Subgroups — C₂×C₃×C₄×C₈² #72181985

Order 1536.

3844 conjugacy classes of subgroups.

The subgroup lattice has 144 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄²×C₈	C₂	1	3	yes	maximal
2	C₂²×C₃×C₈²	C₂	1	4	yes	maximal
3	C₃×C₄×C₈²	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄×C₈²	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
5	C₂²×C₃×C₄×C₈	C₄	1	12	yes	—
6	C₂²×C₃×C₄×C₈	C₂²	1	6	yes	—
7	C₂×C₃×C₄³	C₂²	1	1	yes	characteristic
8	C₃×C₄²×C₈	C₄	1	12	yes	—
9	C₃×C₄²×C₈	C₂²	1	12	yes	—
10	C₂×C₃×C₈²	C₄	1	32	yes	direct product factor
11	C₂×C₃×C₈²	C₂²	1	16	yes	—
12	C₂×C₄²×C₈	C₂×C₃	1	3	yes	—
13	C₂²×C₈²	C₂×C₃	1	4	yes	—
14	C₄×C₈²	C₂×C₃	1	8	yes	direct product factor
15	C₂³×C₃×C₈	C₂×C₄	1	12	yes	—
16	C₂²×C₃×C₄²	C₂×C₄	1	6	yes	—
17	C₂²×C₃×C₄²	C₂³	1	1	yes	characteristic
18	C₂×C₃×C₄×C₈	C₈	1	96	yes	direct product factor
19	C₂×C₃×C₄×C₈	C₂×C₄	1	12	yes	—
20	C₂×C₃×C₄×C₈	C₂×C₄	1	48	yes	—
21	C₂×C₃×C₄×C₈	C₂×C₄	1	48	yes	—
22	C₂×C₃×C₄×C₈	C₂³	1	12	yes	—
23	C₃×C₄³	C₂×C₄	1	6	yes	—
24	C₃×C₄³	C₂³	1	2	yes	—
25	C₃×C₈²	C₂×C₄	1	64	yes	direct product factor
26	C₂²×C₄×C₈	C₃×C₄	1	12	yes	—
27	C₂²×C₄×C₈	C₂²×C₃	1	6	yes	—
28	C₂×C₄³	C₂²×C₃	1	1	yes	characteristic
29	C₄²×C₈	C₃×C₄	1	12	yes	—
30	C₄²×C₈	C₂²×C₃	1	12	yes	—
31	C₂×C₈²	C₃×C₄	1	32	yes	direct product factor
32	C₂×C₈²	C₂²×C₃	1	16	yes	—
33	C₂³×C₃×C₄	C₄²	1	4	yes	—
34	C₂³×C₃×C₄	C₂²×C₄	1	3	yes	—
35	C₂²×C₃×C₈	C₄²	1	48	yes	—
36	C₂²×C₃×C₈	C₂×C₈	1	96	yes	—
37	C₂²×C₃×C₈	C₂²×C₄	1	24	yes	—
38	C₂×C₃×C₄²	C₄²	1	12	yes	—
39	C₂×C₃×C₄²	C₂×C₈	1	48	yes	—
40	C₂×C₃×C₄²	C₂²×C₄	1	3	yes	—
41	C₂×C₃×C₄²	C₂²×C₄	1	8	yes	—
42	C₂×C₃×C₄²	C₂²×C₄	1	12	yes	—
43	C₂×C₃×C₄²	C₂⁴	1	1	yes	characteristic
44	C₃×C₄×C₈	C₄²	1	48	yes	—
45	C₃×C₄×C₈	C₂×C₈	1	192	yes	direct product factor
46	C₃×C₄×C₈	C₂²×C₄	1	48	yes	—
47	C₂³×C₈	C₂×C₃×C₄	1	12	yes	—
48	C₂²×C₄²	C₂×C₃×C₄	1	6	yes	—
49	C₂²×C₄²	C₂³×C₃	1	1	yes	characteristic
50	C₂×C₄×C₈	C₃×C₈	1	96	yes	direct product factor
51	C₂×C₄×C₈	C₂×C₃×C₄	1	12	yes	—
52	C₂×C₄×C₈	C₂×C₃×C₄	1	48	yes	—
53	C₂×C₄×C₈	C₂×C₃×C₄	1	48	yes	—
54	C₂×C₄×C₈	C₂³×C₃	1	12	yes	—
