Subgroups — C₃×#10588443 #13141713

Order 1728.

356 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×D₄×Dic₉	C₂	1	1	yes	maximal, characteristic
2	C₃×#63452	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₉×#102	C₂	1	1	yes	maximal, characteristic
4	C₃×#63392	C₂	1	1	yes	maximal, characteristic
5	C₃×#63377	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×Dic₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#63366	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#4763	—	3	1	no	maximal
9	#10588443	C₃	1	1	yes	maximal, characteristic, direct product factor
10	C₂×C₃×D₄×C₉	C₂²	1	1	yes	characteristic
11	C₂²×C₃×Dic₉	—	2	1	no	—
12	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
13	C₃×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₃×#3676	—	2	1	no	—
15	C₃×C₄⋊Dic₉	—	2	1	no	—
16	C₃×C₄⋊Dic₉	C₂²	1	1	yes	characteristic
17	C₃×C₄⋊Dic₉	C₂²	1	1	yes	characteristic
18	C₃×Dic₉⋊C₄	—	2	1	no	—
19	C₃×C₄×Dic₉	C₂²	1	1	yes	characteristic
20	C₂×C₃×C₉⋊C₈	C₂²	1	1	yes	characteristic
21	C₃×D₄×Dic₃	—	3	1	no	—
22	C₃×#890	—	3	1	no	—
23	C₃²×#102	D₃	1	1	yes	characteristic
24	C₃×#832	—	3	1	no	—
25	C₃×#817	—	3	1	no	—
26	C₃×Dic₃⋊C₈	—	3	1	no	—
27	C₃×#807	—	3	1	no	—
28	D₄×Dic₉	C₂×C₃	1	1	yes	characteristic
29	#63452	C₂×C₃	1	1	yes	characteristic, complemented
30	C₉×#102	—	2	1	no	—
31	C₉×#102	C₂×C₃	1	1	yes	characteristic
32	#63392	C₂×C₃	1	1	yes	characteristic
33	#63377	C₂×C₃	1	1	yes	characteristic, complemented
34	Dic₉⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
35	#63366	C₂×C₃	1	1	yes	characteristic, complemented
36	C₂³×C₃×C₉	—	2	1	no	—
37	C₃×D₄×C₉	—	2	1	no	—
38	C₃×D₄×C₉	Q₈	1	2	yes	—
39	C₂×C₃×C₄×C₉	—	2	1	no	—
40	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
41	C₂×C₃×Dic₉	—	2	1	no	—
42	C₂×C₃×Dic₉	—	2	1	no	—
43	C₂×C₃×Dic₉	—	4	2	no	—
44	C₂×C₃×Dic₉	D₄	1	1	yes	characteristic
45	C₂×C₃×Dic₉	D₄	1	1	yes	characteristic
46	C₃×C₈×C₉	—	2	1	no	—
47	C₃×C₉⋊C₈	—	2	1	no	—
48	C₃×#478	—	9	1	no	—
49	#4763	—	3	1	no	—
50	C₂×C₃²×D₄	C₂×D₃	1	1	yes	characteristic
51	C₂²×C₃×Dic₃	—	6	1	no	—
52	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
53	C₃²×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
