Subgroups — C₉×#4924 #13142375

Order 1728.

486 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄²×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₉×#208	C₂	1	2	yes	maximal
3	C₉×C₄⋊(C₃⋊C₈)	C₂	1	4	yes	maximal, complemented
4	C₃×#4924	C₃	1	1	yes	maximal, characteristic
5	C₉×#423	—	3	1	no	maximal
6	C₂²×C₃×C₄×C₉	C₄	1	2	yes	—
7	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
8	C₃×C₄²×C₉	C₄	1	1	yes	characteristic
9	C₃×C₄²×C₉	C₄	1	1	yes	characteristic
10	C₃×C₄²×C₉	C₂²	1	2	yes	—
11	C₉×#208	—	2	4	no	—
12	C₂×C₉×C₃⋊C₈	C₂²	1	4	yes	—
13	C₂×C₃²×C₄²	C₂×C₃	1	1	yes	characteristic
14	C₂×C₃×#208	C₂×C₃	1	2	yes	—
15	C₃×C₄⋊(C₃⋊C₈)	C₂×C₃	1	4	yes	—
16	C₂×C₉×M₁₆	—	3	2	no	—
17	C₂×C₄²×C₉	—	2	1	no	—
18	C₂×C₄²×C₉	D₃	1	1	yes	characteristic
19	C₉×C₄⋊C₈	—	3	4	no	—
20	C₂³×C₃×C₉	C₂×C₄	1	1	yes	characteristic
21	C₂×C₃×C₄×C₉	—	2	4	no	—
22	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
23	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
25	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
26	C₂×C₃×C₄×C₉	Q₈	1	2	yes	—
27	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
28	C₉×C₃⋊C₈	—	2	4	no	—
29	C₃×#423	—	3	1	no	—
30	#4924	C₉	1	1	yes	characteristic, direct product factor
31	C₂²×C₃²×C₄	C₃×C₄	1	2	yes	—
32	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
33	C₃²×C₄²	C₃×C₄	1	1	yes	characteristic
34	C₃²×C₄²	C₃×C₄	1	1	yes	characteristic
35	C₃²×C₄²	C₂²×C₃	1	2	yes	—
36	C₃×#208	—	2	4	no	—
37	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	4	yes	—
38	C₂²×C₄×C₉	—	2	2	no	—
39	C₂²×C₄×C₉	—	2	1	no	—
40	C₂²×C₄×C₉	Dic₃	1	2	yes	—
41	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
42	C₉×M₁₆	—	6	4	no	—
43	C₂×C₈×C₉	—	3	4	no	—
44	C₄²×C₉	—	2	1	no	—
45	C₄²×C₉	—	2	1	no	—
46	C₄²×C₉	—	2	2	no	—
47	C₄²×C₉	Dic₃	1	1	yes	characteristic
48	C₄²×C₉	Dic₃	1	1	yes	characteristic
49	C₄²×C₉	C₂×D₃	1	2	yes	—
50	C₂²×C₃×C₉	—	2	2	no	—
51	C₂²×C₃×C₉	C₄⋊C₄	1	2	yes	—
52	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
53	C₃×C₄×C₉	—	2	2	no	—
54	C₃×C₄×C₉	C₄⋊C₄	1	2	yes	—
55	C₃×C₄×C₉	M₁₆	1	4	yes	complemented
56	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
57	C₃×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₃×M₁₆	—	3	2	no	—
59	C₂×C₃×C₄²	—	2	1	no	—
60	C₂×C₃×C₄²	C₂×C₉	1	1	yes	characteristic
