Subgroups — C₂⁴×C₃³×C₃₂ #72192016

Order 13824.

44856 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₃³×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₃³×C₃₂	C₂	1	30	yes	maximal, direct product factor
3	C₂⁴×C₃²×C₃₂	C₃	1	13	yes	maximal, direct product factor
4	C₂⁴×C₃³×C₈	C₄	1	1	yes	characteristic
5	C₂³×C₃³×C₁₆	C₄	1	15	yes	—
6	C₂³×C₃³×C₁₆	C₂²	1	15	yes	—
7	C₂²×C₃³×C₃₂	C₂²	1	140	yes	direct product factor
8	C₂⁴×C₃²×C₁₆	C₂×C₃	1	13	yes	—
9	C₂³×C₃²×C₃₂	C₂×C₃	1	390	yes	direct product factor
10	C₂⁴×C₃×C₃₂	C₃²	1	13	yes	direct product factor
11	C₂⁴×C₃³×C₄	C₈	1	1	yes	characteristic
12	C₂³×C₃³×C₈	C₈	1	15	yes	—
13	C₂³×C₃³×C₈	C₂×C₄	1	15	yes	—
14	C₂²×C₃³×C₁₆	C₂×C₄	1	105	yes	—
15	C₂²×C₃³×C₁₆	C₂³	1	35	yes	—
16	C₂×C₃³×C₃₂	C₂³	1	120	yes	direct product factor
17	C₂⁴×C₃²×C₈	C₃×C₄	1	13	yes	—
18	C₂³×C₃²×C₁₆	C₃×C₄	1	195	yes	—
19	C₂³×C₃²×C₁₆	C₂²×C₃	1	195	yes	—
20	C₂²×C₃²×C₃₂	C₂²×C₃	1	1820	yes	direct product factor
21	C₂⁵×C₃³	C₁₆	1	1	yes	characteristic
22	C₂³×C₃³×C₄	C₁₆	1	15	yes	—
23	C₂³×C₃³×C₄	C₂×C₈	1	15	yes	—
24	C₂²×C₃³×C₈	C₂×C₈	1	105	yes	—
25	C₂²×C₃³×C₈	C₂²×C₄	1	35	yes	—
26	C₂×C₃³×C₁₆	C₂²×C₄	1	105	yes	—
27	C₂×C₃³×C₁₆	C₂⁴	1	15	yes	—
28	C₃³×C₃₂	C₂⁴	1	16	yes	direct product factor
29	C₂⁴×C₃×C₁₆	C₂×C₃²	1	13	yes	—
30	C₂³×C₃×C₃₂	C₂×C₃²	1	390	yes	direct product factor
31	C₂⁴×C₃²×C₄	C₃×C₈	1	13	yes	—
32	C₂³×C₃²×C₈	C₃×C₈	1	195	yes	—
33	C₂³×C₃²×C₈	C₂×C₃×C₄	1	195	yes	—
34	C₂²×C₃²×C₁₆	C₂×C₃×C₄	1	1365	yes	—
35	C₂²×C₃²×C₁₆	C₂³×C₃	1	455	yes	—
36	C₂×C₃²×C₃₂	C₂³×C₃	1	1560	yes	direct product factor
37	C₂⁴×C₃₂	C₃³	1	1	yes	characteristic, direct product factor, Sylow
38	C₂⁴×C₃³	C₃₂	1	16	yes	direct product factor
39	C₂⁴×C₃³	C₂×C₁₆	1	15	yes	—
40	C₂²×C₃³×C₄	C₂×C₁₆	1	105	yes	—
41	C₂²×C₃³×C₄	C₂²×C₈	1	35	yes	—
42	C₂×C₃³×C₈	C₂²×C₈	1	105	yes	—
43	C₂×C₃³×C₈	C₂³×C₄	1	15	yes	—
44	C₃³×C₁₆	C₂³×C₄	1	15	yes	—
45	C₃³×C₁₆	C₂⁵	1	1	yes	characteristic
46	C₂⁴×C₃×C₈	C₃²×C₄	1	13	yes	—
47	C₂³×C₃×C₁₆	C₃²×C₄	1	195	yes	—
48	C₂³×C₃×C₁₆	C₂²×C₃²	1	195	yes	—
49	C₂²×C₃×C₃₂	C₂²×C₃²	1	1820	yes	direct product factor
50	C₂⁵×C₃²	C₃×C₁₆	1	13	yes	—
