Subgroups — C₃×C₄×C₈×D₉ #13141410

Order 1728.

530 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₈×D₉	C₂	1	2	yes	maximal
2	C₃×C₄²×D₉	C₂	1	1	yes	maximal, characteristic
3	C₃×C₄×C₈×C₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₈×Dic₉	C₂	1	2	yes	maximal, complemented
5	C₃×C₄×C₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×C₄×D₃×C₈	—	3	1	no	maximal
7	C₄×C₈×D₉	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂×C₃×C₄×D₉	C₄	1	2	yes	—
9	C₂×C₃×C₄×D₉	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₈×C₉	C₂²	1	2	yes	—
11	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
12	C₃×C₄×Dic₉	C₄	1	2	yes	—
13	C₃×C₄×Dic₉	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₉⋊C₈	C₂²	1	2	yes	—
15	C₃×C₈×D₉	C₄	1	8	yes	direct product factor
16	C₂×C₃×D₃×C₈	—	3	2	no	—
17	C₃×C₄²×D₃	—	3	1	no	—
18	C₃²×C₄×C₈	D₃	1	1	yes	characteristic
19	C₃×C₈×Dic₃	—	3	2	no	—
20	C₃×C₄×C₃⋊C₈	—	3	1	no	—
21	C₂×C₈×D₉	C₂×C₃	1	2	yes	—
22	C₄²×D₉	C₂×C₃	1	1	yes	characteristic
23	C₄×C₈×C₉	—	2	1	no	—
24	C₄×C₈×C₉	C₂×C₃	1	1	yes	characteristic, complemented
25	C₈×Dic₉	C₂×C₃	1	2	yes	complemented
26	C₄×C₉⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
27	C₂²×C₃×D₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
29	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
30	C₂×C₃×Dic₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₃×Dic₉	C₂×C₄	1	2	yes	—
32	C₃×C₄×D₉	C₈	1	8	yes	direct product factor
33	C₃×C₄×D₉	C₂×C₄	1	4	yes	—
34	C₃×C₈×C₉	C₂×C₄	1	4	yes	complemented
35	C₃×C₉⋊C₈	C₂×C₄	1	4	yes	complemented
36	C₂×C₃×C₄×C₈	—	9	1	no	—
37	C₄×D₃×C₈	—	3	1	no	—
38	C₂×C₃×C₄×D₃	—	3	2	no	—
39	C₂×C₃×C₄×D₃	—	3	1	no	—
40	C₂×C₃²×C₈	C₂×D₃	1	2	yes	—
41	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
42	C₃×C₄×Dic₃	—	3	2	no	—
43	C₃×C₄×Dic₃	—	3	1	no	—
44	C₂×C₃×C₃⋊C₈	—	3	2	no	—
45	C₃×D₃×C₈	—	3	8	no	—
46	C₂×C₄×D₉	C₃×C₄	1	2	yes	—
47	C₂×C₄×D₉	C₂²×C₃	1	1	yes	characteristic
48	C₂×C₈×C₉	—	2	2	no	—
49	C₂×C₈×C₉	C₂²×C₃	1	2	yes	—
50	C₄²×C₉	—	2	1	no	—
51	C₄²×C₉	C₂²×C₃	1	1	yes	characteristic
52	C₄×Dic₉	C₃×C₄	1	2	yes	—
53	C₄×Dic₉	C₂²×C₃	1	1	yes	characteristic
54	C₂×C₉⋊C₈	C₂²×C₃	1	2	yes	—
