Subgroups — C₃×C₉×#480 #13145198

Order 1728.

570 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₉×#121	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₉×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₉×#107	C₂	1	2	yes	maximal, complemented
4	C₃×C₉×#102	C₂	1	2	yes	maximal, complemented
5	C₃×C₉×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃²×#480	C₃	1	1	yes	maximal, characteristic
7	C₉×#480	C₃	1	3	yes	maximal, direct product factor
8	C₂×C₃×D₄×C₉	—	2	1	no	—
9	C₂×C₃×D₄×C₉	C₂²	1	1	yes	characteristic
10	C₂²×C₃×C₄×C₉	—	2	1	no	—
11	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₈×C₉	C₂²	1	2	yes	—
13	C₃×C₉×C₄⋊C₄	—	2	1	no	—
14	C₃×C₉×C₄⋊C₄	C₂²	1	2	yes	—
15	C₃×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	C₃×C₉×C₂²⋊C₄	—	2	1	no	—
17	C₃²×#121	C₂×C₃	1	1	yes	characteristic
18	C₂×C₃²×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
19	C₃²×#107	C₂×C₃	1	2	yes	—
20	C₃²×#102	C₂×C₃	1	2	yes	—
21	C₃²×C₂²⋊C₈	C₂×C₃	1	1	yes	characteristic
22	C₉×#121	C₂×C₃	1	3	yes	—
23	C₂×C₉×C₄⋊C₄	C₂×C₃	1	3	yes	complemented
24	C₉×#107	C₂×C₃	1	6	yes	complemented
25	C₉×#102	C₂×C₃	1	6	yes	complemented
26	C₉×C₂²⋊C₈	C₂×C₃	1	3	yes	complemented
27	C₂³×C₃×C₉	—	2	1	no	—
28	C₂³×C₃×C₉	D₄	1	1	yes	characteristic
29	C₃×D₄×C₉	—	2	2	no	—
30	C₃×D₄×C₉	—	4	2	no	—
31	C₂×C₃×C₄×C₉	—	2	2	no	—
32	C₂×C₃×C₄×C₉	—	2	2	no	—
33	C₂×C₃×C₄×C₉	—	2	1	no	—
34	C₂×C₃×C₄×C₉	—	4	2	no	—
35	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
36	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
37	C₃×C₈×C₉	—	2	2	no	—
38	C₃×#480	C₉	1	3	yes	direct product factor
39	C₃×#480	C₃²	1	1	yes	characteristic
40	C₂×C₃²×D₄	—	2	1	no	—
41	C₂×C₃²×D₄	C₂²×C₃	1	1	yes	characteristic
42	C₂²×C₃²×C₄	—	2	1	no	—
43	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
44	C₂×C₃²×C₈	C₂²×C₃	1	2	yes	—
45	C₃²×C₄⋊C₄	—	2	1	no	—
46	C₃²×C₄⋊C₄	C₂²×C₃	1	2	yes	—
47	C₃²×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
48	C₃²×C₂²⋊C₄	—	2	1	no	—
49	C₂×D₄×C₉	—	2	3	no	—
50	C₂×D₄×C₉	C₂²×C₃	1	3	yes	—
51	C₂²×C₄×C₉	—	2	3	no	—
52	C₂²×C₄×C₉	C₂²×C₃	1	3	yes	—
53	C₂×C₈×C₉	C₂²×C₃	1	6	yes	—
54	C₉×C₄⋊C₄	—	2	3	no	—
55	C₉×C₄⋊C₄	C₂²×C₃	1	6	yes	—
56	C₉×C₄⋊C₄	C₂²×C₃	1	3	yes	—
57	C₉×C₂²⋊C₄	—	2	3	no	—
58	C₂²×C₃×C₉	—	2	1	no	—
59	C₂²×C₃×C₉	—	2	1	no	—
60	C₂²×C₃×C₉	—	4	1	no	—
61	C₂²×C₃×C₉	—	4	1	no	—
62	C₂²×C₃×C₉	—	4	1	no	—
