Subgroups — C₃×#10588512 #13142010

Order 1728.

422 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#3664	C₂	1	2	yes	maximal, complemented
2	C₃×#63436	C₂	1	1	yes	maximal, characteristic
3	C₃×C₄×Dic₁₈	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×#4848	—	3	1	no	maximal
7	#10588512	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×Dic₁₈	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₈×C₉	C₂²	1	2	yes	—
13	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
14	C₃×C₄⋊Dic₉	C₂²	1	1	yes	characteristic
15	C₃×Dic₉⋊C₄	—	2	1	no	—
16	C₃×C₄×Dic₉	—	2	1	no	—
17	C₃×#3664	—	2	4	no	—
18	C₂×C₃×#204	—	3	2	no	—
19	C₃×#874	—	3	1	no	—
20	C₃×C₄×Dic₆	—	3	1	no	—
21	C₃²×C₄⋊C₈	D₃	1	1	yes	characteristic
22	C₃×#821	—	3	2	no	—
23	C₂×#3664	C₂×C₃	1	2	yes	complemented
24	#63436	C₂×C₃	1	1	yes	characteristic
25	C₄×Dic₁₈	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	2	no	—
30	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
31	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
32	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
33	C₂×C₃×Dic₉	—	2	1	no	—
34	C₂×C₃×Dic₉	—	2	1	no	—
35	C₃×D₃₆	—	2	2	no	—
36	C₃×D₃₆	—	2	2	no	—
37	C₃×D₃₆	—	4	2	no	—
38	C₃×D₃₆	—	4	2	no	—
39	C₃×Dic₁₈	—	2	1	no	—
40	C₃×Dic₁₈	D₄	1	2	yes	complemented
41	C₃×C₈×C₉	—	2	2	no	—
42	C₃×#460	—	9	1	no	—
43	#4848	—	3	1	no	—
44	C₂×C₃×D₁₂	—	3	2	no	—
45	C₂×C₃×D₁₂	—	6	1	no	—
46	C₂×C₃×D₁₂	—	6	1	no	—
47	C₂×C₃×Dic₆	—	3	1	no	—
48	C₂×C₃²×C₈	C₂×D₃	1	2	yes	—
49	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
50	C₃×C₄⋊Dic₃	—	3	1	no	—
51	C₃×Dic₃⋊C₄	—	6	1	no	—
52	C₃×C₄×Dic₃	—	6	1	no	—
53	C₃×#204	—	6	4	no	—
54	C₂×D₃₆	—	2	1	no	—
55	C₂×D₃₆	—	2	1	no	—
56	C₂×D₃₆	C₂²×C₃	1	2	yes	—
57	C₂×Dic₁₈	C₂²×C₃	1	1	yes	characteristic
58	C₂×C₈×C₉	—	2	2	no	—
59	C₂×C₈×C₉	C₂²×C₃	1	2	yes	—
60	C₄²×C₉	—	2	1	no	—
61	C₄²×C₉	C₂²×C₃	1	1	yes	characteristic
62	C₄⋊Dic₉	C₂²×C₃	1	1	yes	characteristic
63	Dic₉⋊C₄	—	2	1	no	—
64	C₄×Dic₉	—	2	1	no	—
65	#3664	—	2	4	no	—
66	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
67	C₂×C₃×D₉	—	4	2	no	—
68	C₂×C₃×D₉	—	4	2	no	—
69	C₂×C₃×D₉	—	4	2	no	—
70	C₃×C₄×C₉	—	2	1	no	—
71	C₃×C₄×C₉	SD₁₆	1	2	yes	complemented
72	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
73	C₃×C₄×C₉	#41	1	1	yes	characteristic
74	C₃×Dic₉	—	2	2	no	—
75	C₃×Dic₉	—	4	1	no	—
76	C₂×C₃×SD₁₆	—	9	2	no	—
77	C₃×#127	—	9	1	no	—
78	C₃×C₄×Q₈	—	9	1	no	—
79	C₂×#204	—	3	2	no	—
