Subgroups — C₃³×C₁₆×C₆₄ #72204635

Order 27648.

4060 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	C₃³×C₁₆×C₃₂	C₂	1	1	yes	maximal, characteristic
2	C₃³×C₈×C₆₄	C₂	1	2	yes	maximal
3	C₃²×C₁₆×C₆₄	C₃	1	13	yes	maximal, direct product factor
4	C₃³×C₁₆²	C₄	1	1	yes	characteristic
5	C₃³×C₈×C₃₂	C₄	1	1	yes	characteristic
6	C₃³×C₈×C₃₂	C₂²	1	1	yes	characteristic
7	C₃³×C₄×C₆₄	C₄	1	4	yes	—
8	C₃²×C₁₆×C₃₂	C₂×C₃	1	13	yes	—
9	C₃²×C₈×C₆₄	C₂×C₃	1	26	yes	—
10	C₃³×C₈×C₁₆	C₈	1	2	yes	—
11	C₃³×C₈×C₁₆	C₂×C₄	1	1	yes	characteristic
12	C₃³×C₄×C₃₂	C₈	1	2	yes	—
13	C₃³×C₄×C₃₂	C₂×C₄	1	2	yes	—
14	C₂×C₃³×C₆₄	C₈	1	8	yes	—
15	C₃×C₁₆×C₆₄	C₃²	1	13	yes	direct product factor
16	C₃²×C₁₆²	C₃×C₄	1	13	yes	—
17	C₃²×C₈×C₃₂	C₃×C₄	1	13	yes	—
18	C₃²×C₈×C₃₂	C₂²×C₃	1	13	yes	—
19	C₃²×C₄×C₆₄	C₃×C₄	1	52	yes	—
20	C₃×C₁₆×C₃₂	C₂×C₃²	1	13	yes	—
21	C₃×C₈×C₆₄	C₂×C₃²	1	26	yes	—
22	C₁₆×C₆₄	C₃³	1	1	yes	characteristic, direct product factor, Sylow
23	C₂×C₃³×C₃₂	C₁₆	1	4	yes	—
24	C₂×C₃³×C₃₂	C₂×C₈	1	4	yes	—
25	C₃³×C₄×C₁₆	C₁₆	1	4	yes	—
26	C₃³×C₄×C₁₆	C₄²	1	1	yes	characteristic
27	C₃³×C₄×C₁₆	C₂×C₈	1	1	yes	characteristic
28	C₃³×C₈²	C₂×C₈	1	1	yes	characteristic
29	C₃³×C₆₄	C₁₆	1	16	yes	direct product factor
30	C₂×C₃²×C₆₄	C₃×C₈	1	104	yes	—
31	C₃²×C₄×C₃₂	C₃×C₈	1	26	yes	—
32	C₃²×C₄×C₃₂	C₂×C₃×C₄	1	26	yes	—
33	C₃²×C₈×C₁₆	C₃×C₈	1	26	yes	—
34	C₃²×C₈×C₁₆	C₂×C₃×C₄	1	13	yes	—
35	C₂×C₃³×C₁₆	C₃₂	1	8	yes	—
36	C₂×C₃³×C₁₆	C₄×C₈	1	2	yes	—
37	C₂×C₃³×C₁₆	C₂×C₁₆	1	2	yes	—
38	C₃³×C₄×C₈	C₄×C₈	1	1	yes	characteristic
39	C₃³×C₄×C₈	C₂×C₁₆	1	2	yes	—
40	C₃³×C₃₂	C₃₂	1	8	yes	—
41	C₃³×C₃₂	C₂×C₁₆	1	8	yes	—
42	C₃×C₄×C₆₄	C₃²×C₄	1	52	yes	—
43	C₃×C₈×C₃₂	C₃²×C₄	1	13	yes	—
44	C₃×C₈×C₃₂	C₂²×C₃²	1	13	yes	—
45	C₃×C₁₆²	C₃²×C₄	1	13	yes	—
46	C₂×C₃²×C₃₂	C₃×C₁₆	1	52	yes	—
47	C₂×C₃²×C₃₂	C₂×C₃×C₈	1	52	yes	—
48	C₃²×C₄×C₁₆	C₃×C₁₆	1	52	yes	—
