Subgroups — C₃×C₉×C₁₆×C₃₂ #72192004

Order 13824.

1140 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₉×C₁₆²	C₂	1	1	yes	maximal, characteristic
2	C₃×C₈×C₉×C₃₂	C₂	1	2	yes	maximal
3	C₃²×C₁₆×C₃₂	C₃	1	1	yes	maximal, characteristic
4	C₉×C₁₆×C₃₂	C₃	1	3	yes	maximal, direct product factor
5	C₃×C₈×C₉×C₁₆	C₄	1	2	yes	—
6	C₃×C₈×C₉×C₁₆	C₂²	1	1	yes	characteristic
7	C₃×C₄×C₉×C₃₂	C₄	1	4	yes	—
8	C₃²×C₁₆²	C₂×C₃	1	1	yes	characteristic
9	C₉×C₁₆²	C₂×C₃	1	3	yes	—
10	C₃²×C₈×C₃₂	C₂×C₃	1	2	yes	—
11	C₈×C₉×C₃₂	C₂×C₃	1	6	yes	—
12	C₃×C₁₆×C₃₂	C₉	1	3	yes	direct product factor
13	C₃×C₁₆×C₃₂	C₃²	1	1	yes	characteristic
14	C₂×C₃×C₉×C₃₂	C₈	1	8	yes	—
15	C₃×C₄×C₉×C₁₆	C₈	1	4	yes	—
16	C₃×C₄×C₉×C₁₆	C₂×C₄	1	2	yes	—
17	C₃×C₈²×C₉	C₂×C₄	1	1	yes	characteristic
18	C₃²×C₄×C₃₂	C₃×C₄	1	4	yes	—
19	C₃²×C₈×C₁₆	C₃×C₄	1	2	yes	—
20	C₃²×C₈×C₁₆	C₂²×C₃	1	1	yes	characteristic
21	C₄×C₉×C₃₂	C₃×C₄	1	12	yes	—
22	C₈×C₉×C₁₆	C₃×C₄	1	6	yes	—
23	C₈×C₉×C₁₆	C₂²×C₃	1	3	yes	—
24	C₂×C₃×C₉×C₁₆	C₁₆	1	8	yes	—
25	C₂×C₃×C₉×C₁₆	C₂×C₈	1	4	yes	—
26	C₃×C₄×C₈×C₉	C₄²	1	1	yes	characteristic
27	C₃×C₄×C₈×C₉	C₂×C₈	1	2	yes	—
28	C₃×C₉×C₃₂	C₁₆	1	16	yes	direct product factor
29	C₃×C₈×C₃₂	C₂×C₉	1	6	yes	—
30	C₃×C₈×C₃₂	C₂×C₃²	1	2	yes	—
31	C₃×C₁₆²	C₂×C₉	1	3	yes	—
32	C₃×C₁₆²	C₂×C₃²	1	1	yes	characteristic
33	C₂×C₃²×C₃₂	C₃×C₈	1	8	yes	—
34	C₃²×C₄×C₁₆	C₃×C₈	1	4	yes	—
35	C₃²×C₄×C₁₆	C₂×C₃×C₄	1	2	yes	—
36	C₃²×C₈²	C₂×C₃×C₄	1	1	yes	characteristic
37	C₂×C₉×C₃₂	C₃×C₈	1	24	yes	—
38	C₄×C₉×C₁₆	C₃×C₈	1	12	yes	—
39	C₄×C₉×C₁₆	C₂×C₃×C₄	1	6	yes	—
40	C₈²×C₉	C₂×C₃×C₄	1	3	yes	—
41	C₁₆×C₃₂	C₃×C₉	1	1	yes	characteristic, direct product factor, Sylow
42	C₂×C₃×C₈×C₉	C₄×C₈	1	2	yes	—
43	C₂×C₃×C₈×C₉	C₂×C₁₆	1	4	yes	—
44	C₃×C₄²×C₉	C₄×C₈	1	1	yes	characteristic
45	C₃×C₉×C₁₆	C₃₂	1	16	yes	direct product factor
46	C₃×C₉×C₁₆	C₂×C₁₆	1	8	yes	—
47	C₃×C₄×C₃₂	C₄×C₉	1	12	yes	—
