Subgroups — C₃²×D₃×2₋¹⁺⁴ #13181726

Order 1728.

5348 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	C₃³×2₋¹⁺⁴	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃²×#1010	C₂	1	10	yes	maximal, complemented
3	C₃²×D₃×#41	C₂	1	10	yes	maximal, complemented
4	C₃²×#1007	C₂	1	5	yes	maximal, complemented
5	C₂×C₃²×D₃×Q₈	C₂	1	5	yes	maximal, complemented
6	C₂×C₃²×2₋¹⁺⁴	—	3	1	no	maximal
7	C₃×D₃×2₋¹⁺⁴	C₃	1	4	yes	maximal, direct product factor
8	C₃³×#41	C₂²	1	10	yes	complemented
9	C₂×C₃³×Q₈	C₂²	1	5	yes	complemented
10	C₃²×#239	C₂²	1	10	yes	complemented
11	C₃²×D₃×Q₈	C₂²	1	10	yes	—
12	C₃²×D₃×Q₈	C₂²	1	20	yes	complemented
13	C₃²×#237	C₂²	1	30	yes	complemented
14	C₃²×D₃×D₄	C₂²	1	10	yes	—
15	C₃²×#235	C₂²	1	30	yes	complemented
16	C₂×C₃²×C₄×D₃	C₂²	1	15	yes	—
17	C₂×C₃²×Dic₆	C₂²	1	15	yes	complemented
18	C₃²×2₋¹⁺⁴	—	2	4	no	—
19	C₃²×2₋¹⁺⁴	—	3	10	no	—
20	C₃²×2₋¹⁺⁴	—	3	5	no	—
21	C₃²×2₋¹⁺⁴	D₃	1	1	yes	characteristic, direct product factor
22	C₃²×2₋¹⁺⁴	C₂×C₃	1	4	yes	complemented
23	C₂×C₃²×#41	—	3	10	no	—
24	C₂²×C₃²×Q₈	—	3	5	no	—
25	C₃×#1010	C₂×C₃	1	40	yes	complemented
26	C₃×D₃×#41	C₂×C₃	1	40	yes	complemented
27	C₃×#1007	C₂×C₃	1	20	yes	complemented
28	C₂×C₃×D₃×Q₈	C₂×C₃	1	20	yes	complemented
29	C₂²×C₃²×D₃	C₂³	1	5	yes	—
30	C₃³×Q₈	C₂³	1	10	yes	—
31	C₃³×D₄	C₂³	1	10	yes	—
32	C₂×C₃³×C₄	C₂³	1	15	yes	—
33	C₃²×#67	C₂³	1	20	yes	—
34	C₂×C₃²×Dic₃	C₂³	1	15	yes	—
35	C₃²×D₁₂	C₂³	1	10	yes	—
36	C₃²×C₄×D₃	C₂³	1	10	yes	—
37	C₃²×C₄×D₃	C₂³	1	30	yes	—
38	C₃²×Dic₆	C₂³	1	30	yes	—
39	C₂×C₃×2₋¹⁺⁴	—	3	4	no	—
40	D₃×2₋¹⁺⁴	C₃²	1	1	yes	characteristic, direct product factor
41	C₃²×#41	—	2	40	no	—
42	C₃²×#41	—	3	10	no	—
43	C₃²×#41	—	3	30	no	—
44	C₃²×#41	—	3	30	no	—
45	C₃²×#41	C₂×D₃	1	10	yes	complemented
46	C₃²×#41	C₂²×C₃	1	40	yes	complemented
47	C₂×C₃²×Q₈	—	2	20	no	—
48	C₂×C₃²×Q₈	—	3	10	no	—
49	C₂×C₃²×Q₈	—	3	20	no	—
50	C₂×C₃²×Q₈	—	3	15	no	—
51	C₂×C₃²×Q₈	C₂×D₃	1	5	yes	complemented
