Subgroups — C₃×#10592812 #13163150

Order 1728.

1252 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#3828	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃³×C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#64135	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#63648	C₂	1	2	yes	maximal, complemented
5	C₃×#63643	C₂	1	2	yes	maximal, complemented
6	#10592812	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₃×#5113	—	3	4	no	maximal
8	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
9	C₂×C₃×#638	—	2	1	no	—
10	C₃×#3828	—	2	2	no	—
11	C₃×#3828	C₄	1	4	yes	complemented
12	C₂×C₃×C₄⋊(C₃⋊D₃)	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₄×C₃⋊D₃	—	2	1	no	—
14	C₂×C₃×#634	C₂²	1	1	yes	characteristic
15	C₂×C₃³×C₈	—	2	2	no	—
16	C₂×C₃³×C₈	C₂²	1	2	yes	—
17	C₃×#3757	—	2	1	no	—
18	C₃×C₄⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
19	C₃×C₄×C₃⋊Dic₃	—	2	1	no	—
20	C₂×#3828	C₂×C₃	1	1	yes	characteristic
21	C₂×C₃×#235	—	3	4	no	—
22	C₂²×C₃²×C₈	—	2	4	no	—
23	C₂²×C₃²×C₈	D₃	1	4	yes	complemented
24	C₂²×C₃²×C₈	C₂×C₃	1	1	yes	characteristic, complemented
25	#64135	C₂×C₃	1	1	yes	characteristic, complemented
26	C₃×#926	—	3	4	no	—
27	#63648	C₂×C₃	1	2	yes	complemented
28	#63643	C₂×C₃	1	2	yes	complemented
29	C₃×#821	—	3	8	no	—
30	C₃×#816	—	3	8	no	—
31	C₂³×C₃³	D₄	1	1	yes	characteristic
32	C₂²×C₃×C₃⋊D₃	—	2	1	no	—
33	C₂×C₃³×C₄	C₂×C₄	1	2	yes	—
34	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
35	C₂×C₃³×C₄	D₄	1	2	yes	—
36	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
37	C₃×#638	—	2	4	no	—
38	C₂×C₃×C₃⋊Dic₃	—	2	1	no	—
39	C₂×C₃×C₃⋊Dic₃	—	2	2	no	—
40	C₃×C₄⋊(C₃⋊D₃)	—	2	1	no	—
41	C₃×C₄⋊(C₃⋊D₃)	C₂×C₄	1	2	yes	—
42	C₃×C₄×C₃⋊D₃	—	2	4	no	—
43	C₃×#634	—	2	1	no	—
44	C₃×#634	C₂×C₄	1	2	yes	—
45	C₃³×C₈	—	2	2	no	—
46	C₃×#416	—	9	1	no	—
47	#5113	—	3	4	no	—
48	C₂²×C₃²×C₄	—	2	4	no	—
49	C₂²×C₃²×C₄	C₂×D₃	1	4	yes	—
50	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
51	C₂×#638	—	2	1	no	—
52	#3828	—	2	2	no	—
53	#3828	C₃×C₄	1	4	yes	complemented
54	C₂×C₄⋊(C₃⋊D₃)	C₂²×C₃	1	1	yes	characteristic
55	C₂×C₄×C₃⋊D₃	—	2	1	no	—
56	C₂×#634	C₂²×C₃	1	1	yes	characteristic
57	C₂×C₃×#67	—	6	4	no	—
58	C₃×#235	—	3	16	no	—
59	C₃×#235	—	6	8	no	—
