Subgroups — #62972

Order 256.

3411 conjugacy classes of subgroups.

The subgroup lattice has 129 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3568	C₂	1	7	yes	maximal, complemented
2	#3560	C₂	1	21	yes	maximal, complemented
3	#3546	C₂	1	21	yes	maximal, complemented
4	#3532	C₂	1	14	yes	maximal, complemented
5	#563	C₂²	1	21	yes	complemented
6	H(3,𝔽₄)	C₂²	1	7	yes	complemented
7	#560	C₂²	1	14	yes	complemented
8	#560	C₂²	1	42	yes	complemented
9	#556	C₂²	1	21	yes	complemented
10	#553	C₂²	1	42	yes	complemented
11	#551	C₂²	1	42	yes	complemented
12	#550	C₂²	1	21	yes	complemented
13	#547	C₂²	1	21	yes	complemented
14	#546	C₂²	1	42	yes	complemented
15	#546	C₂²	1	42	yes	complemented
16	D₄²	C₂²	1	21	yes	complemented
17	#542	C₂²	1	42	yes	complemented
18	#539	C₂²	1	21	yes	complemented
19	#538	C₂²	1	21	yes	complemented
20	#538	C₂²	1	42	yes	complemented
21	#538	C₂²	1	42	yes	complemented
22	#537	C₂²	1	21	yes	complemented
23	#535	C₂²	1	42	yes	complemented
24	#534	C₂²	1	21	yes	complemented
25	#532	C₂²	1	21	yes	complemented
26	#529	C₂²	1	21	yes	complemented
27	C₂×#126	C₂²	1	7	yes	complemented
28	C₂×#120	C₂²	1	14	yes	complemented
29	C₂⁵	C₂³	1	2	yes	complemented
30	C₂×#41	C₂³	1	42	yes	complemented
31	C₂²×D₄	C₂³	1	42	yes	—
32	C₄⋊Q₈	C₂³	1	21	yes	complemented
33	#127	C₂³	1	21	yes	—
34	#127	C₂³	1	14	yes	complemented
35	#126	C₂³	1	14	yes	complemented
36	#126	C₂³	1	42	yes	complemented
37	#126	C₂³	1	42	yes	complemented
38	#126	C₂³	1	14	yes	complemented
39	#126	C₂³	1	14	yes	complemented
40	#125	C₂³	1	21	yes	complemented
41	#124	C₂³	1	21	yes	—
42	#124	C₂³	1	42	yes	—
43	#124	C₂³	1	42	yes	complemented
44	#123	C₂³	1	42	yes	—
45	#123	C₂³	1	42	yes	—
46	#123	C₂³	1	42	yes	complemented
47	#123	C₂³	1	42	yes	complemented
48	C₂²⋊Q₈	C₂³	1	21	yes	complemented
49	C₂²⋊Q₈	C₂³	1	42	yes	complemented
50	C₂²⋊Q₈	C₂³	1	42	yes	complemented
51	#121	C₂³	1	21	yes	—
52	#121	C₂³	1	21	yes	—
53	#121	C₂³	1	42	yes	—
54	#121	C₂³	1	42	yes	—
55	#121	C₂³	1	42	yes	—
56	#121	C₂³	1	42	yes	complemented
57	#121	C₂³	1	21	yes	complemented
58	#121	C₂³	1	42	yes	complemented
59	#120	C₂³	1	14	yes	—
60	#120	C₂³	1	42	yes	—
61	#120	C₂³	1	42	yes	—
62	#120	C₂³	1	14	yes	complemented
63	#120	C₂³	1	42	yes	complemented
64	C₄×Q₈	C₂³	1	21	yes	complemented
65	C₄×D₄	C₂³	1	21	yes	—
66	C₄×D₄	C₂³	1	21	yes	—
67	C₄×D₄	C₂³	1	42	yes	complemented
68	C₄×D₄	C₂³	1	21	yes	complemented
69	C₄×D₄	C₂³	1	42	yes	complemented
70	#117	C₂³	1	42	yes	complemented
71	C₂×C₄⋊C₄	C₂³	1	21	yes	complemented
72	C₂×C₂²⋊C₄	C₂³	1	21	yes	—
73	C₂×C₂²⋊C₄	C₂³	1	42	yes	complemented
74	C₂×C₄²	C₂³	1	7	yes	complemented
75	C₂⁴	—	8	6	no	—
76	C₂⁴	C₂⁴	1	7	yes	—
77	C₂⁴	C₂⁴	1	14	yes	—
78	#41	—	8	42	no	—
79	C₂×Q₈	C₂⁴	1	21	yes	—
80	C₂×D₄	—	8	42	no	—
81	C₂×D₄	C₂⁴	1	21	yes	—
82	C₂×D₄	C₂⁴	1	21	yes	—
83	C₂×D₄	C₂⁴	1	21	yes	—
84	C₂×D₄	C₂⁴	1	42	yes	—
85	C₂×D₄	C₂⁴	1	42	yes	—
86	C₂²×C₄	C₂⁴	1	21	yes	—
87	C₂²×C₄	C₂⁴	1	21	yes	—
88	C₂²×C₄	C₂⁴	1	42	yes	—
89	C₄⋊C₄	C₂⁴	1	21	yes	—
90	C₄⋊C₄	C₂⁴	1	42	yes	—
91	C₄⋊C₄	C₂⁴	1	21	yes	—
92	C₄⋊C₄	C₂⁴	1	21	yes	—
93	C₄⋊C₄	C₂⁴	1	21	yes	—
94	C₂²⋊C₄	C₂⁴	1	42	yes	—
95	C₂²⋊C₄	C₂⁴	1	42	yes	—
96	C₂²⋊C₄	C₂⁴	1	42	yes	—
97	C₂²⋊C₄	C₂⁴	1	42	yes	—
98	C₂²⋊C₄	C₂⁴	1	42	yes	—
99	C₄²	C₂⁴	1	21	yes	—
100	C₄²	C₂⁴	1	7	yes	—
101	C₄²	C₂⁴	1	14	yes	—
102	C₂³	—	4	42	no	—
103	C₂³	—	4	21	no	—
104	C₂³	—	8	16	no	—
105	C₂³	C₂⁵	1	14	yes	—
106	C₂³	C₂⁵	1	7	yes	—
107	Q₈	—	4	21	no	—
108	D₄	—	4	21	no	—
109	D₄	—	4	21	no	—
110	D₄	—	4	42	no	—
111	D₄	—	4	42	no	—
112	D₄	—	4	21	no	—
113	C₂×C₄	—	4	42	no	—
114	C₂×C₄	—	4	21	no	—
115	C₂×C₄	—	4	21	no	—
116	C₂×C₄	C₂⁵	1	21	yes	—
117	C₂×C₄	C₂⁵	1	21	yes	—
118	C₂²	—	2	21	no	—
119	C₂²	—	2	42	no	—
120	C₂²	—	8	28	no	—
121	C₂²	—	16	7	no	—
122	C₂²	C₂⁶	1	1	yes	characteristic
123	C₄	—	2	21	no	—
124	C₄	—	2	21	no	—
125	C₂	—	4	14	no	—
126	C₂	—	4	7	no	—
127	C₂	H(7,𝔽₂)	1	3	yes	—

