Subgroups — C₂×C₄×#131 #60055

Order 256.

1683 conjugacy classes of subgroups.

The subgroup lattice has 119 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#131	C₂	1	2	yes	maximal
2	C₂×C₄×#41	C₂	1	1	yes	maximal, characteristic
3	C₄×#131	C₂	1	16	yes	maximal, direct product factor
4	C₂×#405	C₂	1	6	yes	maximal, complemented
5	C₂×C₄×M₁₆	C₂	1	3	yes	maximal, complemented
6	C₂²×C₄×C₈	C₂	1	3	yes	maximal, complemented
7	C₂²×#41	C₂²	1	1	yes	characteristic
8	C₂×#131	C₄	1	32	yes	direct product factor
9	C₂×#131	C₂²	1	16	yes	—
10	C₂²×M₁₆	C₂²	1	6	yes	—
11	C₂³×C₈	C₂²	1	6	yes	—
12	C₄×#41	C₄	1	8	yes	—
13	C₄×#41	C₂²	1	8	yes	—
14	C₂×C₄×Q₈	C₄	1	2	yes	—
15	C₂×C₄×D₄	C₄	1	6	yes	—
16	C₂×#117	C₂²	1	3	yes	—
17	C₂²×C₄²	C₂²	1	3	yes	—
18	#405	C₂²	1	48	yes	complemented
19	C₄×M₁₆	C₂²	1	24	yes	complemented
20	C₂×#97	C₂²	1	6	yes	—
21	C₂×C₄×C₈	C₂²	1	3	yes	—
22	C₂×C₄×C₈	C₂²	1	6	yes	—
23	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈	C₂²	1	24	yes	complemented
25	C₂×#41	C₂×C₄	1	16	yes	—
26	C₂×#41	C₂×C₄	1	4	yes	—
27	C₂×#41	C₂³	1	4	yes	—
28	C₂²×Q₈	C₂×C₄	1	1	yes	characteristic
29	C₂²×D₄	C₂×C₄	1	3	yes	—
30	C₂³×C₄	C₂×C₄	1	6	yes	—
31	C₂³×C₄	C₂³	1	3	yes	—
32	#131	C₂×C₄	1	128	yes	direct product factor
33	C₂×M₁₆	C₂×C₄	1	48	yes	complemented
34	C₂×M₁₆	C₂³	1	24	yes	—
35	C₂²×C₈	C₂×C₄	1	48	yes	complemented
36	C₂²×C₈	C₂³	1	2	yes	—
37	C₂²×C₈	C₂³	1	6	yes	—
38	C₂²×C₈	C₂³	1	24	yes	—
39	C₄×Q₈	C₂×C₄	1	16	yes	—
40	C₄×D₄	C₂×C₄	1	48	yes	—
41	#117	C₂×C₄	1	12	yes	—
42	#117	C₂³	1	12	yes	—
43	C₂×C₄⋊C₄	C₂×C₄	1	6	yes	—
44	C₂×C₂²⋊C₄	C₂×C₄	1	6	yes	—
45	C₂×C₄²	C₂×C₄	1	6	yes	—
46	C₂×C₄²	C₂×C₄	1	12	yes	—
47	C₂×C₄²	C₂³	1	3	yes	—
48	C₂×C₄²	C₂³	1	12	yes	—
49	C₂×C₄²	C₂³	1	1	yes	characteristic
50	#97	C₂³	1	24	yes	—
51	C₄×C₈	C₂³	1	12	yes	—
52	C₄×C₈	C₂³	1	24	yes	—
53	C₄×C₈	C₂³	1	4	yes	—
54	C₂⁴	C₂²×C₄	1	3	yes	—
55	#41	C₄²	1	32	yes	—
56	#41	C₂²×C₄	1	32	yes	—
57	C₂×Q₈	C₄²	1	8	yes	—
58	C₂×Q₈	C₂²×C₄	1	4	yes	—
59	C₂×D₄	C₄²	1	24	yes	—
60	C₂×D₄	C₂²×C₄	1	12	yes	—
61	C₂²×C₄	—	2	24	no	—
62	C₂²×C₄	C₂²×C₄	1	6	yes	—
63	C₂²×C₄	C₂²×C₄	1	24	yes	—
64	C₂²×C₄	C₂²×C₄	1	3	yes	—
65	C₂²×C₄	C₂²×C₄	1	6	yes	—
66	C₂²×C₄	C₂²×C₄	1	24	yes	—
67	C₂²×C₄	C₂²×C₄	1	2	yes	—
68	C₂²×C₄	C₂⁴	1	1	yes	characteristic
69	C₂²×C₄	C₂⁴	1	6	yes	—