63	C₂²×C₃×C₉	D₈	1	2	yes	—
64	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
65	C₃×C₄×C₉	—	2	1	no	—
66	C₃×C₄×C₉	—	4	2	no	—
67	C₃×C₄×C₉	—	4	1	no	—
68	C₃×C₄×C₉	#41	1	2	yes	—
69	C₃×#121	C₂×C₉	1	3	yes	—
70	C₃×#121	C₂×C₃²	1	1	yes	characteristic
71	C₂×C₃×C₄⋊C₄	C₂×C₉	1	3	yes	complemented
72	C₂×C₃×C₄⋊C₄	C₂×C₃²	1	1	yes	characteristic
73	C₃×#107	C₂×C₉	1	6	yes	complemented
74	C₃×#107	C₂×C₃²	1	2	yes	—
75	C₃×#102	C₂×C₉	1	6	yes	complemented
76	C₃×#102	C₂×C₃²	1	2	yes	—
77	C₃×C₂²⋊C₈	C₂×C₉	1	3	yes	complemented
78	C₃×C₂²⋊C₈	C₂×C₃²	1	1	yes	characteristic
79	C₂³×C₃²	—	2	1	no	—
80	C₂³×C₃²	C₃×D₄	1	1	yes	characteristic
81	C₃²×D₄	—	2	2	no	—
82	C₃²×D₄	—	4	2	no	—
83	C₂×C₃²×C₄	—	2	2	no	—
84	C₂×C₃²×C₄	—	2	1	no	—
85	C₂×C₃²×C₄	—	2	2	no	—
86	C₂×C₃²×C₄	—	4	2	no	—
87	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
88	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
89	C₂³×C₉	—	2	3	no	—
90	C₂³×C₉	C₃×D₄	1	3	yes	—
91	C₃²×C₈	—	2	2	no	—
92	D₄×C₉	—	2	6	no	—
93	D₄×C₉	—	4	6	no	—
94	C₂×C₄×C₉	—	2	6	no	—
95	C₂×C₄×C₉	—	2	3	no	—
96	C₂×C₄×C₉	—	2	6	no	—
97	C₂×C₄×C₉	—	4	6	no	—
98	C₂×C₄×C₉	C₃×D₄	1	3	yes	—
99	C₂×C₄×C₉	C₂³×C₃	1	3	yes	—
100	C₈×C₉	—	2	6	no	—
101	#480	C₃×C₉	1	1	yes	characteristic, direct product factor, Sylow
102	C₂×C₃×C₉	—	2	2	no	—
103	C₂×C₃×C₉	—	8	1	no	—
104	C₂×C₃×C₉	#123	1	1	yes	characteristic
105	C₂×C₃×C₉	C₂×D₈	1	1	yes	characteristic
106	C₂×C₃×C₉	#137	1	1	yes	characteristic
107	C₂×C₃×D₄	—	2	3	no	—
108	C₂×C₃×D₄	—	2	1	no	—
109	C₂×C₃×D₄	C₂²×C₉	1	3	yes	—
110	C₂×C₃×D₄	C₂²×C₃²	1	1	yes	characteristic
111	C₂²×C₃×C₄	—	2	3	no	—
112	C₂²×C₃×C₄	—	2	1	no	—
113	C₂²×C₃×C₄	C₂²×C₉	1	3	yes	—
114	C₂²×C₃×C₄	C₂²×C₃²	1	1	yes	characteristic
115	C₂×C₃×C₈	C₂²×C₉	1	6	yes	—
116	C₂×C₃×C₈	C₂²×C₃²	1	2	yes	—
117	C₃×C₄⋊C₄	—	2	3	no	—
118	C₃×C₄⋊C₄	—	2	1	no	—
119	C₃×C₄⋊C₄	C₂²×C₉	1	6	yes	—
120	C₃×C₄⋊C₄	C₂²×C₉	1	3	yes	—
121	C₃×C₄⋊C₄	C₂²×C₃²	1	2	yes	—
122	C₃×C₄⋊C₄	C₂²×C₃²	1	1	yes	characteristic
123	C₃×C₂²⋊C₄	—	2	3	no	—
124	C₃×C₂²⋊C₄	—	2	1	no	—
125	C₂²×C₃²	—	2	1	no	—
126	C₂²×C₃²	—	2	1	no	—
127	C₂²×C₃²	—	4	1	no	—
128	C₂²×C₃²	—	4	1	no	—
129	C₂²×C₃²	—	4	1	no	—
130	C₂²×C₃²	C₃×D₈	1	2	yes	—
131	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
132	C₃²×C₄	—	2	1	no	—
133	C₃²×C₄	—	4	2	no	—
134	C₃²×C₄	—	4	1	no	—
135	C₃²×C₄	C₃×#41	1	2	yes	—
136	C₂²×C₉	—	2	3	no	—
137	C₂²×C₉	—	2	3	no	—