80	#874	—	3	1	no	—
81	C₄×Dic₆	—	3	1	no	—
82	C₃×C₄⋊C₈	—	2	1	no	—
83	C₃×C₄⋊C₈	D₉	1	1	yes	characteristic, complemented
84	C₃×C₄⋊C₈	C₃×D₃	1	1	yes	characteristic
85	C₃×#102	—	9	2	no	—
86	#821	—	3	2	no	—
87	C₂²×C₃×D₃	—	6	2	no	—
88	C₂×C₃²×C₄	D₁₂	1	1	yes	characteristic
89	C₂×C₃²×C₄	D₁₂	1	1	yes	characteristic
90	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
91	C₂×C₃×Dic₃	—	6	1	no	—
92	C₂×C₃×Dic₃	—	6	1	no	—
93	C₃×D₁₂	—	6	2	no	—
94	C₃×D₁₂	—	6	2	no	—
95	C₃×D₁₂	—	12	2	no	—
96	C₃×D₁₂	—	12	2	no	—
97	C₃×Dic₆	—	3	2	no	—
98	C₃×Dic₆	—	6	1	no	—
99	C₂²×D₉	—	2	2	no	—
100	C₃²×C₈	—	2	2	no	—
101	C₂×C₄×C₉	—	2	1	no	—
102	C₂×C₄×C₉	—	2	1	no	—
103	C₂×C₄×C₉	—	2	1	no	—
104	C₂×C₄×C₉	C₃×D₄	1	1	yes	characteristic
105	C₂×C₄×C₉	C₃×D₄	1	1	yes	characteristic
106	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
107	C₂×Dic₉	—	2	1	no	—
108	C₂×Dic₉	—	2	1	no	—
109	D₃₆	—	2	2	no	—
110	D₃₆	—	2	2	no	—
111	D₃₆	—	4	2	no	—
112	D₃₆	—	4	2	no	—
113	Dic₁₈	—	2	1	no	—
114	Dic₁₈	C₃×D₄	1	2	yes	complemented
115	C₈×C₉	—	2	2	no	—
116	C₈×C₉	—	4	2	no	—
117	#460	—	9	1	no	Sylow
118	C₂×C₃×C₉	#121	1	1	yes	characteristic
119	C₂×C₃×C₉	C₂×SD₁₆	1	1	yes	characteristic
120	C₂×C₃×C₉	AGL(1,ℤ/8)	1	1	yes	characteristic
121	C₃×D₉	—	8	2	no	—
122	C₂×C₃×Q₈	—	9	1	no	—
123	C₂×C₃×D₄	—	9	2	no	—
124	C₂×C₃×D₄	—	18	1	no	—
125	C₂×C₃×D₄	—	18	1	no	—
126	C₂×D₁₂	—	3	2	no	—
127	C₂×D₁₂	—	6	1	no	—
128	C₂×D₁₂	—	6	1	no	—
129	C₂×Dic₆	—	3	1	no	—
130	C₃×SD₁₆	—	18	4	no	—
131	C₂×C₃×C₈	—	2	2	no	—
132	C₂×C₃×C₈	C₂×D₉	1	2	yes	—
133	C₂×C₃×C₈	C₂×C₃×D₃	1	2	yes	—
134	C₃×C₄⋊C₄	—	9	1	no	—
135	C₃×C₄⋊C₄	—	18	1	no	—
136	C₃×C₄²	—	2	1	no	—
137	C₃×C₄²	—	18	1	no	—
138	C₃×C₄²	C₂×D₉	1	1	yes	characteristic
139	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
140	C₄⋊Dic₃	—	3	1	no	—
141	Dic₃⋊C₄	—	6	1	no	—
142	C₄×Dic₃	—	6	1	no	—
143	#204	—	6	4	no	—
144	C₂²×C₃²	C₂×D₁₂	1	1	yes	characteristic
145	C₂×C₃×D₃	—	12	2	no	—
146	C₂×C₃×D₃	—	12	2	no	—
147	C₂×C₃×D₃	—	12	2	no	—
148	C₃²×C₄	—	2	1	no	—
149	C₃²×C₄	#204	1	2	yes	—
150	C₃²×C₄	D₃×D₄	1	1	yes	characteristic
151	C₃²×C₄	#239	1	1	yes	characteristic
152	C₃×Dic₃	—	6	2	no	—
153	C₃×Dic₃	—	12	1	no	—
154	C₂²×C₉	—	2	1	no	—
155	C₂²×C₉	C₂×C₃×D₄	1	1	yes	characteristic
156	C₂×D₉	—	4	2	no	—
157	C₂×D₉	—	4	2	no	—
158	C₂×D₉	—	4	2	no	—
159	C₄×C₉	—	2	1	no	—
160	C₄×C₉	—	2	2	no	—
161	C₄×C₉	—	2	1	no	—
162	C₄×C₉	—	2	1	no	—
163	C₄×C₉	—	4	1	no	—
164	C₄×C₉	C₃×SD₁₆	1	2	yes	complemented