54	C₃×#217	—	6	1	no	—
55	C₃×C₄⋊Dic₃	—	3	1	no	—
56	C₃×C₄⋊Dic₃	—	3	1	no	—
57	C₃×C₄⋊Dic₃	—	6	1	no	—
58	C₃×Dic₃⋊C₄	—	6	1	no	—
59	C₃×C₄×Dic₃	—	3	1	no	—
60	C₂×C₃×C₃⋊C₈	—	3	1	no	—
61	C₂×D₄×C₉	—	2	1	no	—
62	C₂×D₄×C₉	C₂²×C₃	1	1	yes	characteristic
63	C₂²×Dic₉	—	2	1	no	—
64	C₂×C₈×C₉	—	2	1	no	—
65	C₂×C₈×C₉	C₂²×C₃	1	1	yes	characteristic
66	C₉×C₄⋊C₄	—	2	1	no	—
67	C₉×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
68	#3676	—	2	1	no	—
69	C₄⋊Dic₉	—	2	1	no	—
70	C₄⋊Dic₉	C₂²×C₃	1	1	yes	characteristic
71	C₄⋊Dic₉	C₂²×C₃	1	1	yes	characteristic
72	Dic₉⋊C₄	—	2	1	no	—
73	C₄×Dic₉	C₂²×C₃	1	1	yes	characteristic
74	C₂×C₉⋊C₈	C₂²×C₃	1	1	yes	characteristic
75	C₂²×C₃×C₉	—	2	2	no	—
76	C₂²×C₃×C₉	—	4	1	no	—
77	C₂²×C₃×C₉	—	4	1	no	—
78	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
79	C₃×C₄×C₉	—	4	1	no	—
80	C₃×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
81	C₃×C₄×C₉	#41	1	1	yes	characteristic
82	C₃×Dic₉	—	2	1	no	—
83	C₃×Dic₉	—	2	1	no	—
84	C₃×Dic₉	—	4	1	no	—
85	C₃×Dic₉	—	4	1	no	—
86	C₃×#125	—	9	1	no	—
87	C₃×C₄×D₄	—	9	1	no	—
88	D₄×Dic₃	—	3	1	no	—
89	#890	—	3	1	no	—
90	C₃×#107	—	9	1	no	—
91	C₃×#106	—	9	1	no	—
92	C₃×C₄⋊C₈	—	9	1	no	—
93	C₃×#102	—	2	1	no	—
94	C₃×#102	—	9	1	no	—
95	C₃×#102	D₉	1	1	yes	characteristic
96	C₃×#102	C₃×D₃	1	1	yes	characteristic
97	#832	—	3	1	no	—
98	#817	—	3	1	no	—
99	Dic₃⋊C₈	—	3	1	no	—
100	#807	—	3	1	no	—
101	C₂³×C₃²	—	2	1	no	—
102	C₃²×D₄	—	2	1	no	—
103	C₃²×D₄	Dic₆	1	2	yes	—
104	C₂×C₃²×C₄	—	2	1	no	—
105	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
106	C₂×C₃×Dic₃	—	3	1	no	—
107	C₂×C₃×Dic₃	—	3	1	no	—
108	C₂×C₃×Dic₃	—	6	1	no	—
109	C₂×C₃×Dic₃	—	6	1	no	—
110	C₂×C₃×Dic₃	—	12	2	no	—
111	C₂³×C₉	—	2	1	no	—
112	C₂³×C₉	—	4	1	no	—
113	C₃²×C₈	—	2	1	no	—
114	C₃×C₃⋊C₈	—	6	1	no	—
115	D₄×C₉	—	2	1	no	—
116	D₄×C₉	—	2	2	no	—
117	D₄×C₉	—	4	1	no	—
118	D₄×C₉	C₃×Q₈	1	2	yes	—
119	C₂×C₄×C₉	—	2	1	no	—
120	C₂×C₄×C₉	—	2	1	no	—
121	C₂×C₄×C₉	—	4	1	no	—
122	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
123	C₂×Dic₉	—	2	1	no	—
124	C₂×Dic₉	—	2	1	no	—
125	C₂×Dic₉	—	4	2	no	—
126	C₂×Dic₉	C₃×D₄	1	1	yes	characteristic
127	C₂×Dic₉	C₃×D₄	1	1	yes	characteristic
128	C₈×C₉	—	2	1	no	—
