Subgroups — C₃×C₁₆×C₅₁₂ #72201819

Order 24576.

476 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×C₁₆×C₂₅₆	C₂	1	1	yes	maximal, characteristic
2	C₃×C₈×C₅₁₂	C₂	1	2	yes	maximal
3	C₁₆×C₅₁₂	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
4	C₃×C₁₆×C₁₂₈	C₄	1	1	yes	characteristic
5	C₃×C₈×C₂₅₆	C₄	1	1	yes	characteristic
6	C₃×C₈×C₂₅₆	C₂²	1	1	yes	characteristic
7	C₃×C₄×C₅₁₂	C₄	1	4	yes	—
8	C₁₆×C₂₅₆	C₂×C₃	1	1	yes	characteristic
9	C₈×C₅₁₂	C₂×C₃	1	2	yes	—
10	C₃×C₁₆×C₆₄	C₈	1	1	yes	characteristic
11	C₃×C₈×C₁₂₈	C₈	1	1	yes	characteristic
12	C₃×C₈×C₁₂₈	C₂×C₄	1	1	yes	characteristic
13	C₃×C₄×C₂₅₆	C₈	1	2	yes	—
14	C₃×C₄×C₂₅₆	C₂×C₄	1	2	yes	—
15	C₂×C₃×C₅₁₂	C₈	1	8	yes	—
16	C₁₆×C₁₂₈	C₃×C₄	1	1	yes	characteristic
17	C₈×C₂₅₆	C₃×C₄	1	1	yes	characteristic
18	C₈×C₂₅₆	C₂²×C₃	1	1	yes	characteristic
19	C₄×C₅₁₂	C₃×C₄	1	4	yes	—
20	C₃×C₁₆×C₃₂	C₁₆	1	1	yes	characteristic
21	C₃×C₈×C₆₄	C₁₆	1	1	yes	characteristic
22	C₃×C₈×C₆₄	C₂×C₈	1	1	yes	characteristic
23	C₃×C₄×C₁₂₈	C₁₆	1	2	yes	—
24	C₃×C₄×C₁₂₈	C₄²	1	1	yes	characteristic
25	C₃×C₄×C₁₂₈	C₂×C₈	1	1	yes	characteristic
26	C₂×C₃×C₂₅₆	C₁₆	1	4	yes	—
27	C₂×C₃×C₂₅₆	C₂×C₈	1	4	yes	—
28	C₃×C₅₁₂	C₁₆	1	16	yes	direct product factor
29	C₁₆×C₆₄	C₃×C₈	1	1	yes	characteristic
30	C₈×C₁₂₈	C₃×C₈	1	1	yes	characteristic
31	C₈×C₁₂₈	C₂×C₃×C₄	1	1	yes	characteristic
32	C₄×C₂₅₆	C₃×C₈	1	2	yes	—
33	C₄×C₂₅₆	C₂×C₃×C₄	1	2	yes	—
34	C₂×C₅₁₂	C₃×C₈	1	8	yes	—
35	C₂×C₃×C₁₂₈	C₃₂	1	4	yes	—
36	C₂×C₃×C₁₂₈	C₄×C₈	1	2	yes	—
37	C₂×C₃×C₁₂₈	C₂×C₁₆	1	2	yes	—
38	C₃×C₄×C₆₄	C₃₂	1	2	yes	—
39	C₃×C₄×C₆₄	C₄×C₈	1	1	yes	characteristic
40	C₃×C₄×C₆₄	C₂×C₁₆	1	1	yes	characteristic
41	C₃×C₈×C₃₂	C₃₂	1	1	yes	characteristic
42	C₃×C₈×C₃₂	C₂×C₁₆	1	1	yes	characteristic
43	C₃×C₁₆²	C₃₂	1	1	yes	characteristic
44	C₃×C₂₅₆	C₃₂	1	8	yes	—
45	C₃×C₂₅₆	C₂×C₁₆	1	8	yes	—