138	C₂²×C₉	—	4	3	no	—
139	C₂²×C₉	—	4	3	no	—
140	C₂²×C₉	—	4	3	no	—
141	C₂²×C₉	C₃×D₈	1	6	yes	—
142	C₂²×C₉	C₂×C₃×D₄	1	3	yes	—
143	C₄×C₉	—	2	3	no	—
144	C₄×C₉	—	4	6	no	—
145	C₄×C₉	—	4	3	no	—
146	C₄×C₉	C₃×#41	1	6	yes	—
147	#121	C₂×C₃×C₉	1	1	yes	characteristic
148	C₂×C₄⋊C₄	C₂×C₃×C₉	1	1	yes	characteristic, complemented
149	#107	C₂×C₃×C₉	1	2	yes	complemented
150	#102	C₂×C₃×C₉	1	2	yes	complemented
151	C₂²⋊C₈	C₂×C₃×C₉	1	1	yes	characteristic, complemented
152	C₃×C₉	#480	1	1	yes	characteristic, direct product factor, Sylow
153	C₂³×C₃	—	2	3	no	—
154	C₂³×C₃	—	2	1	no	—
155	C₂³×C₃	D₄×C₉	1	3	yes	—
156	C₂³×C₃	C₃²×D₄	1	1	yes	characteristic
157	C₃×D₄	—	2	6	no	—
158	C₃×D₄	—	2	2	no	—
159	C₃×D₄	—	4	6	no	—
160	C₃×D₄	—	4	2	no	—
161	C₂×C₃×C₄	—	2	6	no	—
162	C₂×C₃×C₄	—	2	2	no	—
163	C₂×C₃×C₄	—	2	3	no	—
164	C₂×C₃×C₄	—	2	6	no	—
165	C₂×C₃×C₄	—	2	1	no	—
166	C₂×C₃×C₄	—	2	2	no	—
167	C₂×C₃×C₄	—	4	6	no	—
168	C₂×C₃×C₄	—	4	2	no	—
169	C₂×C₃×C₄	D₄×C₉	1	3	yes	—
170	C₂×C₃×C₄	C₂³×C₉	1	3	yes	—
171	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
172	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
173	C₃×C₈	—	2	6	no	—
174	C₃×C₈	—	2	2	no	—
175	C₂×C₃²	—	2	2	no	—
176	C₂×C₃²	—	8	1	no	—
177	C₂×C₃²	C₃×#123	1	1	yes	characteristic
178	C₂×C₃²	C₂×C₃×D₈	1	1	yes	characteristic
179	C₂×C₃²	C₃×#137	1	1	yes	characteristic
180	C₂×C₉	—	2	6	no	—
181	C₂×C₉	—	8	3	no	—
182	C₂×C₉	C₃×#123	1	3	yes	—
183	C₂×C₉	C₂×C₃×D₈	1	3	yes	—
184	C₂×C₉	C₃×#137	1	3	yes	—
185	C₂×D₄	—	2	1	no	—
186	C₂×D₄	C₂²×C₃×C₉	1	1	yes	characteristic
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	C₂²×C₃×C₉	1	1	yes	characteristic
189	C₂×C₈	C₂²×C₃×C₉	1	2	yes	—
190	C₄⋊C₄	—	2	1	no	—
191	C₄⋊C₄	C₂²×C₃×C₉	1	2	yes	—
192	C₄⋊C₄	C₂²×C₃×C₉	1	1	yes	characteristic
193	C₂²⋊C₄	—	2	1	no	—
194	C₂²×C₃	—	2	3	no	—
195	C₂²×C₃	—	2	3	no	—
196	C₂²×C₃	—	2	1	no	—
197	C₂²×C₃	—	2	1	no	—
198	C₂²×C₃	—	4	3	no	—
199	C₂²×C₃	—	4	3	no	—
200	C₂²×C₃	—	4	3	no	—
201	C₂²×C₃	—	4	1	no	—
202	C₂²×C₃	—	4	1	no	—
203	C₂²×C₃	—	4	1	no	—
204	C₂²×C₃	C₉×D₈	1	6	yes	—
205	C₂²×C₃	C₂×D₄×C₉	1	3	yes	—
206	C₂²×C₃	C₃²×D₈	1	2	yes	—
207	C₂²×C₃	C₂×C₃²×D₄	1	1	yes	characteristic
208	C₃×C₄	—	2	3	no	—
209	C₃×C₄	—	2	1	no	—
210	C₃×C₄	—	4	6	no	—
211	C₃×C₄	—	4	3	no	—
212	C₃×C₄	—	4	2	no	—
213	C₃×C₄	—	4	1	no	—
214	C₃×C₄	C₉×#41	1	6	yes	—
215	C₃×C₄	C₃²×#41	1	2	yes	—
