Subgroups — C₃×C₉×#489 #13145225

Order 1728.

500 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₉×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×C₉×#124	C₂	1	1	yes	maximal, characteristic
3	C₃×C₉×#103	C₂	1	2	yes	maximal, complemented
4	C₃×C₉×#102	C₂	1	2	yes	maximal
5	C₃×C₉×#97	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃²×#489	C₃	1	1	yes	maximal, characteristic
7	C₉×#489	C₃	1	3	yes	maximal, direct product factor
8	C₂×C₃×Q₈×C₉	—	2	1	no	—
9	C₂×C₃×Q₈×C₉	C₂²	1	1	yes	characteristic
10	C₂×C₃×D₄×C₉	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₈×C₉	C₂²	1	2	yes	—
12	C₃×C₉×C₄⋊C₄	—	2	1	no	—
13	C₃×C₉×C₄⋊C₄	C₂²	1	2	yes	—
14	C₃×C₉×C₂²⋊C₄	—	2	1	no	—
15	C₃×C₉×C₂²⋊C₄	—	2	1	no	—
16	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
17	C₃²×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
18	C₃²×#124	C₂×C₃	1	1	yes	characteristic
19	C₃²×#103	C₂×C₃	1	2	yes	—
20	C₃²×#102	C₂×C₃	1	2	yes	—
21	C₃²×#97	C₂×C₃	1	1	yes	characteristic
22	C₉×C₄⋊Q₈	C₂×C₃	1	3	yes	complemented
23	C₉×#124	C₂×C₃	1	3	yes	—
24	C₉×#103	C₂×C₃	1	6	yes	complemented
25	C₉×#102	C₂×C₃	1	6	yes	—
26	C₉×#97	C₂×C₃	1	3	yes	complemented
27	C₂³×C₃×C₉	—	2	1	no	—
28	C₃×Q₈×C₉	—	2	2	no	—
29	C₃×Q₈×C₉	—	4	2	no	—
30	C₃×D₄×C₉	—	2	2	no	—
31	C₂×C₃×C₄×C₉	—	2	2	no	—
32	C₂×C₃×C₄×C₉	—	2	1	no	—
33	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
34	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
35	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
36	C₃×C₈×C₉	—	2	2	no	—
37	C₃×#489	C₉	1	3	yes	direct product factor
38	C₃×#489	C₃²	1	1	yes	characteristic
39	C₂×C₃²×Q₈	—	2	1	no	—
40	C₂×C₃²×Q₈	C₂²×C₃	1	1	yes	characteristic
41	C₂×C₃²×D₄	C₂²×C₃	1	1	yes	characteristic
42	C₂×C₃²×C₈	C₂²×C₃	1	2	yes	—
43	C₃²×C₄⋊C₄	—	2	1	no	—
44	C₃²×C₄⋊C₄	C₂²×C₃	1	2	yes	—
45	C₃²×C₂²⋊C₄	—	2	1	no	—
46	C₃²×C₂²⋊C₄	—	2	1	no	—
47	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
48	C₂×Q₈×C₉	—	2	3	no	—
49	C₂×Q₈×C₉	C₂²×C₃	1	3	yes	—
50	C₂×D₄×C₉	C₂²×C₃	1	3	yes	—
51	C₂×C₈×C₉	C₂²×C₃	1	6	yes	—
52	C₉×C₄⋊C₄	—	2	3	no	—
53	C₉×C₄⋊C₄	C₂²×C₃	1	6	yes	—
54	C₉×C₂²⋊C₄	—	2	3	no	—
55	C₉×C₂²⋊C₄	—	2	3	no	—
56	C₄²×C₉	C₂²×C₃	1	3	yes	—
