Subgroups — C₃×#10592546 #13162885

Order 1728.

764 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×C₄⋊(C₃⋊D₃)	C₂	1	1	yes	maximal, characteristic
2	C₃×#64079	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃³×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#63648	C₂	1	1	yes	maximal, characteristic
5	C₃×#63648	C₂	1	1	yes	maximal, characteristic
6	C₃×#63645	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#63644	C₂	1	1	yes	maximal, characteristic, complemented
8	#10592546	C₃	1	1	yes	maximal, characteristic, direct product factor
9	C₃×#4847	—	3	4	no	maximal
10	C₂×C₃×C₄⋊(C₃⋊D₃)	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₄×C₃⋊D₃	—	2	1	no	—
12	C₂×C₃³×C₈	C₂²	1	1	yes	characteristic
13	C₂×C₃³×C₈	C₂²	1	1	yes	characteristic
14	C₃³×C₄²	C₂²	1	1	yes	characteristic
15	C₃×#3752	—	2	1	no	—
16	C₃×C₄⋊(C₃⋊Dic₃)	C₂²	1	1	yes	characteristic
17	C₃×C₄⋊(C₃⋊Dic₃)	C₂²	1	1	yes	characteristic
18	C₃×C₄⋊(C₃⋊Dic₃)	C₂²	1	1	yes	characteristic
19	C₃×#3750	—	2	1	no	—
20	C₃×#3750	—	2	1	no	—
21	C₄×C₄⋊(C₃⋊D₃)	C₂×C₃	1	1	yes	characteristic
22	#64079	C₂×C₃	1	1	yes	characteristic, complemented
23	C₃×C₄×D₁₂	—	3	4	no	—
24	C₃×#870	—	3	4	no	—
25	C₃²×C₄⋊C₈	—	2	4	no	—
26	C₃²×C₄⋊C₈	D₃	1	4	yes	complemented
27	C₃²×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
28	#63648	C₂×C₃	1	1	yes	characteristic
29	#63648	C₂×C₃	1	1	yes	characteristic
30	#63645	C₂×C₃	1	1	yes	characteristic, complemented
31	#63644	C₂×C₃	1	1	yes	characteristic, complemented
32	C₃×#821	—	3	4	no	—
33	C₃×#821	—	3	4	no	—
34	C₃×#818	—	3	4	no	—
35	C₃×#817	—	3	4	no	—
36	C₂²×C₃×C₃⋊D₃	—	2	1	no	—
37	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
38	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
39	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
40	C₂×C₃×C₃⋊Dic₃	—	2	1	no	—
41	C₂×C₃×C₃⋊Dic₃	—	2	1	no	—
42	C₂×C₃×C₃⋊Dic₃	—	2	1	no	—
43	C₃×C₄⋊(C₃⋊D₃)	—	2	1	no	—
44	C₃×C₄⋊(C₃⋊D₃)	Q₈	1	2	yes	—
45	C₃×C₄×C₃⋊D₃	—	4	2	no	—
46	C₃³×C₈	—	2	1	no	—
47	C₃³×C₈	—	2	1	no	—
48	C₃×#478	—	9	1	no	—
49	#4847	—	3	4	no	—
50	C₂×C₄⋊(C₃⋊D₃)	C₂²×C₃	1	1	yes	characteristic
51	C₂×C₄×C₃⋊D₃	—	2	1	no	—
52	C₂×C₃×D₁₂	—	3	4	no	—
53	C₂×C₃×C₄×D₃	—	6	4	no	—
54	C₂×C₃²×C₈	—	2	4	no	—
55	C₂×C₃²×C₈	—	2	4	no	—
