Subgroups — D₃×C₉×#127 #13153265

Order 1728.

1602 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₃×D₄×C₉	C₂	1	6	yes	maximal, complemented
2	C₃×C₉×#127	C₂	1	1	yes	maximal, characteristic, complemented
3	C₉×#940	C₂	1	6	yes	maximal, complemented
4	C₉×#874	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄²×D₃×C₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×D₃×#127	C₃	1	1	yes	maximal, characteristic
7	C₂×C₉×#127	—	3	1	no	maximal
8	C₂³×D₃×C₉	C₂²	1	4	yes	—
9	C₂×C₃×D₄×C₉	C₂²	1	6	yes	complemented
10	C₂×C₉×#67	C₂²	1	12	yes	complemented
11	D₃×D₄×C₉	—	2	24	no	—
12	C₂×C₉×D₁₂	C₂²	1	6	yes	complemented
13	C₂×C₄×D₃×C₉	C₂²	1	3	yes	—
14	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic, complemented
15	C₄×C₉×Dic₃	C₂²	1	3	yes	complemented
16	C₂×C₃×D₃×D₄	C₂×C₃	1	6	yes	—
17	C₃²×#127	C₂×C₃	1	1	yes	characteristic
18	C₃×#940	C₂×C₃	1	6	yes	—
19	C₃×#874	C₂×C₃	1	1	yes	characteristic
20	C₃×C₄²×D₃	C₂×C₃	1	1	yes	characteristic
21	C₂²×D₄×C₉	—	3	6	no	—
22	C₉×#127	—	2	1	no	—
23	C₉×#127	—	3	6	no	—
24	C₉×#127	—	3	1	no	—
25	C₉×#127	D₃	1	1	yes	characteristic, direct product factor
26	C₂×C₄²×C₉	—	3	1	no	—
27	C₂³×C₃×C₉	C₂³	1	4	yes	—
28	C₂²×D₃×C₉	—	2	24	no	—
29	C₂²×D₃×C₉	C₂³	1	4	yes	—
30	C₂²×D₃×C₉	C₂³	1	1	yes	characteristic
31	C₃×D₄×C₉	—	2	12	no	—
32	C₂×C₃×C₄×C₉	C₂³	1	3	yes	—
33	C₉×#67	—	2	24	no	—
34	C₂×C₉×Dic₃	C₂³	1	3	yes	—
35	C₉×D₁₂	—	2	12	no	—
36	C₄×D₃×C₉	D₄	1	12	yes	complemented
37	C₂×C₃×#127	—	3	1	no	—
38	D₃×#127	C₉	1	1	yes	characteristic, direct product factor
39	C₂³×C₃×D₃	C₂²×C₃	1	4	yes	—
40	C₂×C₃²×D₄	C₂²×C₃	1	6	yes	—
41	C₂×C₃×#67	C₂²×C₃	1	12	yes	—
42	C₃×D₃×D₄	—	2	24	no	—
43	C₂×C₃×D₁₂	C₂²×C₃	1	6	yes	—
44	C₂×C₃×C₄×D₃	C₂²×C₃	1	3	yes	—
45	C₂⁴×C₉	—	3	4	no	—
46	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
47	C₃×C₄×Dic₃	C₂²×C₃	1	3	yes	—
48	C₂×D₄×C₉	—	2	6	no	—
49	C₂×D₄×C₉	—	3	12	no	—
50	C₂×D₄×C₉	—	3	6	no	—
51	C₂×D₄×C₉	—	6	24	no	—
52	C₂×D₄×C₉	C₂×D₃	1	6	yes	complemented
53	C₂²×C₄×C₉	—	3	3	no	—
54	C₄²×C₉	—	2	1	no	—
55	C₄²×C₉	—	3	3	no	—
56	C₄²×C₉	C₂×D₃	1	1	yes	characteristic, complemented
57	C₂²×C₃×C₉	—	2	12	no	—
58	C₂²×C₃×C₉	C₂⁴	1	1	yes	characteristic
59	C₂×D₃×C₉	—	2	12	no	—
60	C₂×D₃×C₉	—	4	16	no	—
61	C₂×D₃×C₉	C₂×D₄	1	6	yes	—
62	C₃×C₄×C₉	C₂×D₄	1	6	yes	complemented
63	C₉×Dic₃	C₂×D₄	1	6	yes	complemented
64	C₂²×C₃×D₄	—	3	6	no	—
65	C₂×D₃×D₄	C₂×C₉	1	6	yes	complemented
66	C₃×#127	—	2	1	no	—
67	C₃×#127	—	3	6	no	—
68	C₃×#127	—	3	1	no	—
