Subgroups — C₂×C₃×C₄²×D₉ #13152279

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×C₃×C₄×D₉	C₂	1	3	yes	maximal
2	C₂×C₃×C₄²×C₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×C₄×Dic₉	C₂	1	3	yes	maximal, complemented
4	C₃×C₄²×D₉	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃×C₄²×D₃	—	3	1	no	maximal
6	C₂×C₄²×D₉	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₂³×C₃×D₉	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₄×C₉	C₂²	1	3	yes	—
9	C₂²×C₃×Dic₉	C₂²	1	3	yes	—
10	C₂×C₃×C₄×D₉	C₄	1	24	yes	direct product factor
11	C₂×C₃×C₄×D₉	C₂²	1	12	yes	—
12	C₃×C₄²×C₉	C₂²	1	4	yes	complemented
13	C₃×C₄×Dic₉	C₂²	1	12	yes	complemented
14	C₂²×C₃×C₄×D₃	—	3	3	no	—
15	C₂×C₃²×C₄²	D₃	1	1	yes	characteristic
16	C₂×C₃×C₄×Dic₃	—	3	3	no	—
17	C₃×C₄²×D₃	—	3	8	no	—
18	C₂²×C₄×D₉	C₂×C₃	1	3	yes	—
19	C₂×C₄²×C₉	—	2	1	no	—
20	C₂×C₄²×C₉	C₂×C₃	1	1	yes	characteristic, complemented
21	C₂×C₄×Dic₉	C₂×C₃	1	3	yes	complemented
22	C₄²×D₉	C₂×C₃	1	8	yes	direct product factor
23	C₂³×C₃×C₉	C₂³	1	1	yes	characteristic
24	C₂²×C₃×D₉	C₂×C₄	1	12	yes	—
25	C₂²×C₃×D₉	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×C₃×Dic₉	C₂×C₄	1	12	yes	complemented
29	C₂×C₃×Dic₉	C₂³	1	6	yes	—
30	C₃×C₄×D₉	C₂×C₄	1	48	yes	direct product factor
31	C₂²×C₃×C₄²	—	9	1	no	—
32	C₂×C₄²×D₃	—	3	1	no	—
33	C₂³×C₃×D₃	—	3	1	no	—
34	C₂²×C₃²×C₄	C₂×D₃	1	3	yes	—
35	C₂²×C₃×Dic₃	—	3	3	no	—
36	C₂×C₃×C₄×D₃	—	3	24	no	—
37	C₂×C₃×C₄×D₃	—	3	12	no	—
38	C₂³×D₉	C₂²×C₃	1	1	yes	characteristic
39	C₃²×C₄²	C₂×D₃	1	4	yes	—
40	C₃×C₄×Dic₃	—	3	12	no	—
41	C₂²×C₄×C₉	—	2	3	no	—
42	C₂²×C₄×C₉	C₂²×C₃	1	3	yes	—
43	C₂²×Dic₉	C₂²×C₃	1	3	yes	—
44	C₂×C₄×D₉	C₃×C₄	1	24	yes	direct product factor
45	C₂×C₄×D₉	C₂²×C₃	1	12	yes	—
46	C₄²×C₉	—	2	4	no	—
47	C₄²×C₉	C₂²×C₃	1	4	yes	complemented
48	C₄×Dic₉	C₂²×C₃	1	12	yes	complemented
49	C₂²×C₃×C₉	C₂²×C₄	1	6	yes	—
50	C₂²×C₃×C₉	C₂⁴	1	1	yes	characteristic
51	C₂×C₃×D₉	C₄²	1	16	yes	direct product factor
52	C₂×C₃×D₉	C₂²×C₄	1	12	yes	—
53	C₃×C₄×C₉	C₂²×C₄	1	12	yes	complemented
54	C₃×Dic₉	C₂²×C₄	1	12	yes	complemented
55	C₂³×C₃×C₄	—	9	3	no	—
