Subgroups — C₉×#4551 #13135650

Order 1728.

418 conjugacy classes of subgroups.

The subgroup lattice has 246 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×C₉×C₄⋊Dic₃	C₂	1	2	yes	maximal
3	C₃×#4551	C₃	1	1	yes	maximal, characteristic
4	C₉×#340	—	3	1	no	maximal
5	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
6	C₂²×C₉×Dic₃	—	2	2	no	—
7	C₂×C₃×C₈×C₉	—	2	2	no	—
8	C₂×C₃×C₈×C₉	C₄	1	2	yes	complemented
9	C₉×C₄⋊Dic₃	—	2	2	no	—
10	C₉×C₄⋊Dic₃	C₄	1	4	yes	complemented
11	C₂²×C₃²×C₈	C₂×C₃	1	1	yes	characteristic
12	C₂×C₃×C₄⋊Dic₃	C₂×C₃	1	2	yes	—
13	C₂²×C₈×C₉	—	2	1	no	—
14	C₂²×C₈×C₉	D₃	1	1	yes	characteristic
15	C₂×C₉×C₄⋊C₄	—	3	2	no	—
16	C₂³×C₃×C₉	D₄	1	1	yes	characteristic
17	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
18	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
19	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
20	C₂×C₃×C₄×C₉	Q₈	1	1	yes	characteristic
21	C₂×C₉×Dic₃	—	2	4	no	—
22	C₂×C₉×Dic₃	—	4	2	no	—
23	C₂×C₉×Dic₃	—	4	2	no	—
24	C₃×C₈×C₉	—	2	2	no	—
25	C₃×#340	—	3	1	no	—
26	#4551	C₉	1	1	yes	characteristic, direct product factor
27	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
28	C₂²×C₃×Dic₃	—	2	2	no	—
29	C₂×C₃²×C₈	—	2	2	no	—
30	C₂×C₃²×C₈	C₃×C₄	1	2	yes	—
31	C₃×C₄⋊Dic₃	—	2	2	no	—
32	C₃×C₄⋊Dic₃	C₃×C₄	1	4	yes	—
33	C₂²×C₄×C₉	—	2	1	no	—
34	C₂²×C₄×C₉	—	6	2	no	—
35	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
36	C₂×C₈×C₉	—	2	4	no	—
37	C₂×C₈×C₉	—	2	2	no	—
38	C₂×C₈×C₉	—	2	2	no	—
39	C₂×C₈×C₉	Dic₃	1	2	yes	complemented
40	C₉×C₄⋊C₄	—	3	4	no	—
41	C₉×C₄⋊C₄	—	6	2	no	—
42	C₂²×C₃×C₉	C₂²⋊C₄	1	2	yes	—
43	C₂²×C₃×C₉	C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₃×C₉	D₈	1	1	yes	characteristic
45	C₂²×C₃×C₉	SD₁₆	1	2	yes	—
46	C₂²×C₃×C₉	Q₁₆	1	1	yes	characteristic
47	C₃×C₄×C₉	C₄²	1	1	yes	characteristic
48	C₃×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
49	C₃×C₄×C₉	C₄⋊C₄	1	2	yes	—
50	C₉×Dic₃	—	4	4	no	—
51	C₂²×C₃×C₈	—	2	1	no	—
52	C₂²×C₃×C₈	C₂×C₉	1	1	yes	characteristic
53	C₂²×C₃×C₈	C₃×D₃	1	1	yes	characteristic
54	C₂×C₃×C₄⋊C₄	—	3	2	no	—
55	C₂×C₄⋊Dic₃	C₂×C₉	1	2	yes	—
56	C₂³×C₃²	C₃×D₄	1	1	yes	characteristic
57	C₂×C₃²×C₄	C₂×C₃×C₄	1	2	yes	—
58	C₂×C₃²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
59	C₂×C₃²×C₄	C₃×D₄	1	2	yes	—
