Subgroups — C₃×#10592771 #13163113

Order 1728.

1272 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×#127	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×#3753	C₂	1	2	yes	maximal, complemented
3	C₃×C₄×C₄⋊(C₃⋊D₃)	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#63659	C₂	1	2	yes	maximal, complemented
5	C₃×C₄⋊(C₃⋊(C₃⋊C₈))	C₂	1	1	yes	maximal, characteristic, complemented
6	#10592771	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₃×#5072	—	3	4	no	maximal
8	C₂×C₃³×D₄	—	2	1	no	—
9	C₂×C₃³×D₄	—	2	1	no	—
10	C₂×C₃³×D₄	C₂²	1	2	yes	—
11	C₂×C₃×C₄⋊(C₃⋊D₃)	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₄×C₃⋊D₃	—	2	1	no	—
13	C₃³×C₄²	C₂²	1	1	yes	characteristic
14	C₃×#3753	—	2	4	no	—
15	C₃×#3752	—	2	1	no	—
16	C₃×C₄⋊(C₃⋊Dic₃)	C₂²	1	1	yes	characteristic
17	C₂×C₃×C₃⋊(C₃⋊C₈)	C₂²	1	2	yes	—
18	C₃²×#127	—	2	4	no	—
19	C₃²×#127	D₃	1	4	yes	complemented
20	C₃²×#127	C₂×C₃	1	1	yes	characteristic, complemented
21	C₂×#3753	C₂×C₃	1	2	yes	complemented
22	C₄×C₄⋊(C₃⋊D₃)	C₂×C₃	1	1	yes	characteristic, complemented
23	C₂×C₃×#213	—	3	8	no	—
24	C₃×C₄×D₁₂	—	3	4	no	—
25	#63659	C₂×C₃	1	2	yes	complemented
26	C₄⋊(C₃⋊(C₃⋊C₈))	C₂×C₃	1	1	yes	characteristic, complemented
27	C₃×#832	—	3	8	no	—
28	C₃×C₄⋊(C₃⋊C₈)	—	3	4	no	—
29	C₂³×C₃³	—	2	2	no	—
30	C₂²×C₃×C₃⋊D₃	—	2	1	no	—
31	C₃³×D₄	—	2	2	no	—
32	C₃³×D₄	—	2	2	no	—
33	C₃³×D₄	—	4	2	no	—
34	C₃³×D₄	—	4	2	no	—
35	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
36	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
37	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
38	C₂×C₃×C₃⋊Dic₃	—	2	1	no	—
39	C₃×C₄⋊(C₃⋊D₃)	—	2	1	no	—
40	C₃×C₄⋊(C₃⋊D₃)	D₄	1	2	yes	complemented
41	C₃×C₄×C₃⋊D₃	—	4	2	no	—
42	C₃×C₃⋊(C₃⋊C₈)	—	2	2	no	—
43	C₃×#459	—	9	1	no	—
44	#5072	—	3	4	no	—
45	C₂×C₃²×D₄	—	2	8	no	—
46	C₂×C₃²×D₄	—	2	8	no	—
47	C₂×C₃²×D₄	—	2	8	no	—
48	C₂×C₃²×D₄	—	2	4	no	—
49	C₂×C₃²×D₄	—	2	4	no	—
50	C₂×C₃²×D₄	—	2	1	no	—
51	C₂×C₃²×D₄	—	2	1	no	—
52	C₂×C₃²×D₄	C₂×D₃	1	8	yes	—
53	C₂×C₃²×D₄	C₂²×C₃	1	2	yes	—
54	C₂×C₄⋊(C₃⋊D₃)	C₂²×C₃	1	1	yes	characteristic
55	C₂×C₄×C₃⋊D₃	—	2	1	no	—
56	C₂×C₃×D₁₂	—	3	4	no	—
57	C₂×C₃×C₄×D₃	—	6	4	no	—
58	C₃²×C₄²	—	2	4	no	—
59	C₃²×C₄²	C₂×D₃	1	4	yes	—
60	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
61	#3753	—	2	4	no	—
62	#3752	—	2	1	no	—