56	C₂²×C₄×D₃	—	3	3	no	—
57	C₂×C₃×C₄²	—	2	1	no	—
58	C₂×C₃×C₄²	—	9	3	no	—
59	C₂×C₃×C₄²	—	9	8	no	—
60	C₂×C₃×C₄²	D₉	1	1	yes	characteristic, direct product factor
61	C₂×C₃×C₄²	C₃×D₃	1	1	yes	characteristic
62	C₂×C₄×Dic₃	—	3	3	no	—
63	C₄²×D₃	—	3	8	no	—
64	C₂³×C₃²	C₂²×D₃	1	1	yes	characteristic
65	C₂²×C₃×D₃	—	3	12	no	—
66	C₂²×C₃×D₃	—	3	2	no	—
67	C₂×C₃²×C₄	C₄×D₃	1	12	yes	—
68	C₂×C₃²×C₄	C₂²×D₃	1	6	yes	—
69	C₂×C₃×Dic₃	—	3	12	no	—
70	C₂×C₃×Dic₃	—	3	6	no	—
71	C₃×C₄×D₃	—	3	48	no	—
72	C₂³×C₉	—	2	1	no	—
73	C₂³×C₉	C₂³×C₃	1	1	yes	characteristic
74	C₂²×D₉	C₂×C₃×C₄	1	12	yes	—
75	C₂²×D₉	C₂³×C₃	1	2	yes	—
76	C₂×C₄×C₉	—	2	12	no	—
77	C₂×C₄×C₉	—	2	6	no	—
78	C₂×C₄×C₉	C₂×C₃×C₄	1	12	yes	complemented
79	C₂×C₄×C₉	C₂³×C₃	1	6	yes	—
80	C₂×Dic₉	C₂×C₃×C₄	1	12	yes	complemented
81	C₂×Dic₉	C₂³×C₃	1	6	yes	—
82	C₄×D₉	C₂×C₃×C₄	1	48	yes	direct product factor
83	C₂²×C₄²	—	9	1	no	Sylow
84	C₂×C₃×C₉	C₂×C₄²	1	4	yes	complemented
85	C₂×C₃×C₉	C₂³×C₄	1	3	yes	—
86	C₃×D₉	C₂×C₄²	1	8	yes	direct product factor
87	C₂⁴×C₃	—	9	1	no	—
88	C₂³×D₃	—	3	1	no	—
89	C₂²×C₃×C₄	—	2	3	no	—
90	C₂²×C₃×C₄	—	9	3	no	—
91	C₂²×C₃×C₄	—	9	24	no	—
92	C₂²×C₃×C₄	—	9	12	no	—
93	C₂²×C₃×C₄	C₂×D₉	1	3	yes	—
94	C₂²×C₃×C₄	C₂×C₃×D₃	1	3	yes	—
95	C₂²×Dic₃	—	3	3	no	—
96	C₂×C₄×D₃	—	3	24	no	—
97	C₂×C₄×D₃	—	3	12	no	—
98	C₃×C₄²	—	2	4	no	—
99	C₃×C₄²	—	9	12	no	—
100	C₃×C₄²	C₂×D₉	1	4	yes	direct product factor
101	C₃×C₄²	C₂×C₃×D₃	1	4	yes	—
102	C₄×Dic₃	—	3	12	no	—
103	C₂²×C₃²	C₂×C₄×D₃	1	6	yes	—
104	C₂²×C₃²	C₂³×D₃	1	1	yes	characteristic
105	C₂×C₃×D₃	—	3	16	no	—
106	C₂×C₃×D₃	—	3	12	no	—
107	C₃²×C₄	C₂×C₄×D₃	1	12	yes	—
108	C₃×Dic₃	—	3	12	no	—
109	C₂²×C₉	—	2	1	no	—
110	C₂²×C₉	—	2	6	no	—
111	C₂²×C₉	C₂²×C₃×C₄	1	6	yes	—
112	C₂²×C₉	C₂⁴×C₃	1	1	yes	characteristic
113	C₂×D₉	C₃×C₄²	1	16	yes	direct product factor
114	C₂×D₉	C₂²×C₃×C₄	1	12	yes	—
115	C₄×C₉	—	2	12	no	—
116	C₄×C₉	C₂²×C₃×C₄	1	12	yes	complemented
117	Dic₉	C₂²×C₃×C₄	1	12	yes	complemented
118	C₂³×C₄	—	9	3	no	—
119	C₂×C₄²	—	9	3	no	—
120	C₂×C₄²	—	9	8	no	—
121	C₂×C₄²	C₃×D₉	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	—	9	12	no	—
125	C₂³×C₃	—	9	2	no	—
