Subgroups — C₃×C₉×#407 #13144979

Order 1728.

950 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	C₂³×C₃×C₄×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₉×M₁₆	C₂	1	2	yes	maximal, complemented
3	C₃×C₉×C₂²⋊C₈	C₂	1	4	yes	maximal, complemented
4	C₃²×#407	C₃	1	1	yes	maximal, characteristic
5	C₉×#407	C₃	1	3	yes	maximal, direct product factor
6	C₂⁴×C₃×C₉	C₄	1	1	yes	characteristic
7	C₂²×C₃×C₄×C₉	—	2	4	no	—
8	C₂²×C₃×C₄×C₉	C₄	1	1	yes	characteristic
9	C₂²×C₃×C₄×C₉	C₄	1	2	yes	—
10	C₂²×C₃×C₄×C₉	C₂²	1	2	yes	—
11	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
12	C₃×C₉×M₁₆	—	2	4	no	—
13	C₂×C₃×C₈×C₉	C₂²	1	4	yes	complemented
14	C₂³×C₃²×C₄	C₂×C₃	1	1	yes	characteristic
15	C₂×C₃²×M₁₆	C₂×C₃	1	2	yes	—
16	C₂³×C₄×C₉	C₂×C₃	1	3	yes	—
17	C₃²×C₂²⋊C₈	C₂×C₃	1	4	yes	—
18	C₂×C₉×M₁₆	C₂×C₃	1	6	yes	complemented
19	C₉×C₂²⋊C₈	C₂×C₃	1	12	yes	complemented
20	C₂³×C₃×C₉	—	2	2	no	—
21	C₂³×C₃×C₉	—	2	4	no	—
22	C₂³×C₃×C₉	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₃×C₉	C₂×C₄	1	2	yes	—
24	C₂×C₃×C₄×C₉	—	2	2	no	—
25	C₂×C₃×C₄×C₉	—	2	4	no	—
26	C₂×C₃×C₄×C₉	—	2	4	no	—
27	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
28	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃×C₄×C₉	D₄	1	4	yes	—
30	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
31	C₃×C₈×C₉	—	2	4	no	—
32	C₃×#407	C₉	1	3	yes	direct product factor
33	C₃×#407	C₃²	1	1	yes	characteristic
34	C₂⁴×C₃²	C₃×C₄	1	1	yes	characteristic
35	C₂²×C₃²×C₄	—	2	4	no	—
36	C₂²×C₃²×C₄	C₃×C₄	1	1	yes	characteristic
37	C₂²×C₃²×C₄	C₃×C₄	1	2	yes	—
38	C₂²×C₃²×C₄	C₂²×C₃	1	2	yes	—
39	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
40	C₂⁴×C₉	C₃×C₄	1	3	yes	—
41	C₃²×M₁₆	—	2	4	no	—
42	C₂×C₃²×C₈	C₂²×C₃	1	4	yes	—
43	C₂²×C₄×C₉	—	2	12	no	—
44	C₂²×C₄×C₉	C₃×C₄	1	3	yes	—
45	C₂²×C₄×C₉	C₃×C₄	1	6	yes	—
46	C₂²×C₄×C₉	C₂²×C₃	1	6	yes	—
47	C₂²×C₄×C₉	C₂²×C₃	1	3	yes	—
48	C₉×M₁₆	—	2	12	no	—
49	C₂×C₈×C₉	C₂²×C₃	1	12	yes	complemented
50	C₂²×C₃×C₉	—	2	8	no	—
51	C₂²×C₃×C₉	—	2	2	no	—
52	C₂²×C₃×C₉	—	2	2	no	—
53	C₂²×C₃×C₉	—	2	2	no	—
54	C₂²×C₃×C₉	C₂²⋊C₄	1	2	yes	—
55	C₂²×C₃×C₉	M₁₆	1	4	yes	complemented
56	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
57	C₃×C₄×C₉	—	2	2	no	—
