Subgroups — C₃×C₉×#473 #13145177

Order 1728.

500 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₉×#137	C₂	1	2	yes	maximal
2	C₂×C₃×C₉×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₉×#131	C₂	1	1	yes	maximal, characteristic
4	C₃×C₉×#108	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₉×#101	C₂	1	2	yes	maximal, complemented
6	C₃²×#473	C₃	1	1	yes	maximal, characteristic
7	C₉×#473	C₃	1	3	yes	maximal, direct product factor
8	C₃×C₉×#41	C₂²	1	1	yes	characteristic
9	C₂×C₃×Q₈×C₉	C₂²	1	2	yes	—
10	C₃×C₉×Q₁₆	—	2	2	no	—
11	C₃×C₉×Q₁₆	—	2	1	no	—
12	C₃×C₉×Q₁₆	—	2	1	no	—
13	C₃×C₉×SD₁₆	—	2	2	no	—
14	C₃×C₉×M₁₆	—	2	1	no	—
15	C₃×C₉×M₁₆	C₂²	1	1	yes	characteristic
16	C₃×C₉×M₁₆	C₂²	1	2	yes	—
17	C₂×C₃×C₈×C₉	—	2	1	no	—
18	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
19	C₃²×#137	C₂×C₃	1	2	yes	—
20	C₂×C₃²×Q₁₆	C₂×C₃	1	1	yes	characteristic
21	C₃²×#131	C₂×C₃	1	1	yes	characteristic
22	C₃²×#108	C₂×C₃	1	1	yes	characteristic
23	C₃²×#101	C₂×C₃	1	2	yes	—
24	C₉×#137	C₂×C₃	1	6	yes	—
25	C₂×C₉×Q₁₆	C₂×C₃	1	3	yes	complemented
26	C₉×#131	C₂×C₃	1	3	yes	—
27	C₉×#108	C₂×C₃	1	3	yes	complemented
28	C₉×#101	C₂×C₃	1	6	yes	complemented
29	C₃×Q₈×C₉	—	2	2	no	—
30	C₃×Q₈×C₉	—	2	2	no	—
31	C₃×Q₈×C₉	D₄	1	1	yes	characteristic
32	C₃×D₄×C₉	—	2	1	no	—
33	C₃×D₄×C₉	D₄	1	1	yes	characteristic
34	C₂×C₃×C₄×C₉	—	2	1	no	—
35	C₂×C₃×C₄×C₉	—	2	2	no	—
36	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
37	C₃×C₈×C₉	—	2	1	no	—
38	C₃×C₈×C₉	—	2	2	no	—
39	C₃×C₈×C₉	D₄	1	1	yes	characteristic
40	C₃×C₈×C₉	D₄	1	1	yes	characteristic
41	C₃×#473	C₉	1	3	yes	direct product factor
42	C₃×#473	C₃²	1	1	yes	characteristic
43	C₃²×#41	C₂²×C₃	1	1	yes	characteristic
44	C₂×C₃²×Q₈	C₂²×C₃	1	2	yes	—
45	C₃²×Q₁₆	—	2	2	no	—
46	C₃²×Q₁₆	—	2	1	no	—
47	C₃²×Q₁₆	—	2	1	no	—
48	C₃²×SD₁₆	—	2	2	no	—
49	C₃²×M₁₆	—	2	1	no	—
50	C₃²×M₁₆	C₂²×C₃	1	1	yes	characteristic
51	C₃²×M₁₆	C₂²×C₃	1	2	yes	—
52	C₂×C₃²×C₈	—	2	1	no	—
53	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
54	C₉×#41	C₂²×C₃	1	3	yes	—
55	C₂×Q₈×C₉	C₂²×C₃	1	6	yes	—
56	C₉×Q₁₆	—	2	6	no	—
57	C₉×Q₁₆	—	2	3	no	—
58	C₉×Q₁₆	—	2	3	no	—
59	C₉×SD₁₆	—	2	6	no	—