126	C₂³×C₃	C₂²×D₉	1	1	yes	characteristic
127	C₂³×C₃	C₂²×C₃×D₃	1	1	yes	characteristic
128	C₂²×D₃	—	3	2	no	—
129	C₂²×D₃	—	3	12	no	—
130	C₂×C₃×C₄	—	2	12	no	—
131	C₂×C₃×C₄	—	2	6	no	—
132	C₂×C₃×C₄	—	9	12	no	—
133	C₂×C₃×C₄	—	9	6	no	—
134	C₂×C₃×C₄	—	9	48	no	—
135	C₂×C₃×C₄	C₄×D₉	1	12	yes	direct product factor
136	C₂×C₃×C₄	C₂²×D₉	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	—	3	6	no	—
140	C₂×Dic₃	—	3	12	no	—
141	C₄×D₃	—	3	48	no	—
142	C₂×C₃²	C₄²×D₃	1	4	yes	—
143	C₂×C₃²	C₂²×C₄×D₃	1	3	yes	—
144	C₃×D₃	—	3	8	no	—
145	C₂×C₉	—	2	3	no	—
146	C₂×C₉	—	2	4	no	—
147	C₂×C₉	C₂×C₃×C₄²	1	4	yes	complemented
148	C₂×C₉	C₂³×C₃×C₄	1	3	yes	—
149	D₉	C₂×C₃×C₄²	1	8	yes	direct product factor
150	C₂⁴	—	9	1	no	—
151	C₂²×C₄	—	9	3	no	—
152	C₂²×C₄	—	9	24	no	—
153	C₂²×C₄	—	9	12	no	—
154	C₂²×C₄	C₂×C₃×D₉	1	3	yes	—
155	C₄²	—	9	12	no	—
156	C₄²	C₂×C₃×D₉	1	4	yes	direct product factor
157	C₂²×C₃	—	2	1	no	—
158	C₂²×C₃	—	2	6	no	—
159	C₂²×C₃	—	9	16	no	—
160	C₂²×C₃	—	9	12	no	—
161	C₂²×C₃	C₂×C₄×D₉	1	6	yes	—
162	C₂²×C₃	C₂³×D₉	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	—	3	12	no	—
166	C₂×D₃	—	3	16	no	—
167	C₃×C₄	—	2	12	no	—
168	C₃×C₄	—	9	12	no	—
169	C₃×C₄	C₂×C₄×D₉	1	12	yes	direct product factor
170	C₃×C₄	C₂×C₃×C₄×D₃	1	12	yes	—
171	Dic₃	—	3	12	no	—
172	C₃²	C₂×C₄²×D₃	1	1	yes	characteristic
173	C₉	—	2	1	no	—
174	C₉	C₂²×C₃×C₄²	1	1	yes	characteristic, complemented
175	C₂³	—	9	2	no	—
176	C₂³	—	9	12	no	—
177	C₂³	C₂²×C₃×D₉	1	1	yes	characteristic
178	C₂×C₄	—	9	6	no	—
179	C₂×C₄	—	9	12	no	—
180	C₂×C₄	—	9	48	no	—
181	C₂×C₄	C₃×C₄×D₉	1	12	yes	direct product factor
182	C₂×C₄	C₂²×C₃×D₉	1	6	yes	—
183	C₂×C₃	—	2	3	no	—
184	C₂×C₃	—	2	4	no	—
185	C₂×C₃	—	9	8	no	—
186	C₂×C₃	C₄²×D₉	1	4	yes	direct product factor
187	C₂×C₃	C₂²×C₄×D₉	1	3	yes	—
188	C₂×C₃	C₃×C₄²×D₃	1	4	yes	—
189	C₂×C₃	C₂²×C₃×C₄×D₃	1	3	yes	—
190	D₃	—	3	8	no	—
191	C₂²	—	9	12	no	—
192	C₂²	—	9	16	no	—
193	C₂²	C₂×C₃×C₄×D₉	1	6	yes	—
194	C₂²	C₂³×C₃×D₉	1	1	yes	characteristic
195	C₄	—	9	12	no	—
196	C₄	C₂×C₃×C₄×D₉	1	12	yes	direct product factor
197	C₃	—	2	1	no	—
