Subgroups — C₃×D₉×2₋¹⁺⁴ #13164955

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	C₃×#63716	C₂	1	10	yes	maximal, complemented
3	C₃×#41×D₉	C₂	1	10	yes	maximal, complemented
4	C₃×#63713	C₂	1	5	yes	maximal, complemented
5	C₂×C₃×Q₈×D₉	C₂	1	5	yes	maximal, complemented
6	C₃×D₃×2₋¹⁺⁴	—	3	1	no	maximal
7	D₉×2₋¹⁺⁴	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₃×C₉×#41	C₂²	1	10	yes	complemented
9	C₂×C₃×Q₈×C₉	C₂²	1	5	yes	complemented
10	C₃×#3701	C₂²	1	10	yes	complemented
11	C₃×Q₈×D₉	C₂²	1	10	yes	—
12	C₃×Q₈×D₉	C₂²	1	20	yes	complemented
13	C₃×#3699	C₂²	1	30	yes	complemented
14	C₃×D₄×D₉	C₂²	1	10	yes	—
15	C₃×#3697	C₂²	1	30	yes	complemented
16	C₂×C₃×C₄×D₉	C₂²	1	15	yes	—
17	C₂×C₃×Dic₁₈	C₂²	1	15	yes	complemented
18	C₃²×2₋¹⁺⁴	D₃	1	1	yes	characteristic
19	C₃×#1010	—	3	10	no	—
20	C₃×D₃×#41	—	3	10	no	—
21	C₃×#1007	—	3	5	no	—
22	C₂×C₃×D₃×Q₈	—	3	5	no	—
23	C₉×2₋¹⁺⁴	—	2	1	no	—
24	C₉×2₋¹⁺⁴	C₂×C₃	1	1	yes	characteristic, complemented
25	#63716	C₂×C₃	1	10	yes	complemented
26	#41×D₉	C₂×C₃	1	10	yes	complemented
27	#63713	C₂×C₃	1	5	yes	complemented
28	C₂×Q₈×D₉	C₂×C₃	1	5	yes	complemented
29	C₂²×C₃×D₉	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	C₃×#611	C₂³	1	20	yes	—
34	C₂×C₃×Dic₉	C₂³	1	15	yes	—
35	C₃×D₃₆	C₂³	1	10	yes	—
36	C₃×C₄×D₉	C₂³	1	10	yes	—
37	C₃×C₄×D₉	C₂³	1	30	yes	—
38	C₃×Dic₁₈	C₂³	1	30	yes	—
39	C₂×C₃×2₋¹⁺⁴	—	9	1	no	—
40	D₃×2₋¹⁺⁴	—	3	1	no	—
41	C₃²×#41	C₂×D₃	1	10	yes	—
42	C₂×C₃²×Q₈	C₂×D₃	1	5	yes	—
43	C₃×#239	—	3	10	no	—
44	C₃×D₃×Q₈	—	3	10	no	—
45	C₃×D₃×Q₈	—	3	20	no	—
46	C₃×#237	—	3	30	no	—
47	C₃×D₃×D₄	—	3	10	no	—
48	C₃×#235	—	3	30	no	—
49	C₂×C₃×C₄×D₃	—	3	15	no	—
50	C₂×C₃×Dic₆	—	3	15	no	—
51	C₉×#41	—	2	10	no	—
52	C₉×#41	C₂²×C₃	1	10	yes	complemented
53	C₂×Q₈×C₉	—	2	5	no	—
54	C₂×Q₈×C₉	C₂²×C₃	1	5	yes	complemented
55	#3701	C₂²×C₃	1	10	yes	complemented
56	Q₈×D₉	C₂²×C₃	1	10	yes	—
57	Q₈×D₉	C₂²×C₃	1	20	yes	complemented
58	#3699	C₂²×C₃	1	30	yes	complemented
59	D₄×D₉	C₂²×C₃	1	10	yes	—
60	#3697	C₂²×C₃	1	30	yes	complemented
61	C₂×C₄×D₉	C₂²×C₃	1	15	yes	—
62	C₂×Dic₁₈	C₂²×C₃	1	15	yes	complemented
63	C₂²×C₃×C₉	C₂⁴	1	5	yes	—
64	C₂×C₃×D₉	—	2	10	no	—
65	C₂×C₃×D₉	C₂⁴	1	5	yes	—
