Subgroups — D₄×H(5,𝔽₂) #62981

Order 256.

6207 conjugacy classes of subgroups.

The subgroup lattice has 205 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	#3498	C₂	1	6	yes	maximal, complemented
3	#3496	C₂	1	9	yes	maximal, complemented
4	#3488	C₂	1	12	yes	maximal, complemented
5	#3484	C₂	1	18	yes	maximal, complemented
6	D₄×#41	C₂	1	6	yes	maximal, complemented
7	C₂×D₄²	C₂	1	9	yes	maximal, complemented
8	C₄×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×H(5,𝔽₂)	—	2	32	no	—
10	C₂×H(5,𝔽₂)	C₂²	1	1	yes	characteristic
11	C₂×H(5,𝔽₂)	C₂²	1	18	yes	complemented
12	C₂×H(5,𝔽₂)	C₂²	1	12	yes	complemented
13	C₂²×#41	C₂²	1	12	yes	complemented
14	C₂³×D₄	C₂²	1	18	yes	complemented
15	C₂³×D₄	C₂²	1	6	yes	complemented
16	#550	C₂²	1	6	yes	complemented
17	#550	C₂²	1	12	yes	complemented
18	D₄×Q₈	C₂²	1	2	yes	complemented
19	#548	C₂²	1	12	yes	complemented
20	#547	C₂²	1	18	yes	complemented
21	#546	C₂²	1	36	yes	complemented
22	#546	C₂²	1	72	yes	complemented
23	D₄²	C₂²	1	18	yes	complemented
24	D₄²	C₂²	1	72	yes	complemented
25	D₄²	C₂²	1	36	yes	complemented
26	D₄²	C₂²	1	36	yes	complemented
27	#537	C₂²	1	18	yes	complemented
28	#535	C₂²	1	36	yes	complemented
29	#534	C₂²	1	36	yes	complemented
30	#532	C₂²	1	18	yes	complemented
31	C₂×#127	C₂²	1	9	yes	complemented
32	#525	C₂²	1	36	yes	complemented
33	C₂×#121	C₂²	1	18	yes	complemented
34	C₂×#121	C₂²	1	18	yes	complemented
35	C₂×#120	C₂²	1	36	yes	complemented
36	#520	C₂²	1	6	yes	complemented
37	#518	C₂²	1	9	yes	complemented
38	C₄×#41	C₂²	1	6	yes	complemented
39	C₂×C₄×D₄	C₂²	1	9	yes	complemented
40	C₂×C₄×D₄	C₂²	1	9	yes	complemented
41	C₂⁵	C₂³	1	12	yes	complemented
42	H(5,𝔽₂)	—	2	96	no	—
43	H(5,𝔽₂)	—	2	144	no	—
44	H(5,𝔽₂)	D₄	1	16	yes	direct product factor
45	C₂×#41	—	2	96	no	—
46	C₂×#41	C₂³	1	6	yes	—
47	C₂×#41	C₂³	1	36	yes	complemented
48	C₂×#41	C₂³	1	36	yes	complemented
49	C₂×#41	C₂³	1	12	yes	complemented
50	C₂²×Q₈	C₂³	1	4	yes	complemented
51	C₂²×D₄	—	2	144	no	—
52	C₂²×D₄	—	4	24	no	—
53	C₂²×D₄	C₂³	1	9	yes	—
54	C₂²×D₄	C₂³	1	9	yes	—
55	C₂²×D₄	C₂³	1	36	yes	complemented
56	C₂²×D₄	C₂³	1	18	yes	complemented
57	C₂²×D₄	C₂³	1	36	yes	complemented
58	C₂²×D₄	C₂³	1	72	yes	complemented
59	C₂²×D₄	C₂³	1	36	yes	complemented