198	C₃	C₂×C₄²×D₉	1	1	yes	characteristic, direct product factor
199	C₃	C₂×C₃×C₄²×D₃	1	1	yes	characteristic
200	C₂	—	9	8	no	—
201	C₂	C₃×C₄²×D₉	1	4	yes	direct product factor
202	C₂	C₂²×C₃×C₄×D₉	1	3	yes	—

	subgroup	G/H
center, hypercenter	C₂×C₃×C₄²	D₉
commutator, hypocenter	C₉	C₂²×C₃×C₄²
Frattini	C₂²×C₃	C₂³×C₃×D₃
Fitting	C₂×C₃×C₄²×C₉	C₂
socle	C₂³×C₃²	C₂²×D₃
prime radical	C₂³×C₃×D₉	C₂²
cosocle	C₂²×C₉	C₂⁴×C₃
LCS	G ▷ C₉
UCS	C₁ ◁ C₂×C₃×C₄²

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×C₃×C₄×Dic₉	C₂	semidirect
4. C₃×C₄²×D₉	C₂	direct (N × H)
6. C₂×C₄²×D₉	C₃	direct (N × H)
10. C₂×C₃×C₄×D₉	C₄	direct (N × H)
12. C₃×C₄²×C₉	C₂²	semidirect
13. C₃×C₄×Dic₉	C₂²	semidirect
20. C₂×C₄²×C₉	C₂×C₃	semidirect
21. C₂×C₄×Dic₉	C₂×C₃	semidirect
22. C₄²×D₉	C₂×C₃	direct (N × H)
26. C₂×C₃×C₄×C₉	C₂×C₄	semidirect
28. C₂×C₃×Dic₉	C₂×C₄	semidirect
30. C₃×C₄×D₉	C₂×C₄	direct (N × H)
44. C₂×C₄×D₉	C₃×C₄	direct (N × H)
47. C₄²×C₉	C₂²×C₃	semidirect
48. C₄×Dic₉	C₂²×C₃	semidirect
51. C₂×C₃×D₉	C₄²	direct (N × H)
53. C₃×C₄×C₉	C₂²×C₄	semidirect
54. C₃×Dic₉	C₂²×C₄	semidirect
60. C₂×C₃×C₄²	D₉	direct (N × H)
78. C₂×C₄×C₉	C₂×C₃×C₄	semidirect
80. C₂×Dic₉	C₂×C₃×C₄	semidirect
82. C₄×D₉	C₂×C₃×C₄	direct (N × H)
84. C₂×C₃×C₉	C₂×C₄²	semidirect
86. C₃×D₉	C₂×C₄²	direct (N × H)
100. C₃×C₄²	C₂×D₉	direct (N × H)
113. C₂×D₉	C₃×C₄²	direct (N × H)
116. C₄×C₉	C₂²×C₃×C₄	semidirect
117. Dic₉	C₂²×C₃×C₄	semidirect
121. C₂×C₄²	C₃×D₉	direct (N × H)
122. C₃×C₉	C₂²×C₄²	semidirect
135. C₂×C₃×C₄	C₄×D₉	direct (N × H)
147. C₂×C₉	C₂×C₃×C₄²	semidirect
149. D₉	C₂×C₃×C₄²	direct (N × H)
156. C₄²	C₂×C₃×D₉	direct (N × H)
169. C₃×C₄	C₂×C₄×D₉	direct (N × H)
174. C₉	C₂²×C₃×C₄²	semidirect
181. C₂×C₄	C₃×C₄×D₉	direct (N × H)
186. C₂×C₃	C₄²×D₉	direct (N × H)
196. C₄	C₂×C₃×C₄×D₉	direct (N × H)
198. C₃	C₂×C₄²×D₉	direct (N × H)
201. C₂	C₃×C₄²×D₉	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×C₃×C₄×D₉	2
C₂×C₃×C₄²×C₉	2
C₂×C₃×C₄×Dic₉	2
C₃×C₄²×D₉	2
C₂×C₃×C₄²×D₃	3
C₂×C₄²×D₉	3

Subgroups — C2×C3×C42×D9 #13152279

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₄²×D₉ #13152279