Subgroups — C₃×#10596006 #13179286

Order 1728.

1744 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×#117	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#64139	C₂	1	2	yes	maximal, complemented
3	C₃×#64103	C₂	1	4	yes	maximal, complemented
4	C₃×#64093	C₂	1	4	yes	maximal, complemented
5	C₃×#64083	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×C₄×C₄⋊(C₃⋊D₃)	C₂	1	2	yes	maximal, complemented
7	C₃×#64079	C₂	1	1	yes	maximal, characteristic, complemented
8	#10596006	C₃	1	1	yes	maximal, characteristic, direct product factor
9	C₃×#5535	—	3	4	no	maximal
10	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
11	C₂×C₃×#638	C₂²	1	4	yes	—
12	C₂×C₃×C₄⋊(C₃⋊D₃)	C₂²	1	2	yes	—
13	C₂×C₃×C₄⋊(C₃⋊D₃)	C₂²	1	4	yes	—
14	C₂×C₃×C₄×C₃⋊D₃	C₂²	1	4	yes	—
15	C₃³×C₄⋊C₄	C₂²	1	2	yes	—
16	C₃³×C₂²⋊C₄	C₂²	1	2	yes	—
17	C₃³×C₄²	C₂²	1	1	yes	characteristic
18	C₃³×C₄²	C₂²	1	1	yes	characteristic
19	C₃×#3752	C₂²	1	4	yes	—
20	C₃×#3752	C₂²	1	4	yes	—
21	C₃×C₄⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
22	C₃×#3750	C₂²	1	4	yes	—
23	C₃²×#117	—	2	4	no	—
24	C₃²×#117	D₃	1	4	yes	complemented
25	C₃²×#117	C₂×C₃	1	1	yes	characteristic, complemented
26	#64139	C₂×C₃	1	2	yes	complemented
27	#64103	C₂×C₃	1	4	yes	complemented
28	#64093	C₂×C₃	1	4	yes	complemented
29	#64083	C₂×C₃	1	1	yes	characteristic, complemented
30	C₄×C₄⋊(C₃⋊D₃)	C₂×C₃	1	2	yes	complemented
31	#64079	C₂×C₃	1	1	yes	characteristic, complemented
32	C₃×#930	—	3	8	no	—
33	C₃×#894	—	3	16	no	—
34	C₃×#884	—	3	16	no	—
35	C₃×#874	—	3	4	no	—
36	C₃×C₄×D₁₂	—	3	8	no	—
37	C₃×#870	—	3	4	no	—
38	C₂³×C₃³	C₂³	1	1	yes	characteristic
39	C₂²×C₃×C₃⋊D₃	C₂³	1	4	yes	—
40	C₂×C₃³×C₄	—	2	2	no	—
41	C₂×C₃³×C₄	C₂³	1	2	yes	—
42	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
43	C₂×C₃³×C₄	C₂³	1	2	yes	—
44	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
45	C₃×#638	—	4	4	no	—
46	C₂×C₃×C₃⋊Dic₃	C₂³	1	4	yes	—
47	C₃×C₄⋊(C₃⋊D₃)	—	2	4	no	—
48	C₃×C₄⋊(C₃⋊D₃)	—	4	4	no	—
49	C₃×C₄×C₃⋊D₃	—	4	4	no	—
50	C₃×#540	—	9	1	no	—
51	#5535	—	3	4	no	—
52	C₂²×C₃²×C₄	—	2	4	no	—
53	C₂²×C₃²×C₄	C₂×D₃	1	4	yes	—
54	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
55	C₂×#638	C₂²×C₃	1	4	yes	—
56	C₂×C₄⋊(C₃⋊D₃)	C₂²×C₃	1	2	yes	—
57	C₂×C₄⋊(C₃⋊D₃)	C₂²×C₃	1	4	yes	—
58	C₂×C₄×C₃⋊D₃	C₂²×C₃	1	4	yes	—
59	C₂×C₃×#67	—	3	16	no	—
60	C₂×C₃×D₁₂	—	3	8	no	—
