Subgroups — C₃×#10592409 #13162749

Order 1728.

880 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×#3744	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×C₄⋊(C₃⋊D₃)	C₂	1	1	yes	maximal, characteristic
3	C₃×C₄×#634	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃³×#97	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#63648	C₂	1	1	yes	maximal, characteristic
6	C₃×#63645	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#63643	C₂	1	1	yes	maximal, characteristic, complemented
8	#10592409	C₃	1	1	yes	maximal, characteristic, direct product factor
9	C₃×#4710	—	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×C₃×#634	C₂²	1	1	yes	characteristic
13	C₂×C₃³×C₈	C₂²	1	1	yes	characteristic
14	C₂×C₃³×C₈	C₂²	1	1	yes	characteristic
15	C₃³×C₄²	C₂²	1	1	yes	characteristic
16	C₃×#3752	—	2	1	no	—
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×C₄×C₃⋊Dic₃	—	2	1	no	—
21	C₃×#3744	C₄	1	4	yes	complemented
22	C₂×#3744	C₂×C₃	1	1	yes	characteristic
23	C₄×C₄⋊(C₃⋊D₃)	C₂×C₃	1	1	yes	characteristic
24	C₄×#634	C₂×C₃	1	1	yes	characteristic, complemented
25	C₂×C₃×#204	—	3	4	no	—
26	C₃×C₄×D₁₂	—	3	4	no	—
27	C₃×C₄×Dic₆	—	3	4	no	—
28	C₃²×#97	—	2	4	no	—
29	C₃²×#97	D₃	1	4	yes	complemented
30	C₃²×#97	C₂×C₃	1	1	yes	characteristic, complemented
31	#63648	C₂×C₃	1	1	yes	characteristic
32	#63645	C₂×C₃	1	1	yes	characteristic, complemented
33	#63643	C₂×C₃	1	1	yes	characteristic, complemented
34	C₃×#821	—	3	4	no	—
35	C₃×#818	—	3	4	no	—
36	C₃×#816	—	3	4	no	—
37	C₂²×C₃×C₃⋊D₃	—	2	1	no	—
38	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
39	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
40	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
41	C₂×C₃×C₃⋊Dic₃	—	2	1	no	—
42	C₂×C₃×C₃⋊Dic₃	—	2	1	no	—
43	C₂×C₃×C₃⋊Dic₃	—	2	1	no	—
44	C₃×C₄⋊(C₃⋊D₃)	—	2	1	no	—
45	C₃×C₄⋊(C₃⋊D₃)	C₂×C₄	1	2	yes	—
46	C₃×C₄×C₃⋊D₃	—	4	2	no	—
47	C₃×#634	—	2	1	no	—
48	C₃×#634	C₂×C₄	1	2	yes	complemented
49	C₃³×C₈	—	2	1	no	—
50	C₃³×C₈	C₂×C₄	1	2	yes	complemented
51	C₃×#440	—	9	1	no	—
52	#4710	—	3	4	no	—
53	C₂×C₄⋊(C₃⋊D₃)	C₂²×C₃	1	1	yes	characteristic
54	C₂×C₄×C₃⋊D₃	—	2	1	no	—
55	C₂×#634	C₂²×C₃	1	1	yes	characteristic
56	C₂×C₃×D₁₂	—	3	4	no	—
57	C₂×C₃×C₄×D₃	—	6	4	no	—
58	C₂×C₃×Dic₆	—	3	4	no	—
59	C₂×C₃²×C₈	—	2	4	no	—
60	C₂×C₃²×C₈	—	2	4	no	—
61	C₂×C₃²×C₈	C₂×D₃	1	4	yes	—
62	C₂×C₃²×C₈	C₂×D₃	1	4	yes	—
63	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
