Subgroups — #62842

Order 256.

5885 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×H(5,𝔽₂)	C₂	1	2	yes	maximal, complemented
2	#3496	C₂	1	16	yes	maximal, complemented
3	#3485	C₂	1	16	yes	maximal, complemented
4	#3484	C₂	1	16	yes	maximal, complemented
5	#3449	C₂	1	4	yes	maximal, complemented
6	C₂×#534	C₂	1	8	yes	maximal, complemented
7	C₂×#518	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×H(5,𝔽₂)	—	2	32	no	—
9	C₂×H(5,𝔽₂)	C₂²	1	16	yes	complemented
10	C₂×H(5,𝔽₂)	C₂²	1	16	yes	complemented
11	C₂²×#41	C₂²	1	4	yes	complemented
12	C₂²×#41	C₂²	1	8	yes	complemented
13	C₂³×D₄	C₂²	1	1	yes	characteristic
14	C₂³×D₄	C₂²	1	8	yes	complemented
15	C₂³×D₄	C₂²	1	8	yes	complemented
16	#547	C₂²	1	64	yes	complemented
17	#546	C₂²	1	128	yes	complemented
18	D₄²	C₂²	1	64	yes	complemented
19	#537	C₂²	1	16	yes	complemented
20	#536	C₂²	1	16	yes	complemented
21	#535	C₂²	1	16	yes	complemented
22	#534	C₂²	1	16	yes	complemented
23	#534	C₂²	1	64	yes	complemented
24	#534	C₂²	1	32	yes	complemented
25	#532	C₂²	1	16	yes	complemented
26	#525	C₂²	1	32	yes	complemented
27	C₂×#123	C₂²	1	32	yes	complemented
28	C₂×#121	C₂²	1	32	yes	complemented
29	C₂×#120	C₂²	1	32	yes	complemented
30	#518	C₂²	1	16	yes	complemented
31	C₂×C₄×D₄	C₂²	1	8	yes	complemented
32	C₂×#117	C₂²	1	2	yes	complemented
33	C₂²×C₂²⋊C₄	C₂²	1	4	yes	complemented
34	C₂⁵	C₂³	1	2	yes	—
35	C₂⁵	C₂³	1	8	yes	complemented
36	H(5,𝔽₂)	—	2	128	no	—
37	H(5,𝔽₂)	—	2	128	no	—
38	C₂×#41	—	2	64	no	—
39	C₂×#41	—	2	32	no	—
40	C₂×#41	C₂³	1	16	yes	—
41	C₂×#41	C₂³	1	16	yes	complemented
42	C₂×#41	C₂³	1	64	yes	complemented
43	C₂²×Q₈	C₂³	1	4	yes	complemented
44	C₂²×D₄	—	2	64	no	—
45	C₂²×D₄	—	2	64	no	—
46	C₂²×D₄	D₄	1	16	yes	complemented
47	C₂²×D₄	C₂³	1	32	yes	—
48	C₂²×D₄	C₂³	1	8	yes	—
49	C₂²×D₄	C₂³	1	4	yes	—
50	C₂²×D₄	C₂³	1	8	yes	—
51	C₂²×D₄	C₂³	1	32	yes	complemented
52	C₂²×D₄	C₂³	1	16	yes	complemented
53	C₂²×D₄	C₂³	1	4	yes	complemented
54	C₂²×D₄	C₂³	1	64	yes	complemented
55	C₂²×D₄	C₂³	1	32	yes	complemented
56	C₂³×C₄	C₂³	1	1	yes	characteristic
57	C₂³×C₄	C₂³	1	8	yes	complemented
58	C₂³×C₄	C₂³	1	8	yes	complemented
59	C₂³×C₄	C₂³	1	8	yes	complemented
60	C₄⋊Q₈	C₂³	1	16	yes	complemented
61	#127	C₂³	1	16	yes	complemented
62	#124	C₂³	1	32	yes	complemented
63	#123	C₂³	1	64	yes	complemented
64	#123	C₂³	1	128	yes	complemented
65	C₂²⋊Q₈	C₂³	1	64	yes	complemented
66	#121	C₂³	1	64	yes	complemented
67	#121	C₂³	1	128	yes	complemented