63	C₄⋊(C₃⋊Dic₃)	C₂²×C₃	1	1	yes	characteristic
64	C₂×C₃⋊(C₃⋊C₈)	C₂²×C₃	1	2	yes	—
65	C₃×#213	—	6	16	no	—
66	C₃×(C₂×D₃)⋊C₄	—	6	4	no	—
67	C₃×C₄⋊Dic₃	—	3	4	no	—
68	C₂×C₃×C₃⋊C₈	—	3	8	no	—
69	C₂²×C₃³	—	4	2	no	—
70	C₂²×C₃³	—	4	2	no	—
71	C₂²×C₃³	—	4	2	no	—
72	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
73	C₂×C₃×C₃⋊D₃	—	2	2	no	—
74	C₂×C₃×C₃⋊D₃	—	4	1	no	—
75	C₂×C₃×C₃⋊D₃	—	4	1	no	—
76	C₃³×C₄	—	2	1	no	—
77	C₃³×C₄	D₈	1	2	yes	complemented
78	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
79	C₃³×C₄	#41	1	1	yes	characteristic
80	C₃×C₃⋊Dic₃	—	4	1	no	—
81	C₂×C₃×D₈	—	9	2	no	—
82	C₃×#127	—	2	4	no	—
83	C₃×#127	C₃×D₃	1	4	yes	complemented
84	C₃×#127	C₃⋊D₃	1	1	yes	characteristic, complemented
85	C₃×C₄×D₄	—	9	1	no	—
86	C₂×#213	—	3	8	no	—
87	C₄×D₁₂	—	3	4	no	—
88	C₃×C₄⋊C₈	—	9	1	no	—
89	C₃×#102	—	9	2	no	—
90	#832	—	3	8	no	—
91	C₄⋊(C₃⋊C₈)	—	3	4	no	—
92	C₂³×C₃²	—	2	16	no	—
93	C₂³×C₃²	—	2	8	no	—
94	C₂³×C₃²	—	2	2	no	—
95	C₂²×C₃⋊D₃	—	2	1	no	—
96	C₂²×C₃×D₃	—	6	4	no	—
97	C₃²×D₄	—	2	2	no	—
98	C₃²×D₄	—	2	2	no	—
99	C₃²×D₄	—	2	8	no	—
100	C₃²×D₄	—	2	8	no	—
101	C₃²×D₄	—	4	8	no	—
102	C₃²×D₄	—	4	8	no	—
103	C₃²×D₄	—	4	16	no	—
104	C₃²×D₄	—	4	8	no	—
105	C₃²×D₄	—	4	2	no	—
106	C₃²×D₄	—	4	16	no	—
107	C₃²×D₄	—	4	8	no	—
108	C₃²×D₄	—	4	2	no	—
109	C₂×C₃²×C₄	—	2	4	no	—
110	C₂×C₃²×C₄	—	2	4	no	—
111	C₂×C₃²×C₄	—	2	4	no	—
112	C₂×C₃²×C₄	#67	1	4	yes	—
113	C₂×C₃²×C₄	#67	1	4	yes	—
114	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
115	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
116	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
117	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
118	C₂×C₃⋊Dic₃	—	2	1	no	—
119	C₄⋊(C₃⋊D₃)	—	2	1	no	—
120	C₄⋊(C₃⋊D₃)	C₃×D₄	1	2	yes	complemented
121	C₄×C₃⋊D₃	—	4	2	no	—
122	C₂×C₃×Dic₃	—	6	4	no	—
123	C₃×D₁₂	—	3	8	no	—
124	C₃×D₁₂	—	6	4	no	—
125	C₃×C₄×D₃	—	12	8	no	—
126	C₃⋊(C₃⋊C₈)	—	2	2	no	—
127	C₃×C₃⋊C₈	—	6	8	no	—
128	#459	—	9	1	no	Sylow
129	C₂×C₃³	—	8	2	no	—
130	C₂×C₃³	#121	1	1	yes	characteristic
131	C₂×C₃³	C₂×D₈	1	1	yes	characteristic
132	C₂×C₃³	AGL(1,ℤ/8)	1	1	yes	characteristic
133	C₃×C₃⋊D₃	—	4	2	no	—
134	C₂×C₃×D₄	—	2	8	no	—
135	C₂×C₃×D₄	—	2	8	no	—
136	C₂×C₃×D₄	—	2	4	no	—
137	C₂×C₃×D₄	—	2	4	no	—
138	C₂×C₃×D₄	—	2	1	no	—
139	C₂×C₃×D₄	—	2	1	no	—
140	C₂×C₃×D₄	—	2	8	no	—