64	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
65	C₃²×C₄²	—	2	4	no	—
66	C₃²×C₄²	C₂×D₃	1	4	yes	—
67	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
68	#3752	—	2	1	no	—
69	C₄⋊(C₃⋊Dic₃)	C₂²×C₃	1	1	yes	characteristic
70	C₄⋊(C₃⋊Dic₃)	C₂²×C₃	1	1	yes	characteristic
71	#3750	—	2	1	no	—
72	C₄×C₃⋊Dic₃	—	2	1	no	—
73	#3744	C₃×C₄	1	4	yes	complemented
74	C₃×(C₂×D₃)⋊C₄	—	6	4	no	—
75	C₃×C₄⋊Dic₃	—	3	4	no	—
76	C₃×C₄⋊Dic₃	—	3	4	no	—
77	C₃×Dic₃⋊C₄	—	6	4	no	—
78	C₃×C₄×Dic₃	—	6	4	no	—
79	C₃×#204	—	3	16	no	—
80	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
81	C₂×C₃×C₃⋊D₃	—	2	2	no	—
82	C₂×C₃×C₃⋊D₃	—	4	1	no	—
83	C₂×C₃×C₃⋊D₃	—	4	1	no	—
84	C₃³×C₄	—	2	1	no	—
85	C₃³×C₄	—	2	1	no	—
86	C₃³×C₄	C₂²×C₄	1	1	yes	characteristic
87	C₃³×C₄	#41	1	1	yes	characteristic
88	C₃×C₃⋊Dic₃	—	2	2	no	—
89	C₃×C₃⋊Dic₃	—	4	1	no	—
90	C₃×C₃⋊Dic₃	—	4	1	no	—
91	C₂×C₃×SD₁₆	—	9	1	no	—
92	C₃×C₄×Q₈	—	9	1	no	—
93	C₃×C₄×D₄	—	9	1	no	—
94	C₂×#204	—	3	4	no	—
95	C₄×D₁₂	—	3	4	no	—
96	C₄×Dic₆	—	3	4	no	—
97	C₃×#107	—	9	1	no	—
98	C₃×#103	—	9	1	no	—
99	C₃×#102	—	9	1	no	—
100	C₃×#97	—	2	4	no	—
101	C₃×#97	C₃×D₃	1	4	yes	complemented
102	C₃×#97	C₃⋊D₃	1	1	yes	characteristic, complemented
103	#821	—	3	4	no	—
104	#818	—	3	4	no	—
105	#816	—	3	4	no	—
106	C₂²×C₃⋊D₃	—	2	1	no	—
107	C₂²×C₃×D₃	—	6	4	no	—
108	C₂×C₃²×C₄	—	2	4	no	—
109	C₂×C₃²×C₄	—	2	4	no	—
110	C₂×C₃²×C₄	—	2	4	no	—
111	C₂×C₃²×C₄	D₁₂	1	4	yes	—
112	C₂×C₃²×C₄	D₁₂	1	4	yes	—
113	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
114	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
115	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
116	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
117	C₂×C₃⋊Dic₃	—	2	1	no	—
118	C₂×C₃⋊Dic₃	—	2	1	no	—
119	C₂×C₃⋊Dic₃	—	2	1	no	—
120	C₄⋊(C₃⋊D₃)	—	2	1	no	—
121	C₄⋊(C₃⋊D₃)	C₂×C₃×C₄	1	2	yes	—
122	C₄×C₃⋊D₃	—	4	2	no	—
123	#634	—	2	1	no	—
124	#634	C₂×C₃×C₄	1	2	yes	complemented
125	C₂×C₃×Dic₃	—	6	4	no	—
126	C₂×C₃×Dic₃	—	6	4	no	—
127	C₂×C₃×Dic₃	—	6	4	no	—
128	C₃×D₁₂	—	3	8	no	—
129	C₃×D₁₂	—	6	4	no	—
130	C₃×C₄×D₃	—	12	8	no	—
131	C₃×Dic₆	—	3	8	no	—
132	C₃×Dic₆	—	6	4	no	—
133	C₃²×C₈	—	2	8	no	—
134	C₃²×C₈	—	2	4	no	—
135	C₃²×C₈	—	2	1	no	—
136	C₃²×C₈	—	2	8	no	—
137	C₃²×C₈	C₄×D₃	1	8	yes	complemented
138	C₃²×C₈	C₂×C₃×C₄	1	2	yes	complemented
139	#440	—	9	1	no	Sylow
140	C₂×C₃³	C₄×D₄	1	1	yes	characteristic
141	C₂×C₃³	AGL(1,ℤ/8)	1	1	yes	characteristic
142	C₂×C₃³	#137	1	1	yes	characteristic
