Subgroups — C₃×#10592516 #13162852

Order 1728.

888 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×Q₈×C₃⋊Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×#3755	C₂	1	1	yes	maximal, characteristic
3	C₃×#64102	C₂	1	1	yes	maximal, characteristic
4	C₃³×#103	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#63648	C₂	1	1	yes	maximal, characteristic
6	C₃×#63642	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#63634	C₂	1	1	yes	maximal, characteristic, complemented
8	#10592516	C₃	1	1	yes	maximal, characteristic, direct product factor
9	C₃×#4817	—	3	4	no	maximal
10	C₂×C₃³×Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₄⋊(C₃⋊D₃)	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₄×C₃⋊D₃	—	2	1	no	—
13	C₂×C₃³×C₈	C₂²	1	1	yes	characteristic
14	C₃³×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₃×#3755	C₄	1	4	yes	complemented
16	C₃×#3752	—	2	1	no	—
17	C₃×C₄⋊(C₃⋊Dic₃)	—	2	1	no	—
18	C₃×C₄⋊(C₃⋊Dic₃)	C₂²	1	1	yes	characteristic
19	C₃×C₄×C₃⋊Dic₃	—	2	1	no	—
20	C₃×C₄×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
21	C₂×C₃×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
22	Q₈×C₃⋊Dic₃	C₂×C₃	1	1	yes	characteristic, complemented
23	C₂×#3755	C₂×C₃	1	1	yes	characteristic
24	#64102	C₂×C₃	1	1	yes	characteristic
25	C₃×Q₈×Dic₃	—	3	4	no	—
26	C₂×C₃×#215	—	3	4	no	—
27	C₃×#893	—	3	4	no	—
28	C₃²×#103	—	2	4	no	—
29	C₃²×#103	D₃	1	4	yes	complemented
30	C₃²×#103	C₂×C₃	1	1	yes	characteristic, complemented
31	#63648	C₂×C₃	1	1	yes	characteristic
32	#63642	C₂×C₃	1	1	yes	characteristic, complemented
33	#63634	C₂×C₃	1	1	yes	characteristic, complemented
34	C₃×#821	—	3	4	no	—
35	C₃×#815	—	3	4	no	—
36	C₃×#807	—	3	4	no	—
37	C₂²×C₃×C₃⋊D₃	—	2	1	no	—
38	C₃³×Q₈	—	2	1	no	—
39	C₃³×Q₈	C₂×C₄	1	2	yes	complemented
40	C₂×C₃³×C₄	—	2	1	no	—
41	C₂×C₃³×C₄	—	2	1	no	—
42	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
43	C₂×C₃×C₃⋊Dic₃	—	2	1	no	—
44	C₂×C₃×C₃⋊Dic₃	D₄	1	1	yes	characteristic
45	C₂×C₃×C₃⋊Dic₃	D₄	1	1	yes	characteristic
46	C₃×C₄⋊(C₃⋊D₃)	—	2	1	no	—
47	C₃×C₄⋊(C₃⋊D₃)	C₂×C₄	1	2	yes	—
48	C₃×C₄×C₃⋊D₃	—	4	2	no	—
49	C₃³×C₈	—	2	1	no	—
50	C₃×C₃⋊(C₃⋊C₈)	C₂×C₄	1	2	yes	complemented
51	C₃×#440	—	9	1	no	—
52	#4817	—	3	4	no	—
53	C₂×C₃²×Q₈	—	2	4	no	—
54	C₂×C₃²×Q₈	C₂×D₃	1	4	yes	—
55	C₂×C₃²×Q₈	C₂²×C₃	1	1	yes	characteristic
56	C₂×C₄⋊(C₃⋊D₃)	C₂²×C₃	1	1	yes	characteristic
