Subgroups — D₃×C₉×2₋¹⁺⁴ #13164957

Order 1728.

2528 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₉×2₋¹⁺⁴	C₂	1	1	yes	maximal, characteristic, complemented
2	C₉×#1010	C₂	1	10	yes	maximal, complemented
3	D₃×C₉×#41	C₂	1	10	yes	maximal, complemented
4	C₉×#1007	C₂	1	5	yes	maximal, complemented
5	C₂×D₃×Q₈×C₉	C₂	1	5	yes	maximal, complemented
6	C₃×D₃×2₋¹⁺⁴	C₃	1	1	yes	maximal, characteristic
7	C₂×C₉×2₋¹⁺⁴	—	3	1	no	maximal
8	C₃×C₉×#41	C₂²	1	10	yes	complemented
9	C₂×C₃×Q₈×C₉	C₂²	1	5	yes	complemented
10	C₉×#239	C₂²	1	10	yes	complemented
11	D₃×Q₈×C₉	C₂²	1	10	yes	—
12	D₃×Q₈×C₉	C₂²	1	20	yes	complemented
13	C₉×#237	C₂²	1	30	yes	complemented
14	D₃×D₄×C₉	C₂²	1	10	yes	—
15	C₉×#235	C₂²	1	30	yes	complemented
16	C₂×C₄×D₃×C₉	C₂²	1	15	yes	—
17	C₂×C₉×Dic₆	C₂²	1	15	yes	complemented
18	C₃²×2₋¹⁺⁴	C₂×C₃	1	1	yes	characteristic
19	C₃×#1010	C₂×C₃	1	10	yes	—
20	C₃×D₃×#41	C₂×C₃	1	10	yes	—
21	C₃×#1007	C₂×C₃	1	5	yes	—
22	C₂×C₃×D₃×Q₈	C₂×C₃	1	5	yes	—
23	C₉×2₋¹⁺⁴	—	2	1	no	—
24	C₉×2₋¹⁺⁴	—	3	10	no	—
25	C₉×2₋¹⁺⁴	—	3	5	no	—
26	C₉×2₋¹⁺⁴	D₃	1	1	yes	characteristic, direct product factor
27	C₂×C₉×#41	—	3	10	no	—
28	C₂²×Q₈×C₉	—	3	5	no	—
29	C₂²×D₃×C₉	C₂³	1	5	yes	—
30	C₃×Q₈×C₉	C₂³	1	10	yes	—
31	C₃×D₄×C₉	C₂³	1	10	yes	—
32	C₂×C₃×C₄×C₉	C₂³	1	15	yes	—
33	C₉×#67	C₂³	1	20	yes	—
34	C₂×C₉×Dic₃	C₂³	1	15	yes	—
35	C₉×D₁₂	C₂³	1	10	yes	—
36	C₄×D₃×C₉	C₂³	1	10	yes	—
37	C₄×D₃×C₉	C₂³	1	30	yes	—
38	C₉×Dic₆	C₂³	1	30	yes	—
39	C₂×C₃×2₋¹⁺⁴	—	3	1	no	—
40	D₃×2₋¹⁺⁴	C₉	1	1	yes	characteristic, direct product factor
41	C₃²×#41	C₂²×C₃	1	10	yes	—
42	C₂×C₃²×Q₈	C₂²×C₃	1	5	yes	—
43	C₃×#239	C₂²×C₃	1	10	yes	—
44	C₃×D₃×Q₈	C₂²×C₃	1	10	yes	—
45	C₃×D₃×Q₈	C₂²×C₃	1	20	yes	—
46	C₃×#237	C₂²×C₃	1	30	yes	—
47	C₃×D₃×D₄	C₂²×C₃	1	10	yes	—
48	C₃×#235	C₂²×C₃	1	30	yes	—
49	C₂×C₃×C₄×D₃	C₂²×C₃	1	15	yes	—