46	C₂×C₂₅₆	C₃×C₁₆	1	4	yes	—
47	C₂×C₂₅₆	C₂×C₃×C₈	1	4	yes	—
48	C₄×C₁₂₈	C₃×C₁₆	1	2	yes	—
49	C₄×C₁₂₈	C₃×C₄²	1	1	yes	characteristic
50	C₄×C₁₂₈	C₂×C₃×C₈	1	1	yes	characteristic
51	C₈×C₆₄	C₃×C₁₆	1	1	yes	characteristic
52	C₈×C₆₄	C₂×C₃×C₈	1	1	yes	characteristic
53	C₁₆×C₃₂	C₃×C₁₆	1	1	yes	characteristic
54	C₅₁₂	C₃×C₁₆	1	16	yes	direct product factor
55	C₂×C₃×C₆₄	C₆₄	1	4	yes	—
56	C₂×C₃×C₆₄	C₈²	1	1	yes	characteristic
57	C₂×C₃×C₆₄	C₄×C₁₆	1	1	yes	characteristic
58	C₂×C₃×C₆₄	C₂×C₃₂	1	2	yes	—
59	C₃×C₄×C₃₂	C₆₄	1	2	yes	—
60	C₃×C₄×C₃₂	C₄×C₁₆	1	1	yes	characteristic
61	C₃×C₄×C₃₂	C₂×C₃₂	1	1	yes	characteristic
62	C₃×C₈×C₁₆	C₆₄	1	2	yes	—
63	C₃×C₈×C₁₆	C₂×C₃₂	1	1	yes	characteristic
64	C₃×C₁₂₈	C₆₄	1	8	yes	—
65	C₃×C₁₂₈	C₄×C₁₆	1	4	yes	—
66	C₃×C₁₂₈	C₂×C₃₂	1	4	yes	—
67	C₂×C₁₂₈	C₃×C₃₂	1	4	yes	—
68	C₂×C₁₂₈	C₃×C₄×C₈	1	2	yes	—
69	C₂×C₁₂₈	C₂×C₃×C₁₆	1	2	yes	—
70	C₄×C₆₄	C₃×C₃₂	1	2	yes	—
71	C₄×C₆₄	C₃×C₄×C₈	1	1	yes	characteristic
72	C₄×C₆₄	C₂×C₃×C₁₆	1	1	yes	characteristic
73	C₈×C₃₂	C₃×C₃₂	1	1	yes	characteristic
74	C₈×C₃₂	C₂×C₃×C₁₆	1	1	yes	characteristic
75	C₁₆²	C₃×C₃₂	1	1	yes	characteristic
76	C₂₅₆	C₃×C₃₂	1	8	yes	—
77	C₂₅₆	C₂×C₃×C₁₆	1	8	yes	—
78	C₂×C₃×C₃₂	C₁₂₈	1	4	yes	—
79	C₂×C₃×C₃₂	C₈×C₁₆	1	1	yes	characteristic
80	C₂×C₃×C₃₂	C₄×C₃₂	1	1	yes	characteristic
81	C₂×C₃×C₃₂	C₂×C₆₄	1	2	yes	—
82	C₃×C₄×C₁₆	C₁₂₈	1	4	yes	—
83	C₃×C₄×C₁₆	C₄×C₃₂	1	1	yes	characteristic
84	C₃×C₄×C₁₆	C₂×C₆₄	1	1	yes	characteristic
85	C₃×C₈²	C₂×C₆₄	1	1	yes	characteristic
86	C₃×C₆₄	C₁₂₈	1	8	yes	—
87	C₃×C₆₄	C₈×C₁₆	1	2	yes	—
88	C₃×C₆₄	C₄×C₃₂	1	2	yes	—
89	C₃×C₆₄	C₂×C₆₄	1	4	yes	—
90	C₂×C₆₄	C₃×C₆₄	1	4	yes	—
91	C₂×C₆₄	C₃×C₈²	1	1	yes	characteristic
92	C₂×C₆₄	C₃×C₄×C₁₆	1	1	yes	characteristic
93	C₂×C₆₄	C₂×C₃×C₃₂	1	2	yes	—
94	C₄×C₃₂	C₃×C₆₄	1	2	yes	—
95	C₄×C₃₂	C₃×C₄×C₁₆	1	1	yes	characteristic
96	C₄×C₃₂	C₂×C₃×C₃₂	1	1	yes	characteristic
