Subgroups — C₃×#10596187 #13179468

Order 1728.

2292 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×#3830	C₂	1	2	yes	maximal, complemented
2	C₃³×#127	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#64147	C₂	1	4	yes	maximal, complemented
4	C₃×#64145	C₂	1	2	yes	maximal, complemented
5	C₃×D₄×C₃⋊Dic₃	C₂	1	4	yes	maximal, complemented
6	C₃×C₄²×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#64078	C₂	1	1	yes	maximal, characteristic, complemented
8	#10596187	C₃	1	1	yes	maximal, characteristic, direct product factor
9	C₃×#5716	—	3	4	no	maximal
10	C₂×C₃³×D₄	C₂²	1	4	yes	—
11	C₂×C₃³×D₄	C₂²	1	2	yes	—
12	C₂×C₃×#638	C₂²	1	4	yes	—
13	C₂²×C₃×C₃⋊Dic₃	C₂²	1	4	yes	—
14	C₃×#3830	—	2	8	no	—
15	C₂×C₃×C₄×C₃⋊D₃	C₂²	1	2	yes	—
16	C₂×C₃×C₄×C₃⋊D₃	C₂²	1	1	yes	characteristic
17	C₂×C₃×#634	C₂²	1	2	yes	—
18	C₃³×C₄²	C₂²	1	1	yes	characteristic
19	C₃×#3757	C₂²	1	8	yes	—
20	C₃×C₄⋊(C₃⋊Dic₃)	C₂²	1	4	yes	—
21	C₃×C₄×C₃⋊Dic₃	C₂²	1	2	yes	—
22	C₃×C₄×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
23	C₂×#3830	C₂×C₃	1	2	yes	complemented
24	C₂×C₃×#237	—	3	8	no	—
25	C₃²×#127	—	2	4	no	—
26	C₃²×#127	D₃	1	4	yes	complemented
27	C₃²×#127	C₂×C₃	1	1	yes	characteristic, complemented
28	#64147	C₂×C₃	1	4	yes	complemented
29	#64145	C₂×C₃	1	2	yes	complemented
30	D₄×C₃⋊Dic₃	C₂×C₃	1	4	yes	complemented
31	C₄²×C₃⋊D₃	C₂×C₃	1	1	yes	characteristic, complemented
32	#64078	C₂×C₃	1	1	yes	characteristic, complemented
33	C₃×#938	—	3	16	no	—
34	C₃×#936	—	3	8	no	—
35	C₃×D₄×Dic₃	—	3	16	no	—
36	C₃×C₄²×D₃	—	3	4	no	—
37	C₃×#869	—	3	4	no	—
38	C₂³×C₃³	C₂³	1	4	yes	—
39	C₂²×C₃×C₃⋊D₃	C₂³	1	1	yes	characteristic
40	C₃³×D₄	—	2	4	no	—
41	C₃³×D₄	—	2	8	no	—
42	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
43	C₂×C₃³×C₄	C₂³	1	2	yes	—
44	C₃×#638	—	2	8	no	—
45	C₂×C₃×C₃⋊Dic₃	—	2	8	no	—
46	C₂×C₃×C₃⋊Dic₃	C₂³	1	2	yes	—
47	C₂×C₃×C₃⋊Dic₃	C₂³	1	4	yes	—
48	C₂×C₃×C₃⋊Dic₃	C₂³	1	1	yes	characteristic
49	C₃×C₄×C₃⋊D₃	—	2	4	no	—
50	C₃×C₄×C₃⋊D₃	D₄	1	4	yes	complemented
51	C₃×#634	—	2	4	no	—
52	C₃×#532	—	9	1	no	—
53	#5716	—	3	4	no	—
54	C₂×C₃²×D₄	—	2	16	no	—
55	C₂×C₃²×D₄	—	2	8	no	—
56	C₂×C₃²×D₄	C₂×D₃	1	16	yes	—
57	C₂×C₃²×D₄	C₂×D₃	1	8	yes	—
58	C₂×C₃²×D₄	C₂²×C₃	1	2	yes	—
59	C₂×C₃²×D₄	C₂²×C₃	1	4	yes	—