66	C₂×C₃×D₉	C₂⁴	1	1	yes	characteristic
67	C₃×C₄×C₉	C₂⁴	1	10	yes	—
68	C₃×Dic₉	C₂⁴	1	10	yes	—
69	C₃×2₋¹⁺⁴	—	2	1	no	—
70	C₃×2₋¹⁺⁴	—	9	10	no	—
71	C₃×2₋¹⁺⁴	—	9	5	no	—
72	C₃×2₋¹⁺⁴	D₉	1	1	yes	characteristic, direct product factor
73	C₃×2₋¹⁺⁴	C₃×D₃	1	1	yes	characteristic
74	C₂×C₃×#41	—	9	10	no	—
75	C₂²×C₃×Q₈	—	9	5	no	—
76	#1010	—	3	10	no	—
77	D₃×#41	—	3	10	no	—
78	#1007	—	3	5	no	—
79	C₂×D₃×Q₈	—	3	5	no	—
80	C₂²×C₃×D₃	—	3	5	no	—
81	C₃²×Q₈	C₂²×D₃	1	10	yes	—
82	C₃²×D₄	C₂²×D₃	1	10	yes	—
83	C₂×C₃²×C₄	C₂²×D₃	1	15	yes	—
84	C₃×#67	—	3	20	no	—
85	C₂×C₃×Dic₃	—	3	15	no	—
86	C₃×D₁₂	—	3	10	no	—
87	C₃×C₄×D₃	—	3	10	no	—
88	C₃×C₄×D₃	—	3	30	no	—
89	C₃×Dic₆	—	3	30	no	—
90	C₂²×D₉	C₂³×C₃	1	5	yes	—
91	Q₈×C₉	—	2	10	no	—
92	Q₈×C₉	C₂³×C₃	1	10	yes	—
93	D₄×C₉	—	2	10	no	—
94	D₄×C₉	C₂³×C₃	1	10	yes	—
95	C₂×C₄×C₉	—	2	15	no	—
96	C₂×C₄×C₉	C₂³×C₃	1	15	yes	—
97	#611	C₂³×C₃	1	20	yes	—
98	C₂×Dic₉	C₂³×C₃	1	15	yes	—
99	D₃₆	C₂³×C₃	1	10	yes	—
100	C₄×D₉	C₂³×C₃	1	10	yes	—
101	C₄×D₉	C₂³×C₃	1	30	yes	—
102	Dic₁₈	C₂³×C₃	1	30	yes	—
103	C₂×2₋¹⁺⁴	—	9	1	no	Sylow
104	C₂×C₃×C₉	—	2	5	no	—
105	C₂×C₃×C₉	C₂⁵	1	1	yes	characteristic
106	C₃×D₉	—	2	5	no	—
107	C₃×D₉	2₋¹⁺⁴	1	2	yes	direct product factor
108	C₃×#41	—	2	10	no	—
109	C₃×#41	—	9	10	no	—
110	C₃×#41	—	9	30	no	—
111	C₃×#41	—	9	30	no	—
112	C₃×#41	C₂×D₉	1	10	yes	complemented
113	C₃×#41	C₂×C₃×D₃	1	10	yes	—
114	C₂×C₃×Q₈	—	2	5	no	—
115	C₂×C₃×Q₈	—	9	10	no	—
116	C₂×C₃×Q₈	—	9	20	no	—
117	C₂×C₃×Q₈	—	9	15	no	—
118	C₂×C₃×Q₈	C₂×D₉	1	5	yes	complemented
119	C₂×C₃×Q₈	C₂×C₃×D₃	1	5	yes	—
120	C₂×C₃×D₄	—	9	10	no	—
121	C₂²×C₃×C₄	—	9	15	no	—
122	#239	—	3	10	no	—
123	D₃×Q₈	—	3	10	no	—
124	D₃×Q₈	—	3	20	no	—
125	#237	—	3	30	no	—
126	D₃×D₄	—	3	10	no	—
127	#235	—	3	30	no	—
128	C₂×C₄×D₃	—	3	15	no	—
129	C₂×Dic₆	—	3	15	no	—
130	C₂²×C₃²	C₂³×D₃	1	5	yes	—
131	C₂×C₃×D₃	—	3	1	no	—
132	C₂×C₃×D₃	—	3	5	no	—
133	C₂×C₃×D₃	—	6	10	no	—
134	C₃²×C₄	C₂³×D₃	1	10	yes	—
135	C₃×Dic₃	—	3	10	no	—
136	C₂²×C₉	—	2	5	no	—
137	C₂²×C₉	C₂⁴×C₃	1	5	yes	—
138	C₂×D₉	—	2	10	no	—
139	C₂×D₉	C₂⁴×C₃	1	5	yes	—
140	C₂×D₉	C₂⁴×C₃	1	1	yes	characteristic
141	C₄×C₉	—	2	10	no	—
142	C₄×C₉	C₂⁴×C₃	1	10	yes	—
143	Dic₉	C₂⁴×C₃	1	10	yes	—