56	C₂×C₃²×C₈	C₂×D₃	1	4	yes	—
57	C₂×C₃²×C₈	C₂×D₃	1	4	yes	—
58	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
59	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
60	C₃²×C₄²	—	2	4	no	—
61	C₃²×C₄²	C₂×D₃	1	4	yes	—
62	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
63	#3752	—	2	1	no	—
64	C₄⋊(C₃⋊Dic₃)	C₂²×C₃	1	1	yes	characteristic
65	C₄⋊(C₃⋊Dic₃)	C₂²×C₃	1	1	yes	characteristic
66	C₄⋊(C₃⋊Dic₃)	C₂²×C₃	1	1	yes	characteristic
67	#3750	—	2	1	no	—
68	#3750	—	2	1	no	—
69	C₃×(C₂×D₃)⋊C₄	—	6	4	no	—
70	C₃×C₄⋊Dic₃	—	3	4	no	—
71	C₃×C₄⋊Dic₃	—	3	4	no	—
72	C₃×C₄⋊Dic₃	—	3	4	no	—
73	C₃×Dic₃⋊C₄	—	6	4	no	—
74	C₃×Dic₃⋊C₄	—	6	4	no	—
75	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
76	C₂×C₃×C₃⋊D₃	—	2	2	no	—
77	C₂×C₃×C₃⋊D₃	—	4	1	no	—
78	C₂×C₃×C₃⋊D₃	—	4	1	no	—
79	C₃³×C₄	—	2	1	no	—
80	C₃³×C₄	—	2	1	no	—
81	C₃³×C₄	C₂×Q₈	1	1	yes	characteristic
82	C₃³×C₄	#41	1	1	yes	characteristic
83	C₃×C₃⋊Dic₃	—	4	1	no	—
84	C₃×C₃⋊Dic₃	—	4	1	no	—
85	C₃×C₃⋊Dic₃	—	4	1	no	—
86	C₃×#125	—	9	1	no	—
87	C₃×C₄×D₄	—	9	1	no	—
88	C₄×D₁₂	—	3	4	no	—
89	#870	—	3	4	no	—
90	C₃×#107	—	9	1	no	—
91	C₃×#106	—	9	1	no	—
92	C₃×C₄⋊C₈	—	2	4	no	—
93	C₃×C₄⋊C₈	C₃×D₃	1	4	yes	complemented
94	C₃×C₄⋊C₈	C₃⋊D₃	1	1	yes	characteristic, complemented
95	C₃×#102	—	9	1	no	—
96	C₃×#102	—	9	1	no	—
97	#821	—	3	4	no	—
98	#821	—	3	4	no	—
99	#818	—	3	4	no	—
100	#817	—	3	4	no	—
101	C₂²×C₃⋊D₃	—	2	1	no	—
102	C₂²×C₃×D₃	—	6	4	no	—
103	C₂×C₃²×C₄	—	2	4	no	—
104	C₂×C₃²×C₄	—	2	4	no	—
105	C₂×C₃²×C₄	—	2	4	no	—
106	C₂×C₃²×C₄	D₁₂	1	4	yes	—
107	C₂×C₃²×C₄	D₁₂	1	4	yes	—
108	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
109	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
110	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
111	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
112	C₂×C₃⋊Dic₃	—	2	1	no	—
113	C₂×C₃⋊Dic₃	—	2	1	no	—
114	C₂×C₃⋊Dic₃	—	2	1	no	—
115	C₄⋊(C₃⋊D₃)	—	2	1	no	—
116	C₄⋊(C₃⋊D₃)	C₃×Q₈	1	2	yes	—
117	C₄×C₃⋊D₃	—	4	2	no	—
118	C₂×C₃×Dic₃	—	6	4	no	—
119	C₂×C₃×Dic₃	—	6	4	no	—
120	C₂×C₃×Dic₃	—	6	4	no	—
121	C₃×D₁₂	—	3	8	no	—
122	C₃×D₁₂	—	6	4	no	—
123	C₃×C₄×D₃	—	12	8	no	—
124	C₃²×C₈	—	2	1	no	—
125	C₃²×C₈	—	2	1	no	—
126	C₃²×C₈	—	2	4	no	—
127	C₃²×C₈	—	2	4	no	—
128	C₃²×C₈	—	4	4	no	—
