Subgroups — C₃×C₉×#467 #13145159

Order 1728.

570 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₉×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃×C₈×C₉	C₂	1	1	yes	maximal, characteristic
3	C₃×C₉×C₂²⋊Q₈	C₂	1	2	yes	maximal
4	C₃×C₉×#107	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₉×#103	C₂	1	2	yes	maximal, complemented
6	C₃²×#467	C₃	1	1	yes	maximal, characteristic
7	C₉×#467	C₃	1	3	yes	maximal, direct product factor
8	C₂×C₃×Q₈×C₉	C₂²	1	2	yes	—
9	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
10	C₃×C₉×Q₁₆	—	2	2	no	—
11	C₂×C₃×C₈×C₉	—	2	2	no	—
12	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
14	C₃×C₉×C₄⋊C₄	—	2	2	no	—
15	C₃×C₉×C₄⋊C₄	C₂²	1	2	yes	—
16	C₃×C₉×C₂²⋊C₄	—	2	2	no	—
17	C₂×C₃²×Q₁₆	C₂×C₃	1	1	yes	characteristic
18	C₂²×C₃²×C₈	C₂×C₃	1	1	yes	characteristic
19	C₃²×C₂²⋊Q₈	C₂×C₃	1	2	yes	—
20	C₃²×#107	C₂×C₃	1	1	yes	characteristic
21	C₃²×#103	C₂×C₃	1	2	yes	—
22	C₂×C₉×Q₁₆	C₂×C₃	1	3	yes	complemented
23	C₂²×C₈×C₉	C₂×C₃	1	3	yes	—
24	C₉×C₂²⋊Q₈	C₂×C₃	1	6	yes	—
25	C₉×#107	C₂×C₃	1	3	yes	complemented
26	C₉×#103	C₂×C₃	1	6	yes	complemented
27	C₂³×C₃×C₉	D₄	1	1	yes	characteristic
28	C₃×Q₈×C₉	—	2	2	no	—
29	C₃×Q₈×C₉	—	2	2	no	—
30	C₂×C₃×C₄×C₉	—	2	1	no	—
31	C₂×C₃×C₄×C₉	—	2	1	no	—
32	C₂×C₃×C₄×C₉	—	2	2	no	—
33	C₂×C₃×C₄×C₉	—	2	2	no	—
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	1	no	—
37	C₃×C₈×C₉	D₄	1	2	yes	—
38	C₃×#467	C₉	1	3	yes	direct product factor
39	C₃×#467	C₃²	1	1	yes	characteristic
40	C₂×C₃²×Q₈	C₂²×C₃	1	2	yes	—
41	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
42	C₃²×Q₁₆	—	2	2	no	—
43	C₂×C₃²×C₈	—	2	2	no	—
44	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
45	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
46	C₃²×C₄⋊C₄	—	2	2	no	—
47	C₃²×C₄⋊C₄	C₂²×C₃	1	2	yes	—
48	C₃²×C₂²⋊C₄	—	2	2	no	—
49	C₂×Q₈×C₉	C₂²×C₃	1	6	yes	—
50	C₂²×C₄×C₉	C₂²×C₃	1	3	yes	—
51	C₉×Q₁₆	—	2	6	no	—
52	C₂×C₈×C₉	—	2	6	no	—
53	C₂×C₈×C₉	C₂²×C₃	1	3	yes	—
54	C₂×C₈×C₉	C₂²×C₃	1	3	yes	—
55	C₉×C₄⋊C₄	—	2	6	no	—
56	C₉×C₄⋊C₄	C₂²×C₃	1	6	yes	—
57	C₉×C₂²⋊C₄	—	2	6	no	—
58	C₂²×C₃×C₉	—	2	1	no	—
59	C₂²×C₃×C₉	—	2	1	no	—
60	C₂²×C₃×C₉	Q₁₆	1	2	yes	complemented
61	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
62	C₃×C₄×C₉	—	2	1	no	—
