Subgroups — C₃×#10592939 #13163271

Order 1728.

1604 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃³×#41	C₂	1	1	yes	maximal, characteristic
2	C₃×#64135	C₂	1	2	yes	maximal, complemented
3	C₃×#63661	C₂	1	4	yes	maximal, complemented
4	#10592939	C₃	1	1	yes	maximal, characteristic, direct product factor
5	C₃×#5240	—	3	4	no	maximal
6	C₃³×#41	—	2	2	no	—
7	C₃³×#41	—	2	2	no	—
8	C₂×C₃³×Q₈	C₄	1	1	yes	characteristic, complemented
9	C₂×C₃³×D₄	C₄	1	1	yes	characteristic, complemented
10	C₂×C₃³×D₄	C₂²	1	2	yes	—
11	C₂²×C₃³×C₄	C₄	1	2	yes	complemented
12	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
13	C₃×#3757	C₂²	1	4	yes	—
14	C₃×C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
15	C₃×C₄×C₃⋊Dic₃	—	2	2	no	—
16	C₂×C₃²×#41	—	2	4	no	—
17	C₂×C₃²×#41	D₃	1	4	yes	—
18	C₂×C₃²×#41	C₂×C₃	1	1	yes	characteristic
19	#64135	C₂×C₃	1	2	yes	complemented
20	C₃×#926	—	3	8	no	—
21	#63661	C₂×C₃	1	4	yes	complemented
22	C₃×#834	—	3	16	no	—
23	C₂³×C₃³	C₂×C₄	1	2	yes	—
24	C₂³×C₃³	C₂³	1	1	yes	characteristic
25	C₃³×Q₈	—	2	2	no	—
26	C₃³×D₄	—	2	4	no	—
27	C₃³×D₄	—	2	2	no	—
28	C₂×C₃³×C₄	—	2	2	no	—
29	C₂×C₃³×C₄	—	2	2	no	—
30	C₂×C₃³×C₄	C₂×C₄	1	2	yes	—
31	C₂×C₃³×C₄	C₂×C₄	1	1	yes	characteristic
32	C₂×C₃³×C₄	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃³×C₄	D₄	1	2	yes	—
34	C₂×C₃³×C₄	D₄	1	2	yes	—
35	C₂×C₃×C₃⋊Dic₃	—	2	4	no	—
36	C₃×#410	—	9	1	no	—
37	#5240	—	3	4	no	—
38	C₃²×#41	—	2	16	no	—
39	C₃²×#41	—	2	16	no	—
40	C₃²×#41	—	2	8	no	—
41	C₃²×#41	—	2	8	no	—
42	C₃²×#41	—	2	2	no	—
43	C₃²×#41	—	2	2	no	—
44	C₂×C₃²×Q₈	—	2	4	no	—
45	C₂×C₃²×Q₈	Dic₃	1	4	yes	complemented
46	C₂×C₃²×Q₈	C₃×C₄	1	1	yes	characteristic, complemented
47	C₂×C₃²×D₄	—	2	4	no	—
48	C₂×C₃²×D₄	—	2	8	no	—
49	C₂×C₃²×D₄	Dic₃	1	4	yes	complemented
50	C₂×C₃²×D₄	C₃×C₄	1	1	yes	characteristic, complemented
51	C₂×C₃²×D₄	C₂×D₃	1	8	yes	—
52	C₂×C₃²×D₄	C₂²×C₃	1	2	yes	—
53	C₂²×C₃²×C₄	—	2	8	no	—
54	C₂²×C₃²×C₄	—	2	4	no	—
55	C₂²×C₃²×C₄	Dic₃	1	8	yes	complemented
56	C₂²×C₃²×C₄	C₃×C₄	1	2	yes	complemented
57	C₂²×C₃²×C₄	C₂×D₃	1	4	yes	—
58	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
59	#3757	C₂²×C₃	1	4	yes	—
60	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
61	C₄×C₃⋊Dic₃	—	2	2	no	—
62	C₃×#217	—	3	16	no	—
63	C₃×C₄⋊Dic₃	—	6	8	no	—