69	C₃×#127	C₂×C₉	1	1	yes	characteristic, complemented
70	C₃×#127	C₃×D₃	1	1	yes	characteristic
71	C₂×C₃×C₄²	—	3	1	no	—
72	#940	C₂×C₉	1	6	yes	complemented
73	#874	C₂×C₉	1	1	yes	characteristic, complemented
74	C₄²×D₃	C₂×C₉	1	1	yes	characteristic, complemented
75	C₂³×C₃²	C₂³×C₃	1	4	yes	—
76	C₂²×C₃×D₃	—	2	24	no	—
77	C₂²×C₃×D₃	C₂³×C₃	1	4	yes	—
78	C₂²×C₃×D₃	C₂³×C₃	1	1	yes	characteristic
79	C₃²×D₄	—	2	12	no	—
80	C₂×C₃²×C₄	C₂³×C₃	1	3	yes	—
81	C₃×#67	—	2	24	no	—
82	C₂×C₃×Dic₃	C₂³×C₃	1	3	yes	—
83	C₃×D₁₂	—	2	12	no	—
84	C₃×C₄×D₃	C₃×D₄	1	12	yes	—
85	C₂³×C₉	—	2	4	no	—
86	C₂³×C₉	—	3	1	no	—
87	C₂³×C₉	—	3	4	no	—
88	C₂³×C₉	—	6	24	no	—
89	C₂³×C₉	C₂²×D₃	1	4	yes	—
90	D₄×C₉	—	2	12	no	—
91	D₄×C₉	—	4	12	no	—
92	D₄×C₉	—	6	24	no	—
93	D₄×C₉	—	6	12	no	—
94	C₂×C₄×C₉	—	2	3	no	—
95	C₂×C₄×C₉	—	3	3	no	—
96	C₂×C₄×C₉	—	3	12	no	—
97	C₂×C₄×C₉	C₂²×D₃	1	3	yes	—
98	C₂×#127	—	3	1	no	Sylow
99	C₂×C₃×C₉	—	4	4	no	—
100	C₂×C₃×C₉	C₂²×D₄	1	3	yes	—
101	D₃×C₉	—	4	4	no	—
102	D₃×C₉	#127	1	4	yes	direct product factor
103	C₂⁴×C₃	—	3	4	no	—
104	C₂³×D₃	C₂²×C₉	1	4	yes	—
105	C₂×C₃×D₄	—	2	6	no	—
106	C₂×C₃×D₄	—	3	12	no	—
107	C₂×C₃×D₄	—	3	6	no	—
108	C₂×C₃×D₄	—	6	24	no	—
109	C₂×C₃×D₄	C₂²×C₉	1	6	yes	complemented
110	C₂×C₃×D₄	C₂×C₃×D₃	1	6	yes	—
111	C₂²×C₃×C₄	—	3	3	no	—
112	C₂×#67	C₂²×C₉	1	12	yes	complemented
113	D₃×D₄	—	2	24	no	—
114	C₂×D₁₂	C₂²×C₉	1	6	yes	complemented
115	C₂×C₄×D₃	C₂²×C₉	1	3	yes	—
116	C₃×C₄²	—	2	1	no	—
117	C₃×C₄²	—	3	3	no	—
118	C₃×C₄²	C₂²×C₉	1	1	yes	characteristic, complemented
119	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
120	C₄×Dic₃	C₂²×C₉	1	3	yes	complemented
121	C₂²×C₃²	—	2	12	no	—
122	C₂²×C₃²	C₂⁴×C₃	1	1	yes	characteristic
123	C₂×C₃×D₃	—	2	12	no	—
124	C₂×C₃×D₃	—	4	16	no	—
125	C₂×C₃×D₃	C₂×C₃×D₄	1	6	yes	—
126	C₃²×C₄	C₂×C₃×D₄	1	6	yes	—
127	C₃×Dic₃	C₂×C₃×D₄	1	6	yes	—
128	C₂²×C₉	—	2	12	no	—
129	C₂²×C₉	—	2	1	no	—
130	C₂²×C₉	—	3	6	no	—
131	C₂²×C₉	—	4	12	no	—
132	C₂²×C₉	—	6	12	no	—
133	C₂²×C₉	—	12	16	no	—
134	C₂²×C₉	C₂³×D₃	1	1	yes	characteristic
135	C₄×C₉	—	2	6	no	—
136	C₄×C₉	—	3	6	no	—
137	C₄×C₉	D₃×D₄	1	6	yes	complemented
138	C₂²×D₄	—	3	6	no	—
139	#127	—	3	6	no	—
140	#127	—	3	1	no	—
141	#127	D₃×C₉	1	1	yes	characteristic, direct product factor
142	C₂×C₄²	—	3	1	no	—
143	C₃×C₉	C₂×#127	1	1	yes	characteristic, complemented, Sylow
144	C₂³×C₃	—	2	4	no	—
145	C₂³×C₃	—	3	1	no	—
146	C₂³×C₃	—	3	4	no	—
147	C₂³×C₃	—	6	24	no	—
148	C₂³×C₃	C₂³×C₉	1	4	yes	—
