Subgroups — C₃×C₉×#423 #13145027

Order 1728.

630 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₉×M₁₆	C₂	1	2	yes	maximal
2	C₂×C₃×C₄²×C₉	C₂	1	1	yes	maximal, characteristic
3	C₃×C₉×C₄⋊C₈	C₂	1	4	yes	maximal, complemented
4	C₃²×#423	C₃	1	1	yes	maximal, characteristic
5	C₉×#423	C₃	1	3	yes	maximal, direct product factor
6	C₂²×C₃×C₄×C₉	C₄	1	2	yes	—
7	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
8	C₃×C₉×M₁₆	—	2	4	no	—
9	C₂×C₃×C₈×C₉	C₂²	1	4	yes	—
10	C₃×C₄²×C₉	C₄	1	1	yes	characteristic
11	C₃×C₄²×C₉	C₄	1	1	yes	characteristic
12	C₃×C₄²×C₉	C₂²	1	2	yes	—
13	C₂×C₃²×M₁₆	C₂×C₃	1	2	yes	—
14	C₂×C₃²×C₄²	C₂×C₃	1	1	yes	characteristic
15	C₃²×C₄⋊C₈	C₂×C₃	1	4	yes	—
16	C₂×C₉×M₁₆	C₂×C₃	1	6	yes	—
17	C₂×C₄²×C₉	C₂×C₃	1	3	yes	—
18	C₉×C₄⋊C₈	C₂×C₃	1	12	yes	complemented
19	C₂³×C₃×C₉	C₂×C₄	1	1	yes	characteristic
20	C₂×C₃×C₄×C₉	—	2	4	no	—
21	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
22	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
24	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
25	C₂×C₃×C₄×C₉	Q₈	1	2	yes	—
26	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
27	C₃×C₈×C₉	—	2	4	no	—
28	C₃×#423	C₉	1	3	yes	direct product factor
29	C₃×#423	C₃²	1	1	yes	characteristic
30	C₂²×C₃²×C₄	C₃×C₄	1	2	yes	—
31	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
32	C₃²×M₁₆	—	2	4	no	—
33	C₂×C₃²×C₈	C₂²×C₃	1	4	yes	—
34	C₃²×C₄²	C₃×C₄	1	1	yes	characteristic
35	C₃²×C₄²	C₃×C₄	1	1	yes	characteristic
36	C₃²×C₄²	C₂²×C₃	1	2	yes	—
37	C₂²×C₄×C₉	C₃×C₄	1	6	yes	—
38	C₂²×C₄×C₉	C₂²×C₃	1	3	yes	—
39	C₉×M₁₆	—	2	12	no	—
40	C₂×C₈×C₉	C₂²×C₃	1	12	yes	—
41	C₄²×C₉	C₃×C₄	1	3	yes	—
42	C₄²×C₉	C₃×C₄	1	3	yes	—
43	C₄²×C₉	C₂²×C₃	1	6	yes	—
44	C₂²×C₃×C₉	—	2	2	no	—
45	C₂²×C₃×C₉	C₄⋊C₄	1	2	yes	—
46	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
47	C₃×C₄×C₉	—	2	2	no	—
48	C₃×C₄×C₉	C₄⋊C₄	1	2	yes	—
49	C₃×C₄×C₉	M₁₆	1	4	yes	complemented
50	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
51	C₃×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₃×M₁₆	C₂×C₉	1	6	yes	—
53	C₂×C₃×M₁₆	C₂×C₃²	1	2	yes	—
54	C₂×C₃×C₄²	C₂×C₉	1	3	yes	—
55	C₂×C₃×C₄²	C₂×C₃²	1	1	yes	characteristic
56	C₃×C₄⋊C₈	C₂×C₉	1	12	yes	complemented
57	C₃×C₄⋊C₈	C₂×C₃²	1	4	yes	—
58	C₂³×C₃²	C₂×C₃×C₄	1	1	yes	characteristic
59	C₂×C₃²×C₄	—	2	4	no	—
60	C₂×C₃²×C₄	C₂×C₃×C₄	1	2	yes	—
61	C₂×C₃²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
62	C₂×C₃²×C₄	C₂×C₃×C₄	1	2	yes	—
63	C₂×C₃²×C₄	C₃×D₄	1	2	yes	—
64	C₂×C₃²×C₄	C₃×Q₈	1	2	yes	—
65	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
