Subgroups — C₃×#10588782 #13143076

Order 1728.

466 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₈×C₉	C₂	1	1	yes	maximal, characteristic
2	C₃×#63488	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#3667	C₂	1	1	yes	maximal, characteristic
4	C₃×#63375	C₂	1	2	yes	maximal, complemented
5	C₃×C₈×Dic₉	C₂	1	2	yes	maximal, complemented
6	C₃×#5101	—	3	1	no	maximal
7	#10588782	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
9	C₂×C₃×C₈×C₉	C₄	1	4	yes	complemented
10	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
12	C₃×#3676	C₄	1	2	yes	—
13	C₃×C₄⋊Dic₉	C₄	1	2	yes	—
14	C₃×C₄×Dic₉	C₂²	1	2	yes	—
15	C₃×#3667	C₄	1	4	yes	complemented
16	C₂×C₃×C₉⋊C₈	C₂²	1	2	yes	—
17	C₂²×C₃²×C₈	D₃	1	1	yes	characteristic
18	C₃×#926	—	3	1	no	—
19	C₂×C₃×#208	—	3	1	no	—
20	C₃×#815	—	3	2	no	—
21	C₃×C₈×Dic₃	—	3	2	no	—
22	C₂²×C₈×C₉	—	2	1	no	—
23	C₂²×C₈×C₉	C₂×C₃	1	1	yes	characteristic
24	#63488	C₂×C₃	1	1	yes	characteristic
25	C₂×#3667	C₂×C₃	1	1	yes	characteristic
26	#63375	C₂×C₃	1	2	yes	complemented
27	C₈×Dic₉	C₂×C₃	1	2	yes	complemented
28	C₂³×C₃×C₉	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
31	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
32	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
33	C₂×C₃×Dic₉	C₂×C₄	1	4	yes	—
34	C₃×C₈×C₉	C₂×C₄	1	2	yes	—
35	C₃×C₈×C₉	C₂×C₄	1	2	yes	—
36	C₃×C₉⋊C₈	C₂×C₄	1	4	yes	—
37	C₃×#405	—	9	1	no	—
38	#5101	—	3	1	no	—
39	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
40	C₂×C₃²×C₈	Dic₃	1	4	yes	—
41	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
42	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
43	C₃×#217	—	3	2	no	—
44	C₃×C₄⋊Dic₃	—	3	2	no	—
45	C₃×C₄×Dic₃	—	3	2	no	—
46	C₃×#208	—	3	4	no	—
47	C₂×C₃×C₃⋊C₈	—	3	2	no	—
48	C₂²×C₄×C₉	—	2	1	no	—
49	C₂²×C₄×C₉	C₂²×C₃	1	1	yes	characteristic
50	C₂×C₈×C₉	—	2	4	no	—
51	C₂×C₈×C₉	—	2	1	no	—
52	C₂×C₈×C₉	—	2	1	no	—
53	C₂×C₈×C₉	C₃×C₄	1	4	yes	complemented
54	C₂×C₈×C₉	C₂²×C₃	1	1	yes	characteristic
55	C₂×C₈×C₉	C₂²×C₃	1	1	yes	characteristic
56	#3676	C₃×C₄	1	2	yes	—
57	C₄⋊Dic₉	C₃×C₄	1	2	yes	—
58	C₄×Dic₉	C₂²×C₃	1	2	yes	—
59	#3667	C₃×C₄	1	4	yes	complemented
60	C₂×C₉⋊C₈	C₂²×C₃	1	2	yes	—
61	C₂²×C₃×C₉	—	2	2	no	—
62	C₂²×C₃×C₉	C₄²	1	2	yes	—
63	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
64	C₃×C₄×C₉	C₄²	1	2	yes	—
65	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