129	C₃²×C₈	—	4	4	no	—
130	#478	—	9	1	no	Sylow
131	C₂×C₃³	C₂²⋊Q₈	1	1	yes	characteristic
132	C₂×C₃³	#135	1	1	yes	characteristic
133	C₂×C₃³	AGL(1,ℤ/8)	1	1	yes	characteristic
134	C₃×C₃⋊D₃	—	4	2	no	—
135	C₂×C₃×D₄	—	9	1	no	—
136	C₂²×C₃×C₄	—	18	1	no	—
137	C₂×D₁₂	—	3	4	no	—
138	C₂×C₄×D₃	—	6	4	no	—
139	C₂×C₃×C₈	—	2	4	no	—
140	C₂×C₃×C₈	—	2	4	no	—
141	C₂×C₃×C₈	C₂×C₃×D₃	1	4	yes	—
142	C₂×C₃×C₈	C₂×C₃×D₃	1	4	yes	—
143	C₂×C₃×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
144	C₂×C₃×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
145	C₃×C₄⋊C₄	—	9	1	no	—
146	C₃×C₄⋊C₄	—	9	1	no	—
147	C₃×C₄⋊C₄	—	9	1	no	—
148	C₃×C₄⋊C₄	—	18	1	no	—
149	C₃×C₄⋊C₄	—	18	1	no	—
150	C₃×C₂²⋊C₄	—	18	1	no	—
151	C₃×C₄²	—	2	4	no	—
152	C₃×C₄²	C₂×C₃×D₃	1	4	yes	—
153	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
154	(C₂×D₃)⋊C₄	—	6	4	no	—
155	C₄⋊Dic₃	—	3	4	no	—
156	C₄⋊Dic₃	—	3	4	no	—
157	C₄⋊Dic₃	—	3	4	no	—
158	Dic₃⋊C₄	—	6	4	no	—
159	Dic₃⋊C₄	—	6	4	no	—
160	C₂²×C₃²	—	2	4	no	—
161	C₂²×C₃²	C₂×D₁₂	1	4	yes	—
162	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
163	C₂×C₃⋊D₃	—	2	2	no	—
164	C₂×C₃⋊D₃	—	4	1	no	—
165	C₂×C₃⋊D₃	—	4	1	no	—
166	C₂×C₃×D₃	—	6	8	no	—
167	C₂×C₃×D₃	—	12	4	no	—
168	C₂×C₃×D₃	—	12	4	no	—
169	C₃²×C₄	—	2	8	no	—
170	C₃²×C₄	—	2	4	no	—
171	C₃²×C₄	—	2	1	no	—
172	C₃²×C₄	—	2	1	no	—
173	C₃²×C₄	—	2	4	no	—
174	C₃²×C₄	—	2	4	no	—
175	C₃²×C₄	—	2	4	no	—
176	C₃²×C₄	—	4	4	no	—
177	C₃²×C₄	#237	1	4	yes	—
178	C₃²×C₄	D₃×Q₈	1	4	yes	—
179	C₃²×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
180	C₃²×C₄	C₃×#41	1	1	yes	characteristic
181	C₃⋊Dic₃	—	4	1	no	—
182	C₃⋊Dic₃	—	4	1	no	—
183	C₃⋊Dic₃	—	4	1	no	—
184	C₃×Dic₃	—	12	4	no	—
185	C₃×Dic₃	—	12	4	no	—
186	C₃×Dic₃	—	12	4	no	—
187	#125	—	9	1	no	—
188	C₄×D₄	—	9	1	no	—
189	#107	—	9	1	no	—
190	#106	—	9	1	no	—
191	C₄⋊C₈	C₃×C₃⋊D₃	1	1	yes	characteristic, complemented
192	#102	—	9	1	no	—
193	#102	—	9	1	no	—
194	C₃³	#478	1	1	yes	characteristic, complemented, Sylow
195	C₂³×C₃	—	18	1	no	—
196	C₂²×D₃	—	6	4	no	—
197	C₃×D₄	—	9	2	no	—
198	C₃×D₄	—	18	1	no	—
199	C₂×C₃×C₄	—	2	4	no	—
200	C₂×C₃×C₄	—	2	4	no	—
201	C₂×C₃×C₄	—	2	4	no	—
202	C₂×C₃×C₄	—	18	1	no	—
203	C₂×C₃×C₄	—	18	1	no	—
204	C₂×C₃×C₄	—	18	1	no	—
205	C₂×C₃×C₄	—	36	2	no	—