61	C₂×C₃×C₄²	C₃×D₃	1	1	yes	characteristic
62	C₂×#208	C₂×C₉	1	2	yes	—
63	C₃×C₄⋊C₈	—	3	4	no	—
64	C₄⋊(C₃⋊C₈)	C₂×C₉	1	4	yes	complemented
65	C₂³×C₃²	C₂×C₃×C₄	1	1	yes	characteristic
66	C₂×C₃²×C₄	—	2	4	no	—
67	C₂×C₃²×C₄	C₂×C₃×C₄	1	2	yes	—
68	C₂×C₃²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
69	C₂×C₃²×C₄	C₂×C₃×C₄	1	2	yes	—
70	C₂×C₃²×C₄	C₃×D₄	1	2	yes	—
71	C₂×C₃²×C₄	C₃×Q₈	1	2	yes	—
72	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
73	C₂³×C₉	—	2	1	no	—
74	C₂³×C₉	C₂×Dic₃	1	1	yes	characteristic
75	C₃×C₃⋊C₈	—	2	4	no	—
76	C₂×C₄×C₉	—	2	8	no	—
77	C₂×C₄×C₉	—	2	4	no	—
78	C₂×C₄×C₉	—	2	2	no	—
79	C₂×C₄×C₉	—	2	2	no	—
80	C₂×C₄×C₉	—	2	2	no	—
81	C₂×C₄×C₉	—	2	2	no	—
82	C₂×C₄×C₉	—	2	1	no	—
83	C₂×C₄×C₉	—	2	1	no	—
84	C₂×C₄×C₉	Dic₆	1	2	yes	—
85	C₂×C₄×C₉	D₁₂	1	2	yes	—
86	C₂×C₄×C₉	C₂×Dic₃	1	2	yes	—
87	C₂×C₄×C₉	C₂×Dic₃	1	2	yes	—
88	C₂×C₄×C₉	C₂×Dic₃	1	1	yes	characteristic
89	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
90	C₈×C₉	—	6	4	no	—
91	#423	—	3	1	no	Sylow
92	C₂×C₃×C₉	—	2	2	no	—
93	C₂×C₃×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂×C₃×C₉	C₂×M₁₆	1	2	yes	—
95	C₂²×C₃×C₄	—	2	2	no	—
96	C₂²×C₃×C₄	—	2	1	no	—
97	C₂²×C₃×C₄	C₄×C₉	1	2	yes	—
98	C₂²×C₃×C₄	C₂²×C₉	1	1	yes	characteristic
99	C₂²×C₃×C₄	C₃×Dic₃	1	2	yes	—
100	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
101	C₃×M₁₆	—	6	4	no	—
102	C₂×C₃×C₈	—	3	4	no	—
103	C₃×C₄²	—	2	1	no	—
104	C₃×C₄²	—	2	1	no	—
105	C₃×C₄²	—	2	2	no	—
106	C₃×C₄²	C₄×C₉	1	1	yes	characteristic
107	C₃×C₄²	C₄×C₉	1	1	yes	characteristic
108	C₃×C₄²	C₂²×C₉	1	2	yes	—
109	C₃×C₄²	C₃×Dic₃	1	1	yes	characteristic
110	C₃×C₄²	C₃×Dic₃	1	1	yes	characteristic
111	C₃×C₄²	C₂×C₃×D₃	1	2	yes	—
112	#208	—	2	4	no	—
113	C₂×C₃⋊C₈	C₂²×C₉	1	4	yes	—
114	C₂²×C₃²	—	2	2	no	—
115	C₂²×C₃²	C₃×C₄⋊C₄	1	2	yes	—
116	C₂²×C₃²	C₂²×C₃×C₄	1	1	yes	characteristic
117	C₃²×C₄	—	2	2	no	—
118	C₃²×C₄	C₃×C₄⋊C₄	1	2	yes	—
119	C₃²×C₄	C₃×M₁₆	1	4	yes	—
120	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
121	C₃²×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
122	C₂²×C₉	—	2	4	no	—
123	C₂²×C₉	—	2	2	no	—
124	C₂²×C₉	—	2	2	no	—
125	C₂²×C₉	—	2	1	no	—
126	C₂²×C₉	C₄⋊Dic₃	1	2	yes	—
127	C₂²×C₉	C₂²×Dic₃	1	1	yes	characteristic
128	C₄×C₉	—	2	4	no	—
129	C₄×C₉	—	2	2	no	—