149	C₂³×C₃	C₂²×C₃×D₃	1	4	yes	—
150	C₂²×D₃	—	2	24	no	—
151	C₂²×D₃	C₂³×C₉	1	4	yes	—
152	C₂²×D₃	C₂³×C₉	1	1	yes	characteristic
153	C₃×D₄	—	2	12	no	—
154	C₃×D₄	—	2	12	no	—
155	C₃×D₄	—	4	12	no	—
156	C₃×D₄	—	6	24	no	—
157	C₃×D₄	—	6	12	no	—
158	C₂×C₃×C₄	—	2	3	no	—
159	C₂×C₃×C₄	—	3	3	no	—
160	C₂×C₃×C₄	—	3	12	no	—
161	C₂×C₃×C₄	C₂³×C₉	1	3	yes	—
162	C₂×C₃×C₄	C₂²×C₃×D₃	1	3	yes	—
163	#67	—	2	24	no	—
164	C₂×Dic₃	C₂³×C₉	1	3	yes	—
165	D₁₂	—	2	12	no	—
166	C₄×D₃	D₄×C₉	1	12	yes	complemented
167	C₂×C₃²	—	4	4	no	—
168	C₂×C₃²	C₂²×C₃×D₄	1	3	yes	—
169	C₃×D₃	—	4	4	no	—
170	C₃×D₃	C₃×#127	1	4	yes	—
171	C₂×C₉	—	2	3	no	—
172	C₂×C₉	—	3	4	no	—
173	C₂×C₉	—	4	4	no	—
174	C₂×C₉	—	8	4	no	—
175	C₂×C₉	—	12	4	no	—
176	C₂×C₉	C₂×D₃×D₄	1	3	yes	—
177	C₂⁴	—	3	4	no	—
178	C₂×D₄	—	3	12	no	—
179	C₂×D₄	—	3	6	no	—
180	C₂×D₄	—	6	24	no	—
181	C₂×D₄	C₂×D₃×C₉	1	6	yes	complemented
182	C₂²×C₄	—	3	3	no	—
183	C₄²	—	3	3	no	—
184	C₄²	C₂×D₃×C₉	1	1	yes	characteristic, complemented
185	C₂²×C₃	—	2	12	no	—
186	C₂²×C₃	—	2	12	no	—
187	C₂²×C₃	—	2	1	no	—
188	C₂²×C₃	—	3	6	no	—
189	C₂²×C₃	—	4	12	no	—
190	C₂²×C₃	—	6	12	no	—
191	C₂²×C₃	—	12	16	no	—
192	C₂²×C₃	C₂⁴×C₉	1	1	yes	characteristic
193	C₂²×C₃	C₂³×C₃×D₃	1	1	yes	characteristic
194	C₂×D₃	—	2	12	no	—
195	C₂×D₃	—	4	16	no	—
196	C₂×D₃	C₂×D₄×C₉	1	6	yes	—
197	C₃×C₄	—	2	6	no	—
198	C₃×C₄	—	3	6	no	—
199	C₃×C₄	C₂×D₄×C₉	1	6	yes	complemented
200	C₃×C₄	C₃×D₃×D₄	1	6	yes	—
201	Dic₃	C₂×D₄×C₉	1	6	yes	complemented
202	C₃²	C₂×C₃×#127	1	1	yes	characteristic
203	C₉	—	2	1	no	—
204	C₉	D₃×#127	1	1	yes	characteristic, direct product factor
205	C₂³	—	3	1	no	—
206	C₂³	—	3	4	no	—
207	C₂³	—	6	24	no	—
208	C₂³	C₂²×D₃×C₉	1	4	yes	—
209	D₄	—	2	12	no	—
210	D₄	—	6	24	no	—
211	D₄	—	6	12	no	—
212	C₂×C₄	—	3	3	no	—
213	C₂×C₄	—	3	12	no	—
214	C₂×C₄	C₂²×D₃×C₉	1	3	yes	—
215	C₂×C₃	—	2	3	no	—
216	C₂×C₃	—	3	4	no	—
217	C₂×C₃	—	4	4	no	—
218	C₂×C₃	—	4	4	no	—
219	C₂×C₃	—	8	4	no	—
220	C₂×C₃	—	12	4	no	—
221	C₂×C₃	C₂²×D₄×C₉	1	3	yes	—
222	C₂×C₃	C₂×C₃×D₃×D₄	1	3	yes	—
223	D₃	—	4	4	no	—
224	D₃	C₉×#127	1	4	yes	direct product factor
225	C₂²	—	2	12	no	—
226	C₂²	—	3	6	no	—
227	C₂²	—	6	12	no	—
228	C₂²	—	12	16	no	—
229	C₂²	C₂³×D₃×C₉	1	1	yes	characteristic
230	C₄	—	3	6	no	—
231	C₄	D₃×D₄×C₉	1	6	yes	complemented
232	C₃	—	2	1	no	—
233	C₃	C₂×C₉×#127	1	1	yes	characteristic, complemented
234	C₃	C₃×D₃×#127	1	1	yes	characteristic
235	C₂	—	3	4	no	—
236	C₂	—	4	4	no	—
237	C₂	—	12	4	no	—
238	C₂	C₂×D₃×D₄×C₉	1	3	yes	—