63	C₃×C₄×C₉	—	4	2	no	—
64	C₃×C₄×C₉	—	4	2	no	—
65	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
66	C₃×C₄×C₉	#41	1	1	yes	characteristic
67	C₂×C₃×Q₁₆	C₂×C₉	1	3	yes	complemented
68	C₂×C₃×Q₁₆	C₂×C₃²	1	1	yes	characteristic
69	C₂²×C₃×C₈	C₂×C₉	1	3	yes	—
70	C₂²×C₃×C₈	C₂×C₃²	1	1	yes	characteristic
71	C₃×C₂²⋊Q₈	C₂×C₉	1	6	yes	—
72	C₃×C₂²⋊Q₈	C₂×C₃²	1	2	yes	—
73	C₃×#107	C₂×C₉	1	3	yes	complemented
74	C₃×#107	C₂×C₃²	1	1	yes	characteristic
75	C₃×#103	C₂×C₉	1	6	yes	complemented
76	C₃×#103	C₂×C₃²	1	2	yes	—
77	C₂³×C₃²	C₃×D₄	1	1	yes	characteristic
78	C₃²×Q₈	—	2	2	no	—
79	C₃²×Q₈	—	2	2	no	—
80	C₂×C₃²×C₄	—	2	1	no	—
81	C₂×C₃²×C₄	—	2	1	no	—
82	C₂×C₃²×C₄	—	2	2	no	—
83	C₂×C₃²×C₄	—	2	2	no	—
84	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
85	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
86	C₂³×C₉	C₃×D₄	1	3	yes	—
87	C₃²×C₈	—	2	1	no	—
88	C₃²×C₈	C₃×D₄	1	2	yes	—
89	Q₈×C₉	—	2	6	no	—
90	Q₈×C₉	—	2	6	no	—
91	C₂×C₄×C₉	—	2	3	no	—
92	C₂×C₄×C₉	—	2	3	no	—
93	C₂×C₄×C₉	—	2	6	no	—
94	C₂×C₄×C₉	—	2	6	no	—
95	C₂×C₄×C₉	C₃×D₄	1	3	yes	—
96	C₂×C₄×C₉	C₂³×C₃	1	3	yes	—
97	C₈×C₉	—	2	3	no	—
98	C₈×C₉	C₃×D₄	1	6	yes	—
99	#467	C₃×C₉	1	1	yes	characteristic, direct product factor, Sylow
100	C₂×C₃×C₉	—	2	2	no	—
101	C₂×C₃×C₉	#121	1	1	yes	characteristic
102	C₂×C₃×C₉	C₂×Q₁₆	1	1	yes	characteristic
103	C₂×C₃×C₉	#135	1	1	yes	characteristic
104	C₂×C₃×Q₈	C₂²×C₉	1	6	yes	—
105	C₂×C₃×Q₈	C₂²×C₃²	1	2	yes	—
106	C₂²×C₃×C₄	C₂²×C₉	1	3	yes	—
107	C₂²×C₃×C₄	C₂²×C₃²	1	1	yes	characteristic
108	C₃×Q₁₆	—	2	6	no	—
109	C₃×Q₁₆	—	2	2	no	—
110	C₂×C₃×C₈	—	2	6	no	—
111	C₂×C₃×C₈	—	2	2	no	—
112	C₂×C₃×C₈	C₂²×C₉	1	3	yes	—
113	C₂×C₃×C₈	C₂²×C₉	1	3	yes	—
114	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
115	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
116	C₃×C₄⋊C₄	—	2	6	no	—
117	C₃×C₄⋊C₄	—	2	2	no	—
118	C₃×C₄⋊C₄	C₂²×C₉	1	6	yes	—
119	C₃×C₄⋊C₄	C₂²×C₃²	1	2	yes	—
120	C₃×C₂²⋊C₄	—	2	6	no	—
121	C₃×C₂²⋊C₄	—	2	2	no	—
122	C₂²×C₃²	—	2	1	no	—
123	C₂²×C₃²	—	2	1	no	—
124	C₂²×C₃²	C₃×Q₁₆	1	2	yes	—
125	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
126	C₃²×C₄	—	2	1	no	—
127	C₃²×C₄	—	4	2	no	—
128	C₃²×C₄	—	4	2	no	—
129	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
