Subgroups — C₂×C₃²×C₈×Dic₃ #13162534

Order 1728.

1276 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃³×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃²×C₄×Dic₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₃²×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₃²×C₈×Dic₃	C₂	1	4	yes	maximal, direct product factor
5	C₂×C₃²×C₄×C₈	—	3	1	no	maximal
6	C₂×C₃×C₈×Dic₃	C₃	1	4	yes	maximal, direct product factor
7	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
8	C₂²×C₃²×Dic₃	C₄	1	2	yes	—
9	C₂×C₃³×C₈	C₄	1	4	yes	complemented
10	C₂×C₃³×C₈	C₂²	1	2	yes	—
11	C₃²×C₄×Dic₃	C₄	1	2	yes	—
12	C₃²×C₄×Dic₃	C₂²	1	2	yes	—
13	C₂×C₃²×C₃⋊C₈	C₄	1	4	yes	complemented
14	C₂×C₃²×C₃⋊C₈	C₂²	1	2	yes	—
15	C₂²×C₃²×C₈	—	2	4	no	—
16	C₂²×C₃²×C₈	—	3	1	no	—
17	C₂²×C₃²×C₈	D₃	1	1	yes	characteristic
18	C₂²×C₃²×C₈	C₂×C₃	1	4	yes	—
19	C₂×C₃²×C₄²	—	3	1	no	—
20	C₂×C₃×C₄×Dic₃	C₂×C₃	1	4	yes	—
21	C₂²×C₃×C₃⋊C₈	C₂×C₃	1	4	yes	—
22	C₃²×C₄×C₈	—	3	4	no	—
23	C₃×C₈×Dic₃	C₂×C₃	1	16	yes	direct product factor
24	C₂³×C₃³	C₂×C₄	1	1	yes	characteristic
25	C₂×C₃³×C₄	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃³×C₄	C₂×C₄	1	2	yes	—
27	C₂×C₃³×C₄	C₂×C₄	1	2	yes	—
28	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
29	C₂×C₃²×Dic₃	C₈	1	8	yes	direct product factor
30	C₂×C₃²×Dic₃	C₂×C₄	1	4	yes	—
31	C₃³×C₈	C₂×C₄	1	4	yes	complemented
32	C₃²×C₃⋊C₈	C₂×C₄	1	4	yes	complemented
33	C₂×C₃×C₄×C₈	—	3	4	no	—
34	C₂×C₈×Dic₃	C₃²	1	1	yes	characteristic, direct product factor
35	C₂²×C₃²×C₄	—	2	4	no	—
36	C₂²×C₃²×C₄	—	3	2	no	—
37	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
38	C₂²×C₃²×C₄	C₂²×C₃	1	4	yes	—
39	C₂²×C₃×Dic₃	C₃×C₄	1	8	yes	—
40	C₂×C₃²×C₈	—	2	16	no	—
41	C₂×C₃²×C₈	—	2	8	no	—
42	C₂×C₃²×C₈	—	3	4	no	—
43	C₂×C₃²×C₈	—	3	2	no	—
44	C₂×C₃²×C₈	Dic₃	1	4	yes	direct product factor
45	C₂×C₃²×C₈	C₃×C₄	1	16	yes	complemented
46	C₂×C₃²×C₈	C₂×D₃	1	2	yes	—
47	C₂×C₃²×C₈	C₂²×C₃	1	8	yes	—
48	C₃²×C₄²	—	3	2	no	—
49	C₃²×C₄²	—	3	2	no	—
50	C₃×C₄×Dic₃	C₃×C₄	1	8	yes	—
51	C₃×C₄×Dic₃	C₂²×C₃	1	8	yes	—
52	C₂×C₃×C₃⋊C₈	C₃×C₄	1	16	yes	complemented