60	C₂×C₃×D₁₂	—	3	4	no	—
61	C₂×C₃×C₄×D₃	—	6	4	no	—
62	C₂×C₃×Dic₆	—	3	4	no	—
63	C₂×C₃²×C₈	—	2	16	no	—
64	C₂×C₃²×C₈	—	2	8	no	—
65	C₂×C₃²×C₈	—	2	8	no	—
66	C₂×C₃²×C₈	—	2	2	no	—
67	C₂×C₃²×C₈	C₂×D₃	1	8	yes	—
68	C₂×C₃²×C₈	C₂²×C₃	1	2	yes	—
69	#3757	—	2	1	no	—
70	C₄⋊(C₃⋊Dic₃)	C₂²×C₃	1	2	yes	—
71	C₄×C₃⋊Dic₃	—	2	1	no	—
72	C₃×#217	—	6	4	no	—
73	C₃×C₄⋊Dic₃	—	3	8	no	—
74	C₃×C₄×Dic₃	—	6	4	no	—
75	C₂²×C₃³	—	2	2	no	—
76	C₂²×C₃³	C₂²⋊C₄	1	2	yes	—
77	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
78	C₂×C₃×C₃⋊D₃	—	2	2	no	—
79	C₂×C₃×C₃⋊D₃	—	4	2	no	—
80	C₃³×C₄	C₂²⋊C₄	1	2	yes	—
81	C₃³×C₄	C₂²×C₄	1	1	yes	characteristic
82	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
83	C₃×C₃⋊Dic₃	—	2	2	no	—
84	C₃×C₃⋊Dic₃	—	4	2	no	—
85	C₂×C₃×#41	—	9	1	no	—
86	C₂×#235	—	3	4	no	—
87	C₂²×C₃×C₈	—	2	4	no	—
88	C₂²×C₃×C₈	C₃×D₃	1	4	yes	complemented
89	C₂²×C₃×C₈	C₃⋊D₃	1	1	yes	characteristic, complemented
90	C₃×#117	—	9	1	no	—
91	#926	—	3	4	no	—
92	C₃×#103	—	9	2	no	—
93	C₃×#102	—	9	2	no	—
94	#821	—	3	8	no	—
95	#816	—	3	8	no	—
96	C₂³×C₃²	—	2	4	no	—
97	C₂³×C₃²	D₁₂	1	4	yes	—
98	C₂³×C₃²	C₃×D₄	1	1	yes	characteristic
99	C₂²×C₃⋊D₃	—	2	1	no	—
100	C₂²×C₃×D₃	—	6	4	no	—
101	C₂×C₃²×C₄	—	2	8	no	—
102	C₂×C₃²×C₄	—	2	4	no	—
103	C₂×C₃²×C₄	—	2	8	no	—
104	C₂×C₃²×C₄	—	2	4	no	—
105	C₂×C₃²×C₄	C₄×D₃	1	8	yes	—
106	C₂×C₃²×C₄	D₁₂	1	4	yes	—
107	C₂×C₃²×C₄	#67	1	8	yes	—
108	C₂×C₃²×C₄	C₂×C₃×C₄	1	2	yes	—
109	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
110	C₂×C₃²×C₄	C₃×D₄	1	2	yes	—
111	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
112	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
113	#638	—	2	4	no	—
114	C₂×C₃⋊Dic₃	—	2	1	no	—
115	C₂×C₃⋊Dic₃	—	2	2	no	—
116	C₄⋊(C₃⋊D₃)	—	2	1	no	—
117	C₄⋊(C₃⋊D₃)	C₂×C₃×C₄	1	2	yes	—
118	C₄×C₃⋊D₃	—	2	4	no	—
119	#634	—	2	1	no	—
120	#634	C₂×C₃×C₄	1	2	yes	—
121	C₃×#67	—	6	16	no	—
122	C₂×C₃×Dic₃	—	6	4	no	—
123	C₂×C₃×Dic₃	—	6	8	no	—
124	C₃×D₁₂	—	3	8	no	—
125	C₃×D₁₂	—	6	4	no	—
126	C₃×C₄×D₃	—	6	16	no	—
127	C₃×Dic₆	—	3	8	no	—
128	C₃×Dic₆	—	6	4	no	—
129	C₃²×C₈	—	2	16	no	—
130	C₃²×C₈	—	2	8	no	—
131	C₃²×C₈	—	2	2	no	—
132	#416	—	9	1	no	Sylow
133	C₂×C₃³	—	2	2	no	—
134	C₂×C₃³	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	C₂×C₃³	#135	1	2	yes	—
136	C₃×C₃⋊D₃	—	4	2	no	—