51	C₂³×C₃²×C₄	C₃×C₁₆	1	195	yes	—
52	C₂³×C₃²×C₄	C₂×C₃×C₈	1	195	yes	—
53	C₂²×C₃²×C₈	C₂×C₃×C₈	1	1365	yes	—
54	C₂²×C₃²×C₈	C₂²×C₃×C₄	1	455	yes	—
55	C₂×C₃²×C₁₆	C₂²×C₃×C₄	1	1365	yes	—
56	C₂×C₃²×C₁₆	C₂⁴×C₃	1	195	yes	—
57	C₃²×C₃₂	C₂⁴×C₃	1	208	yes	direct product factor
58	C₂⁴×C₁₆	C₂×C₃³	1	1	yes	characteristic
59	C₂³×C₃₂	C₂×C₃³	1	30	yes	direct product factor
60	C₂³×C₃³	C₂×C₃₂	1	120	yes	direct product factor
61	C₂³×C₃³	C₂²×C₁₆	1	35	yes	—
62	C₂×C₃³×C₄	C₂²×C₁₆	1	105	yes	—
63	C₂×C₃³×C₄	C₂³×C₈	1	15	yes	—
64	C₃³×C₈	C₂³×C₈	1	15	yes	—
65	C₃³×C₈	C₂⁴×C₄	1	1	yes	characteristic
66	C₂⁴×C₃×C₄	C₃²×C₈	1	13	yes	—
67	C₂³×C₃×C₈	C₃²×C₈	1	195	yes	—
68	C₂³×C₃×C₈	C₂×C₃²×C₄	1	195	yes	—
69	C₂²×C₃×C₁₆	C₂×C₃²×C₄	1	1365	yes	—
70	C₂²×C₃×C₁₆	C₂³×C₃²	1	455	yes	—
71	C₂×C₃×C₃₂	C₂³×C₃²	1	1560	yes	direct product factor
72	C₂⁴×C₃²	C₃×C₃₂	1	208	yes	direct product factor
73	C₂⁴×C₃²	C₂×C₃×C₁₆	1	195	yes	—
74	C₂²×C₃²×C₄	C₂×C₃×C₁₆	1	1365	yes	—
75	C₂²×C₃²×C₄	C₂²×C₃×C₈	1	455	yes	—
76	C₂×C₃²×C₈	C₂²×C₃×C₈	1	1365	yes	—
77	C₂×C₃²×C₈	C₂³×C₃×C₄	1	195	yes	—
78	C₃²×C₁₆	C₂³×C₃×C₄	1	195	yes	—
79	C₃²×C₁₆	C₂⁵×C₃	1	13	yes	—
80	C₂⁴×C₈	C₃³×C₄	1	1	yes	characteristic
81	C₂³×C₁₆	C₃³×C₄	1	15	yes	—
82	C₂³×C₁₆	C₂²×C₃³	1	15	yes	—
83	C₂²×C₃₂	C₂²×C₃³	1	140	yes	direct product factor
84	C₂²×C₃³	C₂²×C₃₂	1	140	yes	direct product factor
85	C₂²×C₃³	C₂³×C₁₆	1	15	yes	—
86	C₃³×C₄	C₂³×C₁₆	1	15	yes	—
87	C₃³×C₄	C₂⁴×C₈	1	1	yes	characteristic
88	C₂⁵×C₃	C₃²×C₁₆	1	13	yes	—
89	C₂³×C₃×C₄	C₃²×C₁₆	1	195	yes	—
90	C₂³×C₃×C₄	C₂×C₃²×C₈	1	195	yes	—
91	C₂²×C₃×C₈	C₂×C₃²×C₈	1	1365	yes	—
92	C₂²×C₃×C₈	C₂²×C₃²×C₄	1	455	yes	—
93	C₂×C₃×C₁₆	C₂²×C₃²×C₄	1	1365	yes	—
94	C₂×C₃×C₁₆	C₂⁴×C₃²	1	195	yes	—
95	C₃×C₃₂	C₂⁴×C₃²	1	208	yes	direct product factor
96	C₂³×C₃²	C₂×C₃×C₃₂	1	1560	yes	direct product factor
97	C₂³×C₃²	C₂²×C₃×C₁₆	1	455	yes	—
98	C₂×C₃²×C₄	C₂²×C₃×C₁₆	1	1365	yes	—
99	C₂×C₃²×C₄	C₂³×C₃×C₈	1	195	yes	—
100	C₃²×C₈	C₂³×C₃×C₈	1	195	yes	—
101	C₃²×C₈	C₂⁴×C₃×C₄	1	13	yes	—
102	C₂⁴×C₄	C₃³×C₈	1	1	yes	characteristic
103	C₂³×C₈	C₃³×C₈	1	15	yes	—
104	C₂³×C₈	C₂×C₃³×C₄	1	15	yes	—
105	C₂²×C₁₆	C₂×C₃³×C₄	1	105	yes	—