53	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	8	yes	—
54	C₂²×C₃³	C₄²	1	2	yes	—
55	C₂²×C₃³	C₂×C₈	1	4	yes	—
56	C₂²×C₃³	C₂²×C₄	1	1	yes	characteristic
57	C₃³×C₄	C₄²	1	2	yes	—
58	C₃³×C₄	C₂²×C₄	1	1	yes	characteristic
59	C₃³×C₄	C₂²×C₄	1	1	yes	characteristic
60	C₃²×Dic₃	C₂×C₈	1	8	yes	direct product factor
61	C₂²×C₃×C₈	—	2	4	no	—
62	C₂²×C₃×C₈	—	3	4	no	—
63	C₂²×C₃×C₈	C₃×D₃	1	4	yes	—
64	C₂²×C₃×C₈	C₂×C₃²	1	1	yes	characteristic
65	C₂×C₃×C₄²	—	3	4	no	—
66	C₂×C₄×Dic₃	C₂×C₃²	1	1	yes	characteristic
67	C₂²×C₃⋊C₈	C₂×C₃²	1	1	yes	characteristic
68	C₃×C₄×C₈	—	3	16	no	—
69	C₈×Dic₃	C₂×C₃²	1	4	yes	direct product factor
70	C₂³×C₃²	—	2	4	no	—
71	C₂³×C₃²	C₄×D₃	1	1	yes	characteristic
72	C₂³×C₃²	C₂×C₃×C₄	1	4	yes	—
73	C₂×C₃²×C₄	—	2	4	no	—
74	C₂×C₃²×C₄	—	2	8	no	—
75	C₂×C₃²×C₄	—	2	8	no	—
76	C₂×C₃²×C₄	—	2	4	no	—
77	C₂×C₃²×C₄	—	3	8	no	—
78	C₂×C₃²×C₄	—	3	4	no	—
79	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
80	C₂×C₃²×C₄	C₄×D₃	1	2	yes	—
81	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
82	C₂×C₃²×C₄	C₂×C₃×C₄	1	4	yes	—
83	C₂×C₃²×C₄	C₂×C₃×C₄	1	8	yes	—
84	C₂×C₃²×C₄	C₂×C₃×C₄	1	8	yes	—
85	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
86	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
87	C₂×C₃×Dic₃	C₃×C₈	1	32	yes	direct product factor
88	C₂×C₃×Dic₃	C₂×C₃×C₄	1	16	yes	—
89	C₃²×C₈	—	2	16	no	—
90	C₃²×C₈	—	3	4	no	—
91	C₃²×C₈	C₂×Dic₃	1	4	yes	direct product factor
92	C₃²×C₈	C₂×C₃×C₄	1	16	yes	complemented
93	C₃×C₃⋊C₈	C₂×C₃×C₄	1	16	yes	complemented
94	C₂×C₄×C₈	—	3	1	no	Sylow
95	C₂×C₃³	C₄×C₈	1	4	yes	complemented
96	C₂×C₃³	C₂×C₄²	1	1	yes	characteristic
97	C₂×C₃³	C₂²×C₈	1	2	yes	—
98	C₂²×C₃×C₄	—	2	4	no	—
99	C₂²×C₃×C₄	—	3	8	no	—
100	C₂²×C₃×C₄	C₂×C₃×D₃	1	4	yes	—
101	C₂²×C₃×C₄	C₂²×C₃²	1	1	yes	characteristic
102	C₂²×Dic₃	C₃²×C₄	1	2	yes	—
103	C₂×C₃×C₈	—	2	16	no	—
104	C₂×C₃×C₈	—	2	8	no	—
105	C₂×C₃×C₈	—	3	16	no	—
106	C₂×C₃×C₈	—	3	8	no	—
107	C₂×C₃×C₈	C₃×Dic₃	1	16	yes	direct product factor
108	C₂×C₃×C₈	C₃²×C₄	1	4	yes	complemented
109	C₂×C₃×C₈	C₂×C₃×D₃	1	8	yes	—
110	C₂×C₃×C₈	C₂²×C₃²	1	2	yes	—