64	C₃×C₄×Dic₃	—	6	8	no	—
65	C₂²×C₃³	—	2	2	no	—
66	C₂²×C₃³	—	4	1	no	—
67	C₂²×C₃³	—	4	2	no	—
68	C₂²×C₃³	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₃³	C₂×D₄	1	2	yes	—
70	C₃³×C₄	—	2	2	no	—
71	C₃³×C₄	C₂²⋊C₄	1	1	yes	characteristic
72	C₃³×C₄	C₂²⋊C₄	1	1	yes	characteristic
73	C₃³×C₄	C₂²⋊C₄	1	2	yes	—
74	C₃×C₃⋊Dic₃	—	4	4	no	—
75	C₂×C₃×#41	—	2	4	no	—
76	C₂×C₃×#41	C₃×D₃	1	4	yes	—
77	C₂×C₃×#41	C₃⋊D₃	1	1	yes	characteristic
78	C₃×#117	—	9	2	no	—
79	#926	—	3	8	no	—
80	C₃×#99	—	9	4	no	—
81	#834	—	3	16	no	—
82	C₂³×C₃²	—	2	8	no	—
83	C₂³×C₃²	—	2	4	no	—
84	C₂³×C₃²	C₂×Dic₃	1	8	yes	—
85	C₂³×C₃²	C₂×C₃×C₄	1	2	yes	—
86	C₂³×C₃²	C₂²×D₃	1	4	yes	—
87	C₂³×C₃²	C₂³×C₃	1	1	yes	characteristic
88	C₃²×Q₈	—	2	16	no	—
89	C₃²×Q₈	—	2	8	no	—
90	C₃²×Q₈	—	2	2	no	—
91	C₃²×D₄	—	2	32	no	—
92	C₃²×D₄	—	2	16	no	—
93	C₃²×D₄	—	2	16	no	—
94	C₃²×D₄	—	2	4	no	—
95	C₃²×D₄	—	2	8	no	—
96	C₃²×D₄	—	2	2	no	—
97	C₂×C₃²×C₄	—	2	16	no	—
98	C₂×C₃²×C₄	—	2	16	no	—
99	C₂×C₃²×C₄	—	2	8	no	—
100	C₂×C₃²×C₄	—	2	8	no	—
101	C₂×C₃²×C₄	—	2	2	no	—
102	C₂×C₃²×C₄	—	2	2	no	—
103	C₂×C₃²×C₄	—	2	4	no	—
104	C₂×C₃²×C₄	—	2	8	no	—
105	C₂×C₃²×C₄	—	2	4	no	—
106	C₂×C₃²×C₄	—	2	8	no	—
107	C₂×C₃²×C₄	—	2	8	no	—
108	C₂×C₃²×C₄	C₂×Dic₃	1	8	yes	—
109	C₂×C₃²×C₄	C₂×Dic₃	1	4	yes	—
110	C₂×C₃²×C₄	C₂×Dic₃	1	4	yes	—
111	C₂×C₃²×C₄	#67	1	8	yes	—
112	C₂×C₃²×C₄	#67	1	8	yes	—
113	C₂×C₃²×C₄	C₂×C₃×C₄	1	2	yes	—
114	C₂×C₃²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
115	C₂×C₃²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
116	C₂×C₃²×C₄	C₃×D₄	1	2	yes	—
117	C₂×C₃²×C₄	C₃×D₄	1	2	yes	—
118	C₂×C₃⋊Dic₃	—	2	4	no	—
119	C₂×C₃×Dic₃	—	6	16	no	—
120	#410	—	9	1	no	Sylow
121	C₂×C₃³	—	2	1	no	—
122	C₂×C₃³	—	2	2	no	—
123	C₂×C₃³	—	4	2	no	—
124	C₂×C₃³	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₃×#41	—	2	16	no	—
126	C₃×#41	—	2	16	no	—
127	C₃×#41	—	2	8	no	—
128	C₃×#41	—	2	8	no	—
129	C₃×#41	—	2	2	no	—
130	C₃×#41	—	2	2	no	—
131	C₂×C₃×Q₈	—	2	4	no	—
132	C₂×C₃×Q₈	C₃×Dic₃	1	4	yes	complemented
133	C₂×C₃×Q₈	C₃⋊Dic₃	1	1	yes	characteristic, complemented
134	C₂×C₃×D₄	—	2	4	no	—
135	C₂×C₃×D₄	—	2	8	no	—
136	C₂×C₃×D₄	C₃×Dic₃	1	4	yes	complemented
137	C₂×C₃×D₄	C₃⋊Dic₃	1	1	yes	characteristic, complemented
138	C₂×C₃×D₄	C₂×C₃×D₃	1	8	yes	—
139	C₂×C₃×D₄	C₂×C₃⋊D₃	1	2	yes	—