66	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
67	C₃×Dic₉	—	2	4	no	—
68	C₂×C₃×M₁₆	—	9	1	no	—
69	C₂²×C₃×C₈	—	2	1	no	—
70	C₂²×C₃×C₈	D₉	1	1	yes	characteristic
71	C₂²×C₃×C₈	C₃×D₃	1	1	yes	characteristic
72	C₃×#117	—	9	1	no	—
73	#926	—	3	1	no	—
74	C₂×#208	—	3	1	no	—
75	C₃×#97	—	9	2	no	—
76	C₃×C₄×C₈	—	9	2	no	—
77	#815	—	3	2	no	—
78	C₈×Dic₃	—	3	2	no	—
79	C₂³×C₃²	C₄×D₃	1	1	yes	characteristic
80	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
81	C₂×C₃²×C₄	C₄×D₃	1	2	yes	—
82	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
83	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
84	C₂×C₃×Dic₃	—	3	4	no	—
85	C₂³×C₉	—	2	1	no	—
86	C₂³×C₉	C₂×C₃×C₄	1	1	yes	characteristic
87	C₃²×C₈	C₂×Dic₃	1	2	yes	—
88	C₃²×C₈	C₂×Dic₃	1	2	yes	—
89	C₃×C₃⋊C₈	—	3	4	no	—
90	C₂×C₄×C₉	—	2	1	no	—
91	C₂×C₄×C₉	—	2	2	no	—
92	C₂×C₄×C₉	—	2	2	no	—
93	C₂×C₄×C₉	—	2	1	no	—
94	C₂×C₄×C₉	C₂×C₃×C₄	1	1	yes	characteristic
95	C₂×C₄×C₉	C₂×C₃×C₄	1	2	yes	—
96	C₂×C₄×C₉	C₂×C₃×C₄	1	2	yes	—
97	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
98	C₂×Dic₉	C₂×C₃×C₄	1	4	yes	—
99	C₈×C₉	—	2	2	no	—
100	C₈×C₉	—	2	2	no	—
101	C₈×C₉	C₂×C₃×C₄	1	2	yes	—
102	C₈×C₉	C₂×C₃×C₄	1	2	yes	—
103	C₉⋊C₈	C₂×C₃×C₄	1	4	yes	—
104	#405	—	9	1	no	Sylow
105	C₂×C₃×C₉	—	2	2	no	—
106	C₂×C₃×C₉	C₂×C₄²	1	1	yes	characteristic
107	C₂×C₃×C₉	#131	1	2	yes	—
108	C₂²×C₃×C₄	—	2	1	no	—
109	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
110	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
111	C₃×M₁₆	—	9	4	no	—
112	C₂×C₃×C₈	—	2	4	no	—
113	C₂×C₃×C₈	—	2	1	no	—
114	C₂×C₃×C₈	—	2	1	no	—
115	C₂×C₃×C₈	—	9	2	no	—
116	C₂×C₃×C₈	Dic₉	1	4	yes	complemented
117	C₂×C₃×C₈	C₂×D₉	1	1	yes	characteristic
118	C₂×C₃×C₈	C₂×D₉	1	1	yes	characteristic
119	C₂×C₃×C₈	C₃×Dic₃	1	4	yes	—
120	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
121	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
122	C₃×C₄⋊C₄	—	9	2	no	—
123	C₃×C₂²⋊C₄	—	9	2	no	—
124	C₃×C₄²	—	9	2	no	—
125	#217	—	3	2	no	—
126	C₄⋊Dic₃	—	3	2	no	—
127	C₄×Dic₃	—	3	2	no	—
128	#208	—	3	4	no	—
129	C₂×C₃⋊C₈	—	3	2	no	—
130	C₂²×C₃²	—	2	2	no	—
131	C₂²×C₃²	C₄×Dic₃	1	2	yes	—
132	C₂²×C₃²	C₂×C₄×D₃	1	1	yes	characteristic
133	C₃²×C₄	C₄×Dic₃	1	2	yes	—
134	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
135	C₃²×C₄	C₂²×Dic₃	1	1	yes	characteristic
136	C₃×Dic₃	—	6	4	no	—