58	C₃×C₄×C₉	C₂²⋊C₄	1	2	yes	—
59	C₃×C₄×C₉	C₂×D₄	1	2	yes	—
60	C₂³×C₃×C₄	C₂×C₉	1	3	yes	—
61	C₂³×C₃×C₄	C₂×C₃²	1	1	yes	characteristic
62	C₂×C₃×M₁₆	C₂×C₉	1	6	yes	complemented
63	C₂×C₃×M₁₆	C₂×C₃²	1	2	yes	—
64	C₃×C₂²⋊C₈	C₂×C₉	1	12	yes	complemented
65	C₃×C₂²⋊C₈	C₂×C₃²	1	4	yes	—
66	C₂³×C₃²	—	2	2	no	—
67	C₂³×C₃²	—	2	4	no	—
68	C₂³×C₃²	C₂×C₃×C₄	1	1	yes	characteristic
69	C₂³×C₃²	C₂×C₃×C₄	1	2	yes	—
70	C₂×C₃²×C₄	—	2	2	no	—
71	C₂×C₃²×C₄	—	2	4	no	—
72	C₂×C₃²×C₄	—	2	4	no	—
73	C₂×C₃²×C₄	C₂×C₃×C₄	1	2	yes	—
74	C₂×C₃²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
75	C₂×C₃²×C₄	C₃×D₄	1	4	yes	—
76	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
77	C₂³×C₉	—	2	6	no	—
78	C₂³×C₉	—	2	12	no	—
79	C₂³×C₉	C₂×C₃×C₄	1	3	yes	—
80	C₂³×C₉	C₂×C₃×C₄	1	6	yes	—
81	C₃²×C₈	—	2	4	no	—
82	C₂×C₄×C₉	—	2	6	no	—
83	C₂×C₄×C₉	—	2	12	no	—
84	C₂×C₄×C₉	—	2	12	no	—
85	C₂×C₄×C₉	C₂×C₃×C₄	1	6	yes	—
86	C₂×C₄×C₉	C₂×C₃×C₄	1	3	yes	—
87	C₂×C₄×C₉	C₃×D₄	1	12	yes	—
88	C₂×C₄×C₉	C₂³×C₃	1	3	yes	—
89	C₈×C₉	—	2	12	no	—
90	#407	C₃×C₉	1	1	yes	characteristic, direct product factor, Sylow
91	C₂×C₃×C₉	—	2	2	no	—
92	C₂×C₃×C₉	—	2	4	no	—
93	C₂×C₃×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₃×C₉	C₂×M₁₆	1	2	yes	—
95	C₂⁴×C₃	C₄×C₉	1	3	yes	—
96	C₂⁴×C₃	C₃²×C₄	1	1	yes	characteristic
97	C₂²×C₃×C₄	—	2	12	no	—
98	C₂²×C₃×C₄	—	2	4	no	—
99	C₂²×C₃×C₄	C₄×C₉	1	3	yes	—
100	C₂²×C₃×C₄	C₄×C₉	1	6	yes	—
101	C₂²×C₃×C₄	C₂²×C₉	1	6	yes	—
102	C₂²×C₃×C₄	C₂²×C₉	1	3	yes	—
103	C₂²×C₃×C₄	C₃²×C₄	1	1	yes	characteristic
104	C₂²×C₃×C₄	C₃²×C₄	1	2	yes	—
105	C₂²×C₃×C₄	C₂²×C₃²	1	2	yes	—
106	C₂²×C₃×C₄	C₂²×C₃²	1	1	yes	characteristic
107	C₃×M₁₆	—	2	12	no	—
108	C₃×M₁₆	—	2	4	no	—
109	C₂×C₃×C₈	C₂²×C₉	1	12	yes	complemented
110	C₂×C₃×C₈	C₂²×C₃²	1	4	yes	—
111	C₂²×C₃²	—	2	8	no	—
112	C₂²×C₃²	—	2	2	no	—
113	C₂²×C₃²	—	2	2	no	—
114	C₂²×C₃²	—	2	2	no	—
115	C₂²×C₃²	C₃×C₂²⋊C₄	1	2	yes	—
116	C₂²×C₃²	C₃×M₁₆	1	4	yes	—
117	C₂²×C₃²	C₂²×C₃×C₄	1	1	yes	characteristic
118	C₃²×C₄	—	2	2	no	—
119	C₃²×C₄	C₃×C₂²⋊C₄	1	2	yes	—
120	C₃²×C₄	C₂×C₃×D₄	1	2	yes	—
121	C₂²×C₉	—	2	24	no	—
122	C₂²×C₉	—	2	6	no	—
123	C₂²×C₉	—	2	6	no	—
124	C₂²×C₉	—	2	6	no	—
125	C₂²×C₉	C₃×C₂²⋊C₄	1	6	yes	—
126	C₂²×C₉	C₃×M₁₆	1	12	yes	complemented
