Subgroups — C₂²×#431 #60114

Order 256.

1543 conjugacy classes of subgroups.

The subgroup lattice has 131 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄²	C₂	1	1	yes	maximal, characteristic
2	C₂×#431	C₂	1	24	yes	maximal, direct product factor
3	C₂²×C₄⋊C₈	C₂	1	2	yes	maximal, complemented
4	C₂²×C₂²⋊C₈	C₂	1	2	yes	maximal
5	C₂²×C₄×C₈	C₂	1	2	yes	maximal, complemented
6	C₂⁴×C₄	C₄	1	1	yes	characteristic
7	C₂⁴×C₄	C₄	1	1	yes	characteristic
8	C₂⁴×C₄	C₂²	1	1	yes	characteristic
9	C₂³×C₈	C₂²	1	4	yes	—
10	C₂²×C₄²	C₄	1	1	yes	characteristic
11	C₂²×C₄²	C₄	1	1	yes	characteristic
12	C₂²×C₄²	C₄	1	12	yes	—
13	C₂²×C₄²	C₂²	1	1	yes	characteristic
14	C₂²×C₄²	C₂²	1	12	yes	—
15	C₂²×C₄²	C₂²	1	1	yes	characteristic
16	#431	C₂²	1	64	yes	direct product factor
17	C₂×C₄⋊C₈	C₂²	1	24	yes	complemented
18	C₂×C₂²⋊C₈	C₂²	1	24	yes	—
19	C₂×C₄×C₈	C₂²	1	24	yes	complemented
20	C₂⁵	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄	—	2	8	no	—
22	C₂³×C₄	—	2	8	no	—
23	C₂³×C₄	C₈	1	16	yes	complemented
24	C₂³×C₄	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄	C₂×C₄	1	12	yes	—
26	C₂³×C₄	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄	C₂×C₄	1	6	yes	—
28	C₂³×C₄	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₄	C₂×C₄	1	12	yes	—
31	C₂³×C₄	C₂×C₄	1	1	yes	characteristic
32	C₂³×C₄	C₂³	1	6	yes	—
33	C₂³×C₄	C₂³	1	1	yes	characteristic
34	C₂²×C₈	—	2	16	no	—
35	C₂²×C₈	C₂³	1	24	yes	—
36	C₂×C₄²	C₂×C₄	1	12	yes	—
37	C₂×C₄²	C₂×C₄	1	48	yes	—
38	C₂×C₄²	C₂×C₄	1	12	yes	—
39	C₂×C₄²	C₂×C₄	1	6	yes	—
40	C₂×C₄²	C₂×C₄	1	6	yes	—
41	C₂×C₄²	C₂³	1	6	yes	—
42	C₂×C₄²	C₂³	1	6	yes	—
43	C₂×C₄²	C₂³	1	16	yes	—
44	C₄⋊C₈	C₂³	1	32	yes	complemented
45	C₂²⋊C₈	C₂³	1	32	yes	—
46	C₄×C₈	C₂³	1	32	yes	complemented
47	C₂⁴	—	2	4	no	—
48	C₂⁴	—	2	4	no	—
49	C₂⁴	C₂×C₈	1	8	yes	—
50	C₂⁴	C₂²×C₄	1	6	yes	—
51	C₂⁴	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄	—	2	4	no	—
53	C₂²×C₄	—	2	4	no	—
54	C₂²×C₄	—	2	24	no	—
55	C₂²×C₄	—	2	48	no	—
56	C₂²×C₄	—	2	24	no	—
57	C₂²×C₄	C₂×C₈	1	8	yes	—
58	C₂²×C₄	C₂×C₈	1	8	yes	—
59	C₂²×C₄	C₂×C₈	1	96	yes	complemented
60	C₂²×C₄	M₁₆	1	8	yes	—
61	C₂²×C₄	C₂²×C₄	1	6	yes	—
62	C₂²×C₄	C₂²×C₄	1	12	yes	—
63	C₂²×C₄	C₂²×C₄	1	6	yes	—
64	C₂²×C₄	C₂²×C₄	1	16	yes	—
65	C₂²×C₄	C₂²×C₄	1	3	yes	—
66	C₂²×C₄	C₂²×C₄	1	6	yes	—
67	C₂²×C₄	C₂²×C₄	1	6	yes	—
68	C₂²×C₄	C₂²×C₄	1	16	yes	—