50	C₂×C₃×Dic₆	C₂²×C₃	1	15	yes	—
51	C₉×#41	—	2	10	no	—
52	C₉×#41	—	3	10	no	—
53	C₉×#41	—	3	30	no	—
54	C₉×#41	—	3	30	no	—
55	C₉×#41	C₂×D₃	1	10	yes	complemented
56	C₂×Q₈×C₉	—	2	5	no	—
57	C₂×Q₈×C₉	—	3	10	no	—
58	C₂×Q₈×C₉	—	3	20	no	—
59	C₂×Q₈×C₉	—	3	15	no	—
60	C₂×Q₈×C₉	C₂×D₃	1	5	yes	complemented
61	C₂×D₄×C₉	—	3	10	no	—
62	C₂²×C₄×C₉	—	3	15	no	—
63	C₂²×C₃×C₉	C₂⁴	1	5	yes	—
64	C₂×D₃×C₉	—	2	10	no	—
65	C₂×D₃×C₉	C₂⁴	1	5	yes	—
66	C₂×D₃×C₉	C₂⁴	1	1	yes	characteristic
67	C₃×C₄×C₉	C₂⁴	1	10	yes	—
68	C₉×Dic₃	C₂⁴	1	10	yes	—
69	C₃×2₋¹⁺⁴	—	2	1	no	—
70	C₃×2₋¹⁺⁴	—	3	10	no	—
71	C₃×2₋¹⁺⁴	—	3	5	no	—
72	C₃×2₋¹⁺⁴	C₂×C₉	1	1	yes	characteristic, complemented
73	C₃×2₋¹⁺⁴	C₃×D₃	1	1	yes	characteristic
74	C₂×C₃×#41	—	3	10	no	—
75	C₂²×C₃×Q₈	—	3	5	no	—
76	#1010	C₂×C₉	1	10	yes	complemented
77	D₃×#41	C₂×C₉	1	10	yes	complemented
78	#1007	C₂×C₉	1	5	yes	complemented
79	C₂×D₃×Q₈	C₂×C₉	1	5	yes	complemented
80	C₂²×C₃×D₃	C₂³×C₃	1	5	yes	—
81	C₃²×Q₈	C₂³×C₃	1	10	yes	—
82	C₃²×D₄	C₂³×C₃	1	10	yes	—
83	C₂×C₃²×C₄	C₂³×C₃	1	15	yes	—
84	C₃×#67	C₂³×C₃	1	20	yes	—
85	C₂×C₃×Dic₃	C₂³×C₃	1	15	yes	—
86	C₃×D₁₂	C₂³×C₃	1	10	yes	—
87	C₃×C₄×D₃	C₂³×C₃	1	10	yes	—
88	C₃×C₄×D₃	C₂³×C₃	1	30	yes	—
89	C₃×Dic₆	C₂³×C₃	1	30	yes	—
90	C₂³×C₉	—	3	5	no	—
91	Q₈×C₉	—	2	10	no	—
92	Q₈×C₉	—	3	30	no	—
93	Q₈×C₉	C₂²×D₃	1	10	yes	—
94	D₄×C₉	—	2	10	no	—
95	D₄×C₉	—	3	20	no	—
96	D₄×C₉	—	3	10	no	—
97	D₄×C₉	C₂²×D₃	1	10	yes	—
98	C₂×C₄×C₉	—	2	15	no	—
99	C₂×C₄×C₉	—	3	15	no	—
100	C₂×C₄×C₉	—	3	10	no	—
101	C₂×C₄×C₉	—	3	30	no	—
102	C₂×C₄×C₉	C₂²×D₃	1	15	yes	—
103	C₂×2₋¹⁺⁴	—	3	1	no	Sylow
104	C₂×C₃×C₉	—	2	5	no	—
105	C₂×C₃×C₉	C₂⁵	1	1	yes	characteristic
106	D₃×C₉	—	2	5	no	—
107	D₃×C₉	2₋¹⁺⁴	1	2	yes	direct product factor
108	C₃×#41	—	2	10	no	—
109	C₃×#41	—	3	10	no	—
110	C₃×#41	—	3	30	no	—
111	C₃×#41	—	3	30	no	—
112	C₃×#41	C₂²×C₉	1	10	yes	complemented