55	C₄³	C₂×C₃×C₄	1	6	yes	—
56	C₄³	C₂³×C₃	1	2	yes	—
57	C₈²	C₂×C₃×C₄	1	64	yes	direct product factor
58	C₂⁴×C₃	C₂×C₄²	1	1	yes	characteristic
59	C₂²×C₃×C₄	C₄×C₈	1	48	yes	—
60	C₂²×C₃×C₄	C₂×C₄²	1	3	yes	—
61	C₂²×C₃×C₄	C₂×C₄²	1	8	yes	—
62	C₂²×C₃×C₄	C₂×C₄²	1	12	yes	—
63	C₂²×C₃×C₄	C₂²×C₈	1	24	yes	—
64	C₂²×C₃×C₄	C₂³×C₄	1	3	yes	—
65	C₂×C₃×C₈	C₄×C₈	1	192	yes	direct product factor
66	C₂×C₃×C₈	C₂×C₄²	1	48	yes	—
67	C₂×C₃×C₈	C₂²×C₈	1	96	yes	—
68	C₃×C₄²	C₄×C₈	1	48	yes	—
69	C₃×C₄²	C₂×C₄²	1	12	yes	—
70	C₃×C₄²	C₂²×C₈	1	48	yes	—
71	C₃×C₄²	C₂³×C₄	1	4	yes	—
72	C₂³×C₄	C₃×C₄²	1	4	yes	—
73	C₂³×C₄	C₂²×C₃×C₄	1	3	yes	—
74	C₂²×C₈	C₃×C₄²	1	48	yes	—
75	C₂²×C₈	C₂×C₃×C₈	1	96	yes	—
76	C₂²×C₈	C₂²×C₃×C₄	1	24	yes	—
77	C₂×C₄²	C₃×C₄²	1	12	yes	—
78	C₂×C₄²	C₂×C₃×C₈	1	48	yes	—
79	C₂×C₄²	C₂²×C₃×C₄	1	3	yes	—
80	C₂×C₄²	C₂²×C₃×C₄	1	8	yes	—
81	C₂×C₄²	C₂²×C₃×C₄	1	12	yes	—
82	C₂×C₄²	C₂⁴×C₃	1	1	yes	characteristic
83	C₄×C₈	C₃×C₄²	1	48	yes	—
84	C₄×C₈	C₂×C₃×C₈	1	192	yes	direct product factor
85	C₄×C₈	C₂²×C₃×C₄	1	48	yes	—
86	C₂³×C₃	C₄³	1	2	yes	—
87	C₂³×C₃	C₂×C₄×C₈	1	12	yes	—
88	C₂³×C₃	C₂²×C₄²	1	1	yes	characteristic
89	C₂×C₃×C₄	C₈²	1	64	yes	direct product factor
90	C₂×C₃×C₄	C₄³	1	6	yes	—
91	C₂×C₃×C₄	C₂×C₄×C₈	1	12	yes	—
92	C₂×C₃×C₄	C₂×C₄×C₈	1	48	yes	—
93	C₂×C₃×C₄	C₂×C₄×C₈	1	48	yes	—
94	C₂×C₃×C₄	C₂²×C₄²	1	6	yes	—
95	C₂×C₃×C₄	C₂³×C₈	1	12	yes	—
96	C₃×C₈	C₂×C₄×C₈	1	96	yes	direct product factor
97	C₂⁴	C₂×C₃×C₄²	1	1	yes	characteristic
98	C₂²×C₄	C₃×C₄×C₈	1	48	yes	—
99	C₂²×C₄	C₂×C₃×C₄²	1	3	yes	—
100	C₂²×C₄	C₂×C₃×C₄²	1	8	yes	—
101	C₂²×C₄	C₂×C₃×C₄²	1	12	yes	—
102	C₂²×C₄	C₂²×C₃×C₈	1	24	yes	—
103	C₂²×C₄	C₂³×C₃×C₄	1	3	yes	—
104	C₂×C₈	C₃×C₄×C₈	1	192	yes	direct product factor
105	C₂×C₈	C₂×C₃×C₄²	1	48	yes	—
106	C₂×C₈	C₂²×C₃×C₈	1	96	yes	—
107	C₄²	C₃×C₄×C₈	1	48	yes	—
108	C₄²	C₂×C₃×C₄²	1	12	yes	—
109	C₄²	C₂²×C₃×C₈	1	48	yes	—
110	C₄²	C₂³×C₃×C₄	1	4	yes	—
111	C₂²×C₃	C₂×C₈²	1	16	yes	—
112	C₂²×C₃	C₄²×C₈	1	12	yes	—
113	C₂²×C₃	C₂×C₄³	1	1	yes	characteristic
114	C₂²×C₃	C₂²×C₄×C₈	1	6	yes	—
115	C₃×C₄	C₂×C₈²	1	32	yes	direct product factor
116	C₃×C₄	C₄²×C₈	1	12	yes	—
