Subgroups — C₂×C₄²×D₃×C₉ #13152288

Order 1728.

1602 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×D₃×C₉	C₂	1	3	yes	maximal
2	C₂×C₃×C₄²×C₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×C₉×Dic₃	C₂	1	3	yes	maximal, complemented
4	C₄²×D₃×C₉	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃×C₄²×D₃	C₃	1	1	yes	maximal, characteristic
6	C₂²×C₄²×C₉	—	3	1	no	maximal
7	C₂³×D₃×C₉	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₄×C₉	C₂²	1	3	yes	—
9	C₂²×C₉×Dic₃	C₂²	1	3	yes	—
10	C₂×C₄×D₃×C₉	C₄	1	24	yes	direct product factor
11	C₂×C₄×D₃×C₉	C₂²	1	12	yes	—
12	C₃×C₄²×C₉	C₂²	1	4	yes	complemented
13	C₄×C₉×Dic₃	C₂²	1	12	yes	complemented
14	C₂²×C₃×C₄×D₃	C₂×C₃	1	3	yes	—
15	C₂×C₃²×C₄²	C₂×C₃	1	1	yes	characteristic
16	C₂×C₃×C₄×Dic₃	C₂×C₃	1	3	yes	—
17	C₃×C₄²×D₃	C₂×C₃	1	8	yes	—
18	C₂³×C₄×C₉	—	3	3	no	—
19	C₂×C₄²×C₉	—	2	1	no	—
20	C₂×C₄²×C₉	—	3	3	no	—
21	C₂×C₄²×C₉	—	3	8	no	—
22	C₂×C₄²×C₉	D₃	1	1	yes	characteristic, direct product factor
23	C₂³×C₃×C₉	C₂³	1	1	yes	characteristic
24	C₂²×D₃×C₉	C₂×C₄	1	12	yes	—
25	C₂²×D₃×C₉	C₂³	1	2	yes	—
26	C₂×C₃×C₄×C₉	C₂×C₄	1	12	yes	complemented
27	C₂×C₃×C₄×C₉	C₂³	1	6	yes	—
28	C₂×C₉×Dic₃	C₂×C₄	1	12	yes	complemented
29	C₂×C₉×Dic₃	C₂³	1	6	yes	—
30	C₄×D₃×C₉	C₂×C₄	1	48	yes	direct product factor
31	C₂²×C₃×C₄²	—	3	1	no	—
32	C₂×C₄²×D₃	C₉	1	1	yes	characteristic, direct product factor
33	C₂³×C₃×D₃	C₂²×C₃	1	1	yes	characteristic
34	C₂²×C₃²×C₄	C₂²×C₃	1	3	yes	—
35	C₂²×C₃×Dic₃	C₂²×C₃	1	3	yes	—
36	C₂×C₃×C₄×D₃	C₃×C₄	1	24	yes	—
37	C₂×C₃×C₄×D₃	C₂²×C₃	1	12	yes	—
38	C₂⁴×C₉	—	3	1	no	—
39	C₃²×C₄²	C₂²×C₃	1	4	yes	—
40	C₃×C₄×Dic₃	C₂²×C₃	1	12	yes	—
41	C₂²×C₄×C₉	—	2	3	no	—
42	C₂²×C₄×C₉	—	3	3	no	—
43	C₂²×C₄×C₉	—	3	24	no	—
44	C₂²×C₄×C₉	—	3	12	no	—
45	C₂²×C₄×C₉	C₂×D₃	1	3	yes	—
46	C₄²×C₉	—	2	4	no	—
47	C₄²×C₉	—	3	12	no	—
48	C₄²×C₉	C₂×D₃	1	4	yes	direct product factor
49	C₂²×C₃×C₉	C₂²×C₄	1	6	yes	—
50	C₂²×C₃×C₉	C₂⁴	1	1	yes	characteristic
51	C₂×D₃×C₉	C₄²	1	16	yes	direct product factor
52	C₂×D₃×C₉	C₂²×C₄	1	12	yes	—
53	C₃×C₄×C₉	C₂²×C₄	1	12	yes	complemented
54	C₉×Dic₃	C₂²×C₄	1	12	yes	complemented
55	C₂³×C₃×C₄	—	3	3	no	—
56	C₂²×C₄×D₃	C₂×C₉	1	3	yes	—
57	C₂×C₃×C₄²	—	2	1	no	—
58	C₂×C₃×C₄²	—	3	3	no	—