137	C₂²×C₉	—	2	4	no	—
138	C₂²×C₉	—	2	2	no	—
139	C₂²×C₉	—	2	2	no	—
140	C₂²×C₉	—	2	1	no	—
141	C₂²×C₉	C₃×C₄²	1	2	yes	—
142	C₂²×C₉	C₂²×C₃×C₄	1	1	yes	characteristic
143	C₄×C₉	—	2	2	no	—
144	C₄×C₉	—	2	1	no	—
145	C₄×C₉	—	2	1	no	—
146	C₄×C₉	C₃×C₄²	1	2	yes	—
147	C₄×C₉	C₂²×C₃×C₄	1	1	yes	characteristic
148	C₄×C₉	C₂²×C₃×C₄	1	1	yes	characteristic
149	Dic₉	—	2	4	no	—
150	C₂×M₁₆	—	9	1	no	—
151	C₂²×C₈	C₃×D₉	1	1	yes	characteristic
152	#117	—	9	1	no	—
153	#97	—	9	2	no	—
154	C₄×C₈	—	9	2	no	—
155	C₃×C₉	#405	1	1	yes	characteristic, complemented, Sylow
156	C₂³×C₃	—	2	1	no	—
157	C₂³×C₃	C₄×D₉	1	1	yes	characteristic
158	C₂³×C₃	C₃×C₄×D₃	1	1	yes	characteristic
159	C₂×C₃×C₄	—	2	2	no	—
160	C₂×C₃×C₄	—	2	1	no	—
161	C₂×C₃×C₄	—	2	2	no	—
162	C₂×C₃×C₄	—	2	1	no	—
163	C₂×C₃×C₄	—	9	4	no	—
164	C₂×C₃×C₄	C₄×D₉	1	1	yes	characteristic
165	C₂×C₃×C₄	C₄×D₉	1	2	yes	—
166	C₂×C₃×C₄	C₂×Dic₉	1	2	yes	—
167	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
168	C₂×C₃×C₄	C₃×C₄×D₃	1	1	yes	characteristic
169	C₂×C₃×C₄	C₃×C₄×D₃	1	2	yes	—
170	C₂×C₃×C₄	C₂×C₃×Dic₃	1	2	yes	—
171	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
172	C₂×Dic₃	—	3	4	no	—
173	C₃×C₈	—	2	2	no	—
174	C₃×C₈	—	2	2	no	—
175	C₃×C₈	—	9	4	no	—
176	C₃×C₈	C₂×Dic₉	1	2	yes	—
177	C₃×C₈	C₂×Dic₉	1	2	yes	—
178	C₃×C₈	C₂×C₃×Dic₃	1	2	yes	—
179	C₃×C₈	C₂×C₃×Dic₃	1	2	yes	—
180	C₃⋊C₈	—	3	4	no	—
181	C₂×C₃²	—	2	2	no	—
182	C₂×C₃²	#901	1	2	yes	—
183	C₂×C₃²	C₂×C₄×Dic₃	1	1	yes	characteristic
184	C₂×C₉	—	2	4	no	—
185	C₂×C₉	—	2	2	no	—
186	C₂×C₉	—	2	2	no	—
187	C₂×C₉	—	2	1	no	—
188	C₂×C₉	C₂×C₃×C₄²	1	1	yes	characteristic
189	C₂×C₉	C₃×#131	1	2	yes	—
190	C₂²×C₄	C₂×C₃×D₉	1	1	yes	characteristic
191	M₁₆	—	9	4	no	—
192	C₂×C₈	—	9	2	no	—
193	C₂×C₈	C₃×Dic₉	1	4	yes	complemented
194	C₂×C₈	C₂×C₃×D₉	1	1	yes	characteristic
195	C₂×C₈	C₂×C₃×D₉	1	1	yes	characteristic
196	C₄⋊C₄	—	9	2	no	—
197	C₂²⋊C₄	—	9	2	no	—
198	C₄²	—	9	2	no	—
199	C₂²×C₃	—	2	4	no	—
200	C₂²×C₃	—	2	2	no	—
201	C₂²×C₃	—	2	2	no	—
202	C₂²×C₃	—	2	1	no	—
203	C₂²×C₃	—	2	2	no	—
204	C₂²×C₃	C₄×Dic₉	1	2	yes	—
205	C₂²×C₃	C₂×C₄×D₉	1	1	yes	characteristic
206	C₂²×C₃	C₃×C₄×Dic₃	1	2	yes	—
207	C₂²×C₃	C₂×C₃×C₄×D₃	1	1	yes	characteristic
208	C₃×C₄	—	2	1	no	—
209	C₃×C₄	—	2	1	no	—
210	C₃×C₄	—	2	2	no	—
211	C₃×C₄	—	18	4	no	—
212	C₃×C₄	C₄×Dic₉	1	2	yes	—
213	C₃×C₄	C₂×C₄×D₉	1	1	yes	characteristic
214	C₃×C₄	C₂²×Dic₉	1	1	yes	characteristic