143	C₃×C₃⋊D₃	—	4	2	no	—
144	C₂×C₃×Q₈	—	9	1	no	—
145	C₂×C₃×D₄	—	9	1	no	—
146	C₂²×C₃×C₄	—	18	1	no	—
147	C₂×D₁₂	—	3	4	no	—
148	C₂×C₄×D₃	—	6	4	no	—
149	C₂×Dic₆	—	3	4	no	—
150	C₃×SD₁₆	—	9	4	no	—
151	C₂×C₃×C₈	—	2	4	no	—
152	C₂×C₃×C₈	—	2	4	no	—
153	C₂×C₃×C₈	C₂×C₃×D₃	1	4	yes	—
154	C₂×C₃×C₈	C₂×C₃×D₃	1	4	yes	—
155	C₂×C₃×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
156	C₂×C₃×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
157	C₃×C₄⋊C₄	—	9	1	no	—
158	C₃×C₄⋊C₄	—	9	1	no	—
159	C₃×C₄⋊C₄	—	18	1	no	—
160	C₃×C₂²⋊C₄	—	18	1	no	—
161	C₃×C₄²	—	2	4	no	—
162	C₃×C₄²	—	18	1	no	—
163	C₃×C₄²	C₂×C₃×D₃	1	4	yes	—
164	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
165	(C₂×D₃)⋊C₄	—	6	4	no	—
166	C₄⋊Dic₃	—	3	4	no	—
167	C₄⋊Dic₃	—	3	4	no	—
168	Dic₃⋊C₄	—	6	4	no	—
169	C₄×Dic₃	—	6	4	no	—
170	#204	—	3	16	no	—
171	C₂²×C₃²	—	2	4	no	—
172	C₂²×C₃²	C₂×D₁₂	1	4	yes	—
173	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
174	C₂×C₃⋊D₃	—	2	2	no	—
175	C₂×C₃⋊D₃	—	4	1	no	—
176	C₂×C₃⋊D₃	—	4	1	no	—
177	C₂×C₃×D₃	—	6	8	no	—
178	C₂×C₃×D₃	—	12	4	no	—
179	C₂×C₃×D₃	—	12	4	no	—
180	C₃²×C₄	—	2	1	no	—
181	C₃²×C₄	—	2	1	no	—
182	C₃²×C₄	—	2	4	no	—
183	C₃²×C₄	—	2	4	no	—
184	C₃²×C₄	—	2	4	no	—
185	C₃²×C₄	—	2	4	no	—
186	C₃²×C₄	—	4	4	no	—
187	C₃²×C₄	—	4	4	no	—
188	C₃²×C₄	C₂×C₄×D₃	1	4	yes	—
189	C₃²×C₄	#235	1	4	yes	—
190	C₃²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
191	C₃²×C₄	C₃×#41	1	1	yes	characteristic
192	C₃⋊Dic₃	—	2	2	no	—
193	C₃⋊Dic₃	—	4	1	no	—
194	C₃⋊Dic₃	—	4	1	no	—
195	C₃×Dic₃	—	6	8	no	—
196	C₃×Dic₃	—	12	4	no	—
197	C₃×Dic₃	—	12	4	no	—
198	C₂×SD₁₆	—	9	1	no	—
199	C₄×Q₈	—	9	1	no	—
200	C₄×D₄	—	9	1	no	—
201	#107	—	9	1	no	—
202	#103	—	9	1	no	—
203	#102	—	9	1	no	—
204	#97	C₃×C₃⋊D₃	1	1	yes	characteristic, complemented
205	C₃³	#440	1	1	yes	characteristic, complemented, Sylow
206	C₂³×C₃	—	18	1	no	—
207	C₂²×D₃	—	6	4	no	—
208	C₃×Q₈	—	9	2	no	—
209	C₃×Q₈	—	18	1	no	—
210	C₃×D₄	—	9	2	no	—
211	C₃×D₄	—	18	1	no	—
212	C₂×C₃×C₄	—	2	4	no	—
213	C₂×C₃×C₄	—	2	4	no	—
214	C₂×C₃×C₄	—	2	4	no	—
215	C₂×C₃×C₄	—	18	1	no	—
216	C₂×C₃×C₄	—	18	1	no	—
217	C₂×C₃×C₄	—	18	1	no	—
218	C₂×C₃×C₄	—	36	2	no	—
219	C₂×C₃×C₄	C₃×D₁₂	1	4	yes	—
220	C₂×C₃×C₄	C₃×D₁₂	1	4	yes	—
221	C₂×C₃×C₄	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
222	C₂×C₃×C₄	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
223	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