68	#121	C₂³	1	64	yes	complemented
69	#120	C₂³	1	64	yes	complemented
70	#120	C₂³	1	128	yes	complemented
71	C₄×D₄	C₂³	1	64	yes	complemented
72	C₄×D₄	C₂³	1	64	yes	complemented
73	#117	C₂³	1	16	yes	complemented
74	C₂×C₄⋊C₄	C₂³	1	4	yes	complemented
75	C₂×C₄⋊C₄	C₂³	1	32	yes	complemented
76	C₂×C₂²⋊C₄	C₂³	1	4	yes	—
77	C₂×C₂²⋊C₄	C₂³	1	8	yes	complemented
78	C₂×C₂²⋊C₄	C₂³	1	32	yes	complemented
79	C₂×C₂²⋊C₄	C₂³	1	64	yes	complemented
80	C₂×C₄²	C₂³	1	4	yes	complemented
81	C₂⁴	—	2	32	no	—
82	C₂⁴	—	8	4	no	—
83	C₂⁴	—	8	16	no	—
84	C₂⁴	C₂×D₄	1	16	yes	complemented
85	C₂⁴	C₂⁴	1	32	yes	—
86	C₂⁴	C₂⁴	1	8	yes	—
87	C₂⁴	C₂⁴	1	1	yes	characteristic
88	C₂⁴	C₂⁴	1	8	yes	—
89	C₂⁴	C₂⁴	1	4	yes	—
90	#41	—	2	256	no	—
91	#41	—	2	64	no	—
92	#41	—	2	64	no	—
93	C₂×Q₈	—	2	16	no	—
94	C₂×Q₈	C₂⁴	1	16	yes	—
95	C₂×D₄	—	2	32	no	—
96	C₂×D₄	—	2	64	no	—
97	C₂×D₄	—	2	128	no	—
98	C₂×D₄	—	2	128	no	—
99	C₂×D₄	—	2	256	no	—
100	C₂×D₄	—	2	16	no	—
101	C₂×D₄	—	8	32	no	—
102	C₂×D₄	C₂×D₄	1	64	yes	complemented
103	C₂×D₄	C₂×D₄	1	32	yes	complemented
104	C₂×D₄	C₂⁴	1	16	yes	—
105	C₂×D₄	C₂⁴	1	64	yes	—
106	C₂×D₄	C₂⁴	1	16	yes	—
107	C₂×D₄	C₂⁴	1	16	yes	—
108	C₂×D₄	C₂⁴	1	64	yes	—
109	C₂×D₄	C₂⁴	1	16	yes	—
110	C₂²×C₄	—	2	32	no	—
111	C₂²×C₄	—	2	32	no	—
112	C₂²×C₄	—	8	8	no	—
113	C₂²×C₄	C₂×D₄	1	8	yes	complemented
114	C₂²×C₄	C₂⁴	1	32	yes	—
115	C₂²×C₄	C₂⁴	1	32	yes	—
116	C₂²×C₄	C₂⁴	1	8	yes	—
117	C₂²×C₄	C₂⁴	1	8	yes	—
118	C₂²×C₄	C₂⁴	1	32	yes	—
119	C₂²×C₄	C₂⁴	1	2	yes	—
120	C₂²×C₄	C₂⁴	1	4	yes	—
121	C₄⋊C₄	C₂⁴	1	64	yes	—
122	C₄⋊C₄	C₂⁴	1	16	yes	—
123	C₂²⋊C₄	C₂⁴	1	16	yes	—
124	C₂²⋊C₄	C₂⁴	1	32	yes	—
125	C₂²⋊C₄	C₂⁴	1	128	yes	—
126	C₄²	C₂⁴	1	16	yes	—
127	C₂³	—	2	16	no	—
128	C₂³	—	2	64	no	—
129	C₂³	—	4	8	no	—
130	C₂³	—	4	2	no	—
131	C₂³	—	4	16	no	—
132	C₂³	—	4	16	no	—
133	C₂³	—	4	64	no	—
134	C₂³	—	8	16	no	—
135	C₂³	—	8	64	no	—
136	C₂³	C₂²×D₄	1	32	yes	—
137	C₂³	C₂²×D₄	1	4	yes	—
138	C₂³	C₂²×D₄	1	16	yes	—
139	C₂³	C₂⁵	1	16	yes	—
140	C₂³	C₂⁵	1	2	yes	—
141	C₂³	C₂⁵	1	8	yes	—
142	C₂³	C₂⁵	1	1	yes	characteristic
143	Q₈	—	2	32	no	—
144	D₄	—	2	32	no	—
145	D₄	—	2	128	no	—
146	D₄	—	2	32	no	—
147	D₄	—	2	32	no	—
148	D₄	—	4	64	no	—
149	D₄	C₂²×D₄	1	64	yes	—
150	C₂×C₄	—	2	16	no	—
151	C₂×C₄	—	2	64	no	—
152	C₂×C₄	—	2	64	no	—
153	C₂×C₄	—	4	8	no	—
154	C₂×C₄	—	4	32	no	—
155	C₂×C₄	C₂²×D₄	1	16	yes	—
156	C₂×C₄	C₂²×D₄	1	8	yes	—
157	C₂×C₄	C₂⁵	1	16	yes	—
158	C₂×C₄	C₂⁵	1	16	yes	—
159	C₂×C₄	C₂⁵	1	4	yes	—
160	C₂²	—	2	2	no	—
161	C₂²	—	2	16	no	—
162	C₂²	—	2	4	no	—
163	C₂²	—	2	16	no	—