140	C₂²×C₃×C₄	—	2	8	no	—
141	C₂²×C₃×C₄	—	2	4	no	—
142	C₂²×C₃×C₄	C₃×Dic₃	1	8	yes	complemented
143	C₂²×C₃×C₄	C₃⋊Dic₃	1	2	yes	complemented
144	C₂²×C₃×C₄	C₂×C₃×D₃	1	4	yes	—
145	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
146	C₃×C₄⋊C₄	—	18	2	no	—
147	C₃×C₂²⋊C₄	—	9	4	no	—
148	C₃×C₄²	—	18	2	no	—
149	#217	—	3	16	no	—
150	C₄⋊Dic₃	—	6	8	no	—
151	C₄×Dic₃	—	6	8	no	—
152	C₂²×C₃²	—	2	16	no	—
153	C₂²×C₃²	—	2	8	no	—
154	C₂²×C₃²	—	2	2	no	—
155	C₂²×C₃²	—	2	4	no	—
156	C₂²×C₃²	—	2	8	no	—
157	C₂²×C₃²	—	4	4	no	—
158	C₂²×C₃²	—	4	1	no	—
159	C₂²×C₃²	—	4	8	no	—
160	C₂²×C₃²	—	4	2	no	—
161	C₂²×C₃²	—	4	8	no	—
162	C₂²×C₃²	—	4	16	no	—
163	C₂²×C₃²	C₂²×Dic₃	1	4	yes	—
164	C₂²×C₃²	C₂×#67	1	8	yes	—
165	C₂²×C₃²	C₂²×C₃×C₄	1	1	yes	characteristic
166	C₂²×C₃²	C₂×C₃×D₄	1	2	yes	—
167	C₃²×C₄	—	2	16	no	—
168	C₃²×C₄	—	2	8	no	—
169	C₃²×C₄	—	2	2	no	—
170	C₃²×C₄	—	2	4	no	—
171	C₃²×C₄	—	2	4	no	—
172	C₃²×C₄	—	2	8	no	—
173	C₃²×C₄	#217	1	4	yes	—
174	C₃²×C₄	#217	1	8	yes	—
175	C₃²×C₄	#217	1	4	yes	—
176	C₃²×C₄	C₃×C₂²⋊C₄	1	1	yes	characteristic
177	C₃²×C₄	C₃×C₂²⋊C₄	1	1	yes	characteristic
178	C₃²×C₄	C₃×C₂²⋊C₄	1	2	yes	—
179	C₃⋊Dic₃	—	4	4	no	—
180	C₃×Dic₃	—	12	16	no	—
181	C₂×#41	C₃×C₃⋊D₃	1	1	yes	characteristic
182	#117	—	9	2	no	—
183	#99	—	9	4	no	—
184	C₃³	#410	1	1	yes	characteristic, complemented, Sylow
185	C₂³×C₃	—	2	8	no	—
186	C₂³×C₃	—	2	4	no	—
187	C₂³×C₃	C₂×C₃×Dic₃	1	8	yes	—
188	C₂³×C₃	C₂×C₃⋊Dic₃	1	2	yes	—
189	C₂³×C₃	C₂²×C₃×D₃	1	4	yes	—
190	C₂³×C₃	C₂²×C₃⋊D₃	1	1	yes	characteristic
191	C₃×Q₈	—	2	16	no	—
192	C₃×Q₈	—	2	8	no	—
193	C₃×Q₈	—	2	2	no	—
194	C₃×D₄	—	2	32	no	—
195	C₃×D₄	—	2	16	no	—
196	C₃×D₄	—	2	16	no	—
197	C₃×D₄	—	2	4	no	—
198	C₃×D₄	—	2	8	no	—
199	C₃×D₄	—	2	2	no	—
200	C₂×C₃×C₄	—	2	16	no	—
201	C₂×C₃×C₄	—	2	16	no	—
202	C₂×C₃×C₄	—	2	8	no	—
203	C₂×C₃×C₄	—	2	8	no	—
204	C₂×C₃×C₄	—	2	2	no	—
205	C₂×C₃×C₄	—	2	2	no	—
206	C₂×C₃×C₄	—	2	4	no	—
207	C₂×C₃×C₄	—	2	8	no	—
208	C₂×C₃×C₄	—	2	4	no	—
209	C₂×C₃×C₄	—	2	8	no	—
210	C₂×C₃×C₄	—	2	8	no	—
211	C₂×C₃×C₄	—	18	4	no	—
212	C₂×C₃×C₄	C₂×C₃×Dic₃	1	8	yes	—
213	C₂×C₃×C₄	C₂×C₃×Dic₃	1	4	yes	—
214	C₂×C₃×C₄	C₂×C₃×Dic₃	1	4	yes	—
215	C₂×C₃×C₄	C₃×#67	1	8	yes	—
216	C₂×C₃×C₄	C₃×#67	1	8	yes	—
217	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	2	yes	—