224	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
225	C₂×Dic₃	—	6	4	no	—
226	C₂×Dic₃	—	6	4	no	—
227	C₂×Dic₃	—	6	4	no	—
228	D₁₂	—	3	8	no	—
229	D₁₂	—	6	4	no	—
230	C₄×D₃	—	12	8	no	—
231	Dic₆	—	3	8	no	—
232	Dic₆	—	6	4	no	—
233	C₃×C₈	—	2	8	no	—
234	C₃×C₈	—	2	4	no	—
235	C₃×C₈	—	2	1	no	—
236	C₃×C₈	—	2	8	no	—
237	C₃×C₈	C₃×C₄×D₃	1	8	yes	complemented
238	C₃×C₈	C₄×C₃⋊D₃	1	2	yes	complemented
239	C₂×C₃²	—	2	4	no	—
240	C₂×C₃²	—	2	4	no	—
241	C₂×C₃²	—	2	4	no	—
242	C₂×C₃²	C₄×D₁₂	1	4	yes	—
243	C₂×C₃²	#908	1	4	yes	—
244	C₂×C₃²	#909	1	4	yes	—
245	C₂×C₃²	C₃×C₄×D₄	1	1	yes	characteristic
246	C₂×C₃²	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
247	C₂×C₃²	C₃×#137	1	1	yes	characteristic
248	C₃⋊D₃	—	4	2	no	—
249	C₃×D₃	—	12	8	no	—
250	C₂×Q₈	—	9	1	no	—
251	C₂×D₄	—	9	1	no	—
252	C₂²×C₄	—	18	1	no	—
253	SD₁₆	—	9	4	no	—
254	C₂×C₈	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
255	C₂×C₈	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
256	C₄⋊C₄	—	9	1	no	—
257	C₄⋊C₄	—	9	1	no	—
258	C₄⋊C₄	—	18	1	no	—
259	C₂²⋊C₄	—	18	1	no	—
260	C₄²	—	18	1	no	—
261	C₄²	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
262	C₂²×C₃	—	2	4	no	—
263	C₂²×C₃	—	18	2	no	—
264	C₂²×C₃	—	36	1	no	—
265	C₂²×C₃	—	36	1	no	—
266	C₂²×C₃	C₂×C₃×D₁₂	1	4	yes	—
267	C₂²×C₃	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
268	C₂×D₃	—	6	8	no	—
269	C₂×D₃	—	12	4	no	—
270	C₂×D₃	—	12	4	no	—
271	C₃×C₄	—	2	4	no	—
272	C₃×C₄	—	2	4	no	—
273	C₃×C₄	—	2	1	no	—
274	C₃×C₄	—	2	1	no	—
275	C₃×C₄	—	2	4	no	—
276	C₃×C₄	—	2	4	no	—
277	C₃×C₄	—	4	4	no	—
278	C₃×C₄	—	4	4	no	—
279	C₃×C₄	—	18	2	no	—
280	C₃×C₄	—	36	1	no	—
281	C₃×C₄	—	36	1	no	—
282	C₃×C₄	C₂×C₃×C₄×D₃	1	4	yes	—
283	C₃×C₄	C₃×#235	1	4	yes	—
284	C₃×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
285	C₃×C₄	#3828	1	1	yes	characteristic
286	Dic₃	—	6	8	no	—
287	Dic₃	—	12	4	no	—
288	Dic₃	—	12	4	no	—
289	C₃²	—	2	4	no	—
290	C₃²	#4710	1	4	yes	complemented
291	C₃²	C₃×#440	1	1	yes	characteristic, complemented
292	C₂³	—	18	1	no	—
293	Q₈	—	9	2	no	—
294	Q₈	—	18	1	no	—
295	D₄	—	9	2	no	—
296	D₄	—	18	1	no	—
297	C₂×C₄	—	18	1	no	—
298	C₂×C₄	—	18	1	no	—
299	C₂×C₄	—	18	1	no	—
300	C₂×C₄	—	36	2	no	—
301	C₂×C₄	C₃×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
302	C₂×C₄	C₃×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
303	C₂×C₄	C₂²×C₃×C₃⋊D₃	1	1	yes	characteristic
304	C₈	—	2	1	no	—
305	C₈	C₃×C₄×C₃⋊D₃	1	2	yes	complemented