206	C₂×C₃×C₄	C₃×D₁₂	1	4	yes	—
207	C₂×C₃×C₄	C₃×D₁₂	1	4	yes	—
208	C₂×C₃×C₄	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
209	C₂×C₃×C₄	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
210	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
211	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
212	C₂×Dic₃	—	6	4	no	—
213	C₂×Dic₃	—	6	4	no	—
214	C₂×Dic₃	—	6	4	no	—
215	D₁₂	—	3	8	no	—
216	D₁₂	—	6	4	no	—
217	C₄×D₃	—	12	8	no	—
218	C₃×C₈	—	2	4	no	—
219	C₃×C₈	—	2	4	no	—
220	C₃×C₈	—	2	1	no	—
221	C₃×C₈	—	2	1	no	—
222	C₃×C₈	—	4	4	no	—
223	C₃×C₈	—	4	4	no	—
224	C₂×C₃²	—	2	4	no	—
225	C₂×C₃²	—	2	4	no	—
226	C₂×C₃²	—	2	4	no	—
227	C₂×C₃²	#897	1	4	yes	—
228	C₂×C₃²	#904	1	4	yes	—
229	C₂×C₃²	#908	1	4	yes	—
230	C₂×C₃²	C₃×C₂²⋊Q₈	1	1	yes	characteristic
231	C₂×C₃²	C₃×#135	1	1	yes	characteristic
232	C₂×C₃²	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
233	C₃⋊D₃	—	4	2	no	—
234	C₃×D₃	—	12	8	no	—
235	C₂×D₄	—	9	1	no	—
236	C₂²×C₄	—	18	1	no	—
237	C₂×C₈	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
238	C₂×C₈	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
239	C₄⋊C₄	—	9	1	no	—
240	C₄⋊C₄	—	9	1	no	—
241	C₄⋊C₄	—	9	1	no	—
242	C₄⋊C₄	—	18	1	no	—
243	C₄⋊C₄	—	18	1	no	—
244	C₂²⋊C₄	—	18	1	no	—
245	C₄²	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
246	C₂²×C₃	—	2	4	no	—
247	C₂²×C₃	—	18	2	no	—
248	C₂²×C₃	—	36	1	no	—
249	C₂²×C₃	—	36	1	no	—
250	C₂²×C₃	C₂×C₃×D₁₂	1	4	yes	—
251	C₂²×C₃	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
252	C₂×D₃	—	6	8	no	—
253	C₂×D₃	—	12	4	no	—
254	C₂×D₃	—	12	4	no	—
255	C₃×C₄	—	2	8	no	—
256	C₃×C₄	—	2	4	no	—
257	C₃×C₄	—	2	4	no	—
258	C₃×C₄	—	2	1	no	—
259	C₃×C₄	—	2	1	no	—
260	C₃×C₄	—	2	4	no	—
261	C₃×C₄	—	2	4	no	—
262	C₃×C₄	—	4	4	no	—
263	C₃×C₄	—	36	1	no	—
264	C₃×C₄	—	36	1	no	—
265	C₃×C₄	—	36	1	no	—
266	C₃×C₄	C₃×#237	1	4	yes	—
267	C₃×C₄	C₃×D₃×Q₈	1	4	yes	—
268	C₃×C₄	#3830	1	1	yes	characteristic
269	C₃×C₄	Q₈×C₃⋊D₃	1	1	yes	characteristic
270	Dic₃	—	12	4	no	—
271	Dic₃	—	12	4	no	—
272	Dic₃	—	12	4	no	—
273	C₃²	—	2	4	no	—
274	C₃²	#4847	1	4	yes	complemented
275	C₃²	C₃×#478	1	1	yes	characteristic, complemented
276	C₂³	—	18	1	no	—
277	D₄	—	9	2	no	—
278	D₄	—	18	1	no	—
279	C₂×C₄	—	18	1	no	—
280	C₂×C₄	—	18	1	no	—