52	C₂×C₃²×Q₈	C₂²×C₃	1	20	yes	complemented
53	C₂×C₃²×D₄	—	3	10	no	—
54	C₂²×C₃²×C₄	—	3	15	no	—
55	C₃×#239	C₂²×C₃	1	40	yes	complemented
56	C₃×D₃×Q₈	C₂²×C₃	1	40	yes	—
57	C₃×D₃×Q₈	C₂²×C₃	1	80	yes	complemented
58	C₃×#237	C₂²×C₃	1	120	yes	complemented
59	C₃×D₃×D₄	C₂²×C₃	1	40	yes	—
60	C₃×#235	C₂²×C₃	1	120	yes	complemented
61	C₂×C₃×C₄×D₃	C₂²×C₃	1	60	yes	—
62	C₂×C₃×Dic₆	C₂²×C₃	1	60	yes	complemented
63	C₂²×C₃³	C₂⁴	1	5	yes	—
64	C₂×C₃²×D₃	—	2	10	no	—
65	C₂×C₃²×D₃	C₂⁴	1	5	yes	—
66	C₂×C₃²×D₃	C₂⁴	1	1	yes	characteristic
67	C₃³×C₄	C₂⁴	1	10	yes	—
68	C₃²×Dic₃	C₂⁴	1	10	yes	—
69	C₃×2₋¹⁺⁴	—	2	4	no	—
70	C₃×2₋¹⁺⁴	—	3	40	no	—
71	C₃×2₋¹⁺⁴	—	3	20	no	—
72	C₃×2₋¹⁺⁴	C₃×D₃	1	4	yes	direct product factor
73	C₃×2₋¹⁺⁴	C₂×C₃²	1	1	yes	characteristic, complemented
74	C₂×C₃×#41	—	3	40	no	—
75	C₂²×C₃×Q₈	—	3	20	no	—
76	#1010	C₂×C₃²	1	10	yes	complemented
77	D₃×#41	C₂×C₃²	1	10	yes	complemented
78	#1007	C₂×C₃²	1	5	yes	complemented
79	C₂×D₃×Q₈	C₂×C₃²	1	5	yes	complemented
80	C₂³×C₃²	—	3	5	no	—
81	C₂²×C₃×D₃	C₂³×C₃	1	20	yes	—
82	C₃²×Q₈	—	2	40	no	—
83	C₃²×Q₈	—	3	30	no	—
84	C₃²×Q₈	C₂²×D₃	1	10	yes	—
85	C₃²×Q₈	C₂³×C₃	1	40	yes	—
86	C₃²×D₄	—	2	40	no	—
87	C₃²×D₄	—	3	20	no	—
88	C₃²×D₄	—	3	10	no	—
89	C₃²×D₄	C₂²×D₃	1	10	yes	—
90	C₃²×D₄	C₂³×C₃	1	40	yes	—
91	C₂×C₃²×C₄	—	2	60	no	—
92	C₂×C₃²×C₄	—	3	15	no	—
93	C₂×C₃²×C₄	—	3	10	no	—
94	C₂×C₃²×C₄	—	3	30	no	—
95	C₂×C₃²×C₄	C₂²×D₃	1	15	yes	—
96	C₂×C₃²×C₄	C₂³×C₃	1	60	yes	—
97	C₃×#67	C₂³×C₃	1	80	yes	—
98	C₂×C₃×Dic₃	C₂³×C₃	1	60	yes	—
99	C₃×D₁₂	C₂³×C₃	1	40	yes	—
100	C₃×C₄×D₃	C₂³×C₃	1	40	yes	—
101	C₃×C₄×D₃	C₂³×C₃	1	120	yes	—
102	C₃×Dic₆	C₂³×C₃	1	120	yes	—
103	C₂×2₋¹⁺⁴	—	3	1	no	Sylow
104	C₂×C₃³	—	2	5	no	—
105	C₂×C₃³	C₂⁵	1	1	yes	characteristic
106	C₃²×D₃	—	2	5	no	—
107	C₃²×D₃	2₋¹⁺⁴	1	2	yes	direct product factor
108	C₃×#41	—	2	40	no	—
109	C₃×#41	—	3	40	no	—
110	C₃×#41	—	3	120	no	—
111	C₃×#41	—	3	120	no	—
112	C₃×#41	C₂×C₃×D₃	1	40	yes	complemented
113	C₃×#41	C₂²×C₃²	1	10	yes	complemented