60	C₂²×D₄	C₂³	1	18	yes	complemented
61	C₂²×D₄	C₂³	1	18	yes	complemented
62	C₂³×C₄	C₂³	1	18	yes	complemented
63	C₂³×C₄	C₂³	1	6	yes	complemented
64	C₄⋊Q₈	C₂³	1	6	yes	complemented
65	#127	C₂³	1	18	yes	complemented
66	#127	C₂³	1	36	yes	complemented
67	#124	C₂³	1	36	yes	complemented
68	#123	C₂³	1	72	yes	complemented
69	C₂²⋊Q₈	C₂³	1	12	yes	complemented
70	C₂²⋊Q₈	C₂³	1	36	yes	complemented
71	#121	C₂³	1	36	yes	complemented
72	#121	C₂³	1	36	yes	complemented
73	#121	C₂³	1	72	yes	complemented
74	#121	C₂³	1	72	yes	complemented
75	#121	C₂³	1	36	yes	complemented
76	#121	C₂³	1	36	yes	complemented
77	#120	C₂³	1	72	yes	complemented
78	#120	C₂³	1	144	yes	complemented
79	C₄×Q₈	C₂³	1	2	yes	complemented
80	C₄×Q₈	C₂³	1	6	yes	complemented
81	C₄×D₄	C₂³	1	6	yes	—
82	C₄×D₄	C₂³	1	18	yes	complemented
83	C₄×D₄	C₂³	1	36	yes	complemented
84	C₄×D₄	C₂³	1	36	yes	complemented
85	C₄×D₄	C₂³	1	18	yes	complemented
86	C₄×D₄	C₂³	1	18	yes	complemented
87	C₄×D₄	C₂³	1	36	yes	complemented
88	#117	C₂³	1	18	yes	complemented
89	C₂×C₄⋊C₄	C₂³	1	9	yes	complemented
90	C₂×C₄⋊C₄	C₂³	1	9	yes	complemented
91	C₂×C₂²⋊C₄	C₂³	1	18	yes	complemented
92	C₂×C₂²⋊C₄	C₂³	1	18	yes	complemented
93	C₂×C₂²⋊C₄	C₂³	1	36	yes	complemented
94	C₂×C₄²	C₂³	1	9	yes	complemented
95	C₂⁴	—	2	48	no	—
96	C₂⁴	—	4	24	no	—
97	C₂⁴	—	8	12	no	—
98	C₂⁴	C₂⁴	1	18	yes	—
99	C₂⁴	C₂⁴	1	36	yes	—
100	C₂⁴	C₂⁴	1	6	yes	—
101	#41	—	2	48	no	—
102	#41	—	2	144	no	—
103	#41	—	2	144	no	—
104	#41	C₂×D₄	1	48	yes	complemented
105	C₂×Q₈	—	2	16	no	—
106	C₂×Q₈	C₂⁴	1	12	yes	—
107	C₂×Q₈	C₂⁴	1	2	yes	—
108	C₂×D₄	—	2	144	no	—
109	C₂×D₄	—	2	144	no	—
110	C₂×D₄	—	2	288	no	—
111	C₂×D₄	—	2	36	no	—
112	C₂×D₄	—	2	72	no	—
113	C₂×D₄	—	4	36	no	—
114	C₂×D₄	—	4	72	no	—
115	C₂×D₄	—	8	18	no	—
116	C₂×D₄	C₂×D₄	1	72	yes	complemented
117	C₂×D₄	C₂⁴	1	18	yes	—
118	C₂×D₄	C₂⁴	1	9	yes	—
119	C₂×D₄	C₂⁴	1	72	yes	—
120	C₂×D₄	C₂⁴	1	36	yes	—
121	C₂×D₄	C₂⁴	1	36	yes	—
122	C₂×D₄	C₂⁴	1	36	yes	—
123	C₂×D₄	C₂⁴	1	18	yes	—
124	C₂×D₄	C₂⁴	1	1	yes	characteristic
125	C₂²×C₄	—	2	72	no	—
126	C₂²×C₄	—	4	12	no	—
127	C₂²×C₄	C₂⁴	1	9	yes	—
128	C₂²×C₄	C₂⁴	1	18	yes	—
129	C₂²×C₄	C₂⁴	1	12	yes	—
130	C₂²×C₄	C₂⁴	1	36	yes	—
131	C₂²×C₄	C₂⁴	1	9	yes	—
132	C₂²×C₄	C₂⁴	1	18	yes	—
133	C₂²×C₄	C₂⁴	1	9	yes	—
