Subgroups — C₃×C₉×#488 #13145222

Order 1728.

480 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₉×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×C₉×#124	C₂	1	1	yes	maximal, characteristic
3	C₃×C₉×#103	C₂	1	2	yes	maximal, complemented
4	C₃×C₉×#102	C₂	1	2	yes	maximal
5	C₃×C₄×C₈×C₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃²×#488	C₃	1	1	yes	maximal, characteristic
7	C₉×#488	C₃	1	3	yes	maximal, direct product factor
8	C₂×C₃×Q₈×C₉	C₂²	1	1	yes	characteristic
9	C₂×C₃×D₄×C₉	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₈×C₉	C₂²	1	2	yes	—
11	C₃×C₉×C₄⋊C₄	—	2	2	no	—
12	C₃×C₉×C₄⋊C₄	C₂²	1	2	yes	—
13	C₃×C₉×C₂²⋊C₄	—	2	2	no	—
14	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
15	C₃²×#125	C₂×C₃	1	1	yes	characteristic
16	C₃²×#124	C₂×C₃	1	1	yes	characteristic
17	C₃²×#103	C₂×C₃	1	2	yes	—
18	C₃²×#102	C₂×C₃	1	2	yes	—
19	C₃²×C₄×C₈	C₂×C₃	1	1	yes	characteristic
20	C₉×#125	C₂×C₃	1	3	yes	complemented
21	C₉×#124	C₂×C₃	1	3	yes	—
22	C₉×#103	C₂×C₃	1	6	yes	complemented
23	C₉×#102	C₂×C₃	1	6	yes	—
24	C₄×C₈×C₉	C₂×C₃	1	3	yes	complemented
25	C₂³×C₃×C₉	—	2	1	no	—
26	C₃×Q₈×C₉	—	2	2	no	—
27	C₃×D₄×C₉	—	2	2	no	—
28	C₂×C₃×C₄×C₉	—	2	1	no	—
29	C₂×C₃×C₄×C₉	—	2	2	no	—
30	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
31	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
32	C₃×C₈×C₉	—	2	2	no	—
33	C₃×#488	C₉	1	3	yes	direct product factor
34	C₃×#488	C₃²	1	1	yes	characteristic
35	C₂×C₃²×Q₈	C₂²×C₃	1	1	yes	characteristic
36	C₂×C₃²×D₄	C₂²×C₃	1	1	yes	characteristic
37	C₂×C₃²×C₈	C₂²×C₃	1	2	yes	—
38	C₃²×C₄⋊C₄	—	2	2	no	—
39	C₃²×C₄⋊C₄	C₂²×C₃	1	2	yes	—
40	C₃²×C₂²⋊C₄	—	2	2	no	—
41	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
42	C₂×Q₈×C₉	C₂²×C₃	1	3	yes	—
43	C₂×D₄×C₉	C₂²×C₃	1	3	yes	—
44	C₂×C₈×C₉	C₂²×C₃	1	6	yes	—
45	C₉×C₄⋊C₄	—	2	6	no	—
46	C₉×C₄⋊C₄	C₂²×C₃	1	6	yes	—
47	C₉×C₂²⋊C₄	—	2	6	no	—
48	C₄²×C₉	C₂²×C₃	1	3	yes	—
49	C₂²×C₃×C₉	—	4	1	no	—
50	C₂²×C₃×C₉	—	4	2	no	—
51	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
52	C₃×C₄×C₉	—	2	2	no	—
53	C₃×C₄×C₉	—	4	1	no	—
54	C₃×C₄×C₉	—	4	2	no	—
55	C₃×C₄×C₉	#41	1	2	yes	—
56	C₃×#125	C₂×C₉	1	3	yes	complemented
57	C₃×#125	C₂×C₃²	1	1	yes	characteristic
58	C₃×#124	C₂×C₉	1	3	yes	—
59	C₃×#124	C₂×C₃²	1	1	yes	characteristic
60	C₃×#103	C₂×C₉	1	6	yes	complemented
61	C₃×#103	C₂×C₃²	1	2	yes	—
62	C₃×#102	C₂×C₉	1	6	yes	—