114	C₂×C₃×Q₈	—	2	20	no	—
115	C₂×C₃×Q₈	—	3	40	no	—
116	C₂×C₃×Q₈	—	3	80	no	—
117	C₂×C₃×Q₈	—	3	60	no	—
118	C₂×C₃×Q₈	C₂×C₃×D₃	1	20	yes	complemented
119	C₂×C₃×Q₈	C₂²×C₃²	1	5	yes	complemented
120	C₂×C₃×D₄	—	3	40	no	—
121	C₂²×C₃×C₄	—	3	60	no	—
122	#239	C₂²×C₃²	1	10	yes	complemented
123	D₃×Q₈	C₂²×C₃²	1	10	yes	—
124	D₃×Q₈	C₂²×C₃²	1	20	yes	complemented
125	#237	C₂²×C₃²	1	30	yes	complemented
126	D₃×D₄	C₂²×C₃²	1	10	yes	—
127	#235	C₂²×C₃²	1	30	yes	complemented
128	C₂×C₄×D₃	C₂²×C₃²	1	15	yes	—
129	C₂×Dic₆	C₂²×C₃²	1	15	yes	complemented
130	C₂²×C₃²	—	2	20	no	—
131	C₂²×C₃²	—	3	1	no	—
132	C₂²×C₃²	—	3	5	no	—
133	C₂²×C₃²	—	6	10	no	—
134	C₂²×C₃²	C₂³×D₃	1	5	yes	—
135	C₂²×C₃²	C₂⁴×C₃	1	20	yes	—
136	C₂×C₃×D₃	—	2	40	no	—
137	C₂×C₃×D₃	C₂⁴×C₃	1	20	yes	—
138	C₂×C₃×D₃	C₂⁴×C₃	1	4	yes	—
139	C₃²×C₄	—	2	40	no	—
140	C₃²×C₄	—	3	10	no	—
141	C₃²×C₄	C₂³×D₃	1	10	yes	—
142	C₃²×C₄	C₂⁴×C₃	1	40	yes	—
143	C₃×Dic₃	C₂⁴×C₃	1	40	yes	—
144	2₋¹⁺⁴	—	3	10	no	—
145	2₋¹⁺⁴	—	3	5	no	—
146	2₋¹⁺⁴	C₃²×D₃	1	1	yes	characteristic, direct product factor
147	C₂×#41	—	3	10	no	—
148	C₂²×Q₈	—	3	5	no	—
149	C₃³	C₂×2₋¹⁺⁴	1	1	yes	characteristic, complemented, Sylow
150	C₂³×C₃	—	3	20	no	—
151	C₂²×D₃	C₂³×C₃²	1	5	yes	—
152	C₃×Q₈	—	2	40	no	—
153	C₃×Q₈	—	3	120	no	—
154	C₃×Q₈	C₂²×C₃×D₃	1	40	yes	—
155	C₃×Q₈	C₂³×C₃²	1	10	yes	—
156	C₃×D₄	—	2	40	no	—
157	C₃×D₄	—	3	80	no	—
158	C₃×D₄	—	3	40	no	—
159	C₃×D₄	C₂²×C₃×D₃	1	40	yes	—
160	C₃×D₄	C₂³×C₃²	1	10	yes	—
161	C₂×C₃×C₄	—	2	60	no	—
162	C₂×C₃×C₄	—	3	60	no	—
163	C₂×C₃×C₄	—	3	40	no	—
164	C₂×C₃×C₄	—	3	120	no	—
165	C₂×C₃×C₄	C₂²×C₃×D₃	1	60	yes	—
166	C₂×C₃×C₄	C₂³×C₃²	1	15	yes	—
167	#67	C₂³×C₃²	1	20	yes	—
168	C₂×Dic₃	C₂³×C₃²	1	15	yes	—
169	D₁₂	C₂³×C₃²	1	10	yes	—
170	C₄×D₃	C₂³×C₃²	1	10	yes	—
171	C₄×D₃	C₂³×C₃²	1	30	yes	—
172	Dic₆	C₂³×C₃²	1	30	yes	—
173	C₂×C₃²	—	2	5	no	—
174	C₂×C₃²	—	2	4	no	—
175	C₂×C₃²	—	2	20	no	—
176	C₂×C₃²	—	3	2	no	—
177	C₂×C₃²	—	4	20	no	—
178	C₂×C₃²	—	6	5	no	—