57	C₂×C₄×C₃⋊D₃	—	2	1	no	—
58	C₂×C₃×D₁₂	—	3	4	no	—
59	C₂×C₃×C₄×D₃	—	6	4	no	—
60	C₂×C₃²×C₈	—	2	4	no	—
61	C₂×C₃²×C₈	C₂×D₃	1	4	yes	—
62	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
63	C₃²×C₄⋊C₄	—	2	4	no	—
64	C₃²×C₄⋊C₄	C₂×D₃	1	4	yes	—
65	C₃²×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
66	#3755	C₃×C₄	1	4	yes	complemented
67	#3752	—	2	1	no	—
68	C₄⋊(C₃⋊Dic₃)	—	2	1	no	—
69	C₄⋊(C₃⋊Dic₃)	C₂²×C₃	1	1	yes	characteristic
70	C₄×C₃⋊Dic₃	—	2	1	no	—
71	C₄×C₃⋊Dic₃	C₂²×C₃	1	1	yes	characteristic
72	C₂×C₃⋊(C₃⋊C₈)	C₂²×C₃	1	1	yes	characteristic
73	C₃×#215	—	3	16	no	—
74	C₃×(C₂×D₃)⋊C₄	—	6	4	no	—
75	C₃×C₄⋊Dic₃	—	3	4	no	—
76	C₃×C₄⋊Dic₃	—	6	4	no	—
77	C₃×C₄×Dic₃	—	3	4	no	—
78	C₃×C₄×Dic₃	—	6	4	no	—
79	C₂×C₃×C₃⋊C₈	—	3	4	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	2	no	—
85	C₃³×C₄	—	4	1	no	—
86	C₃³×C₄	C₂²×C₄	1	1	yes	characteristic
87	C₃³×C₄	#41	1	1	yes	characteristic
88	C₃×C₃⋊Dic₃	—	2	1	no	—
89	C₃×C₃⋊Dic₃	—	2	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	Q₈×Dic₃	—	3	4	no	—
95	C₂×#215	—	3	4	no	—
96	#893	—	3	4	no	—
97	C₃×#107	—	9	1	no	—
98	C₃×#103	—	2	4	no	—
99	C₃×#103	C₃×D₃	1	4	yes	complemented
100	C₃×#103	C₃⋊D₃	1	1	yes	characteristic, complemented
101	C₃×#102	—	9	1	no	—
102	C₃×#97	—	9	1	no	—
103	#821	—	3	4	no	—
104	#815	—	3	4	no	—
105	#807	—	3	4	no	—
106	C₂²×C₃⋊D₃	—	2	1	no	—
107	C₂²×C₃×D₃	—	6	4	no	—
108	C₃²×Q₈	—	2	1	no	—
109	C₃²×Q₈	—	2	8	no	—
110	C₃²×Q₈	—	2	4	no	—
111	C₃²×Q₈	—	4	4	no	—
112	C₃²×Q₈	C₄×D₃	1	8	yes	complemented
113	C₃²×Q₈	C₂×C₃×C₄	1	2	yes	complemented
114	C₂×C₃²×C₄	—	2	1	no	—
115	C₂×C₃²×C₄	—	2	1	no	—
116	C₂×C₃²×C₄	—	2	4	no	—
117	C₂×C₃²×C₄	—	2	4	no	—
118	C₂×C₃²×C₄	—	2	4	no	—
119	C₂×C₃²×C₄	—	4	4	no	—
120	C₂×C₃²×C₄	—	4	4	no	—
121	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
122	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
123	C₂×C₃⋊Dic₃	—	2	1	no	—
124	C₂×C₃⋊Dic₃	C₃×D₄	1	1	yes	characteristic
125	C₂×C₃⋊Dic₃	C₃×D₄	1	1	yes	characteristic
126	C₄⋊(C₃⋊D₃)	—	2	1	no	—
127	C₄⋊(C₃⋊D₃)	C₂×C₃×C₄	1	2	yes	—
128	C₄×C₃⋊D₃	—	4	2	no	—
129	C₂×C₃×Dic₃	—	3	4	no	—
130	C₂×C₃×Dic₃	—	3	4	no	—
131	C₂×C₃×Dic₃	—	6	4	no	—
132	C₃×D₁₂	—	3	8	no	—
133	C₃×D₁₂	—	6	4	no	—
134	C₃×C₄×D₃	—	12	8	no	—
135	C₃²×C₈	—	2	1	no	—
136	C₃²×C₈	—	2	4	no	—
137	C₃²×C₈	—	4	4	no	—
138	C₃⋊(C₃⋊C₈)	C₂×C₃×C₄	1	2	yes	complemented
139	C₃×C₃⋊C₈	—	3	8	no	—
140	#440	—	9	1	no	Sylow
141	C₂×C₃³	C₄×D₄	1	1	yes	characteristic