48	C₃×C₄×C₃₂	C₃²×C₄	1	4	yes	—
49	C₃×C₈×C₁₆	C₄×C₉	1	6	yes	—
50	C₃×C₈×C₁₆	C₂²×C₉	1	3	yes	—
51	C₃×C₈×C₁₆	C₃²×C₄	1	2	yes	—
52	C₃×C₈×C₁₆	C₂²×C₃²	1	1	yes	characteristic
53	C₂×C₃²×C₁₆	C₃×C₁₆	1	8	yes	—
54	C₂×C₃²×C₁₆	C₂×C₃×C₈	1	4	yes	—
55	C₃²×C₄×C₈	C₃×C₄²	1	1	yes	characteristic
56	C₃²×C₄×C₈	C₂×C₃×C₈	1	2	yes	—
57	C₃²×C₃₂	C₃×C₁₆	1	16	yes	—
58	C₂×C₉×C₁₆	C₃×C₁₆	1	24	yes	—
59	C₂×C₉×C₁₆	C₂×C₃×C₈	1	12	yes	—
60	C₄×C₈×C₉	C₃×C₄²	1	3	yes	—
61	C₄×C₈×C₉	C₂×C₃×C₈	1	6	yes	—
62	C₉×C₃₂	C₃×C₁₆	1	48	yes	direct product factor
63	C₈×C₃₂	C₂×C₃×C₉	1	2	yes	—
64	C₁₆²	C₂×C₃×C₉	1	1	yes	characteristic
65	C₂×C₃×C₄×C₉	C₈²	1	1	yes	characteristic
66	C₂×C₃×C₄×C₉	C₄×C₁₆	1	2	yes	—
67	C₃×C₈×C₉	C₄×C₁₆	1	4	yes	—
68	C₃×C₈×C₉	C₂×C₃₂	1	8	yes	—
69	C₂×C₃×C₃₂	C₈×C₉	1	24	yes	—
70	C₂×C₃×C₃₂	C₃²×C₈	1	8	yes	—
71	C₃×C₄×C₁₆	C₈×C₉	1	12	yes	—
72	C₃×C₄×C₁₆	C₂×C₄×C₉	1	6	yes	—
73	C₃×C₄×C₁₆	C₃²×C₈	1	4	yes	—
74	C₃×C₄×C₁₆	C₂×C₃²×C₄	1	2	yes	—
75	C₃×C₈²	C₂×C₄×C₉	1	3	yes	—
76	C₃×C₈²	C₂×C₃²×C₄	1	1	yes	characteristic
77	C₂×C₃²×C₈	C₃×C₄×C₈	1	2	yes	—
78	C₂×C₃²×C₈	C₂×C₃×C₁₆	1	4	yes	—
79	C₃²×C₄²	C₃×C₄×C₈	1	1	yes	characteristic
80	C₃²×C₁₆	C₃×C₃₂	1	16	yes	—
81	C₃²×C₁₆	C₂×C₃×C₁₆	1	8	yes	—
82	C₂×C₈×C₉	C₃×C₄×C₈	1	6	yes	—
83	C₂×C₈×C₉	C₂×C₃×C₁₆	1	12	yes	—
84	C₄²×C₉	C₃×C₄×C₈	1	3	yes	—
85	C₉×C₁₆	C₃×C₃₂	1	48	yes	direct product factor
86	C₉×C₁₆	C₂×C₃×C₁₆	1	24	yes	—
87	C₄×C₃₂	C₃×C₄×C₉	1	4	yes	—
88	C₈×C₁₆	C₃×C₄×C₉	1	2	yes	—
89	C₈×C₁₆	C₂²×C₃×C₉	1	1	yes	characteristic
90	C₂²×C₃×C₉	C₈×C₁₆	1	1	yes	characteristic
91	C₃×C₄×C₉	C₈×C₁₆	1	2	yes	—
92	C₃×C₄×C₉	C₄×C₃₂	1	4	yes	—
93	C₂×C₃×C₁₆	C₉×C₁₆	1	24	yes	—
94	C₂×C₃×C₁₆	C₂×C₈×C₉	1	12	yes	—
95	C₂×C₃×C₁₆	C₃²×C₁₆	1	8	yes	—
96	C₂×C₃×C₁₆	C₂×C₃²×C₈	1	4	yes	—
97	C₃×C₄×C₈	C₄²×C₉	1	3	yes	—
98	C₃×C₄×C₈	C₂×C₈×C₉	1	6	yes	—
99	C₃×C₄×C₈	C₃²×C₄²	1	1	yes	characteristic
100	C₃×C₄×C₈	C₂×C₃²×C₈	1	2	yes	—