141	C₂×C₃×D₄	—	9	1	no	—
142	C₂×C₃×D₄	C₂×C₃×D₃	1	8	yes	—
143	C₂×C₃×D₄	C₂×C₃⋊D₃	1	2	yes	—
144	C₂²×C₃×C₄	—	18	1	no	—
145	C₂×D₁₂	—	3	4	no	—
146	C₂×C₄×D₃	—	6	4	no	—
147	C₃×D₈	—	18	4	no	—
148	C₂×C₃×C₈	—	9	2	no	—
149	C₃×C₄⋊C₄	—	9	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	#213	—	6	16	no	—
155	(C₂×D₃)⋊C₄	—	6	4	no	—
156	C₄⋊Dic₃	—	3	4	no	—
157	C₂×C₃⋊C₈	—	3	8	no	—
158	C₂²×C₃²	—	2	4	no	—
159	C₂²×C₃²	—	4	16	no	—
160	C₂²×C₃²	—	4	16	no	—
161	C₂²×C₃²	—	4	16	no	—
162	C₂²×C₃²	—	4	8	no	—
163	C₂²×C₃²	—	4	8	no	—
164	C₂²×C₃²	—	4	8	no	—
165	C₂²×C₃²	—	4	2	no	—
166	C₂²×C₃²	—	4	2	no	—
167	C₂²×C₃²	—	4	2	no	—
168	C₂²×C₃²	C₂×#67	1	4	yes	—
169	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
170	C₂×C₃⋊D₃	—	2	2	no	—
171	C₂×C₃⋊D₃	—	4	1	no	—
172	C₂×C₃⋊D₃	—	4	1	no	—
173	C₂×C₃×D₃	—	6	8	no	—
174	C₂×C₃×D₃	—	12	4	no	—
175	C₂×C₃×D₃	—	12	4	no	—
176	C₃²×C₄	—	2	8	no	—
177	C₃²×C₄	—	2	4	no	—
178	C₃²×C₄	—	2	1	no	—
179	C₃²×C₄	—	2	8	no	—
180	C₃²×C₄	—	2	4	no	—
181	C₃²×C₄	—	2	4	no	—
182	C₃²×C₄	#213	1	8	yes	complemented
183	C₃²×C₄	C₃×D₈	1	2	yes	complemented
184	C₃²×C₄	D₃×D₄	1	4	yes	—
185	C₃²×C₄	#237	1	4	yes	—
186	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
187	C₃²×C₄	C₃×#41	1	1	yes	characteristic
188	C₃⋊Dic₃	—	4	1	no	—
189	C₃×Dic₃	—	12	4	no	—
190	C₂×D₈	—	9	2	no	—
191	#127	C₃×C₃⋊D₃	1	1	yes	characteristic, complemented
192	C₄×D₄	—	9	1	no	—
193	C₄⋊C₈	—	9	1	no	—
194	#102	—	9	2	no	—
195	C₃³	#459	1	1	yes	characteristic, complemented, Sylow
196	C₂³×C₃	—	2	16	no	—
197	C₂³×C₃	—	2	8	no	—
198	C₂³×C₃	—	2	2	no	—
199	C₂³×C₃	—	18	1	no	—
200	C₂²×D₃	—	6	4	no	—
201	C₃×D₄	—	2	8	no	—
202	C₃×D₄	—	2	8	no	—
203	C₃×D₄	—	2	2	no	—
204	C₃×D₄	—	2	2	no	—
205	C₃×D₄	—	4	8	no	—
206	C₃×D₄	—	4	8	no	—
207	C₃×D₄	—	4	16	no	—
208	C₃×D₄	—	4	8	no	—
209	C₃×D₄	—	4	2	no	—
210	C₃×D₄	—	4	16	no	—
211	C₃×D₄	—	4	8	no	—
212	C₃×D₄	—	4	2	no	—
213	C₃×D₄	—	9	2	no	—
214	C₃×D₄	—	18	1	no	—
215	C₂×C₃×C₄	—	2	4	no	—
216	C₂×C₃×C₄	—	2	4	no	—
217	C₂×C₃×C₄	—	2	4	no	—
218	C₂×C₃×C₄	—	18	1	no	—
219	C₂×C₃×C₄	—	36	2	no	—
220	C₂×C₃×C₄	C₃×#67	1	4	yes	—
221	C₂×C₃×C₄	C₃×#67	1	4	yes	—
222	C₂×C₃×C₄	#638	1	1	yes	characteristic
223	C₂×C₃×C₄	#638	1	1	yes	characteristic
224	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
225	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
