Subgroups — #62832

Order 256.

4325 conjugacy classes of subgroups.

The subgroup lattice has 143 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3543	C₂	1	24	yes	maximal, complemented
2	#3508	C₂	1	12	yes	maximal, complemented
3	#3498	C₂	1	12	yes	maximal, complemented
4	#3484	C₂	1	12	yes	maximal, complemented
5	#3444	C₂	1	2	yes	maximal, complemented
6	C₂²×#127	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×H(5,𝔽₂)	C₂²	1	12	yes	complemented
8	C₂³×D₄	C₂²	1	6	yes	complemented
9	#560	C₂²	1	32	yes	complemented
10	#559	C₂²	1	24	yes	complemented
11	#550	C₂²	1	48	yes	complemented
12	#546	C₂²	1	96	yes	complemented
13	D₄²	C₂²	1	96	yes	complemented
14	D₄²	C₂²	1	24	yes	complemented
15	#540	C₂²	1	48	yes	complemented
16	#535	C₂²	1	24	yes	complemented
17	#535	C₂²	1	48	yes	complemented
18	#534	C₂²	1	48	yes	complemented
19	#533	C₂²	1	6	yes	complemented
20	#532	C₂²	1	6	yes	complemented
21	#532	C₂²	1	24	yes	complemented
22	C₂×#127	C₂²	1	12	yes	complemented
23	C₂×#127	C₂²	1	12	yes	complemented
24	C₂×#127	C₂²	1	8	yes	complemented
25	#529	C₂²	1	24	yes	complemented
26	#525	C₂²	1	12	yes	complemented
27	C₂×#120	C₂²	1	16	yes	complemented
28	#518	C₂²	1	12	yes	complemented
29	C₄×#41	C₂²	1	12	yes	complemented
30	C₂²×C₄²	C₂²	1	1	yes	characteristic, complemented
31	C₂⁵	C₂³	1	4	yes	—
32	H(5,𝔽₂)	—	4	48	no	—
33	C₂×#41	C₂³	1	12	yes	—
34	C₂×#41	C₂³	1	48	yes	complemented
35	C₂²×D₄	—	4	24	no	—
36	C₂²×D₄	C₂³	1	12	yes	—
37	C₂²×D₄	C₂³	1	48	yes	—
38	C₂²×D₄	C₂³	1	12	yes	—
39	C₂²×D₄	C₂³	1	48	yes	—
40	C₂²×D₄	C₂³	1	48	yes	complemented
41	C₂³×C₄	C₂³	1	3	yes	—
42	C₄⋊Q₈	C₂³	1	6	yes	complemented
43	#127	C₂³	1	8	yes	—
44	#127	C₂³	1	24	yes	—
45	#127	C₂³	1	24	yes	—
46	#127	C₂³	1	8	yes	—
47	#127	C₂³	1	2	yes	complemented
48	#127	C₂³	1	48	yes	complemented
49	#126	C₂³	1	16	yes	complemented
50	#125	C₂³	1	12	yes	complemented
51	#124	C₂³	1	12	yes	complemented
52	#124	C₂³	1	48	yes	complemented
53	#123	C₂³	1	24	yes	—
54	#123	C₂³	1	96	yes	complemented
55	C₂²⋊Q₈	C₂³	1	24	yes	—
56	#121	C₂³	1	24	yes	—
57	#121	C₂³	1	96	yes	complemented
58	#121	C₂³	1	192	yes	complemented
59	#120	C₂³	1	8	yes	—
60	#120	C₂³	1	96	yes	—
61	#120	C₂³	1	96	yes	complemented
62	C₄×Q₈	C₂³	1	24	yes	complemented
63	C₄×D₄	C₂³	1	48	yes	—
64	C₄×D₄	C₂³	1	24	yes	complemented
65	C₄×D₄	C₂³	1	96	yes	complemented
66	#117	C₂³	1	12	yes	—
67	#117	C₂³	1	24	yes	complemented
68	C₂×C₂²⋊C₄	C₂³	1	48	yes	—
69	C₂×C₄²	C₂³	1	6	yes	—
70	C₂×C₄²	C₂³	1	6	yes	—
71	C₂×C₄²	C₂³	1	8	yes	complemented
72	C₂⁴	—	4	24	no	—
73	C₂⁴	C₂⁴	1	1	yes	characteristic
74	C₂⁴	C₂⁴	1	24	yes	—
75	C₂⁴	C₂⁴	1	16	yes	—
76	#41	—	2	48	no	—
77	#41	—	4	96	no	—
78	C₂×Q₈	C₂⁴	1	12	yes	—
79	C₂×D₄	—	2	48	no	—
80	C₂×D₄	—	4	96	no	—
81	C₂×D₄	—	4	96	no	—
82	C₂×D₄	—	4	24	no	—
83	C₂×D₄	—	4	96	no	—