164	C₂²	—	2	32	no	—
165	C₂²	—	4	4	no	—
166	C₂²	—	4	32	no	—
167	C₂²	—	4	16	no	—
168	C₂²	—	4	32	no	—
169	C₂²	—	8	64	no	—
170	C₂²	C₂³×D₄	1	4	yes	—
171	C₂²	C₂³×D₄	1	16	yes	—
172	C₂²	C₂³×D₄	1	2	yes	—
173	C₂²	C₂⁶	1	1	yes	characteristic
174	C₄	—	2	16	no	—
175	C₄	—	2	16	no	—
176	C₄	C₂³×D₄	1	8	yes	—
177	C₂	—	2	4	no	—
178	C₂	—	2	2	no	—
179	C₂	—	2	16	no	—
180	C₂	—	4	16	no	—
181	C₂	C₂⁴×D₄	1	1	yes	characteristic
182	C₂	H(7,𝔽₂)	1	2	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. C₂²×H(5,𝔽₂)	C₂	semidirect
2. #3496	C₂	semidirect
3. #3485	C₂	semidirect
4. #3484	C₂	semidirect
5. #3449	C₂	semidirect
6. C₂×#534	C₂	semidirect
7. C₂×#518	C₂	semidirect
9. C₂×H(5,𝔽₂)	C₂²	semidirect
10. C₂×H(5,𝔽₂)	C₂²	semidirect
11. C₂²×#41	C₂²	semidirect
12. C₂²×#41	C₂²	semidirect
14. C₂³×D₄	C₂²	semidirect
15. C₂³×D₄	C₂²	semidirect
16. #547	C₂²	semidirect
17. #546	C₂²	semidirect
18. D₄²	C₂²	semidirect
19. #537	C₂²	semidirect
20. #536	C₂²	semidirect
21. #535	C₂²	semidirect
22. #534	C₂²	semidirect
23. #534	C₂²	semidirect
24. #534	C₂²	semidirect
25. #532	C₂²	semidirect
26. #525	C₂²	semidirect
27. C₂×#123	C₂²	semidirect
28. C₂×#121	C₂²	semidirect
29. C₂×#120	C₂²	semidirect
30. #518	C₂²	semidirect
31. C₂×C₄×D₄	C₂²	semidirect
32. C₂×#117	C₂²	semidirect
33. C₂²×C₂²⋊C₄	C₂²	semidirect
35. C₂⁵	C₂³	semidirect
41. C₂×#41	C₂³	semidirect
42. C₂×#41	C₂³	semidirect
43. C₂²×Q₈	C₂³	semidirect
46. C₂²×D₄	D₄	semidirect
51. C₂²×D₄	C₂³	semidirect
52. C₂²×D₄	C₂³	semidirect
53. C₂²×D₄	C₂³	semidirect
54. C₂²×D₄	C₂³	semidirect
55. C₂²×D₄	C₂³	semidirect
57. C₂³×C₄	C₂³	semidirect
58. C₂³×C₄	C₂³	semidirect
59. C₂³×C₄	C₂³	semidirect
60. C₄⋊Q₈	C₂³	semidirect
61. #127	C₂³	semidirect
62. #124	C₂³	semidirect
63. #123	C₂³	semidirect
64. #123	C₂³	semidirect
65. C₂²⋊Q₈	C₂³	semidirect
66. #121	C₂³	semidirect
67. #121	C₂³	semidirect
68. #121	C₂³	semidirect
69. #120	C₂³	semidirect
70. #120	C₂³	semidirect
71. C₄×D₄	C₂³	semidirect
72. C₄×D₄	C₂³	semidirect
73. #117	C₂³	semidirect
74. C₂×C₄⋊C₄	C₂³	semidirect
75. C₂×C₄⋊C₄	C₂³	semidirect
77. C₂×C₂²⋊C₄	C₂³	semidirect
78. C₂×C₂²⋊C₄	C₂³	semidirect
79. C₂×C₂²⋊C₄	C₂³	semidirect
80. C₂×C₄²	C₂³	semidirect
84. C₂⁴	C₂×D₄	semidirect
102. C₂×D₄	C₂×D₄	semidirect
103. C₂×D₄	C₂×D₄	semidirect
113. C₂²×C₄	C₂×D₄	semidirect

Maximal subgroups

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

subgroup	index
C₂²×H(5,𝔽₂)	2
#3496	2
#3485	2
#3484	2
#3449	2
C₂×#534	2
C₂×#518	2