Subgroups — C₂×C₃×C₉×SD₃₂ #13145276

Order 1728.

450 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×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×C₉×D₈	C₂	1	1	yes	maximal, characteristic
3	C₃×C₉×SD₃₂	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₃×C₉×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃²×SD₃₂	C₃	1	1	yes	maximal, characteristic
6	C₂×C₉×SD₃₂	C₃	1	3	yes	maximal, direct product factor
7	C₂×C₃×Q₈×C₉	—	2	1	no	—
8	C₂×C₃×D₄×C₉	—	2	1	no	—
9	C₃×C₉×Q₁₆	—	2	1	no	—
10	C₃×C₉×Q₁₆	C₂²	1	2	yes	complemented
11	C₃×C₉×D₈	—	2	1	no	—
12	C₃×C₉×D₈	C₂²	1	2	yes	—
13	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
14	C₃×C₉×C₁₆	C₂²	1	2	yes	complemented
15	C₂×C₃²×Q₁₆	C₂×C₃	1	1	yes	characteristic
16	C₂×C₃²×D₈	C₂×C₃	1	1	yes	characteristic
17	C₃²×SD₃₂	C₂×C₃	1	4	yes	—
18	C₂×C₃²×C₁₆	C₂×C₃	1	1	yes	characteristic
19	C₂×C₉×Q₁₆	C₂×C₃	1	3	yes	complemented
20	C₂×C₉×D₈	C₂×C₃	1	3	yes	—
21	C₉×SD₃₂	C₂×C₃	1	12	yes	direct product factor
22	C₂×C₉×C₁₆	C₂×C₃	1	3	yes	complemented
23	C₂³×C₃×C₉	—	4	1	no	—
24	C₃×Q₈×C₉	—	2	2	no	—
25	C₃×Q₈×C₉	—	4	1	no	—
26	C₃×D₄×C₉	—	2	2	no	—
27	C₃×D₄×C₉	—	4	1	no	—
28	C₂×C₃×C₄×C₉	—	4	1	no	—
29	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
30	C₃×C₈×C₉	D₄	1	1	yes	characteristic
31	C₃×C₈×C₉	C₂³	1	1	yes	characteristic
32	C₂×C₃×SD₃₂	C₉	1	3	yes	direct product factor
33	C₂×C₃×SD₃₂	C₃²	1	1	yes	characteristic
34	C₂×C₃²×Q₈	—	2	1	no	—
35	C₂×C₃²×D₄	—	2	1	no	—
36	C₃²×Q₁₆	—	2	1	no	—
37	C₃²×Q₁₆	C₂²×C₃	1	2	yes	—
38	C₃²×D₈	—	2	1	no	—
39	C₃²×D₈	C₂²×C₃	1	2	yes	—
40	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
41	C₂×Q₈×C₉	—	2	3	no	—
42	C₂×D₄×C₉	—	2	3	no	—
43	C₃²×C₁₆	C₂²×C₃	1	2	yes	—
44	C₉×Q₁₆	—	2	3	no	—
45	C₉×Q₁₆	C₂²×C₃	1	6	yes	complemented
46	C₉×D₈	—	2	3	no	—
47	C₉×D₈	C₂²×C₃	1	6	yes	—
48	C₂×C₈×C₉	C₂²×C₃	1	3	yes	—
49	C₉×C₁₆	C₂²×C₃	1	6	yes	complemented
50	C₂²×C₃×C₉	—	4	2	no	—
51	C₂²×C₃×C₉	—	8	2	no	—
52	C₂²×C₃×C₉	D₈	1	1	yes	characteristic
53	C₃×C₄×C₉	—	4	2	no	—
54	C₃×C₄×C₉	D₈	1	1	yes	characteristic
55	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
56	C₂×C₃×Q₁₆	C₂×C₉	1	3	yes	complemented
57	C₂×C₃×Q₁₆	C₂×C₃²	1	1	yes	characteristic
58	C₂×C₃×D₈	C₂×C₉	1	3	yes	—
59	C₂×C₃×D₈	C₂×C₃²	1	1	yes	characteristic
60	C₃×SD₃₂	C₂×C₉	1	12	yes	direct product factor
61	C₃×SD₃₂	C₂×C₃²	1	4	yes	—
62	C₂×C₃×C₁₆	C₂×C₉	1	3	yes	complemented
63	C₂×C₃×C₁₆	C₂×C₃²	1	1	yes	characteristic
64	C₂³×C₃²	—	4	1	no	—
65	C₃²×Q₈	—	2	2	no	—
66	C₃²×Q₈	—	4	1	no	—
67	C₃²×D₄	—	2	2	no	—
68	C₃²×D₄	—	4	1	no	—
69	C₂×C₃²×C₄	—	4	1	no	—
70	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
71	C₂³×C₉	—	4	3	no	—