218	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
219	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
220	C₂×C₃×C₄	#638	1	2	yes	—
221	C₂×C₃×C₄	#638	1	2	yes	—
222	C₂×Dic₃	—	6	16	no	—
223	C₂×C₃²	—	2	8	no	—
224	C₂×C₃²	—	2	4	no	—
225	C₂×C₃²	—	2	2	no	—
226	C₂×C₃²	—	2	1	no	—
227	C₂×C₃²	—	2	4	no	—
228	C₂×C₃²	—	2	8	no	—
229	C₂×C₃²	—	4	8	no	—
230	C₂×C₃²	—	4	16	no	—
231	C₂×C₃²	—	4	8	no	—
232	C₂×C₃²	—	4	2	no	—
233	C₂×C₃²	C₂×#217	1	4	yes	—
234	C₂×C₃²	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
235	#41	—	2	2	no	—
236	#41	—	2	2	no	—
237	C₂×Q₈	C₃×C₃⋊Dic₃	1	1	yes	characteristic, complemented
238	C₂×D₄	C₃×C₃⋊Dic₃	1	1	yes	characteristic, complemented
239	C₂×D₄	C₂×C₃×C₃⋊D₃	1	2	yes	—
240	C₂²×C₄	C₃×C₃⋊Dic₃	1	2	yes	complemented
241	C₂²×C₄	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
242	C₄⋊C₄	—	18	2	no	—
243	C₂²⋊C₄	—	9	4	no	—
244	C₄²	—	18	2	no	—
245	C₂²×C₃	—	2	16	no	—
246	C₂²×C₃	—	2	8	no	—
247	C₂²×C₃	—	2	2	no	—
248	C₂²×C₃	—	2	4	no	—
249	C₂²×C₃	—	2	8	no	—
250	C₂²×C₃	—	4	4	no	—
251	C₂²×C₃	—	4	1	no	—
252	C₂²×C₃	—	4	8	no	—
253	C₂²×C₃	—	4	2	no	—
254	C₂²×C₃	—	4	8	no	—
255	C₂²×C₃	—	4	16	no	—
256	C₂²×C₃	C₂²×C₃×Dic₃	1	4	yes	—
257	C₂²×C₃	C₂×C₃×#67	1	8	yes	—
258	C₂²×C₃	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
259	C₂²×C₃	C₂×#638	1	2	yes	—
260	C₃×C₄	—	2	16	no	—
261	C₃×C₄	—	2	8	no	—
262	C₃×C₄	—	2	2	no	—
263	C₃×C₄	—	2	4	no	—
264	C₃×C₄	—	2	4	no	—
265	C₃×C₄	—	2	8	no	—
266	C₃×C₄	—	36	4	no	—
267	C₃×C₄	C₃×#217	1	4	yes	—
268	C₃×C₄	C₃×#217	1	4	yes	—
269	C₃×C₄	C₃×#217	1	8	yes	—
270	C₃×C₄	#3757	1	1	yes	characteristic
271	C₃×C₄	#3757	1	2	yes	—
272	C₃×C₄	#3757	1	1	yes	characteristic
273	Dic₃	—	12	16	no	—
274	C₃²	—	2	4	no	—
275	C₃²	#5240	1	4	yes	complemented
276	C₃²	C₃×#410	1	1	yes	characteristic, complemented
277	C₂³	C₂×C₃×C₃⋊Dic₃	1	2	yes	—
278	C₂³	C₂²×C₃×C₃⋊D₃	1	1	yes	characteristic
279	Q₈	—	2	2	no	—
280	D₄	—	2	4	no	—
281	D₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	18	4	no	—
285	C₂×C₄	C₂×C₃×C₃⋊Dic₃	1	2	yes	—
286	C₂×C₄	C₂×C₃×C₃⋊Dic₃	1	1	yes	characteristic
287	C₂×C₄	C₂×C₃×C₃⋊Dic₃	1	1	yes	characteristic
288	C₂×C₄	C₃×#638	1	2	yes	—
289	C₂×C₄	C₃×#638	1	2	yes	—
290	C₂×C₃	—	2	8	no	—
291	C₂×C₃	—	2	8	no	—
292	C₂×C₃	—	2	4	no	—
293	C₂×C₃	—	2	2	no	—