101	C₃×C₃₂	C₉×C₁₆	1	48	yes	direct product factor
102	C₃×C₃₂	C₃²×C₁₆	1	16	yes	—
103	C₂×C₃²×C₄	C₃×C₈²	1	1	yes	characteristic
104	C₂×C₃²×C₄	C₃×C₄×C₁₆	1	2	yes	—
105	C₃²×C₈	C₃×C₄×C₁₆	1	4	yes	—
106	C₃²×C₈	C₂×C₃×C₃₂	1	8	yes	—
107	C₂×C₄×C₉	C₃×C₈²	1	3	yes	—
108	C₂×C₄×C₉	C₃×C₄×C₁₆	1	6	yes	—
109	C₈×C₉	C₃×C₄×C₁₆	1	12	yes	—
110	C₈×C₉	C₂×C₃×C₃₂	1	24	yes	—
111	C₂×C₃₂	C₃×C₈×C₉	1	8	yes	—
112	C₄×C₁₆	C₃×C₈×C₉	1	4	yes	—
113	C₄×C₁₆	C₂×C₃×C₄×C₉	1	2	yes	—
114	C₈²	C₂×C₃×C₄×C₉	1	1	yes	characteristic
115	C₂×C₃×C₉	C₁₆²	1	1	yes	characteristic
116	C₂×C₃×C₉	C₈×C₃₂	1	2	yes	—
117	C₂×C₃×C₈	C₄×C₈×C₉	1	6	yes	—
118	C₂×C₃×C₈	C₂×C₉×C₁₆	1	12	yes	—
119	C₂×C₃×C₈	C₃²×C₄×C₈	1	2	yes	—
120	C₂×C₃×C₈	C₂×C₃²×C₁₆	1	4	yes	—
121	C₃×C₄²	C₄×C₈×C₉	1	3	yes	—
122	C₃×C₄²	C₃²×C₄×C₈	1	1	yes	characteristic
123	C₃×C₁₆	C₉×C₃₂	1	48	yes	direct product factor
124	C₃×C₁₆	C₂×C₉×C₁₆	1	24	yes	—
125	C₃×C₁₆	C₃²×C₃₂	1	16	yes	—
126	C₃×C₁₆	C₂×C₃²×C₁₆	1	8	yes	—
127	C₂²×C₃²	C₃×C₈×C₁₆	1	1	yes	characteristic
128	C₃²×C₄	C₃×C₈×C₁₆	1	2	yes	—
129	C₃²×C₄	C₃×C₄×C₃₂	1	4	yes	—
130	C₂²×C₉	C₃×C₈×C₁₆	1	3	yes	—
131	C₄×C₉	C₃×C₈×C₁₆	1	6	yes	—
132	C₄×C₉	C₃×C₄×C₃₂	1	12	yes	—
133	C₂×C₁₆	C₃×C₉×C₁₆	1	8	yes	—
134	C₂×C₁₆	C₂×C₃×C₈×C₉	1	4	yes	—
135	C₄×C₈	C₃×C₄²×C₉	1	1	yes	characteristic
136	C₄×C₈	C₂×C₃×C₈×C₉	1	2	yes	—
137	C₃₂	C₃×C₉×C₁₆	1	16	yes	direct product factor
138	C₃×C₉	C₁₆×C₃₂	1	1	yes	characteristic, direct product factor, Sylow
139	C₂×C₃×C₄	C₈²×C₉	1	3	yes	—
140	C₂×C₃×C₄	C₄×C₉×C₁₆	1	6	yes	—
141	C₂×C₃×C₄	C₃²×C₈²	1	1	yes	characteristic
142	C₂×C₃×C₄	C₃²×C₄×C₁₆	1	2	yes	—
143	C₃×C₈	C₄×C₉×C₁₆	1	12	yes	—
144	C₃×C₈	C₂×C₉×C₃₂	1	24	yes	—
145	C₃×C₈	C₃²×C₄×C₁₆	1	4	yes	—
146	C₃×C₈	C₂×C₃²×C₃₂	1	8	yes	—
147	C₂×C₃²	C₃×C₁₆²	1	1	yes	characteristic
148	C₂×C₃²	C₃×C₈×C₃₂	1	2	yes	—
149	C₂×C₉	C₃×C₁₆²	1	3	yes	—
150	C₂×C₉	C₃×C₈×C₃₂	1	6	yes	—
151	C₂×C₈	C₃×C₄×C₈×C₉	1	2	yes	—