97	C₈×C₁₆	C₃×C₆₄	1	2	yes	—
98	C₈×C₁₆	C₂×C₃×C₃₂	1	1	yes	characteristic
99	C₁₂₈	C₃×C₆₄	1	8	yes	—
100	C₁₂₈	C₃×C₄×C₁₆	1	4	yes	—
101	C₁₂₈	C₂×C₃×C₃₂	1	4	yes	—
102	C₂×C₃×C₁₆	C₂₅₆	1	8	yes	—
103	C₂×C₃×C₁₆	C₈×C₃₂	1	1	yes	characteristic
104	C₂×C₃×C₁₆	C₄×C₆₄	1	1	yes	characteristic
105	C₂×C₃×C₁₆	C₂×C₁₂₈	1	2	yes	—
106	C₃×C₄×C₈	C₄×C₆₄	1	1	yes	characteristic
107	C₃×C₄×C₈	C₂×C₁₂₈	1	2	yes	—
108	C₃×C₃₂	C₂₅₆	1	8	yes	—
109	C₃×C₃₂	C₁₆²	1	1	yes	characteristic
110	C₃×C₃₂	C₈×C₃₂	1	1	yes	characteristic
111	C₃×C₃₂	C₄×C₆₄	1	2	yes	—
112	C₃×C₃₂	C₂×C₁₂₈	1	4	yes	—
113	C₂×C₃₂	C₃×C₁₂₈	1	4	yes	—
114	C₂×C₃₂	C₃×C₈×C₁₆	1	1	yes	characteristic
115	C₂×C₃₂	C₃×C₄×C₃₂	1	1	yes	characteristic
116	C₂×C₃₂	C₂×C₃×C₆₄	1	2	yes	—
117	C₄×C₁₆	C₃×C₁₂₈	1	4	yes	—
118	C₄×C₁₆	C₃×C₄×C₃₂	1	1	yes	characteristic
119	C₄×C₁₆	C₂×C₃×C₆₄	1	1	yes	characteristic
120	C₈²	C₂×C₃×C₆₄	1	1	yes	characteristic
121	C₆₄	C₃×C₁₂₈	1	8	yes	—
122	C₆₄	C₃×C₈×C₁₆	1	2	yes	—
123	C₆₄	C₃×C₄×C₃₂	1	2	yes	—
124	C₆₄	C₂×C₃×C₆₄	1	4	yes	—
125	C₂×C₃×C₈	C₈×C₆₄	1	1	yes	characteristic
126	C₂×C₃×C₈	C₄×C₁₂₈	1	1	yes	characteristic
127	C₂×C₃×C₈	C₂×C₂₅₆	1	4	yes	—
128	C₃×C₄²	C₄×C₁₂₈	1	1	yes	characteristic
129	C₃×C₁₆	C₅₁₂	1	16	yes	direct product factor
130	C₃×C₁₆	C₁₆×C₃₂	1	1	yes	characteristic
131	C₃×C₁₆	C₈×C₆₄	1	1	yes	characteristic
132	C₃×C₁₆	C₄×C₁₂₈	1	2	yes	—
133	C₃×C₁₆	C₂×C₂₅₆	1	4	yes	—
134	C₂×C₁₆	C₃×C₂₅₆	1	8	yes	—
135	C₂×C₁₆	C₃×C₈×C₃₂	1	1	yes	characteristic
136	C₂×C₁₆	C₃×C₄×C₆₄	1	1	yes	characteristic
137	C₂×C₁₆	C₂×C₃×C₁₂₈	1	2	yes	—
138	C₄×C₈	C₃×C₄×C₆₄	1	1	yes	characteristic
139	C₄×C₈	C₂×C₃×C₁₂₈	1	2	yes	—
140	C₃₂	C₃×C₂₅₆	1	8	yes	—
141	C₃₂	C₃×C₁₆²	1	1	yes	characteristic
142	C₃₂	C₃×C₈×C₃₂	1	1	yes	characteristic
143	C₃₂	C₃×C₄×C₆₄	1	2	yes	—
144	C₃₂	C₂×C₃×C₁₂₈	1	4	yes	—
145	C₂×C₃×C₄	C₄×C₂₅₆	1	2	yes	—