127	C₂²×C₉	C₂²×C₃×C₄	1	3	yes	—
128	C₄×C₉	—	2	6	no	—
129	C₄×C₉	C₃×C₂²⋊C₄	1	6	yes	—
130	C₄×C₉	C₂×C₃×D₄	1	6	yes	—
131	C₂³×C₄	C₂×C₃×C₉	1	1	yes	characteristic
132	C₂×M₁₆	C₂×C₃×C₉	1	2	yes	complemented
133	C₂²⋊C₈	C₂×C₃×C₉	1	4	yes	complemented
134	C₃×C₉	#407	1	1	yes	characteristic, direct product factor, Sylow
135	C₂³×C₃	—	2	6	no	—
136	C₂³×C₃	—	2	12	no	—
137	C₂³×C₃	—	2	2	no	—
138	C₂³×C₃	—	2	4	no	—
139	C₂³×C₃	C₂×C₄×C₉	1	3	yes	—
140	C₂³×C₃	C₂×C₄×C₉	1	6	yes	—
141	C₂³×C₃	C₂×C₃²×C₄	1	1	yes	characteristic
142	C₂³×C₃	C₂×C₃²×C₄	1	2	yes	—
143	C₂×C₃×C₄	—	2	6	no	—
144	C₂×C₃×C₄	—	2	2	no	—
145	C₂×C₃×C₄	—	2	12	no	—
146	C₂×C₃×C₄	—	2	12	no	—
147	C₂×C₃×C₄	—	2	4	no	—
148	C₂×C₃×C₄	—	2	4	no	—
149	C₂×C₃×C₄	C₂×C₄×C₉	1	6	yes	—
150	C₂×C₃×C₄	C₂×C₄×C₉	1	3	yes	—
151	C₂×C₃×C₄	D₄×C₉	1	12	yes	—
152	C₂×C₃×C₄	C₂³×C₉	1	3	yes	—
153	C₂×C₃×C₄	C₂×C₃²×C₄	1	2	yes	—
154	C₂×C₃×C₄	C₂×C₃²×C₄	1	1	yes	characteristic
155	C₂×C₃×C₄	C₃²×D₄	1	4	yes	—
156	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
157	C₃×C₈	—	2	12	no	—
158	C₃×C₈	—	2	4	no	—
159	C₂×C₃²	—	2	2	no	—
160	C₂×C₃²	—	2	4	no	—
161	C₂×C₃²	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
162	C₂×C₃²	C₂×C₃×M₁₆	1	2	yes	—
163	C₂×C₉	—	2	6	no	—
164	C₂×C₉	—	2	12	no	—
165	C₂×C₉	C₂×C₃×C₂²⋊C₄	1	3	yes	—
166	C₂×C₉	C₂×C₃×M₁₆	1	6	yes	—
167	C₂⁴	C₃×C₄×C₉	1	1	yes	characteristic
168	C₂²×C₄	—	2	4	no	—
169	C₂²×C₄	C₃×C₄×C₉	1	1	yes	characteristic
170	C₂²×C₄	C₃×C₄×C₉	1	2	yes	—
171	C₂²×C₄	C₂²×C₃×C₉	1	2	yes	—
172	C₂²×C₄	C₂²×C₃×C₉	1	1	yes	characteristic
173	M₁₆	—	2	4	no	—
174	C₂×C₈	C₂²×C₃×C₉	1	4	yes	complemented
175	C₂²×C₃	—	2	8	no	—
176	C₂²×C₃	—	2	24	no	—
177	C₂²×C₃	—	2	6	no	—
178	C₂²×C₃	—	2	6	no	—
179	C₂²×C₃	—	2	6	no	—
180	C₂²×C₃	—	2	2	no	—
181	C₂²×C₃	—	2	2	no	—
182	C₂²×C₃	—	2	2	no	—
183	C₂²×C₃	C₉×C₂²⋊C₄	1	6	yes	—
184	C₂²×C₃	C₉×M₁₆	1	12	yes	complemented
185	C₂²×C₃	C₂²×C₄×C₉	1	3	yes	—
186	C₂²×C₃	C₃²×C₂²⋊C₄	1	2	yes	—
187	C₂²×C₃	C₃²×M₁₆	1	4	yes	—
188	C₂²×C₃	C₂²×C₃²×C₄	1	1	yes	characteristic
189	C₃×C₄	—	2	6	no	—
190	C₃×C₄	—	2	2	no	—
191	C₃×C₄	C₉×C₂²⋊C₄	1	6	yes	—
192	C₃×C₄	C₂×D₄×C₉	1	6	yes	—
193	C₃×C₄	C₃²×C₂²⋊C₄	1	2	yes	—
194	C₃×C₄	C₂×C₃²×D₄	1	2	yes	—
195	C₃²	C₃×#407	1	1	yes	characteristic