306	C₂×C₃	—	2	4	no	—
307	C₂×C₃	—	2	4	no	—
308	C₂×C₃	—	2	4	no	—
309	C₂×C₃	—	36	2	no	—
310	C₂×C₃	C₃×C₄×D₁₂	1	4	yes	—
311	C₂×C₃	C₃×#908	1	4	yes	—
312	C₂×C₃	C₃×#909	1	4	yes	—
313	C₂×C₃	C₄×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
314	C₂×C₃	#64117	1	1	yes	characteristic
315	C₂×C₃	#64118	1	1	yes	characteristic
316	D₃	—	12	8	no	—
317	C₂²	—	18	2	no	—
318	C₂²	—	36	1	no	—
319	C₂²	—	36	1	no	—
320	C₂²	C₂×C₃×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	18	2	no	—
324	C₄	—	36	1	no	—
325	C₄	—	36	1	no	—
326	C₄	C₂×C₃×C₄×C₃⋊D₃	1	1	yes	characteristic
327	C₄	C₃×#3828	1	1	yes	characteristic
328	C₃	—	2	4	no	—
329	C₃	C₃×#4710	1	4	yes	complemented
330	C₃	#10592409	1	1	yes	characteristic, direct product factor
331	C₂	—	36	2	no	—
332	C₂	C₃×C₄×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
333	C₂	C₃×#64117	1	1	yes	characteristic
334	C₂	C₃×#64118	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂×C₄⋊(C₃⋊D₃)
commutator	C₃²×C₄	C₂²×C₃×C₄
Frattini	C₂×C₄	C₂²×C₃×C₃⋊D₃
Fitting	C₃³×#97	C₂
socle	C₂²×C₃³	C₂×D₄
prime radical	C₂×C₃×C₄⋊(C₃⋊D₃)	C₂²
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₃×#97	C₃⋊D₃
hypocenter	C₃²	C₃×#440
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×#97

Complements

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

N	H ≅ G/N	type
3. C₃×C₄×#634	C₂	semidirect
4. C₃³×#97	C₂	semidirect
6. C₃×#63645	C₂	semidirect
7. C₃×#63643	C₂	semidirect
8. #10592409	C₃	direct (N × H)
21. C₃×#3744	C₄	semidirect
24. C₄×#634	C₂×C₃	semidirect
29. C₃²×#97	D₃	semidirect
30. C₃²×#97	C₂×C₃	semidirect
32. #63645	C₂×C₃	semidirect
33. #63643	C₂×C₃	semidirect
48. C₃×#634	C₂×C₄	semidirect
50. C₃³×C₈	C₂×C₄	semidirect
73. #3744	C₃×C₄	semidirect
101. C₃×#97	C₃×D₃	semidirect
102. C₃×#97	C₃⋊D₃	semidirect
124. #634	C₂×C₃×C₄	semidirect
137. C₃²×C₈	C₄×D₃	semidirect
138. C₃²×C₈	C₂×C₃×C₄	semidirect
204. #97	C₃×C₃⋊D₃	semidirect
205. C₃³	#440	semidirect
237. C₃×C₈	C₃×C₄×D₃	semidirect
238. C₃×C₈	C₄×C₃⋊D₃	semidirect
290. C₃²	#4710	semidirect
291. C₃²	C₃×#440	semidirect
305. C₈	C₃×C₄×C₃⋊D₃	semidirect
329. C₃	C₃×#4710	semidirect
330. C₃	#10592409	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×#3744	2
C₃×C₄×C₄⋊(C₃⋊D₃)	2
C₃×C₄×#634	2
C₃³×#97	2
C₃×#63648	2
C₃×#63645	2
C₃×#63643	2
#10592409	3
C₃×#4710	3

Subgroups — C3×#10592409 #13162749

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10592409 #13162749