111	C₃×C₄²	—	3	8	no	—
112	C₃×C₄²	—	3	8	no	—
113	C₄×Dic₃	C₃²×C₄	1	2	yes	—
114	C₄×Dic₃	C₂²×C₃²	1	2	yes	—
115	C₂×C₃⋊C₈	C₃²×C₄	1	4	yes	complemented
116	C₂×C₃⋊C₈	C₂²×C₃²	1	2	yes	—
117	C₂²×C₃²	—	2	16	no	—
118	C₂²×C₃²	—	2	8	no	—
119	C₂²×C₃²	—	2	4	no	—
120	C₂²×C₃²	D₃×C₈	1	4	yes	—
121	C₂²×C₃²	C₄×Dic₃	1	2	yes	—
122	C₂²×C₃²	C₃×C₄²	1	8	yes	—
123	C₂²×C₃²	C₂×C₃×C₈	1	16	yes	—
124	C₂²×C₃²	C₂×C₄×D₃	1	1	yes	characteristic
125	C₂²×C₃²	C₂²×C₃×C₄	1	4	yes	—
126	C₃²×C₄	—	2	8	no	—
127	C₃²×C₄	—	2	4	no	—
128	C₃²×C₄	—	2	4	no	—
129	C₃²×C₄	—	3	8	no	—
130	C₃²×C₄	C₄×Dic₃	1	2	yes	—
131	C₃²×C₄	C₃×C₄²	1	8	yes	—
132	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
133	C₃²×C₄	C₂²×Dic₃	1	1	yes	characteristic
134	C₃²×C₄	C₂²×C₃×C₄	1	4	yes	—
135	C₃²×C₄	C₂²×C₃×C₄	1	4	yes	—
136	C₃×Dic₃	C₂×C₃×C₈	1	32	yes	direct product factor
137	C₂²×C₈	—	3	1	no	—
138	C₂²×C₈	C₃²×D₃	1	1	yes	characteristic
139	C₂×C₄²	—	3	1	no	—
140	C₄×C₈	—	3	4	no	—
141	C₃³	C₂×C₄×C₈	1	1	yes	characteristic, complemented, Sylow
142	C₂³×C₃	—	2	4	no	—
143	C₂³×C₃	C₃×C₄×D₃	1	4	yes	—
144	C₂³×C₃	C₂×C₃²×C₄	1	1	yes	characteristic
145	C₂×C₃×C₄	—	2	8	no	—
146	C₂×C₃×C₄	—	2	4	no	—
147	C₂×C₃×C₄	—	2	8	no	—
148	C₂×C₃×C₄	—	2	4	no	—
149	C₂×C₃×C₄	—	3	32	no	—
150	C₂×C₃×C₄	—	3	16	no	—
151	C₂×C₃×C₄	C₃×C₄×D₃	1	4	yes	—
152	C₂×C₃×C₄	C₃×C₄×D₃	1	8	yes	—
153	C₂×C₃×C₄	C₂×C₃×Dic₃	1	8	yes	—
154	C₂×C₃×C₄	C₂×C₃²×C₄	1	1	yes	characteristic
155	C₂×C₃×C₄	C₂×C₃²×C₄	1	2	yes	—
156	C₂×C₃×C₄	C₂×C₃²×C₄	1	2	yes	—
157	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
158	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
159	C₂×Dic₃	C₃²×C₈	1	8	yes	direct product factor
160	C₂×Dic₃	C₂×C₃²×C₄	1	4	yes	—
161	C₃×C₈	—	2	16	no	—
162	C₃×C₈	—	3	16	no	—
163	C₃×C₈	C₂×C₃×Dic₃	1	16	yes	direct product factor
164	C₃×C₈	C₂×C₃²×C₄	1	4	yes	complemented
165	C₃⋊C₈	C₂×C₃²×C₄	1	4	yes	complemented
166	C₂×C₃²	—	2	8	no	—
167	C₂×C₃²	—	2	4	no	—
168	C₂×C₃²	—	2	16	no	—
169	C₂×C₃²	C₈×Dic₃	1	4	yes	direct product factor
170	C₂×C₃²	C₃×C₄×C₈	1	16	yes	complemented
171	C₂×C₃²	C₂×D₃×C₈	1	2	yes	—
172	C₂×C₃²	C₂×C₄×Dic₃	1	1	yes	characteristic