59	C₂×C₃×C₄²	—	3	8	no	—
60	C₂×C₃×C₄²	C₂×C₉	1	1	yes	characteristic, complemented
61	C₂×C₃×C₄²	C₃×D₃	1	1	yes	characteristic
62	C₂×C₄×Dic₃	C₂×C₉	1	3	yes	complemented
63	C₄²×D₃	C₂×C₉	1	8	yes	direct product factor
64	C₂³×C₃²	C₂³×C₃	1	1	yes	characteristic
65	C₂²×C₃×D₃	C₂×C₃×C₄	1	12	yes	—
66	C₂²×C₃×D₃	C₂³×C₃	1	2	yes	—
67	C₂×C₃²×C₄	C₂×C₃×C₄	1	12	yes	—
68	C₂×C₃²×C₄	C₂³×C₃	1	6	yes	—
69	C₂×C₃×Dic₃	C₂×C₃×C₄	1	12	yes	—
70	C₂×C₃×Dic₃	C₂³×C₃	1	6	yes	—
71	C₃×C₄×D₃	C₂×C₃×C₄	1	48	yes	—
72	C₂³×C₉	—	2	1	no	—
73	C₂³×C₉	—	3	12	no	—
74	C₂³×C₉	—	3	2	no	—
75	C₂³×C₉	C₂²×D₃	1	1	yes	characteristic
76	C₂×C₄×C₉	—	2	12	no	—
77	C₂×C₄×C₉	—	2	6	no	—
78	C₂×C₄×C₉	—	3	12	no	—
79	C₂×C₄×C₉	—	3	6	no	—
80	C₂×C₄×C₉	—	3	48	no	—
81	C₂×C₄×C₉	C₄×D₃	1	12	yes	direct product factor
82	C₂×C₄×C₉	C₂²×D₃	1	6	yes	—
83	C₂²×C₄²	—	3	1	no	Sylow
84	C₂×C₃×C₉	C₂×C₄²	1	4	yes	complemented
85	C₂×C₃×C₉	C₂³×C₄	1	3	yes	—
86	D₃×C₉	C₂×C₄²	1	8	yes	direct product factor
87	C₂⁴×C₃	—	3	1	no	—
88	C₂³×D₃	C₂²×C₉	1	1	yes	characteristic
89	C₂²×C₃×C₄	—	2	3	no	—
90	C₂²×C₃×C₄	—	3	3	no	—
91	C₂²×C₃×C₄	—	3	24	no	—
92	C₂²×C₃×C₄	—	3	12	no	—
93	C₂²×C₃×C₄	C₂²×C₉	1	3	yes	—
94	C₂²×C₃×C₄	C₂×C₃×D₃	1	3	yes	—
95	C₂²×Dic₃	C₂²×C₉	1	3	yes	—
96	C₂×C₄×D₃	C₄×C₉	1	24	yes	direct product factor
97	C₂×C₄×D₃	C₂²×C₉	1	12	yes	—
98	C₃×C₄²	—	2	4	no	—
99	C₃×C₄²	—	3	12	no	—
100	C₃×C₄²	C₂²×C₉	1	4	yes	complemented
101	C₃×C₄²	C₂×C₃×D₃	1	4	yes	—
102	C₄×Dic₃	C₂²×C₉	1	12	yes	complemented
103	C₂²×C₃²	C₂²×C₃×C₄	1	6	yes	—
104	C₂²×C₃²	C₂⁴×C₃	1	1	yes	characteristic
105	C₂×C₃×D₃	C₃×C₄²	1	16	yes	—
106	C₂×C₃×D₃	C₂²×C₃×C₄	1	12	yes	—
107	C₃²×C₄	C₂²×C₃×C₄	1	12	yes	—
108	C₃×Dic₃	C₂²×C₃×C₄	1	12	yes	—
109	C₂²×C₉	—	2	1	no	—
110	C₂²×C₉	—	2	6	no	—
111	C₂²×C₉	—	3	16	no	—
112	C₂²×C₉	—	3	12	no	—
113	C₂²×C₉	C₂×C₄×D₃	1	6	yes	—
114	C₂²×C₉	C₂³×D₃	1	1	yes	characteristic
115	C₄×C₉	—	2	12	no	—
116	C₄×C₉	—	3	12	no	—
117	C₄×C₉	C₂×C₄×D₃	1	12	yes	direct product factor
118	C₂³×C₄	—	3	3	no	—
119	C₂×C₄²	—	3	3	no	—
120	C₂×C₄²	—	3	8	no	—
121	C₂×C₄²	D₃×C₉	1	1	yes	characteristic, direct product factor
122	C₃×C₉	C₂²×C₄²	1	1	yes	characteristic, complemented, Sylow
123	C₂³×C₃	—	2	1	no	—
124	C₂³×C₃	—	3	2	no	—
125	C₂³×C₃	—	3	12	no	—
126	C₂³×C₃	C₂³×C₉	1	1	yes	characteristic