129	C₈×C₉	—	4	1	no	—
130	C₉⋊C₈	—	2	1	no	—
131	#478	—	9	1	no	Sylow
132	C₂×C₃×C₉	—	4	2	no	—
133	C₂×C₃×C₉	C₂²⋊Q₈	1	1	yes	characteristic
134	C₂×C₃×C₉	#135	1	1	yes	characteristic
135	C₂×C₃×C₉	AGL(1,ℤ/8)	1	1	yes	characteristic
136	C₂×C₃×D₄	—	2	1	no	—
137	C₂×C₃×D₄	C₂×D₉	1	1	yes	characteristic
138	C₂×C₃×D₄	C₂×C₃×D₃	1	1	yes	characteristic
139	C₂²×C₃×C₄	—	18	1	no	—
140	C₂²×Dic₃	—	6	1	no	—
141	C₂×C₃×C₈	—	2	1	no	—
142	C₂×C₃×C₈	—	9	1	no	—
143	C₂×C₃×C₈	C₂×D₉	1	1	yes	characteristic
144	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
145	C₃×C₄⋊C₄	—	2	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×C₄⋊C₄	C₂×D₉	1	1	yes	characteristic
151	C₃×C₄⋊C₄	C₂×C₃×D₃	1	1	yes	characteristic
152	C₃×C₂²⋊C₄	—	18	1	no	—
153	C₃×C₄²	—	9	1	no	—
154	#217	—	6	1	no	—
155	C₄⋊Dic₃	—	3	1	no	—
156	C₄⋊Dic₃	—	3	1	no	—
157	C₄⋊Dic₃	—	6	1	no	—
158	Dic₃⋊C₄	—	6	1	no	—
159	C₄×Dic₃	—	3	1	no	—
160	C₂×C₃⋊C₈	—	3	1	no	—
161	C₂²×C₃²	—	2	2	no	—
162	C₂²×C₃²	—	4	1	no	—
163	C₂²×C₃²	—	4	1	no	—
164	C₂²×C₃²	D₃×D₄	1	1	yes	characteristic
165	C₃²×C₄	—	4	1	no	—
166	C₃²×C₄	C₂×Dic₆	1	1	yes	characteristic
167	C₃²×C₄	#237	1	1	yes	characteristic
168	C₃×Dic₃	—	6	1	no	—
169	C₃×Dic₃	—	6	1	no	—
170	C₃×Dic₃	—	12	1	no	—
171	C₃×Dic₃	—	12	1	no	—
172	C₂²×C₉	—	2	2	no	—
173	C₂²×C₉	—	2	1	no	—
174	C₂²×C₉	—	4	2	no	—
175	C₂²×C₉	—	4	1	no	—
176	C₂²×C₉	—	4	1	no	—
177	C₂²×C₉	—	8	1	no	—
178	C₂²×C₉	—	8	1	no	—
179	C₂²×C₉	C₂×C₃×D₄	1	1	yes	characteristic
180	C₄×C₉	—	2	1	no	—
181	C₄×C₉	—	2	1	no	—
182	C₄×C₉	—	4	1	no	—
183	C₄×C₉	—	8	1	no	—
184	C₄×C₉	C₂×C₃×Q₈	1	1	yes	characteristic
185	C₄×C₉	C₃×#41	1	1	yes	characteristic
186	Dic₉	—	2	1	no	—
187	Dic₉	—	2	1	no	—
188	Dic₉	—	4	1	no	—
189	Dic₉	—	4	1	no	—
190	#125	—	9	1	no	—
191	C₄×D₄	—	9	1	no	—
192	#107	—	9	1	no	—
193	#106	—	9	1	no	—
194	C₄⋊C₈	—	9	1	no	—
195	#102	—	9	1	no	—
196	#102	C₃×D₉	1	1	yes	characteristic
197	C₃×C₉	#478	1	1	yes	characteristic, complemented, Sylow
198	C₂³×C₃	—	2	1	no	—
199	C₂³×C₃	—	2	1	no	—
200	C₂³×C₃	—	4	1	no	—
201	C₃×D₄	—	2	1	no	—
202	C₃×D₄	—	2	1	no	—
203	C₃×D₄	—	2	2	no	—
204	C₃×D₄	—	4	1	no	—
205	C₃×D₄	Dic₁₈	1	2	yes	—
206	C₃×D₄	C₃×Dic₆	1	2	yes	—
207	C₂×C₃×C₄	—	2	1	no	—
208	C₂×C₃×C₄	—	2	1	no	—
