Subgroups — C₂³×C₄×D₃×C₉ #13164896

Order 1728.

4322 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	C₂⁴×D₃×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₃×C₄×C₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂³×C₉×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₄×D₃×C₉	C₂	1	28	yes	maximal, direct product factor
5	C₂³×C₃×C₄×D₃	C₃	1	1	yes	maximal, characteristic
6	C₂⁴×C₄×C₉	—	3	1	no	maximal
7	C₂⁴×C₃×C₉	C₂²	1	1	yes	characteristic
8	C₂³×D₃×C₉	C₄	1	16	yes	direct product factor
9	C₂³×D₃×C₉	C₂²	1	14	yes	—
10	C₂²×C₃×C₄×C₉	C₂²	1	14	yes	complemented
11	C₂²×C₉×Dic₃	C₂²	1	14	yes	complemented
12	C₂×C₄×D₃×C₉	C₂²	1	112	yes	direct product factor
13	C₂⁴×C₃×D₃	C₂×C₃	1	1	yes	characteristic
14	C₂³×C₃²×C₄	C₂×C₃	1	1	yes	characteristic
15	C₂³×C₃×Dic₃	C₂×C₃	1	1	yes	characteristic
16	C₂²×C₃×C₄×D₃	C₂×C₃	1	28	yes	—
17	C₂⁵×C₉	—	3	1	no	—
18	C₂³×C₄×C₉	—	2	1	no	—
19	C₂³×C₄×C₉	—	3	1	no	—
20	C₂³×C₄×C₉	—	3	28	no	—
21	C₂³×C₄×C₉	D₃	1	1	yes	characteristic, direct product factor
22	C₂³×C₃×C₉	C₂×C₄	1	8	yes	complemented
23	C₂³×C₃×C₉	C₂³	1	7	yes	—
24	C₂²×D₃×C₉	C₂×C₄	1	112	yes	direct product factor
25	C₂²×D₃×C₉	C₂³	1	28	yes	—
26	C₂×C₃×C₄×C₉	C₂³	1	28	yes	complemented
27	C₂×C₉×Dic₃	C₂³	1	28	yes	complemented
28	C₄×D₃×C₉	C₂³	1	64	yes	direct product factor
29	C₂⁴×C₃×C₄	—	3	1	no	—
30	C₂³×C₄×D₃	C₉	1	1	yes	characteristic, direct product factor
31	C₂⁴×C₃²	C₂²×C₃	1	1	yes	characteristic
32	C₂³×C₃×D₃	C₃×C₄	1	16	yes	—
33	C₂³×C₃×D₃	C₂²×C₃	1	14	yes	—
34	C₂²×C₃²×C₄	C₂²×C₃	1	14	yes	—
35	C₂²×C₃×Dic₃	C₂²×C₃	1	14	yes	—
36	C₂×C₃×C₄×D₃	C₂²×C₃	1	112	yes	—
37	C₂⁴×C₉	—	2	1	no	—
38	C₂⁴×C₉	—	3	16	no	—
39	C₂⁴×C₉	—	3	14	no	—
40	C₂⁴×C₉	C₂×D₃	1	1	yes	characteristic
41	C₂²×C₄×C₉	—	2	14	no	—
42	C₂²×C₄×C₉	—	3	14	no	—
43	C₂²×C₄×C₉	—	3	112	no	—
44	C₂²×C₄×C₉	C₂×D₃	1	14	yes	direct product factor
45	C₂²×C₃×C₉	C₂²×C₄	1	28	yes	complemented
46	C₂²×C₃×C₉	C₂⁴	1	7	yes	—
47	C₂×D₃×C₉	C₂²×C₄	1	112	yes	direct product factor
48	C₂×D₃×C₉	C₂⁴	1	8	yes	—
49	C₃×C₄×C₉	C₂⁴	1	8	yes	complemented
50	C₉×Dic₃	C₂⁴	1	8	yes	complemented
51	C₂⁵×C₃	—	3	1	no	—
52	C₂⁴×D₃	C₂×C₉	1	1	yes	characteristic
53	C₂³×C₃×C₄	—	2	1	no	—
54	C₂³×C₃×C₄	—	3	1	no	—
55	C₂³×C₃×C₄	—	3	28	no	—
56	C₂³×C₃×C₄	C₂×C₉	1	1	yes	characteristic, complemented
57	C₂³×C₃×C₄	C₃×D₃	1	1	yes	characteristic