127	C₂³×C₃	C₂²×C₃×D₃	1	1	yes	characteristic
128	C₂²×D₃	C₂×C₄×C₉	1	12	yes	—
129	C₂²×D₃	C₂³×C₉	1	2	yes	—
130	C₂×C₃×C₄	—	2	12	no	—
131	C₂×C₃×C₄	—	2	6	no	—
132	C₂×C₃×C₄	—	3	6	no	—
133	C₂×C₃×C₄	—	3	12	no	—
134	C₂×C₃×C₄	—	3	48	no	—
135	C₂×C₃×C₄	C₂×C₄×C₉	1	12	yes	complemented
136	C₂×C₃×C₄	C₂³×C₉	1	6	yes	—
137	C₂×C₃×C₄	C₃×C₄×D₃	1	12	yes	—
138	C₂×C₃×C₄	C₂²×C₃×D₃	1	6	yes	—
139	C₂×Dic₃	C₂×C₄×C₉	1	12	yes	complemented
140	C₂×Dic₃	C₂³×C₉	1	6	yes	—
141	C₄×D₃	C₂×C₄×C₉	1	48	yes	direct product factor
142	C₂×C₃²	C₂×C₃×C₄²	1	4	yes	—
143	C₂×C₃²	C₂³×C₃×C₄	1	3	yes	—
144	C₃×D₃	C₂×C₃×C₄²	1	8	yes	—
145	C₂×C₉	—	2	3	no	—
146	C₂×C₉	—	2	4	no	—
147	C₂×C₉	—	3	8	no	—
148	C₂×C₉	C₄²×D₃	1	4	yes	direct product factor
149	C₂×C₉	C₂²×C₄×D₃	1	3	yes	—
150	C₂⁴	—	3	1	no	—
151	C₂²×C₄	—	3	3	no	—
152	C₂²×C₄	—	3	24	no	—
153	C₂²×C₄	—	3	12	no	—
154	C₂²×C₄	C₂×D₃×C₉	1	3	yes	—
155	C₄²	—	3	12	no	—
156	C₄²	C₂×D₃×C₉	1	4	yes	direct product factor
157	C₂²×C₃	—	2	1	no	—
158	C₂²×C₃	—	2	6	no	—
159	C₂²×C₃	—	3	12	no	—
160	C₂²×C₃	—	3	16	no	—
161	C₂²×C₃	C₂²×C₄×C₉	1	6	yes	—
162	C₂²×C₃	C₂⁴×C₉	1	1	yes	characteristic
163	C₂²×C₃	C₂×C₃×C₄×D₃	1	6	yes	—
164	C₂²×C₃	C₂³×C₃×D₃	1	1	yes	characteristic
165	C₂×D₃	C₄²×C₉	1	16	yes	direct product factor
166	C₂×D₃	C₂²×C₄×C₉	1	12	yes	—
167	C₃×C₄	—	2	12	no	—
168	C₃×C₄	—	3	12	no	—
169	C₃×C₄	C₂²×C₄×C₉	1	12	yes	complemented
170	C₃×C₄	C₂×C₃×C₄×D₃	1	12	yes	—
171	Dic₃	C₂²×C₄×C₉	1	12	yes	complemented
172	C₃²	C₂²×C₃×C₄²	1	1	yes	characteristic
173	C₉	—	2	1	no	—
174	C₉	C₂×C₄²×D₃	1	1	yes	characteristic, direct product factor
175	C₂³	—	3	2	no	—
176	C₂³	—	3	12	no	—
177	C₂³	C₂²×D₃×C₉	1	1	yes	characteristic
178	C₂×C₄	—	3	6	no	—
179	C₂×C₄	—	3	12	no	—
180	C₂×C₄	—	3	48	no	—
181	C₂×C₄	C₄×D₃×C₉	1	12	yes	direct product factor
182	C₂×C₄	C₂²×D₃×C₉	1	6	yes	—
183	C₂×C₃	—	2	3	no	—
184	C₂×C₃	—	2	4	no	—
185	C₂×C₃	—	3	8	no	—
186	C₂×C₃	C₂×C₄²×C₉	1	4	yes	complemented
187	C₂×C₃	C₂³×C₄×C₉	1	3	yes	—
188	C₂×C₃	C₃×C₄²×D₃	1	4	yes	—
189	C₂×C₃	C₂²×C₃×C₄×D₃	1	3	yes	—
190	D₃	C₂×C₄²×C₉	1	8	yes	direct product factor
191	C₂²	—	3	12	no	—
192	C₂²	—	3	16	no	—
193	C₂²	C₂×C₄×D₃×C₉	1	6	yes	—
194	C₂²	C₂³×D₃×C₉	1	1	yes	characteristic
195	C₄	—	3	12	no	—