72	C₃²×C₈	C₃×D₄	1	1	yes	characteristic
73	C₃²×C₈	C₂³×C₃	1	1	yes	characteristic
74	Q₈×C₉	—	2	6	no	—
75	Q₈×C₉	—	4	3	no	—
76	D₄×C₉	—	2	6	no	—
77	D₄×C₉	—	4	3	no	—
78	C₂×C₄×C₉	—	4	3	no	—
79	C₂×C₄×C₉	C₃×D₄	1	3	yes	—
80	C₈×C₉	C₃×D₄	1	3	yes	—
81	C₈×C₉	C₂³×C₃	1	3	yes	—
82	C₂×SD₃₂	C₃×C₉	1	1	yes	characteristic, direct product factor, Sylow
83	C₂×C₃×C₉	—	8	2	no	—
84	C₂×C₃×C₉	SD₃₂	1	2	yes	direct product factor
85	C₂×C₃×C₉	C₂×D₈	1	1	yes	characteristic
86	C₂×C₃×Q₈	—	2	3	no	—
87	C₂×C₃×Q₈	—	2	1	no	—
88	C₂×C₃×D₄	—	2	3	no	—
89	C₂×C₃×D₄	—	2	1	no	—
90	C₃×Q₁₆	—	2	3	no	—
91	C₃×Q₁₆	—	2	1	no	—
92	C₃×Q₁₆	C₂²×C₉	1	6	yes	complemented
93	C₃×Q₁₆	C₂²×C₃²	1	2	yes	—
94	C₃×D₈	—	2	3	no	—
95	C₃×D₈	—	2	1	no	—
96	C₃×D₈	C₂²×C₉	1	6	yes	—
97	C₃×D₈	C₂²×C₃²	1	2	yes	—
98	C₂×C₃×C₈	C₂²×C₉	1	3	yes	—
99	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
100	C₃×C₁₆	C₂²×C₉	1	6	yes	complemented
101	C₃×C₁₆	C₂²×C₃²	1	2	yes	—
102	C₂²×C₃²	—	4	2	no	—
103	C₂²×C₃²	—	8	2	no	—
104	C₂²×C₃²	C₃×D₈	1	1	yes	characteristic
105	C₃²×C₄	—	4	2	no	—
106	C₃²×C₄	C₃×D₈	1	1	yes	characteristic
107	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
108	C₂²×C₉	—	4	6	no	—
109	C₂²×C₉	—	8	6	no	—
110	C₂²×C₉	C₃×D₈	1	3	yes	—
111	C₄×C₉	—	4	6	no	—
112	C₄×C₉	C₃×D₈	1	3	yes	—
113	C₄×C₉	C₂×C₃×D₄	1	3	yes	—
114	C₂×Q₁₆	C₂×C₃×C₉	1	1	yes	characteristic, complemented
115	C₂×D₈	C₂×C₃×C₉	1	1	yes	characteristic
116	SD₃₂	C₂×C₃×C₉	1	4	yes	direct product factor
117	C₂×C₁₆	C₂×C₃×C₉	1	1	yes	characteristic, complemented
118	C₃×C₉	C₂×SD₃₂	1	1	yes	characteristic, direct product factor, Sylow
119	C₂³×C₃	—	4	3	no	—
120	C₂³×C₃	—	4	1	no	—
121	C₃×Q₈	—	2	6	no	—
122	C₃×Q₈	—	2	2	no	—
123	C₃×Q₈	—	4	3	no	—
124	C₃×Q₈	—	4	1	no	—
125	C₃×D₄	—	2	6	no	—
126	C₃×D₄	—	2	2	no	—
127	C₃×D₄	—	4	3	no	—
128	C₃×D₄	—	4	1	no	—
129	C₂×C₃×C₄	—	4	3	no	—
130	C₂×C₃×C₄	—	4	1	no	—
131	C₂×C₃×C₄	D₄×C₉	1	3	yes	—
132	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
133	C₃×C₈	D₄×C₉	1	3	yes	—
134	C₃×C₈	C₂³×C₉	1	3	yes	—
135	C₃×C₈	C₃²×D₄	1	1	yes	characteristic
136	C₃×C₈	C₂³×C₃²	1	1	yes	characteristic
137	C₂×C₃²	—	8	2	no	—
138	C₂×C₃²	C₃×SD₃₂	1	2	yes	—
139	C₂×C₃²	C₂×C₃×D₈	1	1	yes	characteristic
140	C₂×C₉	—	8	6	no	—
141	C₂×C₉	C₃×SD₃₂	1	6	yes	direct product factor
142	C₂×C₉	C₂×C₃×D₈	1	3	yes	—
143	C₂×Q₈	—	2	1	no	—
144	C₂×D₄	—	2	1	no	—
145	Q₁₆	—	2	1	no	—
146	Q₁₆	C₂²×C₃×C₉	1	2	yes	complemented
147	D₈	—	2	1	no	—
148	D₈	C₂²×C₃×C₉	1	2	yes	—
149	C₂×C₈	C₂²×C₃×C₉	1	1	yes	characteristic
150	C₁₆	C₂²×C₃×C₉	1	2	yes	complemented
151	C₂²×C₃	—	4	6	no	—
152	C₂²×C₃	—	4	2	no	—
153	C₂²×C₃	—	8	6	no	—
154	C₂²×C₃	—	8	2	no	—
155	C₂²×C₃	C₉×D₈	1	3	yes	—