113	C₃×#41	C₂×C₃×D₃	1	10	yes	—
114	C₂×C₃×Q₈	—	2	5	no	—
115	C₂×C₃×Q₈	—	3	10	no	—
116	C₂×C₃×Q₈	—	3	20	no	—
117	C₂×C₃×Q₈	—	3	15	no	—
118	C₂×C₃×Q₈	C₂²×C₉	1	5	yes	complemented
119	C₂×C₃×Q₈	C₂×C₃×D₃	1	5	yes	—
120	C₂×C₃×D₄	—	3	10	no	—
121	C₂²×C₃×C₄	—	3	15	no	—
122	#239	C₂²×C₉	1	10	yes	complemented
123	D₃×Q₈	C₂²×C₉	1	10	yes	—
124	D₃×Q₈	C₂²×C₉	1	20	yes	complemented
125	#237	C₂²×C₉	1	30	yes	complemented
126	D₃×D₄	C₂²×C₉	1	10	yes	—
127	#235	C₂²×C₉	1	30	yes	complemented
128	C₂×C₄×D₃	C₂²×C₉	1	15	yes	—
129	C₂×Dic₆	C₂²×C₉	1	15	yes	complemented
130	C₂²×C₃²	C₂⁴×C₃	1	5	yes	—
131	C₂×C₃×D₃	—	2	10	no	—
132	C₂×C₃×D₃	C₂⁴×C₃	1	5	yes	—
133	C₂×C₃×D₃	C₂⁴×C₃	1	1	yes	characteristic
134	C₃²×C₄	C₂⁴×C₃	1	10	yes	—
135	C₃×Dic₃	C₂⁴×C₃	1	10	yes	—
136	C₂²×C₉	—	2	5	no	—
137	C₂²×C₉	—	3	1	no	—
138	C₂²×C₉	—	3	5	no	—
139	C₂²×C₉	—	6	10	no	—
140	C₂²×C₉	C₂³×D₃	1	5	yes	—
141	C₄×C₉	—	2	10	no	—
142	C₄×C₉	—	3	10	no	—
143	C₄×C₉	C₂³×D₃	1	10	yes	—
144	2₋¹⁺⁴	—	3	10	no	—
145	2₋¹⁺⁴	—	3	5	no	—
146	2₋¹⁺⁴	D₃×C₉	1	1	yes	characteristic, direct product factor
147	C₂×#41	—	3	10	no	—
148	C₂²×Q₈	—	3	5	no	—
149	C₃×C₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic, complemented, Sylow
150	C₂³×C₃	—	3	5	no	—
151	C₂²×D₃	C₂³×C₉	1	5	yes	—
152	C₃×Q₈	—	2	10	no	—
153	C₃×Q₈	—	3	30	no	—
154	C₃×Q₈	C₂³×C₉	1	10	yes	—
155	C₃×Q₈	C₂²×C₃×D₃	1	10	yes	—
156	C₃×D₄	—	2	10	no	—
157	C₃×D₄	—	3	20	no	—
158	C₃×D₄	—	3	10	no	—
159	C₃×D₄	C₂³×C₉	1	10	yes	—
160	C₃×D₄	C₂²×C₃×D₃	1	10	yes	—
161	C₂×C₃×C₄	—	2	15	no	—
162	C₂×C₃×C₄	—	3	15	no	—
163	C₂×C₃×C₄	—	3	10	no	—
164	C₂×C₃×C₄	—	3	30	no	—
165	C₂×C₃×C₄	C₂³×C₉	1	15	yes	—
166	C₂×C₃×C₄	C₂²×C₃×D₃	1	15	yes	—
167	#67	C₂³×C₉	1	20	yes	—
168	C₂×Dic₃	C₂³×C₉	1	15	yes	—
169	D₁₂	C₂³×C₉	1	10	yes	—
170	C₄×D₃	C₂³×C₉	1	10	yes	—
171	C₄×D₃	C₂³×C₉	1	30	yes	—
172	Dic₆	C₂³×C₉	1	30	yes	—