	subgroup	G/H
center	C₂²×C₉	C₂³×D₃
commutator	C₂²×C₃	C₂⁴×C₉
Frattini	C₂²×C₃	C₂³×C₃×D₃
Fitting	C₃×C₉×#127	C₂
socle, cosocle	C₂²×C₃²	C₂⁴×C₃
prime radical	C₃×D₃×#127	C₃
hypercenter	C₉×#127	D₃
hypocenter	C₃	C₂×C₉×#127
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₉ ◁ C₉×#127

Complements

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

N	H ≅ G/N	type
1. C₂×D₃×D₄×C₉	C₂	semidirect
2. C₃×C₉×#127	C₂	semidirect
3. C₉×#940	C₂	semidirect
4. C₉×#874	C₂	semidirect
5. C₄²×D₃×C₉	C₂	semidirect
9. C₂×C₃×D₄×C₉	C₂²	semidirect
10. C₂×C₉×#67	C₂²	semidirect
12. C₂×C₉×D₁₂	C₂²	semidirect
14. C₃×C₄²×C₉	C₂²	semidirect
15. C₄×C₉×Dic₃	C₂²	semidirect
25. C₉×#127	D₃	direct (N × H)
36. C₄×D₃×C₉	D₄	semidirect
38. D₃×#127	C₉	direct (N × H)
52. C₂×D₄×C₉	C₂×D₃	semidirect
56. C₄²×C₉	C₂×D₃	semidirect
62. C₃×C₄×C₉	C₂×D₄	semidirect
63. C₉×Dic₃	C₂×D₄	semidirect
65. C₂×D₃×D₄	C₂×C₉	semidirect
69. C₃×#127	C₂×C₉	semidirect
72. #940	C₂×C₉	semidirect
73. #874	C₂×C₉	semidirect
74. C₄²×D₃	C₂×C₉	semidirect
102. D₃×C₉	#127	direct (N × H)
109. C₂×C₃×D₄	C₂²×C₉	semidirect
112. C₂×#67	C₂²×C₉	semidirect
114. C₂×D₁₂	C₂²×C₉	semidirect
118. C₃×C₄²	C₂²×C₉	semidirect
120. C₄×Dic₃	C₂²×C₉	semidirect
137. C₄×C₉	D₃×D₄	semidirect
141. #127	D₃×C₉	direct (N × H)
143. C₃×C₉	C₂×#127	semidirect
166. C₄×D₃	D₄×C₉	semidirect
181. C₂×D₄	C₂×D₃×C₉	semidirect
184. C₄²	C₂×D₃×C₉	semidirect
199. C₃×C₄	C₂×D₄×C₉	semidirect
201. Dic₃	C₂×D₄×C₉	semidirect
204. C₉	D₃×#127	direct (N × H)
224. D₃	C₉×#127	direct (N × H)
231. C₄	D₃×D₄×C₉	semidirect
233. C₃	C₂×C₉×#127	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×D₃×D₄×C₉	2
C₃×C₉×#127	2
C₉×#940	2
C₉×#874	2
C₄²×D₃×C₉	2
C₃×D₃×#127	3
C₂×C₉×#127	3

Subgroups — D3×C9×#127 #13153265

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₃×C₉×#127 #13153265