61	C₂×C₃×D₁₂	—	3	16	no	—
62	C₂×C₃×C₄×D₃	—	3	16	no	—
63	C₃²×C₄⋊C₄	—	2	8	no	—
64	C₃²×C₄⋊C₄	C₂×D₃	1	8	yes	—
65	C₃²×C₄⋊C₄	C₂²×C₃	1	2	yes	—
66	C₃²×C₂²⋊C₄	—	2	8	no	—
67	C₃²×C₂²⋊C₄	C₂×D₃	1	8	yes	—
68	C₃²×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
69	C₃²×C₄²	—	2	4	no	—
70	C₃²×C₄²	—	2	4	no	—
71	C₃²×C₄²	C₂×D₃	1	4	yes	—
72	C₃²×C₄²	C₂×D₃	1	4	yes	—
73	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
74	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
75	#3752	C₂²×C₃	1	4	yes	—
76	#3752	C₂²×C₃	1	4	yes	—
77	C₄⋊(C₃⋊Dic₃)	C₂²×C₃	1	2	yes	—
78	#3750	C₂²×C₃	1	4	yes	—
79	C₃×(C₂×D₃)⋊C₄	—	3	16	no	—
80	C₃×(C₂×D₃)⋊C₄	—	3	16	no	—
81	C₃×C₄⋊Dic₃	—	3	8	no	—
82	C₃×Dic₃⋊C₄	—	3	16	no	—
83	C₂²×C₃³	—	2	1	no	—
84	C₂²×C₃³	—	2	2	no	—
85	C₂²×C₃³	C₂⁴	1	1	yes	characteristic
86	C₂×C₃×C₃⋊D₃	—	2	4	no	—
87	C₂×C₃×C₃⋊D₃	—	2	4	no	—
88	C₂×C₃×C₃⋊D₃	—	2	4	no	—
89	C₃³×C₄	—	2	1	no	—
90	C₃³×C₄	—	2	2	no	—
91	C₃³×C₄	—	2	2	no	—
92	C₃³×C₄	#41	1	2	yes	—
93	C₃×C₃⋊Dic₃	—	2	4	no	—
94	C₃×#127	—	9	1	no	—
95	C₃×#125	—	9	1	no	—
96	C₃×#123	—	9	4	no	—
97	C₃×#121	—	9	2	no	—
98	C₃×#121	—	9	4	no	—
99	C₃×C₄×D₄	—	9	2	no	—
100	C₃×#117	—	2	4	no	—
101	C₃×#117	C₃×D₃	1	4	yes	complemented
102	C₃×#117	C₃⋊D₃	1	1	yes	characteristic, complemented
103	#930	—	3	8	no	—
104	#894	—	3	16	no	—
105	#884	—	3	16	no	—
106	#874	—	3	4	no	—
107	C₄×D₁₂	—	3	8	no	—
108	#870	—	3	4	no	—
109	C₂³×C₃²	—	2	4	no	—
110	C₂³×C₃²	C₂²×D₃	1	4	yes	—
111	C₂³×C₃²	C₂³×C₃	1	1	yes	characteristic
112	C₂²×C₃⋊D₃	C₂³×C₃	1	4	yes	—
113	C₂²×C₃×D₃	—	3	16	no	—
114	C₂×C₃²×C₄	—	2	16	no	—
115	C₂×C₃²×C₄	—	2	8	no	—
116	C₂×C₃²×C₄	—	2	2	no	—
117	C₂×C₃²×C₄	—	2	4	no	—
118	C₂×C₃²×C₄	—	2	4	no	—
119	C₂×C₃²×C₄	—	2	8	no	—
120	C₂×C₃²×C₄	—	2	8	no	—
121	C₂×C₃²×C₄	C₂²×D₃	1	8	yes	—
122	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
123	C₂×C₃²×C₄	C₂²×D₃	1	8	yes	—
124	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
125	C₂×C₃²×C₄	C₂³×C₃	1	2	yes	—
126	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
127	C₂×C₃²×C₄	C₂³×C₃	1	2	yes	—
128	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
129	#638	—	4	4	no	—
130	C₂×C₃⋊Dic₃	C₂³×C₃	1	4	yes	—
131	C₄⋊(C₃⋊D₃)	—	2	4	no	—
132	C₄⋊(C₃⋊D₃)	—	4	4	no	—
133	C₄×C₃⋊D₃	—	4	4	no	—
134	C₃×#67	—	12	16	no	—
135	C₂×C₃×Dic₃	—	3	16	no	—
136	C₃×D₁₂	—	6	16	no	—
137	C₃×D₁₂	—	12	16	no	—
138	C₃×C₄×D₃	—	12	16	no	—