137	C₃×#41	—	9	4	no	—
138	C₃×#41	—	18	2	no	—
139	C₂×C₃×Q₈	—	9	1	no	—
140	C₂×C₃×D₄	—	9	1	no	—
141	C₂×C₃×D₄	—	18	1	no	—
142	C₂²×C₃×C₄	—	2	4	no	—
143	C₂²×C₃×C₄	—	18	1	no	—
144	C₂²×C₃×C₄	C₂×C₃×D₃	1	4	yes	—
145	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
146	C₂×#67	—	6	4	no	—
147	#235	—	3	16	no	—
148	#235	—	6	8	no	—
149	C₂×D₁₂	—	3	4	no	—
150	C₂×C₄×D₃	—	6	4	no	—
151	C₂×Dic₆	—	3	4	no	—
152	C₂×C₃×C₈	—	2	16	no	—
153	C₂×C₃×C₈	—	2	8	no	—
154	C₂×C₃×C₈	—	2	2	no	—
155	C₂×C₃×C₈	—	2	8	no	—
156	C₂×C₃×C₈	C₂×C₃×D₃	1	8	yes	—
157	C₂×C₃×C₈	C₂×C₃⋊D₃	1	2	yes	—
158	C₃×C₄⋊C₄	—	9	2	no	—
159	C₃×C₂²⋊C₄	—	18	1	no	—
160	C₃×C₄²	—	18	1	no	—
161	#217	—	6	4	no	—
162	C₄⋊Dic₃	—	3	8	no	—
163	C₄×Dic₃	—	6	4	no	—
164	C₂²×C₃²	—	2	16	no	—
165	C₂²×C₃²	—	2	8	no	—
166	C₂²×C₃²	—	2	2	no	—
167	C₂²×C₃²	—	2	8	no	—
168	C₂²×C₃²	—	2	4	no	—
169	C₂²×C₃²	(C₂×D₃)⋊C₄	1	8	yes	—
170	C₂²×C₃²	C₃×C₂²⋊C₄	1	2	yes	—
171	C₂²×C₃²	C₂×D₁₂	1	4	yes	—
172	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
173	C₂×C₃⋊D₃	—	2	2	no	—
174	C₂×C₃⋊D₃	—	4	2	no	—
175	C₂×C₃×D₃	—	6	8	no	—
176	C₂×C₃×D₃	—	12	8	no	—
177	C₃²×C₄	—	2	8	no	—
178	C₃²×C₄	—	2	4	no	—
179	C₃²×C₄	—	2	4	no	—
180	C₃²×C₄	(C₂×D₃)⋊C₄	1	8	yes	—
181	C₃²×C₄	C₃×C₂²⋊C₄	1	2	yes	—
182	C₃²×C₄	C₂×C₄×D₃	1	4	yes	—
183	C₃²×C₄	C₂×#67	1	4	yes	—
184	C₃²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
185	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
186	C₃⋊Dic₃	—	2	2	no	—
187	C₃⋊Dic₃	—	4	2	no	—
188	C₃×Dic₃	—	6	8	no	—
189	C₃×Dic₃	—	12	8	no	—
190	C₂×#41	—	9	1	no	—
191	C₂²×C₈	C₃×C₃⋊D₃	1	1	yes	characteristic, complemented
192	#117	—	9	1	no	—
193	#103	—	9	2	no	—
194	#102	—	9	2	no	—
195	C₃³	#416	1	1	yes	characteristic, complemented, Sylow
196	C₂³×C₃	—	2	4	no	—
197	C₂³×C₃	—	18	1	no	—
198	C₂³×C₃	C₃×D₁₂	1	4	yes	—
199	C₂³×C₃	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
200	C₂²×D₃	—	6	4	no	—
201	C₃×Q₈	—	9	2	no	—
202	C₃×Q₈	—	18	1	no	—
203	C₃×D₄	—	9	2	no	—
204	C₃×D₄	—	18	4	no	—
205	C₃×D₄	—	18	1	no	—
206	C₂×C₃×C₄	—	2	8	no	—
207	C₂×C₃×C₄	—	2	4	no	—
208	C₂×C₃×C₄	—	2	8	no	—
209	C₂×C₃×C₄	—	2	4	no	—
210	C₂×C₃×C₄	—	18	1	no	—
211	C₂×C₃×C₄	—	18	4	no	—
212	C₂×C₃×C₄	—	18	2	no	—
213	C₂×C₃×C₄	C₃×C₄×D₃	1	8	yes	—
214	C₂×C₃×C₄	C₃×D₁₂	1	4	yes	—
