Subgroups — C₃×C₉×#495 #13145243

Order 1728.

650 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₉×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×C₉×SD₁₆	C₂	1	2	yes	maximal, complemented
3	C₂×C₃×C₉×D₈	C₂	1	1	yes	maximal, characteristic
4	C₃×C₉×#124	C₂	1	2	yes	maximal, complemented
5	C₃×C₄×C₈×C₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃²×#495	C₃	1	1	yes	maximal, characteristic
7	C₉×#495	C₃	1	3	yes	maximal, direct product factor
8	C₂×C₃×Q₈×C₉	C₂²	1	2	yes	—
9	C₂×C₃×D₄×C₉	C₂²	1	2	yes	—
10	C₃×C₉×Q₁₆	—	2	2	no	—
11	C₃×C₉×SD₁₆	—	2	4	no	—
12	C₃×C₉×D₈	—	2	2	no	—
13	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
15	C₃×C₉×C₂²⋊C₄	—	2	4	no	—
16	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
17	C₂×C₃²×Q₁₆	C₂×C₃	1	1	yes	characteristic
18	C₂×C₃²×SD₁₆	C₂×C₃	1	2	yes	—
19	C₂×C₃²×D₈	C₂×C₃	1	1	yes	characteristic
20	C₃²×#124	C₂×C₃	1	2	yes	—
21	C₃²×C₄×C₈	C₂×C₃	1	1	yes	characteristic
22	C₂×C₉×Q₁₆	C₂×C₃	1	3	yes	complemented
23	C₂×C₉×SD₁₆	C₂×C₃	1	6	yes	complemented
24	C₂×C₉×D₈	C₂×C₃	1	3	yes	—
25	C₉×#124	C₂×C₃	1	6	yes	complemented
26	C₄×C₈×C₉	C₂×C₃	1	3	yes	complemented
27	C₂³×C₃×C₉	—	2	2	no	—
28	C₃×Q₈×C₉	—	2	2	no	—
29	C₃×Q₈×C₉	—	2	2	no	—
30	C₃×D₄×C₉	—	2	2	no	—
31	C₃×D₄×C₉	—	2	2	no	—
32	C₂×C₃×C₄×C₉	—	2	2	no	—
33	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
34	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
35	C₃×C₈×C₉	D₄	1	2	yes	—
36	C₃×C₈×C₉	D₄	1	2	yes	—
37	C₃×#495	C₉	1	3	yes	direct product factor
38	C₃×#495	C₃²	1	1	yes	characteristic
39	C₂×C₃²×Q₈	C₂²×C₃	1	2	yes	—
40	C₂×C₃²×D₄	C₂²×C₃	1	2	yes	—
41	C₃²×Q₁₆	—	2	2	no	—
42	C₃²×SD₁₆	—	2	4	no	—
43	C₃²×D₈	—	2	2	no	—
44	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
45	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
46	C₃²×C₂²⋊C₄	—	2	4	no	—
47	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
48	C₂×Q₈×C₉	C₂²×C₃	1	6	yes	—
49	C₂×D₄×C₉	C₂²×C₃	1	6	yes	—
50	C₉×Q₁₆	—	2	6	no	—
51	C₉×SD₁₆	—	2	12	no	—
52	C₉×D₈	—	2	6	no	—
53	C₂×C₈×C₉	C₂²×C₃	1	3	yes	—
54	C₂×C₈×C₉	C₂²×C₃	1	3	yes	—
55	C₉×C₂²⋊C₄	—	2	12	no	—
56	C₄²×C₉	C₂²×C₃	1	3	yes	—
57	C₂²×C₃×C₉	—	4	2	no	—
58	C₂²×C₃×C₉	—	4	4	no	—
59	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
60	C₃×C₄×C₉	—	2	2	no	—
61	C₃×C₄×C₉	—	4	2	no	—
62	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
63	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
64	C₂×C₃×Q₁₆	C₂×C₉	1	3	yes	complemented
65	C₂×C₃×Q₁₆	C₂×C₃²	1	1	yes	characteristic
66	C₂×C₃×SD₁₆	C₂×C₉	1	6	yes	complemented
67	C₂×C₃×SD₁₆	C₂×C₃²	1	2	yes	—
68	C₂×C₃×D₈	C₂×C₉	1	3	yes	—
69	C₂×C₃×D₈	C₂×C₃²	1	1	yes	characteristic
70	C₃×#124	C₂×C₉	1	6	yes	complemented
71	C₃×#124	C₂×C₃²	1	2	yes	—
72	C₃×C₄×C₈	C₂×C₉	1	3	yes	complemented
73	C₃×C₄×C₈	C₂×C₃²	1	1	yes	characteristic
74	C₂³×C₃²	—	2	2	no	—
75	C₃²×Q₈	—	2	2	no	—
76	C₃²×Q₈	—	2	2	no	—
77	C₃²×D₄	—	2	2	no	—
78	C₃²×D₄	—	2	2	no	—
79	C₂×C₃²×C₄	—	2	2	no	—
80	C₂×C₃²×C₄	C₃×D₄	1	2	yes	—
81	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
82	C₂³×C₉	—	2	6	no	—
83	C₃²×C₈	C₃×D₄	1	2	yes	—
84	C₃²×C₈	C₃×D₄	1	2	yes	—
85	Q₈×C₉	—	2	6	no	—
86	Q₈×C₉	—	2	6	no	—
87	D₄×C₉	—	2	6	no	—
88	D₄×C₉	—	2	6	no	—
89	C₂×C₄×C₉	—	2	6	no	—
90	C₂×C₄×C₉	C₃×D₄	1	6	yes	—
91	C₂×C₄×C₉	C₂³×C₃	1	3	yes	—
92	C₈×C₉	C₃×D₄	1	6	yes	—
93	C₈×C₉	C₃×D₄	1	6	yes	—
94	#495	C₃×C₉	1	1	yes	characteristic, direct product factor, Sylow
95	C₂×C₃×C₉	—	8	2	no	—
96	C₂×C₃×C₉	#127	1	1	yes	characteristic
97	C₂×C₃×C₉	#135	1	2	yes	—
98	C₂×C₃×Q₈	C₂²×C₉	1	6	yes	—
99	C₂×C₃×Q₈	C₂²×C₃²	1	2	yes	—
100	C₂×C₃×D₄	C₂²×C₉	1	6	yes	—