55	C₈×D₉	C₃×C₄	1	8	yes	direct product factor
56	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₃×D₉	C₄²	1	2	yes	—
58	C₂×C₃×D₉	C₂×C₈	1	4	yes	—
59	C₃×C₄×C₉	C₂×C₈	1	4	yes	complemented
60	C₃×C₄×C₉	C₂²×C₄	1	2	yes	—
61	C₃×Dic₉	C₄²	1	2	yes	—
62	C₃×Dic₉	C₂×C₈	1	4	yes	complemented
63	C₂²×C₃×C₈	—	9	2	no	—
64	C₂×C₃×C₄²	—	9	1	no	—
65	C₂×D₃×C₈	—	3	2	no	—
66	C₄²×D₃	—	3	1	no	—
67	C₃×C₄×C₈	—	2	1	no	—
68	C₃×C₄×C₈	—	9	2	no	—
69	C₃×C₄×C₈	—	9	1	no	—
70	C₃×C₄×C₈	D₉	1	1	yes	characteristic, direct product factor
71	C₃×C₄×C₈	C₃×D₃	1	1	yes	characteristic
72	C₈×Dic₃	—	3	2	no	—
73	C₄×C₃⋊C₈	—	3	1	no	—
74	C₂²×C₃×D₃	—	3	1	no	—
75	C₂×C₃²×C₄	C₄×D₃	1	2	yes	—
76	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
77	C₂×C₃×Dic₃	—	3	1	no	—
78	C₂×C₃×Dic₃	—	3	2	no	—
79	C₃×C₄×D₃	—	3	8	no	—
80	C₃×C₄×D₃	—	3	4	no	—
81	C₂²×D₉	C₂×C₃×C₄	1	1	yes	characteristic
82	C₃²×C₈	C₄×D₃	1	4	yes	—
83	C₃×C₃⋊C₈	—	3	4	no	—
84	C₂×C₄×C₉	—	2	2	no	—
85	C₂×C₄×C₉	—	2	1	no	—
86	C₂×C₄×C₉	C₂×C₃×C₄	1	2	yes	—
87	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
88	C₂×Dic₉	C₂×C₃×C₄	1	1	yes	characteristic
89	C₂×Dic₉	C₂×C₃×C₄	1	2	yes	—
90	C₄×D₉	C₃×C₈	1	8	yes	direct product factor
91	C₄×D₉	C₂×C₃×C₄	1	4	yes	—
92	C₈×C₉	—	2	4	no	—
93	C₈×C₉	C₂×C₃×C₄	1	4	yes	complemented
94	C₉⋊C₈	C₂×C₃×C₄	1	4	yes	complemented
95	C₂×C₄×C₈	—	9	1	no	Sylow
96	C₂×C₃×C₉	C₂×C₄²	1	1	yes	characteristic
97	C₂×C₃×C₉	C₂²×C₈	1	2	yes	—
98	C₃×D₉	C₄×C₈	1	4	yes	direct product factor
99	C₂²×C₃×C₄	—	9	2	no	—
100	C₂²×C₃×C₄	—	9	1	no	—
101	C₂×C₄×D₃	—	3	2	no	—
102	C₂×C₄×D₃	—	3	1	no	—
103	C₂×C₃×C₈	—	2	2	no	—
104	C₂×C₃×C₈	—	9	2	no	—
105	C₂×C₃×C₈	—	9	8	no	—
106	C₂×C₃×C₈	C₂×D₉	1	2	yes	—
107	C₂×C₃×C₈	C₂×C₃×D₃	1	2	yes	—
108	C₃×C₄²	—	2	1	no	—
109	C₃×C₄²	—	9	2	no	—
110	C₃×C₄²	—	9	1	no	—
111	C₃×C₄²	C₂×D₉	1	1	yes	characteristic
112	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
113	C₄×Dic₃	—	3	2	no	—
114	C₄×Dic₃	—	3	1	no	—
115	C₂×C₃⋊C₈	—	3	2	no	—
116	D₃×C₈	—	3	8	no	—
117	C₂²×C₃²	C₂×C₄×D₃	1	1	yes	characteristic
118	C₂×C₃×D₃	—	3	2	no	—
119	C₂×C₃×D₃	—	3	4	no	—
120	C₃²×C₄	D₃×C₈	1	4	yes	—
121	C₃²×C₄	C₂×C₄×D₃	1	2	yes	—
122	C₃×Dic₃	—	3	2	no	—
123	C₃×Dic₃	—	3	4	no	—
