Subgroups — C₂⁴×C₄³×C₉ #72188024

Order 9216.

1135359 conjugacy classes of subgroups.

The subgroup lattice has 150 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×C₄²×C₉	C₂	1	7	yes	maximal
2	C₂³×C₄³×C₉	C₂	1	120	yes	maximal, direct product factor
3	C₂⁴×C₃×C₄³	C₃	1	1	yes	maximal, characteristic
4	C₂⁶×C₄×C₉	C₂²	1	7	yes	—
5	C₂⁴×C₄²×C₉	C₄	1	448	yes	direct product factor
6	C₂⁴×C₄²×C₉	C₂²	1	420	yes	—
7	C₂²×C₄³×C₉	C₂²	1	2240	yes	direct product factor
8	C₂⁵×C₃×C₄²	C₂×C₃	1	7	yes	—
9	C₂³×C₃×C₄³	C₂×C₃	1	120	yes	—
10	C₂⁴×C₄³	C₉	1	1	yes	characteristic, direct product factor, Sylow
11	C₂⁷×C₉	C₂³	1	1	yes	characteristic
12	C₂⁵×C₄×C₉	C₂×C₄	1	672	yes	—
13	C₂⁵×C₄×C₉	C₂³	1	210	yes	—
14	C₂³×C₄²×C₉	C₂×C₄	1	13440	yes	direct product factor
15	C₂³×C₄²×C₉	C₂³	1	3920	yes	—
16	C₂×C₄³×C₉	C₂³	1	7680	yes	direct product factor
17	C₂⁶×C₃×C₄	C₂²×C₃	1	7	yes	—
18	C₂⁴×C₃×C₄²	C₃×C₄	1	448	yes	—
19	C₂⁴×C₃×C₄²	C₂²×C₃	1	420	yes	—
20	C₂²×C₃×C₄³	C₂²×C₃	1	2240	yes	—
21	C₂⁶×C₉	C₂²×C₄	1	112	yes	—
22	C₂⁶×C₉	C₂⁴	1	15	yes	—
23	C₂⁴×C₄×C₉	C₄²	1	7168	yes	direct product factor
24	C₂⁴×C₄×C₉	C₂²×C₄	1	10080	yes	—
25	C₂⁴×C₄×C₉	C₂⁴	1	980	yes	—
26	C₂²×C₄²×C₉	C₂²×C₄	1	62720	yes	direct product factor
27	C₂²×C₄²×C₉	C₂⁴	1	6720	yes	—
28	C₄³×C₉	C₂⁴	1	4096	yes	direct product factor
29	C₂⁵×C₄²	C₂×C₉	1	7	yes	—
30	C₂³×C₄³	C₂×C₉	1	120	yes	direct product factor
31	C₂⁷×C₃	C₂³×C₃	1	1	yes	characteristic
32	C₂⁵×C₃×C₄	C₂×C₃×C₄	1	672	yes	—
33	C₂⁵×C₃×C₄	C₂³×C₃	1	210	yes	—
34	C₂³×C₃×C₄²	C₂×C₃×C₄	1	13440	yes	—
35	C₂³×C₃×C₄²	C₂³×C₃	1	3920	yes	—
36	C₂×C₃×C₄³	C₂³×C₃	1	7680	yes	—
37	C₂⁵×C₉	C₂×C₄²	1	1792	yes	—
38	C₂⁵×C₉	C₂³×C₄	1	840	yes	—
39	C₂⁵×C₉	C₂⁵	1	35	yes	—
40	C₂³×C₄×C₉	C₂×C₄²	1	53760	yes	direct product factor
41	C₂³×C₄×C₉	C₂³×C₄	1	23520	yes	—
42	C₂³×C₄×C₉	C₂⁵	1	840	yes	—
43	C₂×C₄²×C₉	C₂³×C₄	1	53760	yes	direct product factor
44	C₂×C₄²×C₉	C₂⁵	1	1792	yes	—