196	C₉	C₃×#407	1	3	yes	direct product factor
197	C₂³	—	2	2	no	—
198	C₂³	—	2	4	no	—
199	C₂³	C₂×C₃×C₄×C₉	1	1	yes	characteristic
200	C₂³	C₂×C₃×C₄×C₉	1	2	yes	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	C₂×C₃×C₄×C₉	1	2	yes	—
205	C₂×C₄	C₂×C₃×C₄×C₉	1	1	yes	characteristic
206	C₂×C₄	C₃×D₄×C₉	1	4	yes	—
207	C₂×C₄	C₂³×C₃×C₉	1	1	yes	characteristic
208	C₈	—	2	4	no	—
209	C₂×C₃	—	2	6	no	—
210	C₂×C₃	—	2	12	no	—
211	C₂×C₃	—	2	2	no	—
212	C₂×C₃	—	2	4	no	—
213	C₂×C₃	C₂×C₉×C₂²⋊C₄	1	3	yes	—
214	C₂×C₃	C₂×C₉×M₁₆	1	6	yes	—
215	C₂×C₃	C₂×C₃²×C₂²⋊C₄	1	1	yes	characteristic
216	C₂×C₃	C₂×C₃²×M₁₆	1	2	yes	—
217	C₂²	—	2	8	no	—
218	C₂²	—	2	2	no	—
219	C₂²	—	2	2	no	—
220	C₂²	—	2	2	no	—
221	C₂²	C₃×C₉×C₂²⋊C₄	1	2	yes	—
222	C₂²	C₃×C₉×M₁₆	1	4	yes	complemented
223	C₂²	C₂²×C₃×C₄×C₉	1	1	yes	characteristic
224	C₄	—	2	2	no	—
225	C₄	C₃×C₉×C₂²⋊C₄	1	2	yes	—
226	C₄	C₂×C₃×D₄×C₉	1	2	yes	—
227	C₃	C₉×#407	1	3	yes	direct product factor
228	C₃	C₃²×#407	1	1	yes	characteristic
229	C₂	—	2	2	no	—
230	C₂	—	2	4	no	—
231	C₂	C₂×C₃×C₉×C₂²⋊C₄	1	1	yes	characteristic
232	C₂	C₂×C₃×C₉×M₁₆	1	2	yes	—

	subgroup	G/H
center	C₂×C₃×C₄×C₉	C₂³
commutator	C₂²	C₂²×C₃×C₄×C₉
Frattini, cosocle	C₂×C₃×C₄	C₂³×C₃²
socle	C₂²×C₃²	C₂²×C₃×C₄
prime radical	C₂⁴×C₃²	C₃×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₉×M₁₆	C₂	semidirect
3. C₃×C₉×C₂²⋊C₈	C₂	semidirect
5. C₉×#407	C₃	direct (N × H)
13. C₂×C₃×C₈×C₉	C₂²	semidirect
18. C₂×C₉×M₁₆	C₂×C₃	semidirect
19. C₉×C₂²⋊C₈	C₂×C₃	semidirect
32. C₃×#407	C₉	direct (N × H)
49. C₂×C₈×C₉	C₂²×C₃	semidirect
55. C₂²×C₃×C₉	M₁₆	semidirect
62. C₂×C₃×M₁₆	C₂×C₉	semidirect
64. C₃×C₂²⋊C₈	C₂×C₉	semidirect
90. #407	C₃×C₉	direct (N × H)
109. C₂×C₃×C₈	C₂²×C₉	semidirect
126. C₂²×C₉	C₃×M₁₆	semidirect
132. C₂×M₁₆	C₂×C₃×C₉	semidirect
133. C₂²⋊C₈	C₂×C₃×C₉	semidirect
134. C₃×C₉	#407	direct (N × H)
174. C₂×C₈	C₂²×C₃×C₉	semidirect
184. C₂²×C₃	C₉×M₁₆	semidirect
196. C₉	C₃×#407	direct (N × H)
222. C₂²	C₃×C₉×M₁₆	semidirect
227. C₃	C₉×#407	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃×C₄×C₉	2
C₂×C₃×C₉×M₁₆	2
C₃×C₉×C₂²⋊C₈	2
C₃²×#407	3
C₉×#407	3

Subgroups — C3×C9×#407 #13144979

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₉×#407 #13144979