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂²	C₂⁶
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ G
LECS	G ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #3568	C₂	semidirect
2. #3560	C₂	semidirect
3. #3546	C₂	semidirect
4. #3532	C₂	semidirect
5. #563	C₂²	semidirect
6. H(3,𝔽₄)	C₂²	semidirect
7. #560	C₂²	semidirect
8. #560	C₂²	semidirect
9. #556	C₂²	semidirect
10. #553	C₂²	semidirect
11. #551	C₂²	semidirect
12. #550	C₂²	semidirect
13. #547	C₂²	semidirect
14. #546	C₂²	semidirect
15. #546	C₂²	semidirect
16. D₄²	C₂²	semidirect
17. #542	C₂²	semidirect
18. #539	C₂²	semidirect
19. #538	C₂²	semidirect
20. #538	C₂²	semidirect
21. #538	C₂²	semidirect
22. #537	C₂²	semidirect
23. #535	C₂²	semidirect
24. #534	C₂²	semidirect
25. #532	C₂²	semidirect
26. #529	C₂²	semidirect
27. C₂×#126	C₂²	semidirect
28. C₂×#120	C₂²	semidirect
29. C₂⁵	C₂³	semidirect
30. C₂×#41	C₂³	semidirect
32. C₄⋊Q₈	C₂³	semidirect
34. #127	C₂³	semidirect
35. #126	C₂³	semidirect
36. #126	C₂³	semidirect
37. #126	C₂³	semidirect
38. #126	C₂³	semidirect
39. #126	C₂³	semidirect
40. #125	C₂³	semidirect
43. #124	C₂³	semidirect
46. #123	C₂³	semidirect
47. #123	C₂³	semidirect
48. C₂²⋊Q₈	C₂³	semidirect
49. C₂²⋊Q₈	C₂³	semidirect
50. C₂²⋊Q₈	C₂³	semidirect
56. #121	C₂³	semidirect
57. #121	C₂³	semidirect
58. #121	C₂³	semidirect
62. #120	C₂³	semidirect
63. #120	C₂³	semidirect
64. C₄×Q₈	C₂³	semidirect
67. C₄×D₄	C₂³	semidirect
68. C₄×D₄	C₂³	semidirect
69. C₄×D₄	C₂³	semidirect
70. #117	C₂³	semidirect
71. C₂×C₄⋊C₄	C₂³	semidirect
73. C₂×C₂²⋊C₄	C₂³	semidirect
74. C₂×C₄²	C₂³	semidirect

Maximal subgroups

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

subgroup	index
#3568	2
#3560	2
#3546	2
#3532	2