209	C₂×C₃×C₄	—	2	1	no	—
210	C₂×C₃×C₄	—	4	1	no	—
211	C₂×C₃×C₄	—	9	1	no	—
212	C₂×C₃×C₄	—	9	1	no	—
213	C₂×C₃×C₄	—	18	1	no	—
214	C₂×C₃×C₄	—	18	1	no	—
215	C₂×C₃×C₄	—	36	2	no	—
216	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
217	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
218	C₂×Dic₃	—	3	1	no	—
219	C₂×Dic₃	—	3	1	no	—
220	C₂×Dic₃	—	6	1	no	—
221	C₂×Dic₃	—	6	1	no	—
222	C₂×Dic₃	—	12	2	no	—
223	C₃×C₈	—	2	1	no	—
224	C₃×C₈	—	2	1	no	—
225	C₃×C₈	—	4	1	no	—
226	C₃×C₈	—	18	1	no	—
227	C₃⋊C₈	—	6	1	no	—
228	C₂×C₃²	—	4	2	no	—
229	C₂×C₃²	#878	1	1	yes	characteristic
230	C₂×C₃²	#911	1	1	yes	characteristic
231	C₂×C₃²	#916	1	1	yes	characteristic
232	C₂×C₉	—	2	1	no	—
233	C₂×C₉	—	2	1	no	—
234	C₂×C₉	—	2	1	no	—
235	C₂×C₉	—	4	2	no	—
236	C₂×C₉	—	8	2	no	—
237	C₂×C₉	C₃×C₂²⋊Q₈	1	1	yes	characteristic
238	C₂×C₉	C₃×#135	1	1	yes	characteristic
239	C₂×C₉	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
240	C₂×D₄	C₂×C₃×D₉	1	1	yes	characteristic
241	C₂²×C₄	—	18	1	no	—
242	C₂×C₈	—	9	1	no	—
243	C₂×C₈	C₂×C₃×D₉	1	1	yes	characteristic
244	C₄⋊C₄	—	9	1	no	—
245	C₄⋊C₄	—	9	1	no	—
246	C₄⋊C₄	—	18	1	no	—
247	C₄⋊C₄	—	18	1	no	—
248	C₄⋊C₄	C₂×C₃×D₉	1	1	yes	characteristic
249	C₂²⋊C₄	—	18	1	no	—
250	C₄²	—	9	1	no	—
251	C₂²×C₃	—	2	2	no	—
252	C₂²×C₃	—	2	2	no	—
253	C₂²×C₃	—	2	1	no	—
254	C₂²×C₃	—	4	2	no	—
255	C₂²×C₃	—	4	1	no	—
256	C₂²×C₃	—	4	1	no	—
257	C₂²×C₃	—	4	1	no	—
258	C₂²×C₃	—	4	1	no	—
259	C₂²×C₃	—	8	1	no	—
260	C₂²×C₃	—	8	1	no	—
261	C₂²×C₃	D₄×D₉	1	1	yes	characteristic
262	C₂²×C₃	C₃×D₃×D₄	1	1	yes	characteristic
263	C₃×C₄	—	2	1	no	—
264	C₃×C₄	—	2	1	no	—
265	C₃×C₄	—	4	1	no	—
266	C₃×C₄	—	4	1	no	—
267	C₃×C₄	—	8	1	no	—
268	C₃×C₄	—	18	1	no	—
269	C₃×C₄	—	18	1	no	—
270	C₃×C₄	—	36	1	no	—
271	C₃×C₄	—	36	1	no	—
272	C₃×C₄	C₂×Dic₁₈	1	1	yes	characteristic
273	C₃×C₄	#3699	1	1	yes	characteristic
274	C₃×C₄	C₂×C₃×Dic₆	1	1	yes	characteristic
275	C₃×C₄	C₃×#237	1	1	yes	characteristic
276	Dic₃	—	6	1	no	—
277	Dic₃	—	6	1	no	—
278	Dic₃	—	12	1	no	—
279	Dic₃	—	12	1	no	—
280	C₃²	#4763	1	1	yes	characteristic
281	C₉	—	2	1	no	—
282	C₉	C₃×#478	1	1	yes	characteristic, complemented
283	C₂³	—	2	1	no	—
284	D₄	—	2	1	no	—
