Subgroups — C₃×#10588527 #13142023

Order 1728.

458 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×D₇₂	C₂	1	2	yes	maximal, complemented
2	C₃×#63436	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₄×D₃₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₉×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#63380	C₂	1	2	yes	maximal, complemented
6	C₃×#4843	—	3	1	no	maximal
7	#10588527	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂×C₃×D₃₆	—	2	1	no	—
9	C₂×C₃×D₃₆	—	2	1	no	—
10	C₂×C₃×D₃₆	C₂²	1	2	yes	—
11	C₂×C₃×D₃₆	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₄×D₉	—	2	1	no	—
13	C₂×C₃×C₈×C₉	C₂²	1	2	yes	—
14	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
15	C₃×(C₂×D₉)⋊C₄	—	2	1	no	—
16	C₃×C₄⋊Dic₉	C₂²	1	1	yes	characteristic
17	C₃×D₇₂	—	2	4	no	—
18	C₂×C₃×D₂₄	—	3	2	no	—
19	C₃×#874	—	3	1	no	—
20	C₃×C₄×D₁₂	—	3	1	no	—
21	C₃²×C₄⋊C₈	D₃	1	1	yes	characteristic
22	C₃×#821	—	3	2	no	—
23	C₂×D₇₂	C₂×C₃	1	2	yes	complemented
24	#63436	C₂×C₃	1	1	yes	characteristic, complemented
25	C₄×D₃₆	C₂×C₃	1	1	yes	characteristic, complemented
26	C₉×C₄⋊C₈	—	2	1	no	—
27	C₉×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
28	#63380	C₂×C₃	1	2	yes	complemented
29	C₂²×C₃×D₉	—	2	1	no	—
30	C₂²×C₃×D₉	—	2	2	no	—
31	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
32	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
33	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
34	C₂×C₃×Dic₉	—	2	1	no	—
35	C₃×D₃₆	—	2	2	no	—
36	C₃×D₃₆	—	2	1	no	—
37	C₃×D₃₆	—	2	2	no	—
38	C₃×D₃₆	—	4	2	no	—
39	C₃×D₃₆	—	4	2	no	—
40	C₃×D₃₆	D₄	1	2	yes	complemented
41	C₃×C₄×D₉	—	4	2	no	—
42	C₃×C₈×C₉	—	2	2	no	—
43	C₃×#459	—	9	1	no	—
44	#4843	—	3	1	no	—
45	C₂×C₃×D₁₂	—	3	1	no	—
46	C₂×C₃×D₁₂	—	3	2	no	—
47	C₂×C₃×D₁₂	—	6	1	no	—
48	C₂×C₃×D₁₂	—	6	1	no	—
49	C₂×C₃×C₄×D₃	—	6	1	no	—
50	C₂×C₃²×C₈	C₂×D₃	1	2	yes	—
51	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
52	C₃×(C₂×D₃)⋊C₄	—	6	1	no	—
53	C₃×C₄⋊Dic₃	—	3	1	no	—
54	C₃×D₂₄	—	6	4	no	—
55	C₂×D₃₆	—	2	1	no	—
56	C₂×D₃₆	—	2	1	no	—
57	C₂×D₃₆	C₂²×C₃	1	2	yes	—
58	C₂×D₃₆	C₂²×C₃	1	1	yes	characteristic
59	C₂×C₄×D₉	—	2	1	no	—