142	C₂×C₃³	AGL(1,ℤ/8)	1	1	yes	characteristic
143	C₂×C₃³	#137	1	1	yes	characteristic
144	C₃×C₃⋊D₃	—	4	2	no	—
145	C₂×C₃×Q₈	—	2	4	no	—
146	C₂×C₃×Q₈	C₂×C₃×D₃	1	4	yes	—
147	C₂×C₃×Q₈	C₂×C₃⋊D₃	1	1	yes	characteristic
148	C₂×C₃×D₄	—	9	1	no	—
149	C₂²×C₃×C₄	—	18	1	no	—
150	C₂×D₁₂	—	3	4	no	—
151	C₂×C₄×D₃	—	6	4	no	—
152	C₃×SD₁₆	—	9	4	no	—
153	C₂×C₃×C₈	—	2	4	no	—
154	C₂×C₃×C₈	—	9	1	no	—
155	C₂×C₃×C₈	C₂×C₃×D₃	1	4	yes	—
156	C₂×C₃×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
157	C₃×C₄⋊C₄	—	2	4	no	—
158	C₃×C₄⋊C₄	—	9	1	no	—
159	C₃×C₄⋊C₄	—	18	1	no	—
160	C₃×C₄⋊C₄	C₂×C₃×D₃	1	4	yes	—
161	C₃×C₄⋊C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
162	C₃×C₂²⋊C₄	—	18	1	no	—
163	C₃×C₄²	—	9	1	no	—
164	C₃×C₄²	—	18	1	no	—
165	#215	—	3	16	no	—
166	(C₂×D₃)⋊C₄	—	6	4	no	—
167	C₄⋊Dic₃	—	3	4	no	—
168	C₄⋊Dic₃	—	6	4	no	—
169	C₄×Dic₃	—	3	4	no	—
170	C₄×Dic₃	—	6	4	no	—
171	C₂×C₃⋊C₈	—	3	4	no	—
172	C₂²×C₃²	—	2	4	no	—
173	C₂²×C₃²	D₃×D₄	1	4	yes	—
174	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
175	C₂×C₃⋊D₃	—	2	2	no	—
176	C₂×C₃⋊D₃	—	4	1	no	—
177	C₂×C₃⋊D₃	—	4	1	no	—
178	C₂×C₃×D₃	—	6	8	no	—
179	C₂×C₃×D₃	—	12	4	no	—
180	C₂×C₃×D₃	—	12	4	no	—
181	C₃²×C₄	—	2	2	no	—
182	C₃²×C₄	—	2	4	no	—
183	C₃²×C₄	—	2	4	no	—
184	C₃²×C₄	—	2	8	no	—
185	C₃²×C₄	—	4	8	no	—
186	C₃²×C₄	—	4	1	no	—
187	C₃²×C₄	—	4	4	no	—
188	C₃²×C₄	—	8	4	no	—
189	C₃²×C₄	C₂×C₄×D₃	1	4	yes	—
190	C₃²×C₄	#237	1	4	yes	—
191	C₃²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
192	C₃²×C₄	C₃×#41	1	1	yes	characteristic
193	C₃⋊Dic₃	—	2	1	no	—
194	C₃⋊Dic₃	—	2	1	no	—
195	C₃⋊Dic₃	—	4	1	no	—
196	C₃×Dic₃	—	6	4	no	—
197	C₃×Dic₃	—	6	4	no	—
198	C₃×Dic₃	—	12	4	no	—
199	C₂×SD₁₆	—	9	1	no	—
200	C₄×Q₈	—	9	1	no	—
201	C₄×D₄	—	9	1	no	—
202	#107	—	9	1	no	—
203	#103	C₃×C₃⋊D₃	1	1	yes	characteristic, complemented
204	#102	—	9	1	no	—
205	#97	—	9	1	no	—
206	C₃³	#440	1	1	yes	characteristic, complemented, Sylow
207	C₂³×C₃	—	18	1	no	—
208	C₂²×D₃	—	6	4	no	—
209	C₃×Q₈	—	2	4	no	—
210	C₃×Q₈	—	2	1	no	—
211	C₃×Q₈	—	2	8	no	—
212	C₃×Q₈	—	4	4	no	—
213	C₃×Q₈	C₃×C₄×D₃	1	8	yes	complemented
214	C₃×Q₈	C₄×C₃⋊D₃	1	2	yes	complemented
215	C₃×D₄	—	9	2	no	—
216	C₃×D₄	—	18	1	no	—
217	C₂×C₃×C₄	—	2	4	no	—
218	C₂×C₃×C₄	—	2	1	no	—
219	C₂×C₃×C₄	—	2	4	no	—
220	C₂×C₃×C₄	—	2	1	no	—
221	C₂×C₃×C₄	—	2	4	no	—
222	C₂×C₃×C₄	—	4	4	no	—
223	C₂×C₃×C₄	—	4	4	no	—
224	C₂×C₃×C₄	—	9	1	no	—