196	C₄	C₂×C₄×D₃×C₉	1	12	yes	direct product factor
197	C₃	—	2	1	no	—
198	C₃	C₂²×C₄²×C₉	1	1	yes	characteristic, complemented
199	C₃	C₂×C₃×C₄²×D₃	1	1	yes	characteristic
200	C₂	—	3	8	no	—
201	C₂	C₄²×D₃×C₉	1	4	yes	direct product factor
202	C₂	C₂²×C₄×D₃×C₉	1	3	yes	—

	subgroup	G/H
center, hypercenter	C₂×C₄²×C₉	D₃
commutator, hypocenter	C₃	C₂²×C₄²×C₉
Frattini	C₂²×C₃	C₂³×C₃×D₃
Fitting	C₂×C₃×C₄²×C₉	C₂
socle	C₂³×C₃²	C₂³×C₃
prime radical	C₂³×C₃×D₃	C₂²×C₃
cosocle	C₂²×C₃²	C₂⁴×C₃
LCS	G ▷ C₃
UCS	C₁ ◁ C₂×C₄²×C₉

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₄²×C₉	C₂	semidirect
3. C₂×C₄×C₉×Dic₃	C₂	semidirect
4. C₄²×D₃×C₉	C₂	direct (N × H)
10. C₂×C₄×D₃×C₉	C₄	direct (N × H)
12. C₃×C₄²×C₉	C₂²	semidirect
13. C₄×C₉×Dic₃	C₂²	semidirect
22. C₂×C₄²×C₉	D₃	direct (N × H)
26. C₂×C₃×C₄×C₉	C₂×C₄	semidirect
28. C₂×C₉×Dic₃	C₂×C₄	semidirect
30. C₄×D₃×C₉	C₂×C₄	direct (N × H)
32. C₂×C₄²×D₃	C₉	direct (N × H)
48. C₄²×C₉	C₂×D₃	direct (N × H)
51. C₂×D₃×C₉	C₄²	direct (N × H)
53. C₃×C₄×C₉	C₂²×C₄	semidirect
54. C₉×Dic₃	C₂²×C₄	semidirect
60. C₂×C₃×C₄²	C₂×C₉	semidirect
62. C₂×C₄×Dic₃	C₂×C₉	semidirect
63. C₄²×D₃	C₂×C₉	direct (N × H)
81. C₂×C₄×C₉	C₄×D₃	direct (N × H)
84. C₂×C₃×C₉	C₂×C₄²	semidirect
86. D₃×C₉	C₂×C₄²	direct (N × H)
96. C₂×C₄×D₃	C₄×C₉	direct (N × H)
100. C₃×C₄²	C₂²×C₉	semidirect
102. C₄×Dic₃	C₂²×C₉	semidirect
117. C₄×C₉	C₂×C₄×D₃	direct (N × H)
121. C₂×C₄²	D₃×C₉	direct (N × H)
122. C₃×C₉	C₂²×C₄²	semidirect
135. C₂×C₃×C₄	C₂×C₄×C₉	semidirect
139. C₂×Dic₃	C₂×C₄×C₉	semidirect
141. C₄×D₃	C₂×C₄×C₉	direct (N × H)
148. C₂×C₉	C₄²×D₃	direct (N × H)
156. C₄²	C₂×D₃×C₉	direct (N × H)
165. C₂×D₃	C₄²×C₉	direct (N × H)
169. C₃×C₄	C₂²×C₄×C₉	semidirect
171. Dic₃	C₂²×C₄×C₉	semidirect
174. C₉	C₂×C₄²×D₃	direct (N × H)
181. C₂×C₄	C₄×D₃×C₉	direct (N × H)
186. C₂×C₃	C₂×C₄²×C₉	semidirect
190. D₃	C₂×C₄²×C₉	direct (N × H)
196. C₄	C₂×C₄×D₃×C₉	direct (N × H)
198. C₃	C₂²×C₄²×C₉	semidirect
201. C₂	C₄²×D₃×C₉	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×C₄×D₃×C₉	2
C₂×C₃×C₄²×C₉	2
C₂×C₄×C₉×Dic₃	2
C₄²×D₃×C₉	2
C₂×C₃×C₄²×D₃	3
C₂²×C₄²×C₉	3

Subgroups — C2×C42×D3×C9 #13152288

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄²×D₃×C₉ #13152288