281	C₂×C₄	—	18	1	no	—
282	C₂×C₄	—	36	2	no	—
283	C₂×C₄	C₃×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
284	C₂×C₄	C₃×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
285	C₂×C₄	C₂²×C₃×C₃⋊D₃	1	1	yes	characteristic
286	C₈	—	2	1	no	—
287	C₈	—	2	1	no	—
288	C₂×C₃	—	2	4	no	—
289	C₂×C₃	—	2	4	no	—
290	C₂×C₃	—	2	4	no	—
291	C₂×C₃	—	36	2	no	—
292	C₂×C₃	C₃×#897	1	4	yes	—
293	C₂×C₃	C₃×#904	1	4	yes	—
294	C₂×C₃	C₃×#908	1	4	yes	—
295	C₂×C₃	#64106	1	1	yes	characteristic
296	C₂×C₃	#64113	1	1	yes	characteristic
297	C₂×C₃	#64117	1	1	yes	characteristic
298	D₃	—	12	8	no	—
299	C₂²	—	18	2	no	—
300	C₂²	—	36	1	no	—
301	C₂²	—	36	1	no	—
302	C₂²	C₂×C₃×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
303	C₄	—	2	1	no	—
304	C₄	—	2	1	no	—
305	C₄	—	36	1	no	—
306	C₄	—	36	1	no	—
307	C₄	—	36	1	no	—
308	C₄	C₃×#3830	1	1	yes	characteristic
309	C₄	C₃×Q₈×C₃⋊D₃	1	1	yes	characteristic
310	C₃	—	2	4	no	—
311	C₃	C₃×#4847	1	4	yes	complemented
312	C₃	#10592546	1	1	yes	characteristic, direct product factor
313	C₂	—	36	2	no	—
314	C₂	C₃×#64106	1	1	yes	characteristic
315	C₂	C₃×#64113	1	1	yes	characteristic
316	C₂	C₃×#64117	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂×C₄⋊(C₃⋊D₃)
commutator, cosocle	C₂×C₃²×C₄	C₂³×C₃
Frattini	C₂×C₄	C₂²×C₃×C₃⋊D₃
Fitting	C₃³×C₄⋊C₈	C₂
socle	C₂²×C₃³	C₂×D₄
prime radical	C₂×C₃×C₄⋊(C₃⋊D₃)	C₂²
hypercenter	C₃×C₄⋊C₈	C₃⋊D₃
hypocenter	C₃²	C₃×#478
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₃×#64079	C₂	semidirect
3. C₃³×C₄⋊C₈	C₂	semidirect
6. C₃×#63645	C₂	semidirect
7. C₃×#63644	C₂	semidirect
8. #10592546	C₃	direct (N × H)
22. #64079	C₂×C₃	semidirect
26. C₃²×C₄⋊C₈	D₃	semidirect
27. C₃²×C₄⋊C₈	C₂×C₃	semidirect
30. #63645	C₂×C₃	semidirect
31. #63644	C₂×C₃	semidirect
93. C₃×C₄⋊C₈	C₃×D₃	semidirect
94. C₃×C₄⋊C₈	C₃⋊D₃	semidirect
191. C₄⋊C₈	C₃×C₃⋊D₃	semidirect
194. C₃³	#478	semidirect
274. C₃²	#4847	semidirect
275. C₃²	C₃×#478	semidirect
311. C₃	C₃×#4847	semidirect
312. C₃	#10592546	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₄×C₄⋊(C₃⋊D₃)	2
C₃×#64079	2
C₃³×C₄⋊C₈	2
C₃×#63648	2
C₃×#63645	2
C₃×#63644	2
#10592546	3
C₃×#4847	3

Subgroups — C3×#10592546 #13162885

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10592546 #13162885