Subgroups — C₃×C₄⁵×C₈ #72201899

Order 24576.

10828572 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	C₃×C₄⁶	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄⁴×C₈	C₂	1	62	yes	maximal
3	C₄⁵×C₈	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
4	C₂×C₃×C₄⁵	C₄	1	32	yes	—
5	C₂×C₃×C₄⁵	C₂²	1	31	yes	—
6	C₂²×C₃×C₄³×C₈	C₂²	1	620	yes	—
7	C₃×C₄⁴×C₈	C₄	1	1984	yes	direct product factor
8	C₄⁶	C₂×C₃	1	1	yes	characteristic
9	C₂×C₄⁴×C₈	C₂×C₃	1	62	yes	—
10	C₂²×C₃×C₄⁴	C₂×C₄	1	496	yes	—
11	C₂²×C₃×C₄⁴	C₂³	1	155	yes	—
12	C₃×C₄⁵	C₈	1	1024	yes	direct product factor
13	C₃×C₄⁵	C₂×C₄	1	992	yes	—
14	C₂³×C₃×C₄²×C₈	C₂³	1	1240	yes	—
15	C₂×C₃×C₄³×C₈	C₂×C₄	1	29760	yes	—
16	C₂×C₄⁵	C₃×C₄	1	32	yes	—
17	C₂×C₄⁵	C₂²×C₃	1	31	yes	—
18	C₂²×C₄³×C₈	C₂²×C₃	1	620	yes	—
19	C₄⁴×C₈	C₃×C₄	1	1984	yes	direct product factor
20	C₂³×C₃×C₄³	C₂²×C₄	1	1240	yes	—
21	C₂³×C₃×C₄³	C₂⁴	1	155	yes	—
22	C₂×C₃×C₄⁴	C₄²	1	7936	yes	—
23	C₂×C₃×C₄⁴	C₂×C₈	1	15872	yes	—
24	C₂×C₃×C₄⁴	C₂²×C₄	1	7440	yes	—
25	C₂⁴×C₃×C₄×C₈	C₂⁴	1	496	yes	—
26	C₂²×C₃×C₄²×C₈	C₂²×C₄	1	69440	yes	—
27	C₃×C₄³×C₈	C₄²	1	158720	yes	direct product factor
28	C₂²×C₄⁴	C₂×C₃×C₄	1	496	yes	—
29	C₂²×C₄⁴	C₂³×C₃	1	155	yes	—
30	C₄⁵	C₃×C₈	1	1024	yes	direct product factor
31	C₄⁵	C₂×C₃×C₄	1	992	yes	—
32	C₂³×C₄²×C₈	C₂³×C₃	1	1240	yes	—
33	C₂×C₄³×C₈	C₂×C₃×C₄	1	29760	yes	—
34	C₂⁵×C₃×C₈	C₂⁵	1	32	yes	—
35	C₂⁴×C₃×C₄²	C₂³×C₄	1	620	yes	—
36	C₂⁴×C₃×C₄²	C₂⁵	1	31	yes	—
37	C₂³×C₃×C₄×C₈	C₂³×C₄	1	29760	yes	—
38	C₂²×C₃×C₄³	C₂×C₄²	1	29760	yes	—
39	C₂²×C₃×C₄³	C₂²×C₈	1	39680	yes	—
40	C₂²×C₃×C₄³	C₂³×C₄	1	8680	yes	—
41	C₂×C₃×C₄²×C₈	C₂×C₄²	1	555520	yes	—
42	C₃×C₄⁴	C₄×C₈	1	126976	yes	direct product factor
43	C₃×C₄⁴	C₂×C₄²	1	39680	yes	—
44	C₂⁴×C₄×C₈	C₂⁴×C₃	1	496	yes	—
45	C₂³×C₄³	C₂²×C₃×C₄	1	1240	yes	—
46	C₂³×C₄³	C₂⁴×C₃	1	155	yes	—
47	C₂²×C₄²×C₈	C₂²×C₃×C₄	1	69440	yes	—