106	C₂²×C₁₆	C₂³×C₃³	1	35	yes	—
107	C₂×C₃₂	C₂³×C₃³	1	120	yes	direct product factor
108	C₂×C₃³	C₂³×C₃₂	1	30	yes	direct product factor
109	C₂×C₃³	C₂⁴×C₁₆	1	1	yes	characteristic
110	C₂⁴×C₃	C₃²×C₃₂	1	208	yes	direct product factor
111	C₂⁴×C₃	C₂×C₃²×C₁₆	1	195	yes	—
112	C₂²×C₃×C₄	C₂×C₃²×C₁₆	1	1365	yes	—
113	C₂²×C₃×C₄	C₂²×C₃²×C₈	1	455	yes	—
114	C₂×C₃×C₈	C₂²×C₃²×C₈	1	1365	yes	—
115	C₂×C₃×C₈	C₂³×C₃²×C₄	1	195	yes	—
116	C₃×C₁₆	C₂³×C₃²×C₄	1	195	yes	—
117	C₃×C₁₆	C₂⁵×C₃²	1	13	yes	—
118	C₂²×C₃²	C₂²×C₃×C₃₂	1	1820	yes	direct product factor
119	C₂²×C₃²	C₂³×C₃×C₁₆	1	195	yes	—
120	C₃²×C₄	C₂³×C₃×C₁₆	1	195	yes	—
121	C₃²×C₄	C₂⁴×C₃×C₈	1	13	yes	—
122	C₂⁵	C₃³×C₁₆	1	1	yes	characteristic
123	C₂³×C₄	C₃³×C₁₆	1	15	yes	—
124	C₂³×C₄	C₂×C₃³×C₈	1	15	yes	—
125	C₂²×C₈	C₂×C₃³×C₈	1	105	yes	—
126	C₂²×C₈	C₂²×C₃³×C₄	1	35	yes	—
127	C₂×C₁₆	C₂²×C₃³×C₄	1	105	yes	—
128	C₂×C₁₆	C₂⁴×C₃³	1	15	yes	—
129	C₃₂	C₂⁴×C₃³	1	16	yes	direct product factor
130	C₃³	C₂⁴×C₃₂	1	1	yes	characteristic, direct product factor, Sylow
131	C₂³×C₃	C₂×C₃²×C₃₂	1	1560	yes	direct product factor
132	C₂³×C₃	C₂²×C₃²×C₁₆	1	455	yes	—
133	C₂×C₃×C₄	C₂²×C₃²×C₁₆	1	1365	yes	—
134	C₂×C₃×C₄	C₂³×C₃²×C₈	1	195	yes	—
135	C₃×C₈	C₂³×C₃²×C₈	1	195	yes	—
136	C₃×C₈	C₂⁴×C₃²×C₄	1	13	yes	—
137	C₂×C₃²	C₂³×C₃×C₃₂	1	390	yes	direct product factor
138	C₂×C₃²	C₂⁴×C₃×C₁₆	1	13	yes	—
139	C₂⁴	C₃³×C₃₂	1	16	yes	direct product factor
140	C₂⁴	C₂×C₃³×C₁₆	1	15	yes	—
141	C₂²×C₄	C₂×C₃³×C₁₆	1	105	yes	—
142	C₂²×C₄	C₂²×C₃³×C₈	1	35	yes	—
143	C₂×C₈	C₂²×C₃³×C₈	1	105	yes	—
144	C₂×C₈	C₂³×C₃³×C₄	1	15	yes	—
145	C₁₆	C₂³×C₃³×C₄	1	15	yes	—
146	C₁₆	C₂⁵×C₃³	1	1	yes	characteristic
147	C₂²×C₃	C₂²×C₃²×C₃₂	1	1820	yes	direct product factor
148	C₂²×C₃	C₂³×C₃²×C₁₆	1	195	yes	—
149	C₃×C₄	C₂³×C₃²×C₁₆	1	195	yes	—
150	C₃×C₄	C₂⁴×C₃²×C₈	1	13	yes	—
151	C₃²	C₂⁴×C₃×C₃₂	1	13	yes	direct product factor
152	C₂³	C₂×C₃³×C₃₂	1	120	yes	direct product factor
153	C₂³	C₂²×C₃³×C₁₆	1	35	yes	—
154	C₂×C₄	C₂²×C₃³×C₁₆	1	105	yes	—
155	C₂×C₄	C₂³×C₃³×C₈	1	15	yes	—
156	C₈	C₂³×C₃³×C₈	1	15	yes	—
157	C₈	C₂⁴×C₃³×C₄	1	1	yes	characteristic
