Subgroups — C₂²×#535 #62674

Order 256.

7207 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#41	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂⁴×D₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#535	C₂	1	48	yes	maximal, direct product factor
4	C₂²×#127	C₂	1	2	yes	maximal, complemented
5	C₂²×#124	C₂	1	2	yes	maximal, complemented
6	C₂²×#121	C₂	1	4	yes	maximal, complemented
7	C₂²×#120	C₂	1	4	yes	maximal, complemented
8	C₂²×#117	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂⁶	C₂²	1	2	yes	—
10	C₂²×#41	—	2	16	no	—
11	C₂²×#41	C₂²	1	24	yes	complemented
12	C₂³×Q₈	C₂²	1	1	yes	characteristic, complemented
13	C₂³×D₄	—	2	16	no	—
14	C₂³×D₄	C₂²	1	2	yes	—
15	C₂³×D₄	C₂²	1	2	yes	—
16	C₂³×D₄	C₂²	1	2	yes	—
17	C₂³×D₄	C₂²	1	24	yes	complemented
18	C₂³×D₄	C₂²	1	1	yes	characteristic, complemented
19	C₂³×D₄	C₂²	1	8	yes	complemented
20	C₂⁴×C₄	C₂²	1	1	yes	characteristic
21	C₂⁴×C₄	C₂²	1	2	yes	complemented
22	#535	C₂²	1	256	yes	direct product factor
23	C₂×#127	C₂²	1	48	yes	complemented
24	C₂×#124	C₂²	1	48	yes	complemented
25	C₂×#121	C₂²	1	96	yes	complemented
26	C₂×#120	C₂²	1	96	yes	complemented
27	C₂×#117	C₂²	1	24	yes	complemented
28	C₂²×C₄⋊C₄	C₂²	1	2	yes	complemented
29	C₂²×C₂²⋊C₄	C₂²	1	2	yes	—
30	C₂²×C₂²⋊C₄	C₂²	1	8	yes	complemented
31	C₂²×C₄²	C₂²	1	2	yes	complemented
32	C₂⁵	—	2	16	no	—
33	C₂⁵	—	4	16	no	—
34	C₂⁵	C₂³	1	4	yes	—
35	C₂⁵	C₂³	1	2	yes	—
36	C₂⁵	C₂³	1	1	yes	characteristic
37	C₂⁵	C₂³	1	24	yes	—
38	C₂×#41	—	2	192	no	—
39	C₂×#41	C₂³	1	64	yes	complemented
40	C₂²×Q₈	—	2	8	no	—
41	C₂²×Q₈	C₂³	1	12	yes	complemented
42	C₂²×D₄	—	2	16	no	—
43	C₂²×D₄	—	2	16	no	—
44	C₂²×D₄	—	2	16	no	—
45	C₂²×D₄	—	2	8	no	—
46	C₂²×D₄	—	2	192	no	—
47	C₂²×D₄	—	4	32	no	—
48	C₂²×D₄	C₂³	1	24	yes	—
49	C₂²×D₄	C₂³	1	24	yes	—
50	C₂²×D₄	C₂³	1	24	yes	—
51	C₂²×D₄	C₂³	1	64	yes	complemented
52	C₂²×D₄	C₂³	1	12	yes	complemented
53	C₂²×D₄	C₂³	1	96	yes	complemented
54	C₂³×C₄	—	2	16	no	—
55	C₂³×C₄	D₄	1	16	yes	complemented
56	C₂³×C₄	C₂³	1	4	yes	—
57	C₂³×C₄	C₂³	1	2	yes	—
58	C₂³×C₄	C₂³	1	1	yes	characteristic
59	C₂³×C₄	C₂³	1	1	yes	characteristic
60	C₂³×C₄	C₂³	1	12	yes	—
61	C₂³×C₄	C₂³	1	24	yes	complemented
62	#127	C₂³	1	128	yes	complemented
63	#124	C₂³	1	128	yes	complemented