130	C₄×C₉	—	2	4	no	—
131	C₄×C₉	—	2	2	no	—
132	C₄×C₉	—	2	1	no	—
133	C₄×C₉	—	2	1	no	—
134	C₄×C₉	#208	1	4	yes	complemented
135	C₄×C₉	C₄⋊Dic₃	1	2	yes	—
136	C₄×C₉	C₂×Dic₆	1	1	yes	characteristic
137	C₄×C₉	C₂×D₁₂	1	1	yes	characteristic
138	C₂×M₁₆	—	3	2	no	—
139	C₂×C₄²	D₃×C₉	1	1	yes	characteristic
140	C₄⋊C₈	—	3	4	no	—
141	C₃×C₉	#423	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	C₂×C₃×Dic₃	1	1	yes	characteristic
145	C₂×C₃×C₄	—	2	8	no	—
146	C₂×C₃×C₄	—	2	4	no	—
147	C₂×C₃×C₄	—	2	4	no	—
148	C₂×C₃×C₄	—	2	2	no	—
149	C₂×C₃×C₄	—	2	2	no	—
150	C₂×C₃×C₄	—	2	2	no	—
151	C₂×C₃×C₄	—	2	2	no	—
152	C₂×C₃×C₄	—	2	1	no	—
153	C₂×C₃×C₄	—	2	1	no	—
154	C₂×C₃×C₄	C₂×C₄×C₉	1	2	yes	—
155	C₂×C₃×C₄	C₂×C₄×C₉	1	1	yes	characteristic
156	C₂×C₃×C₄	C₂×C₄×C₉	1	2	yes	—
157	C₂×C₃×C₄	D₄×C₉	1	2	yes	—
158	C₂×C₃×C₄	Q₈×C₉	1	2	yes	—
159	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
160	C₂×C₃×C₄	C₃×Dic₆	1	2	yes	—
161	C₂×C₃×C₄	C₃×D₁₂	1	2	yes	—
162	C₂×C₃×C₄	C₂×C₃×Dic₃	1	2	yes	—
163	C₂×C₃×C₄	C₂×C₃×Dic₃	1	2	yes	—
164	C₂×C₃×C₄	C₂×C₃×Dic₃	1	1	yes	characteristic
165	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
166	C₃×C₈	—	6	4	no	—
167	C₃⋊C₈	—	2	4	no	—
168	C₂×C₃²	—	2	2	no	—
169	C₂×C₃²	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
170	C₂×C₃²	C₂×C₃×M₁₆	1	2	yes	—
171	C₂×C₉	—	2	4	no	—
172	C₂×C₉	—	2	2	no	—
173	C₂×C₉	—	2	2	no	—
174	C₂×C₉	—	2	1	no	—
175	C₂×C₉	C₂×#208	1	2	yes	—
176	C₂×C₉	C₂×C₄⋊Dic₃	1	1	yes	characteristic
177	C₂²×C₄	C₉×Dic₃	1	2	yes	—
178	C₂²×C₄	C₂×D₃×C₉	1	1	yes	characteristic
179	M₁₆	—	6	4	no	—
180	C₂×C₈	—	3	4	no	—
181	C₄²	C₉×Dic₃	1	1	yes	characteristic
182	C₄²	C₉×Dic₃	1	1	yes	characteristic
183	C₄²	C₂×D₃×C₉	1	2	yes	—
184	C₂²×C₃	—	2	4	no	—
185	C₂²×C₃	—	2	2	no	—
186	C₂²×C₃	—	2	2	no	—
187	C₂²×C₃	—	2	1	no	—
188	C₂²×C₃	—	2	2	no	—
189	C₂²×C₃	C₉×C₄⋊C₄	1	2	yes	—
190	C₂²×C₃	C₂²×C₄×C₉	1	1	yes	characteristic
191	C₂²×C₃	C₃×C₄⋊Dic₃	1	2	yes	—
192	C₂²×C₃	C₂²×C₃×Dic₃	1	1	yes	characteristic
193	C₃×C₄	—	2	4	no	—
194	C₃×C₄	—	2	2	no	—
195	C₃×C₄	—	2	2	no	—
196	C₃×C₄	—	2	1	no	—
197	C₃×C₄	—	2	1	no	—
198	C₃×C₄	—	2	4	no	—
199	C₃×C₄	—	2	2	no	—
200	C₃×C₄	C₉×C₄⋊C₄	1	2	yes	—
201	C₃×C₄	C₉×M₁₆	1	4	yes	complemented
202	C₃×C₄	C₂×D₄×C₉	1	1	yes	characteristic
203	C₃×C₄	C₂×Q₈×C₉	1	1	yes	characteristic