63	C₃×#102	C₂×C₃²	1	2	yes	—
64	C₃×C₄×C₈	C₂×C₉	1	3	yes	complemented
65	C₃×C₄×C₈	C₂×C₃²	1	1	yes	characteristic
66	C₂³×C₃²	—	2	1	no	—
67	C₃²×Q₈	—	2	2	no	—
68	C₃²×D₄	—	2	2	no	—
69	C₂×C₃²×C₄	—	2	1	no	—
70	C₂×C₃²×C₄	—	2	2	no	—
71	C₂×C₃²×C₄	C₃×D₄	1	2	yes	—
72	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
73	C₂³×C₉	—	2	3	no	—
74	C₃²×C₈	—	2	2	no	—
75	Q₈×C₉	—	2	6	no	—
76	D₄×C₉	—	2	6	no	—
77	C₂×C₄×C₉	—	2	3	no	—
78	C₂×C₄×C₉	—	2	6	no	—
79	C₂×C₄×C₉	C₃×D₄	1	6	yes	—
80	C₂×C₄×C₉	C₂³×C₃	1	3	yes	—
81	C₈×C₉	—	2	6	no	—
82	#488	C₃×C₉	1	1	yes	characteristic, direct product factor, Sylow
83	C₂×C₃×C₉	—	8	1	no	—
84	C₂×C₃×C₉	#124	1	1	yes	characteristic
85	C₂×C₃×C₉	#135	1	2	yes	—
86	C₂×C₃×Q₈	C₂²×C₉	1	3	yes	—
87	C₂×C₃×Q₈	C₂²×C₃²	1	1	yes	characteristic
88	C₂×C₃×D₄	C₂²×C₉	1	3	yes	—
89	C₂×C₃×D₄	C₂²×C₃²	1	1	yes	characteristic
90	C₂×C₃×C₈	C₂²×C₉	1	6	yes	—
91	C₂×C₃×C₈	C₂²×C₃²	1	2	yes	—
92	C₃×C₄⋊C₄	—	2	6	no	—
93	C₃×C₄⋊C₄	—	2	2	no	—
94	C₃×C₄⋊C₄	C₂²×C₉	1	6	yes	—
95	C₃×C₄⋊C₄	C₂²×C₃²	1	2	yes	—
96	C₃×C₂²⋊C₄	—	2	6	no	—
97	C₃×C₂²⋊C₄	—	2	2	no	—
98	C₃×C₄²	C₂²×C₉	1	3	yes	—
99	C₃×C₄²	C₂²×C₃²	1	1	yes	characteristic
100	C₂²×C₃²	—	4	1	no	—
101	C₂²×C₃²	—	4	2	no	—
102	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
103	C₃²×C₄	—	2	2	no	—
104	C₃²×C₄	—	4	1	no	—
105	C₃²×C₄	—	4	2	no	—
106	C₃²×C₄	C₃×#41	1	2	yes	—
107	C₂²×C₉	—	4	3	no	—
108	C₂²×C₉	—	4	6	no	—
109	C₂²×C₉	C₂×C₃×D₄	1	3	yes	—
110	C₄×C₉	—	2	6	no	—
111	C₄×C₉	—	4	3	no	—
112	C₄×C₉	—	4	6	no	—
113	C₄×C₉	C₃×#41	1	6	yes	—
114	#125	C₂×C₃×C₉	1	1	yes	characteristic, complemented
115	#124	C₂×C₃×C₉	1	1	yes	characteristic
116	#103	C₂×C₃×C₉	1	2	yes	complemented
117	#102	C₂×C₃×C₉	1	2	yes	—
118	C₄×C₈	C₂×C₃×C₉	1	1	yes	characteristic, complemented
119	C₃×C₉	#488	1	1	yes	characteristic, direct product factor, Sylow
120	C₂³×C₃	—	2	3	no	—
121	C₂³×C₃	—	2	1	no	—
122	C₃×Q₈	—	2	6	no	—
123	C₃×Q₈	—	2	2	no	—
124	C₃×D₄	—	2	6	no	—
125	C₃×D₄	—	2	2	no	—
126	C₂×C₃×C₄	—	2	3	no	—
127	C₂×C₃×C₄	—	2	6	no	—
128	C₂×C₃×C₄	—	2	1	no	—
129	C₂×C₃×C₄	—	2	2	no	—
130	C₂×C₃×C₄	D₄×C₉	1	6	yes	—
131	C₂×C₃×C₄	C₂³×C₉	1	3	yes	—
132	C₂×C₃×C₄	C₃²×D₄	1	2	yes	—
133	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
134	C₃×C₈	—	2	6	no	—
135	C₃×C₈	—	2	2	no	—
136	C₂×C₃²	—	8	1	no	—
137	C₂×C₃²	C₃×#124	1	1	yes	characteristic
138	C₂×C₃²	C₃×#135	1	2	yes	—
139	C₂×C₉	—	8	3	no	—