48	C₂×C₄⁴	C₃×C₄²	1	7936	yes	—
49	C₂×C₄⁴	C₂×C₃×C₈	1	15872	yes	—
50	C₂×C₄⁴	C₂²×C₃×C₄	1	7440	yes	—
51	C₄³×C₈	C₃×C₄²	1	158720	yes	direct product factor
52	C₂⁵×C₃×C₄	C₂⁴×C₄	1	62	yes	—
53	C₂⁵×C₃×C₄	C₂⁶	1	1	yes	characteristic
54	C₂⁴×C₃×C₈	C₂⁴×C₄	1	1984	yes	—
55	C₂³×C₃×C₄²	C₂²×C₄²	1	17360	yes	—
56	C₂³×C₃×C₄²	C₂³×C₈	1	19840	yes	—
57	C₂³×C₃×C₄²	C₂⁴×C₄	1	1860	yes	—
58	C₂²×C₃×C₄×C₈	C₂²×C₄²	1	277760	yes	—
59	C₂×C₃×C₄³	C₄³	1	79360	yes	—
60	C₂×C₃×C₄³	C₂×C₄×C₈	1	476160	yes	—
61	C₂×C₃×C₄³	C₂²×C₄²	1	69440	yes	—
62	C₃×C₄²×C₈	C₄³	1	634880	yes	direct product factor
63	C₂⁵×C₈	C₂⁵×C₃	1	32	yes	—
64	C₂⁴×C₄²	C₂³×C₃×C₄	1	620	yes	—
65	C₂⁴×C₄²	C₂⁵×C₃	1	31	yes	—
66	C₂³×C₄×C₈	C₂³×C₃×C₄	1	29760	yes	—
67	C₂²×C₄³	C₂×C₃×C₄²	1	29760	yes	—
68	C₂²×C₄³	C₂²×C₃×C₈	1	39680	yes	—
69	C₂²×C₄³	C₂³×C₃×C₄	1	8680	yes	—
70	C₂×C₄²×C₈	C₂×C₃×C₄²	1	555520	yes	—
71	C₄⁴	C₃×C₄×C₈	1	126976	yes	direct product factor
72	C₄⁴	C₂×C₃×C₄²	1	39680	yes	—
73	C₂⁶×C₃	C₂⁵×C₄	1	1	yes	characteristic
74	C₂⁴×C₃×C₄	C₂³×C₄²	1	1860	yes	—
75	C₂⁴×C₃×C₄	C₂⁴×C₈	1	1984	yes	—
76	C₂⁴×C₃×C₄	C₂⁵×C₄	1	62	yes	—
77	C₂³×C₃×C₈	C₂³×C₄²	1	19840	yes	—
78	C₂²×C₃×C₄²	C₂×C₄³	1	69440	yes	—
79	C₂²×C₃×C₄²	C₂²×C₄×C₈	1	277760	yes	—
80	C₂²×C₃×C₄²	C₂³×C₄²	1	17360	yes	—
81	C₂×C₃×C₄×C₈	C₂×C₄³	1	476160	yes	—
82	C₃×C₄³	C₄²×C₈	1	634880	yes	direct product factor
83	C₃×C₄³	C₂×C₄³	1	79360	yes	—
84	C₂⁵×C₄	C₂⁴×C₃×C₄	1	62	yes	—
85	C₂⁵×C₄	C₂⁶×C₃	1	1	yes	characteristic
86	C₂⁴×C₈	C₂⁴×C₃×C₄	1	1984	yes	—
87	C₂³×C₄²	C₂²×C₃×C₄²	1	17360	yes	—
88	C₂³×C₄²	C₂³×C₃×C₈	1	19840	yes	—
89	C₂³×C₄²	C₂⁴×C₃×C₄	1	1860	yes	—
90	C₂²×C₄×C₈	C₂²×C₃×C₄²	1	277760	yes	—
91	C₂×C₄³	C₃×C₄³	1	79360	yes	—
92	C₂×C₄³	C₂×C₃×C₄×C₈	1	476160	yes	—
93	C₂×C₄³	C₂²×C₃×C₄²	1	69440	yes	—
94	C₄²×C₈	C₃×C₄³	1	634880	yes	direct product factor
95	C₂⁵×C₃	C₂⁴×C₄²	1	31	yes	—
96	C₂⁵×C₃	C₂⁵×C₈	1	32	yes	—