179	C₂×C₃²	C₂⁴×D₃	1	1	yes	characteristic
180	C₂×C₃²	C₂⁵×C₃	1	4	yes	—
181	C₃×D₃	—	2	20	no	—
182	C₃×D₃	C₃×2₋¹⁺⁴	1	8	yes	direct product factor
183	#41	—	3	10	no	—
184	#41	—	3	30	no	—
185	#41	—	3	30	no	—
186	#41	C₂×C₃²×D₃	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×C₃²×D₃	1	5	yes	complemented
191	C₂×D₄	—	3	10	no	—
192	C₂²×C₄	—	3	15	no	—
193	C₂²×C₃	—	2	20	no	—
194	C₂²×C₃	—	3	4	no	—
195	C₂²×C₃	—	3	20	no	—
196	C₂²×C₃	—	6	40	no	—
197	C₂²×C₃	C₂³×C₃×D₃	1	20	yes	—
198	C₂²×C₃	C₂⁴×C₃²	1	5	yes	—
199	C₂×D₃	—	2	10	no	—
200	C₂×D₃	C₂⁴×C₃²	1	5	yes	—
201	C₂×D₃	C₂⁴×C₃²	1	1	yes	characteristic
202	C₃×C₄	—	2	40	no	—
203	C₃×C₄	—	3	40	no	—
204	C₃×C₄	C₂³×C₃×D₃	1	40	yes	—
205	C₃×C₄	C₂⁴×C₃²	1	10	yes	—
206	Dic₃	C₂⁴×C₃²	1	10	yes	—
207	C₃²	—	2	4	no	—
208	C₃²	D₃×2₋¹⁺⁴	1	1	yes	characteristic, direct product factor
209	C₃²	C₂×C₃×2₋¹⁺⁴	1	4	yes	complemented
210	C₂³	—	3	5	no	—
211	Q₈	—	3	30	no	—
212	Q₈	C₂²×C₃²×D₃	1	10	yes	—
213	D₄	—	3	20	no	—
214	D₄	—	3	10	no	—
215	D₄	C₂²×C₃²×D₃	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×C₃²×D₃	1	15	yes	—
220	C₂×C₃	—	2	20	no	—
221	C₂×C₃	—	2	5	no	—
222	C₂×C₃	—	2	4	no	—
223	C₂×C₃	—	3	8	no	—
224	C₂×C₃	—	4	20	no	—
225	C₂×C₃	—	6	20	no	—
226	C₂×C₃	C₂⁴×C₃×D₃	1	4	yes	—
227	C₂×C₃	C₂⁵×C₃²	1	1	yes	characteristic
228	D₃	—	2	5	no	—
229	D₃	C₃²×2₋¹⁺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×C₃²×D₃	1	5	yes	—
234	C₄	—	3	10	no	—
235	C₄	C₂³×C₃²×D₃	1	10	yes	—
236	C₃	—	2	4	no	—
237	C₃	C₃×D₃×2₋¹⁺⁴	1	4	yes	direct product factor
238	C₃	C₂×C₃²×2₋¹⁺⁴	1	1	yes	characteristic, complemented
239	C₂	—	2	5	no	—
240	C₂	—	3	2	no	—
241	C₂	—	6	5	no	—
242	C₂	C₂⁴×C₃²×D₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃²	C₂⁴×D₃
commutator, cosocle	C₂×C₃	C₂⁵×C₃²
Frattini	C₂	C₂⁴×C₃²×D₃
Fitting	C₃³×2₋¹⁺⁴	C₂
socle	C₂×C₃³	C₂⁵
hypercenter	C₃²×2₋¹⁺⁴	D₃
hypocenter	C₃	C₂×C₃²×2₋¹⁺⁴
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃² ◁ C₃²×2₋¹⁺⁴