140	C₂×C₉	C₃×#124	1	3	yes	—
141	C₂×C₉	C₃×#135	1	6	yes	—
142	C₂×Q₈	C₂²×C₃×C₉	1	1	yes	characteristic
143	C₂×D₄	C₂²×C₃×C₉	1	1	yes	characteristic
144	C₂×C₈	C₂²×C₃×C₉	1	2	yes	—
145	C₄⋊C₄	—	2	2	no	—
146	C₄⋊C₄	C₂²×C₃×C₉	1	2	yes	—
147	C₂²⋊C₄	—	2	2	no	—
148	C₄²	C₂²×C₃×C₉	1	1	yes	characteristic
149	C₂²×C₃	—	4	3	no	—
150	C₂²×C₃	—	4	6	no	—
151	C₂²×C₃	—	4	1	no	—
152	C₂²×C₃	—	4	2	no	—
153	C₂²×C₃	C₂×D₄×C₉	1	3	yes	—
154	C₂²×C₃	C₂×C₃²×D₄	1	1	yes	characteristic
155	C₃×C₄	—	2	6	no	—
156	C₃×C₄	—	2	2	no	—
157	C₃×C₄	—	4	3	no	—
158	C₃×C₄	—	4	6	no	—
159	C₃×C₄	—	4	1	no	—
160	C₃×C₄	—	4	2	no	—
161	C₃×C₄	C₉×#41	1	6	yes	—
162	C₃×C₄	C₃²×#41	1	2	yes	—
163	C₃²	C₃×#488	1	1	yes	characteristic
164	C₉	C₃×#488	1	3	yes	direct product factor
165	C₂³	—	2	1	no	—
166	Q₈	—	2	2	no	—
167	D₄	—	2	2	no	—
168	C₂×C₄	—	2	1	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	C₃×D₄×C₉	1	2	yes	—
171	C₂×C₄	C₂³×C₃×C₉	1	1	yes	characteristic
172	C₈	—	2	2	no	—
173	C₂×C₃	—	8	1	no	—
174	C₂×C₃	—	8	3	no	—
175	C₂×C₃	C₉×#124	1	3	yes	—
176	C₂×C₃	C₉×#135	1	6	yes	—
177	C₂×C₃	C₃²×#124	1	1	yes	characteristic
178	C₂×C₃	C₃²×#135	1	2	yes	—
179	C₂²	—	4	1	no	—
180	C₂²	—	4	2	no	—
181	C₂²	C₂×C₃×D₄×C₉	1	1	yes	characteristic
182	C₄	—	2	2	no	—
183	C₄	—	4	1	no	—
184	C₄	—	4	2	no	—
185	C₄	C₃×C₉×#41	1	2	yes	—
186	C₃	C₉×#488	1	3	yes	direct product factor
187	C₃	C₃²×#488	1	1	yes	characteristic
188	C₂	—	8	1	no	—
189	C₂	C₃×C₉×#124	1	1	yes	characteristic
190	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×D₄	C₂²×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. C₃×C₉×#125	C₂	semidirect
3. C₃×C₉×#103	C₂	semidirect
5. C₃×C₄×C₈×C₉	C₂	semidirect
7. C₉×#488	C₃	direct (N × H)
20. C₉×#125	C₂×C₃	semidirect
22. C₉×#103	C₂×C₃	semidirect
24. C₄×C₈×C₉	C₂×C₃	semidirect
33. C₃×#488	C₉	direct (N × H)
56. C₃×#125	C₂×C₉	semidirect
60. C₃×#103	C₂×C₉	semidirect
64. C₃×C₄×C₈	C₂×C₉	semidirect
82. #488	C₃×C₉	direct (N × H)
114. #125	C₂×C₃×C₉	semidirect
116. #103	C₂×C₃×C₉	semidirect
118. C₄×C₈	C₂×C₃×C₉	semidirect
119. C₃×C₉	#488	direct (N × H)
164. C₉	C₃×#488	direct (N × H)
186. C₃	C₉×#488	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×#125	2
C₃×C₉×#124	2
C₃×C₉×#103	2
C₃×C₉×#102	2
C₃×C₄×C₈×C₉	2
C₃²×#488	3
C₉×#488	3

Subgroups — C3×C9×#488 #13145222

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₉×#488 #13145222