173	C₂×C₃²	—	2	5	no	—
174	C₂×C₃²	C₂⁵×C₃	1	1	yes	characteristic
175	C₃×D₃	—	2	5	no	—
176	C₃×D₃	C₃×2₋¹⁺⁴	1	2	yes	—
177	C₂×C₉	—	2	5	no	—
178	C₂×C₉	—	2	1	no	—
179	C₂×C₉	—	3	2	no	—
180	C₂×C₉	—	4	5	no	—
181	C₂×C₉	—	6	5	no	—
182	C₂×C₉	C₂⁴×D₃	1	1	yes	characteristic
183	#41	—	3	10	no	—
184	#41	—	3	30	no	—
185	#41	—	3	30	no	—
186	#41	C₂×D₃×C₉	1	10	yes	complemented
187	C₂×Q₈	—	3	10	no	—
188	C₂×Q₈	—	3	20	no	—
189	C₂×Q₈	—	3	15	no	—
190	C₂×Q₈	C₂×D₃×C₉	1	5	yes	complemented
191	C₂×D₄	—	3	10	no	—
192	C₂²×C₄	—	3	15	no	—
193	C₂²×C₃	—	2	5	no	—
194	C₂²×C₃	—	3	1	no	—
195	C₂²×C₃	—	3	5	no	—
196	C₂²×C₃	—	6	10	no	—
197	C₂²×C₃	C₂⁴×C₉	1	5	yes	—
198	C₂²×C₃	C₂³×C₃×D₃	1	5	yes	—
199	C₂×D₃	—	2	10	no	—
200	C₂×D₃	C₂⁴×C₉	1	5	yes	—
201	C₂×D₃	C₂⁴×C₉	1	1	yes	characteristic
202	C₃×C₄	—	2	10	no	—
203	C₃×C₄	—	3	10	no	—
204	C₃×C₄	C₂⁴×C₉	1	10	yes	—
205	C₃×C₄	C₂³×C₃×D₃	1	10	yes	—
206	Dic₃	C₂⁴×C₉	1	10	yes	—
207	C₃²	C₂×C₃×2₋¹⁺⁴	1	1	yes	characteristic
208	C₉	—	2	1	no	—
209	C₉	D₃×2₋¹⁺⁴	1	1	yes	characteristic, direct product factor
210	C₂³	—	3	5	no	—
211	Q₈	—	3	30	no	—
212	Q₈	C₂²×D₃×C₉	1	10	yes	—
213	D₄	—	3	20	no	—
214	D₄	—	3	10	no	—
215	D₄	C₂²×D₃×C₉	1	10	yes	—
216	C₂×C₄	—	3	15	no	—
217	C₂×C₄	—	3	10	no	—
218	C₂×C₄	—	3	30	no	—
219	C₂×C₄	C₂²×D₃×C₉	1	15	yes	—
220	C₂×C₃	—	2	5	no	—
221	C₂×C₃	—	2	5	no	—
222	C₂×C₃	—	2	1	no	—
223	C₂×C₃	—	3	2	no	—
224	C₂×C₃	—	4	5	no	—
225	C₂×C₃	—	6	5	no	—
226	C₂×C₃	C₂⁵×C₉	1	1	yes	characteristic
227	C₂×C₃	C₂⁴×C₃×D₃	1	1	yes	characteristic
228	D₃	—	2	5	no	—
229	D₃	C₉×2₋¹⁺⁴	1	2	yes	direct product factor
230	C₂²	—	3	1	no	—
231	C₂²	—	3	5	no	—
232	C₂²	—	6	10	no	—
233	C₂²	C₂³×D₃×C₉	1	5	yes	—
234	C₄	—	3	10	no	—
235	C₄	C₂³×D₃×C₉	1	10	yes	—
236	C₃	—	2	1	no	—
237	C₃	C₂×C₉×2₋¹⁺⁴	1	1	yes	characteristic, complemented
238	C₃	C₃×D₃×2₋¹⁺⁴	1	1	yes	characteristic