45	C₂⁶×C₄	C₂²×C₉	1	7	yes	—
46	C₂⁴×C₄²	C₄×C₉	1	448	yes	direct product factor
47	C₂⁴×C₄²	C₂²×C₉	1	420	yes	—
48	C₂²×C₄³	C₂²×C₉	1	2240	yes	direct product factor
49	C₂⁶×C₃	C₂²×C₃×C₄	1	112	yes	—
50	C₂⁶×C₃	C₂⁴×C₃	1	15	yes	—
51	C₂⁴×C₃×C₄	C₃×C₄²	1	7168	yes	—
52	C₂⁴×C₃×C₄	C₂²×C₃×C₄	1	10080	yes	—
53	C₂⁴×C₃×C₄	C₂⁴×C₃	1	980	yes	—
54	C₂²×C₃×C₄²	C₂²×C₃×C₄	1	62720	yes	—
55	C₂²×C₃×C₄²	C₂⁴×C₃	1	6720	yes	—
56	C₃×C₄³	C₂⁴×C₃	1	4096	yes	—
57	C₂⁴×C₉	C₄³	1	4096	yes	direct product factor
58	C₂⁴×C₉	C₂²×C₄²	1	6720	yes	—
59	C₂⁴×C₉	C₂⁴×C₄	1	980	yes	—
60	C₂⁴×C₉	C₂⁶	1	15	yes	—
61	C₂²×C₄×C₉	C₂²×C₄²	1	62720	yes	direct product factor
62	C₂²×C₄×C₉	C₂⁴×C₄	1	10080	yes	—
63	C₂²×C₄×C₉	C₂⁶	1	112	yes	—
64	C₄²×C₉	C₂⁴×C₄	1	7168	yes	direct product factor
65	C₂⁷	C₂³×C₉	1	1	yes	characteristic
66	C₂⁵×C₄	C₂×C₄×C₉	1	672	yes	—
67	C₂⁵×C₄	C₂³×C₉	1	210	yes	—
68	C₂³×C₄²	C₂×C₄×C₉	1	13440	yes	direct product factor
69	C₂³×C₄²	C₂³×C₉	1	3920	yes	—
70	C₂×C₄³	C₂³×C₉	1	7680	yes	direct product factor
71	C₂⁵×C₃	C₂×C₃×C₄²	1	1792	yes	—
72	C₂⁵×C₃	C₂³×C₃×C₄	1	840	yes	—
73	C₂⁵×C₃	C₂⁵×C₃	1	35	yes	—
74	C₂³×C₃×C₄	C₂×C₃×C₄²	1	53760	yes	—
75	C₂³×C₃×C₄	C₂³×C₃×C₄	1	23520	yes	—
76	C₂³×C₃×C₄	C₂⁵×C₃	1	840	yes	—
77	C₂×C₃×C₄²	C₂³×C₃×C₄	1	53760	yes	—
78	C₂×C₃×C₄²	C₂⁵×C₃	1	1792	yes	—
79	C₂³×C₉	C₂×C₄³	1	7680	yes	direct product factor
80	C₂³×C₉	C₂³×C₄²	1	3920	yes	—
81	C₂³×C₉	C₂⁵×C₄	1	210	yes	—
82	C₂³×C₉	C₂⁷	1	1	yes	characteristic
83	C₂×C₄×C₉	C₂³×C₄²	1	13440	yes	direct product factor
84	C₂×C₄×C₉	C₂⁵×C₄	1	672	yes	—
85	C₂⁶	C₂²×C₄×C₉	1	112	yes	—
86	C₂⁶	C₂⁴×C₉	1	15	yes	—
87	C₂⁴×C₄	C₄²×C₉	1	7168	yes	direct product factor
88	C₂⁴×C₄	C₂²×C₄×C₉	1	10080	yes	—
89	C₂⁴×C₄	C₂⁴×C₉	1	980	yes	—
90	C₂²×C₄²	C₂²×C₄×C₉	1	62720	yes	direct product factor
91	C₂²×C₄²	C₂⁴×C₉	1	6720	yes	—
92	C₄³	C₂⁴×C₉	1	4096	yes	direct product factor
93	C₂⁴×C₃	C₃×C₄³	1	4096	yes	—
94	C₂⁴×C₃	C₂²×C₃×C₄²	1	6720	yes	—
