	C₂×D₃×D₄	1	4	yes	—
241	C₂×C₃²	C₂×#239	1	8	yes	—
242	C₂×C₃²	C₂²×C₃×D₄	1	1	yes	characteristic
243	C₂×C₃²	C₂×C₃×#41	1	2	yes	—
244	C₃⋊D₃	—	2	2	no	—
245	C₃⋊D₃	—	4	4	no	—
246	C₃×D₃	—	6	8	no	—
247	C₃×D₃	—	12	16	no	—
248	#41	—	18	8	no	—
249	C₂×Q₈	C₂×C₃×C₃⋊D₃	1	2	yes	—
250	C₂×D₄	—	9	4	no	—
251	C₂×D₄	—	9	4	no	—
252	C₂×D₄	—	9	2	no	—
253	C₂²×C₄	—	9	1	no	—
254	C₂²×C₄	—	9	2	no	—
255	C₂²×C₄	—	9	4	no	—
256	C₄⋊C₄	C₂×C₃×C₃⋊D₃	1	4	yes	—
257	C₂²⋊C₄	—	9	8	no	—
258	C₄²	—	9	2	no	—
259	C₄²	—	9	1	no	—
260	C₄²	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
261	C₂²×C₃	—	2	4	no	—
262	C₂²×C₃	—	9	2	no	—
263	C₂²×C₃	—	18	2	no	—
264	C₂²×C₃	—	18	4	no	—
265	C₂²×C₃	—	18	8	no	—
266	C₂²×C₃	C₂³×C₃×D₃	1	4	yes	—
267	C₂²×C₃	C₂³×C₃⋊D₃	1	1	yes	characteristic
268	C₂×D₃	—	3	8	no	—
269	C₂×D₃	—	6	8	no	—
270	C₂×D₃	—	6	16	no	—
271	C₂×D₃	—	6	32	no	—
272	C₃×C₄	—	2	16	no	—
273	C₃×C₄	—	2	4	no	—
274	C₃×C₄	—	2	8	no	—
275	C₃×C₄	—	2	16	no	—
276	C₃×C₄	—	4	16	no	—
277	C₃×C₄	—	9	2	no	—
278	C₃×C₄	—	18	2	no	—
279	C₃×C₄	C₃×D₃×D₄	1	8	yes	—
280	C₃×C₄	C₃×#239	1	16	yes	complemented
281	C₃×C₄	D₄×C₃⋊D₃	1	2	yes	—
282	C₃×C₄	#3832	1	4	yes	complemented
283	Dic₃	—	3	8	no	—
284	Dic₃	—	6	8	no	—
285	C₃²	—	2	4	no	—
286	C₃²	#5725	1	4	yes	complemented
287	C₃²	C₃×#532	1	1	yes	characteristic, complemented
288	C₂³	—	9	1	no	—
289	C₂³	—	9	4	no	—
290	Q₈	—	2	4	no	—
291	D₄	—	18	8	no	—
292	D₄	—	18	4	no	—
293	D₄	—	18	8	no	—
294	C₂×C₄	—	9	1	no	—
295	C₂×C₄	—	9	4	no	—
296	C₂×C₄	—	9	2	no	—
297	C₂×C₄	—	18	8	no	—
298	C₂×C₄	—	18	4	no	—
299	C₂×C₄	C₂²×C₃×C₃⋊D₃	1	4	yes	—
300	C₂×C₄	C₂²×C₃×C₃⋊D₃	1	1	yes	characteristic
301	C₂×C₄	C₂²×C₃×C₃⋊D₃	1	2	yes	—
302	C₂×C₃	—	2	4	no	—
303	C₂×C₃	—	2	8	no	—
304	C₂×C₃	—	18	2	no	—
305	C₂×C₃	—	36	4	no	—
306	C₂×C₃	C₂×C₃×D₃×D₄	1	4	yes	—
307	C₂×C₃	C₂×C₃×#239	1	8	yes	—
308	C₂×C₃	C₂×D₄×C₃⋊D₃	1	1	yes	characteristic
309	C₂×C₃	C₂×#3832	1	2	yes	—
310	D₃	—	6	8	no	—
311	D₃	—	12	16	no	—
312	C₂²	—	9	2	no	—
313	C₂²	—	18	2	no	—
314	C₂²	—	18	4	no	—
315	C₂²	—	18	8	no	—
316	C₂²	C₂³×C₃×C₃⋊D₃	1	1	yes	characteristic
317	C₄	—	2	4	no	—
318	C₄	—	9	2	no	—
319	C₄	—	18	2	no	—
320	C₄	C₃×D₄×C₃⋊D₃	1	2	yes	—
321	C₄	C₃×#3832	1	4	yes	complemented
322	C₃	—	2	4	no	—
323	C₃	C₃×#5725	1	4	yes	complemented
324	C₃	#10596196	1	1	yes	characteristic, direct product factor
325	C₂	—	18	2	no	—
326	C₂	—	36	4	no	—
327	C₂	C₂×C₃×D₄×C₃⋊D₃	1	1	yes	characteristic
328	C₂	C₂×C₃×#3832	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃	C₂³×C₃⋊D₃
commutator, cosocle	C₂²×C₃²	C₂⁴×C₃
Frattini	C₂²	C₂³×C₃×C₃⋊D₃
Fitting	C₃³×C₄⋊Q₈	C₂
socle	C₂²×C₃³	C₂⁴
hypercenter	C₃×C₄⋊Q₈	C₃⋊D₃
hypocenter	C₃²	C₃×#532
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#3832	C₂	semidirect
2. C₃³×C₄⋊Q₈	C₂	semidirect
3. C₃×#64156	C₂	semidirect
4. C₃×#64104	C₂	semidirect
5. C₃×#64102	C₂	semidirect
6. C₃×#64083	C₂	semidirect
7. C₃×C₄²×C₃⋊D₃	C₂	semidirect
8. #10596196	C₃	direct (N × H)
23. C₂×#3832	C₂×C₃	semidirect
26. C₃²×C₄⋊Q₈	D₃	semidirect
27. C₃²×C₄⋊Q₈	C₂×C₃	semidirect
28. #64156	C₂×C₃	semidirect
29. #64104	C₂×C₃	semidirect
30. #64102	C₂×C₃	semidirect
31. #64083	C₂×C₃	semidirect
32. C₄²×C₃⋊D₃	C₂×C₃	semidirect
51. C₃×C₄×C₃⋊D₃	D₄	semidirect
90. C₃³×C₄	#41	semidirect
96. C₃×C₄⋊Q₈	C₃×D₃	semidirect
97. C₃×C₄⋊Q₈	C₃⋊D₃	semidirect
131. C₄×C₃⋊D₃	C₃×D₄	semidirect
191. C₃²×C₄	#239	semidirect
193. C₃²×C₄	C₃×#41	semidirect
199. C₄⋊Q₈	C₃×C₃⋊D₃	semidirect
205. C₃³	#532	semidirect
280. C₃×C₄	C₃×#239	semidirect
282. C₃×C₄	#3832	semidirect
286. C₃²	#5725	semidirect
287. C₃²	C₃×#532	semidirect
321. C₄	C₃×#3832	semidirect
323. C₃	C₃×#5725	semidirect
324. C₃	#10596196	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×#3832	2
C₃³×C₄⋊Q₈	2
C₃×#64156	2
C₃×#64104	2
C₃×#64102	2
C₃×#64083	2
C₃×C₄²×C₃⋊D₃	2
#10596196	3
C₃×#5725	3

Subgroups — C3×#10596196 #13179477

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10596196 #13179477