144	2₋¹⁺⁴	—	9	10	no	—
145	2₋¹⁺⁴	—	9	5	no	—
146	2₋¹⁺⁴	C₃×D₉	1	1	yes	characteristic, direct product factor
147	C₂×#41	—	9	10	no	—
148	C₂²×Q₈	—	9	5	no	—
149	C₃×C₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic, complemented, Sylow
150	C₂³×C₃	—	9	5	no	—
151	C₂²×D₃	—	3	5	no	—
152	C₃×Q₈	—	2	10	no	—
153	C₃×Q₈	—	9	30	no	—
154	C₃×Q₈	C₂²×D₉	1	10	yes	—
155	C₃×Q₈	C₂²×C₃×D₃	1	10	yes	—
156	C₃×D₄	—	2	10	no	—
157	C₃×D₄	—	9	20	no	—
158	C₃×D₄	—	9	10	no	—
159	C₃×D₄	C₂²×D₉	1	10	yes	—
160	C₃×D₄	C₂²×C₃×D₃	1	10	yes	—
161	C₂×C₃×C₄	—	2	15	no	—
162	C₂×C₃×C₄	—	9	15	no	—
163	C₂×C₃×C₄	—	9	10	no	—
164	C₂×C₃×C₄	—	9	30	no	—
165	C₂×C₃×C₄	C₂²×D₉	1	15	yes	—
166	C₂×C₃×C₄	C₂²×C₃×D₃	1	15	yes	—
167	#67	—	3	20	no	—
168	C₂×Dic₃	—	3	15	no	—
169	D₁₂	—	3	10	no	—
170	C₄×D₃	—	3	10	no	—
171	C₄×D₃	—	3	30	no	—
172	Dic₆	—	3	30	no	—
173	C₂×C₃²	—	2	5	no	—
174	C₂×C₃²	C₂⁴×D₃	1	1	yes	characteristic
175	C₃×D₃	—	3	2	no	—
176	C₃×D₃	—	6	5	no	—
177	C₂×C₉	—	2	5	no	—
178	C₂×C₉	—	2	1	no	—
179	C₂×C₉	—	4	5	no	—
180	C₂×C₉	C₂⁵×C₃	1	1	yes	characteristic
181	D₉	—	2	5	no	—
182	D₉	C₃×2₋¹⁺⁴	1	2	yes	direct product factor
183	#41	—	9	10	no	—
184	#41	—	9	30	no	—
185	#41	—	9	30	no	—
186	#41	C₂×C₃×D₉	1	10	yes	complemented
187	C₂×Q₈	—	9	10	no	—
188	C₂×Q₈	—	9	20	no	—
189	C₂×Q₈	—	9	15	no	—
190	C₂×Q₈	C₂×C₃×D₉	1	5	yes	complemented
191	C₂×D₄	—	9	10	no	—
192	C₂²×C₄	—	9	15	no	—
193	C₂²×C₃	—	2	5	no	—
194	C₂²×C₃	—	9	1	no	—
195	C₂²×C₃	—	9	5	no	—
196	C₂²×C₃	—	18	10	no	—
197	C₂²×C₃	C₂³×D₉	1	5	yes	—
198	C₂²×C₃	C₂³×C₃×D₃	1	5	yes	—
199	C₂×D₃	—	3	1	no	—
200	C₂×D₃	—	3	5	no	—
201	C₂×D₃	—	6	10	no	—
202	C₃×C₄	—	2	10	no	—
203	C₃×C₄	—	9	10	no	—
204	C₃×C₄	C₂³×D₉	1	10	yes	—
205	C₃×C₄	C₂³×C₃×D₃	1	10	yes	—
206	Dic₃	—	3	10	no	—
207	C₃²	D₃×2₋¹⁺⁴	1	1	yes	characteristic
208	C₉	—	2	1	no	—
209	C₉	C₂×C₃×2₋¹⁺⁴	1	1	yes	characteristic, complemented
210	C₂³	—	9	5	no	—
211	Q₈	—	9	30	no	—
212	Q₈	C₂²×C₃×D₉	1	10	yes	—
213	D₄	—	9	20	no	—
214	D₄	—	9	10	no	—
215	D₄	C₂²×C₃×D₉	1	10	yes	—
216	C₂×C₄	—	9	15	no	—
217	C₂×C₄	—	9	10	no	—
218	C₂×C₄	—	9	30	no	—
219	C₂×C₄	C₂²×C₃×D₉	1	15	yes	—
220	C₂×C₃	—	2	5	no	—
221	C₂×C₃	—	2	5	no	—
222	C₂×C₃	—	2	1	no	—
223	C₂×C₃	—	4	5	no	—
