Subgroups — C₂×#1565 #17786

Order 256.

417 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#431	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×M₁₆	C₂	1	1	yes	maximal, characteristic
4	#1565	C₂	1	8	yes	maximal, direct product factor
5	C₂×#335	C₂	1	2	yes	maximal, complemented
6	C₂×#334	C₂	1	2	yes	maximal, complemented
7	C₂²×M₁₆	C₄	1	1	yes	characteristic
8	C₂²×M₁₆	C₄	1	1	yes	characteristic
9	C₂³×C₈	—	2	1	no	—
10	C₂²×C₄²	C₂²	1	1	yes	characteristic
11	#431	—	2	2	no	—
12	#431	C₂²	1	4	yes	—
13	C₂×C₄⋊C₈	—	2	2	no	—
14	C₂×C₄⋊C₈	—	2	1	no	—
15	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₈	C₂²	1	2	yes	—
17	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₈	C₂²	1	4	yes	—
19	C₂×C₂²⋊C₈	—	2	1	no	—
20	C₄×M₁₆	C₄	1	4	yes	—
21	C₄×M₁₆	C₂²	1	4	yes	—
22	C₂×#97	C₄	1	2	yes	—
23	C₂×C₄×C₈	—	2	1	no	—
24	C₂×C₄×C₈	C₂²	1	2	yes	—
25	#335	C₂²	1	8	yes	complemented
26	#334	C₂²	1	8	yes	complemented
27	C₂³×C₄	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄	D₄	1	1	yes	characteristic
29	C₂³×C₄	Q₈	1	1	yes	characteristic
30	C₂×M₁₆	—	2	4	no	—
31	C₂×M₁₆	C₈	1	8	yes	complemented
32	C₂×M₁₆	C₂×C₄	1	4	yes	—
33	C₂×M₁₆	C₂×C₄	1	4	yes	—
34	C₂²×C₈	—	2	4	no	—
35	C₂²×C₈	—	2	2	no	—
36	C₂²×C₈	—	2	1	no	—
37	C₂²×C₈	—	2	1	no	—
38	C₂²×C₈	—	4	4	no	—
39	C₂²×C₈	C₂×C₄	1	2	yes	—
40	C₂²×C₈	C₂×C₄	1	2	yes	—
41	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²	C₂×C₄	1	2	yes	—
43	C₂×C₄²	D₄	1	2	yes	—
44	C₂×C₄²	D₄	1	1	yes	characteristic
45	C₂×C₄²	Q₈	1	1	yes	characteristic
46	C₂×C₄²	Q₈	1	2	yes	—
47	C₂×C₄²	C₂³	1	2	yes	—
48	C₂×C₄²	C₂³	1	1	yes	characteristic
49	C₄⋊C₈	—	2	4	no	—
50	C₄⋊C₈	—	2	2	no	—
51	C₄⋊C₈	—	2	1	no	—
52	C₄⋊C₈	—	2	1	no	—
53	C₄⋊C₈	C₂³	1	2	yes	—
54	C₄⋊C₈	C₂³	1	4	yes	—
55	C₄⋊C₈	C₂³	1	2	yes	—
56	C₂²⋊C₈	—	2	4	no	—
57	#97	C₂×C₄	1	8	yes	—
58	C₄×C₈	—	2	4	no	—
59	C₄×C₈	C₂×C₄	1	4	yes	—
60	C₄×C₈	C₂³	1	4	yes	—
61	C₂⁴	C₄⋊C₄	1	1	yes	characteristic
62	C₂²×C₄	—	2	2	no	—
63	C₂²×C₄	—	2	2	no	—
64	C₂²×C₄	—	4	2	no	—
65	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
66	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
67	C₂²×C₄	C₄⋊C₄	1	2	yes	—
68	C₂²×C₄	C₄⋊C₄	1	2	yes	—
69	C₂²×C₄	C₄⋊C₄	1	2	yes	—
70	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
71	C₂²×C₄	C₂×C₈	1	4	yes	—
72	C₂²×C₄	M₁₆	1	4	yes	—
73	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄	C₂²×C₄	1	2	yes	—
75	C₂²×C₄	C₂×D₄	1	2	yes	—
76	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
78	C₂²×C₄	C₂×Q₈	1	2	yes	—
79	M₁₆	—	2	8	no	—
80	M₁₆	C₂×C₈	1	16	yes	complemented
81	C₂×C₈	—	2	4	no	—
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	2	2	no	—
84	C₂×C₈	—	2	4	no	—
85	C₂×C₈	—	4	8	no	—