124	C₂²×C₉	—	2	1	no	—
125	C₂²×C₉	C₂²×C₃×C₄	1	1	yes	characteristic
126	C₂×D₉	C₃×C₄²	1	2	yes	—
127	C₂×D₉	C₂×C₃×C₈	1	4	yes	—
128	C₄×C₉	—	2	4	no	—
129	C₄×C₉	—	2	2	no	—
130	C₄×C₉	C₂×C₃×C₈	1	4	yes	complemented
131	C₄×C₉	C₂²×C₃×C₄	1	2	yes	—
132	Dic₉	C₃×C₄²	1	2	yes	—
133	Dic₉	C₂×C₃×C₈	1	4	yes	complemented
134	C₂²×C₈	—	9	2	no	—
135	C₂×C₄²	—	9	1	no	—
136	C₄×C₈	—	9	2	no	—
137	C₄×C₈	—	9	1	no	—
138	C₄×C₈	C₃×D₉	1	1	yes	characteristic, direct product factor
139	C₃×C₉	C₂×C₄×C₈	1	1	yes	characteristic, complemented, Sylow
140	C₂³×C₃	—	9	1	no	—
141	C₂²×D₃	—	3	1	no	—
142	C₂×C₃×C₄	—	2	2	no	—
143	C₂×C₃×C₄	—	2	1	no	—
144	C₂×C₃×C₄	—	9	1	no	—
145	C₂×C₃×C₄	—	9	2	no	—
146	C₂×C₃×C₄	—	9	8	no	—
147	C₂×C₃×C₄	—	9	4	no	—
148	C₂×C₃×C₄	C₄×D₉	1	2	yes	—
149	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
150	C₂×C₃×C₄	C₃×C₄×D₃	1	2	yes	—
151	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
152	C₂×Dic₃	—	3	2	no	—
153	C₂×Dic₃	—	3	1	no	—
154	C₄×D₃	—	3	8	no	—
155	C₄×D₃	—	3	4	no	—
156	C₃×C₈	—	2	4	no	—
157	C₃×C₈	—	9	4	no	—
158	C₃×C₈	C₄×D₉	1	4	yes	direct product factor
159	C₃×C₈	C₃×C₄×D₃	1	4	yes	—
160	C₃⋊C₈	—	3	4	no	—
161	C₂×C₃²	C₄²×D₃	1	1	yes	characteristic
162	C₂×C₃²	C₂×D₃×C₈	1	2	yes	—
163	C₃×D₃	—	3	4	no	—
164	C₂×C₉	—	2	2	no	—
165	C₂×C₉	—	2	1	no	—
166	C₂×C₉	C₂×C₃×C₄²	1	1	yes	characteristic
167	C₂×C₉	C₂²×C₃×C₈	1	2	yes	—
168	D₉	C₃×C₄×C₈	1	4	yes	direct product factor
169	C₂²×C₄	—	9	2	no	—
170	C₂²×C₄	—	9	1	no	—
171	C₂×C₈	—	9	2	no	—
172	C₂×C₈	—	9	8	no	—
173	C₂×C₈	C₂×C₃×D₉	1	2	yes	—
174	C₄²	—	9	2	no	—
175	C₄²	—	9	1	no	—
176	C₄²	C₂×C₃×D₉	1	1	yes	characteristic
177	C₂²×C₃	—	2	1	no	—
178	C₂²×C₃	—	9	2	no	—
179	C₂²×C₃	—	9	4	no	—
180	C₂²×C₃	C₂×C₄×D₉	1	1	yes	characteristic
181	C₂²×C₃	C₂×C₃×C₄×D₃	1	1	yes	characteristic
182	C₂×D₃	—	3	4	no	—
183	C₂×D₃	—	3	2	no	—
184	C₃×C₄	—	2	2	no	—
185	C₃×C₄	—	2	4	no	—
186	C₃×C₄	—	9	2	no	—
187	C₃×C₄	—	9	4	no	—
188	C₃×C₄	C₈×D₉	1	4	yes	direct product factor
189	C₃×C₄	C₂×C₄×D₉	1	2	yes	—
190	C₃×C₄	C₃×D₃×C₈	1	4	yes	—
191	C₃×C₄	C₂×C₃×C₄×D₃	1	2	yes	—
192	Dic₃	—	3	4	no	—
193	Dic₃	—	3	2	no	—
194	C₃²	C₄×D₃×C₈	1	1	yes	characteristic
195	C₉	—	2	1	no	—