60	C₂×C₃²×C₄	C₃×Q₈	1	1	yes	characteristic
61	C₂×C₃×Dic₃	—	2	4	no	—
62	C₂×C₃×Dic₃	—	4	2	no	—
63	C₂×C₃×Dic₃	—	4	2	no	—
64	C₂³×C₉	—	2	1	no	—
65	C₂³×C₉	D₁₂	1	1	yes	characteristic
66	C₃²×C₈	—	2	2	no	—
67	C₂×C₄×C₉	—	2	1	no	—
68	C₂×C₄×C₉	—	2	2	no	—
69	C₂×C₄×C₉	—	2	1	no	—
70	C₂×C₄×C₉	—	2	2	no	—
71	C₂×C₄×C₉	—	6	4	no	—
72	C₂×C₄×C₉	—	12	2	no	—
73	C₂×C₄×C₉	—	12	2	no	—
74	C₂×C₄×C₉	Dic₆	1	1	yes	characteristic
75	C₂×C₄×C₉	C₄×D₃	1	2	yes	—
76	C₂×C₄×C₉	C₂×Dic₃	1	1	yes	characteristic
77	C₂×C₄×C₉	#67	1	2	yes	—
78	C₈×C₉	—	2	4	no	—
79	C₈×C₉	—	2	2	no	—
80	#340	—	3	1	no	Sylow
81	C₂×C₃×C₉	#95	1	1	yes	characteristic
82	C₂×C₃×C₉	#102	1	2	yes	—
83	C₂×C₃×C₉	#103	1	2	yes	—
84	C₂×C₃×C₉	#106	1	1	yes	characteristic
85	C₂×C₃×C₉	#107	1	1	yes	characteristic
86	C₂²×C₃×C₄	—	2	1	no	—
87	C₂²×C₃×C₄	—	6	2	no	—
88	C₂²×C₃×C₄	C₂²×C₉	1	1	yes	characteristic
89	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
90	C₂²×Dic₃	—	2	2	no	—
91	C₂×C₃×C₈	—	2	4	no	—
92	C₂×C₃×C₈	—	2	2	no	—
93	C₂×C₃×C₈	—	2	2	no	—
94	C₂×C₃×C₈	—	2	2	no	—
95	C₂×C₃×C₈	C₄×C₉	1	2	yes	complemented
96	C₂×C₃×C₈	C₃×Dic₃	1	2	yes	—
97	C₃×C₄⋊C₄	—	3	4	no	—
98	C₃×C₄⋊C₄	—	6	2	no	—
99	C₄⋊Dic₃	—	2	2	no	—
100	C₄⋊Dic₃	C₄×C₉	1	4	yes	complemented
101	C₂²×C₃²	C₃×C₂²⋊C₄	1	2	yes	—
102	C₂²×C₃²	C₃×C₄⋊C₄	1	1	yes	characteristic
103	C₂²×C₃²	C₃×D₈	1	1	yes	characteristic
104	C₂²×C₃²	C₃×SD₁₆	1	2	yes	—
105	C₂²×C₃²	C₃×Q₁₆	1	1	yes	characteristic
106	C₃²×C₄	C₃×C₄²	1	1	yes	characteristic
107	C₃²×C₄	C₃×C₂²⋊C₄	1	1	yes	characteristic
108	C₃²×C₄	C₃×C₄⋊C₄	1	2	yes	—
109	C₃×Dic₃	—	4	4	no	—
110	C₂²×C₉	—	2	2	no	—
111	C₂²×C₉	—	2	1	no	—
112	C₂²×C₉	—	2	1	no	—
113	C₂²×C₉	—	2	1	no	—
114	C₂²×C₉	—	2	2	no	—
115	C₂²×C₉	#204	1	2	yes	—
116	C₂²×C₉	D₂₄	1	1	yes	characteristic
117	C₂²×C₉	Dic₁₂	1	1	yes	characteristic
118	C₂²×C₉	C₄⋊Dic₃	1	1	yes	characteristic
119	C₂²×C₉	(C₂×D₃)⋊C₄	1	2	yes	—
120	C₄×C₉	—	2	1	no	—
121	C₄×C₉	—	2	2	no	—
122	C₄×C₉	—	2	1	no	—
123	C₄×C₉	—	12	4	no	—
124	C₄×C₉	C₄×Dic₃	1	1	yes	characteristic
125	C₄×C₉	Dic₃⋊C₄	1	2	yes	—
126	C₄×C₉	#217	1	1	yes	characteristic
127	C₂²×C₈	D₃×C₉	1	1	yes	characteristic
128	C₂×C₄⋊C₄	—	3	2	no	—
129	C₃×C₉	#340	1	1	yes	characteristic, complemented, Sylow
130	C₂³×C₃	—	2	1	no	—
131	C₂³×C₃	D₄×C₉	1	1	yes	characteristic