294	C₂×C₃	—	2	1	no	—
295	C₂×C₃	—	2	4	no	—
296	C₂×C₃	—	4	8	no	—
297	C₂×C₃	—	4	16	no	—
298	C₂×C₃	—	4	8	no	—
299	C₂×C₃	—	4	2	no	—
300	C₂×C₃	C₂×C₃×#217	1	4	yes	—
301	C₂×C₃	C₂×#3757	1	1	yes	characteristic
302	C₂²	—	2	2	no	—
303	C₂²	—	4	1	no	—
304	C₂²	—	4	2	no	—
305	C₂²	C₂²×C₃×C₃⋊Dic₃	1	1	yes	characteristic
306	C₂²	C₂×C₃×#638	1	2	yes	—
307	C₄	—	2	2	no	—
308	C₄	—	36	4	no	—
309	C₄	C₃×#3757	1	1	yes	characteristic
310	C₄	C₃×#3757	1	1	yes	characteristic
311	C₄	C₃×#3757	1	2	yes	—
312	C₃	—	2	4	no	—
313	C₃	C₃×#5240	1	4	yes	complemented
314	C₃	#10592939	1	1	yes	characteristic, direct product factor
315	C₂	—	2	2	no	—
316	C₂	—	2	1	no	—
317	C₂	—	4	2	no	—
318	C₂	C₂×C₃×#3757	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄	#3757
commutator	C₂²×C₃²	C₂²×C₃×C₄
Frattini	C₂³	C₂²×C₃×C₃⋊D₃
Fitting, prime radical	C₂×C₃³×#41	C₂
socle	C₂×C₃³	C₂×C₂²⋊C₄
cosocle	C₂³×C₃²	C₂³×C₃
hypercenter	C₂×C₃×#41	C₃⋊D₃
hypocenter	C₃²	C₃×#410
LCS	G ▷ C₂²×C₃² ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₃×C₄ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×#41

Complements

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

N	H ≅ G/N	type
2. C₃×#64135	C₂	semidirect
3. C₃×#63661	C₂	semidirect
4. #10592939	C₃	direct (N × H)
8. C₂×C₃³×Q₈	C₄	semidirect
9. C₂×C₃³×D₄	C₄	semidirect
11. C₂²×C₃³×C₄	C₄	semidirect
19. #64135	C₂×C₃	semidirect
21. #63661	C₂×C₃	semidirect
45. C₂×C₃²×Q₈	Dic₃	semidirect
46. C₂×C₃²×Q₈	C₃×C₄	semidirect
49. C₂×C₃²×D₄	Dic₃	semidirect
50. C₂×C₃²×D₄	C₃×C₄	semidirect
55. C₂²×C₃²×C₄	Dic₃	semidirect
56. C₂²×C₃²×C₄	C₃×C₄	semidirect
132. C₂×C₃×Q₈	C₃×Dic₃	semidirect
133. C₂×C₃×Q₈	C₃⋊Dic₃	semidirect
136. C₂×C₃×D₄	C₃×Dic₃	semidirect
137. C₂×C₃×D₄	C₃⋊Dic₃	semidirect
142. C₂²×C₃×C₄	C₃×Dic₃	semidirect
143. C₂²×C₃×C₄	C₃⋊Dic₃	semidirect
184. C₃³	#410	semidirect
237. C₂×Q₈	C₃×C₃⋊Dic₃	semidirect
238. C₂×D₄	C₃×C₃⋊Dic₃	semidirect
240. C₂²×C₄	C₃×C₃⋊Dic₃	semidirect
275. C₃²	#5240	semidirect
276. C₃²	C₃×#410	semidirect
313. C₃	C₃×#5240	semidirect
314. C₃	#10592939	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×C₃³×#41	2
C₃×#64135	2
C₃×#63661	2
#10592939	3
C₃×#5240	3

Subgroups — C3×#10592939 #13163271

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10592939 #13163271