64	#121	C₂³	1	256	yes	complemented
65	#120	C₂³	1	256	yes	complemented
66	#117	C₂³	1	64	yes	complemented
67	C₂×C₄⋊C₄	C₂³	1	24	yes	complemented
68	C₂×C₂²⋊C₄	C₂³	1	24	yes	—
69	C₂×C₂²⋊C₄	C₂³	1	96	yes	complemented
70	C₂×C₄²	C₂³	1	24	yes	complemented
71	C₂⁴	—	2	8	no	—
72	C₂⁴	—	2	8	no	—
73	C₂⁴	—	2	96	no	—
74	C₂⁴	—	2	16	no	—
75	C₂⁴	—	2	16	no	—
76	C₂⁴	—	2	32	no	—
77	C₂⁴	—	4	96	no	—
78	C₂⁴	—	4	128	no	—
79	C₂⁴	C₂×D₄	1	8	yes	—
80	C₂⁴	C₂⁴	1	24	yes	—
81	C₂⁴	C₂⁴	1	12	yes	—
82	C₂⁴	C₂⁴	1	6	yes	—
83	C₂⁴	C₂⁴	1	32	yes	—
84	C₂⁴	C₂⁴	1	1	yes	characteristic
85	#41	—	2	256	no	—
86	C₂×Q₈	—	2	48	no	—
87	C₂×Q₈	C₂⁴	1	16	yes	complemented
88	C₂×D₄	—	2	96	no	—
89	C₂×D₄	—	2	96	no	—
90	C₂×D₄	—	2	96	no	—
91	C₂×D₄	—	2	48	no	—
92	C₂×D₄	—	2	256	no	—
93	C₂×D₄	—	4	192	no	—
94	C₂×D₄	C₂⁴	1	32	yes	—
95	C₂×D₄	C₂⁴	1	32	yes	—
96	C₂×D₄	C₂⁴	1	32	yes	—
97	C₂×D₄	C₂⁴	1	16	yes	complemented
98	C₂×D₄	C₂⁴	1	128	yes	complemented
99	C₂²×C₄	—	2	8	no	—
100	C₂²×C₄	—	2	16	no	—
101	C₂²×C₄	—	2	96	no	—
102	C₂²×C₄	C₂×D₄	1	8	yes	—
103	C₂²×C₄	C₂×D₄	1	8	yes	—
104	C₂²×C₄	C₂×D₄	1	96	yes	complemented
105	C₂²×C₄	C₂⁴	1	24	yes	—
106	C₂²×C₄	C₂⁴	1	12	yes	—
107	C₂²×C₄	C₂⁴	1	6	yes	—
108	C₂²×C₄	C₂⁴	1	6	yes	—
109	C₂²×C₄	C₂⁴	1	16	yes	—
110	C₂²×C₄	C₂⁴	1	32	yes	complemented
111	C₄⋊C₄	C₂⁴	1	32	yes	complemented
112	C₂²⋊C₄	C₂⁴	1	32	yes	—
113	C₂²⋊C₄	C₂⁴	1	128	yes	complemented
114	C₄²	C₂⁴	1	32	yes	complemented
115	C₂³	—	2	32	no	—
116	C₂³	—	2	24	no	—
117	C₂³	—	2	24	no	—
118	C₂³	—	2	64	no	—
119	C₂³	—	2	48	no	—
120	C₂³	—	2	48	no	—
121	C₂³	—	2	96	no	—
122	C₂³	—	4	64	no	—
123	C₂³	—	4	384	no	—
124	C₂³	—	4	32	no	—
125	C₂³	—	4	64	no	—
126	C₂³	C₂²×D₄	1	24	yes	—
127	C₂³	C₂²×D₄	1	4	yes	—
128	C₂³	H(5,𝔽₂)	1	8	yes	—
129	C₂³	C₂⁵	1	16	yes	—
130	C₂³	C₂⁵	1	8	yes	—
131	C₂³	C₂⁵	1	4	yes	—
132	C₂³	C₂⁵	1	3	yes	—
133	Q₈	—	2	32	no	—
134	D₄	—	2	64	no	—
135	D₄	—	2	64	no	—
136	D₄	—	2	64	no	—
137	D₄	—	2	32	no	—
138	D₄	—	4	128	no	—
139	C₂×C₄	—	2	24	no	—
140	C₂×C₄	—	2	48	no	—
141	C₂×C₄	—	2	64	no	—
142	C₂×C₄	C₂²×D₄	1	24	yes	—
143	C₂×C₄	C₂²×D₄	1	24	yes	—
144	C₂×C₄	C₂²×D₄	1	64	yes	complemented
145	C₂×C₄	C₂⁵	1	16	yes	—
146	C₂×C₄	C₂⁵	1	8	yes	—
147	C₂×C₄	C₂⁵	1	4	yes	—
148	C₂×C₄	C₂⁵	1	4	yes	—
149	C₂²	—	2	8	no	—
150	C₂²	—	2	48	no	—