226	C₂×Dic₃	—	6	4	no	—
227	D₁₂	—	3	8	no	—
228	D₁₂	—	6	4	no	—
229	C₄×D₃	—	12	8	no	—
230	C₃×C₈	—	18	2	no	—
231	C₃⋊C₈	—	6	8	no	—
232	C₂×C₃²	—	2	4	no	—
233	C₂×C₃²	—	2	4	no	—
234	C₂×C₃²	—	2	4	no	—
235	C₂×C₃²	—	8	16	no	—
236	C₂×C₃²	—	8	8	no	—
237	C₂×C₃²	—	8	2	no	—
238	C₂×C₃²	C₂×#213	1	4	yes	—
239	C₂×C₃²	#932	1	4	yes	—
240	C₂×C₃²	#938	1	4	yes	—
241	C₂×C₃²	C₃×#121	1	1	yes	characteristic
242	C₂×C₃²	C₂×C₃×D₈	1	1	yes	characteristic
243	C₂×C₃²	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
244	C₃⋊D₃	—	4	2	no	—
245	C₃×D₃	—	12	8	no	—
246	C₂×D₄	—	2	1	no	—
247	C₂×D₄	—	2	1	no	—
248	C₂×D₄	—	9	1	no	—
249	C₂×D₄	C₂×C₃×C₃⋊D₃	1	2	yes	—
250	C₂²×C₄	—	18	1	no	—
251	D₈	—	18	4	no	—
252	C₂×C₈	—	9	2	no	—
253	C₄⋊C₄	—	9	1	no	—
254	C₂²⋊C₄	—	18	1	no	—
255	C₄²	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
256	C₂²×C₃	—	2	4	no	—
257	C₂²×C₃	—	4	16	no	—
258	C₂²×C₃	—	4	16	no	—
259	C₂²×C₃	—	4	16	no	—
260	C₂²×C₃	—	4	8	no	—
261	C₂²×C₃	—	4	8	no	—
262	C₂²×C₃	—	4	8	no	—
263	C₂²×C₃	—	4	2	no	—
264	C₂²×C₃	—	4	2	no	—
265	C₂²×C₃	—	4	2	no	—
266	C₂²×C₃	—	18	2	no	—
267	C₂²×C₃	—	36	1	no	—
268	C₂²×C₃	—	36	1	no	—
269	C₂²×C₃	C₂×C₃×#67	1	4	yes	—
270	C₂²×C₃	C₂×#638	1	1	yes	characteristic
271	C₂×D₃	—	6	8	no	—
272	C₂×D₃	—	12	4	no	—
273	C₂×D₃	—	12	4	no	—
274	C₃×C₄	—	2	8	no	—
275	C₃×C₄	—	2	4	no	—
276	C₃×C₄	—	2	1	no	—
277	C₃×C₄	—	2	8	no	—
278	C₃×C₄	—	2	4	no	—
279	C₃×C₄	—	2	4	no	—
280	C₃×C₄	—	36	1	no	—
281	C₃×C₄	C₃×#213	1	8	yes	complemented
282	C₃×C₄	#3753	1	2	yes	complemented
283	C₃×C₄	C₃×D₃×D₄	1	4	yes	—
284	C₃×C₄	C₃×#237	1	4	yes	—
285	C₃×C₄	D₄×C₃⋊D₃	1	1	yes	characteristic
286	C₃×C₄	#3830	1	1	yes	characteristic
287	Dic₃	—	12	4	no	—
288	C₃²	—	2	4	no	—
289	C₃²	#5072	1	4	yes	complemented
290	C₃²	C₃×#459	1	1	yes	characteristic, complemented
291	C₂³	—	2	2	no	—
292	C₂³	—	18	1	no	—
293	D₄	—	2	2	no	—
294	D₄	—	2	2	no	—
295	D₄	—	4	2	no	—
296	D₄	—	4	2	no	—
297	D₄	—	9	2	no	—
298	D₄	—	18	1	no	—
299	C₂×C₄	—	18	1	no	—
300	C₂×C₄	—	36	2	no	—
301	C₂×C₄	C₃×#638	1	1	yes	characteristic
302	C₂×C₄	C₃×#638	1	1	yes	characteristic
303	C₂×C₄	C₂²×C₃×C₃⋊D₃	1	1	yes	characteristic
304	C₈	—	18	2	no	—
305	C₂×C₃	—	2	4	no	—
306	C₂×C₃	—	2	4	no	—
307	C₂×C₃	—	2	4	no	—
308	C₂×C₃	—	8	16	no	—
309	C₂×C₃	—	8	2	no	—
310	C₂×C₃	—	8	8	no	—
311	C₂×C₃	—	36	2	no	—
312	C₂×C₃	C₂×C₃×#213	1	4	yes	—