285	D₄	C₃×Dic₁₈	1	2	yes	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	9	1	no	—
288	C₂×C₄	—	9	1	no	—
289	C₂×C₄	—	18	1	no	—
290	C₂×C₄	—	18	1	no	—
291	C₂×C₄	—	36	2	no	—
292	C₂×C₄	C₂²×C₃×D₉	1	1	yes	characteristic
293	C₈	—	2	1	no	—
294	C₈	—	18	1	no	—
295	C₂×C₃	—	2	1	no	—
296	C₂×C₃	—	2	1	no	—
297	C₂×C₃	—	2	1	no	—
298	C₂×C₃	—	4	2	no	—
299	C₂×C₃	—	4	2	no	—
300	C₂×C₃	—	8	2	no	—
301	C₂×C₃	#63440	1	1	yes	characteristic
302	C₂×C₃	#63473	1	1	yes	characteristic
303	C₂×C₃	#63478	1	1	yes	characteristic
304	C₂×C₃	C₃×#878	1	1	yes	characteristic
305	C₂×C₃	C₃×#911	1	1	yes	characteristic
306	C₂×C₃	C₃×#916	1	1	yes	characteristic
307	C₂²	—	2	2	no	—
308	C₂²	—	4	1	no	—
309	C₂²	—	4	1	no	—
310	C₂²	C₃×D₄×D₉	1	1	yes	characteristic
311	C₄	—	4	1	no	—
312	C₄	—	18	1	no	—
313	C₄	—	18	1	no	—
314	C₄	—	36	1	no	—
315	C₄	—	36	1	no	—
316	C₄	C₂×C₃×Dic₁₈	1	1	yes	characteristic
317	C₄	C₃×#3699	1	1	yes	characteristic
318	C₃	—	2	1	no	—
319	C₃	#10588443	1	1	yes	characteristic, direct product factor
320	C₃	C₃×#4763	1	1	yes	characteristic
321	C₂	—	4	2	no	—
322	C₂	C₃×#63440	1	1	yes	characteristic
323	C₂	C₃×#63473	1	1	yes	characteristic
324	C₂	C₃×#63478	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	D₄×D₉
commutator, cosocle	C₂×C₄×C₉	C₂³×C₃
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₃×C₉×#102	C₂
socle	C₂²×C₃²	D₃×D₄
prime radical	C₂×C₃²×D₄	C₂×D₃
hypercenter	C₃×#102	D₉
hypocenter	C₉	C₃×#478
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄ ◁ C₃×#102

Complements

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

N	H ≅ G/N	type
2. C₃×#63452	C₂	semidirect
5. C₃×#63377	C₂	semidirect
6. C₃×Dic₉⋊C₈	C₂	semidirect
7. C₃×#63366	C₂	semidirect
9. #10588443	C₃	direct (N × H)
29. #63452	C₂×C₃	semidirect
33. #63377	C₂×C₃	semidirect
34. Dic₉⋊C₈	C₂×C₃	semidirect
35. #63366	C₂×C₃	semidirect
197. C₃×C₉	#478	semidirect
282. C₉	C₃×#478	semidirect
319. C₃	#10588443	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×D₄×Dic₉	2
C₃×#63452	2
C₃×C₉×#102	2
C₃×#63392	2
C₃×#63377	2
C₃×Dic₉⋊C₈	2
C₃×#63366	2
C₃×#4763	3
#10588443	3

Subgroups — C3×#10588443 #13141713

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10588443 #13141713