130	C₃²×C₄	C₃×#41	1	1	yes	characteristic
131	C₂²×C₉	—	2	3	no	—
132	C₂²×C₉	—	2	3	no	—
133	C₂²×C₉	C₃×Q₁₆	1	6	yes	complemented
134	C₂²×C₉	C₂×C₃×D₄	1	3	yes	—
135	C₄×C₉	—	2	3	no	—
136	C₄×C₉	—	4	6	no	—
137	C₄×C₉	—	4	6	no	—
138	C₄×C₉	C₂×C₃×D₄	1	3	yes	—
139	C₄×C₉	C₃×#41	1	3	yes	—
140	C₂×Q₁₆	C₂×C₃×C₉	1	1	yes	characteristic, complemented
141	C₂²×C₈	C₂×C₃×C₉	1	1	yes	characteristic
142	C₂²⋊Q₈	C₂×C₃×C₉	1	2	yes	—
143	#107	C₂×C₃×C₉	1	1	yes	characteristic, complemented
144	#103	C₂×C₃×C₉	1	2	yes	complemented
145	C₃×C₉	#467	1	1	yes	characteristic, direct product factor, Sylow
146	C₂³×C₃	D₄×C₉	1	3	yes	—
147	C₂³×C₃	C₃²×D₄	1	1	yes	characteristic
148	C₃×Q₈	—	2	6	no	—
149	C₃×Q₈	—	2	6	no	—
150	C₃×Q₈	—	2	2	no	—
151	C₃×Q₈	—	2	2	no	—
152	C₂×C₃×C₄	—	2	3	no	—
153	C₂×C₃×C₄	—	2	3	no	—
154	C₂×C₃×C₄	—	2	1	no	—
155	C₂×C₃×C₄	—	2	1	no	—
156	C₂×C₃×C₄	—	2	6	no	—
157	C₂×C₃×C₄	—	2	6	no	—
158	C₂×C₃×C₄	—	2	2	no	—
159	C₂×C₃×C₄	—	2	2	no	—
160	C₂×C₃×C₄	D₄×C₉	1	3	yes	—
161	C₂×C₃×C₄	C₂³×C₉	1	3	yes	—
162	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
163	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
164	C₃×C₈	—	2	3	no	—
165	C₃×C₈	—	2	1	no	—
166	C₃×C₈	D₄×C₉	1	6	yes	—
167	C₃×C₈	C₃²×D₄	1	2	yes	—
168	C₂×C₃²	—	2	2	no	—
169	C₂×C₃²	C₃×#121	1	1	yes	characteristic
170	C₂×C₃²	C₂×C₃×Q₁₆	1	1	yes	characteristic
171	C₂×C₃²	C₃×#135	1	1	yes	characteristic
172	C₂×C₉	—	2	6	no	—
173	C₂×C₉	C₃×#121	1	3	yes	—
174	C₂×C₉	C₂×C₃×Q₁₆	1	3	yes	—
175	C₂×C₉	C₃×#135	1	3	yes	—
176	C₂×Q₈	C₂²×C₃×C₉	1	2	yes	—
177	C₂²×C₄	C₂²×C₃×C₉	1	1	yes	characteristic
178	Q₁₆	—	2	2	no	—
179	C₂×C₈	—	2	2	no	—
180	C₂×C₈	C₂²×C₃×C₉	1	1	yes	characteristic
181	C₂×C₈	C₂²×C₃×C₉	1	1	yes	characteristic
182	C₄⋊C₄	—	2	2	no	—
183	C₄⋊C₄	C₂²×C₃×C₉	1	2	yes	—
184	C₂²⋊C₄	—	2	2	no	—
185	C₂²×C₃	—	2	3	no	—
186	C₂²×C₃	—	2	3	no	—
187	C₂²×C₃	—	2	1	no	—
188	C₂²×C₃	—	2	1	no	—
189	C₂²×C₃	C₉×Q₁₆	1	6	yes	complemented
190	C₂²×C₃	C₂×D₄×C₉	1	3	yes	—
191	C₂²×C₃	C₃²×Q₁₆	1	2	yes	—
192	C₂²×C₃	C₂×C₃²×D₄	1	1	yes	characteristic
193	C₃×C₄	—	2	3	no	—
194	C₃×C₄	—	2	1	no	—
195	C₃×C₄	—	4	6	no	—
196	C₃×C₄	—	4	6	no	—
197	C₃×C₄	—	4	2	no	—
198	C₃×C₄	—	4	2	no	—
199	C₃×C₄	C₂×D₄×C₉	1	3	yes	—
200	C₃×C₄	C₉×#41	1	3	yes	—
201	C₃×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