215	C₂×C₃×C₄	C₃×#67	1	8	yes	—
216	C₂×C₃×C₄	C₄×C₃⋊D₃	1	2	yes	—
217	C₂×C₃×C₄	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
218	C₂×C₃×C₄	#638	1	2	yes	—
219	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
220	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
221	#67	—	6	16	no	—
222	C₂×Dic₃	—	6	4	no	—
223	C₂×Dic₃	—	6	8	no	—
224	D₁₂	—	3	8	no	—
225	D₁₂	—	6	4	no	—
226	C₄×D₃	—	6	16	no	—
227	Dic₆	—	3	8	no	—
228	Dic₆	—	6	4	no	—
229	C₃×C₈	—	2	16	no	—
230	C₃×C₈	—	2	8	no	—
231	C₃×C₈	—	2	2	no	—
232	C₂×C₃²	—	2	16	no	—
233	C₂×C₃²	—	2	2	no	—
234	C₂×C₃²	—	2	8	no	—
235	C₂×C₃²	—	2	8	no	—
236	C₂×C₃²	—	2	4	no	—
237	C₂×C₃²	#904	1	8	yes	—
238	C₂×C₃²	C₂×(C₂×D₃)⋊C₄	1	4	yes	—
239	C₂×C₃²	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
240	C₂×C₃²	C₃×#135	1	2	yes	—
241	C₃⋊D₃	—	4	2	no	—
242	C₃×D₃	—	12	8	no	—
243	#41	—	9	4	no	—
244	#41	—	18	2	no	—
245	C₂×Q₈	—	9	1	no	—
246	C₂×D₄	—	9	1	no	—
247	C₂×D₄	—	18	1	no	—
248	C₂²×C₄	—	18	1	no	—
249	C₂²×C₄	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	C₂×C₃×C₃⋊D₃	1	2	yes	—
252	C₄⋊C₄	—	9	2	no	—
253	C₂²⋊C₄	—	18	1	no	—
254	C₄²	—	18	1	no	—
255	C₂²×C₃	—	2	16	no	—
256	C₂²×C₃	—	2	8	no	—
257	C₂²×C₃	—	2	2	no	—
258	C₂²×C₃	—	2	8	no	—
259	C₂²×C₃	—	2	4	no	—
260	C₂²×C₃	—	18	2	no	—
261	C₂²×C₃	—	36	2	no	—
262	C₂²×C₃	C₃×(C₂×D₃)⋊C₄	1	8	yes	—
263	C₂²×C₃	#3752	1	2	yes	—
264	C₂²×C₃	C₂×C₃×D₁₂	1	4	yes	—
265	C₂²×C₃	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
266	C₂×D₃	—	6	8	no	—
267	C₂×D₃	—	12	8	no	—
268	C₃×C₄	—	2	8	no	—
269	C₃×C₄	—	2	4	no	—
270	C₃×C₄	—	2	4	no	—
271	C₃×C₄	—	18	2	no	—
272	C₃×C₄	—	36	2	no	—
273	C₃×C₄	C₃×(C₂×D₃)⋊C₄	1	8	yes	—
274	C₃×C₄	#3752	1	2	yes	—
275	C₃×C₄	C₂×C₃×C₄×D₃	1	4	yes	—
276	C₃×C₄	C₂×C₃×#67	1	4	yes	—
277	C₃×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
278	C₃×C₄	C₂×#638	1	1	yes	characteristic
279	Dic₃	—	6	8	no	—
280	Dic₃	—	12	8	no	—
281	C₃²	—	2	4	no	—
282	C₃²	#5113	1	4	yes	complemented
283	C₃²	C₃×#416	1	1	yes	characteristic, complemented
284	C₂³	—	18	1	no	—
285	C₂³	C₃×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
286	Q₈	—	9	2	no	—
287	Q₈	—	18	1	no	—
288	D₄	—	9	2	no	—
289	D₄	—	18	4	no	—
290	D₄	—	18	1	no	—
291	C₂×C₄	—	18	1	no	—
292	C₂×C₄	—	18	4	no	—
293	C₂×C₄	—	18	2	no	—