86	C₂×C₈	—	4	2	no	—
87	C₂×C₈	—	4	4	no	—
88	C₂×C₈	—	4	2	no	—
89	C₂×C₈	C₂×C₈	1	8	yes	complemented
90	C₂×C₈	C₂²×C₄	1	4	yes	—
91	C₂×C₈	C₂²×C₄	1	4	yes	—
92	C₄²	C₄⋊C₄	1	2	yes	—
93	C₄²	C₄⋊C₄	1	2	yes	—
94	C₄²	C₄⋊C₄	1	2	yes	—
95	C₄²	C₂²×C₄	1	1	yes	characteristic
96	C₄²	C₂²×C₄	1	2	yes	—
97	C₄²	C₂×D₄	1	2	yes	—
98	C₄²	C₂×D₄	1	1	yes	characteristic
99	C₄²	C₂×Q₈	1	2	yes	—
100	C₄²	C₂×Q₈	1	1	yes	characteristic
101	C₄²	C₂⁴	1	1	yes	characteristic
102	C₂³	—	2	2	no	—
103	C₂³	—	2	2	no	—
104	C₂³	C₄⋊C₈	1	4	yes	—
105	C₂³	C₂×C₄⋊C₄	1	2	yes	—
106	C₂³	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	4	no	—
113	C₂×C₄	—	4	2	no	—
114	C₂×C₄	—	4	2	no	—
115	C₂×C₄	C₄⋊C₈	1	8	yes	—
116	C₂×C₄	C₄⋊C₈	1	4	yes	—
117	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₄	C₂×C₄⋊C₄	1	2	yes	—
119	C₂×C₄	C₂×C₄⋊C₄	1	2	yes	—
120	C₂×C₄	C₂×C₄⋊C₄	1	2	yes	—
121	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₄	C₂²×C₈	1	2	yes	—
124	C₂×C₄	C₂²×C₈	1	4	yes	—
125	C₂×C₄	C₂×M₁₆	1	2	yes	—
126	C₂×C₄	C₂×M₁₆	1	4	yes	—
127	C₂×C₄	AGL(1,ℤ/8)	1	2	yes	—
128	C₂×C₄	#137	1	2	yes	—
129	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
130	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
132	C₈	—	2	4	no	—
133	C₈	—	4	2	no	—
134	C₈	—	4	4	no	—
135	C₈	—	4	2	no	—
136	C₈	C₂²×C₈	1	8	yes	complemented
137	C₂²	—	2	8	no	—
138	C₂²	—	2	2	no	—
139	C₂²	—	2	2	no	—
140	C₂²	C₂×C₄⋊C₈	1	4	yes	—
141	C₂²	C₂×C₄⋊C₈	1	2	yes	—
142	C₂²	#428	1	2	yes	—
143	C₂²	#430	1	2	yes	—
144	C₂²	C₂²×C₄⋊C₄	1	1	yes	characteristic
145	C₄	—	2	2	no	—
146	C₄	—	2	2	no	—
147	C₄	—	2	1	no	—
148	C₄	—	2	1	no	—
149	C₄	C₂×C₄⋊C₈	1	2	yes	—
150	C₄	C₂×C₄⋊C₈	1	4	yes	—
151	C₄	#428	1	2	yes	—
152	C₄	C₂³×C₈	1	1	yes	characteristic
153	C₄	C₂²×M₁₆	1	1	yes	characteristic
154	C₄	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
155	C₄	C₂×#137	1	1	yes	characteristic
156	C₂	—	2	4	no	—
157	C₂	#1565	1	4	yes	direct product factor
158	C₂	C₂²×C₄⋊C₈	1	1	yes	characteristic
159	C₂	C₂×#428	1	1	yes	characteristic
160	C₂	C₂×#430	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₄
commutator	C₄	C₂³×C₈
Frattini, cosocle	C₄²	C₂⁴
socle	C₂³	C₂×C₄⋊C₄
prime radical	C₂⁴	C₄⋊C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
4. #1565	C₂	direct (N × H)
5. C₂×#335	C₂	semidirect
6. C₂×#334	C₂	semidirect
25. #335	C₂²	semidirect
26. #334	C₂²	semidirect
31. C₂×M₁₆	C₈	semidirect
80. M₁₆	C₂×C₈	semidirect
89. C₂×C₈	C₂×C₈	semidirect
136. C₈	C₂²×C₈	semidirect
157. C₂	#1565	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#431	2
C₂²×C₄⋊C₈	2
C₂×C₄×M₁₆	2
#1565	2
C₂×#335	2
C₂×#334	2

Subgroups — C2×#1565 #17786

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#1565 #17786