146	C₂×C₃×C₄	C₈×C₁₂₈	1	1	yes	characteristic
147	C₃×C₈	C₂×C₅₁₂	1	8	yes	—
148	C₃×C₈	C₄×C₂₅₆	1	2	yes	—
149	C₃×C₈	C₈×C₁₂₈	1	1	yes	characteristic
150	C₃×C₈	C₁₆×C₆₄	1	1	yes	characteristic
151	C₂×C₈	C₂×C₃×C₂₅₆	1	4	yes	—
152	C₂×C₈	C₃×C₄×C₁₂₈	1	1	yes	characteristic
153	C₂×C₈	C₃×C₈×C₆₄	1	1	yes	characteristic
154	C₄²	C₃×C₄×C₁₂₈	1	1	yes	characteristic
155	C₁₆	C₃×C₅₁₂	1	16	yes	direct product factor
156	C₁₆	C₂×C₃×C₂₅₆	1	4	yes	—
157	C₁₆	C₃×C₄×C₁₂₈	1	2	yes	—
158	C₁₆	C₃×C₈×C₆₄	1	1	yes	characteristic
159	C₁₆	C₃×C₁₆×C₃₂	1	1	yes	characteristic
160	C₂²×C₃	C₈×C₂₅₆	1	1	yes	characteristic
161	C₃×C₄	C₄×C₅₁₂	1	4	yes	—
162	C₃×C₄	C₈×C₂₅₆	1	1	yes	characteristic
163	C₃×C₄	C₁₆×C₁₂₈	1	1	yes	characteristic
164	C₂×C₄	C₃×C₄×C₂₅₆	1	2	yes	—
165	C₂×C₄	C₃×C₈×C₁₂₈	1	1	yes	characteristic
166	C₈	C₂×C₃×C₅₁₂	1	8	yes	—
167	C₈	C₃×C₄×C₂₅₆	1	2	yes	—
168	C₈	C₃×C₈×C₁₂₈	1	1	yes	characteristic
169	C₈	C₃×C₁₆×C₆₄	1	1	yes	characteristic
170	C₂×C₃	C₈×C₅₁₂	1	2	yes	—
171	C₂×C₃	C₁₆×C₂₅₆	1	1	yes	characteristic
172	C₂²	C₃×C₈×C₂₅₆	1	1	yes	characteristic
173	C₄	C₃×C₄×C₅₁₂	1	4	yes	—
174	C₄	C₃×C₈×C₂₅₆	1	1	yes	characteristic
175	C₄	C₃×C₁₆×C₁₂₈	1	1	yes	characteristic
176	C₃	C₁₆×C₅₁₂	1	1	yes	characteristic, direct product factor, Sylow
177	C₂	C₃×C₈×C₅₁₂	1	2	yes	—
178	C₂	C₃×C₁₆×C₂₅₆	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₈×C₂₅₆	C₂²×C₃
socle, prime radical	C₂²×C₃	C₈×C₂₅₆
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₆×C₅₁₂	C₃	direct (N × H)
28. C₃×C₅₁₂	C₁₆	direct (N × H)
54. C₅₁₂	C₃×C₁₆	direct (N × H)
129. C₃×C₁₆	C₅₁₂	direct (N × H)
155. C₁₆	C₃×C₅₁₂	direct (N × H)
176. C₃	C₁₆×C₅₁₂	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×C₁₆×C₂₅₆	2
C₃×C₈×C₅₁₂	2
C₁₆×C₅₁₂	3

Subgroups — C3×C16×C512 #72201819

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₁₆×C₅₁₂ #72201819