57	C₂²×C₃×C₉	—	4	1	no	—
58	C₂²×C₃×C₉	—	4	1	no	—
59	C₂²×C₃×C₉	—	4	1	no	—
60	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
61	C₃×C₄×C₉	—	2	1	no	—
62	C₃×C₄×C₉	—	2	1	no	—
63	C₃×C₄×C₉	—	4	1	no	—
64	C₃×C₄×C₉	—	4	2	no	—
65	C₃×C₄×C₉	#41	1	2	yes	—
66	C₃×C₄⋊Q₈	C₂×C₉	1	3	yes	complemented
67	C₃×C₄⋊Q₈	C₂×C₃²	1	1	yes	characteristic
68	C₃×#124	C₂×C₉	1	3	yes	—
69	C₃×#124	C₂×C₃²	1	1	yes	characteristic
70	C₃×#103	C₂×C₉	1	6	yes	complemented
71	C₃×#103	C₂×C₃²	1	2	yes	—
72	C₃×#102	C₂×C₉	1	6	yes	—
73	C₃×#102	C₂×C₃²	1	2	yes	—
74	C₃×#97	C₂×C₉	1	3	yes	complemented
75	C₃×#97	C₂×C₃²	1	1	yes	characteristic
76	C₂³×C₃²	—	2	1	no	—
77	C₃²×Q₈	—	2	2	no	—
78	C₃²×Q₈	—	4	2	no	—
79	C₃²×D₄	—	2	2	no	—
80	C₂×C₃²×C₄	—	2	1	no	—
81	C₂×C₃²×C₄	—	2	2	no	—
82	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
83	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
84	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
85	C₂³×C₉	—	2	3	no	—
86	C₃²×C₈	—	2	2	no	—
87	Q₈×C₉	—	2	6	no	—
88	Q₈×C₉	—	4	6	no	—
89	D₄×C₉	—	2	6	no	—
90	C₂×C₄×C₉	—	2	3	no	—
91	C₂×C₄×C₉	—	2	6	no	—
92	C₂×C₄×C₉	C₃×D₄	1	3	yes	—
93	C₂×C₄×C₉	C₃×D₄	1	3	yes	—
94	C₂×C₄×C₉	C₂³×C₃	1	3	yes	—
95	C₈×C₉	—	2	6	no	—
96	#489	C₃×C₉	1	1	yes	characteristic, direct product factor, Sylow
97	C₂×C₃×C₉	—	8	1	no	—
98	C₂×C₃×C₉	#124	1	1	yes	characteristic
99	C₂×C₃×C₉	AGL(1,ℤ/8)	1	1	yes	characteristic
100	C₂×C₃×C₉	#137	1	1	yes	characteristic
101	C₂×C₃×Q₈	—	2	3	no	—
102	C₂×C₃×Q₈	—	2	1	no	—
103	C₂×C₃×Q₈	C₂²×C₉	1	3	yes	—
104	C₂×C₃×Q₈	C₂²×C₃²	1	1	yes	characteristic
105	C₂×C₃×D₄	C₂²×C₉	1	3	yes	—
106	C₂×C₃×D₄	C₂²×C₃²	1	1	yes	characteristic
107	C₂×C₃×C₈	C₂²×C₉	1	6	yes	—
108	C₂×C₃×C₈	C₂²×C₃²	1	2	yes	—
109	C₃×C₄⋊C₄	—	2	3	no	—
110	C₃×C₄⋊C₄	—	2	1	no	—
111	C₃×C₄⋊C₄	C₂²×C₉	1	6	yes	—
112	C₃×C₄⋊C₄	C₂²×C₃²	1	2	yes	—
113	C₃×C₂²⋊C₄	—	2	3	no	—
114	C₃×C₂²⋊C₄	—	2	3	no	—
115	C₃×C₂²⋊C₄	—	2	1	no	—
116	C₃×C₂²⋊C₄	—	2	1	no	—
117	C₃×C₄²	C₂²×C₉	1	3	yes	—
118	C₃×C₄²	C₂²×C₃²	1	1	yes	characteristic
119	C₂²×C₃²	—	4	1	no	—
120	C₂²×C₃²	—	4	1	no	—
121	C₂²×C₃²	—	4	1	no	—
122	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
123	C₃²×C₄	—	2	1	no	—
124	C₃²×C₄	—	2	1	no	—