139	#540	—	9	1	no	Sylow
140	C₂×C₃³	—	4	1	no	—
141	C₂×C₃³	C₂×#41	1	1	yes	characteristic
142	C₂×C₃³	H(5,𝔽₂)	1	2	yes	—
143	C₃×C₃⋊D₃	—	4	4	no	—
144	C₂×C₃×D₄	—	9	4	no	—
145	C₂×C₃×D₄	—	9	2	no	—
146	C₂×C₃×D₄	—	9	4	no	—
147	C₂²×C₃×C₄	—	2	4	no	—
148	C₂²×C₃×C₄	—	9	4	no	—
149	C₂²×C₃×C₄	C₂×C₃×D₃	1	4	yes	—
150	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
151	C₂×#67	—	3	16	no	—
152	C₂×D₁₂	—	3	8	no	—
153	C₂×D₁₂	—	3	16	no	—
154	C₂×C₄×D₃	—	3	16	no	—
155	C₃×C₄⋊C₄	—	2	8	no	—
156	C₃×C₄⋊C₄	—	9	4	no	—
157	C₃×C₄⋊C₄	—	9	2	no	—
158	C₃×C₄⋊C₄	C₂×C₃×D₃	1	8	yes	—
159	C₃×C₄⋊C₄	C₂×C₃⋊D₃	1	2	yes	—
160	C₃×C₂²⋊C₄	—	2	8	no	—
161	C₃×C₂²⋊C₄	—	9	4	no	—
162	C₃×C₂²⋊C₄	—	9	4	no	—
163	C₃×C₂²⋊C₄	C₂×C₃×D₃	1	8	yes	—
164	C₃×C₂²⋊C₄	C₂×C₃⋊D₃	1	2	yes	—
165	C₃×C₄²	—	2	4	no	—
166	C₃×C₄²	—	2	4	no	—
167	C₃×C₄²	C₂×C₃×D₃	1	4	yes	—
168	C₃×C₄²	C₂×C₃×D₃	1	4	yes	—
169	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
170	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
171	(C₂×D₃)⋊C₄	—	3	16	no	—
172	(C₂×D₃)⋊C₄	—	3	16	no	—
173	C₄⋊Dic₃	—	3	8	no	—
174	Dic₃⋊C₄	—	3	16	no	—
175	C₂²×C₃²	—	2	8	no	—
176	C₂²×C₃²	—	2	4	no	—
177	C₂²×C₃²	—	2	1	no	—
178	C₂²×C₃²	—	2	2	no	—
179	C₂²×C₃²	—	2	4	no	—
180	C₂²×C₃²	—	2	8	no	—
181	C₂²×C₃²	—	4	8	no	—
182	C₂²×C₃²	C₂³×D₃	1	4	yes	—
183	C₂²×C₃²	C₂⁴×C₃	1	1	yes	characteristic
184	C₂×C₃⋊D₃	—	2	4	no	—
185	C₂×C₃⋊D₃	—	2	4	no	—
186	C₂×C₃⋊D₃	—	2	4	no	—
187	C₂×C₃×D₃	—	6	16	no	—
188	C₂×C₃×D₃	—	6	16	no	—
189	C₂×C₃×D₃	—	6	16	no	—
190	C₃²×C₄	—	2	8	no	—
191	C₃²×C₄	—	2	4	no	—
192	C₃²×C₄	—	2	1	no	—
193	C₃²×C₄	—	2	2	no	—
194	C₃²×C₄	—	2	2	no	—
195	C₃²×C₄	—	2	8	no	—
196	C₃²×C₄	—	2	8	no	—
197	C₃²×C₄	—	2	8	no	—
198	C₃²×C₄	—	4	8	no	—
199	C₃²×C₄	—	4	8	no	—
200	C₃²×C₄	#235	1	8	yes	—
201	C₃²×C₄	C₃×#41	1	2	yes	—
202	C₃⋊Dic₃	—	2	4	no	—
203	C₃×Dic₃	—	6	16	no	—
204	#127	—	9	1	no	—
205	#125	—	9	1	no	—
206	#123	—	9	4	no	—
207	#121	—	9	2	no	—
208	#121	—	9	4	no	—
209	C₄×D₄	—	9	2	no	—
210	#117	C₃×C₃⋊D₃	1	1	yes	characteristic, complemented
211	C₃³	#540	1	1	yes	characteristic, complemented, Sylow
212	C₂³×C₃	—	2	4	no	—
213	C₂³×C₃	—	9	4	no	—
214	C₂³×C₃	C₂²×C₃×D₃	1	4	yes	—
215	C₂³×C₃	C₂²×C₃⋊D₃	1	1	yes	characteristic
216	C₂²×D₃	—	3	16	no	—
217	C₃×D₄	—	18	4	no	—
218	C₃×D₄	—	36	4	no	—
219	C₃×D₄	—	36	4	no	—
220	C₂×C₃×C₄	—	2	16	no	—
221	C₂×C₃×C₄	—	2	8	no	—