294	C₂×C₄	C₃×C₄×C₃⋊D₃	1	2	yes	—
295	C₂×C₄	C₃×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
296	C₂×C₄	C₃×#638	1	2	yes	—
297	C₂×C₄	C₂²×C₃×C₃⋊D₃	1	1	yes	characteristic
298	C₈	—	2	2	no	—
299	C₂×C₃	—	2	16	no	—
300	C₂×C₃	—	2	2	no	—
301	C₂×C₃	—	2	8	no	—
302	C₂×C₃	—	2	8	no	—
303	C₂×C₃	—	2	4	no	—
304	C₂×C₃	—	36	2	no	—
305	C₂×C₃	C₃×#904	1	8	yes	—
306	C₂×C₃	C₂×C₃×(C₂×D₃)⋊C₄	1	4	yes	—
307	C₂×C₃	#64113	1	2	yes	—
308	C₂×C₃	C₂×#3752	1	1	yes	characteristic
309	D₃	—	12	8	no	—
310	C₂²	—	2	2	no	—
311	C₂²	—	18	2	no	—
312	C₂²	—	36	2	no	—
313	C₂²	C₃×#3752	1	2	yes	—
314	C₂²	C₂×C₃×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
315	C₄	—	18	2	no	—
316	C₄	—	36	2	no	—
317	C₄	C₃×#3752	1	2	yes	—
318	C₄	C₂×C₃×C₄×C₃⋊D₃	1	1	yes	characteristic
319	C₄	C₂×C₃×#638	1	1	yes	characteristic
320	C₃	—	2	4	no	—
321	C₃	C₃×#5113	1	4	yes	complemented
322	C₃	#10592812	1	1	yes	characteristic, direct product factor
323	C₂	—	2	2	no	—
324	C₂	—	36	2	no	—
325	C₂	C₃×#64113	1	2	yes	—
326	C₂	C₂×C₃×#3752	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₄⋊(C₃⋊D₃)
commutator	C₃²×C₄	C₂²×C₃×C₄
Frattini	C₂×C₄	C₂²×C₃×C₃⋊D₃
Fitting	C₂²×C₃³×C₈	C₂
socle	C₂²×C₃³	C₂×D₄
prime radical	C₂×C₃×#3828	C₂
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₂²×C₃×C₈	C₃⋊D₃
hypocenter	C₃²	C₃×#416
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂²×C₃×C₄ ◁ C₂²×C₃×C₈

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃³×C₈	C₂	semidirect
3. C₃×#64135	C₂	semidirect
4. C₃×#63648	C₂	semidirect
5. C₃×#63643	C₂	semidirect
6. #10592812	C₃	direct (N × H)
11. C₃×#3828	C₄	semidirect
23. C₂²×C₃²×C₈	D₃	semidirect
24. C₂²×C₃²×C₈	C₂×C₃	semidirect
25. #64135	C₂×C₃	semidirect
27. #63648	C₂×C₃	semidirect
28. #63643	C₂×C₃	semidirect
53. #3828	C₃×C₄	semidirect
88. C₂²×C₃×C₈	C₃×D₃	semidirect
89. C₂²×C₃×C₈	C₃⋊D₃	semidirect
191. C₂²×C₈	C₃×C₃⋊D₃	semidirect
195. C₃³	#416	semidirect
282. C₃²	#5113	semidirect
283. C₃²	C₃×#416	semidirect
321. C₃	C₃×#5113	semidirect
322. C₃	#10592812	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×#3828	2
C₂²×C₃³×C₈	2
C₃×#64135	2
C₃×#63648	2
C₃×#63643	2
#10592812	3
C₃×#5113	3

Subgroups — C3×#10592812 #13163150

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10592812 #13163150