97	C₂³×C₃×C₄	C₂²×C₄³	1	8680	yes	—
98	C₂³×C₃×C₄	C₂³×C₄×C₈	1	29760	yes	—
99	C₂³×C₃×C₄	C₂⁴×C₄²	1	620	yes	—
100	C₂²×C₃×C₈	C₂²×C₄³	1	39680	yes	—
101	C₂×C₃×C₄²	C₄⁴	1	39680	yes	—
102	C₂×C₃×C₄²	C₂×C₄²×C₈	1	555520	yes	—
103	C₂×C₃×C₄²	C₂²×C₄³	1	29760	yes	—
104	C₃×C₄×C₈	C₄⁴	1	126976	yes	direct product factor
105	C₂⁶	C₂⁵×C₃×C₄	1	1	yes	characteristic
106	C₂⁴×C₄	C₂³×C₃×C₄²	1	1860	yes	—
107	C₂⁴×C₄	C₂⁴×C₃×C₈	1	1984	yes	—
108	C₂⁴×C₄	C₂⁵×C₃×C₄	1	62	yes	—
109	C₂³×C₈	C₂³×C₃×C₄²	1	19840	yes	—
110	C₂²×C₄²	C₂×C₃×C₄³	1	69440	yes	—
111	C₂²×C₄²	C₂²×C₃×C₄×C₈	1	277760	yes	—
112	C₂²×C₄²	C₂³×C₃×C₄²	1	17360	yes	—
113	C₂×C₄×C₈	C₂×C₃×C₄³	1	476160	yes	—
114	C₄³	C₃×C₄²×C₈	1	634880	yes	direct product factor
115	C₄³	C₂×C₃×C₄³	1	79360	yes	—
116	C₂⁴×C₃	C₂³×C₄³	1	155	yes	—
117	C₂⁴×C₃	C₂⁴×C₄×C₈	1	496	yes	—
118	C₂²×C₃×C₄	C₂×C₄⁴	1	7440	yes	—
119	C₂²×C₃×C₄	C₂²×C₄²×C₈	1	69440	yes	—
120	C₂²×C₃×C₄	C₂³×C₄³	1	1240	yes	—
121	C₂×C₃×C₈	C₂×C₄⁴	1	15872	yes	—
122	C₃×C₄²	C₄³×C₈	1	158720	yes	direct product factor
123	C₃×C₄²	C₂×C₄⁴	1	7936	yes	—
124	C₂⁵	C₂⁴×C₃×C₄²	1	31	yes	—
125	C₂⁵	C₂⁵×C₃×C₈	1	32	yes	—
126	C₂³×C₄	C₂²×C₃×C₄³	1	8680	yes	—
127	C₂³×C₄	C₂³×C₃×C₄×C₈	1	29760	yes	—
128	C₂³×C₄	C₂⁴×C₃×C₄²	1	620	yes	—
129	C₂²×C₈	C₂²×C₃×C₄³	1	39680	yes	—
130	C₂×C₄²	C₃×C₄⁴	1	39680	yes	—
131	C₂×C₄²	C₂×C₃×C₄²×C₈	1	555520	yes	—
132	C₂×C₄²	C₂²×C₃×C₄³	1	29760	yes	—
133	C₄×C₈	C₃×C₄⁴	1	126976	yes	direct product factor
134	C₂³×C₃	C₂³×C₄²×C₈	1	1240	yes	—
135	C₂³×C₃	C₂²×C₄⁴	1	155	yes	—
136	C₂×C₃×C₄	C₂×C₄³×C₈	1	29760	yes	—
137	C₂×C₃×C₄	C₄⁵	1	992	yes	—
138	C₂×C₃×C₄	C₂²×C₄⁴	1	496	yes	—
139	C₃×C₈	C₄⁵	1	1024	yes	direct product factor
140	C₂⁴	C₂⁴×C₃×C₄×C₈	1	496	yes	—
141	C₂⁴	C₂³×C₃×C₄³	1	155	yes	—
142	C₂²×C₄	C₂²×C₃×C₄²×C₈	1	69440	yes	—
143	C₂²×C₄	C₂×C₃×C₄⁴	1	7440	yes	—
144	C₂²×C₄	C₂³×C₃×C₄³	1	1240	yes	—
145	C₂×C₈	C₂×C₃×C₄⁴	1	15872	yes	—
