Subgroups — C₂×C₃×C₈×Dic₉ #13143073

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×C₃×C₄×Dic₉	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₃×C₉⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₃×C₈×Dic₉	C₂	1	4	yes	maximal, direct product factor
5	C₂×C₃×C₈×Dic₃	—	3	1	no	maximal
6	C₂×C₈×Dic₉	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
8	C₂²×C₃×Dic₉	C₄	1	2	yes	—
9	C₂×C₃×C₈×C₉	C₄	1	4	yes	complemented
10	C₂×C₃×C₈×C₉	C₂²	1	2	yes	—
11	C₃×C₄×Dic₉	C₄	1	2	yes	—
12	C₃×C₄×Dic₉	C₂²	1	2	yes	—
13	C₂×C₃×C₉⋊C₈	C₄	1	4	yes	complemented
14	C₂×C₃×C₉⋊C₈	C₂²	1	2	yes	—
15	C₂²×C₃²×C₈	D₃	1	1	yes	characteristic
16	C₂×C₃×C₄×Dic₃	—	3	1	no	—
17	C₂²×C₃×C₃⋊C₈	—	3	1	no	—
18	C₃×C₈×Dic₃	—	3	4	no	—
19	C₂²×C₈×C₉	—	2	1	no	—
20	C₂²×C₈×C₉	C₂×C₃	1	1	yes	characteristic
21	C₂×C₄×Dic₉	C₂×C₃	1	1	yes	characteristic
22	C₂²×C₉⋊C₈	C₂×C₃	1	1	yes	characteristic
23	C₈×Dic₉	C₂×C₃	1	4	yes	direct product factor
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×C₃×Dic₉	C₈	1	8	yes	direct product factor
30	C₂×C₃×Dic₉	C₂×C₄	1	4	yes	—
31	C₃×C₈×C₉	C₂×C₄	1	4	yes	complemented
32	C₃×C₉⋊C₈	C₂×C₄	1	4	yes	complemented
33	C₂×C₃×C₄×C₈	—	9	1	no	—
34	C₂×C₈×Dic₃	—	3	1	no	—
35	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
36	C₂²×C₃×Dic₃	—	3	2	no	—
37	C₂×C₃²×C₈	Dic₃	1	4	yes	—
38	C₂×C₃²×C₈	C₂×D₃	1	2	yes	—
39	C₃×C₄×Dic₃	—	3	2	no	—
40	C₃×C₄×Dic₃	—	3	2	no	—
41	C₂×C₃×C₃⋊C₈	—	3	4	no	—
42	C₂×C₃×C₃⋊C₈	—	3	2	no	—
43	C₂²×C₄×C₉	—	2	1	no	—
44	C₂²×C₄×C₉	C₂²×C₃	1	1	yes	characteristic
45	C₂²×Dic₉	C₃×C₄	1	2	yes	—
46	C₂×C₈×C₉	—	2	4	no	—
47	C₂×C₈×C₉	—	2	2	no	—
48	C₂×C₈×C₉	C₃×C₄	1	4	yes	complemented
49	C₂×C₈×C₉	C₂²×C₃	1	2	yes	—
50	C₄×Dic₉	C₃×C₄	1	2	yes	—
51	C₄×Dic₉	C₂²×C₃	1	2	yes	—
52	C₂×C₉⋊C₈	C₃×C₄	1	4	yes	complemented
53	C₂×C₉⋊C₈	C₂²×C₃	1	2	yes	—
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	C₃×Dic₉	C₂×C₈	1	8	yes	direct product factor
61	C₂²×C₃×C₈	—	2	1	no	—
62	C₂²×C₃×C₈	—	9	1	no	—
63	C₂²×C₃×C₈	D₉	1	1	yes	characteristic
64	C₂²×C₃×C₈	C₃×D₃	1	1	yes	characteristic
65	C₂×C₃×C₄²	—	9	1	no	—
66	C₂×C₄×Dic₃	—	3	1	no	—
67	C₂²×C₃⋊C₈	—	3	1	no	—
68	C₃×C₄×C₈	—	9	4	no	—
69	C₈×Dic₃	—	3	4	no	—
70	C₂³×C₃²	C₄×D₃	1	1	yes	characteristic
71	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
72	C₂×C₃²×C₄	C₄×D₃	1	2	yes	—
73	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
74	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
75	C₂×C₃×Dic₃	—	3	8	no	—
76	C₂×C₃×Dic₃	—	3	4	no	—
77	C₂³×C₉	—	2	1	no	—
78	C₂³×C₉	C₂×C₃×C₄	1	1	yes	characteristic
79	C₃²×C₈	C₂×Dic₃	1	4	yes	—
80	C₃×C₃⋊C₈	—	3	4	no	—
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	C₂×C₃×C₄	1	1	yes	characteristic
86	C₂×C₄×C₉	C₂×C₃×C₄	1	2	yes	—
87	C₂×C₄×C₉	C₂×C₃×C₄	1	2	yes	—
88	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
89	C₂×Dic₉	C₃×C₈	1	8	yes	direct product factor
90	C₂×Dic₉	C₂×C₃×C₄	1	4	yes	—
91	C₈×C₉	—	2	4	no	—
92	C₈×C₉	C₂×C₃×C₄	1	4	yes	complemented
93	C₉⋊C₈	C₂×C₃×C₄	1	4	yes	complemented
94	C₂×C₄×C₈	—	9	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	—	9	2	no	—
100	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
101	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
102	C₂²×Dic₃	—	3	2	no	—
103	C₂×C₃×C₈	—	2	4	no	—
104	C₂×C₃×C₈	—	2	2	no	—
105	C₂×C₃×C₈	—	9	4	no	—
106	C₂×C₃×C₈	—	9	2	no	—
107	C₂×C₃×C₈	Dic₉	1	4	yes	direct product factor
108	C₂×C₃×C₈	C₂×D₉	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	—	9	2	no	—
112	C₃×C₄²	—	9	2	no	—
113	C₄×Dic₃	—	3	2	no	—
114	C₄×Dic₃	—	3	2	no	—
115	C₂×C₃⋊C₈	—	3	4	no	—
116	C₂×C₃⋊C₈	—	3	2	no	—