313	C₂×C₃	C₃×#932	1	4	yes	—
314	C₂×C₃	C₃×#938	1	4	yes	—
315	C₂×C₃	C₂×#3753	1	1	yes	characteristic
316	C₂×C₃	#64141	1	1	yes	characteristic
317	C₂×C₃	#64147	1	1	yes	characteristic
318	D₃	—	12	8	no	—
319	C₂²	—	4	2	no	—
320	C₂²	—	4	2	no	—
321	C₂²	—	4	2	no	—
322	C₂²	—	18	2	no	—
323	C₂²	—	36	1	no	—
324	C₂²	—	36	1	no	—
325	C₂²	C₂×C₃×#638	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	36	1	no	—
328	C₄	C₃×#3753	1	2	yes	complemented
329	C₄	C₃×D₄×C₃⋊D₃	1	1	yes	characteristic
330	C₄	C₃×#3830	1	1	yes	characteristic
331	C₃	—	2	4	no	—
332	C₃	C₃×#5072	1	4	yes	complemented
333	C₃	#10592771	1	1	yes	characteristic, direct product factor
334	C₂	—	8	2	no	—
335	C₂	—	36	2	no	—
336	C₂	C₂×C₃×#3753	1	1	yes	characteristic
337	C₂	C₃×#64141	1	1	yes	characteristic
338	C₂	C₃×#64147	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂×#638
commutator, cosocle	C₂×C₃²×C₄	C₂³×C₃
Frattini	C₂×C₄	C₂²×C₃×C₃⋊D₃
Fitting	C₃³×#127	C₂
socle	C₂²×C₃³	C₂×D₄
hypercenter	C₃×#127	C₃⋊D₃
hypocenter	C₃²	C₃×#459
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×#127

Complements

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

N	H ≅ G/N	type
1. C₃³×#127	C₂	semidirect
2. C₂×C₃×#3753	C₂	semidirect
3. C₃×C₄×C₄⋊(C₃⋊D₃)	C₂	semidirect
4. C₃×#63659	C₂	semidirect
5. C₃×C₄⋊(C₃⋊(C₃⋊C₈))	C₂	semidirect
6. #10592771	C₃	direct (N × H)
19. C₃²×#127	D₃	semidirect
20. C₃²×#127	C₂×C₃	semidirect
21. C₂×#3753	C₂×C₃	semidirect
22. C₄×C₄⋊(C₃⋊D₃)	C₂×C₃	semidirect
25. #63659	C₂×C₃	semidirect
26. C₄⋊(C₃⋊(C₃⋊C₈))	C₂×C₃	semidirect
40. C₃×C₄⋊(C₃⋊D₃)	D₄	semidirect
77. C₃³×C₄	D₈	semidirect
83. C₃×#127	C₃×D₃	semidirect
84. C₃×#127	C₃⋊D₃	semidirect
120. C₄⋊(C₃⋊D₃)	C₃×D₄	semidirect
182. C₃²×C₄	#213	semidirect
183. C₃²×C₄	C₃×D₈	semidirect
191. #127	C₃×C₃⋊D₃	semidirect
195. C₃³	#459	semidirect
281. C₃×C₄	C₃×#213	semidirect
282. C₃×C₄	#3753	semidirect
289. C₃²	#5072	semidirect
290. C₃²	C₃×#459	semidirect
328. C₄	C₃×#3753	semidirect
332. C₃	C₃×#5072	semidirect
333. C₃	#10592771	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃³×#127	2
C₂×C₃×#3753	2
C₃×C₄×C₄⋊(C₃⋊D₃)	2
C₃×#63659	2
C₃×C₄⋊(C₃⋊(C₃⋊C₈))	2
#10592771	3
C₃×#5072	3

Subgroups — C3×#10592771 #13163113

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10592771 #13163113