60	C₉×M₁₆	—	2	3	no	—
61	C₉×M₁₆	C₂²×C₃	1	3	yes	—
62	C₉×M₁₆	C₂²×C₃	1	6	yes	—
63	C₂×C₈×C₉	—	2	3	no	—
64	C₂×C₈×C₉	C₂²×C₃	1	3	yes	—
65	C₂²×C₃×C₉	—	2	1	no	—
66	C₂²×C₃×C₉	#41	1	1	yes	characteristic
67	C₃×C₄×C₉	—	2	1	no	—
68	C₃×C₄×C₉	—	4	2	no	—
69	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
70	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
71	C₃×#137	C₂×C₉	1	6	yes	—
72	C₃×#137	C₂×C₃²	1	2	yes	—
73	C₂×C₃×Q₁₆	C₂×C₉	1	3	yes	complemented
74	C₂×C₃×Q₁₆	C₂×C₃²	1	1	yes	characteristic
75	C₃×#131	C₂×C₉	1	3	yes	—
76	C₃×#131	C₂×C₃²	1	1	yes	characteristic
77	C₃×#108	C₂×C₉	1	3	yes	complemented
78	C₃×#108	C₂×C₃²	1	1	yes	characteristic
79	C₃×#101	C₂×C₉	1	6	yes	complemented
80	C₃×#101	C₂×C₃²	1	2	yes	—
81	C₃²×Q₈	—	2	2	no	—
82	C₃²×Q₈	—	2	2	no	—
83	C₃²×Q₈	C₃×D₄	1	1	yes	characteristic
84	C₃²×D₄	—	2	1	no	—
85	C₃²×D₄	C₃×D₄	1	1	yes	characteristic
86	C₂×C₃²×C₄	—	2	1	no	—
87	C₂×C₃²×C₄	—	2	2	no	—
88	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
89	C₃²×C₈	—	2	1	no	—
90	C₃²×C₈	—	2	2	no	—
91	C₃²×C₈	C₃×D₄	1	1	yes	characteristic
92	C₃²×C₈	C₃×D₄	1	1	yes	characteristic
93	Q₈×C₉	—	2	6	no	—
94	Q₈×C₉	—	2	6	no	—
95	Q₈×C₉	C₃×D₄	1	3	yes	—
96	D₄×C₉	—	2	3	no	—
97	D₄×C₉	C₃×D₄	1	3	yes	—
98	C₂×C₄×C₉	—	2	3	no	—
99	C₂×C₄×C₉	—	2	6	no	—
100	C₂×C₄×C₉	C₂³×C₃	1	3	yes	—
101	C₈×C₉	—	2	3	no	—
102	C₈×C₉	—	2	6	no	—
103	C₈×C₉	C₃×D₄	1	3	yes	—
104	C₈×C₉	C₃×D₄	1	3	yes	—
105	#473	C₃×C₉	1	1	yes	characteristic, direct product factor, Sylow
106	C₂×C₃×C₉	—	2	1	no	—
107	C₂×C₃×C₉	—	4	1	no	—
108	C₂×C₃×C₉	#121	1	1	yes	characteristic
109	C₃×#41	C₂²×C₉	1	3	yes	—
110	C₃×#41	C₂²×C₃²	1	1	yes	characteristic
111	C₂×C₃×Q₈	C₂²×C₉	1	6	yes	—
112	C₂×C₃×Q₈	C₂²×C₃²	1	2	yes	—
113	C₃×Q₁₆	—	2	6	no	—
114	C₃×Q₁₆	—	2	2	no	—
115	C₃×Q₁₆	—	2	3	no	—
116	C₃×Q₁₆	—	2	3	no	—
117	C₃×Q₁₆	—	2	1	no	—
118	C₃×Q₁₆	—	2	1	no	—
119	C₃×SD₁₆	—	2	6	no	—
120	C₃×SD₁₆	—	2	2	no	—
121	C₃×M₁₆	—	2	3	no	—
122	C₃×M₁₆	—	2	1	no	—
123	C₃×M₁₆	C₂²×C₉	1	3	yes	—
124	C₃×M₁₆	C₂²×C₉	1	6	yes	—
125	C₃×M₁₆	C₂²×C₃²	1	1	yes	characteristic
126	C₃×M₁₆	C₂²×C₃²	1	2	yes	—
127	C₂×C₃×C₈	—	2	3	no	—
128	C₂×C₃×C₈	—	2	1	no	—
129	C₂×C₃×C₈	C₂²×C₉	1	3	yes	—
