Subgroups — C₂×C₈×C₉×Dic₃ #13143081

Order 1728.

562 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×C₃×C₈×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₉×Dic₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₉×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₈×C₉×Dic₃	C₂	1	4	yes	maximal, direct product factor
5	C₂×C₃×C₈×Dic₃	C₃	1	1	yes	maximal, characteristic
6	C₂×C₄×C₈×C₉	—	3	1	no	maximal
7	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
8	C₂²×C₉×Dic₃	C₄	1	2	yes	—
9	C₂×C₃×C₈×C₉	C₄	1	4	yes	complemented
10	C₂×C₃×C₈×C₉	C₂²	1	2	yes	—
11	C₄×C₉×Dic₃	C₄	1	2	yes	—
12	C₄×C₉×Dic₃	C₂²	1	2	yes	—
13	C₂×C₉×C₃⋊C₈	C₄	1	4	yes	complemented
14	C₂×C₉×C₃⋊C₈	C₂²	1	2	yes	—
15	C₂²×C₃²×C₈	C₂×C₃	1	1	yes	characteristic
16	C₂×C₃×C₄×Dic₃	C₂×C₃	1	1	yes	characteristic
17	C₂²×C₃×C₃⋊C₈	C₂×C₃	1	1	yes	characteristic
18	C₃×C₈×Dic₃	C₂×C₃	1	4	yes	—
19	C₂²×C₈×C₉	—	2	1	no	—
20	C₂²×C₈×C₉	—	3	1	no	—
21	C₂²×C₈×C₉	D₃	1	1	yes	characteristic
22	C₂×C₄²×C₉	—	3	1	no	—
23	C₄×C₈×C₉	—	3	4	no	—
24	C₂³×C₃×C₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
27	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
28	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
29	C₂×C₉×Dic₃	C₈	1	8	yes	direct product factor
30	C₂×C₉×Dic₃	C₂×C₄	1	4	yes	—
31	C₃×C₈×C₉	C₂×C₄	1	4	yes	complemented
32	C₉×C₃⋊C₈	C₂×C₄	1	4	yes	complemented
33	C₂×C₃×C₄×C₈	—	3	1	no	—
34	C₂×C₈×Dic₃	C₉	1	1	yes	characteristic, direct product factor
35	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
36	C₂²×C₃×Dic₃	C₃×C₄	1	2	yes	—
37	C₂×C₃²×C₈	C₃×C₄	1	4	yes	—
38	C₂×C₃²×C₈	C₂²×C₃	1	2	yes	—
39	C₃×C₄×Dic₃	C₃×C₄	1	2	yes	—
40	C₃×C₄×Dic₃	C₂²×C₃	1	2	yes	—
41	C₂×C₃×C₃⋊C₈	C₃×C₄	1	4	yes	—
42	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	2	yes	—
43	C₂²×C₄×C₉	—	2	1	no	—
44	C₂²×C₄×C₉	—	3	2	no	—
45	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
46	C₂×C₈×C₉	—	2	4	no	—
47	C₂×C₈×C₉	—	2	2	no	—
48	C₂×C₈×C₉	—	3	4	no	—
49	C₂×C₈×C₉	—	3	2	no	—
50	C₂×C₈×C₉	Dic₃	1	4	yes	direct product factor
51	C₂×C₈×C₉	C₂×D₃	1	2	yes	—
52	C₄²×C₉	—	3	2	no	—
53	C₄²×C₉	—	3	2	no	—
54	C₂²×C₃×C₉	C₄²	1	2	yes	—
55	C₂²×C₃×C₉	C₂×C₈	1	4	yes	—
56	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
57	C₃×C₄×C₉	C₄²	1	2	yes	—
58	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
59	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
60	C₉×Dic₃	C₂×C₈	1	8	yes	direct product factor
61	C₂²×C₃×C₈	—	2	1	no	—
62	C₂²×C₃×C₈	—	3	1	no	—
63	C₂²×C₃×C₈	C₂×C₉	1	1	yes	characteristic
64	C₂²×C₃×C₈	C₃×D₃	1	1	yes	characteristic
65	C₂×C₃×C₄²	—	3	1	no	—
66	C₂×C₄×Dic₃	C₂×C₉	1	1	yes	characteristic
67	C₂²×C₃⋊C₈	C₂×C₉	1	1	yes	characteristic
68	C₃×C₄×C₈	—	3	4	no	—
69	C₈×Dic₃	C₂×C₉	1	4	yes	direct product factor
70	C₂³×C₃²	C₂×C₃×C₄	1	1	yes	characteristic
71	C₂×C₃²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
72	C₂×C₃²×C₄	C₂×C₃×C₄	1	2	yes	—
73	C₂×C₃²×C₄	C₂×C₃×C₄	1	2	yes	—