60	C₂×C₈×C₉	—	2	2	no	—
61	C₂×C₈×C₉	C₂²×C₃	1	2	yes	—
62	C₄²×C₉	—	2	1	no	—
63	C₄²×C₉	C₂²×C₃	1	1	yes	characteristic
64	(C₂×D₉)⋊C₄	—	2	1	no	—
65	C₄⋊Dic₉	C₂²×C₃	1	1	yes	characteristic
66	D₇₂	—	2	4	no	—
67	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
68	C₂×C₃×D₉	—	2	2	no	—
69	C₂×C₃×D₉	—	4	1	no	—
70	C₂×C₃×D₉	—	4	1	no	—
71	C₂×C₃×D₉	—	4	2	no	—
72	C₂×C₃×D₉	—	4	2	no	—
73	C₂×C₃×D₉	—	4	2	no	—
74	C₃×C₄×C₉	—	2	1	no	—
75	C₃×C₄×C₉	D₈	1	2	yes	complemented
76	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
77	C₃×C₄×C₉	#41	1	1	yes	characteristic
78	C₃×Dic₉	—	4	1	no	—
79	C₂×C₃×D₈	—	9	2	no	—
80	C₃×#127	—	9	1	no	—
81	C₃×C₄×D₄	—	9	1	no	—
82	C₂×D₂₄	—	3	2	no	—
83	#874	—	3	1	no	—
84	C₄×D₁₂	—	3	1	no	—
85	C₃×C₄⋊C₈	—	2	1	no	—
86	C₃×C₄⋊C₈	D₉	1	1	yes	characteristic, complemented
87	C₃×C₄⋊C₈	C₃×D₃	1	1	yes	characteristic
88	C₃×#102	—	9	2	no	—
89	#821	—	3	2	no	—
90	C₂²×C₃×D₃	—	6	1	no	—
91	C₂²×C₃×D₃	—	6	2	no	—
92	C₂×C₃²×C₄	D₁₂	1	1	yes	characteristic
93	C₂×C₃²×C₄	D₁₂	1	1	yes	characteristic
94	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
95	C₂×C₃×Dic₃	—	6	1	no	—
96	C₃×D₁₂	—	3	2	no	—
97	C₃×D₁₂	—	6	2	no	—
98	C₃×D₁₂	—	6	1	no	—
99	C₃×D₁₂	—	6	2	no	—
100	C₃×D₁₂	—	12	2	no	—
101	C₃×D₁₂	—	12	2	no	—
102	C₃×C₄×D₃	—	12	2	no	—
103	C₂²×D₉	—	2	1	no	—
104	C₂²×D₉	—	2	2	no	—
105	C₃²×C₈	—	2	2	no	—
106	C₂×C₄×C₉	—	2	1	no	—
107	C₂×C₄×C₉	—	2	1	no	—
108	C₂×C₄×C₉	—	2	1	no	—
109	C₂×C₄×C₉	C₃×D₄	1	1	yes	characteristic
110	C₂×C₄×C₉	C₃×D₄	1	1	yes	characteristic
111	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
112	C₂×Dic₉	—	2	1	no	—
113	D₃₆	—	2	2	no	—
114	D₃₆	—	2	1	no	—
115	D₃₆	—	2	2	no	—
116	D₃₆	—	4	2	no	—
117	D₃₆	—	4	2	no	—
118	D₃₆	C₃×D₄	1	2	yes	complemented
119	C₄×D₉	—	4	2	no	—
120	C₈×C₉	—	2	2	no	—
121	C₈×C₉	—	4	2	no	—
122	#459	—	9	1	no	Sylow
123	C₂×C₃×C₉	#121	1	1	yes	characteristic
124	C₂×C₃×C₉	C₂×D₈	1	1	yes	characteristic
125	C₂×C₃×C₉	AGL(1,ℤ/8)	1	1	yes	characteristic
126	C₃×D₉	—	4	2	no	—
127	C₃×D₉	—	8	2	no	—
128	C₂×C₃×D₄	—	9	1	no	—
129	C₂×C₃×D₄	—	9	2	no	—
130	C₂×C₃×D₄	—	18	1	no	—
131	C₂×C₃×D₄	—	18	1	no	—
132	C₂²×C₃×C₄	—	18	1	no	—
133	C₂×D₁₂	—	3	1	no	—
134	C₂×D₁₂	—	3	2	no	—