152	C₂×C₈	C₂×C₃×C₉×C₁₆	1	4	yes	—
153	C₄²	C₃×C₄×C₈×C₉	1	1	yes	characteristic
154	C₁₆	C₃×C₉×C₃₂	1	16	yes	direct product factor
155	C₁₆	C₂×C₃×C₉×C₁₆	1	8	yes	—
156	C₂²×C₃	C₈×C₉×C₁₆	1	3	yes	—
157	C₂²×C₃	C₃²×C₈×C₁₆	1	1	yes	characteristic
158	C₃×C₄	C₈×C₉×C₁₆	1	6	yes	—
159	C₃×C₄	C₄×C₉×C₃₂	1	12	yes	—
160	C₃×C₄	C₃²×C₈×C₁₆	1	2	yes	—
161	C₃×C₄	C₃²×C₄×C₃₂	1	4	yes	—
162	C₃²	C₃×C₁₆×C₃₂	1	1	yes	characteristic
163	C₉	C₃×C₁₆×C₃₂	1	3	yes	direct product factor
164	C₂×C₄	C₃×C₈²×C₉	1	1	yes	characteristic
165	C₂×C₄	C₃×C₄×C₉×C₁₆	1	2	yes	—
166	C₈	C₃×C₄×C₉×C₁₆	1	4	yes	—
167	C₈	C₂×C₃×C₉×C₃₂	1	8	yes	—
168	C₂×C₃	C₈×C₉×C₃₂	1	6	yes	—
169	C₂×C₃	C₃²×C₈×C₃₂	1	2	yes	—
170	C₂×C₃	C₉×C₁₆²	1	3	yes	—
171	C₂×C₃	C₃²×C₁₆²	1	1	yes	characteristic
172	C₂²	C₃×C₈×C₉×C₁₆	1	1	yes	characteristic
173	C₄	C₃×C₄×C₉×C₃₂	1	4	yes	—
174	C₄	C₃×C₈×C₉×C₁₆	1	2	yes	—
175	C₃	C₉×C₁₆×C₃₂	1	3	yes	direct product factor
176	C₃	C₃²×C₁₆×C₃₂	1	1	yes	characteristic
177	C₂	C₃×C₈×C₉×C₃₂	1	2	yes	—
178	C₂	C₃×C₉×C₁₆²	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₃×C₈×C₁₆	C₂²×C₃²
socle, prime radical	C₂²×C₃²	C₃×C₈×C₁₆
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₉×C₁₆×C₃₂	C₃	direct (N × H)
12. C₃×C₁₆×C₃₂	C₉	direct (N × H)
28. C₃×C₉×C₃₂	C₁₆	direct (N × H)
41. C₁₆×C₃₂	C₃×C₉	direct (N × H)
45. C₃×C₉×C₁₆	C₃₂	direct (N × H)
62. C₉×C₃₂	C₃×C₁₆	direct (N × H)
85. C₉×C₁₆	C₃×C₃₂	direct (N × H)
101. C₃×C₃₂	C₉×C₁₆	direct (N × H)
123. C₃×C₁₆	C₉×C₃₂	direct (N × H)
137. C₃₂	C₃×C₉×C₁₆	direct (N × H)
138. C₃×C₉	C₁₆×C₃₂	direct (N × H)
154. C₁₆	C₃×C₉×C₃₂	direct (N × H)
163. C₉	C₃×C₁₆×C₃₂	direct (N × H)
175. C₃	C₉×C₁₆×C₃₂	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×C₁₆²	2
C₃×C₈×C₉×C₃₂	2
C₃²×C₁₆×C₃₂	3
C₉×C₁₆×C₃₂	3

Subgroups — C3×C9×C16×C32 #72192004

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₉×C₁₆×C₃₂ #72192004