225	C₂×C₃×C₄	—	9	1	no	—
226	C₂×C₃×C₄	—	18	1	no	—
227	C₂×C₃×C₄	—	36	2	no	—
228	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
229	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
230	C₂×Dic₃	—	3	4	no	—
231	C₂×Dic₃	—	3	4	no	—
232	C₂×Dic₃	—	6	4	no	—
233	D₁₂	—	3	8	no	—
234	D₁₂	—	6	4	no	—
235	C₄×D₃	—	12	8	no	—
236	C₃×C₈	—	2	4	no	—
237	C₃×C₈	—	2	1	no	—
238	C₃×C₈	—	4	4	no	—
239	C₃×C₈	—	9	2	no	—
240	C₃⋊C₈	—	3	8	no	—
241	C₂×C₃²	—	2	4	no	—
242	C₂×C₃²	—	2	4	no	—
243	C₂×C₃²	—	2	4	no	—
244	C₂×C₃²	#67⋊C₄	1	4	yes	—
245	C₂×C₃²	#914	1	4	yes	—
246	C₂×C₃²	#918	1	4	yes	—
247	C₂×C₃²	C₃×C₄×D₄	1	1	yes	characteristic
248	C₂×C₃²	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
249	C₂×C₃²	C₃×#137	1	1	yes	characteristic
250	C₃⋊D₃	—	4	2	no	—
251	C₃×D₃	—	12	8	no	—
252	C₂×Q₈	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
253	C₂×D₄	—	9	1	no	—
254	C₂²×C₄	—	18	1	no	—
255	SD₁₆	—	9	4	no	—
256	C₂×C₈	—	9	1	no	—
257	C₂×C₈	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
258	C₄⋊C₄	—	9	1	no	—
259	C₄⋊C₄	—	18	1	no	—
260	C₄⋊C₄	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
261	C₂²⋊C₄	—	18	1	no	—
262	C₄²	—	9	1	no	—
263	C₄²	—	18	1	no	—
264	C₂²×C₃	—	2	4	no	—
265	C₂²×C₃	—	18	2	no	—
266	C₂²×C₃	—	36	1	no	—
267	C₂²×C₃	—	36	1	no	—
268	C₂²×C₃	C₃×D₃×D₄	1	4	yes	—
269	C₂²×C₃	D₄×C₃⋊D₃	1	1	yes	characteristic
270	C₂×D₃	—	6	8	no	—
271	C₂×D₃	—	12	4	no	—
272	C₂×D₃	—	12	4	no	—
273	C₃×C₄	—	2	8	no	—
274	C₃×C₄	—	2	2	no	—
275	C₃×C₄	—	2	4	no	—
276	C₃×C₄	—	2	4	no	—
277	C₃×C₄	—	4	8	no	—
278	C₃×C₄	—	4	4	no	—
279	C₃×C₄	—	4	1	no	—
280	C₃×C₄	—	8	4	no	—
281	C₃×C₄	—	18	1	no	—
282	C₃×C₄	—	18	1	no	—
283	C₃×C₄	—	36	1	no	—
284	C₃×C₄	C₂×C₃×C₄×D₃	1	4	yes	—
285	C₃×C₄	C₃×#237	1	4	yes	—
286	C₃×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
287	C₃×C₄	#3830	1	1	yes	characteristic
288	Dic₃	—	6	4	no	—
289	Dic₃	—	6	4	no	—
290	Dic₃	—	12	4	no	—
291	C₃²	—	2	4	no	—
292	C₃²	#4817	1	4	yes	complemented
293	C₃²	C₃×#440	1	1	yes	characteristic, complemented
294	C₂³	—	18	1	no	—
295	Q₈	—	2	1	no	—
296	Q₈	C₃×C₄×C₃⋊D₃	1	2	yes	complemented
297	D₄	—	9	2	no	—
298	D₄	—	18	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	9	1	no	—
302	C₂×C₄	—	9	1	no	—
303	C₂×C₄	—	18	1	no	—
304	C₂×C₄	—	36	2	no	—
305	C₂×C₄	C₂²×C₃×C₃⋊D₃	1	1	yes	characteristic
306	C₈	—	2	1	no	—
307	C₈	—	9	2	no	—
308	C₂×C₃	—	2	4	no	—
309	C₂×C₃	—	2	4	no	—