49	C₃²×C₄×C₁₆	C₃×C₄²	1	13	yes	—
50	C₃²×C₄×C₁₆	C₂×C₃×C₈	1	13	yes	—
51	C₃²×C₈²	C₂×C₃×C₈	1	13	yes	—
52	C₃²×C₆₄	C₃×C₁₆	1	208	yes	direct product factor
53	C₈×C₆₄	C₂×C₃³	1	2	yes	—
54	C₁₆×C₃₂	C₂×C₃³	1	1	yes	characteristic
55	C₂×C₃³×C₈	C₈²	1	1	yes	characteristic
56	C₂×C₃³×C₈	C₄×C₁₆	1	1	yes	characteristic
57	C₂×C₃³×C₈	C₂×C₃₂	1	4	yes	—
58	C₃³×C₄²	C₄×C₁₆	1	1	yes	characteristic
59	C₃³×C₁₆	C₆₄	1	16	yes	direct product factor
60	C₃³×C₁₆	C₄×C₁₆	1	4	yes	—
61	C₃³×C₁₆	C₂×C₃₂	1	4	yes	—
62	C₂×C₃×C₆₄	C₃²×C₈	1	104	yes	—
63	C₃×C₄×C₃₂	C₃²×C₈	1	26	yes	—
64	C₃×C₄×C₃₂	C₂×C₃²×C₄	1	26	yes	—
65	C₃×C₈×C₁₆	C₃²×C₈	1	26	yes	—
66	C₃×C₈×C₁₆	C₂×C₃²×C₄	1	13	yes	—
67	C₂×C₃²×C₁₆	C₃×C₃₂	1	104	yes	—
68	C₂×C₃²×C₁₆	C₃×C₄×C₈	1	26	yes	—
69	C₂×C₃²×C₁₆	C₂×C₃×C₁₆	1	26	yes	—
70	C₃²×C₄×C₈	C₃×C₄×C₈	1	13	yes	—
71	C₃²×C₄×C₈	C₂×C₃×C₁₆	1	26	yes	—
72	C₃²×C₃₂	C₃×C₃₂	1	104	yes	—
73	C₃²×C₃₂	C₂×C₃×C₁₆	1	104	yes	—
74	C₄×C₆₄	C₃³×C₄	1	4	yes	—
75	C₈×C₃₂	C₃³×C₄	1	1	yes	characteristic
76	C₈×C₃₂	C₂²×C₃³	1	1	yes	characteristic
77	C₁₆²	C₃³×C₄	1	1	yes	characteristic
78	C₂×C₃³×C₄	C₈×C₁₆	1	1	yes	characteristic
79	C₂×C₃³×C₄	C₄×C₃₂	1	2	yes	—
80	C₃³×C₈	C₈×C₁₆	1	2	yes	—
81	C₃³×C₈	C₄×C₃₂	1	2	yes	—
82	C₃³×C₈	C₂×C₆₄	1	8	yes	—
83	C₂×C₃×C₃₂	C₃²×C₁₆	1	52	yes	—
84	C₂×C₃×C₃₂	C₂×C₃²×C₈	1	52	yes	—
85	C₃×C₄×C₁₆	C₃²×C₁₆	1	52	yes	—
86	C₃×C₄×C₁₆	C₃²×C₄²	1	13	yes	—
87	C₃×C₄×C₁₆	C₂×C₃²×C₈	1	13	yes	—
88	C₃×C₈²	C₂×C₃²×C₈	1	13	yes	—
89	C₃×C₆₄	C₃²×C₁₆	1	208	yes	direct product factor
90	C₂×C₃²×C₈	C₃×C₈²	1	13	yes	—
91	C₂×C₃²×C₈	C₃×C₄×C₁₆	1	13	yes	—
92	C₂×C₃²×C₈	C₂×C₃×C₃₂	1	52	yes	—
93	C₃²×C₄²	C₃×C₄×C₁₆	1	13	yes	—
94	C₃²×C₁₆	C₃×C₆₄	1	208	yes	direct product factor
95	C₃²×C₁₆	C₃×C₄×C₁₆	1	52	yes	—
96	C₃²×C₁₆	C₂×C₃×C₃₂	1	52	yes	—
97	C₂×C₆₄	C₃³×C₈	1	8	yes	—
98	C₄×C₃₂	C₃³×C₈	1	2	yes	—
99	C₄×C₃₂	C₂×C₃³×C₄	1	2	yes	—
100	C₈×C₁₆	C₃³×C₈	1	2	yes	—