58	C₂³×Dic₃	C₂×C₉	1	1	yes	characteristic, complemented
59	C₂²×C₄×D₃	C₂×C₉	1	28	yes	direct product factor
60	C₂³×C₃²	C₂×C₃×C₄	1	8	yes	—
61	C₂³×C₃²	C₂³×C₃	1	7	yes	—
62	C₂²×C₃×D₃	C₂×C₃×C₄	1	112	yes	—
63	C₂²×C₃×D₃	C₂³×C₃	1	28	yes	—
64	C₂×C₃²×C₄	C₂³×C₃	1	28	yes	—
65	C₂×C₃×Dic₃	C₂³×C₃	1	28	yes	—
66	C₃×C₄×D₃	C₂³×C₃	1	64	yes	—
67	C₂³×C₉	—	2	8	no	—
68	C₂³×C₉	—	2	7	no	—
69	C₂³×C₉	—	3	112	no	—
70	C₂³×C₉	—	3	28	no	—
71	C₂³×C₉	C₄×D₃	1	8	yes	direct product factor
72	C₂³×C₉	C₂²×D₃	1	7	yes	—
73	C₂×C₄×C₉	—	2	28	no	—
74	C₂×C₄×C₉	—	3	28	no	—
75	C₂×C₄×C₉	—	3	64	no	—
76	C₂×C₄×C₉	C₂²×D₃	1	28	yes	direct product factor
77	C₂⁴×C₄	—	3	1	no	Sylow
78	C₂×C₃×C₉	C₂³×C₄	1	14	yes	complemented
79	C₂×C₃×C₉	C₂⁵	1	1	yes	characteristic
80	D₃×C₉	C₂³×C₄	1	16	yes	direct product factor
81	C₂⁴×C₃	—	2	1	no	—
82	C₂⁴×C₃	—	3	16	no	—
83	C₂⁴×C₃	—	3	14	no	—
84	C₂⁴×C₃	C₂²×C₉	1	1	yes	characteristic
85	C₂⁴×C₃	C₂×C₃×D₃	1	1	yes	characteristic
86	C₂³×D₃	C₄×C₉	1	16	yes	direct product factor
87	C₂³×D₃	C₂²×C₉	1	14	yes	—
88	C₂²×C₃×C₄	—	2	14	no	—
89	C₂²×C₃×C₄	—	3	14	no	—
90	C₂²×C₃×C₄	—	3	112	no	—
91	C₂²×C₃×C₄	C₂²×C₉	1	14	yes	complemented
92	C₂²×C₃×C₄	C₂×C₃×D₃	1	14	yes	—
93	C₂²×Dic₃	C₂²×C₉	1	14	yes	complemented
94	C₂×C₄×D₃	C₂²×C₉	1	112	yes	direct product factor
95	C₂²×C₃²	C₂²×C₃×C₄	1	28	yes	—
96	C₂²×C₃²	C₂⁴×C₃	1	7	yes	—
97	C₂×C₃×D₃	C₂²×C₃×C₄	1	112	yes	—
98	C₂×C₃×D₃	C₂⁴×C₃	1	8	yes	—
99	C₃²×C₄	C₂⁴×C₃	1	8	yes	—
100	C₃×Dic₃	C₂⁴×C₃	1	8	yes	—
101	C₂²×C₉	—	2	7	no	—
102	C₂²×C₉	—	2	28	no	—
103	C₂²×C₉	—	3	112	no	—
104	C₂²×C₉	—	3	8	no	—
105	C₂²×C₉	C₂×C₄×D₃	1	28	yes	direct product factor
106	C₂²×C₉	C₂³×D₃	1	7	yes	—
107	C₄×C₉	—	2	8	no	—
108	C₄×C₉	—	3	8	no	—
109	C₄×C₉	C₂³×D₃	1	8	yes	direct product factor
110	C₂⁵	—	3	1	no	—
111	C₂³×C₄	—	3	1	no	—
112	C₂³×C₄	—	3	28	no	—
113	C₂³×C₄	D₃×C₉	1	1	yes	characteristic, direct product factor
114	C₃×C₉	C₂⁴×C₄	1	1	yes	characteristic, complemented, Sylow
115	C₂³×C₃	—	2	8	no	—
116	C₂³×C₃	—	2	7	no	—
117	C₂³×C₃	—	3	28	no	—
118	C₂³×C₃	—	3	112	no	—
119	C₂³×C₃	C₂×C₄×C₉	1	8	yes	complemented
120	C₂³×C₃	C₂³×C₉	1	7	yes	—
121	C₂³×C₃	C₃×C₄×D₃	1	8	yes	—
122	C₂³×C₃	C₂²×C₃×D₃	1	7	yes	—
123	C₂²×D₃	C₂×C₄×C₉	1	112	yes	direct product factor
124	C₂²×D₃	C₂³×C₉	1	28	yes	—
125	C₂×C₃×C₄	—	2	28	no	—
126	C₂×C₃×C₄	—	3	28	no	—
127	C₂×C₃×C₄	—	3	64	no	—
128	C₂×C₃×C₄	C₂³×C₉	1	28	yes	complemented