60	C₂×#638	C₂²×C₃	1	4	yes	—
61	C₂²×C₃⋊Dic₃	C₂²×C₃	1	4	yes	—
62	#3830	—	2	8	no	—
63	C₂×C₄×C₃⋊D₃	C₂²×C₃	1	2	yes	—
64	C₂×C₄×C₃⋊D₃	C₂²×C₃	1	1	yes	characteristic
65	C₂×#634	C₂²×C₃	1	2	yes	—
66	C₂×C₃×#67	—	3	16	no	—
67	C₂²×C₃×Dic₃	—	3	16	no	—
68	C₃×#237	—	6	32	no	—
69	C₂×C₃×C₄×D₃	—	3	4	no	—
70	C₂×C₃×C₄×D₃	—	3	8	no	—
71	C₂×C₃×Dic₆	—	3	8	no	—
72	C₃²×C₄²	—	2	4	no	—
73	C₃²×C₄²	C₂×D₃	1	4	yes	—
74	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
75	#3757	C₂²×C₃	1	8	yes	—
76	C₄⋊(C₃⋊Dic₃)	C₂²×C₃	1	4	yes	—
77	C₄×C₃⋊Dic₃	C₂²×C₃	1	2	yes	—
78	C₄×C₃⋊Dic₃	C₂²×C₃	1	1	yes	characteristic
79	C₃×#217	—	3	32	no	—
80	C₃×C₄⋊Dic₃	—	3	16	no	—
81	C₃×C₄×Dic₃	—	3	8	no	—
82	C₃×C₄×Dic₃	—	3	4	no	—
83	C₂²×C₃³	—	2	4	no	—
84	C₂²×C₃³	—	2	8	no	—
85	C₂²×C₃³	C₂⁴	1	1	yes	characteristic
86	C₂×C₃×C₃⋊D₃	—	2	2	no	—
87	C₂×C₃×C₃⋊D₃	C₂×D₄	1	2	yes	—
88	C₃³×C₄	C₂×D₄	1	2	yes	—
89	C₃³×C₄	#41	1	4	yes	complemented
90	C₃×C₃⋊Dic₃	—	2	4	no	—
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×#237	—	3	8	no	—
95	C₃×C₄⋊Q₈	—	9	1	no	—
96	C₃×#127	—	2	4	no	—
97	C₃×#127	C₃×D₃	1	4	yes	complemented
98	C₃×#127	C₃⋊D₃	1	1	yes	characteristic, complemented
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	#938	—	3	16	no	—
104	#936	—	3	8	no	—
105	D₄×Dic₃	—	3	16	no	—
106	C₄²×D₃	—	3	4	no	—
107	#869	—	3	4	no	—
108	C₂³×C₃²	—	2	16	no	—
109	C₂³×C₃²	C₂²×D₃	1	16	yes	—
110	C₂³×C₃²	C₂³×C₃	1	4	yes	—
111	C₂²×C₃⋊D₃	C₂³×C₃	1	1	yes	characteristic
112	C₂²×C₃×D₃	—	3	4	no	—
113	C₃²×D₄	—	2	4	no	—
114	C₃²×D₄	—	2	8	no	—
115	C₃²×D₄	—	2	16	no	—
116	C₃²×D₄	—	2	32	no	—
117	C₃²×D₄	—	4	16	no	—
118	C₃²×D₄	—	4	32	no	—
119	C₂×C₃²×C₄	—	2	4	no	—
120	C₂×C₃²×C₄	—	2	8	no	—
121	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
122	C₂×C₃²×C₄	C₂²×D₃	1	8	yes	—
123	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
124	C₂×C₃²×C₄	C₂³×C₃	1	2	yes	—
125	#638	—	2	8	no	—
126	C₂×C₃⋊Dic₃	—	2	8	no	—
127	C₂×C₃⋊Dic₃	C₂³×C₃	1	2	yes	—
128	C₂×C₃⋊Dic₃	C₂³×C₃	1	4	yes	—
129	C₂×C₃⋊Dic₃	C₂³×C₃	1	1	yes	characteristic
130	C₄×C₃⋊D₃	—	2	4	no	—
131	C₄×C₃⋊D₃	C₃×D₄	1	4	yes	complemented
132	#634	—	2	4	no	—
133	C₃×#67	—	6	32	no	—
134	C₂×C₃×Dic₃	—	3	4	no	—
135	C₂×C₃×Dic₃	—	3	8	no	—
136	C₂×C₃×Dic₃	—	3	16	no	—
137	C₂×C₃×Dic₃	—	6	32	no	—