125	C₃²×C₄	—	4	1	no	—
126	C₃²×C₄	—	4	2	no	—
127	C₃²×C₄	C₃×#41	1	2	yes	—
128	C₂²×C₉	—	4	3	no	—
129	C₂²×C₉	—	4	3	no	—
130	C₂²×C₉	—	4	3	no	—
131	C₂²×C₉	C₂×C₃×D₄	1	3	yes	—
132	C₄×C₉	—	2	3	no	—
133	C₄×C₉	—	2	3	no	—
134	C₄×C₉	—	4	3	no	—
135	C₄×C₉	—	4	6	no	—
136	C₄×C₉	C₃×#41	1	6	yes	—
137	C₄⋊Q₈	C₂×C₃×C₉	1	1	yes	characteristic, complemented
138	#124	C₂×C₃×C₉	1	1	yes	characteristic
139	#103	C₂×C₃×C₉	1	2	yes	complemented
140	#102	C₂×C₃×C₉	1	2	yes	—
141	#97	C₂×C₃×C₉	1	1	yes	characteristic, complemented
142	C₃×C₉	#489	1	1	yes	characteristic, direct product factor, Sylow
143	C₂³×C₃	—	2	3	no	—
144	C₂³×C₃	—	2	1	no	—
145	C₃×Q₈	—	2	6	no	—
146	C₃×Q₈	—	2	2	no	—
147	C₃×Q₈	—	4	2	no	—
148	C₃×Q₈	—	4	6	no	—
149	C₃×D₄	—	2	6	no	—
150	C₃×D₄	—	2	2	no	—
151	C₂×C₃×C₄	—	2	3	no	—
152	C₂×C₃×C₄	—	2	6	no	—
153	C₂×C₃×C₄	—	2	1	no	—
154	C₂×C₃×C₄	—	2	2	no	—
155	C₂×C₃×C₄	D₄×C₉	1	3	yes	—
156	C₂×C₃×C₄	D₄×C₉	1	3	yes	—
157	C₂×C₃×C₄	C₂³×C₉	1	3	yes	—
158	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
159	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
160	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
161	C₃×C₈	—	2	6	no	—
162	C₃×C₈	—	2	2	no	—
163	C₂×C₃²	—	8	1	no	—
164	C₂×C₃²	C₃×#124	1	1	yes	characteristic
165	C₂×C₃²	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
166	C₂×C₃²	C₃×#137	1	1	yes	characteristic
167	C₂×C₉	—	8	3	no	—
168	C₂×C₉	C₃×#124	1	3	yes	—
169	C₂×C₉	C₃×AGL(1,ℤ/8)	1	3	yes	—
170	C₂×C₉	C₃×#137	1	3	yes	—
171	C₂×Q₈	—	2	1	no	—
172	C₂×Q₈	C₂²×C₃×C₉	1	1	yes	characteristic
173	C₂×D₄	C₂²×C₃×C₉	1	1	yes	characteristic
174	C₂×C₈	C₂²×C₃×C₉	1	2	yes	—
175	C₄⋊C₄	—	2	1	no	—
176	C₄⋊C₄	C₂²×C₃×C₉	1	2	yes	—
177	C₂²⋊C₄	—	2	1	no	—
178	C₂²⋊C₄	—	2	1	no	—
179	C₄²	C₂²×C₃×C₉	1	1	yes	characteristic
180	C₂²×C₃	—	4	3	no	—
181	C₂²×C₃	—	4	3	no	—
182	C₂²×C₃	—	4	3	no	—
183	C₂²×C₃	—	4	1	no	—
184	C₂²×C₃	—	4	1	no	—
185	C₂²×C₃	—	4	1	no	—
186	C₂²×C₃	C₂×D₄×C₉	1	3	yes	—
187	C₂²×C₃	C₂×C₃²×D₄	1	1	yes	characteristic
188	C₃×C₄	—	2	3	no	—
189	C₃×C₄	—	2	3	no	—
190	C₃×C₄	—	2	1	no	—
191	C₃×C₄	—	2	1	no	—
192	C₃×C₄	—	4	3	no	—
193	C₃×C₄	—	4	6	no	—
194	C₃×C₄	—	4	1	no	—
195	C₃×C₄	—	4	2	no	—
196	C₃×C₄	C₉×#41	1	6	yes	—