310	C₂×C₃	—	2	4	no	—
311	C₂×C₃	—	36	2	no	—
312	C₂×C₃	C₃×#67⋊C₄	1	4	yes	—
313	C₂×C₃	C₃×#914	1	4	yes	—
314	C₂×C₃	C₃×#918	1	4	yes	—
315	C₂×C₃	#64090	1	1	yes	characteristic
316	C₂×C₃	#64123	1	1	yes	characteristic
317	C₂×C₃	#64127	1	1	yes	characteristic
318	D₃	—	12	8	no	—
319	C₂²	—	18	2	no	—
320	C₂²	—	36	1	no	—
321	C₂²	—	36	1	no	—
322	C₂²	C₃×D₄×C₃⋊D₃	1	1	yes	characteristic
323	C₄	—	2	2	no	—
324	C₄	—	4	1	no	—
325	C₄	—	18	1	no	—
326	C₄	—	18	1	no	—
327	C₄	—	36	1	no	—
328	C₄	C₂×C₃×C₄×C₃⋊D₃	1	1	yes	characteristic
329	C₄	C₃×#3830	1	1	yes	characteristic
330	C₃	—	2	4	no	—
331	C₃	C₃×#4817	1	4	yes	complemented
332	C₃	#10592516	1	1	yes	characteristic, direct product factor
333	C₂	—	36	2	no	—
334	C₂	C₃×#64090	1	1	yes	characteristic
335	C₂	C₃×#64123	1	1	yes	characteristic
336	C₂	C₃×#64127	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	D₄×C₃⋊D₃
commutator	C₃²×C₄	C₂²×C₃×C₄
Frattini	C₂×C₄	C₂²×C₃×C₃⋊D₃
Fitting	C₃³×#103	C₂
socle	C₂²×C₃³	C₂×D₄
prime radical	C₂×C₃×C₄⋊(C₃⋊D₃)	C₂²
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₃×#103	C₃⋊D₃
hypocenter	C₃²	C₃×#440
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄ ◁ C₃×#103

Complements

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

N	H ≅ G/N	type
1. C₃×Q₈×C₃⋊Dic₃	C₂	semidirect
4. C₃³×#103	C₂	semidirect
6. C₃×#63642	C₂	semidirect
7. C₃×#63634	C₂	semidirect
8. #10592516	C₃	direct (N × H)
15. C₃×#3755	C₄	semidirect
22. Q₈×C₃⋊Dic₃	C₂×C₃	semidirect
29. C₃²×#103	D₃	semidirect
30. C₃²×#103	C₂×C₃	semidirect
32. #63642	C₂×C₃	semidirect
33. #63634	C₂×C₃	semidirect
39. C₃³×Q₈	C₂×C₄	semidirect
50. C₃×C₃⋊(C₃⋊C₈)	C₂×C₄	semidirect
66. #3755	C₃×C₄	semidirect
99. C₃×#103	C₃×D₃	semidirect
100. C₃×#103	C₃⋊D₃	semidirect
112. C₃²×Q₈	C₄×D₃	semidirect
113. C₃²×Q₈	C₂×C₃×C₄	semidirect
138. C₃⋊(C₃⋊C₈)	C₂×C₃×C₄	semidirect
203. #103	C₃×C₃⋊D₃	semidirect
206. C₃³	#440	semidirect
213. C₃×Q₈	C₃×C₄×D₃	semidirect
214. C₃×Q₈	C₄×C₃⋊D₃	semidirect
292. C₃²	#4817	semidirect
293. C₃²	C₃×#440	semidirect
296. Q₈	C₃×C₄×C₃⋊D₃	semidirect
331. C₃	C₃×#4817	semidirect
332. C₃	#10592516	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×Q₈×C₃⋊Dic₃	2
C₂×C₃×#3755	2
C₃×#64102	2
C₃³×#103	2
C₃×#63648	2
C₃×#63642	2
C₃×#63634	2
#10592516	3
C₃×#4817	3

Subgroups — C3×#10592516 #13162852

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10592516 #13162852