204	C₃×C₄	C₃×#208	1	4	yes	—
205	C₃×C₄	C₃×C₄⋊Dic₃	1	2	yes	—
206	C₃×C₄	C₂×C₃×Dic₆	1	1	yes	characteristic
207	C₃×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
208	C₃²	C₃×#423	1	1	yes	characteristic
209	C₉	—	2	1	no	—
210	C₉	#4924	1	1	yes	characteristic, direct product factor
211	C₂³	C₂×C₉×Dic₃	1	1	yes	characteristic
212	C₂×C₄	—	2	4	no	—
213	C₂×C₄	C₉×Dic₆	1	2	yes	—
214	C₂×C₄	C₉×D₁₂	1	2	yes	—
215	C₂×C₄	C₂×C₉×Dic₃	1	2	yes	—
216	C₂×C₄	C₂×C₉×Dic₃	1	2	yes	—
217	C₂×C₄	C₂×C₉×Dic₃	1	1	yes	characteristic
218	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
219	C₈	—	6	4	no	—
220	C₂×C₃	—	2	4	no	—
221	C₂×C₃	—	2	2	no	—
222	C₂×C₃	—	2	2	no	—
223	C₂×C₃	—	2	2	no	—
224	C₂×C₃	—	2	1	no	—
225	C₂×C₃	C₂×C₉×C₄⋊C₄	1	1	yes	characteristic
226	C₂×C₃	C₂×C₉×M₁₆	1	2	yes	—
227	C₂×C₃	C₂×C₃×#208	1	2	yes	—
228	C₂×C₃	C₂×C₃×C₄⋊Dic₃	1	1	yes	characteristic
229	C₂²	—	2	2	no	—
230	C₂²	C₉×C₄⋊Dic₃	1	2	yes	—
231	C₂²	C₂²×C₉×Dic₃	1	1	yes	characteristic
232	C₄	—	2	2	no	—
233	C₄	C₉×#208	1	4	yes	complemented
234	C₄	C₉×C₄⋊Dic₃	1	2	yes	—
235	C₄	C₂×C₉×Dic₆	1	1	yes	characteristic
236	C₄	C₂×C₉×D₁₂	1	1	yes	characteristic
237	C₃	—	2	1	no	—
238	C₃	C₉×#423	1	1	yes	characteristic, complemented
239	C₃	C₃×#4924	1	1	yes	characteristic
240	C₂	—	2	2	no	—
241	C₂	C₂×C₉×#208	1	2	yes	—
242	C₂	C₂×C₉×C₄⋊Dic₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₉	C₂²×D₃
commutator	C₂²×C₃	C₂²×C₄×C₉
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₂×C₃×C₄²×C₉	C₂
socle	C₂²×C₃²	C₂²×C₃×C₄
prime radical	C₂³×C₃²	C₂×C₃×C₄
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₂×C₄²×C₉	D₃
hypocenter	C₃	C₉×#423
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₉ ◁ C₂×C₄²×C₉

Complements

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

N	H ≅ G/N	type
3. C₉×C₄⋊(C₃⋊C₈)	C₂	semidirect
30. #4924	C₉	direct (N × H)
55. C₃×C₄×C₉	M₁₆	semidirect
64. C₄⋊(C₃⋊C₈)	C₂×C₉	semidirect
134. C₄×C₉	#208	semidirect
141. C₃×C₉	#423	semidirect
201. C₃×C₄	C₉×M₁₆	semidirect
210. C₉	#4924	direct (N × H)
233. C₄	C₉×#208	semidirect
238. C₃	C₉×#423	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄²×C₉	2
C₂×C₉×#208	2
C₉×C₄⋊(C₃⋊C₈)	2
C₃×#4924	3
C₉×#423	3

Subgroups — C9×#4924 #13142375

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#4924 #13142375