138	C₃×C₄×D₃	—	3	16	no	—
139	C₃×C₄×D₃	—	6	16	no	—
140	C₃×Dic₆	—	6	16	no	—
141	#532	—	9	1	no	Sylow
142	C₂×C₃³	—	4	4	no	—
143	C₂×C₃³	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₃³	C₂×#41	1	2	yes	—
145	C₃×C₃⋊D₃	—	2	2	no	—
146	C₃×#41	—	18	8	no	—
147	C₂×C₃×Q₈	—	9	2	no	—
148	C₂×C₃×D₄	—	2	16	no	—
149	C₂×C₃×D₄	—	2	8	no	—
150	C₂×C₃×D₄	—	9	4	no	—
151	C₂×C₃×D₄	C₂×C₃×D₃	1	16	yes	—
152	C₂×C₃×D₄	C₂×C₃×D₃	1	8	yes	—
153	C₂×C₃×D₄	C₂×C₃⋊D₃	1	4	yes	—
154	C₂×C₃×D₄	C₂×C₃⋊D₃	1	2	yes	—
155	C₂²×C₃×C₄	—	9	1	no	—
156	C₂²×C₃×C₄	—	9	4	no	—
157	C₂²×C₃×C₄	—	9	2	no	—
158	C₂×#67	—	3	16	no	—
159	C₂²×Dic₃	—	3	16	no	—
160	#237	—	6	32	no	—
161	C₂×C₄×D₃	—	3	4	no	—
162	C₂×C₄×D₃	—	3	8	no	—
163	C₂×Dic₆	—	3	8	no	—
164	C₃×C₄⋊C₄	—	9	4	no	—
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	#217	—	3	32	no	—
172	C₄⋊Dic₃	—	3	16	no	—
173	C₄×Dic₃	—	3	8	no	—
174	C₄×Dic₃	—	3	4	no	—
175	C₂²×C₃²	—	2	4	no	—
176	C₂²×C₃²	—	2	8	no	—
177	C₂²×C₃²	—	2	4	no	—
178	C₂²×C₃²	—	2	16	no	—
179	C₂²×C₃²	—	2	32	no	—
180	C₂²×C₃²	—	4	16	no	—
181	C₂²×C₃²	—	4	32	no	—
182	C₂²×C₃²	C₂³×D₃	1	4	yes	—
183	C₂²×C₃²	C₂⁴×C₃	1	1	yes	characteristic
184	C₂×C₃⋊D₃	—	2	2	no	—
185	C₂×C₃⋊D₃	C₂×C₃×D₄	1	2	yes	—
186	C₂×C₃×D₃	—	3	8	no	—
187	C₂×C₃×D₃	—	6	8	no	—
188	C₃²×C₄	—	2	8	no	—
189	C₃²×C₄	—	2	16	no	—
190	C₃²×C₄	D₃×D₄	1	8	yes	—
191	C₃²×C₄	#237	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	4	no	—
195	C₃⋊Dic₃	—	2	2	no	—
196	C₃⋊Dic₃	C₂×C₃×D₄	1	2	yes	—
197	C₃×Dic₃	—	3	8	no	—
198	C₃×Dic₃	—	6	16	no	—
199	C₃×Dic₃	—	6	8	no	—
200	C₂×#41	—	9	2	no	—
201	C₄⋊Q₈	—	9	1	no	—
202	#127	C₃×C₃⋊D₃	1	1	yes	characteristic, complemented
203	#124	—	9	2	no	—
204	#121	—	9	4	no	—
205	C₄×D₄	—	9	4	no	—
206	C₂×C₄²	—	9	1	no	—
207	C₃³	#532	1	1	yes	characteristic, complemented, Sylow
208	C₂³×C₃	—	2	16	no	—
209	C₂³×C₃	—	9	1	no	—
210	C₂³×C₃	C₂²×C₃×D₃	1	16	yes	—
211	C₂³×C₃	C₂²×C₃⋊D₃	1	4	yes	—
212	C₂²×D₃	—	3	4	no	—
213	C₃×Q₈	—	18	4	no	—
214	C₃×D₄	—	2	16	no	—
215	C₃×D₄	—	2	32	no	—
216	C₃×D₄	—	2	4	no	—
217	C₃×D₄	—	2	8	no	—
218	C₃×D₄	—	4	16	no	—
219	C₃×D₄	—	4	32	no	—
220	C₃×D₄	—	18	8	no	—
221	C₂×C₃×C₄	—	2	4	no	—
222	C₂×C₃×C₄	—	2	8	no	—
223	C₂×C₃×C₄	—	9	1	no	—
224	C₂×C₃×C₄	—	9	4	no	—
225	C₂×C₃×C₄	—	9	2	no	—
