Subgroups — C₂²×#405 #60053

Order 256.

1759 conjugacy classes of subgroups.

The subgroup lattice has 117 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

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

	subgroup	G/H
center	C₂³×C₈	C₂²
commutator	C₂	C₂³×C₄²
Frattini, cosocle	C₂×C₄	C₂⁵
socle	C₂⁴	C₂²×C₄
prime radical	C₂⁵	C₂×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
4. C₂×#405	C₂	direct (N × H)
5. C₂²×#97	C₂	semidirect
6. C₂²×C₄×C₈	C₂	semidirect
8. C₂²×M₁₆	C₄	semidirect
10. C₂³×C₈	C₄	semidirect
20. #405	C₂²	direct (N × H)
21. C₂×#97	C₂²	semidirect
22. C₂×C₄×C₈	C₂²	semidirect
31. C₂×M₁₆	C₂×C₄	semidirect
36. C₂²×C₈	C₂×C₄	semidirect
47. #97	C₂³	semidirect
48. C₄×C₈	C₂³	semidirect
66. M₁₆	C₂²×C₄	semidirect
70. C₂×C₈	C₂²×C₄	semidirect
102. C₂²	#405	direct (N × H)
113. C₂	C₂×#405	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×M₁₆	2
C₂⁴×C₈	2
C₂²×#117	2
C₂×#405	2
C₂²×#97	2
C₂²×C₄×C₈	2

Subgroups — C22×#405 #60053

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#405 #60053