129	C₂×C₃×C₄	C₂²×C₃×D₃	1	28	yes	—
130	C₂×Dic₃	C₂³×C₉	1	28	yes	complemented
131	C₄×D₃	C₂³×C₉	1	64	yes	direct product factor
132	C₂×C₃²	C₂³×C₃×C₄	1	14	yes	—
133	C₂×C₃²	C₂⁵×C₃	1	1	yes	characteristic
134	C₃×D₃	C₂³×C₃×C₄	1	16	yes	—
135	C₂×C₉	—	2	14	no	—
136	C₂×C₉	—	2	1	no	—
137	C₂×C₉	—	3	16	no	—
138	C₂×C₉	C₂²×C₄×D₃	1	14	yes	direct product factor
139	C₂×C₉	C₂⁴×D₃	1	1	yes	characteristic
140	C₂⁴	—	3	16	no	—
141	C₂⁴	—	3	14	no	—
142	C₂⁴	C₂×D₃×C₉	1	1	yes	characteristic
143	C₂²×C₄	—	3	14	no	—
144	C₂²×C₄	—	3	112	no	—
145	C₂²×C₄	C₂×D₃×C₉	1	14	yes	direct product factor
146	C₂²×C₃	—	2	7	no	—
147	C₂²×C₃	—	2	28	no	—
148	C₂²×C₃	—	3	112	no	—
149	C₂²×C₃	—	3	8	no	—
150	C₂²×C₃	C₂²×C₄×C₉	1	28	yes	complemented
151	C₂²×C₃	C₂⁴×C₉	1	7	yes	—
152	C₂²×C₃	C₂×C₃×C₄×D₃	1	28	yes	—
153	C₂²×C₃	C₂³×C₃×D₃	1	7	yes	—
154	C₂×D₃	C₂²×C₄×C₉	1	112	yes	direct product factor
155	C₂×D₃	C₂⁴×C₉	1	8	yes	—
156	C₃×C₄	—	2	8	no	—
157	C₃×C₄	—	3	8	no	—
158	C₃×C₄	C₂⁴×C₉	1	8	yes	complemented
159	C₃×C₄	C₂³×C₃×D₃	1	8	yes	—
160	Dic₃	C₂⁴×C₉	1	8	yes	complemented
161	C₃²	C₂⁴×C₃×C₄	1	1	yes	characteristic
162	C₉	—	2	1	no	—
163	C₉	C₂³×C₄×D₃	1	1	yes	characteristic, direct product factor
164	C₂³	—	3	28	no	—
165	C₂³	—	3	112	no	—
166	C₂³	C₄×D₃×C₉	1	8	yes	direct product factor
167	C₂³	C₂²×D₃×C₉	1	7	yes	—
168	C₂×C₄	—	3	28	no	—
169	C₂×C₄	—	3	64	no	—
170	C₂×C₄	C₂²×D₃×C₉	1	28	yes	direct product factor
171	C₂×C₃	—	2	1	no	—
172	C₂×C₃	—	2	14	no	—
173	C₂×C₃	—	3	16	no	—
174	C₂×C₃	C₂³×C₄×C₉	1	14	yes	complemented
175	C₂×C₃	C₂⁵×C₉	1	1	yes	characteristic
176	C₂×C₃	C₂²×C₃×C₄×D₃	1	14	yes	—
177	C₂×C₃	C₂⁴×C₃×D₃	1	1	yes	characteristic
178	D₃	C₂³×C₄×C₉	1	16	yes	direct product factor
179	C₂²	—	3	8	no	—
180	C₂²	—	3	112	no	—
181	C₂²	C₂×C₄×D₃×C₉	1	28	yes	direct product factor
182	C₂²	C₂³×D₃×C₉	1	7	yes	—
183	C₄	—	3	8	no	—
184	C₄	C₂³×D₃×C₉	1	8	yes	direct product factor
185	C₃	—	2	1	no	—
186	C₃	C₂⁴×C₄×C₉	1	1	yes	characteristic, complemented
187	C₃	C₂³×C₃×C₄×D₃	1	1	yes	characteristic
188	C₂	—	3	16	no	—
189	C₂	C₂²×C₄×D₃×C₉	1	14	yes	direct product factor
190	C₂	C₂⁴×D₃×C₉	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂³×C₄×C₉	D₃
commutator, hypocenter	C₃	C₂⁴×C₄×C₉
Frattini	C₂×C₃	C₂⁴×C₃×D₃
Fitting	C₂³×C₃×C₄×C₉	C₂
socle	C₂⁴×C₃²	C₂²×C₃
prime radical	C₂⁴×C₃×D₃	C₂×C₃
cosocle	C₂×C₃²	C₂⁵×C₃
LCS	G ▷ C₃
UCS	C₁ ◁ C₂³×C₄×C₉

