Subgroups — C₃×C₄⁴×C₈ #72185337

Order 6144.

196944 conjugacy classes of subgroups.

The subgroup lattice has 120 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄⁵	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄³×C₈	C₂	1	30	yes	maximal
3	C₄⁴×C₈	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
4	C₂×C₃×C₄⁴	C₄	1	16	yes	—
5	C₂×C₃×C₄⁴	C₂²	1	15	yes	—
6	C₂²×C₃×C₄²×C₈	C₂²	1	140	yes	—
7	C₃×C₄³×C₈	C₄	1	480	yes	direct product factor
8	C₄⁵	C₂×C₃	1	1	yes	characteristic
9	C₂×C₄³×C₈	C₂×C₃	1	30	yes	—
10	C₂³×C₃×C₄×C₈	C₂³	1	120	yes	—
11	C₂²×C₃×C₄³	C₂×C₄	1	120	yes	—
12	C₂²×C₃×C₄³	C₂³	1	35	yes	—
13	C₂×C₃×C₄²×C₈	C₂×C₄	1	3360	yes	—
14	C₃×C₄⁴	C₈	1	256	yes	direct product factor
15	C₃×C₄⁴	C₂×C₄	1	240	yes	—
16	C₂²×C₄²×C₈	C₂²×C₃	1	140	yes	—
17	C₂×C₄⁴	C₃×C₄	1	16	yes	—
18	C₂×C₄⁴	C₂²×C₃	1	15	yes	—
19	C₄³×C₈	C₃×C₄	1	480	yes	direct product factor
20	C₂⁴×C₃×C₈	C₂⁴	1	16	yes	—
21	C₂³×C₃×C₄²	C₂²×C₄	1	140	yes	—
22	C₂³×C₃×C₄²	C₂⁴	1	15	yes	—
23	C₂²×C₃×C₄×C₈	C₂²×C₄	1	3360	yes	—
24	C₂×C₃×C₄³	C₄²	1	960	yes	—
25	C₂×C₃×C₄³	C₂×C₈	1	1920	yes	—
26	C₂×C₃×C₄³	C₂²×C₄	1	840	yes	—
27	C₃×C₄²×C₈	C₄²	1	8960	yes	direct product factor
28	C₂³×C₄×C₈	C₂³×C₃	1	120	yes	—
29	C₂²×C₄³	C₂×C₃×C₄	1	120	yes	—
30	C₂²×C₄³	C₂³×C₃	1	35	yes	—
31	C₂×C₄²×C₈	C₂×C₃×C₄	1	3360	yes	—
32	C₄⁴	C₃×C₈	1	256	yes	direct product factor
33	C₄⁴	C₂×C₃×C₄	1	240	yes	—
34	C₂⁴×C₃×C₄	C₂³×C₄	1	30	yes	—
35	C₂⁴×C₃×C₄	C₂⁵	1	1	yes	characteristic
36	C₂³×C₃×C₈	C₂³×C₄	1	480	yes	—
37	C₂²×C₃×C₄²	C₂×C₄²	1	1680	yes	—
38	C₂²×C₃×C₄²	C₂²×C₈	1	2240	yes	—
39	C₂²×C₃×C₄²	C₂³×C₄	1	420	yes	—
40	C₂×C₃×C₄×C₈	C₂×C₄²	1	13440	yes	—
41	C₃×C₄³	C₄×C₈	1	7680	yes	direct product factor
42	C₃×C₄³	C₂×C₄²	1	2240	yes	—
43	C₂⁴×C₈	C₂⁴×C₃	1	16	yes	—
44	C₂³×C₄²	C₂²×C₃×C₄	1	140	yes	—
45	C₂³×C₄²	C₂⁴×C₃	1	15	yes	—
46	C₂²×C₄×C₈	C₂²×C₃×C₄	1	3360	yes	—
47	C₂×C₄³	C₃×C₄²	1	960	yes	—
48	C₂×C₄³	C₂×C₃×C₈	1	1920	yes	—
49	C₂×C₄³	C₂²×C₃×C₄	1	840	yes	—
50	C₄²×C₈	C₃×C₄²	1	8960	yes	direct product factor
51	C₂⁵×C₃	C₂⁴×C₄	1	1	yes	characteristic
52	C₂³×C₃×C₄	C₂²×C₄²	1	420	yes	—
53	C₂³×C₃×C₄	C₂³×C₈	1	480	yes	—
54	C₂³×C₃×C₄	C₂⁴×C₄	1	30	yes	—
55	C₂²×C₃×C₈	C₂²×C₄²	1	2240	yes	—
56	C₂×C₃×C₄²	C₄³	1	2240	yes	—
57	C₂×C₃×C₄²	C₂×C₄×C₈	1	13440	yes	—
58	C₂×C₃×C₄²	C₂²×C₄²	1	1680	yes	—
59	C₃×C₄×C₈	C₄³	1	7680	yes	direct product factor
60	C₂⁴×C₄	C₂³×C₃×C₄	1	30	yes	—
61	C₂⁴×C₄	C₂⁵×C₃	1	1	yes	characteristic
62	C₂³×C₈	C₂³×C₃×C₄	1	480	yes	—
63	C₂²×C₄²	C₂×C₃×C₄²	1	1680	yes	—
64	C₂²×C₄²	C₂²×C₃×C₈	1	2240	yes	—
65	C₂²×C₄²	C₂³×C₃×C₄	1	420	yes	—
66	C₂×C₄×C₈	C₂×C₃×C₄²	1	13440	yes	—
67	C₄³	C₃×C₄×C₈	1	7680	yes	direct product factor
68	C₄³	C₂×C₃×C₄²	1	2240	yes	—
69	C₂⁴×C₃	C₂³×C₄²	1	15	yes	—