151	C₂²	—	2	8	no	—
152	C₂²	—	2	16	no	—
153	C₂²	—	2	16	no	—
154	C₂²	—	2	32	no	—
155	C₂²	—	4	128	no	—
156	C₂²	—	4	48	no	—
157	C₂²	—	4	96	no	—
158	C₂²	#535	1	16	yes	direct product factor
159	C₂²	C₂³×D₄	1	8	yes	—
160	C₂²	C₂³×D₄	1	6	yes	—
161	C₂²	C₂×H(5,𝔽₂)	1	12	yes	—
162	C₂²	C₂⁶	1	1	yes	characteristic
163	C₄	—	2	8	no	—
164	C₄	—	2	16	no	—
165	C₄	C₂³×D₄	1	8	yes	—
166	C₄	C₂³×D₄	1	8	yes	—
167	C₂	—	2	8	no	—
168	C₂	—	4	8	no	—
169	C₂	—	4	16	no	—
170	C₂	C₂×#535	1	12	yes	direct product factor
171	C₂	C₂⁴×D₄	1	1	yes	characteristic
172	C₂	C₂²×H(5,𝔽₂)	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁶
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂³×#41	C₂	semidirect
2. C₂⁴×D₄	C₂	semidirect
3. C₂×#535	C₂	direct (N × H)
4. C₂²×#127	C₂	semidirect
5. C₂²×#124	C₂	semidirect
6. C₂²×#121	C₂	semidirect
7. C₂²×#120	C₂	semidirect
8. C₂²×#117	C₂	semidirect
11. C₂²×#41	C₂²	semidirect
12. C₂³×Q₈	C₂²	semidirect
17. C₂³×D₄	C₂²	semidirect
18. C₂³×D₄	C₂²	semidirect
19. C₂³×D₄	C₂²	semidirect
21. C₂⁴×C₄	C₂²	semidirect
22. #535	C₂²	direct (N × H)
23. C₂×#127	C₂²	semidirect
24. C₂×#124	C₂²	semidirect
25. C₂×#121	C₂²	semidirect
26. C₂×#120	C₂²	semidirect
27. C₂×#117	C₂²	semidirect
28. C₂²×C₄⋊C₄	C₂²	semidirect
30. C₂²×C₂²⋊C₄	C₂²	semidirect
31. C₂²×C₄²	C₂²	semidirect
39. C₂×#41	C₂³	semidirect
41. C₂²×Q₈	C₂³	semidirect
51. C₂²×D₄	C₂³	semidirect
52. C₂²×D₄	C₂³	semidirect
53. C₂²×D₄	C₂³	semidirect
55. C₂³×C₄	D₄	semidirect
61. C₂³×C₄	C₂³	semidirect
62. #127	C₂³	semidirect
63. #124	C₂³	semidirect
64. #121	C₂³	semidirect
65. #120	C₂³	semidirect
66. #117	C₂³	semidirect
67. C₂×C₄⋊C₄	C₂³	semidirect
69. C₂×C₂²⋊C₄	C₂³	semidirect
70. C₂×C₄²	C₂³	semidirect
87. C₂×Q₈	C₂⁴	semidirect
97. C₂×D₄	C₂⁴	semidirect
98. C₂×D₄	C₂⁴	semidirect
104. C₂²×C₄	C₂×D₄	semidirect
110. C₂²×C₄	C₂⁴	semidirect
111. C₄⋊C₄	C₂⁴	semidirect
113. C₂²⋊C₄	C₂⁴	semidirect
114. C₄²	C₂⁴	semidirect
144. C₂×C₄	C₂²×D₄	semidirect
158. C₂²	#535	direct (N × H)
170. C₂	C₂×#535	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×#41	2
C₂⁴×D₄	2
C₂×#535	2
C₂²×#127	2
C₂²×#124	2
C₂²×#121	2
C₂²×#120	2
C₂²×#117	2

Subgroups — C22×#535 #62674

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#535 #62674