132	C₂³×C₃	C₃×D₁₂	1	1	yes	characteristic
133	C₂×C₃×C₄	—	2	1	no	—
134	C₂×C₃×C₄	—	2	2	no	—
135	C₂×C₃×C₄	—	2	1	no	—
136	C₂×C₃×C₄	—	2	2	no	—
137	C₂×C₃×C₄	—	6	4	no	—
138	C₂×C₃×C₄	—	12	2	no	—
139	C₂×C₃×C₄	—	12	2	no	—
140	C₂×C₃×C₄	C₂×C₄×C₉	1	2	yes	—
141	C₂×C₃×C₄	C₂×C₄×C₉	1	1	yes	characteristic
142	C₂×C₃×C₄	D₄×C₉	1	2	yes	—
143	C₂×C₃×C₄	Q₈×C₉	1	1	yes	characteristic
144	C₂×C₃×C₄	C₃×Dic₆	1	1	yes	characteristic
145	C₂×C₃×C₄	C₃×C₄×D₃	1	2	yes	—
146	C₂×C₃×C₄	C₂×C₃×Dic₃	1	1	yes	characteristic
147	C₂×C₃×C₄	C₃×#67	1	2	yes	—
148	C₂×Dic₃	—	2	4	no	—
149	C₂×Dic₃	—	4	2	no	—
150	C₂×Dic₃	—	4	2	no	—
151	C₃×C₈	—	2	4	no	—
152	C₃×C₈	—	2	2	no	—
153	C₃×C₈	—	2	2	no	—
154	C₂×C₃²	C₃×#95	1	1	yes	characteristic
155	C₂×C₃²	C₃×#102	1	2	yes	—
156	C₂×C₃²	C₃×#103	1	2	yes	—
157	C₂×C₃²	C₃×#106	1	1	yes	characteristic
158	C₂×C₃²	C₃×#107	1	1	yes	characteristic
159	C₂×C₉	—	2	2	no	—
160	C₂×C₉	—	2	1	no	—
161	C₂×C₉	—	2	1	no	—
162	C₂×C₉	—	2	2	no	—
163	C₂×C₉	—	2	1	no	—
164	C₂×C₉	#816	1	2	yes	—
165	C₂×C₉	#817	1	1	yes	characteristic
166	C₂×C₉	#818	1	1	yes	characteristic
167	C₂×C₉	#821	1	2	yes	—
168	C₂×C₉	C₃⋊#95	1	1	yes	characteristic
169	C₂²×C₄	—	6	2	no	—
170	C₂²×C₄	C₂×D₃×C₉	1	1	yes	characteristic
171	C₂×C₈	—	2	2	no	—
172	C₂×C₈	C₉×Dic₃	1	2	yes	complemented
173	C₄⋊C₄	—	3	4	no	—
174	C₄⋊C₄	—	6	2	no	—
175	C₂²×C₃	—	2	2	no	—
176	C₂²×C₃	—	2	1	no	—
177	C₂²×C₃	—	2	1	no	—
178	C₂²×C₃	—	2	1	no	—
179	C₂²×C₃	—	2	2	no	—
180	C₂²×C₃	C₉×C₂²⋊C₄	1	2	yes	—
181	C₂²×C₃	C₉×C₄⋊C₄	1	1	yes	characteristic
182	C₂²×C₃	C₉×D₈	1	1	yes	characteristic
183	C₂²×C₃	C₉×SD₁₆	1	2	yes	—
184	C₂²×C₃	C₉×Q₁₆	1	1	yes	characteristic
185	C₂²×C₃	C₃×#204	1	2	yes	—
186	C₂²×C₃	C₃×D₂₄	1	1	yes	characteristic
187	C₂²×C₃	C₃×Dic₁₂	1	1	yes	characteristic
188	C₂²×C₃	C₃×C₄⋊Dic₃	1	1	yes	characteristic
189	C₂²×C₃	C₃×(C₂×D₃)⋊C₄	1	2	yes	—
190	C₃×C₄	—	2	1	no	—
191	C₃×C₄	—	2	2	no	—
192	C₃×C₄	—	2	1	no	—
193	C₃×C₄	—	12	4	no	—
194	C₃×C₄	C₄²×C₉	1	1	yes	characteristic
195	C₃×C₄	C₉×C₂²⋊C₄	1	1	yes	characteristic
196	C₃×C₄	C₉×C₄⋊C₄	1	2	yes	—
197	C₃×C₄	C₃×C₄×Dic₃	1	1	yes	characteristic
198	C₃×C₄	C₃×Dic₃⋊C₄	1	2	yes	—
199	C₃×C₄	C₃×#217	1	1	yes	characteristic
200	Dic₃	—	4	4	no	—
201	C₃²	C₃×#340	1	1	yes	characteristic
202	C₉	—	2	1	no	—
203	C₉	#4551	1	1	yes	characteristic, direct product factor