173	C₂×C₃²	C₂×C₃×C₄²	1	4	yes	—
174	C₂×C₃²	C₂²×C₃×C₈	1	8	yes	—
175	C₂²×C₄	—	3	2	no	—
176	C₂²×C₄	C₂×C₃²×D₃	1	1	yes	characteristic
177	C₂×C₈	—	3	4	no	—
178	C₂×C₈	—	3	2	no	—
179	C₂×C₈	C₃²×Dic₃	1	4	yes	direct product factor
180	C₂×C₈	C₂×C₃²×D₃	1	2	yes	—
181	C₄²	—	3	2	no	—
182	C₄²	—	3	2	no	—
183	C₂²×C₃	—	2	4	no	—
184	C₂²×C₃	—	2	16	no	—
185	C₂²×C₃	—	2	8	no	—
186	C₂²×C₃	C₃×D₃×C₈	1	16	yes	—
187	C₂²×C₃	C₃×C₄×Dic₃	1	8	yes	—
188	C₂²×C₃	C₃²×C₄²	1	2	yes	—
189	C₂²×C₃	C₂×C₃²×C₈	1	4	yes	—
190	C₂²×C₃	C₂×C₃×C₄×D₃	1	4	yes	—
191	C₂²×C₃	C₂²×C₃²×C₄	1	1	yes	characteristic
192	C₃×C₄	—	2	4	no	—
193	C₃×C₄	—	2	4	no	—
194	C₃×C₄	—	2	8	no	—
195	C₃×C₄	—	3	32	no	—
196	C₃×C₄	C₃×C₄×Dic₃	1	8	yes	—
197	C₃×C₄	C₃²×C₄²	1	2	yes	—
198	C₃×C₄	C₂×C₃×C₄×D₃	1	4	yes	—
199	C₃×C₄	C₂²×C₃×Dic₃	1	4	yes	—
200	C₃×C₄	C₂²×C₃²×C₄	1	1	yes	characteristic
201	C₃×C₄	C₂²×C₃²×C₄	1	1	yes	characteristic
202	Dic₃	C₂×C₃²×C₈	1	8	yes	direct product factor
203	C₃²	—	2	4	no	—
204	C₃²	C₂×C₈×Dic₃	1	1	yes	characteristic, direct product factor
205	C₃²	C₂×C₃×C₄×C₈	1	4	yes	complemented
206	C₂³	C₃²×C₄×D₃	1	1	yes	characteristic
207	C₂×C₄	—	3	4	no	—
208	C₂×C₄	—	3	8	no	—
209	C₂×C₄	C₃²×C₄×D₃	1	2	yes	—
210	C₂×C₄	C₃²×C₄×D₃	1	1	yes	characteristic
211	C₂×C₄	C₂×C₃²×Dic₃	1	2	yes	—
212	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
213	C₈	—	3	4	no	—
214	C₈	C₂×C₃²×Dic₃	1	4	yes	direct product factor
215	C₂×C₃	—	2	4	no	—
216	C₂×C₃	—	2	16	no	—
217	C₂×C₃	—	2	8	no	—
218	C₂×C₃	C₃×C₈×Dic₃	1	16	yes	direct product factor
219	C₂×C₃	C₃²×C₄×C₈	1	4	yes	complemented
220	C₂×C₃	C₂×C₃×D₃×C₈	1	8	yes	—
221	C₂×C₃	C₂×C₃×C₄×Dic₃	1	4	yes	—
222	C₂×C₃	C₂×C₃²×C₄²	1	1	yes	characteristic
223	C₂×C₃	C₂²×C₃²×C₈	1	2	yes	—
224	C₂²	C₃²×D₃×C₈	1	4	yes	—
225	C₂²	C₃²×C₄×Dic₃	1	2	yes	—
226	C₂²	C₂×C₃²×C₄×D₃	1	1	yes	characteristic
227	C₄	—	3	8	no	—
228	C₄	C₃²×C₄×Dic₃	1	2	yes	—
229	C₄	C₂×C₃²×C₄×D₃	1	1	yes	characteristic
230	C₄	C₂²×C₃²×Dic₃	1	1	yes	characteristic
231	C₃	—	2	4	no	—
232	C₃	C₂×C₃×C₈×Dic₃	1	4	yes	direct product factor
233	C₃	C₂×C₃²×C₄×C₈	1	1	yes	characteristic, complemented
