Subgroups — C₄×D₃×C₈×C₉ #13141425

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

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

Complements

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

N	H ≅ G/N	type
3. C₃×C₄×C₈×C₉	C₂	semidirect
4. C₈×C₉×Dic₃	C₂	semidirect
5. C₄×C₉×C₃⋊C₈	C₂	semidirect
15. D₃×C₈×C₉	C₄	direct (N × H)
26. C₄×C₈×C₉	D₃	direct (N × H)
32. C₄×D₃×C₉	C₈	direct (N × H)
34. C₃×C₈×C₉	C₂×C₄	semidirect
35. C₉×C₃⋊C₈	C₂×C₄	semidirect
37. C₄×D₃×C₈	C₉	direct (N × H)
59. C₃×C₄×C₉	C₂×C₈	semidirect
62. C₉×Dic₃	C₂×C₈	semidirect
70. C₃×C₄×C₈	C₂×C₉	semidirect
72. C₈×Dic₃	C₂×C₉	semidirect
73. C₄×C₃⋊C₈	C₂×C₉	semidirect
94. C₈×C₉	C₄×D₃	direct (N × H)
98. D₃×C₉	C₄×C₈	direct (N × H)
116. D₃×C₈	C₄×C₉	direct (N × H)
132. C₄×C₉	D₃×C₈	direct (N × H)
138. C₄×C₈	D₃×C₉	direct (N × H)
139. C₃×C₉	C₂×C₄×C₈	semidirect
154. C₄×D₃	C₈×C₉	direct (N × H)
158. C₃×C₈	C₂×C₄×C₉	semidirect
160. C₃⋊C₈	C₂×C₄×C₉	semidirect
188. C₃×C₄	C₂×C₈×C₉	semidirect
193. Dic₃	C₂×C₈×C₉	semidirect
196. C₉	C₄×D₃×C₈	direct (N × H)
205. C₈	C₄×D₃×C₉	direct (N × H)
213. D₃	C₄×C₈×C₉	direct (N × H)
219. C₄	D₃×C₈×C₉	direct (N × H)
222. C₃	C₂×C₄×C₈×C₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₃×C₈×C₉	2
C₄²×D₃×C₉	2
C₃×C₄×C₈×C₉	2
C₈×C₉×Dic₃	2
C₄×C₉×C₃⋊C₈	2
C₃×C₄×D₃×C₈	3
C₂×C₄×C₈×C₉	3

Subgroups — C4×D3×C8×C9 #13141425

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×D₃×C₈×C₉ #13141425