222	C₂×C₃×C₄	—	2	2	no	—
223	C₂×C₃×C₄	—	2	4	no	—
224	C₂×C₃×C₄	—	2	4	no	—
225	C₂×C₃×C₄	—	2	8	no	—
226	C₂×C₃×C₄	—	2	8	no	—
227	C₂×C₃×C₄	—	9	4	no	—
228	C₂×C₃×C₄	—	36	4	no	—
229	C₂×C₃×C₄	C₂²×C₃×D₃	1	8	yes	—
230	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
231	C₂×C₃×C₄	C₂²×C₃×D₃	1	8	yes	—
232	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
233	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	2	yes	—
234	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	2	yes	—
235	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
236	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
237	#67	—	12	16	no	—
238	C₂×Dic₃	—	3	16	no	—
239	D₁₂	—	6	16	no	—
240	D₁₂	—	12	16	no	—
241	C₄×D₃	—	12	16	no	—
242	C₂×C₃²	—	2	4	no	—
243	C₂×C₃²	—	2	8	no	—
244	C₂×C₃²	—	4	8	no	—
245	C₂×C₃²	—	4	4	no	—
246	C₂×C₃²	—	4	1	no	—
247	C₂×C₃²	C₂×#235	1	4	yes	—
248	C₂×C₃²	#1009	1	8	yes	—
249	C₂×C₃²	C₂×C₃×#41	1	1	yes	characteristic
250	C₂×C₃²	C₃×H(5,𝔽₂)	1	2	yes	—
251	C₃⋊D₃	—	4	4	no	—
252	C₃×D₃	—	12	16	no	—
253	C₂×D₄	—	9	4	no	—
254	C₂×D₄	—	9	2	no	—
255	C₂×D₄	—	9	4	no	—
256	C₂²×C₄	—	9	4	no	—
257	C₂²×C₄	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
258	C₄⋊C₄	—	9	2	no	—
259	C₄⋊C₄	—	9	4	no	—
260	C₄⋊C₄	C₂×C₃×C₃⋊D₃	1	2	yes	—
261	C₂²⋊C₄	—	9	4	no	—
262	C₂²⋊C₄	—	9	4	no	—
263	C₂²⋊C₄	C₂×C₃×C₃⋊D₃	1	2	yes	—
264	C₄²	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
265	C₄²	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
266	C₂²×C₃	—	2	8	no	—
267	C₂²×C₃	—	2	4	no	—
268	C₂²×C₃	—	2	8	no	—
269	C₂²×C₃	—	2	1	no	—
270	C₂²×C₃	—	2	2	no	—
271	C₂²×C₃	—	2	4	no	—
272	C₂²×C₃	—	4	8	no	—
273	C₂²×C₃	—	18	4	no	—
274	C₂²×C₃	—	18	4	no	—
275	C₂²×C₃	—	18	4	no	—
276	C₂²×C₃	C₂³×C₃×D₃	1	4	yes	—
277	C₂²×C₃	C₂³×C₃⋊D₃	1	1	yes	characteristic
278	C₂×D₃	—	6	16	no	—
279	C₂×D₃	—	6	16	no	—
280	C₂×D₃	—	6	16	no	—
281	C₃×C₄	—	2	8	no	—
282	C₃×C₄	—	2	4	no	—
283	C₃×C₄	—	2	8	no	—
284	C₃×C₄	—	2	8	no	—
285	C₃×C₄	—	2	1	no	—
286	C₃×C₄	—	2	2	no	—
287	C₃×C₄	—	2	2	no	—
288	C₃×C₄	—	2	8	no	—
289	C₃×C₄	—	4	8	no	—
290	C₃×C₄	—	4	8	no	—
291	C₃×C₄	—	18	4	no	—
292	C₃×C₄	C₃×#235	1	8	yes	—
293	C₃×C₄	#3828	1	2	yes	—
294	Dic₃	—	6	16	no	—
295	C₃²	—	2	4	no	—
296	C₃²	#5535	1	4	yes	complemented
297	C₃²	C₃×#540	1	1	yes	characteristic, complemented
298	C₂³	—	9	4	no	—
299	C₂³	C₂²×C₃×C₃⋊D₃	1	1	yes	characteristic