197	C₃×C₄	C₃²×#41	1	2	yes	—
198	C₃²	C₃×#489	1	1	yes	characteristic
199	C₉	C₃×#489	1	3	yes	direct product factor
200	C₂³	—	2	1	no	—
201	Q₈	—	2	2	no	—
202	Q₈	—	4	2	no	—
203	D₄	—	2	2	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	C₃×D₄×C₉	1	1	yes	characteristic
207	C₂×C₄	C₃×D₄×C₉	1	1	yes	characteristic
208	C₂×C₄	C₂³×C₃×C₉	1	1	yes	characteristic
209	C₈	—	2	2	no	—
210	C₂×C₃	—	8	1	no	—
211	C₂×C₃	—	8	3	no	—
212	C₂×C₃	C₉×#124	1	3	yes	—
213	C₂×C₃	C₉×AGL(1,ℤ/8)	1	3	yes	—
214	C₂×C₃	C₉×#137	1	3	yes	—
215	C₂×C₃	C₃²×#124	1	1	yes	characteristic
216	C₂×C₃	C₃²×AGL(1,ℤ/8)	1	1	yes	characteristic
217	C₂×C₃	C₃²×#137	1	1	yes	characteristic
218	C₂²	—	4	1	no	—
219	C₂²	—	4	1	no	—
220	C₂²	—	4	1	no	—
221	C₂²	C₂×C₃×D₄×C₉	1	1	yes	characteristic
222	C₄	—	2	1	no	—
223	C₄	—	2	1	no	—
224	C₄	—	4	1	no	—
225	C₄	—	4	2	no	—
226	C₄	C₃×C₉×#41	1	2	yes	—
227	C₃	C₉×#489	1	3	yes	direct product factor
228	C₃	C₃²×#489	1	1	yes	characteristic
229	C₂	—	8	1	no	—
230	C₂	C₃×C₉×#124	1	1	yes	characteristic
231	C₂	C₃×C₉×AGL(1,ℤ/8)	1	1	yes	characteristic
232	C₂	C₃×C₉×#137	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃×C₉	C₂×D₄
commutator	C₂×C₄	C₂³×C₃×C₉
Frattini, cosocle	C₂×C₃×C₄	C₂³×C₃²
socle	C₂²×C₃²	C₂×C₃×D₄
prime radical	C₂×C₃²×D₄	C₂²×C₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₉ ◁ C₃×C₄²×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×C₉×C₄⋊Q₈	C₂	semidirect
3. C₃×C₉×#103	C₂	semidirect
5. C₃×C₉×#97	C₂	semidirect
7. C₉×#489	C₃	direct (N × H)
22. C₉×C₄⋊Q₈	C₂×C₃	semidirect
24. C₉×#103	C₂×C₃	semidirect
26. C₉×#97	C₂×C₃	semidirect
37. C₃×#489	C₉	direct (N × H)
66. C₃×C₄⋊Q₈	C₂×C₉	semidirect
70. C₃×#103	C₂×C₉	semidirect
74. C₃×#97	C₂×C₉	semidirect
96. #489	C₃×C₉	direct (N × H)
137. C₄⋊Q₈	C₂×C₃×C₉	semidirect
139. #103	C₂×C₃×C₉	semidirect
141. #97	C₂×C₃×C₉	semidirect
142. C₃×C₉	#489	direct (N × H)
199. C₉	C₃×#489	direct (N × H)
227. C₃	C₉×#489	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₉×C₄⋊Q₈	2
C₃×C₉×#124	2
C₃×C₉×#103	2
C₃×C₉×#102	2
C₃×C₉×#97	2
C₃²×#489	3
C₉×#489	3

Subgroups — C3×C9×#489 #13145225

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₉×#489 #13145225