95	C₂⁴×C₃	C₂⁴×C₃×C₄	1	980	yes	—
96	C₂⁴×C₃	C₂⁶×C₃	1	15	yes	—
97	C₂²×C₃×C₄	C₂²×C₃×C₄²	1	62720	yes	—
98	C₂²×C₃×C₄	C₂⁴×C₃×C₄	1	10080	yes	—
99	C₂²×C₃×C₄	C₂⁶×C₃	1	112	yes	—
100	C₃×C₄²	C₂⁴×C₃×C₄	1	7168	yes	—
101	C₂²×C₉	C₂²×C₄³	1	2240	yes	direct product factor
102	C₂²×C₉	C₂⁴×C₄²	1	420	yes	—
103	C₂²×C₉	C₂⁶×C₄	1	7	yes	—
104	C₄×C₉	C₂⁴×C₄²	1	448	yes	direct product factor
105	C₂⁵	C₂×C₄²×C₉	1	1792	yes	—
106	C₂⁵	C₂³×C₄×C₉	1	840	yes	—
107	C₂⁵	C₂⁵×C₉	1	35	yes	—
108	C₂³×C₄	C₂×C₄²×C₉	1	53760	yes	direct product factor
109	C₂³×C₄	C₂³×C₄×C₉	1	23520	yes	—
110	C₂³×C₄	C₂⁵×C₉	1	840	yes	—
111	C₂×C₄²	C₂³×C₄×C₉	1	53760	yes	direct product factor
112	C₂×C₄²	C₂⁵×C₉	1	1792	yes	—
113	C₂³×C₃	C₂×C₃×C₄³	1	7680	yes	—
114	C₂³×C₃	C₂³×C₃×C₄²	1	3920	yes	—
115	C₂³×C₃	C₂⁵×C₃×C₄	1	210	yes	—
116	C₂³×C₃	C₂⁷×C₃	1	1	yes	characteristic
117	C₂×C₃×C₄	C₂³×C₃×C₄²	1	13440	yes	—
118	C₂×C₃×C₄	C₂⁵×C₃×C₄	1	672	yes	—
119	C₂×C₉	C₂³×C₄³	1	120	yes	direct product factor
120	C₂×C₉	C₂⁵×C₄²	1	7	yes	—
121	C₂⁴	C₄³×C₉	1	4096	yes	direct product factor
122	C₂⁴	C₂²×C₄²×C₉	1	6720	yes	—
123	C₂⁴	C₂⁴×C₄×C₉	1	980	yes	—
124	C₂⁴	C₂⁶×C₉	1	15	yes	—
125	C₂²×C₄	C₂²×C₄²×C₉	1	62720	yes	direct product factor
126	C₂²×C₄	C₂⁴×C₄×C₉	1	10080	yes	—
127	C₂²×C₄	C₂⁶×C₉	1	112	yes	—
128	C₄²	C₂⁴×C₄×C₉	1	7168	yes	direct product factor
129	C₂²×C₃	C₂²×C₃×C₄³	1	2240	yes	—
130	C₂²×C₃	C₂⁴×C₃×C₄²	1	420	yes	—
131	C₂²×C₃	C₂⁶×C₃×C₄	1	7	yes	—
132	C₃×C₄	C₂⁴×C₃×C₄²	1	448	yes	—
133	C₉	C₂⁴×C₄³	1	1	yes	characteristic, direct product factor, Sylow
134	C₂³	C₂×C₄³×C₉	1	7680	yes	direct product factor
135	C₂³	C₂³×C₄²×C₉	1	3920	yes	—
136	C₂³	C₂⁵×C₄×C₉	1	210	yes	—
137	C₂³	C₂⁷×C₉	1	1	yes	characteristic
138	C₂×C₄	C₂³×C₄²×C₉	1	13440	yes	direct product factor
139	C₂×C₄	C₂⁵×C₄×C₉	1	672	yes	—
140	C₂×C₃	C₂³×C₃×C₄³	1	120	yes	—
141	C₂×C₃	C₂⁵×C₃×C₄²	1	7	yes	—
142	C₂²	C₂²×C₄³×C₉	1	2240	yes	direct product factor
143	C₂²	C₂⁴×C₄²×C₉	1	420	yes	—