66	C₂³×C₉	C₂×C₃×C₄	1	3	yes	—
67	C₃²×C₈	—	2	4	no	—
68	C₂×C₄×C₉	—	2	12	no	—
69	C₂×C₄×C₉	C₂×C₃×C₄	1	6	yes	—
70	C₂×C₄×C₉	C₂×C₃×C₄	1	3	yes	—
71	C₂×C₄×C₉	C₂×C₃×C₄	1	6	yes	—
72	C₂×C₄×C₉	C₃×D₄	1	6	yes	—
73	C₂×C₄×C₉	C₃×Q₈	1	6	yes	—
74	C₂×C₄×C₉	C₂³×C₃	1	3	yes	—
75	C₈×C₉	—	2	12	no	—
76	#423	C₃×C₉	1	1	yes	characteristic, direct product factor, Sylow
77	C₂×C₃×C₉	—	2	2	no	—
78	C₂×C₃×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₃×C₉	C₂×M₁₆	1	2	yes	—
80	C₂²×C₃×C₄	C₄×C₉	1	6	yes	—
81	C₂²×C₃×C₄	C₂²×C₉	1	3	yes	—
82	C₂²×C₃×C₄	C₃²×C₄	1	2	yes	—
83	C₂²×C₃×C₄	C₂²×C₃²	1	1	yes	characteristic
84	C₃×M₁₆	—	2	12	no	—
85	C₃×M₁₆	—	2	4	no	—
86	C₂×C₃×C₈	C₂²×C₉	1	12	yes	—
87	C₂×C₃×C₈	C₂²×C₃²	1	4	yes	—
88	C₃×C₄²	C₄×C₉	1	3	yes	—
89	C₃×C₄²	C₄×C₉	1	3	yes	—
90	C₃×C₄²	C₂²×C₉	1	6	yes	—
91	C₃×C₄²	C₃²×C₄	1	1	yes	characteristic
92	C₃×C₄²	C₃²×C₄	1	1	yes	characteristic
93	C₃×C₄²	C₂²×C₃²	1	2	yes	—
94	C₂²×C₃²	—	2	2	no	—
95	C₂²×C₃²	C₃×C₄⋊C₄	1	2	yes	—
96	C₂²×C₃²	C₂²×C₃×C₄	1	1	yes	characteristic
97	C₃²×C₄	—	2	2	no	—
98	C₃²×C₄	C₃×C₄⋊C₄	1	2	yes	—
99	C₃²×C₄	C₃×M₁₆	1	4	yes	—
100	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
101	C₃²×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
102	C₂²×C₉	—	2	6	no	—
103	C₂²×C₉	C₃×C₄⋊C₄	1	6	yes	—
104	C₂²×C₉	C₂²×C₃×C₄	1	3	yes	—
105	C₄×C₉	—	2	6	no	—
106	C₄×C₉	C₃×C₄⋊C₄	1	6	yes	—
107	C₄×C₉	C₃×M₁₆	1	12	yes	complemented
108	C₄×C₉	C₂×C₃×D₄	1	3	yes	—
109	C₄×C₉	C₂×C₃×Q₈	1	3	yes	—
110	C₂×M₁₆	C₂×C₃×C₉	1	2	yes	—
111	C₂×C₄²	C₂×C₃×C₉	1	1	yes	characteristic
112	C₄⋊C₈	C₂×C₃×C₉	1	4	yes	complemented
113	C₃×C₉	#423	1	1	yes	characteristic, direct product factor, Sylow
114	C₂³×C₃	C₂×C₄×C₉	1	3	yes	—
115	C₂³×C₃	C₂×C₃²×C₄	1	1	yes	characteristic
116	C₂×C₃×C₄	—	2	12	no	—
117	C₂×C₃×C₄	—	2	4	no	—
118	C₂×C₃×C₄	C₂×C₄×C₉	1	6	yes	—
119	C₂×C₃×C₄	C₂×C₄×C₉	1	3	yes	—
120	C₂×C₃×C₄	C₂×C₄×C₉	1	6	yes	—
121	C₂×C₃×C₄	D₄×C₉	1	6	yes	—
122	C₂×C₃×C₄	Q₈×C₉	1	6	yes	—
123	C₂×C₃×C₄	C₂³×C₉	1	3	yes	—
124	C₂×C₃×C₄	C₂×C₃²×C₄	1	2	yes	—
125	C₂×C₃×C₄	C₂×C₃²×C₄	1	1	yes	characteristic
126	C₂×C₃×C₄	C₂×C₃²×C₄	1	2	yes	—
127	C₂×C₃×C₄	C₃²×D₄	1	2	yes	—
128	C₂×C₃×C₄	C₃²×Q₈	1	2	yes	—
129	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
130	C₃×C₈	—	2	12	no	—
131	C₃×C₈	—	2	4	no	—
132	C₂×C₃²	—	2	2	no	—
133	C₂×C₃²	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
134	C₂×C₃²	C₂×C₃×M₁₆	1	2	yes	—
135	C₂×C₉	—	2	6	no	—
136	C₂×C₉	C₂×C₃×C₄⋊C₄	1	3	yes	—
137	C₂×C₉	C₂×C₃×M₁₆	1	6	yes	—
138	C₂²×C₄	C₃×C₄×C₉	1	2	yes	—