117	C₂²×C₃²	D₃×C₈	1	4	yes	—
118	C₂²×C₃²	C₄×Dic₃	1	2	yes	—
119	C₂²×C₃²	C₂×C₄×D₃	1	1	yes	characteristic
120	C₃²×C₄	C₄×Dic₃	1	2	yes	—
121	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
122	C₃²×C₄	C₂²×Dic₃	1	1	yes	characteristic
123	C₃×Dic₃	—	3	8	no	—
124	C₂²×C₉	—	2	4	no	—
125	C₂²×C₉	—	2	2	no	—
126	C₂²×C₉	—	2	1	no	—
127	C₂²×C₉	C₃×C₄²	1	2	yes	—
128	C₂²×C₉	C₂×C₃×C₈	1	4	yes	—
129	C₂²×C₉	C₂²×C₃×C₄	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	C₃×C₄²	1	2	yes	—
134	C₄×C₉	C₂²×C₃×C₄	1	1	yes	characteristic
135	C₄×C₉	C₂²×C₃×C₄	1	1	yes	characteristic
136	Dic₉	C₂×C₃×C₈	1	8	yes	direct product factor
137	C₂²×C₈	—	9	1	no	—
138	C₂²×C₈	C₃×D₉	1	1	yes	characteristic
139	C₂×C₄²	—	9	1	no	—
140	C₄×C₈	—	9	4	no	—
141	C₃×C₉	C₂×C₄×C₈	1	1	yes	characteristic, complemented, Sylow
142	C₂³×C₃	—	2	1	no	—
143	C₂³×C₃	C₄×D₉	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	—	9	8	no	—
150	C₂×C₃×C₄	—	9	4	no	—
151	C₂×C₃×C₄	C₄×D₉	1	1	yes	characteristic
152	C₂×C₃×C₄	C₄×D₉	1	2	yes	—
153	C₂×C₃×C₄	C₂×Dic₉	1	2	yes	—
154	C₂×C₃×C₄	C₂²×D₉	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	—	3	8	no	—
160	C₂×Dic₃	—	3	4	no	—
161	C₃×C₈	—	2	4	no	—
162	C₃×C₈	—	9	4	no	—
163	C₃×C₈	C₂×Dic₉	1	4	yes	direct product factor
164	C₃×C₈	C₂×C₃×Dic₃	1	4	yes	—
165	C₃⋊C₈	—	3	4	no	—
166	C₂×C₃²	C₈×Dic₃	1	4	yes	—
167	C₂×C₃²	C₂×D₃×C₈	1	2	yes	—
168	C₂×C₃²	C₂×C₄×Dic₃	1	1	yes	characteristic
169	C₂×C₉	—	2	2	no	—
170	C₂×C₉	—	2	1	no	—
171	C₂×C₉	—	2	4	no	—
172	C₂×C₉	C₃×C₄×C₈	1	4	yes	complemented
173	C₂×C₉	C₂×C₃×C₄²	1	1	yes	characteristic
174	C₂×C₉	C₂²×C₃×C₈	1	2	yes	—
175	C₂²×C₄	—	9	2	no	—
176	C₂²×C₄	C₂×C₃×D₉	1	1	yes	characteristic
177	C₂×C₈	—	9	4	no	—
178	C₂×C₈	—	9	2	no	—
179	C₂×C₈	C₃×Dic₉	1	4	yes	direct product factor
180	C₂×C₈	C₂×C₃×D₉	1	2	yes	—
181	C₄²	—	9	2	no	—
182	C₄²	—	9	2	no	—
183	C₂²×C₃	—	2	1	no	—
184	C₂²×C₃	—	2	4	no	—
185	C₂²×C₃	—	2	2	no	—
186	C₂²×C₃	C₈×D₉	1	4	yes	—
187	C₂²×C₃	C₄×Dic₉	1	2	yes	—
188	C₂²×C₃	C₂×C₄×D₉	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	—	9	8	no	—
196	C₃×C₄	C₄×Dic₉	1	2	yes	—
197	C₃×C₄	C₂×C₄×D₉	1	1	yes	characteristic
198	C₃×C₄	C₂²×Dic₉	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	—	3	8	no	—
203	C₃²	C₂×C₈×Dic₃	1	1	yes	characteristic
204	C₉	—	2	1	no	—
205	C₉	C₂×C₃×C₄×C₈	1	1	yes	characteristic, complemented
206	C₂³	C₃×C₄×D₉	1	1	yes	characteristic
207	C₂×C₄	—	9	4	no	—
208	C₂×C₄	—	9	8	no	—
209	C₂×C₄	C₃×C₄×D₉	1	2	yes	—
210	C₂×C₄	C₃×C₄×D₉	1	1	yes	characteristic
211	C₂×C₄	C₂×C₃×Dic₉	1	2	yes	—
212	C₂×C₄	C₂²×C₃×D₉	1	1	yes	characteristic
213	C₈	—	9	4	no	—
214	C₈	C₂×C₃×Dic₉	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	C₈×Dic₉	1	4	yes	direct product factor
219	C₂×C₃	C₂×C₈×D₉	1	2	yes	—
220	C₂×C₃	C₂×C₄×Dic₉	1	1	yes	characteristic
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	C₃×C₈×D₉	1	4	yes	—
225	C₂²	C₃×C₄×Dic₉	1	2	yes	—
226	C₂²	C₂×C₃×C₄×D₉	1	1	yes	characteristic
227	C₄	—	9	8	no	—
228	C₄	C₃×C₄×Dic₉	1	2	yes	—
229	C₄	C₂×C₃×C₄×D₉	1	1	yes	characteristic
230	C₄	C₂²×C₃×Dic₉	1	1	yes	characteristic
231	C₃	—	2	1	no	—
232	C₃	C₂×C₈×Dic₉	1	1	yes	characteristic, direct product factor
233	C₃	C₂×C₃×C₈×Dic₃	1	1	yes	characteristic
234	C₂	C₃×C₈×Dic₉	1	4	yes	direct product factor
235	C₂	C₂×C₃×C₈×D₉	1	2	yes	—
236	C₂	C₂×C₃×C₄×Dic₉	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²×C₃×C₈	D₉
commutator, hypocenter	C₉	C₂×C₃×C₄×C₈
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₂²×C₃×C₈×C₉	C₂
socle, prime radical	C₂³×C₃²	C₄×D₃
cosocle	C₂×C₄×C₉	C₂³×C₃
LCS	G ▷ C₉
UCS	C₁ ◁ C₂²×C₃×C₈