101	C₂×C₃×D₄	C₂²×C₃²	1	2	yes	—
102	C₃×Q₁₆	—	2	6	no	—
103	C₃×Q₁₆	—	2	2	no	—
104	C₃×SD₁₆	—	2	12	no	—
105	C₃×SD₁₆	—	2	4	no	—
106	C₃×D₈	—	2	6	no	—
107	C₃×D₈	—	2	2	no	—
108	C₂×C₃×C₈	C₂²×C₉	1	3	yes	—
109	C₂×C₃×C₈	C₂²×C₉	1	3	yes	—
110	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
111	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
112	C₃×C₂²⋊C₄	—	2	12	no	—
113	C₃×C₂²⋊C₄	—	2	4	no	—
114	C₃×C₄²	C₂²×C₉	1	3	yes	—
115	C₃×C₄²	C₂²×C₃²	1	1	yes	characteristic
116	C₂²×C₃²	—	4	2	no	—
117	C₂²×C₃²	—	4	4	no	—
118	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
119	C₃²×C₄	—	2	2	no	—
120	C₃²×C₄	—	4	2	no	—
121	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
122	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
123	C₂²×C₉	—	4	6	no	—
124	C₂²×C₉	—	4	12	no	—
125	C₂²×C₉	C₂×C₃×D₄	1	3	yes	—
126	C₄×C₉	—	2	6	no	—
127	C₄×C₉	—	4	6	no	—
128	C₄×C₉	C₂×C₃×D₄	1	3	yes	—
129	C₄×C₉	C₂×C₃×D₄	1	3	yes	—
130	C₂×Q₁₆	C₂×C₃×C₉	1	1	yes	characteristic, complemented
131	C₂×SD₁₆	C₂×C₃×C₉	1	2	yes	complemented
132	C₂×D₈	C₂×C₃×C₉	1	1	yes	characteristic
133	#124	C₂×C₃×C₉	1	2	yes	complemented
134	C₄×C₈	C₂×C₃×C₉	1	1	yes	characteristic, complemented
135	C₃×C₉	#495	1	1	yes	characteristic, direct product factor, Sylow
136	C₂³×C₃	—	2	6	no	—
137	C₂³×C₃	—	2	2	no	—
138	C₃×Q₈	—	2	6	no	—
139	C₃×Q₈	—	2	6	no	—
140	C₃×Q₈	—	2	2	no	—
141	C₃×Q₈	—	2	2	no	—
142	C₃×D₄	—	2	6	no	—
143	C₃×D₄	—	2	6	no	—
144	C₃×D₄	—	2	2	no	—
145	C₃×D₄	—	2	2	no	—
146	C₂×C₃×C₄	—	2	6	no	—
147	C₂×C₃×C₄	—	2	2	no	—
148	C₂×C₃×C₄	D₄×C₉	1	6	yes	—
149	C₂×C₃×C₄	C₂³×C₉	1	3	yes	—
150	C₂×C₃×C₄	C₃²×D₄	1	2	yes	—
151	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
152	C₃×C₈	D₄×C₉	1	6	yes	—
153	C₃×C₈	D₄×C₉	1	6	yes	—
154	C₃×C₈	C₃²×D₄	1	2	yes	—
155	C₃×C₈	C₃²×D₄	1	2	yes	—
156	C₂×C₃²	—	8	2	no	—
157	C₂×C₃²	C₃×#127	1	1	yes	characteristic
158	C₂×C₃²	C₃×#135	1	2	yes	—
159	C₂×C₉	—	8	6	no	—
160	C₂×C₉	C₃×#127	1	3	yes	—
161	C₂×C₉	C₃×#135	1	6	yes	—
162	C₂×Q₈	C₂²×C₃×C₉	1	2	yes	—
163	C₂×D₄	C₂²×C₃×C₉	1	2	yes	—
164	Q₁₆	—	2	2	no	—
165	SD₁₆	—	2	4	no	—
166	D₈	—	2	2	no	—
167	C₂×C₈	C₂²×C₃×C₉	1	1	yes	characteristic
168	C₂×C₈	C₂²×C₃×C₉	1	1	yes	characteristic
169	C₂²⋊C₄	—	2	4	no	—
170	C₄²	C₂²×C₃×C₉	1	1	yes	characteristic
171	C₂²×C₃	—	4	6	no	—
172	C₂²×C₃	—	4	12	no	—
173	C₂²×C₃	—	4	2	no	—
174	C₂²×C₃	—	4	4	no	—
175	C₂²×C₃	C₂×D₄×C₉	1	3	yes	—
176	C₂²×C₃	C₂×C₃²×D₄	1	1	yes	characteristic
177	C₃×C₄	—	2	6	no	—
178	C₃×C₄	—	2	2	no	—
179	C₃×C₄	—	4	6	no	—
180	C₃×C₄	—	4	2	no	—
181	C₃×C₄	C₂×D₄×C₉	1	3	yes	—
182	C₃×C₄	C₂×D₄×C₉	1	3	yes	—
183	C₃×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
184	C₃×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
185	C₃²	C₃×#495	1	1	yes	characteristic
186	C₉	C₃×#495	1	3	yes	direct product factor
187	C₂³	—	2	2	no	—
188	Q₈	—	2	2	no	—
189	Q₈	—	2	2	no	—
190	D₄	—	2	2	no	—
191	D₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	C₃×D₄×C₉	1	2	yes	—
194	C₂×C₄	C₂³×C₃×C₉	1	1	yes	characteristic
195	C₈	C₃×D₄×C₉	1	2	yes	—
196	C₈	C₃×D₄×C₉	1	2	yes	—
197	C₂×C₃	—	8	2	no	—
198	C₂×C₃	—	8	6	no	—
199	C₂×C₃	C₉×#127	1	3	yes	—
200	C₂×C₃	C₉×#135	1	6	yes	—
201	C₂×C₃	C₃²×#127	1	1	yes	characteristic
202	C₂×C₃	C₃²×#135	1	2	yes	—
203	C₂²	—	4	2	no	—
204	C₂²	—	4	4	no	—
205	C₂²	C₂×C₃×D₄×C₉	1	1	yes	characteristic
206	C₄	—	2	2	no	—
207	C₄	—	4	2	no	—
208	C₄	C₂×C₃×D₄×C₉	1	1	yes	characteristic
209	C₄	C₂×C₃×D₄×C₉	1	1	yes	characteristic
210	C₃	C₉×#495	1	3	yes	direct product factor
211	C₃	C₃²×#495	1	1	yes	characteristic
212	C₂	—	8	2	no	—
213	C₂	C₃×C₉×#127	1	1	yes	characteristic
214	C₂	C₃×C₉×#135	1	2	yes	—