216	C₃²	C₃×#480	1	1	yes	characteristic
217	C₉	C₃×#480	1	3	yes	direct product factor
218	C₂³	—	2	1	no	—
219	C₂³	C₃×D₄×C₉	1	1	yes	characteristic
220	D₄	—	2	2	no	—
221	D₄	—	4	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	C₃×D₄×C₉	1	1	yes	characteristic
227	C₂×C₄	C₂³×C₃×C₉	1	1	yes	characteristic
228	C₈	—	2	2	no	—
229	C₂×C₃	—	2	2	no	—
230	C₂×C₃	—	2	6	no	—
231	C₂×C₃	—	8	1	no	—
232	C₂×C₃	—	8	3	no	—
233	C₂×C₃	C₉×#123	1	3	yes	—
234	C₂×C₃	C₂×C₉×D₈	1	3	yes	—
235	C₂×C₃	C₉×#137	1	3	yes	—
236	C₂×C₃	C₃²×#123	1	1	yes	characteristic
237	C₂×C₃	C₂×C₃²×D₈	1	1	yes	characteristic
238	C₂×C₃	C₃²×#137	1	1	yes	characteristic
239	C₂²	—	2	1	no	—
240	C₂²	—	2	1	no	—
241	C₂²	—	4	1	no	—
242	C₂²	—	4	1	no	—
243	C₂²	—	4	1	no	—
244	C₂²	C₃×C₉×D₈	1	2	yes	—
245	C₂²	C₂×C₃×D₄×C₉	1	1	yes	characteristic
246	C₄	—	2	1	no	—
247	C₄	—	4	2	no	—
248	C₄	—	4	1	no	—
249	C₄	C₃×C₉×#41	1	2	yes	—
250	C₃	C₉×#480	1	3	yes	direct product factor
251	C₃	C₃²×#480	1	1	yes	characteristic
252	C₂	—	2	2	no	—
253	C₂	—	8	1	no	—
254	C₂	C₃×C₉×#123	1	1	yes	characteristic
255	C₂	C₂×C₃×C₉×D₈	1	1	yes	characteristic
256	C₂	C₃×C₉×#137	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃×C₉	C₂×D₄
commutator	C₂×C₄	C₂³×C₃×C₉
Frattini, cosocle	C₂×C₃×C₄	C₂³×C₃²
socle	C₂²×C₃²	C₂×C₃×D₄
prime radical	C₃²×#121	C₂×C₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₉ ◁ C₂²×C₃×C₄×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₉×C₄⋊C₄	C₂	semidirect
3. C₃×C₉×#107	C₂	semidirect
4. C₃×C₉×#102	C₂	semidirect
5. C₃×C₉×C₂²⋊C₈	C₂	semidirect
7. C₉×#480	C₃	direct (N × H)
23. C₂×C₉×C₄⋊C₄	C₂×C₃	semidirect
24. C₉×#107	C₂×C₃	semidirect
25. C₉×#102	C₂×C₃	semidirect
26. C₉×C₂²⋊C₈	C₂×C₃	semidirect
38. C₃×#480	C₉	direct (N × H)
71. C₂×C₃×C₄⋊C₄	C₂×C₉	semidirect
73. C₃×#107	C₂×C₉	semidirect
75. C₃×#102	C₂×C₉	semidirect
77. C₃×C₂²⋊C₈	C₂×C₉	semidirect
101. #480	C₃×C₉	direct (N × H)
148. C₂×C₄⋊C₄	C₂×C₃×C₉	semidirect
149. #107	C₂×C₃×C₉	semidirect
150. #102	C₂×C₃×C₉	semidirect
151. C₂²⋊C₈	C₂×C₃×C₉	semidirect
152. C₃×C₉	#480	direct (N × H)
217. C₉	C₃×#480	direct (N × H)
250. C₃	C₉×#480	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×C₉×#121	2
C₂×C₃×C₉×C₄⋊C₄	2
C₃×C₉×#107	2
C₃×C₉×#102	2
C₃×C₉×C₂²⋊C₈	2
C₃²×#480	3
C₉×#480	3

Subgroups — C3×C9×#480 #13145198

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₉×#480 #13145198