204	C₂³	C₉×D₁₂	1	1	yes	characteristic
205	C₂×C₄	—	6	4	no	—
206	C₂×C₄	—	12	2	no	—
207	C₂×C₄	—	12	2	no	—
208	C₂×C₄	C₉×Dic₆	1	1	yes	characteristic
209	C₂×C₄	C₄×D₃×C₉	1	2	yes	—
210	C₂×C₄	C₂×C₉×Dic₃	1	1	yes	characteristic
211	C₂×C₄	C₉×#67	1	2	yes	—
212	C₈	—	2	2	no	—
213	C₂×C₃	—	2	1	no	—
214	C₂×C₃	—	2	2	no	—
215	C₂×C₃	—	2	1	no	—
216	C₂×C₃	—	2	1	no	—
217	C₂×C₃	—	2	2	no	—
218	C₂×C₃	C₉×#95	1	1	yes	characteristic
219	C₂×C₃	C₉×#102	1	2	yes	—
220	C₂×C₃	C₉×#103	1	2	yes	—
221	C₂×C₃	C₉×#106	1	1	yes	characteristic
222	C₂×C₃	C₉×#107	1	1	yes	characteristic
223	C₂×C₃	C₃×#816	1	2	yes	—
224	C₂×C₃	C₃×#817	1	1	yes	characteristic
225	C₂×C₃	C₃×#818	1	1	yes	characteristic
226	C₂×C₃	C₃×#821	1	2	yes	—
227	C₂×C₃	C₃×C₃⋊#95	1	1	yes	characteristic
228	C₂²	C₉×#204	1	2	yes	—
229	C₂²	C₉×D₂₄	1	1	yes	characteristic
230	C₂²	C₉×Dic₁₂	1	1	yes	characteristic
231	C₂²	C₉×C₄⋊Dic₃	1	1	yes	characteristic
232	C₂²	C₉×(C₂×D₃)⋊C₄	1	2	yes	—
233	C₄	—	12	4	no	—
234	C₄	C₄×C₉×Dic₃	1	1	yes	characteristic
235	C₄	C₉×Dic₃⋊C₄	1	2	yes	—
236	C₄	C₉×#217	1	1	yes	characteristic
237	C₃	—	2	1	no	—
238	C₃	C₉×#340	1	1	yes	characteristic, complemented
239	C₃	C₃×#4551	1	1	yes	characteristic
240	C₂	C₉×#816	1	2	yes	—
241	C₂	C₉×#817	1	1	yes	characteristic
242	C₂	C₉×#818	1	1	yes	characteristic
243	C₂	C₉×#821	1	2	yes	—
244	C₂	C₉×C₃⋊#95	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₉	D₁₂
commutator	C₃×C₄	C₄²×C₉
Frattini	C₂²×C₃×C₄	C₂×C₃×D₃
Fitting	C₂²×C₃×C₈×C₉	C₂
socle, prime radical	C₂³×C₃²	C₃×D₄
cosocle	C₂²×C₃²×C₄	C₂²×C₃
hypercenter	C₂²×C₈×C₉	D₃
hypocenter	C₃	C₉×#340
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₉ ◁ C₂²×C₄×C₉ ◁ C₂²×C₈×C₉

Complements

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

N	H ≅ G/N	type
8. C₂×C₃×C₈×C₉	C₄	semidirect
10. C₉×C₄⋊Dic₃	C₄	semidirect
26. #4551	C₉	direct (N × H)
39. C₂×C₈×C₉	Dic₃	semidirect
95. C₂×C₃×C₈	C₄×C₉	semidirect
100. C₄⋊Dic₃	C₄×C₉	semidirect
129. C₃×C₉	#340	semidirect
172. C₂×C₈	C₉×Dic₃	semidirect
203. C₉	#4551	direct (N × H)
238. C₃	C₉×#340	semidirect

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×C₉×C₄⋊Dic₃	2
C₃×#4551	3
C₉×#340	3

Subgroups — C9×#4551 #13135650

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#4551 #13135650