234	C₂	C₃²×C₈×Dic₃	1	4	yes	direct product factor
235	C₂	C₂×C₃²×D₃×C₈	1	2	yes	—
236	C₂	C₂×C₃²×C₄×Dic₃	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²×C₃²×C₈	D₃
commutator, hypocenter	C₃	C₂×C₃²×C₄×C₈
Frattini	C₂×C₄	C₂²×C₃²×D₃
Fitting	C₂²×C₃³×C₈	C₂
socle, prime radical	C₂³×C₃³	C₂×C₄
cosocle	C₂×C₃×C₄	C₂³×C₃²
LCS	G ▷ C₃
UCS	C₁ ◁ C₂²×C₃²×C₈

Complements

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

N	H ≅ G/N	type
4. C₃²×C₈×Dic₃	C₂	direct (N × H)
6. C₂×C₃×C₈×Dic₃	C₃	direct (N × H)
9. C₂×C₃³×C₈	C₄	semidirect
13. C₂×C₃²×C₃⋊C₈	C₄	semidirect
23. C₃×C₈×Dic₃	C₂×C₃	direct (N × H)
29. C₂×C₃²×Dic₃	C₈	direct (N × H)
31. C₃³×C₈	C₂×C₄	semidirect
32. C₃²×C₃⋊C₈	C₂×C₄	semidirect
34. C₂×C₈×Dic₃	C₃²	direct (N × H)
44. C₂×C₃²×C₈	Dic₃	direct (N × H)
45. C₂×C₃²×C₈	C₃×C₄	semidirect
52. C₂×C₃×C₃⋊C₈	C₃×C₄	semidirect
60. C₃²×Dic₃	C₂×C₈	direct (N × H)
69. C₈×Dic₃	C₂×C₃²	direct (N × H)
87. C₂×C₃×Dic₃	C₃×C₈	direct (N × H)
91. C₃²×C₈	C₂×Dic₃	direct (N × H)
92. C₃²×C₈	C₂×C₃×C₄	semidirect
93. C₃×C₃⋊C₈	C₂×C₃×C₄	semidirect
95. C₂×C₃³	C₄×C₈	semidirect
107. C₂×C₃×C₈	C₃×Dic₃	direct (N × H)
108. C₂×C₃×C₈	C₃²×C₄	semidirect
115. C₂×C₃⋊C₈	C₃²×C₄	semidirect
136. C₃×Dic₃	C₂×C₃×C₈	direct (N × H)
141. C₃³	C₂×C₄×C₈	semidirect
159. C₂×Dic₃	C₃²×C₈	direct (N × H)
163. C₃×C₈	C₂×C₃×Dic₃	direct (N × H)
164. C₃×C₈	C₂×C₃²×C₄	semidirect
165. C₃⋊C₈	C₂×C₃²×C₄	semidirect
169. C₂×C₃²	C₈×Dic₃	direct (N × H)
170. C₂×C₃²	C₃×C₄×C₈	semidirect
179. C₂×C₈	C₃²×Dic₃	direct (N × H)
202. Dic₃	C₂×C₃²×C₈	direct (N × H)
204. C₃²	C₂×C₈×Dic₃	direct (N × H)
205. C₃²	C₂×C₃×C₄×C₈	semidirect
214. C₈	C₂×C₃²×Dic₃	direct (N × H)
218. C₂×C₃	C₃×C₈×Dic₃	direct (N × H)
219. C₂×C₃	C₃²×C₄×C₈	semidirect
232. C₃	C₂×C₃×C₈×Dic₃	direct (N × H)
233. C₃	C₂×C₃²×C₄×C₈	semidirect
234. C₂	C₃²×C₈×Dic₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃³×C₈	2
C₂×C₃²×C₄×Dic₃	2
C₂²×C₃²×C₃⋊C₈	2
C₃²×C₈×Dic₃	2
C₂×C₃²×C₄×C₈	3
C₂×C₃×C₈×Dic₃	3

Subgroups — C2×C32×C8×Dic3 #13162534

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃²×C₈×Dic₃ #13162534