135	C₂×D₁₂	—	6	1	no	—
136	C₂×D₁₂	—	6	1	no	—
137	C₂×C₄×D₃	—	6	1	no	—
138	C₃×D₈	—	18	4	no	—
139	C₂×C₃×C₈	—	2	2	no	—
140	C₂×C₃×C₈	C₂×D₉	1	2	yes	—
141	C₂×C₃×C₈	C₂×C₃×D₃	1	2	yes	—
142	C₃×C₄⋊C₄	—	9	1	no	—
143	C₃×C₂²⋊C₄	—	18	1	no	—
144	C₃×C₄²	—	2	1	no	—
145	C₃×C₄²	C₂×D₉	1	1	yes	characteristic
146	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
147	(C₂×D₃)⋊C₄	—	6	1	no	—
148	C₄⋊Dic₃	—	3	1	no	—
149	D₂₄	—	6	4	no	—
150	C₂²×C₃²	C₂×D₁₂	1	1	yes	characteristic
151	C₂×C₃×D₃	—	6	2	no	—
152	C₂×C₃×D₃	—	12	1	no	—
153	C₂×C₃×D₃	—	12	1	no	—
154	C₂×C₃×D₃	—	12	2	no	—
155	C₂×C₃×D₃	—	12	2	no	—
156	C₂×C₃×D₃	—	12	2	no	—
157	C₃²×C₄	—	2	1	no	—
158	C₃²×C₄	D₂₄	1	2	yes	—
159	C₃²×C₄	D₃×D₄	1	1	yes	characteristic
160	C₃²×C₄	#239	1	1	yes	characteristic
161	C₃×Dic₃	—	12	1	no	—
162	C₂²×C₉	—	2	1	no	—
163	C₂²×C₉	C₂×C₃×D₄	1	1	yes	characteristic
164	C₂×D₉	—	2	2	no	—
165	C₂×D₉	—	4	1	no	—
166	C₂×D₉	—	4	1	no	—
167	C₂×D₉	—	4	2	no	—
168	C₂×D₉	—	4	2	no	—
169	C₂×D₉	—	4	2	no	—
170	C₄×C₉	—	2	1	no	—
171	C₄×C₉	—	2	2	no	—
172	C₄×C₉	—	2	1	no	—
173	C₄×C₉	—	2	1	no	—
174	C₄×C₉	—	4	1	no	—
175	C₄×C₉	C₃×D₈	1	2	yes	complemented
176	C₄×C₉	C₂×C₃×D₄	1	1	yes	characteristic
177	C₄×C₉	C₃×#41	1	1	yes	characteristic
178	Dic₉	—	4	1	no	—
179	C₂×D₈	—	9	2	no	—
180	#127	—	9	1	no	—
181	C₄×D₄	—	9	1	no	—
182	C₄⋊C₈	C₃×D₉	1	1	yes	characteristic, complemented
183	#102	—	9	2	no	—
184	C₃×C₉	#459	1	1	yes	characteristic, complemented, Sylow
185	C₂³×C₃	—	18	1	no	—
186	C₂³×C₃	—	18	2	no	—
187	C₂²×D₃	—	6	1	no	—
188	C₂²×D₃	—	6	2	no	—
189	C₃×D₄	—	9	2	no	—
190	C₃×D₄	—	18	2	no	—
191	C₃×D₄	—	18	1	no	—
192	C₃×D₄	—	18	2	no	—
193	C₃×D₄	—	36	2	no	—
194	C₃×D₄	—	36	2	no	—
195	C₂×C₃×C₄	—	2	1	no	—
196	C₂×C₃×C₄	—	2	1	no	—
197	C₂×C₃×C₄	—	2	1	no	—
198	C₂×C₃×C₄	—	18	1	no	—
199	C₂×C₃×C₄	—	36	2	no	—
200	C₂×C₃×C₄	D₃₆	1	1	yes	characteristic
201	C₂×C₃×C₄	D₃₆	1	1	yes	characteristic
202	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
203	C₂×C₃×C₄	C₃×D₁₂	1	1	yes	characteristic
204	C₂×C₃×C₄	C₃×D₁₂	1	1	yes	characteristic
205	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
206	C₂×Dic₃	—	6	1	no	—
207	D₁₂	—	3	2	no	—
208	D₁₂	—	6	2	no	—