Complements

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

N	H ≅ G/N	type
1. C₃³×2₋¹⁺⁴	C₂	semidirect
2. C₃²×#1010	C₂	semidirect
3. C₃²×D₃×#41	C₂	semidirect
4. C₃²×#1007	C₂	semidirect
5. C₂×C₃²×D₃×Q₈	C₂	semidirect
7. C₃×D₃×2₋¹⁺⁴	C₃	direct (N × H)
8. C₃³×#41	C₂²	semidirect
9. C₂×C₃³×Q₈	C₂²	semidirect
10. C₃²×#239	C₂²	semidirect
12. C₃²×D₃×Q₈	C₂²	semidirect
13. C₃²×#237	C₂²	semidirect
15. C₃²×#235	C₂²	semidirect
17. C₂×C₃²×Dic₆	C₂²	semidirect
21. C₃²×2₋¹⁺⁴	D₃	direct (N × H)
22. C₃²×2₋¹⁺⁴	C₂×C₃	semidirect
25. C₃×#1010	C₂×C₃	semidirect
26. C₃×D₃×#41	C₂×C₃	semidirect
27. C₃×#1007	C₂×C₃	semidirect
28. C₂×C₃×D₃×Q₈	C₂×C₃	semidirect
40. D₃×2₋¹⁺⁴	C₃²	direct (N × H)
45. C₃²×#41	C₂×D₃	semidirect
46. C₃²×#41	C₂²×C₃	semidirect
51. C₂×C₃²×Q₈	C₂×D₃	semidirect
52. C₂×C₃²×Q₈	C₂²×C₃	semidirect
55. C₃×#239	C₂²×C₃	semidirect
57. C₃×D₃×Q₈	C₂²×C₃	semidirect
58. C₃×#237	C₂²×C₃	semidirect
60. C₃×#235	C₂²×C₃	semidirect
62. C₂×C₃×Dic₆	C₂²×C₃	semidirect
72. C₃×2₋¹⁺⁴	C₃×D₃	direct (N × H)
73. C₃×2₋¹⁺⁴	C₂×C₃²	semidirect
76. #1010	C₂×C₃²	semidirect
77. D₃×#41	C₂×C₃²	semidirect
78. #1007	C₂×C₃²	semidirect
79. C₂×D₃×Q₈	C₂×C₃²	semidirect
107. C₃²×D₃	2₋¹⁺⁴	direct (N × H)
112. C₃×#41	C₂×C₃×D₃	semidirect
113. C₃×#41	C₂²×C₃²	semidirect
118. C₂×C₃×Q₈	C₂×C₃×D₃	semidirect
119. C₂×C₃×Q₈	C₂²×C₃²	semidirect
122. #239	C₂²×C₃²	semidirect
124. D₃×Q₈	C₂²×C₃²	semidirect
125. #237	C₂²×C₃²	semidirect
127. #235	C₂²×C₃²	semidirect
129. C₂×Dic₆	C₂²×C₃²	semidirect
146. 2₋¹⁺⁴	C₃²×D₃	direct (N × H)
149. C₃³	C₂×2₋¹⁺⁴	semidirect
182. C₃×D₃	C₃×2₋¹⁺⁴	direct (N × H)
186. #41	C₂×C₃²×D₃	semidirect
190. C₂×Q₈	C₂×C₃²×D₃	semidirect
208. C₃²	D₃×2₋¹⁺⁴	direct (N × H)
209. C₃²	C₂×C₃×2₋¹⁺⁴	semidirect
229. D₃	C₃²×2₋¹⁺⁴	direct (N × H)
237. C₃	C₃×D₃×2₋¹⁺⁴	direct (N × H)
238. C₃	C₂×C₃²×2₋¹⁺⁴	semidirect

Maximal subgroups

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

subgroup	index
C₃³×2₋¹⁺⁴	2
C₃²×#1010	2
C₃²×D₃×#41	2
C₃²×#1007	2
C₂×C₃²×D₃×Q₈	2
C₂×C₃²×2₋¹⁺⁴	3
C₃×D₃×2₋¹⁺⁴	3

Subgroups — C32×D3×2−1+4 #13181726

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×D₃×2₋¹⁺⁴ #13181726