101	C₈×C₁₆	C₂×C₃³×C₄	1	1	yes	characteristic
102	C₂²×C₃³	C₈×C₃₂	1	1	yes	characteristic
103	C₃³×C₄	C₁₆²	1	1	yes	characteristic
104	C₃³×C₄	C₈×C₃₂	1	1	yes	characteristic
105	C₃³×C₄	C₄×C₆₄	1	4	yes	—
106	C₂×C₃×C₁₆	C₃²×C₃₂	1	104	yes	—
107	C₂×C₃×C₁₆	C₃²×C₄×C₈	1	26	yes	—
108	C₂×C₃×C₁₆	C₂×C₃²×C₁₆	1	26	yes	—
109	C₃×C₄×C₈	C₃²×C₄×C₈	1	13	yes	—
110	C₃×C₄×C₈	C₂×C₃²×C₁₆	1	26	yes	—
111	C₃×C₃₂	C₃²×C₃₂	1	104	yes	—
112	C₃×C₃₂	C₂×C₃²×C₁₆	1	104	yes	—
113	C₂×C₃²×C₄	C₃×C₈×C₁₆	1	13	yes	—
114	C₂×C₃²×C₄	C₃×C₄×C₃₂	1	26	yes	—
115	C₃²×C₈	C₃×C₈×C₁₆	1	26	yes	—
116	C₃²×C₈	C₃×C₄×C₃₂	1	26	yes	—
117	C₃²×C₈	C₂×C₃×C₆₄	1	104	yes	—
118	C₂×C₃₂	C₃³×C₁₆	1	4	yes	—
119	C₂×C₃₂	C₂×C₃³×C₈	1	4	yes	—
120	C₄×C₁₆	C₃³×C₁₆	1	4	yes	—
121	C₄×C₁₆	C₃³×C₄²	1	1	yes	characteristic
122	C₄×C₁₆	C₂×C₃³×C₈	1	1	yes	characteristic
123	C₈²	C₂×C₃³×C₈	1	1	yes	characteristic
124	C₆₄	C₃³×C₁₆	1	16	yes	direct product factor
125	C₂×C₃³	C₁₆×C₃₂	1	1	yes	characteristic
126	C₂×C₃³	C₈×C₆₄	1	2	yes	—
127	C₂×C₃×C₈	C₃²×C₈²	1	13	yes	—
128	C₂×C₃×C₈	C₃²×C₄×C₁₆	1	13	yes	—
129	C₂×C₃×C₈	C₂×C₃²×C₃₂	1	52	yes	—
130	C₃×C₄²	C₃²×C₄×C₁₆	1	13	yes	—
131	C₃×C₁₆	C₃²×C₆₄	1	208	yes	direct product factor
132	C₃×C₁₆	C₃²×C₄×C₁₆	1	52	yes	—
133	C₃×C₁₆	C₂×C₃²×C₃₂	1	52	yes	—
134	C₂²×C₃²	C₃×C₈×C₃₂	1	13	yes	—
135	C₃²×C₄	C₃×C₁₆²	1	13	yes	—
136	C₃²×C₄	C₃×C₈×C₃₂	1	13	yes	—
137	C₃²×C₄	C₃×C₄×C₆₄	1	52	yes	—
138	C₂×C₁₆	C₃³×C₃₂	1	8	yes	—
139	C₂×C₁₆	C₃³×C₄×C₈	1	2	yes	—
140	C₂×C₁₆	C₂×C₃³×C₁₆	1	2	yes	—
141	C₄×C₈	C₃³×C₄×C₈	1	1	yes	characteristic
142	C₄×C₈	C₂×C₃³×C₁₆	1	2	yes	—
143	C₃₂	C₃³×C₃₂	1	8	yes	—
144	C₃₂	C₂×C₃³×C₁₆	1	8	yes	—
145	C₃³	C₁₆×C₆₄	1	1	yes	characteristic, direct product factor, Sylow
146	C₂×C₃×C₄	C₃²×C₈×C₁₆	1	13	yes	—
147	C₂×C₃×C₄	C₃²×C₄×C₃₂	1	26	yes	—
148	C₃×C₈	C₃²×C₈×C₁₆	1	26	yes	—
149	C₃×C₈	C₃²×C₄×C₃₂	1	26	yes	—
150	C₃×C₈	C₂×C₃²×C₆₄	1	104	yes	—