224	C₂×C₃	—	9	2	no	—
225	C₂×C₃	—	18	5	no	—
226	C₂×C₃	C₂⁴×D₉	1	1	yes	characteristic
227	C₂×C₃	C₂⁴×C₃×D₃	1	1	yes	characteristic
228	D₃	—	3	2	no	—
229	D₃	—	6	5	no	—
230	C₂²	—	9	1	no	—
231	C₂²	—	9	5	no	—
232	C₂²	—	18	10	no	—
233	C₂²	C₂³×C₃×D₉	1	5	yes	—
234	C₄	—	9	10	no	—
235	C₄	C₂³×C₃×D₉	1	10	yes	—
236	C₃	—	2	1	no	—
237	C₃	D₉×2₋¹⁺⁴	1	1	yes	characteristic, direct product factor
238	C₃	C₃×D₃×2₋¹⁺⁴	1	1	yes	characteristic
239	C₂	—	2	5	no	—
240	C₂	—	9	2	no	—
241	C₂	—	18	5	no	—
242	C₂	C₂⁴×C₃×D₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃	C₂⁴×D₉
commutator, cosocle	C₂×C₉	C₂⁵×C₃
Frattini	C₂×C₃	C₂⁴×C₃×D₃
Fitting	C₃×C₉×2₋¹⁺⁴	C₂
socle	C₂×C₃²	C₂⁴×D₃
hypercenter	C₃×2₋¹⁺⁴	D₉
hypocenter	C₉	C₂×C₃×2₋¹⁺⁴
LCS	G ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₃ ◁ C₃×2₋¹⁺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. C₃×#63716	C₂	semidirect
3. C₃×#41×D₉	C₂	semidirect
4. C₃×#63713	C₂	semidirect
5. C₂×C₃×Q₈×D₉	C₂	semidirect
7. D₉×2₋¹⁺⁴	C₃	direct (N × H)
8. C₃×C₉×#41	C₂²	semidirect
9. C₂×C₃×Q₈×C₉	C₂²	semidirect
10. C₃×#3701	C₂²	semidirect
12. C₃×Q₈×D₉	C₂²	semidirect
13. C₃×#3699	C₂²	semidirect
15. C₃×#3697	C₂²	semidirect
17. C₂×C₃×Dic₁₈	C₂²	semidirect
24. C₉×2₋¹⁺⁴	C₂×C₃	semidirect
25. #63716	C₂×C₃	semidirect
26. #41×D₉	C₂×C₃	semidirect
27. #63713	C₂×C₃	semidirect
28. C₂×Q₈×D₉	C₂×C₃	semidirect
52. C₉×#41	C₂²×C₃	semidirect
54. C₂×Q₈×C₉	C₂²×C₃	semidirect
55. #3701	C₂²×C₃	semidirect
57. Q₈×D₉	C₂²×C₃	semidirect
58. #3699	C₂²×C₃	semidirect
60. #3697	C₂²×C₃	semidirect
62. C₂×Dic₁₈	C₂²×C₃	semidirect
72. C₃×2₋¹⁺⁴	D₉	direct (N × H)
107. C₃×D₉	2₋¹⁺⁴	direct (N × H)
112. C₃×#41	C₂×D₉	semidirect
118. C₂×C₃×Q₈	C₂×D₉	semidirect
146. 2₋¹⁺⁴	C₃×D₉	direct (N × H)
149. C₃×C₉	C₂×2₋¹⁺⁴	semidirect
182. D₉	C₃×2₋¹⁺⁴	direct (N × H)
186. #41	C₂×C₃×D₉	semidirect
190. C₂×Q₈	C₂×C₃×D₉	semidirect
209. C₉	C₂×C₃×2₋¹⁺⁴	semidirect
237. C₃	D₉×2₋¹⁺⁴	direct (N × H)

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
C₃×#63716	2
C₃×#41×D₉	2
C₃×#63713	2
C₂×C₃×Q₈×D₉	2
C₃×D₃×2₋¹⁺⁴	3
D₉×2₋¹⁺⁴	3

Subgroups — C3×D9×2−1+4 #13164955

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×D₉×2₋¹⁺⁴ #13164955