74	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
75	C₂×C₃×Dic₃	C₃×C₈	1	8	yes	—
76	C₂×C₃×Dic₃	C₂×C₃×C₄	1	4	yes	—
77	C₂³×C₉	—	2	1	no	—
78	C₂³×C₉	C₄×D₃	1	1	yes	characteristic
79	C₃²×C₈	C₂×C₃×C₄	1	4	yes	—
80	C₃×C₃⋊C₈	C₂×C₃×C₄	1	4	yes	—
81	C₂×C₄×C₉	—	2	1	no	—
82	C₂×C₄×C₉	—	2	2	no	—
83	C₂×C₄×C₉	—	2	2	no	—
84	C₂×C₄×C₉	—	2	1	no	—
85	C₂×C₄×C₉	—	3	8	no	—
86	C₂×C₄×C₉	—	3	4	no	—
87	C₂×C₄×C₉	C₄×D₃	1	1	yes	characteristic
88	C₂×C₄×C₉	C₄×D₃	1	2	yes	—
89	C₂×C₄×C₉	C₂×Dic₃	1	2	yes	—
90	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
91	C₈×C₉	—	2	4	no	—
92	C₈×C₉	—	3	4	no	—
93	C₈×C₉	C₂×Dic₃	1	4	yes	direct product factor
94	C₂×C₄×C₈	—	3	1	no	Sylow
95	C₂×C₃×C₉	C₄×C₈	1	4	yes	complemented
96	C₂×C₃×C₉	C₂×C₄²	1	1	yes	characteristic
97	C₂×C₃×C₉	C₂²×C₈	1	2	yes	—
98	C₂²×C₃×C₄	—	2	1	no	—
99	C₂²×C₃×C₄	—	3	2	no	—
100	C₂²×C₃×C₄	C₂²×C₉	1	1	yes	characteristic
101	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
102	C₂²×Dic₃	C₄×C₉	1	2	yes	—
103	C₂×C₃×C₈	—	2	4	no	—
104	C₂×C₃×C₈	—	2	2	no	—
105	C₂×C₃×C₈	—	3	4	no	—
106	C₂×C₃×C₈	—	3	2	no	—
107	C₂×C₃×C₈	C₄×C₉	1	4	yes	complemented
108	C₂×C₃×C₈	C₂²×C₉	1	2	yes	—
109	C₂×C₃×C₈	C₃×Dic₃	1	4	yes	—
110	C₂×C₃×C₈	C₂×C₃×D₃	1	2	yes	—
111	C₃×C₄²	—	3	2	no	—
112	C₃×C₄²	—	3	2	no	—
113	C₄×Dic₃	C₄×C₉	1	2	yes	—
114	C₄×Dic₃	C₂²×C₉	1	2	yes	—
115	C₂×C₃⋊C₈	C₄×C₉	1	4	yes	complemented
116	C₂×C₃⋊C₈	C₂²×C₉	1	2	yes	—
117	C₂²×C₃²	C₃×C₄²	1	2	yes	—
118	C₂²×C₃²	C₂×C₃×C₈	1	4	yes	—
119	C₂²×C₃²	C₂²×C₃×C₄	1	1	yes	characteristic
120	C₃²×C₄	C₃×C₄²	1	2	yes	—
121	C₃²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
122	C₃²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
123	C₃×Dic₃	C₂×C₃×C₈	1	8	yes	—
124	C₂²×C₉	—	2	4	no	—
125	C₂²×C₉	—	2	2	no	—
126	C₂²×C₉	—	2	1	no	—
127	C₂²×C₉	D₃×C₈	1	4	yes	—
128	C₂²×C₉	C₄×Dic₃	1	2	yes	—
129	C₂²×C₉	C₂×C₄×D₃	1	1	yes	characteristic
130	C₄×C₉	—	2	2	no	—
131	C₄×C₉	—	2	1	no	—
132	C₄×C₉	—	2	1	no	—
133	C₄×C₉	—	3	8	no	—
134	C₄×C₉	C₄×Dic₃	1	2	yes	—
135	C₄×C₉	C₂×C₄×D₃	1	1	yes	characteristic
136	C₄×C₉	C₂²×Dic₃	1	1	yes	characteristic
137	C₂²×C₈	—	3	1	no	—
138	C₂²×C₈	D₃×C₉	1	1	yes	characteristic
139	C₂×C₄²	—	3	1	no	—
140	C₄×C₈	—	3	4	no	—
141	C₃×C₉	C₂×C₄×C₈	1	1	yes	characteristic, complemented, Sylow
142	C₂³×C₃	—	2	1	no	—
143	C₂³×C₃	C₂×C₄×C₉	1	1	yes	characteristic
144	C₂³×C₃	C₃×C₄×D₃	1	1	yes	characteristic
145	C₂×C₃×C₄	—	2	2	no	—
146	C₂×C₃×C₄	—	2	1	no	—
147	C₂×C₃×C₄	—	2	2	no	—
148	C₂×C₃×C₄	—	2	1	no	—
149	C₂×C₃×C₄	—	3	8	no	—
150	C₂×C₃×C₄	—	3	4	no	—
151	C₂×C₃×C₄	C₂×C₄×C₉	1	1	yes	characteristic
152	C₂×C₃×C₄	C₂×C₄×C₉	1	2	yes	—
153	C₂×C₃×C₄	C₂×C₄×C₉	1	2	yes	—
154	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
155	C₂×C₃×C₄	C₃×C₄×D₃	1	1	yes	characteristic
156	C₂×C₃×C₄	C₃×C₄×D₃	1	2	yes	—
157	C₂×C₃×C₄	C₂×C₃×Dic₃	1	2	yes	—
158	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
159	C₂×Dic₃	C₈×C₉	1	8	yes	direct product factor