84	C₂×D₄	C₂⁴	1	24	yes	—
85	C₂×D₄	C₂⁴	1	12	yes	—
86	C₂×D₄	C₂⁴	1	48	yes	—
87	C₂×D₄	C₂⁴	1	96	yes	—
88	C₂×D₄	C₂⁴	1	48	yes	—
89	C₂²×C₄	—	2	12	no	—
90	C₂²×C₄	C₂⁴	1	12	yes	—
91	C₂²×C₄	C₂⁴	1	3	yes	—
92	C₂²×C₄	C₂⁴	1	24	yes	—
93	C₂²×C₄	C₂⁴	1	48	yes	—
94	C₂²×C₄	C₂⁴	1	3	yes	—
95	C₄⋊C₄	C₂⁴	1	24	yes	—
96	C₄⋊C₄	C₂⁴	1	48	yes	—
97	C₂²⋊C₄	C₂⁴	1	24	yes	—
98	C₂²⋊C₄	C₂⁴	1	96	yes	—
99	C₂²⋊C₄	C₂⁴	1	48	yes	—
100	C₄²	C₂⁴	1	2	yes	—
101	C₄²	C₂⁴	1	6	yes	—
102	C₄²	C₂⁴	1	6	yes	—
103	C₄²	C₂⁴	1	2	yes	—
104	C₄²	C₂⁴	1	24	yes	—
105	C₂³	—	2	6	no	—
106	C₂³	—	2	48	no	—
107	C₂³	—	4	48	no	—
108	C₂³	—	4	48	no	—
109	C₂³	—	8	32	no	—
110	C₂³	C₂⁵	1	8	yes	—
111	C₂³	C₂⁵	1	16	yes	—
112	C₂³	C₂⁵	1	3	yes	—
113	Q₈	—	2	24	no	—
114	D₄	—	2	48	no	—
115	D₄	—	2	96	no	—
116	D₄	—	2	24	no	—
117	D₄	—	4	48	no	—
118	D₄	—	4	96	no	—
119	C₂×C₄	—	2	6	no	—
120	C₂×C₄	—	2	24	no	—
121	C₂×C₄	—	2	48	no	—
122	C₂×C₄	—	4	48	no	—
123	C₂×C₄	H(5,𝔽₂)	1	12	yes	complemented
124	C₂×C₄	C₂⁵	1	6	yes	—
125	C₂×C₄	C₂⁵	1	24	yes	—
126	C₂×C₄	C₂⁵	1	6	yes	—
127	C₂²	—	2	8	no	—
128	C₂²	—	2	6	no	—
129	C₂²	—	2	24	no	—
130	C₂²	—	2	48	no	—
131	C₂²	—	8	48	no	—
132	C₂²	—	16	16	no	—
133	C₂²	C₂×H(5,𝔽₂)	1	6	yes	—
134	C₂²	C₂⁶	1	1	yes	characteristic
135	C₄	—	2	6	no	—
136	C₄	—	2	24	no	—
137	C₄	C₂×H(5,𝔽₂)	1	12	yes	—
138	C₂	—	2	6	no	—
139	C₂	—	4	16	no	—
140	C₂	—	4	8	no	—
141	C₂	C₂²×H(5,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. #3543	C₂	semidirect
2. #3508	C₂	semidirect
3. #3498	C₂	semidirect
4. #3484	C₂	semidirect
5. #3444	C₂	semidirect
6. C₂²×#127	C₂	semidirect
7. C₂×H(5,𝔽₂)	C₂²	semidirect
8. C₂³×D₄	C₂²	semidirect
9. #560	C₂²	semidirect
10. #559	C₂²	semidirect
11. #550	C₂²	semidirect
12. #546	C₂²	semidirect
13. D₄²	C₂²	semidirect
14. D₄²	C₂²	semidirect
15. #540	C₂²	semidirect
16. #535	C₂²	semidirect
17. #535	C₂²	semidirect
18. #534	C₂²	semidirect
19. #533	C₂²	semidirect
20. #532	C₂²	semidirect
21. #532	C₂²	semidirect
22. C₂×#127	C₂²	semidirect
23. C₂×#127	C₂²	semidirect
24. C₂×#127	C₂²	semidirect
25. #529	C₂²	semidirect
26. #525	C₂²	semidirect
27. C₂×#120	C₂²	semidirect
28. #518	C₂²	semidirect
29. C₄×#41	C₂²	semidirect
30. C₂²×C₄²	C₂²	semidirect
34. C₂×#41	C₂³	semidirect
40. C₂²×D₄	C₂³	semidirect
42. C₄⋊Q₈	C₂³	semidirect
47. #127	C₂³	semidirect
48. #127	C₂³	semidirect
49. #126	C₂³	semidirect
50. #125	C₂³	semidirect
51. #124	C₂³	semidirect
52. #124	C₂³	semidirect
54. #123	C₂³	semidirect
57. #121	C₂³	semidirect
58. #121	C₂³	semidirect
61. #120	C₂³	semidirect
62. C₄×Q₈	C₂³	semidirect
64. C₄×D₄	C₂³	semidirect
65. C₄×D₄	C₂³	semidirect
67. #117	C₂³	semidirect
71. C₂×C₄²	C₂³	semidirect
123. C₂×C₄	H(5,𝔽₂)	semidirect

Maximal subgroups

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

subgroup	index
#3543	2
#3508	2
#3498	2
#3484	2
#3444	2
C₂²×#127	2