Complements

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

N	H ≅ G/N	type
2. C₂³×C₃×C₄×C₉	C₂	semidirect
3. C₂³×C₉×Dic₃	C₂	semidirect
4. C₂²×C₄×D₃×C₉	C₂	direct (N × H)
8. C₂³×D₃×C₉	C₄	direct (N × H)
10. C₂²×C₃×C₄×C₉	C₂²	semidirect
11. C₂²×C₉×Dic₃	C₂²	semidirect
12. C₂×C₄×D₃×C₉	C₂²	direct (N × H)
21. C₂³×C₄×C₉	D₃	direct (N × H)
22. C₂³×C₃×C₉	C₂×C₄	semidirect
24. C₂²×D₃×C₉	C₂×C₄	direct (N × H)
26. C₂×C₃×C₄×C₉	C₂³	semidirect
27. C₂×C₉×Dic₃	C₂³	semidirect
28. C₄×D₃×C₉	C₂³	direct (N × H)
30. C₂³×C₄×D₃	C₉	direct (N × H)
44. C₂²×C₄×C₉	C₂×D₃	direct (N × H)
45. C₂²×C₃×C₉	C₂²×C₄	semidirect
47. C₂×D₃×C₉	C₂²×C₄	direct (N × H)
49. C₃×C₄×C₉	C₂⁴	semidirect
50. C₉×Dic₃	C₂⁴	semidirect
56. C₂³×C₃×C₄	C₂×C₉	semidirect
58. C₂³×Dic₃	C₂×C₉	semidirect
59. C₂²×C₄×D₃	C₂×C₉	direct (N × H)
71. C₂³×C₉	C₄×D₃	direct (N × H)
76. C₂×C₄×C₉	C₂²×D₃	direct (N × H)
78. C₂×C₃×C₉	C₂³×C₄	semidirect
80. D₃×C₉	C₂³×C₄	direct (N × H)
86. C₂³×D₃	C₄×C₉	direct (N × H)
91. C₂²×C₃×C₄	C₂²×C₉	semidirect
93. C₂²×Dic₃	C₂²×C₉	semidirect
94. C₂×C₄×D₃	C₂²×C₉	direct (N × H)
105. C₂²×C₉	C₂×C₄×D₃	direct (N × H)
109. C₄×C₉	C₂³×D₃	direct (N × H)
113. C₂³×C₄	D₃×C₉	direct (N × H)
114. C₃×C₉	C₂⁴×C₄	semidirect
119. C₂³×C₃	C₂×C₄×C₉	semidirect
123. C₂²×D₃	C₂×C₄×C₉	direct (N × H)
128. C₂×C₃×C₄	C₂³×C₉	semidirect
130. C₂×Dic₃	C₂³×C₉	semidirect
131. C₄×D₃	C₂³×C₉	direct (N × H)
138. C₂×C₉	C₂²×C₄×D₃	direct (N × H)
145. C₂²×C₄	C₂×D₃×C₉	direct (N × H)
150. C₂²×C₃	C₂²×C₄×C₉	semidirect
154. C₂×D₃	C₂²×C₄×C₉	direct (N × H)
158. C₃×C₄	C₂⁴×C₉	semidirect
160. Dic₃	C₂⁴×C₉	semidirect
163. C₉	C₂³×C₄×D₃	direct (N × H)
166. C₂³	C₄×D₃×C₉	direct (N × H)
170. C₂×C₄	C₂²×D₃×C₉	direct (N × H)
174. C₂×C₃	C₂³×C₄×C₉	semidirect
178. D₃	C₂³×C₄×C₉	direct (N × H)
181. C₂²	C₂×C₄×D₃×C₉	direct (N × H)
184. C₄	C₂³×D₃×C₉	direct (N × H)
186. C₃	C₂⁴×C₄×C₉	semidirect
189. C₂	C₂²×C₄×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
C₂⁴×D₃×C₉	2
C₂³×C₃×C₄×C₉	2
C₂³×C₉×Dic₃	2
C₂²×C₄×D₃×C₉	2
C₂³×C₃×C₄×D₃	3
C₂⁴×C₄×C₉	3

Subgroups — C23×C4×D3×C9 #13164896

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₄×D₃×C₉ #13164896