130	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
131	C₂²×C₃²	—	2	1	no	—
132	C₂²×C₃²	C₃×#41	1	1	yes	characteristic
133	C₃²×C₄	—	2	1	no	—
134	C₃²×C₄	—	4	2	no	—
135	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
136	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
137	C₂²×C₉	—	2	3	no	—
138	C₂²×C₉	C₃×#41	1	3	yes	—
139	C₄×C₉	—	2	3	no	—
140	C₄×C₉	—	4	6	no	—
141	C₄×C₉	C₂×C₃×D₄	1	3	yes	—
142	C₄×C₉	C₂×C₃×D₄	1	3	yes	—
143	#137	C₂×C₃×C₉	1	2	yes	—
144	C₂×Q₁₆	C₂×C₃×C₉	1	1	yes	characteristic, complemented
145	#131	C₂×C₃×C₉	1	1	yes	characteristic
146	#108	C₂×C₃×C₉	1	1	yes	characteristic, complemented
147	#101	C₂×C₃×C₉	1	2	yes	complemented
148	C₃×C₉	#473	1	1	yes	characteristic, direct product factor, Sylow
149	C₃×Q₈	—	2	6	no	—
150	C₃×Q₈	—	2	6	no	—
151	C₃×Q₈	—	2	2	no	—
152	C₃×Q₈	—	2	2	no	—
153	C₃×Q₈	D₄×C₉	1	3	yes	—
154	C₃×Q₈	C₃²×D₄	1	1	yes	characteristic
155	C₃×D₄	—	2	3	no	—
156	C₃×D₄	—	2	1	no	—
157	C₃×D₄	D₄×C₉	1	3	yes	—
158	C₃×D₄	C₃²×D₄	1	1	yes	characteristic
159	C₂×C₃×C₄	—	2	3	no	—
160	C₂×C₃×C₄	—	2	1	no	—
161	C₂×C₃×C₄	—	2	6	no	—
162	C₂×C₃×C₄	—	2	2	no	—
163	C₂×C₃×C₄	C₂³×C₉	1	3	yes	—
164	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
165	C₃×C₈	—	2	3	no	—
166	C₃×C₈	—	2	6	no	—
167	C₃×C₈	—	2	1	no	—
168	C₃×C₈	—	2	2	no	—
169	C₃×C₈	D₄×C₉	1	3	yes	—
170	C₃×C₈	D₄×C₉	1	3	yes	—
171	C₃×C₈	C₃²×D₄	1	1	yes	characteristic
172	C₃×C₈	C₃²×D₄	1	1	yes	characteristic
173	C₂×C₃²	—	2	1	no	—
174	C₂×C₃²	—	4	1	no	—
175	C₂×C₃²	C₃×#121	1	1	yes	characteristic
176	C₂×C₉	—	2	3	no	—
177	C₂×C₉	—	4	3	no	—
178	C₂×C₉	C₃×#121	1	3	yes	—
179	#41	C₂²×C₃×C₉	1	1	yes	characteristic
180	C₂×Q₈	C₂²×C₃×C₉	1	2	yes	—
181	Q₁₆	—	2	2	no	—
182	Q₁₆	—	2	1	no	—
183	Q₁₆	—	2	1	no	—
184	SD₁₆	—	2	2	no	—
185	M₁₆	—	2	1	no	—
186	M₁₆	C₂²×C₃×C₉	1	1	yes	characteristic
187	M₁₆	C₂²×C₃×C₉	1	2	yes	—
188	C₂×C₈	—	2	1	no	—
189	C₂×C₈	C₂²×C₃×C₉	1	1	yes	characteristic
190	C₂²×C₃	—	2	3	no	—
191	C₂²×C₃	—	2	1	no	—
192	C₂²×C₃	C₉×#41	1	3	yes	—
193	C₂²×C₃	C₃²×#41	1	1	yes	characteristic
194	C₃×C₄	—	2	3	no	—
195	C₃×C₄	—	2	1	no	—
196	C₃×C₄	—	4	6	no	—
197	C₃×C₄	—	4	2	no	—
198	C₃×C₄	C₂×D₄×C₉	1	3	yes	—
199	C₃×C₄	C₂×D₄×C₉	1	3	yes	—