160	C₂×Dic₃	C₂×C₄×C₉	1	4	yes	—
161	C₃×C₈	—	2	4	no	—
162	C₃×C₈	—	3	4	no	—
163	C₃×C₈	C₂×C₄×C₉	1	4	yes	complemented
164	C₃×C₈	C₂×C₃×Dic₃	1	4	yes	—
165	C₃⋊C₈	C₂×C₄×C₉	1	4	yes	complemented
166	C₂×C₃²	C₃×C₄×C₈	1	4	yes	—
167	C₂×C₃²	C₂×C₃×C₄²	1	1	yes	characteristic
168	C₂×C₃²	C₂²×C₃×C₈	1	2	yes	—
169	C₂×C₉	—	2	2	no	—
170	C₂×C₉	—	2	1	no	—
171	C₂×C₉	—	2	4	no	—
172	C₂×C₉	C₈×Dic₃	1	4	yes	direct product factor
173	C₂×C₉	C₂×D₃×C₈	1	2	yes	—
174	C₂×C₉	C₂×C₄×Dic₃	1	1	yes	characteristic
175	C₂²×C₄	—	3	2	no	—
176	C₂²×C₄	C₂×D₃×C₉	1	1	yes	characteristic
177	C₂×C₈	—	3	4	no	—
178	C₂×C₈	—	3	2	no	—
179	C₂×C₈	C₉×Dic₃	1	4	yes	direct product factor
180	C₂×C₈	C₂×D₃×C₉	1	2	yes	—
181	C₄²	—	3	2	no	—
182	C₄²	—	3	2	no	—
183	C₂²×C₃	—	2	1	no	—
184	C₂²×C₃	—	2	4	no	—
185	C₂²×C₃	—	2	2	no	—
186	C₂²×C₃	C₄²×C₉	1	2	yes	—
187	C₂²×C₃	C₂×C₈×C₉	1	4	yes	—
188	C₂²×C₃	C₂²×C₄×C₉	1	1	yes	characteristic
189	C₂²×C₃	C₃×D₃×C₈	1	4	yes	—
190	C₂²×C₃	C₃×C₄×Dic₃	1	2	yes	—
191	C₂²×C₃	C₂×C₃×C₄×D₃	1	1	yes	characteristic
192	C₃×C₄	—	2	1	no	—
193	C₃×C₄	—	2	1	no	—
194	C₃×C₄	—	2	2	no	—
195	C₃×C₄	—	3	8	no	—
196	C₃×C₄	C₄²×C₉	1	2	yes	—
197	C₃×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
198	C₃×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
199	C₃×C₄	C₃×C₄×Dic₃	1	2	yes	—
200	C₃×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
201	C₃×C₄	C₂²×C₃×Dic₃	1	1	yes	characteristic
202	Dic₃	C₂×C₈×C₉	1	8	yes	direct product factor
203	C₃²	C₂×C₃×C₄×C₈	1	1	yes	characteristic
204	C₉	—	2	1	no	—
205	C₉	C₂×C₈×Dic₃	1	1	yes	characteristic, direct product factor
206	C₂³	C₄×D₃×C₉	1	1	yes	characteristic
207	C₂×C₄	—	3	4	no	—
208	C₂×C₄	—	3	8	no	—
209	C₂×C₄	C₄×D₃×C₉	1	2	yes	—
210	C₂×C₄	C₄×D₃×C₉	1	1	yes	characteristic
211	C₂×C₄	C₂×C₉×Dic₃	1	2	yes	—
212	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
213	C₈	—	3	4	no	—
214	C₈	C₂×C₉×Dic₃	1	4	yes	direct product factor
215	C₂×C₃	—	2	2	no	—
216	C₂×C₃	—	2	1	no	—
217	C₂×C₃	—	2	4	no	—
218	C₂×C₃	C₄×C₈×C₉	1	4	yes	complemented
219	C₂×C₃	C₂×C₄²×C₉	1	1	yes	characteristic
220	C₂×C₃	C₂²×C₈×C₉	1	2	yes	—
221	C₂×C₃	C₃×C₈×Dic₃	1	4	yes	—
222	C₂×C₃	C₂×C₃×D₃×C₈	1	2	yes	—
223	C₂×C₃	C₂×C₃×C₄×Dic₃	1	1	yes	characteristic
224	C₂²	D₃×C₈×C₉	1	4	yes	—
225	C₂²	C₄×C₉×Dic₃	1	2	yes	—
226	C₂²	C₂×C₄×D₃×C₉	1	1	yes	characteristic
227	C₄	—	3	8	no	—
228	C₄	C₄×C₉×Dic₃	1	2	yes	—
229	C₄	C₂×C₄×D₃×C₉	1	1	yes	characteristic
230	C₄	C₂²×C₉×Dic₃	1	1	yes	characteristic
231	C₃	—	2	1	no	—
232	C₃	C₂×C₄×C₈×C₉	1	1	yes	characteristic, complemented
233	C₃	C₂×C₃×C₈×Dic₃	1	1	yes	characteristic
234	C₂	C₈×C₉×Dic₃	1	4	yes	direct product factor
235	C₂	C₂×D₃×C₈×C₉	1	2	yes	—
236	C₂	C₂×C₄×C₉×Dic₃	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²×C₈×C₉	D₃
commutator, hypocenter	C₃	C₂×C₄×C₈×C₉
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₂²×C₃×C₈×C₉	C₂
socle, prime radical	C₂³×C₃²	C₂×C₃×C₄
cosocle	C₂×C₃²×C₄	C₂³×C₃
LCS	G ▷ C₃
UCS	C₁ ◁ C₂²×C₈×C₉