202	C₃×C₄	C₃²×#41	1	1	yes	characteristic
203	C₃²	C₃×#467	1	1	yes	characteristic
204	C₉	C₃×#467	1	3	yes	direct product factor
205	C₂³	C₃×D₄×C₉	1	1	yes	characteristic
206	Q₈	—	2	2	no	—
207	Q₈	—	2	2	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	C₃×D₄×C₉	1	1	yes	characteristic
213	C₂×C₄	C₂³×C₃×C₉	1	1	yes	characteristic
214	C₈	—	2	1	no	—
215	C₈	C₃×D₄×C₉	1	2	yes	—
216	C₂×C₃	—	2	2	no	—
217	C₂×C₃	—	2	6	no	—
218	C₂×C₃	C₉×#121	1	3	yes	—
219	C₂×C₃	C₂×C₉×Q₁₆	1	3	yes	—
220	C₂×C₃	C₉×#135	1	3	yes	—
221	C₂×C₃	C₃²×#121	1	1	yes	characteristic
222	C₂×C₃	C₂×C₃²×Q₁₆	1	1	yes	characteristic
223	C₂×C₃	C₃²×#135	1	1	yes	characteristic
224	C₂²	—	2	1	no	—
225	C₂²	—	2	1	no	—
226	C₂²	C₃×C₉×Q₁₆	1	2	yes	complemented
227	C₂²	C₂×C₃×D₄×C₉	1	1	yes	characteristic
228	C₄	—	2	1	no	—
229	C₄	—	4	2	no	—
230	C₄	—	4	2	no	—
231	C₄	C₂×C₃×D₄×C₉	1	1	yes	characteristic
232	C₄	C₃×C₉×#41	1	1	yes	characteristic
233	C₃	C₉×#467	1	3	yes	direct product factor
234	C₃	C₃²×#467	1	1	yes	characteristic
235	C₂	—	2	2	no	—
236	C₂	C₃×C₉×#121	1	1	yes	characteristic
237	C₂	C₂×C₃×C₉×Q₁₆	1	1	yes	characteristic
238	C₂	C₃×C₉×#135	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	C₂³×C₃²	C₃×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₉ ◁ C₂²×C₃×C₄×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₉×Q₁₆	C₂	semidirect
4. C₃×C₉×#107	C₂	semidirect
5. C₃×C₉×#103	C₂	semidirect
7. C₉×#467	C₃	direct (N × H)
22. C₂×C₉×Q₁₆	C₂×C₃	semidirect
25. C₉×#107	C₂×C₃	semidirect
26. C₉×#103	C₂×C₃	semidirect
38. C₃×#467	C₉	direct (N × H)
60. C₂²×C₃×C₉	Q₁₆	semidirect
67. C₂×C₃×Q₁₆	C₂×C₉	semidirect
73. C₃×#107	C₂×C₉	semidirect
75. C₃×#103	C₂×C₉	semidirect
99. #467	C₃×C₉	direct (N × H)
133. C₂²×C₉	C₃×Q₁₆	semidirect
140. C₂×Q₁₆	C₂×C₃×C₉	semidirect
143. #107	C₂×C₃×C₉	semidirect
144. #103	C₂×C₃×C₉	semidirect
145. C₃×C₉	#467	direct (N × H)
189. C₂²×C₃	C₉×Q₁₆	semidirect
204. C₉	C₃×#467	direct (N × H)
226. C₂²	C₃×C₉×Q₁₆	semidirect
233. C₃	C₉×#467	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×C₉×Q₁₆	2
C₂²×C₃×C₈×C₉	2
C₃×C₉×C₂²⋊Q₈	2
C₃×C₉×#107	2
C₃×C₉×#103	2
C₃²×#467	3
C₉×#467	3

Subgroups — C3×C9×#467 #13145159

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₉×#467 #13145159