200	C₃×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
201	C₃×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
202	C₃²	C₃×#473	1	1	yes	characteristic
203	C₉	C₃×#473	1	3	yes	direct product factor
204	Q₈	—	2	2	no	—
205	Q₈	—	2	2	no	—
206	Q₈	C₃×D₄×C₉	1	1	yes	characteristic
207	D₄	—	2	1	no	—
208	D₄	C₃×D₄×C₉	1	1	yes	characteristic
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	C₂³×C₃×C₉	1	1	yes	characteristic
212	C₈	—	2	1	no	—
213	C₈	—	2	2	no	—
214	C₈	C₃×D₄×C₉	1	1	yes	characteristic
215	C₈	C₃×D₄×C₉	1	1	yes	characteristic
216	C₂×C₃	—	2	1	no	—
217	C₂×C₃	—	2	3	no	—
218	C₂×C₃	—	4	1	no	—
219	C₂×C₃	—	4	3	no	—
220	C₂×C₃	C₉×#121	1	3	yes	—
221	C₂×C₃	C₃²×#121	1	1	yes	characteristic
222	C₂²	—	2	1	no	—
223	C₂²	C₃×C₉×#41	1	1	yes	characteristic
224	C₄	—	2	1	no	—
225	C₄	—	4	2	no	—
226	C₄	C₂×C₃×D₄×C₉	1	1	yes	characteristic
227	C₄	C₂×C₃×D₄×C₉	1	1	yes	characteristic
228	C₃	C₉×#473	1	3	yes	direct product factor
229	C₃	C₃²×#473	1	1	yes	characteristic
230	C₂	—	2	1	no	—
231	C₂	—	4	1	no	—
232	C₂	C₃×C₉×#121	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₉	#121
commutator	C₂×C₄	C₂³×C₃×C₉
Frattini, cosocle	C₂×C₃×C₄	C₂³×C₃²
socle	C₂×C₃²	C₃×#121
prime radical	C₃²×#41	C₂²×C₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₉ ◁ C₂×C₃×C₄×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₉×Q₁₆	C₂	semidirect
4. C₃×C₉×#108	C₂	semidirect
5. C₃×C₉×#101	C₂	semidirect
7. C₉×#473	C₃	direct (N × H)
25. C₂×C₉×Q₁₆	C₂×C₃	semidirect
27. C₉×#108	C₂×C₃	semidirect
28. C₉×#101	C₂×C₃	semidirect
41. C₃×#473	C₉	direct (N × H)
73. C₂×C₃×Q₁₆	C₂×C₉	semidirect
77. C₃×#108	C₂×C₉	semidirect
79. C₃×#101	C₂×C₉	semidirect
105. #473	C₃×C₉	direct (N × H)
144. C₂×Q₁₆	C₂×C₃×C₉	semidirect
146. #108	C₂×C₃×C₉	semidirect
147. #101	C₂×C₃×C₉	semidirect
148. C₃×C₉	#473	direct (N × H)
203. C₉	C₃×#473	direct (N × H)
228. C₃	C₉×#473	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×#137	2
C₂×C₃×C₉×Q₁₆	2
C₃×C₉×#131	2
C₃×C₉×#108	2
C₃×C₉×#101	2
C₃²×#473	3
C₉×#473	3

Subgroups — C3×C9×#473 #13145177

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₉×#473 #13145177