209	D₁₂	—	6	1	no	—
210	D₁₂	—	6	2	no	—
211	D₁₂	—	12	2	no	—
212	D₁₂	—	12	2	no	—
213	C₄×D₃	—	12	2	no	—
214	C₃×C₈	—	2	2	no	—
215	C₃×C₈	—	2	2	no	—
216	C₃×C₈	—	4	2	no	—
217	C₂×C₃²	#895	1	1	yes	characteristic
218	C₂×C₃²	C₂×D₂₄	1	1	yes	characteristic
219	C₂×C₃²	#908	1	1	yes	characteristic
220	C₃×D₃	—	12	2	no	—
221	C₃×D₃	—	24	2	no	—
222	C₂×C₉	—	2	1	no	—
223	C₂×C₉	—	2	1	no	—
224	C₂×C₉	—	2	1	no	—
225	C₂×C₉	C₃×#121	1	1	yes	characteristic
226	C₂×C₉	C₂×C₃×D₈	1	1	yes	characteristic
227	C₂×C₉	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
228	D₉	—	4	2	no	—
229	D₉	—	8	2	no	—
230	C₂×D₄	—	9	1	no	—
231	C₂×D₄	—	9	2	no	—
232	C₂×D₄	—	18	1	no	—
233	C₂×D₄	—	18	1	no	—
234	C₂²×C₄	—	18	1	no	—
235	D₈	—	18	4	no	—
236	C₂×C₈	C₂×C₃×D₉	1	2	yes	—
237	C₄⋊C₄	—	9	1	no	—
238	C₂²⋊C₄	—	18	1	no	—
239	C₄²	C₂×C₃×D₉	1	1	yes	characteristic
240	C₂²×C₃	—	2	1	no	—
241	C₂²×C₃	—	18	2	no	—
242	C₂²×C₃	—	36	1	no	—
243	C₂²×C₃	—	36	1	no	—
244	C₂²×C₃	—	36	2	no	—
245	C₂²×C₃	—	36	2	no	—
246	C₂²×C₃	—	36	2	no	—
247	C₂²×C₃	C₂×D₃₆	1	1	yes	characteristic
248	C₂²×C₃	C₂×C₃×D₁₂	1	1	yes	characteristic
249	C₂×D₃	—	6	2	no	—
250	C₂×D₃	—	12	1	no	—
251	C₂×D₃	—	12	1	no	—
252	C₂×D₃	—	12	2	no	—
253	C₂×D₃	—	12	2	no	—
254	C₂×D₃	—	12	2	no	—
255	C₃×C₄	—	2	1	no	—
256	C₃×C₄	—	2	1	no	—
257	C₃×C₄	—	2	1	no	—
258	C₃×C₄	—	2	2	no	—
259	C₃×C₄	—	2	1	no	—
260	C₃×C₄	—	4	1	no	—
261	C₃×C₄	—	36	1	no	—
262	C₃×C₄	D₇₂	1	2	yes	complemented
263	C₃×C₄	D₄×D₉	1	1	yes	characteristic
264	C₃×C₄	#3701	1	1	yes	characteristic
265	C₃×C₄	C₃×D₂₄	1	2	yes	—
266	C₃×C₄	C₃×D₃×D₄	1	1	yes	characteristic
267	C₃×C₄	C₃×#239	1	1	yes	characteristic
268	Dic₃	—	12	1	no	—
269	C₃²	#4843	1	1	yes	characteristic
270	C₉	—	2	1	no	—
271	C₉	C₃×#459	1	1	yes	characteristic, complemented
272	C₂³	—	18	1	no	—
273	C₂³	—	18	2	no	—
274	D₄	—	9	2	no	—
275	D₄	—	18	2	no	—
276	D₄	—	18	1	no	—
277	D₄	—	18	2	no	—
278	D₄	—	36	2	no	—
279	D₄	—	36	2	no	—
280	C₂×C₄	—	18	1	no	—
281	C₂×C₄	—	36	2	no	—
282	C₂×C₄	C₃×D₃₆	1	1	yes	characteristic
283	C₂×C₄	C₃×D₃₆	1	1	yes	characteristic
284	C₂×C₄	C₂²×C₃×D₉	1	1	yes	characteristic
285	C₈	—	2	2	no	—
286	C₂×C₃	—	2	1	no	—
287	C₂×C₃	—	2	1	no	—
288	C₂×C₃	—	2	1	no	—