165	C₄×C₉	C₂×C₃×D₄	1	1	yes	characteristic
166	C₄×C₉	C₃×#41	1	1	yes	characteristic
167	Dic₉	—	2	2	no	—
168	Dic₉	—	4	1	no	—
169	C₂×SD₁₆	—	9	2	no	—
170	#127	—	9	1	no	—
171	C₄×Q₈	—	9	1	no	—
172	C₄⋊C₈	C₃×D₉	1	1	yes	characteristic, complemented
173	#102	—	9	2	no	—
174	C₃×C₉	#460	1	1	yes	characteristic, complemented, Sylow
175	C₂³×C₃	—	18	2	no	—
176	C₂²×D₃	—	6	2	no	—
177	C₃×Q₈	—	9	2	no	—
178	C₃×Q₈	—	18	1	no	—
179	C₃×D₄	—	18	2	no	—
180	C₃×D₄	—	18	2	no	—
181	C₃×D₄	—	36	2	no	—
182	C₃×D₄	—	36	2	no	—
183	C₂×C₃×C₄	—	2	1	no	—
184	C₂×C₃×C₄	—	2	1	no	—
185	C₂×C₃×C₄	—	2	1	no	—
186	C₂×C₃×C₄	—	18	1	no	—
187	C₂×C₃×C₄	—	18	1	no	—
188	C₂×C₃×C₄	D₃₆	1	1	yes	characteristic
189	C₂×C₃×C₄	D₃₆	1	1	yes	characteristic
190	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
191	C₂×C₃×C₄	C₃×D₁₂	1	1	yes	characteristic
192	C₂×C₃×C₄	C₃×D₁₂	1	1	yes	characteristic
193	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
194	C₂×Dic₃	—	6	1	no	—
195	C₂×Dic₃	—	6	1	no	—
196	D₁₂	—	6	2	no	—
197	D₁₂	—	6	2	no	—
198	D₁₂	—	12	2	no	—
199	D₁₂	—	12	2	no	—
200	Dic₆	—	3	2	no	—
201	Dic₆	—	6	1	no	—
202	C₃×C₈	—	2	2	no	—
203	C₃×C₈	—	2	2	no	—
204	C₃×C₈	—	4	2	no	—
205	C₂×C₃²	#895	1	1	yes	characteristic
206	C₂×C₃²	C₂×#204	1	1	yes	characteristic
207	C₂×C₃²	#908	1	1	yes	characteristic
208	C₃×D₃	—	24	2	no	—
209	C₂×C₉	—	2	1	no	—
210	C₂×C₉	—	2	1	no	—
211	C₂×C₉	—	2	1	no	—
212	C₂×C₉	C₃×#121	1	1	yes	characteristic
213	C₂×C₉	C₂×C₃×SD₁₆	1	1	yes	characteristic
214	C₂×C₉	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
215	D₉	—	8	2	no	—
216	C₂×Q₈	—	9	1	no	—
217	C₂×D₄	—	9	2	no	—
218	C₂×D₄	—	18	1	no	—
219	C₂×D₄	—	18	1	no	—
220	SD₁₆	—	18	4	no	—
221	C₂×C₈	C₂×C₃×D₉	1	2	yes	—
222	C₄⋊C₄	—	9	1	no	—
223	C₄⋊C₄	—	18	1	no	—
224	C₄²	—	18	1	no	—
225	C₄²	C₂×C₃×D₉	1	1	yes	characteristic
226	C₂²×C₃	—	2	1	no	—
227	C₂²×C₃	—	36	2	no	—
228	C₂²×C₃	—	36	2	no	—
229	C₂²×C₃	—	36	2	no	—
230	C₂²×C₃	C₂×D₃₆	1	1	yes	characteristic
231	C₂²×C₃	C₂×C₃×D₁₂	1	1	yes	characteristic
232	C₂×D₃	—	12	2	no	—
233	C₂×D₃	—	12	2	no	—
234	C₂×D₃	—	12	2	no	—
235	C₃×C₄	—	2	1	no	—
236	C₃×C₄	—	2	1	no	—
237	C₃×C₄	—	2	1	no	—
238	C₃×C₄	—	2	2	no	—
239	C₃×C₄	—	2	1	no	—
240	C₃×C₄	—	4	1	no	—
241	C₃×C₄	—	18	2	no	—
242	C₃×C₄	—	36	1	no	—
243	C₃×C₄	#3664	1	2	yes	complemented
244	C₃×C₄	D₄×D₉	1	1	yes	characteristic
245	C₃×C₄	#3701	1	1	yes	characteristic
246	C₃×C₄	C₃×#204	1	2	yes	—
247	C₃×C₄	C₃×D₃×D₄	1	1	yes	characteristic