300	D₄	—	18	4	no	—
301	D₄	—	36	4	no	—
302	D₄	—	36	4	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	9	4	no	—
305	C₂×C₄	—	36	4	no	—
306	C₂×C₄	C₂²×C₃×C₃⋊D₃	1	2	yes	—
307	C₂×C₄	C₂²×C₃×C₃⋊D₃	1	2	yes	—
308	C₂×C₄	C₂²×C₃×C₃⋊D₃	1	1	yes	characteristic
309	C₂×C₄	C₂²×C₃×C₃⋊D₃	1	1	yes	characteristic
310	C₂×C₃	—	2	4	no	—
311	C₂×C₃	—	2	8	no	—
312	C₂×C₃	—	4	8	no	—
313	C₂×C₃	—	4	4	no	—
314	C₂×C₃	—	4	1	no	—
315	C₂×C₃	—	36	4	no	—
316	C₂×C₃	C₂×C₃×#235	1	4	yes	—
317	C₂×C₃	C₃×#1009	1	8	yes	—
318	C₂×C₃	C₂×#3828	1	1	yes	characteristic
319	C₂×C₃	#64366	1	2	yes	—
320	D₃	—	12	16	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	2	2	no	—
323	C₂²	—	18	4	no	—
324	C₂²	—	18	4	no	—
325	C₂²	—	18	4	no	—
326	C₂²	C₂³×C₃×C₃⋊D₃	1	1	yes	characteristic
327	C₄	—	2	1	no	—
328	C₄	—	2	2	no	—
329	C₄	—	2	2	no	—
330	C₄	—	18	4	no	—
331	C₄	C₃×#3828	1	2	yes	—
332	C₃	—	2	4	no	—
333	C₃	C₃×#5535	1	4	yes	complemented
334	C₃	#10596006	1	1	yes	characteristic, direct product factor
335	C₂	—	4	1	no	—
336	C₂	—	36	4	no	—
337	C₂	C₂×C₃×#3828	1	1	yes	characteristic
338	C₂	C₃×#64366	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×#117	C₂
socle	C₂²×C₃³	C₂⁴
hypercenter	C₃×#117	C₃⋊D₃
hypocenter	C₃²	C₃×#540
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×#117

Complements

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

N	H ≅ G/N	type
1. C₃³×#117	C₂	semidirect
2. C₃×#64139	C₂	semidirect
3. C₃×#64103	C₂	semidirect
4. C₃×#64093	C₂	semidirect
5. C₃×#64083	C₂	semidirect
6. C₃×C₄×C₄⋊(C₃⋊D₃)	C₂	semidirect
7. C₃×#64079	C₂	semidirect
8. #10596006	C₃	direct (N × H)
24. C₃²×#117	D₃	semidirect
25. C₃²×#117	C₂×C₃	semidirect
26. #64139	C₂×C₃	semidirect
27. #64103	C₂×C₃	semidirect
28. #64093	C₂×C₃	semidirect
29. #64083	C₂×C₃	semidirect
30. C₄×C₄⋊(C₃⋊D₃)	C₂×C₃	semidirect
31. #64079	C₂×C₃	semidirect
101. C₃×#117	C₃×D₃	semidirect
102. C₃×#117	C₃⋊D₃	semidirect
210. #117	C₃×C₃⋊D₃	semidirect
211. C₃³	#540	semidirect
296. C₃²	#5535	semidirect
297. C₃²	C₃×#540	semidirect
333. C₃	C₃×#5535	semidirect
334. C₃	#10596006	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃³×#117	2
C₃×#64139	2
C₃×#64103	2
C₃×#64093	2
C₃×#64083	2
C₃×C₄×C₄⋊(C₃⋊D₃)	2
C₃×#64079	2
#10596006	3
C₃×#5535	3

Subgroups — C3×#10596006 #13179286

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10596006 #13179286