70	C₂⁴×C₃	C₂⁴×C₈	1	16	yes	—
71	C₂²×C₃×C₄	C₂×C₄³	1	840	yes	—
72	C₂²×C₃×C₄	C₂²×C₄×C₈	1	3360	yes	—
73	C₂²×C₃×C₄	C₂³×C₄²	1	140	yes	—
74	C₂×C₃×C₈	C₂×C₄³	1	1920	yes	—
75	C₃×C₄²	C₄²×C₈	1	8960	yes	direct product factor
76	C₃×C₄²	C₂×C₄³	1	960	yes	—
77	C₂⁵	C₂⁴×C₃×C₄	1	1	yes	characteristic
78	C₂³×C₄	C₂²×C₃×C₄²	1	420	yes	—
79	C₂³×C₄	C₂³×C₃×C₈	1	480	yes	—
80	C₂³×C₄	C₂⁴×C₃×C₄	1	30	yes	—
81	C₂²×C₈	C₂²×C₃×C₄²	1	2240	yes	—
82	C₂×C₄²	C₃×C₄³	1	2240	yes	—
83	C₂×C₄²	C₂×C₃×C₄×C₈	1	13440	yes	—
84	C₂×C₄²	C₂²×C₃×C₄²	1	1680	yes	—
85	C₄×C₈	C₃×C₄³	1	7680	yes	direct product factor
86	C₂³×C₃	C₂²×C₄³	1	35	yes	—
87	C₂³×C₃	C₂³×C₄×C₈	1	120	yes	—
88	C₂×C₃×C₄	C₄⁴	1	240	yes	—
89	C₂×C₃×C₄	C₂×C₄²×C₈	1	3360	yes	—
90	C₂×C₃×C₄	C₂²×C₄³	1	120	yes	—
91	C₃×C₈	C₄⁴	1	256	yes	direct product factor
92	C₂⁴	C₂³×C₃×C₄²	1	15	yes	—
93	C₂⁴	C₂⁴×C₃×C₈	1	16	yes	—
94	C₂²×C₄	C₂×C₃×C₄³	1	840	yes	—
95	C₂²×C₄	C₂²×C₃×C₄×C₈	1	3360	yes	—
96	C₂²×C₄	C₂³×C₃×C₄²	1	140	yes	—
97	C₂×C₈	C₂×C₃×C₄³	1	1920	yes	—
98	C₄²	C₃×C₄²×C₈	1	8960	yes	direct product factor
99	C₄²	C₂×C₃×C₄³	1	960	yes	—
100	C₂²×C₃	C₂×C₄⁴	1	15	yes	—
101	C₂²×C₃	C₂²×C₄²×C₈	1	140	yes	—
102	C₃×C₄	C₄³×C₈	1	480	yes	direct product factor
103	C₃×C₄	C₂×C₄⁴	1	16	yes	—
104	C₂³	C₂²×C₃×C₄³	1	35	yes	—
105	C₂³	C₂³×C₃×C₄×C₈	1	120	yes	—
106	C₂×C₄	C₃×C₄⁴	1	240	yes	—
107	C₂×C₄	C₂×C₃×C₄²×C₈	1	3360	yes	—
108	C₂×C₄	C₂²×C₃×C₄³	1	120	yes	—
109	C₈	C₃×C₄⁴	1	256	yes	direct product factor
110	C₂×C₃	C₂×C₄³×C₈	1	30	yes	—
111	C₂×C₃	C₄⁵	1	1	yes	characteristic
112	C₂²	C₂²×C₃×C₄²×C₈	1	140	yes	—
113	C₂²	C₂×C₃×C₄⁴	1	15	yes	—
114	C₄	C₃×C₄³×C₈	1	480	yes	direct product factor
115	C₄	C₂×C₃×C₄⁴	1	16	yes	—
116	C₃	C₄⁴×C₈	1	1	yes	characteristic, direct product factor, Sylow
117	C₂	C₂×C₃×C₄³×C₈	1	30	yes	—
118	C₂	C₃×C₄⁵	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂⁴×C₄	C₂⁵×C₃
socle, prime radical	C₂⁵×C₃	C₂⁴×C₄
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄⁴×C₈	C₃	direct (N × H)
7. C₃×C₄³×C₈	C₄	direct (N × H)
14. C₃×C₄⁴	C₈	direct (N × H)
19. C₄³×C₈	C₃×C₄	direct (N × H)
27. C₃×C₄²×C₈	C₄²	direct (N × H)
32. C₄⁴	C₃×C₈	direct (N × H)
41. C₃×C₄³	C₄×C₈	direct (N × H)
50. C₄²×C₈	C₃×C₄²	direct (N × H)
59. C₃×C₄×C₈	C₄³	direct (N × H)
67. C₄³	C₃×C₄×C₈	direct (N × H)
75. C₃×C₄²	C₄²×C₈	direct (N × H)
85. C₄×C₈	C₃×C₄³	direct (N × H)
91. C₃×C₈	C₄⁴	direct (N × H)
98. C₄²	C₃×C₄²×C₈	direct (N × H)
102. C₃×C₄	C₄³×C₈	direct (N × H)
109. C₈	C₃×C₄⁴	direct (N × H)
114. C₄	C₃×C₄³×C₈	direct (N × H)
116. C₃	C₄⁴×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×C₄⁵	2
C₂×C₃×C₄³×C₈	2
C₄⁴×C₈	3

Subgroups — C3×C44×C8 #72185337

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₄⁴×C₈ #72185337