Complements

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

N	H ≅ G/N	type
4. C₈×C₉×Dic₃	C₂	direct (N × H)
9. C₂×C₃×C₈×C₉	C₄	semidirect
13. C₂×C₉×C₃⋊C₈	C₄	semidirect
29. C₂×C₉×Dic₃	C₈	direct (N × H)
31. C₃×C₈×C₉	C₂×C₄	semidirect
32. C₉×C₃⋊C₈	C₂×C₄	semidirect
34. C₂×C₈×Dic₃	C₉	direct (N × H)
50. C₂×C₈×C₉	Dic₃	direct (N × H)
60. C₉×Dic₃	C₂×C₈	direct (N × H)
69. C₈×Dic₃	C₂×C₉	direct (N × H)
93. C₈×C₉	C₂×Dic₃	direct (N × H)
95. C₂×C₃×C₉	C₄×C₈	semidirect
107. C₂×C₃×C₈	C₄×C₉	semidirect
115. C₂×C₃⋊C₈	C₄×C₉	semidirect
141. C₃×C₉	C₂×C₄×C₈	semidirect
159. C₂×Dic₃	C₈×C₉	direct (N × H)
163. C₃×C₈	C₂×C₄×C₉	semidirect
165. C₃⋊C₈	C₂×C₄×C₉	semidirect
172. C₂×C₉	C₈×Dic₃	direct (N × H)
179. C₂×C₈	C₉×Dic₃	direct (N × H)
202. Dic₃	C₂×C₈×C₉	direct (N × H)
205. C₉	C₂×C₈×Dic₃	direct (N × H)
214. C₈	C₂×C₉×Dic₃	direct (N × H)
218. C₂×C₃	C₄×C₈×C₉	semidirect
232. C₃	C₂×C₄×C₈×C₉	semidirect
234. C₂	C₈×C₉×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×C₃×C₈×C₉	2
C₂×C₄×C₉×Dic₃	2
C₂²×C₉×C₃⋊C₈	2
C₈×C₉×Dic₃	2
C₂×C₃×C₈×Dic₃	3
C₂×C₄×C₈×C₉	3

Subgroups — C2×C8×C9×Dic3 #13143081

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₈×C₉×Dic₃ #13143081