146	C₄²	C₃×C₄³×C₈	1	158720	yes	direct product factor
147	C₄²	C₂×C₃×C₄⁴	1	7936	yes	—
148	C₂²×C₃	C₂²×C₄³×C₈	1	620	yes	—
149	C₂²×C₃	C₂×C₄⁵	1	31	yes	—
150	C₃×C₄	C₄⁴×C₈	1	1984	yes	direct product factor
151	C₃×C₄	C₂×C₄⁵	1	32	yes	—
152	C₂³	C₂³×C₃×C₄²×C₈	1	1240	yes	—
153	C₂³	C₂²×C₃×C₄⁴	1	155	yes	—
154	C₂×C₄	C₂×C₃×C₄³×C₈	1	29760	yes	—
155	C₂×C₄	C₃×C₄⁵	1	992	yes	—
156	C₂×C₄	C₂²×C₃×C₄⁴	1	496	yes	—
157	C₈	C₃×C₄⁵	1	1024	yes	direct product factor
158	C₂×C₃	C₂×C₄⁴×C₈	1	62	yes	—
159	C₂×C₃	C₄⁶	1	1	yes	characteristic
160	C₂²	C₂²×C₃×C₄³×C₈	1	620	yes	—
161	C₂²	C₂×C₃×C₄⁵	1	31	yes	—
162	C₄	C₃×C₄⁴×C₈	1	1984	yes	direct product factor
163	C₄	C₂×C₃×C₄⁵	1	32	yes	—
164	C₃	C₄⁵×C₈	1	1	yes	characteristic, direct product factor, Sylow
165	C₂	C₂×C₃×C₄⁴×C₈	1	62	yes	—
166	C₂	C₃×C₄⁶	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂⁵×C₄	C₂⁶×C₃
socle, prime radical	C₂⁶×C₃	C₂⁵×C₄
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄⁵×C₈	C₃	direct (N × H)
7. C₃×C₄⁴×C₈	C₄	direct (N × H)
12. C₃×C₄⁵	C₈	direct (N × H)
19. C₄⁴×C₈	C₃×C₄	direct (N × H)
27. C₃×C₄³×C₈	C₄²	direct (N × H)
30. C₄⁵	C₃×C₈	direct (N × H)
42. C₃×C₄⁴	C₄×C₈	direct (N × H)
51. C₄³×C₈	C₃×C₄²	direct (N × H)
62. C₃×C₄²×C₈	C₄³	direct (N × H)
71. C₄⁴	C₃×C₄×C₈	direct (N × H)
82. C₃×C₄³	C₄²×C₈	direct (N × H)
94. C₄²×C₈	C₃×C₄³	direct (N × H)
104. C₃×C₄×C₈	C₄⁴	direct (N × H)
114. C₄³	C₃×C₄²×C₈	direct (N × H)
122. C₃×C₄²	C₄³×C₈	direct (N × H)
133. C₄×C₈	C₃×C₄⁴	direct (N × H)
139. C₃×C₈	C₄⁵	direct (N × H)
146. C₄²	C₃×C₄³×C₈	direct (N × H)
150. C₃×C₄	C₄⁴×C₈	direct (N × H)
157. C₈	C₃×C₄⁵	direct (N × H)
162. C₄	C₃×C₄⁴×C₈	direct (N × H)
164. C₃	C₄⁵×C₈	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×C₄⁶	2
C₂×C₃×C₄⁴×C₈	2
C₄⁵×C₈	3

Subgroups — C3×C45×C8 #72201899

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₄⁵×C₈ #72201899