226	C₂×C₃×C₄	—	9	4	no	—
227	C₂×C₃×C₄	—	18	8	no	—
228	C₂×C₃×C₄	—	18	4	no	—
229	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
230	C₂×C₃×C₄	C₂²×C₃×D₃	1	8	yes	—
231	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
232	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	2	yes	—
233	#67	—	6	32	no	—
234	C₂×Dic₃	—	3	4	no	—
235	C₂×Dic₃	—	3	8	no	—
236	C₂×Dic₃	—	3	16	no	—
237	C₂×Dic₃	—	6	32	no	—
238	C₄×D₃	—	3	16	no	—
239	C₄×D₃	—	6	16	no	—
240	Dic₆	—	6	16	no	—
241	C₂×C₃²	—	2	4	no	—
242	C₂×C₃²	—	2	8	no	—
243	C₂×C₃²	—	4	4	no	—
244	C₂×C₃²	—	4	16	no	—
245	C₂×C₃²	—	8	16	no	—
246	C₂×C₃²	C₂×D₃×D₄	1	4	yes	—
247	C₂×C₃²	C₂×#237	1	8	yes	—
248	C₂×C₃²	C₂²×C₃×D₄	1	1	yes	characteristic
249	C₂×C₃²	C₂×C₃×#41	1	2	yes	—
250	C₃⋊D₃	—	2	2	no	—
251	C₃×D₃	—	6	8	no	—
252	#41	—	18	8	no	—
253	C₂×Q₈	—	9	2	no	—
254	C₂×D₄	—	9	4	no	—
255	C₂×D₄	C₂×C₃×C₃⋊D₃	1	4	yes	—
256	C₂×D₄	C₂×C₃×C₃⋊D₃	1	2	yes	—
257	C₂²×C₄	—	9	1	no	—
258	C₂²×C₄	—	9	4	no	—
259	C₂²×C₄	—	9	2	no	—
260	C₄⋊C₄	—	9	4	no	—
261	C₂²⋊C₄	—	9	8	no	—
262	C₄²	—	9	2	no	—
263	C₄²	—	9	1	no	—
264	C₄²	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
265	C₂²×C₃	—	2	16	no	—
266	C₂²×C₃	—	2	32	no	—
267	C₂²×C₃	—	2	4	no	—
268	C₂²×C₃	—	2	8	no	—
269	C₂²×C₃	—	2	4	no	—
270	C₂²×C₃	—	4	16	no	—
271	C₂²×C₃	—	4	32	no	—
272	C₂²×C₃	—	9	2	no	—
273	C₂²×C₃	—	18	2	no	—
274	C₂²×C₃	C₂³×C₃×D₃	1	4	yes	—
275	C₂²×C₃	C₂³×C₃⋊D₃	1	1	yes	characteristic
276	C₂×D₃	—	3	8	no	—
277	C₂×D₃	—	6	8	no	—
278	C₃×C₄	—	2	8	no	—
279	C₃×C₄	—	2	16	no	—
280	C₃×C₄	—	9	2	no	—
281	C₃×C₄	—	18	4	no	—
282	C₃×C₄	—	18	2	no	—
283	C₃×C₄	C₃×D₃×D₄	1	8	yes	—
284	C₃×C₄	C₃×#237	1	16	yes	complemented
285	C₃×C₄	D₄×C₃⋊D₃	1	2	yes	—
286	C₃×C₄	#3830	1	4	yes	complemented
287	Dic₃	—	3	8	no	—
288	Dic₃	—	6	16	no	—
289	Dic₃	—	6	8	no	—
290	C₃²	—	2	4	no	—
291	C₃²	#5716	1	4	yes	complemented
292	C₃²	C₃×#532	1	1	yes	characteristic, complemented
293	C₂³	—	9	1	no	—
294	C₂³	C₂²×C₃×C₃⋊D₃	1	4	yes	—
295	Q₈	—	18	4	no	—
296	D₄	—	2	4	no	—
297	D₄	—	2	8	no	—
298	D₄	—	18	8	no	—
299	C₂×C₄	—	9	1	no	—
300	C₂×C₄	—	9	4	no	—
301	C₂×C₄	—	9	2	no	—
302	C₂×C₄	—	9	4	no	—
303	C₂×C₄	—	18	8	no	—
304	C₂×C₄	—	18	4	no	—
305	C₂×C₄	C₂²×C₃×C₃⋊D₃	1	1	yes	characteristic
306	C₂×C₄	C₂²×C₃×C₃⋊D₃	1	2	yes	—