	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	C₂×C₃²×D₈	C₂×C₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₉ ◁ C₃×C₄²×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₉×Q₁₆	C₂	semidirect
2. C₂×C₃×C₉×SD₁₆	C₂	semidirect
4. C₃×C₉×#124	C₂	semidirect
5. C₃×C₄×C₈×C₉	C₂	semidirect
7. C₉×#495	C₃	direct (N × H)
22. C₂×C₉×Q₁₆	C₂×C₃	semidirect
23. C₂×C₉×SD₁₆	C₂×C₃	semidirect
25. C₉×#124	C₂×C₃	semidirect
26. C₄×C₈×C₉	C₂×C₃	semidirect
37. C₃×#495	C₉	direct (N × H)
64. C₂×C₃×Q₁₆	C₂×C₉	semidirect
66. C₂×C₃×SD₁₆	C₂×C₉	semidirect
70. C₃×#124	C₂×C₉	semidirect
72. C₃×C₄×C₈	C₂×C₉	semidirect
94. #495	C₃×C₉	direct (N × H)
130. C₂×Q₁₆	C₂×C₃×C₉	semidirect
131. C₂×SD₁₆	C₂×C₃×C₉	semidirect
133. #124	C₂×C₃×C₉	semidirect
134. C₄×C₈	C₂×C₃×C₉	semidirect
135. C₃×C₉	#495	direct (N × H)
186. C₉	C₃×#495	direct (N × H)
210. C₃	C₉×#495	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×C₉×Q₁₆	2
C₂×C₃×C₉×SD₁₆	2
C₂×C₃×C₉×D₈	2
C₃×C₉×#124	2
C₃×C₄×C₈×C₉	2
C₃²×#495	3
C₉×#495	3

Subgroups — C3×C9×#495 #13145243

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₉×#495 #13145243