196	C₉	C₂×C₃×C₄×C₈	1	1	yes	characteristic, complemented
197	C₂³	—	9	1	no	—
198	C₂×C₄	—	9	2	no	—
199	C₂×C₄	—	9	1	no	—
200	C₂×C₄	—	9	4	no	—
201	C₂×C₄	—	9	8	no	—
202	C₂×C₄	C₃×C₄×D₉	1	2	yes	—
203	C₂×C₄	C₂²×C₃×D₉	1	1	yes	characteristic
204	C₈	—	9	4	no	—
205	C₈	C₃×C₄×D₉	1	4	yes	direct product factor
206	C₂×C₃	—	2	2	no	—
207	C₂×C₃	—	2	1	no	—
208	C₂×C₃	—	9	4	no	—
209	C₂×C₃	C₄²×D₉	1	1	yes	characteristic
210	C₂×C₃	C₂×C₈×D₉	1	2	yes	—
211	C₂×C₃	C₃×C₄²×D₃	1	1	yes	characteristic
212	C₂×C₃	C₂×C₃×D₃×C₈	1	2	yes	—
213	D₃	—	3	4	no	—
214	C₂²	—	9	4	no	—
215	C₂²	—	9	2	no	—
216	C₂²	C₂×C₃×C₄×D₉	1	1	yes	characteristic
217	C₄	—	9	4	no	—
218	C₄	—	9	2	no	—
219	C₄	C₃×C₈×D₉	1	4	yes	direct product factor
220	C₄	C₂×C₃×C₄×D₉	1	2	yes	—
221	C₃	—	2	1	no	—
222	C₃	C₄×C₈×D₉	1	1	yes	characteristic, direct product factor
223	C₃	C₃×C₄×D₃×C₈	1	1	yes	characteristic
224	C₂	—	9	4	no	—
225	C₂	C₃×C₄²×D₉	1	1	yes	characteristic
226	C₂	C₂×C₃×C₈×D₉	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₃×C₄×C₈×C₉	C₂	semidirect
4. C₃×C₈×Dic₉	C₂	semidirect
5. C₃×C₄×C₉⋊C₈	C₂	semidirect
7. C₄×C₈×D₉	C₃	direct (N × H)
15. C₃×C₈×D₉	C₄	direct (N × H)
24. C₄×C₈×C₉	C₂×C₃	semidirect
25. C₈×Dic₉	C₂×C₃	semidirect
26. C₄×C₉⋊C₈	C₂×C₃	semidirect
32. C₃×C₄×D₉	C₈	direct (N × H)
34. C₃×C₈×C₉	C₂×C₄	semidirect
35. C₃×C₉⋊C₈	C₂×C₄	semidirect
55. C₈×D₉	C₃×C₄	direct (N × H)
59. C₃×C₄×C₉	C₂×C₈	semidirect
62. C₃×Dic₉	C₂×C₈	semidirect
70. C₃×C₄×C₈	D₉	direct (N × H)
90. C₄×D₉	C₃×C₈	direct (N × H)
93. C₈×C₉	C₂×C₃×C₄	semidirect
94. C₉⋊C₈	C₂×C₃×C₄	semidirect
98. C₃×D₉	C₄×C₈	direct (N × H)
130. C₄×C₉	C₂×C₃×C₈	semidirect
133. Dic₉	C₂×C₃×C₈	semidirect
138. C₄×C₈	C₃×D₉	direct (N × H)
139. C₃×C₉	C₂×C₄×C₈	semidirect
158. C₃×C₈	C₄×D₉	direct (N × H)
168. D₉	C₃×C₄×C₈	direct (N × H)
188. C₃×C₄	C₈×D₉	direct (N × H)
196. C₉	C₂×C₃×C₄×C₈	semidirect
205. C₈	C₃×C₄×D₉	direct (N × H)
219. C₄	C₃×C₈×D₉	direct (N × H)
222. C₃	C₄×C₈×D₉	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×C₈×D₉	2
C₃×C₄²×D₉	2
C₃×C₄×C₈×C₉	2
C₃×C₈×Dic₉	2
C₃×C₄×C₉⋊C₈	2
C₃×C₄×D₃×C₈	3
C₄×C₈×D₉	3

Subgroups — C3×C4×C8×D9 #13141410

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₄×C₈×D₉ #13141410