215	C₃×C₄	C₃×C₄×Dic₃	1	2	yes	—
216	C₃×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
217	C₃×C₄	C₂²×C₃×Dic₃	1	1	yes	characteristic
218	Dic₃	—	6	4	no	—
219	C₃²	#5101	1	1	yes	characteristic
220	C₉	—	2	1	no	—
221	C₉	C₃×#405	1	1	yes	characteristic, complemented
222	C₂³	C₃×C₄×D₉	1	1	yes	characteristic
223	C₂×C₄	—	9	4	no	—
224	C₂×C₄	C₃×C₄×D₉	1	2	yes	—
225	C₂×C₄	C₃×C₄×D₉	1	1	yes	characteristic
226	C₂×C₄	C₂×C₃×Dic₉	1	2	yes	—
227	C₂×C₄	C₂²×C₃×D₉	1	1	yes	characteristic
228	C₈	—	9	4	no	—
229	C₈	C₂×C₃×Dic₉	1	2	yes	—
230	C₈	C₂×C₃×Dic₉	1	2	yes	—
231	C₂×C₃	—	2	4	no	—
232	C₂×C₃	—	2	2	no	—
233	C₂×C₃	—	2	2	no	—
234	C₂×C₃	—	2	2	no	—
235	C₂×C₃	—	2	1	no	—
236	C₂×C₃	#63464	1	2	yes	—
237	C₂×C₃	C₂×C₄×Dic₉	1	1	yes	characteristic
238	C₂×C₃	C₃×#901	1	2	yes	—
239	C₂×C₃	C₂×C₃×C₄×Dic₃	1	1	yes	characteristic
240	C₂²	—	2	2	no	—
241	C₂²	C₃×C₄×Dic₉	1	2	yes	—
242	C₂²	C₂×C₃×C₄×D₉	1	1	yes	characteristic
243	C₄	—	18	4	no	—
244	C₄	C₃×C₄×Dic₉	1	2	yes	—
245	C₄	C₂×C₃×C₄×D₉	1	1	yes	characteristic
246	C₄	C₂²×C₃×Dic₉	1	1	yes	characteristic
247	C₃	—	2	1	no	—
248	C₃	#10588782	1	1	yes	characteristic, direct product factor
249	C₃	C₃×#5101	1	1	yes	characteristic
250	C₂	—	2	2	no	—
251	C₂	C₃×#63464	1	2	yes	—
252	C₂	C₂×C₃×C₄×Dic₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₈	C₂×D₉
commutator	C₂×C₉	C₂×C₃×C₄²
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₂²×C₃×C₈×C₉	C₂
socle	C₂²×C₃²	C₂×C₄×D₃
prime radical	C₂³×C₃²	C₄×D₃
cosocle	C₂×C₄×C₉	C₂³×C₃
hypercenter	C₂²×C₃×C₈	D₉
hypocenter	C₉	C₃×#405
LCS	G ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₃×C₈ ◁ C₂²×C₃×C₈

Complements

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

N	H ≅ G/N	type
4. C₃×#63375	C₂	semidirect
5. C₃×C₈×Dic₉	C₂	semidirect
7. #10588782	C₃	direct (N × H)
9. C₂×C₃×C₈×C₉	C₄	semidirect
15. C₃×#3667	C₄	semidirect
26. #63375	C₂×C₃	semidirect
27. C₈×Dic₉	C₂×C₃	semidirect
53. C₂×C₈×C₉	C₃×C₄	semidirect
59. #3667	C₃×C₄	semidirect
116. C₂×C₃×C₈	Dic₉	semidirect
155. C₃×C₉	#405	semidirect
193. C₂×C₈	C₃×Dic₉	semidirect
221. C₉	C₃×#405	semidirect
248. C₃	#10588782	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₈×C₉	2
C₃×#63488	2
C₂×C₃×#3667	2
C₃×#63375	2
C₃×C₈×Dic₉	2
C₃×#5101	3
#10588782	3

Subgroups — C3×#10588782 #13143076

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10588782 #13143076