144	C₂²	C₂⁶×C₄×C₉	1	7	yes	—
145	C₄	C₂⁴×C₄²×C₉	1	448	yes	direct product factor
146	C₃	C₂⁴×C₃×C₄³	1	1	yes	characteristic
147	C₂	C₂³×C₄³×C₉	1	120	yes	direct product factor
148	C₂	C₂⁵×C₄²×C₉	1	7	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂³×C₃	C₂⁷×C₃
socle, prime radical	C₂⁷×C₃	C₂³×C₃
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂³×C₄³×C₉	C₂	direct (N × H)
5. C₂⁴×C₄²×C₉	C₄	direct (N × H)
7. C₂²×C₄³×C₉	C₂²	direct (N × H)
10. C₂⁴×C₄³	C₉	direct (N × H)
14. C₂³×C₄²×C₉	C₂×C₄	direct (N × H)
16. C₂×C₄³×C₉	C₂³	direct (N × H)
23. C₂⁴×C₄×C₉	C₄²	direct (N × H)
26. C₂²×C₄²×C₉	C₂²×C₄	direct (N × H)
28. C₄³×C₉	C₂⁴	direct (N × H)
30. C₂³×C₄³	C₂×C₉	direct (N × H)
40. C₂³×C₄×C₉	C₂×C₄²	direct (N × H)
43. C₂×C₄²×C₉	C₂³×C₄	direct (N × H)
46. C₂⁴×C₄²	C₄×C₉	direct (N × H)
48. C₂²×C₄³	C₂²×C₉	direct (N × H)
57. C₂⁴×C₉	C₄³	direct (N × H)
61. C₂²×C₄×C₉	C₂²×C₄²	direct (N × H)
64. C₄²×C₉	C₂⁴×C₄	direct (N × H)
68. C₂³×C₄²	C₂×C₄×C₉	direct (N × H)
70. C₂×C₄³	C₂³×C₉	direct (N × H)
79. C₂³×C₉	C₂×C₄³	direct (N × H)
83. C₂×C₄×C₉	C₂³×C₄²	direct (N × H)
87. C₂⁴×C₄	C₄²×C₉	direct (N × H)
90. C₂²×C₄²	C₂²×C₄×C₉	direct (N × H)
92. C₄³	C₂⁴×C₉	direct (N × H)
101. C₂²×C₉	C₂²×C₄³	direct (N × H)
104. C₄×C₉	C₂⁴×C₄²	direct (N × H)
108. C₂³×C₄	C₂×C₄²×C₉	direct (N × H)
111. C₂×C₄²	C₂³×C₄×C₉	direct (N × H)
119. C₂×C₉	C₂³×C₄³	direct (N × H)
121. C₂⁴	C₄³×C₉	direct (N × H)
125. C₂²×C₄	C₂²×C₄²×C₉	direct (N × H)
128. C₄²	C₂⁴×C₄×C₉	direct (N × H)
133. C₉	C₂⁴×C₄³	direct (N × H)
134. C₂³	C₂×C₄³×C₉	direct (N × H)
138. C₂×C₄	C₂³×C₄²×C₉	direct (N × H)
142. C₂²	C₂²×C₄³×C₉	direct (N × H)
145. C₄	C₂⁴×C₄²×C₉	direct (N × H)
147. C₂	C₂³×C₄³×C₉	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁵×C₄²×C₉	2
C₂³×C₄³×C₉	2
C₂⁴×C₃×C₄³	3

Subgroups — C24×C43×C9 #72188024

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×C₄³×C₉ #72188024