307	C₂×C₃	—	2	4	no	—
308	C₂×C₃	—	2	8	no	—
309	C₂×C₃	—	4	4	no	—
310	C₂×C₃	—	4	16	no	—
311	C₂×C₃	—	8	16	no	—
312	C₂×C₃	—	18	2	no	—
313	C₂×C₃	C₂×C₃×D₃×D₄	1	4	yes	—
314	C₂×C₃	C₂×C₃×#237	1	8	yes	—
315	C₂×C₃	C₂×D₄×C₃⋊D₃	1	1	yes	characteristic
316	C₂×C₃	C₂×#3830	1	2	yes	—
317	D₃	—	6	8	no	—
318	C₂²	—	2	4	no	—
319	C₂²	—	2	8	no	—
320	C₂²	—	9	2	no	—
321	C₂²	—	18	2	no	—
322	C₂²	C₂³×C₃×C₃⋊D₃	1	1	yes	characteristic
323	C₄	—	9	2	no	—
324	C₄	—	18	4	no	—
325	C₄	—	18	2	no	—
326	C₄	C₃×D₄×C₃⋊D₃	1	2	yes	—
327	C₄	C₃×#3830	1	4	yes	complemented
328	C₃	—	2	4	no	—
329	C₃	C₃×#5716	1	4	yes	complemented
330	C₃	#10596187	1	1	yes	characteristic, direct product factor
331	C₂	—	4	4	no	—
332	C₂	—	18	2	no	—
333	C₂	C₂×C₃×D₄×C₃⋊D₃	1	1	yes	characteristic
334	C₂	C₂×C₃×#3830	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×#127	C₂
socle	C₂²×C₃³	C₂⁴
hypercenter	C₃×#127	C₃⋊D₃
hypocenter	C₃²	C₃×#532
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×#127

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#3830	C₂	semidirect
2. C₃³×#127	C₂	semidirect
3. C₃×#64147	C₂	semidirect
4. C₃×#64145	C₂	semidirect
5. C₃×D₄×C₃⋊Dic₃	C₂	semidirect
6. C₃×C₄²×C₃⋊D₃	C₂	semidirect
7. C₃×#64078	C₂	semidirect
8. #10596187	C₃	direct (N × H)
23. C₂×#3830	C₂×C₃	semidirect
26. C₃²×#127	D₃	semidirect
27. C₃²×#127	C₂×C₃	semidirect
28. #64147	C₂×C₃	semidirect
29. #64145	C₂×C₃	semidirect
30. D₄×C₃⋊Dic₃	C₂×C₃	semidirect
31. C₄²×C₃⋊D₃	C₂×C₃	semidirect
32. #64078	C₂×C₃	semidirect
50. C₃×C₄×C₃⋊D₃	D₄	semidirect
89. C₃³×C₄	#41	semidirect
97. C₃×#127	C₃×D₃	semidirect
98. C₃×#127	C₃⋊D₃	semidirect
131. C₄×C₃⋊D₃	C₃×D₄	semidirect
191. C₃²×C₄	#237	semidirect
193. C₃²×C₄	C₃×#41	semidirect
202. #127	C₃×C₃⋊D₃	semidirect
207. C₃³	#532	semidirect
284. C₃×C₄	C₃×#237	semidirect
286. C₃×C₄	#3830	semidirect
291. C₃²	#5716	semidirect
292. C₃²	C₃×#532	semidirect
327. C₄	C₃×#3830	semidirect
329. C₃	C₃×#5716	semidirect
330. C₃	#10596187	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×#3830	2
C₃³×#127	2
C₃×#64147	2
C₃×#64145	2
C₃×D₄×C₃⋊Dic₃	2
C₃×C₄²×C₃⋊D₃	2
C₃×#64078	2
#10596187	3
C₃×#5716	3

Subgroups — C3×#10596187 #13179468

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10596187 #13179468