134	C₄⋊C₄	C₂⁴	1	6	yes	—
135	C₄⋊C₄	C₂⁴	1	18	yes	—
136	C₄⋊C₄	C₂⁴	1	18	yes	—
137	C₄⋊C₄	C₂⁴	1	6	yes	—
138	C₄⋊C₄	C₂⁴	1	12	yes	—
139	C₂²⋊C₄	C₂⁴	1	12	yes	—
140	C₂²⋊C₄	C₂⁴	1	36	yes	—
141	C₂²⋊C₄	C₂⁴	1	72	yes	—
142	C₂²⋊C₄	C₂⁴	1	36	yes	—
143	C₄²	C₂⁴	1	6	yes	—
144	C₄²	C₂⁴	1	18	yes	—
145	C₂³	—	2	36	no	—
146	C₂³	—	2	36	no	—
147	C₂³	—	2	72	no	—
148	C₂³	—	4	6	no	—
149	C₂³	—	4	6	no	—
150	C₂³	—	4	18	no	—
151	C₂³	—	4	36	no	—
152	C₂³	—	4	36	no	—
153	C₂³	—	8	96	no	—
154	C₂³	C₂²×D₄	1	24	yes	complemented
155	C₂³	C₂⁵	1	18	yes	—
156	C₂³	C₂⁵	1	9	yes	—
157	C₂³	C₂⁵	1	2	yes	—
158	Q₈	—	2	24	no	—
159	Q₈	C₂²×D₄	1	8	yes	complemented
160	D₄	—	2	144	no	—
161	D₄	—	2	36	no	—
162	D₄	—	2	72	no	—
163	D₄	—	2	18	no	—
164	D₄	—	4	36	no	—
165	D₄	—	4	36	no	—
166	D₄	C₂²×D₄	1	72	yes	complemented
167	D₄	H(5,𝔽₂)	1	4	yes	direct product factor
168	C₂×C₄	—	2	36	no	—
169	C₂×C₄	—	2	24	no	—
170	C₂×C₄	—	2	72	no	—
171	C₂×C₄	—	2	18	no	—
172	C₂×C₄	—	4	18	no	—
173	C₂×C₄	—	4	9	no	—
174	C₂×C₄	C₂²×D₄	1	36	yes	complemented
175	C₂×C₄	C₂⁵	1	6	yes	—
176	C₂×C₄	C₂⁵	1	9	yes	—
177	C₂×C₄	C₂⁵	1	12	yes	—
178	C₂×C₄	C₂⁵	1	6	yes	—
179	C₂×C₄	C₂⁵	1	1	yes	characteristic
180	C₂²	—	2	2	no	—
181	C₂²	—	2	9	no	—
182	C₂²	—	2	9	no	—
183	C₂²	—	2	2	no	—
184	C₂²	—	2	18	no	—
185	C₂²	—	2	18	no	—
186	C₂²	—	2	18	no	—
187	C₂²	—	4	24	no	—
188	C₂²	—	4	72	no	—
189	C₂²	—	8	72	no	—
190	C₂²	C₂³×D₄	1	18	yes	—
191	C₂²	C₂×H(5,𝔽₂)	1	4	yes	complemented
192	C₂²	C₂⁶	1	1	yes	characteristic
193	C₄	—	2	12	no	—
194	C₄	—	2	9	no	—
195	C₄	—	2	6	no	—
196	C₄	C₂³×D₄	1	12	yes	—
197	C₄	C₂×H(5,𝔽₂)	1	2	yes	complemented
198	C₂	—	2	4	no	—
199	C₂	—	2	18	no	—
200	C₂	—	4	18	no	—
201	C₂	C₂⁴×D₄	1	1	yes	characteristic
202	C₂	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
203	C₂	H(7,𝔽₂)	1	1	yes	characteristic

	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. #3498	C₂	semidirect
3. #3496	C₂	semidirect
4. #3488	C₂	semidirect
5. #3484	C₂	semidirect
6. D₄×#41	C₂	semidirect
7. C₂×D₄²	C₂	semidirect
8. C₄×H(5,𝔽₂)	C₂	semidirect
11. C₂×H(5,𝔽₂)	C₂²	semidirect
12. C₂×H(5,𝔽₂)	C₂²	semidirect
13. C₂²×#41	C₂²	semidirect
14. C₂³×D₄	C₂²	semidirect
15. C₂³×D₄	C₂²	semidirect
16. #550	C₂²	semidirect
17. #550	C₂²	semidirect
18. D₄×Q₈	C₂²	semidirect