248	C₃×C₄	C₃×#239	1	1	yes	characteristic
249	Dic₃	—	6	2	no	—
250	Dic₃	—	12	1	no	—
251	C₃²	#4848	1	1	yes	characteristic
252	C₉	—	2	1	no	—
253	C₉	C₃×#460	1	1	yes	characteristic, complemented
254	C₂³	—	18	2	no	—
255	Q₈	—	9	2	no	—
256	Q₈	—	18	1	no	—
257	D₄	—	18	2	no	—
258	D₄	—	18	2	no	—
259	D₄	—	36	2	no	—
260	D₄	—	36	2	no	—
261	C₂×C₄	—	18	1	no	—
262	C₂×C₄	—	18	1	no	—
263	C₂×C₄	C₃×D₃₆	1	1	yes	characteristic
264	C₂×C₄	C₃×D₃₆	1	1	yes	characteristic
265	C₂×C₄	C₂²×C₃×D₉	1	1	yes	characteristic
266	C₈	—	2	2	no	—
267	C₂×C₃	—	2	1	no	—
268	C₂×C₃	—	2	1	no	—
269	C₂×C₃	—	2	1	no	—
270	C₂×C₃	—	72	2	no	—
271	C₂×C₃	#63457	1	1	yes	characteristic
272	C₂×C₃	C₂×#3664	1	1	yes	characteristic
273	C₂×C₃	#63470	1	1	yes	characteristic
274	C₂×C₃	C₃×#895	1	1	yes	characteristic
275	C₂×C₃	C₂×C₃×#204	1	1	yes	characteristic
276	C₂×C₃	C₃×#908	1	1	yes	characteristic
277	D₃	—	24	2	no	—
278	C₂²	—	36	2	no	—
279	C₂²	—	36	2	no	—
280	C₂²	—	36	2	no	—
281	C₂²	C₂×C₃×D₃₆	1	1	yes	characteristic
282	C₄	—	2	1	no	—
283	C₄	—	18	2	no	—
284	C₄	—	36	1	no	—
285	C₄	C₃×#3664	1	2	yes	complemented
286	C₄	C₃×D₄×D₉	1	1	yes	characteristic
287	C₄	C₃×#3701	1	1	yes	characteristic
288	C₃	—	2	1	no	—
289	C₃	#10588512	1	1	yes	characteristic, direct product factor
290	C₃	C₃×#4848	1	1	yes	characteristic
291	C₂	—	72	2	no	—
292	C₂	C₃×#63457	1	1	yes	characteristic
293	C₂	C₂×C₃×#3664	1	1	yes	characteristic
294	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
prime radical	C₃×#63436	C₂
hypercenter	C₃×C₄⋊C₈	D₉
hypocenter	C₉	C₃×#460
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×#3664	C₂	semidirect
3. C₃×C₄×Dic₁₈	C₂	semidirect
4. C₃×C₉×C₄⋊C₈	C₂	semidirect
5. C₃×#63380	C₂	semidirect
7. #10588512	C₃	direct (N × H)
23. C₂×#3664	C₂×C₃	semidirect
25. C₄×Dic₁₈	C₂×C₃	semidirect
27. C₉×C₄⋊C₈	C₂×C₃	semidirect
28. #63380	C₂×C₃	semidirect
40. C₃×Dic₁₈	D₄	semidirect
71. C₃×C₄×C₉	SD₁₆	semidirect
83. C₃×C₄⋊C₈	D₉	semidirect
114. Dic₁₈	C₃×D₄	semidirect
164. C₄×C₉	C₃×SD₁₆	semidirect
172. C₄⋊C₈	C₃×D₉	semidirect
174. C₃×C₉	#460	semidirect
243. C₃×C₄	#3664	semidirect
253. C₉	C₃×#460	semidirect
285. C₄	C₃×#3664	semidirect
289. C₃	#10588512	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×#3664	2
C₃×#63436	2
C₃×C₄×Dic₁₈	2
C₃×C₉×C₄⋊C₈	2
C₃×#63380	2
C₃×#4848	3
#10588512	3

Subgroups — C3×#10588512 #13142010

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10588512 #13142010