Complements

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

N	H ≅ G/N	type
4. C₃×C₈×Dic₉	C₂	direct (N × H)
6. C₂×C₈×Dic₉	C₃	direct (N × H)
9. C₂×C₃×C₈×C₉	C₄	semidirect
13. C₂×C₃×C₉⋊C₈	C₄	semidirect
23. C₈×Dic₉	C₂×C₃	direct (N × H)
29. C₂×C₃×Dic₉	C₈	direct (N × H)
31. C₃×C₈×C₉	C₂×C₄	semidirect
32. C₃×C₉⋊C₈	C₂×C₄	semidirect
48. C₂×C₈×C₉	C₃×C₄	semidirect
52. C₂×C₉⋊C₈	C₃×C₄	semidirect
60. C₃×Dic₉	C₂×C₈	direct (N × H)
89. C₂×Dic₉	C₃×C₈	direct (N × H)
92. C₈×C₉	C₂×C₃×C₄	semidirect
93. C₉⋊C₈	C₂×C₃×C₄	semidirect
95. C₂×C₃×C₉	C₄×C₈	semidirect
107. C₂×C₃×C₈	Dic₉	direct (N × H)
136. Dic₉	C₂×C₃×C₈	direct (N × H)
141. C₃×C₉	C₂×C₄×C₈	semidirect
163. C₃×C₈	C₂×Dic₉	direct (N × H)
172. C₂×C₉	C₃×C₄×C₈	semidirect
179. C₂×C₈	C₃×Dic₉	direct (N × H)
205. C₉	C₂×C₃×C₄×C₈	semidirect
214. C₈	C₂×C₃×Dic₉	direct (N × H)
218. C₂×C₃	C₈×Dic₉	direct (N × H)
232. C₃	C₂×C₈×Dic₉	direct (N × H)
234. C₂	C₃×C₈×Dic₉	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×C₃×C₄×Dic₉	2
C₂²×C₃×C₉⋊C₈	2
C₃×C₈×Dic₉	2
C₂×C₃×C₈×Dic₃	3
C₂×C₈×Dic₉	3

Subgroups — C2×C3×C8×Dic9 #13143073

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₈×Dic₉ #13143073