117	C₃×C₄	C₂²×C₄×C₈	1	12	yes	—
118	C₂³	C₃×C₄³	1	2	yes	—
119	C₂³	C₂×C₃×C₄×C₈	1	12	yes	—
120	C₂³	C₂²×C₃×C₄²	1	1	yes	characteristic
121	C₂×C₄	C₃×C₈²	1	64	yes	direct product factor
122	C₂×C₄	C₃×C₄³	1	6	yes	—
123	C₂×C₄	C₂×C₃×C₄×C₈	1	12	yes	—
124	C₂×C₄	C₂×C₃×C₄×C₈	1	48	yes	—
125	C₂×C₄	C₂×C₃×C₄×C₈	1	48	yes	—
126	C₂×C₄	C₂²×C₃×C₄²	1	6	yes	—
127	C₂×C₄	C₂³×C₃×C₈	1	12	yes	—
128	C₈	C₂×C₃×C₄×C₈	1	96	yes	direct product factor
129	C₂×C₃	C₄×C₈²	1	8	yes	direct product factor
130	C₂×C₃	C₂²×C₈²	1	4	yes	—
131	C₂×C₃	C₂×C₄²×C₈	1	3	yes	—
132	C₂²	C₂×C₃×C₈²	1	16	yes	—
133	C₂²	C₃×C₄²×C₈	1	12	yes	—
134	C₂²	C₂×C₃×C₄³	1	1	yes	characteristic
135	C₂²	C₂²×C₃×C₄×C₈	1	6	yes	—
136	C₄	C₂×C₃×C₈²	1	32	yes	direct product factor
137	C₄	C₃×C₄²×C₈	1	12	yes	—
138	C₄	C₂²×C₃×C₄×C₈	1	12	yes	—
139	C₃	C₂×C₄×C₈²	1	1	yes	characteristic, direct product factor, Sylow
140	C₂	C₃×C₄×C₈²	1	8	yes	direct product factor
141	C₂	C₂²×C₃×C₈²	1	4	yes	—
142	C₂	C₂×C₃×C₄²×C₈	1	3	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₄²	C₂⁴×C₃
socle, prime radical	C₂⁴×C₃	C₂×C₄²
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃×C₄×C₈²	C₂	direct (N × H)
4. C₂×C₄×C₈²	C₃	direct (N × H)
10. C₂×C₃×C₈²	C₄	direct (N × H)
14. C₄×C₈²	C₂×C₃	direct (N × H)
18. C₂×C₃×C₄×C₈	C₈	direct (N × H)
25. C₃×C₈²	C₂×C₄	direct (N × H)
31. C₂×C₈²	C₃×C₄	direct (N × H)
45. C₃×C₄×C₈	C₂×C₈	direct (N × H)
50. C₂×C₄×C₈	C₃×C₈	direct (N × H)
57. C₈²	C₂×C₃×C₄	direct (N × H)
65. C₂×C₃×C₈	C₄×C₈	direct (N × H)
84. C₄×C₈	C₂×C₃×C₈	direct (N × H)
89. C₂×C₃×C₄	C₈²	direct (N × H)
96. C₃×C₈	C₂×C₄×C₈	direct (N × H)
104. C₂×C₈	C₃×C₄×C₈	direct (N × H)
115. C₃×C₄	C₂×C₈²	direct (N × H)
121. C₂×C₄	C₃×C₈²	direct (N × H)
128. C₈	C₂×C₃×C₄×C₈	direct (N × H)
129. C₂×C₃	C₄×C₈²	direct (N × H)
136. C₄	C₂×C₃×C₈²	direct (N × H)
139. C₃	C₂×C₄×C₈²	direct (N × H)
140. C₂	C₃×C₄×C₈²	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄²×C₈	2
C₂²×C₃×C₈²	2
C₃×C₄×C₈²	2
C₂×C₄×C₈²	3

Subgroups — C2×C3×C4×C82 #72181985

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₄×C₈² #72181985