69	C₂²×C₄	C₂²×C₄	1	6	yes	—
70	C₂²×C₄	#41	1	8	yes	—
71	C₂²×C₄	C₂⁴	1	4	yes	—
72	C₂²×C₄	C₂⁴	1	3	yes	—
73	C₂×C₈	—	2	48	no	—
74	C₂×C₈	C₂⁴	1	16	yes	—
75	C₄²	C₂²×C₄	1	16	yes	—
76	C₄²	C₂²×C₄	1	16	yes	—
77	C₄²	C₂²×C₄	1	12	yes	—
78	C₄²	C₂²×C₄	1	12	yes	—
79	C₄²	C₂⁴	1	4	yes	—
80	C₄²	C₂⁴	1	4	yes	—
81	C₂³	—	2	12	no	—
82	C₂³	—	2	12	no	—
83	C₂³	—	2	32	no	—
84	C₂³	#117	1	4	yes	—
85	C₂³	C₂²×C₈	1	24	yes	—
86	C₂³	C₂²×C₈	1	4	yes	—
87	C₂³	C₂×M₁₆	1	4	yes	—
88	C₂³	C₂³×C₄	1	4	yes	—
89	C₂³	C₂³×C₄	1	3	yes	—
90	C₂×C₄	—	2	12	no	—
91	C₂×C₄	—	2	12	no	—
92	C₂×C₄	—	2	24	no	—
93	C₂×C₄	—	2	32	no	—
94	C₂×C₄	—	2	8	no	—
95	C₂×C₄	#117	1	12	yes	—
96	C₂×C₄	C₂²×C₈	1	24	yes	—
97	C₂×C₄	C₂²×C₈	1	24	yes	—
98	C₂×C₄	C₂²×C₈	1	64	yes	complemented
99	C₂×C₄	C₂×M₁₆	1	24	yes	—
100	C₂×C₄	C₂³×C₄	1	4	yes	—
101	C₂×C₄	C₂³×C₄	1	4	yes	—
102	C₂×C₄	C₂³×C₄	1	3	yes	—
103	C₂×C₄	C₂³×C₄	1	4	yes	—
104	C₂×C₄	C₂³×C₄	1	4	yes	—
105	C₂×C₄	C₂³×C₄	1	4	yes	—
106	C₂×C₄	C₂×#41	1	12	yes	—
107	C₂×C₄	C₂⁵	1	1	yes	characteristic
108	C₈	—	2	16	no	—
109	C₂²	—	2	4	no	—
110	C₂²	—	2	48	no	—
111	C₂²	—	2	4	no	—
112	C₂²	#431	1	16	yes	direct product factor
113	C₂²	C₂×#117	1	6	yes	—
114	C₂²	C₂³×C₈	1	8	yes	—
115	C₂²	C₂³×C₈	1	6	yes	—
116	C₂²	C₂²×M₁₆	1	6	yes	—
117	C₂²	C₂⁴×C₄	1	1	yes	characteristic
118	C₄	—	2	4	no	—
119	C₄	—	2	4	no	—
120	C₄	C₂×#117	1	6	yes	—
121	C₄	C₂³×C₈	1	8	yes	—
122	C₄	C₂³×C₈	1	8	yes	—
123	C₄	C₂²×M₁₆	1	8	yes	—
124	C₄	C₂²×#41	1	2	yes	—
125	C₂	—	2	8	no	—
126	C₂	C₂×#431	1	12	yes	direct product factor
127	C₂	C₂²×#117	1	1	yes	characteristic
128	C₂	C₂⁴×C₈	1	1	yes	characteristic
129	C₂	C₂³×M₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄²	C₂²
commutator	C₂	C₂⁴×C₈
Frattini, cosocle	C₂×C₄	C₂⁵
socle	C₂⁴	C₂²×C₄
prime radical	C₂⁵	C₂×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₂×#431	C₂	direct (N × H)
3. C₂²×C₄⋊C₈	C₂	semidirect
5. C₂²×C₄×C₈	C₂	semidirect
16. #431	C₂²	direct (N × H)
17. C₂×C₄⋊C₈	C₂²	semidirect
19. C₂×C₄×C₈	C₂²	semidirect
23. C₂³×C₄	C₈	semidirect
44. C₄⋊C₈	C₂³	semidirect
46. C₄×C₈	C₂³	semidirect
59. C₂²×C₄	C₂×C₈	semidirect
98. C₂×C₄	C₂²×C₈	semidirect
112. C₂²	#431	direct (N × H)
126. C₂	C₂×#431	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₄²	2
C₂×#431	2
C₂²×C₄⋊C₈	2
C₂²×C₂²⋊C₈	2
C₂²×C₄×C₈	2

Subgroups — C22×#431 #60114

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#431 #60114