289	C₂×C₃	—	36	2	no	—
290	C₂×C₃	—	72	2	no	—
291	C₂×C₃	#63457	1	1	yes	characteristic
292	C₂×C₃	C₂×D₇₂	1	1	yes	characteristic
293	C₂×C₃	#63470	1	1	yes	characteristic
294	C₂×C₃	C₃×#895	1	1	yes	characteristic
295	C₂×C₃	C₂×C₃×D₂₄	1	1	yes	characteristic
296	C₂×C₃	C₃×#908	1	1	yes	characteristic
297	D₃	—	12	2	no	—
298	D₃	—	24	2	no	—
299	C₂²	—	18	2	no	—
300	C₂²	—	36	1	no	—
301	C₂²	—	36	1	no	—
302	C₂²	—	36	2	no	—
303	C₂²	—	36	2	no	—
304	C₂²	—	36	2	no	—
305	C₂²	C₂×C₃×D₃₆	1	1	yes	characteristic
306	C₄	—	2	1	no	—
307	C₄	—	36	1	no	—
308	C₄	C₃×D₇₂	1	2	yes	complemented
309	C₄	C₃×D₄×D₉	1	1	yes	characteristic
310	C₄	C₃×#3701	1	1	yes	characteristic
311	C₃	—	2	1	no	—
312	C₃	#10588527	1	1	yes	characteristic, direct product factor
313	C₃	C₃×#4843	1	1	yes	characteristic
314	C₂	—	36	2	no	—
315	C₂	—	72	2	no	—
316	C₂	C₃×#63457	1	1	yes	characteristic
317	C₂	C₂×C₃×D₇₂	1	1	yes	characteristic
318	C₂	C₃×#63470	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂×D₃₆
commutator, cosocle	C₂×C₄×C₉	C₂³×C₃
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₃×C₉×C₄⋊C₈	C₂
socle	C₂²×C₃²	C₂×D₁₂
hypercenter	C₃×C₄⋊C₈	D₉
hypocenter	C₉	C₃×#459
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
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
1. C₂×C₃×D₇₂	C₂	semidirect
2. C₃×#63436	C₂	semidirect
3. C₃×C₄×D₃₆	C₂	semidirect
4. C₃×C₉×C₄⋊C₈	C₂	semidirect
5. C₃×#63380	C₂	semidirect
7. #10588527	C₃	direct (N × H)
23. C₂×D₇₂	C₂×C₃	semidirect
24. #63436	C₂×C₃	semidirect
25. C₄×D₃₆	C₂×C₃	semidirect
27. C₉×C₄⋊C₈	C₂×C₃	semidirect
28. #63380	C₂×C₃	semidirect
40. C₃×D₃₆	D₄	semidirect
75. C₃×C₄×C₉	D₈	semidirect
86. C₃×C₄⋊C₈	D₉	semidirect
118. D₃₆	C₃×D₄	semidirect
175. C₄×C₉	C₃×D₈	semidirect
182. C₄⋊C₈	C₃×D₉	semidirect
184. C₃×C₉	#459	semidirect
262. C₃×C₄	D₇₂	semidirect
271. C₉	C₃×#459	semidirect
308. C₄	C₃×D₇₂	semidirect
312. C₃	#10588527	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×D₇₂	2
C₃×#63436	2
C₃×C₄×D₃₆	2
C₃×C₉×C₄⋊C₈	2
C₃×#63380	2
C₃×#4843	3
#10588527	3

Subgroups — C3×#10588527 #13142023

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10588527 #13142023