156	C₂²×C₃	C₃²×D₈	1	1	yes	characteristic
157	C₃×C₄	—	4	6	no	—
158	C₃×C₄	—	4	2	no	—
159	C₃×C₄	C₉×D₈	1	3	yes	—
160	C₃×C₄	C₂×D₄×C₉	1	3	yes	—
161	C₃×C₄	C₃²×D₈	1	1	yes	characteristic
162	C₃×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
163	C₃²	C₂×C₃×SD₃₂	1	1	yes	characteristic
164	C₉	C₂×C₃×SD₃₂	1	3	yes	direct product factor
165	C₂³	—	4	1	no	—
166	Q₈	—	2	2	no	—
167	Q₈	—	4	1	no	—
168	D₄	—	2	2	no	—
169	D₄	—	4	1	no	—
170	C₂×C₄	—	4	1	no	—
171	C₂×C₄	C₃×D₄×C₉	1	1	yes	characteristic
172	C₈	C₃×D₄×C₉	1	1	yes	characteristic
173	C₈	C₂³×C₃×C₉	1	1	yes	characteristic
174	C₂×C₃	—	8	2	no	—
175	C₂×C₃	—	8	6	no	—
176	C₂×C₃	C₉×SD₃₂	1	6	yes	direct product factor
177	C₂×C₃	C₂×C₉×D₈	1	3	yes	—
178	C₂×C₃	C₃²×SD₃₂	1	2	yes	—
179	C₂×C₃	C₂×C₃²×D₈	1	1	yes	characteristic
180	C₂²	—	4	2	no	—
181	C₂²	—	8	2	no	—
182	C₂²	C₃×C₉×D₈	1	1	yes	characteristic
183	C₄	—	4	2	no	—
184	C₄	C₃×C₉×D₈	1	1	yes	characteristic
185	C₄	C₂×C₃×D₄×C₉	1	1	yes	characteristic
186	C₃	C₂×C₉×SD₃₂	1	3	yes	direct product factor
187	C₃	C₂×C₃²×SD₃₂	1	1	yes	characteristic
188	C₂	—	8	2	no	—
189	C₂	C₃×C₉×SD₃₂	1	2	yes	direct product factor
190	C₂	C₂×C₃×C₉×D₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃×C₉	D₈
commutator	C₈	C₂³×C₃×C₉
Frattini, cosocle	C₃×C₈	C₂³×C₃²
socle	C₂²×C₃²	C₃×D₈
prime radical	C₂×C₃²×D₈	C₂×C₃
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₉ ◁ C₂×C₃×C₄×C₉ ◁ C₂×C₃×C₈×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
3. C₃×C₉×SD₃₂	C₂	direct (N × H)
4. C₂×C₃×C₉×C₁₆	C₂	semidirect
6. C₂×C₉×SD₃₂	C₃	direct (N × H)
10. C₃×C₉×Q₁₆	C₂²	semidirect
14. C₃×C₉×C₁₆	C₂²	semidirect
19. C₂×C₉×Q₁₆	C₂×C₃	semidirect
21. C₉×SD₃₂	C₂×C₃	direct (N × H)
22. C₂×C₉×C₁₆	C₂×C₃	semidirect
32. C₂×C₃×SD₃₂	C₉	direct (N × H)
45. C₉×Q₁₆	C₂²×C₃	semidirect
49. C₉×C₁₆	C₂²×C₃	semidirect
56. C₂×C₃×Q₁₆	C₂×C₉	semidirect
60. C₃×SD₃₂	C₂×C₉	direct (N × H)
62. C₂×C₃×C₁₆	C₂×C₉	semidirect
82. C₂×SD₃₂	C₃×C₉	direct (N × H)
84. C₂×C₃×C₉	SD₃₂	direct (N × H)
92. C₃×Q₁₆	C₂²×C₉	semidirect
100. C₃×C₁₆	C₂²×C₉	semidirect
114. C₂×Q₁₆	C₂×C₃×C₉	semidirect
116. SD₃₂	C₂×C₃×C₉	direct (N × H)
117. C₂×C₁₆	C₂×C₃×C₉	semidirect
118. C₃×C₉	C₂×SD₃₂	direct (N × H)
141. C₂×C₉	C₃×SD₃₂	direct (N × H)
146. Q₁₆	C₂²×C₃×C₉	semidirect
150. C₁₆	C₂²×C₃×C₉	semidirect
164. C₉	C₂×C₃×SD₃₂	direct (N × H)
176. C₂×C₃	C₉×SD₃₂	direct (N × H)
186. C₃	C₂×C₉×SD₃₂	direct (N × H)
189. C₂	C₃×C₉×SD₃₂	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
C₂×C₃×C₉×D₈	2
C₃×C₉×SD₃₂	2
C₂×C₃×C₉×C₁₆	2
C₂×C₃²×SD₃₂	3
C₂×C₉×SD₃₂	3

Subgroups — C2×C3×C9×SD32 #13145276

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₉×SD₃₂ #13145276