151	C₂×C₃²	C₃×C₈×C₆₄	1	26	yes	—
152	C₂×C₃²	C₃×C₁₆×C₃₂	1	13	yes	—
153	C₂×C₈	C₃³×C₈²	1	1	yes	characteristic
154	C₂×C₈	C₃³×C₄×C₁₆	1	1	yes	characteristic
155	C₂×C₈	C₂×C₃³×C₃₂	1	4	yes	—
156	C₄²	C₃³×C₄×C₁₆	1	1	yes	characteristic
157	C₁₆	C₃³×C₆₄	1	16	yes	direct product factor
158	C₁₆	C₃³×C₄×C₁₆	1	4	yes	—
159	C₁₆	C₂×C₃³×C₃₂	1	4	yes	—
160	C₂²×C₃	C₃²×C₈×C₃₂	1	13	yes	—
161	C₃×C₄	C₃²×C₄×C₆₄	1	52	yes	—
162	C₃×C₄	C₃²×C₈×C₃₂	1	13	yes	—
163	C₃×C₄	C₃²×C₁₆²	1	13	yes	—
164	C₃²	C₃×C₁₆×C₆₄	1	13	yes	direct product factor
165	C₂×C₄	C₃³×C₄×C₃₂	1	2	yes	—
166	C₂×C₄	C₃³×C₈×C₁₆	1	1	yes	characteristic
167	C₈	C₂×C₃³×C₆₄	1	8	yes	—
168	C₈	C₃³×C₄×C₃₂	1	2	yes	—
169	C₈	C₃³×C₈×C₁₆	1	2	yes	—
170	C₂×C₃	C₃²×C₈×C₆₄	1	26	yes	—
171	C₂×C₃	C₃²×C₁₆×C₃₂	1	13	yes	—
172	C₂²	C₃³×C₈×C₃₂	1	1	yes	characteristic
173	C₄	C₃³×C₄×C₆₄	1	4	yes	—
174	C₄	C₃³×C₈×C₃₂	1	1	yes	characteristic
175	C₄	C₃³×C₁₆²	1	1	yes	characteristic
176	C₃	C₃²×C₁₆×C₆₄	1	13	yes	direct product factor
177	C₂	C₃³×C₈×C₆₄	1	2	yes	—
178	C₂	C₃³×C₁₆×C₃₂	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₈×C₃₂	C₂²×C₃³
socle, prime radical	C₂²×C₃³	C₈×C₃₂
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃²×C₁₆×C₆₄	C₃	direct (N × H)
15. C₃×C₁₆×C₆₄	C₃²	direct (N × H)
22. C₁₆×C₆₄	C₃³	direct (N × H)
29. C₃³×C₆₄	C₁₆	direct (N × H)
52. C₃²×C₆₄	C₃×C₁₆	direct (N × H)
59. C₃³×C₁₆	C₆₄	direct (N × H)
89. C₃×C₆₄	C₃²×C₁₆	direct (N × H)
94. C₃²×C₁₆	C₃×C₆₄	direct (N × H)
124. C₆₄	C₃³×C₁₆	direct (N × H)
131. C₃×C₁₆	C₃²×C₆₄	direct (N × H)
145. C₃³	C₁₆×C₆₄	direct (N × H)
157. C₁₆	C₃³×C₆₄	direct (N × H)
164. C₃²	C₃×C₁₆×C₆₄	direct (N × H)
176. C₃	C₃²×C₁₆×C₆₄	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃³×C₁₆×C₃₂	2
C₃³×C₈×C₆₄	2
C₃²×C₁₆×C₆₄	3

Subgroups — C33×C16×C64 #72204635

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃³×C₁₆×C₆₄ #72204635