239	C₂	—	2	5	no	—
240	C₂	—	3	2	no	—
241	C₂	—	6	5	no	—
242	C₂	C₂⁴×D₃×C₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₉	C₂⁴×D₃
commutator	C₂×C₃	C₂⁵×C₉
Frattini	C₂×C₃	C₂⁴×C₃×D₃
Fitting	C₃×C₉×2₋¹⁺⁴	C₂
socle, cosocle	C₂×C₃²	C₂⁵×C₃
prime radical	C₃×D₃×2₋¹⁺⁴	C₃
hypercenter	C₉×2₋¹⁺⁴	D₃
hypocenter	C₃	C₂×C₉×2₋¹⁺⁴
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₉ ◁ C₉×2₋¹⁺⁴

Complements

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

N	H ≅ G/N	type
1. C₃×C₉×2₋¹⁺⁴	C₂	semidirect
2. C₉×#1010	C₂	semidirect
3. D₃×C₉×#41	C₂	semidirect
4. C₉×#1007	C₂	semidirect
5. C₂×D₃×Q₈×C₉	C₂	semidirect
8. C₃×C₉×#41	C₂²	semidirect
9. C₂×C₃×Q₈×C₉	C₂²	semidirect
10. C₉×#239	C₂²	semidirect
12. D₃×Q₈×C₉	C₂²	semidirect
13. C₉×#237	C₂²	semidirect
15. C₉×#235	C₂²	semidirect
17. C₂×C₉×Dic₆	C₂²	semidirect
26. C₉×2₋¹⁺⁴	D₃	direct (N × H)
40. D₃×2₋¹⁺⁴	C₉	direct (N × H)
55. C₉×#41	C₂×D₃	semidirect
60. C₂×Q₈×C₉	C₂×D₃	semidirect
72. C₃×2₋¹⁺⁴	C₂×C₉	semidirect
76. #1010	C₂×C₉	semidirect
77. D₃×#41	C₂×C₉	semidirect
78. #1007	C₂×C₉	semidirect
79. C₂×D₃×Q₈	C₂×C₉	semidirect
107. D₃×C₉	2₋¹⁺⁴	direct (N × H)
112. C₃×#41	C₂²×C₉	semidirect
118. C₂×C₃×Q₈	C₂²×C₉	semidirect
122. #239	C₂²×C₉	semidirect
124. D₃×Q₈	C₂²×C₉	semidirect
125. #237	C₂²×C₉	semidirect
127. #235	C₂²×C₉	semidirect
129. C₂×Dic₆	C₂²×C₉	semidirect
146. 2₋¹⁺⁴	D₃×C₉	direct (N × H)
149. C₃×C₉	C₂×2₋¹⁺⁴	semidirect
186. #41	C₂×D₃×C₉	semidirect
190. C₂×Q₈	C₂×D₃×C₉	semidirect
209. C₉	D₃×2₋¹⁺⁴	direct (N × H)
229. D₃	C₉×2₋¹⁺⁴	direct (N × H)
237. C₃	C₂×C₉×2₋¹⁺⁴	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₉×2₋¹⁺⁴	2
C₉×#1010	2
D₃×C₉×#41	2
C₉×#1007	2
C₂×D₃×Q₈×C₉	2
C₃×D₃×2₋¹⁺⁴	3
C₂×C₉×2₋¹⁺⁴	3

Subgroups — D3×C9×2−1+4 #13164957

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₃×C₉×2₋¹⁺⁴ #13164957