19. #548	C₂²	semidirect
20. #547	C₂²	semidirect
21. #546	C₂²	semidirect
22. #546	C₂²	semidirect
23. D₄²	C₂²	semidirect
24. D₄²	C₂²	semidirect
25. D₄²	C₂²	semidirect
26. D₄²	C₂²	semidirect
27. #537	C₂²	semidirect
28. #535	C₂²	semidirect
29. #534	C₂²	semidirect
30. #532	C₂²	semidirect
31. C₂×#127	C₂²	semidirect
32. #525	C₂²	semidirect
33. C₂×#121	C₂²	semidirect
34. C₂×#121	C₂²	semidirect
35. C₂×#120	C₂²	semidirect
36. #520	C₂²	semidirect
37. #518	C₂²	semidirect
38. C₄×#41	C₂²	semidirect
39. C₂×C₄×D₄	C₂²	semidirect
40. C₂×C₄×D₄	C₂²	semidirect
41. C₂⁵	C₂³	semidirect
44. H(5,𝔽₂)	D₄	direct (N × H)
47. C₂×#41	C₂³	semidirect
48. C₂×#41	C₂³	semidirect
49. C₂×#41	C₂³	semidirect
50. C₂²×Q₈	C₂³	semidirect
55. C₂²×D₄	C₂³	semidirect
56. C₂²×D₄	C₂³	semidirect
57. C₂²×D₄	C₂³	semidirect
58. C₂²×D₄	C₂³	semidirect
59. C₂²×D₄	C₂³	semidirect
60. C₂²×D₄	C₂³	semidirect
61. C₂²×D₄	C₂³	semidirect
62. C₂³×C₄	C₂³	semidirect
63. C₂³×C₄	C₂³	semidirect
64. C₄⋊Q₈	C₂³	semidirect
65. #127	C₂³	semidirect
66. #127	C₂³	semidirect
67. #124	C₂³	semidirect
68. #123	C₂³	semidirect
69. C₂²⋊Q₈	C₂³	semidirect
70. C₂²⋊Q₈	C₂³	semidirect
71. #121	C₂³	semidirect
72. #121	C₂³	semidirect
73. #121	C₂³	semidirect
74. #121	C₂³	semidirect
75. #121	C₂³	semidirect
76. #121	C₂³	semidirect
77. #120	C₂³	semidirect
78. #120	C₂³	semidirect
79. C₄×Q₈	C₂³	semidirect
80. C₄×Q₈	C₂³	semidirect
82. C₄×D₄	C₂³	semidirect
83. C₄×D₄	C₂³	semidirect
84. C₄×D₄	C₂³	semidirect
85. C₄×D₄	C₂³	semidirect
86. C₄×D₄	C₂³	semidirect
87. C₄×D₄	C₂³	semidirect
88. #117	C₂³	semidirect
89. C₂×C₄⋊C₄	C₂³	semidirect
90. C₂×C₄⋊C₄	C₂³	semidirect
91. C₂×C₂²⋊C₄	C₂³	semidirect
92. C₂×C₂²⋊C₄	C₂³	semidirect
93. C₂×C₂²⋊C₄	C₂³	semidirect
94. C₂×C₄²	C₂³	semidirect
104. #41	C₂×D₄	semidirect
116. C₂×D₄	C₂×D₄	semidirect
154. C₂³	C₂²×D₄	semidirect
159. Q₈	C₂²×D₄	semidirect
166. D₄	C₂²×D₄	semidirect
167. D₄	H(5,𝔽₂)	direct (N × H)
174. C₂×C₄	C₂²×D₄	semidirect
191. C₂²	C₂×H(5,𝔽₂)	semidirect
197. C₄	C₂×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
C₂²×H(5,𝔽₂)	2
#3498	2
#3496	2
#3488	2
#3484	2
D₄×#41	2
C₂×D₄²	2
C₄×H(5,𝔽₂)	2

Subgroups — D4×H(5,𝔽2) #62981

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×H(5,𝔽₂) #62981