158	C₂×C₃	C₂³×C₃²×C₃₂	1	390	yes	direct product factor
159	C₂×C₃	C₂⁴×C₃²×C₁₆	1	13	yes	—
160	C₂²	C₂²×C₃³×C₃₂	1	140	yes	direct product factor
161	C₂²	C₂³×C₃³×C₁₆	1	15	yes	—
162	C₄	C₂³×C₃³×C₁₆	1	15	yes	—
163	C₄	C₂⁴×C₃³×C₈	1	1	yes	characteristic
164	C₃	C₂⁴×C₃²×C₃₂	1	13	yes	direct product factor
165	C₂	C₂³×C₃³×C₃₂	1	30	yes	direct product factor
166	C₂	C₂⁴×C₃³×C₁₆	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₁₆	C₂⁵×C₃³
socle, prime radical	C₂⁵×C₃³	C₁₆
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂³×C₃³×C₃₂	C₂	direct (N × H)
3. C₂⁴×C₃²×C₃₂	C₃	direct (N × H)
7. C₂²×C₃³×C₃₂	C₂²	direct (N × H)
9. C₂³×C₃²×C₃₂	C₂×C₃	direct (N × H)
10. C₂⁴×C₃×C₃₂	C₃²	direct (N × H)
16. C₂×C₃³×C₃₂	C₂³	direct (N × H)
20. C₂²×C₃²×C₃₂	C₂²×C₃	direct (N × H)
28. C₃³×C₃₂	C₂⁴	direct (N × H)
30. C₂³×C₃×C₃₂	C₂×C₃²	direct (N × H)
36. C₂×C₃²×C₃₂	C₂³×C₃	direct (N × H)
37. C₂⁴×C₃₂	C₃³	direct (N × H)
38. C₂⁴×C₃³	C₃₂	direct (N × H)
49. C₂²×C₃×C₃₂	C₂²×C₃²	direct (N × H)
57. C₃²×C₃₂	C₂⁴×C₃	direct (N × H)
59. C₂³×C₃₂	C₂×C₃³	direct (N × H)
60. C₂³×C₃³	C₂×C₃₂	direct (N × H)
71. C₂×C₃×C₃₂	C₂³×C₃²	direct (N × H)
72. C₂⁴×C₃²	C₃×C₃₂	direct (N × H)
83. C₂²×C₃₂	C₂²×C₃³	direct (N × H)
84. C₂²×C₃³	C₂²×C₃₂	direct (N × H)
95. C₃×C₃₂	C₂⁴×C₃²	direct (N × H)
96. C₂³×C₃²	C₂×C₃×C₃₂	direct (N × H)
107. C₂×C₃₂	C₂³×C₃³	direct (N × H)
108. C₂×C₃³	C₂³×C₃₂	direct (N × H)
110. C₂⁴×C₃	C₃²×C₃₂	direct (N × H)
118. C₂²×C₃²	C₂²×C₃×C₃₂	direct (N × H)
129. C₃₂	C₂⁴×C₃³	direct (N × H)
130. C₃³	C₂⁴×C₃₂	direct (N × H)
131. C₂³×C₃	C₂×C₃²×C₃₂	direct (N × H)
137. C₂×C₃²	C₂³×C₃×C₃₂	direct (N × H)
139. C₂⁴	C₃³×C₃₂	direct (N × H)
147. C₂²×C₃	C₂²×C₃²×C₃₂	direct (N × H)
151. C₃²	C₂⁴×C₃×C₃₂	direct (N × H)
152. C₂³	C₂×C₃³×C₃₂	direct (N × H)
158. C₂×C₃	C₂³×C₃²×C₃₂	direct (N × H)
160. C₂²	C₂²×C₃³×C₃₂	direct (N × H)
164. C₃	C₂⁴×C₃²×C₃₂	direct (N × H)
165. C₂	C₂³×C₃³×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
C₂⁴×C₃³×C₁₆	2
C₂³×C₃³×C₃₂	2
C₂⁴×C₃²×C₃₂	3

Subgroups — C24×C33×C32 #72192016

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×C₃³×C₃₂ #72192016