70	M₁₆	C₂²×C₄	1	96	yes	complemented
71	C₂×C₈	C₂²×C₄	1	8	yes	—
72	C₂×C₈	C₂²×C₄	1	24	yes	—
73	C₂×C₈	C₂²×C₄	1	96	yes	complemented
74	C₂×C₈	C₂⁴	1	4	yes	—
75	C₂×C₈	C₂⁴	1	12	yes	—
76	C₄⋊C₄	C₂²×C₄	1	24	yes	—
77	C₂²⋊C₄	C₂²×C₄	1	24	yes	—
78	C₄²	C₂²×C₄	1	2	yes	—
79	C₄²	C₂²×C₄	1	6	yes	—
80	C₄²	C₂²×C₄	1	24	yes	—
81	C₄²	C₂⁴	1	2	yes	—
82	C₄²	C₂⁴	1	6	yes	—
83	C₂³	—	2	12	no	—
84	C₂³	C₂×C₄²	1	12	yes	—
85	C₂³	C₂³×C₄	1	6	yes	—
86	C₂³	C₂³×C₄	1	1	yes	characteristic
87	Q₈	C₂×C₄²	1	16	yes	—
88	D₄	C₂×C₄²	1	48	yes	—
89	C₂×C₄	—	2	12	no	—
90	C₂×C₄	—	2	48	no	—
91	C₂×C₄	C₂×C₄²	1	12	yes	—
92	C₂×C₄	C₂×C₄²	1	24	yes	—
93	C₂×C₄	#131	1	8	yes	direct product factor
94	C₂×C₄	C₂³×C₄	1	12	yes	—
95	C₂×C₄	C₂³×C₄	1	4	yes	—
96	C₂×C₄	C₂³×C₄	1	4	yes	—
97	C₂×C₄	C₂³×C₄	1	24	yes	—
98	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
99	C₂×C₄	C₂³×C₄	1	6	yes	—
100	C₂×C₄	C₂⁵	1	1	yes	characteristic
101	C₈	C₂³×C₄	1	8	yes	—
102	C₈	C₂³×C₄	1	24	yes	—
103	C₂²	—	2	24	no	—
104	C₂²	—	2	12	no	—
105	C₂²	C₂²×C₄²	1	12	yes	—
106	C₂²	C₂²×C₄²	1	2	yes	—
107	C₂²	C₂×#131	1	4	yes	—
108	C₂²	C₂⁴×C₄	1	1	yes	characteristic
109	C₄	—	2	12	no	—
110	C₄	C₂²×C₄²	1	2	yes	—
111	C₄	C₂²×C₄²	1	12	yes	—
112	C₄	C₂×#131	1	8	yes	direct product factor
113	C₄	C₂⁴×C₄	1	2	yes	—
114	C₂	—	2	12	no	—
115	C₂	C₄×#131	1	4	yes	direct product factor
116	C₂	C₂³×C₄²	1	1	yes	characteristic
117	C₂	C₂²×#131	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₈	C₂²
commutator	C₂	C₂³×C₄²
Frattini, cosocle	C₂×C₄	C₂⁵
socle	C₂³	C₂³×C₄
prime radical	C₂²×#41	C₂²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. C₄×#131	C₂	direct (N × H)
4. C₂×#405	C₂	semidirect
5. C₂×C₄×M₁₆	C₂	semidirect
6. C₂²×C₄×C₈	C₂	semidirect
8. C₂×#131	C₄	direct (N × H)
18. #405	C₂²	semidirect
19. C₄×M₁₆	C₂²	semidirect
24. C₂×C₄×C₈	C₂²	semidirect
32. #131	C₂×C₄	direct (N × H)
33. C₂×M₁₆	C₂×C₄	semidirect
35. C₂²×C₈	C₂×C₄	semidirect
70. M₁₆	C₂²×C₄	semidirect
73. C₂×C₈	C₂²×C₄	semidirect
93. C₂×C₄	#131	direct (N × H)
112. C₄	C₂×#131	direct (N × H)
115. C₂	C₄×#131	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#131	2
C₂×C₄×#41	2
C₄×#131	2
C₂×#405	2
C₂×C₄×M₁₆	2
C₂²×C₄×C₈	2

Subgroups — C2×C4×#131 #60055

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#131 #60055