139	C₂²×C₄	C₂²×C₃×C₉	1	1	yes	characteristic
140	M₁₆	—	2	4	no	—
141	C₂×C₈	C₂²×C₃×C₉	1	4	yes	—
142	C₄²	C₃×C₄×C₉	1	1	yes	characteristic
143	C₄²	C₃×C₄×C₉	1	1	yes	characteristic
144	C₄²	C₂²×C₃×C₉	1	2	yes	—
145	C₂²×C₃	—	2	6	no	—
146	C₂²×C₃	—	2	2	no	—
147	C₂²×C₃	C₉×C₄⋊C₄	1	6	yes	—
148	C₂²×C₃	C₂²×C₄×C₉	1	3	yes	—
149	C₂²×C₃	C₃²×C₄⋊C₄	1	2	yes	—
150	C₂²×C₃	C₂²×C₃²×C₄	1	1	yes	characteristic
151	C₃×C₄	—	2	6	no	—
152	C₃×C₄	—	2	2	no	—
153	C₃×C₄	C₉×C₄⋊C₄	1	6	yes	—
154	C₃×C₄	C₉×M₁₆	1	12	yes	complemented
155	C₃×C₄	C₂×D₄×C₉	1	3	yes	—
156	C₃×C₄	C₂×Q₈×C₉	1	3	yes	—
157	C₃×C₄	C₃²×C₄⋊C₄	1	2	yes	—
158	C₃×C₄	C₃²×M₁₆	1	4	yes	—
159	C₃×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
160	C₃×C₄	C₂×C₃²×Q₈	1	1	yes	characteristic
161	C₃²	C₃×#423	1	1	yes	characteristic
162	C₉	C₃×#423	1	3	yes	direct product factor
163	C₂³	C₂×C₃×C₄×C₉	1	1	yes	characteristic
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	C₂×C₃×C₄×C₉	1	2	yes	—
166	C₂×C₄	C₂×C₃×C₄×C₉	1	1	yes	characteristic
167	C₂×C₄	C₂×C₃×C₄×C₉	1	2	yes	—
168	C₂×C₄	C₃×D₄×C₉	1	2	yes	—
169	C₂×C₄	C₃×Q₈×C₉	1	2	yes	—
170	C₂×C₄	C₂³×C₃×C₉	1	1	yes	characteristic
171	C₈	—	2	4	no	—
172	C₂×C₃	—	2	6	no	—
173	C₂×C₃	—	2	2	no	—
174	C₂×C₃	C₂×C₉×C₄⋊C₄	1	3	yes	—
175	C₂×C₃	C₂×C₉×M₁₆	1	6	yes	—
176	C₂×C₃	C₂×C₃²×C₄⋊C₄	1	1	yes	characteristic
177	C₂×C₃	C₂×C₃²×M₁₆	1	2	yes	—
178	C₂²	—	2	2	no	—
179	C₂²	C₃×C₉×C₄⋊C₄	1	2	yes	—
180	C₂²	C₂²×C₃×C₄×C₉	1	1	yes	characteristic
181	C₄	—	2	2	no	—
182	C₄	C₃×C₉×C₄⋊C₄	1	2	yes	—
183	C₄	C₃×C₉×M₁₆	1	4	yes	complemented
184	C₄	C₂×C₃×D₄×C₉	1	1	yes	characteristic
185	C₄	C₂×C₃×Q₈×C₉	1	1	yes	characteristic
186	C₃	C₉×#423	1	3	yes	direct product factor
187	C₃	C₃²×#423	1	1	yes	characteristic
188	C₂	—	2	2	no	—
189	C₂	C₂×C₃×C₉×C₄⋊C₄	1	1	yes	characteristic
190	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	C₂³×C₃²	C₂×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
3. C₃×C₉×C₄⋊C₈	C₂	semidirect
5. C₉×#423	C₃	direct (N × H)
18. C₉×C₄⋊C₈	C₂×C₃	semidirect
28. C₃×#423	C₉	direct (N × H)
49. C₃×C₄×C₉	M₁₆	semidirect
56. C₃×C₄⋊C₈	C₂×C₉	semidirect
76. #423	C₃×C₉	direct (N × H)
107. C₄×C₉	C₃×M₁₆	semidirect
112. C₄⋊C₈	C₂×C₃×C₉	semidirect
113. C₃×C₉	#423	direct (N × H)
154. C₃×C₄	C₉×M₁₆	semidirect
162. C₉	C₃×#423	direct (N × H)
183. C₄	C₃×C₉×M₁₆	semidirect
186. C₃	C₉×#423	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×M₁₆	2
C₂×C₃×C₄²×C₉	2
C₃×C₉×C₄⋊C₈	2
C₃²×#423	3
C₉×#423	3

Subgroups — C3×C9×#423 #13145027

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₉×#423 #13145027