Subgroups — C₂×#2162 #33415

Order 256.

363 conjugacy classes of subgroups.

The subgroup lattice has 130 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₈	C₂	1	1	yes	maximal, characteristic
2	#2162	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄⋊C₁₆	C₂	1	2	yes	maximal, complemented
4	C₂×C₂²⋊C₁₆	C₂	1	2	yes	maximal
5	C₂×C₄×C₁₆	C₂	1	2	yes	maximal, complemented
6	C₂³×C₈	C₄	1	1	yes	characteristic
7	C₂³×C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₄²	C₄	1	1	yes	characteristic
9	C₂²×C₁₆	C₂²	1	4	yes	—
10	C₂×C₄×C₈	C₄	1	1	yes	characteristic
11	C₂×C₄×C₈	C₄	1	1	yes	characteristic
12	C₂×C₄×C₈	C₄	1	4	yes	—
13	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈	C₂²	1	4	yes	—
15	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
16	C₄⋊C₁₆	C₂²	1	8	yes	complemented
17	C₂²⋊C₁₆	C₂²	1	8	yes	—
18	C₄×C₁₆	C₂²	1	8	yes	complemented
19	C₂³×C₄	C₈	1	1	yes	characteristic
20	C₂³×C₄	C₈	1	1	yes	characteristic
21	C₂³×C₄	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈	—	2	4	no	—
23	C₂²×C₈	—	2	4	no	—
24	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈	C₂×C₄	1	4	yes	—
26	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈	C₂×C₄	1	2	yes	—
28	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈	C₂³	1	2	yes	—
30	C₂²×C₈	C₂³	1	1	yes	characteristic
31	C₂×C₄²	C₈	1	1	yes	characteristic
32	C₂×C₄²	C₈	1	4	yes	—
33	C₂×C₄²	C₈	1	1	yes	characteristic
34	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²	C₂×C₄	1	4	yes	—
36	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
37	C₂×C₁₆	—	2	8	no	—
38	C₂×C₁₆	C₂³	1	8	yes	—
39	C₄×C₈	C₂×C₄	1	4	yes	—
40	C₄×C₈	C₂×C₄	1	4	yes	—
41	C₄×C₈	C₂×C₄	1	2	yes	—
42	C₄×C₈	C₂×C₄	1	2	yes	—
43	C₄×C₈	C₂³	1	2	yes	—
44	C₄×C₈	C₂³	1	2	yes	—
45	C₂⁴	C₂×C₈	1	1	yes	characteristic
46	C₂²×C₄	—	2	4	no	—
47	C₂²×C₄	—	2	2	no	—
48	C₂²×C₄	—	2	2	no	—
49	C₂²×C₄	C₁₆	1	8	yes	complemented
50	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
51	C₂²×C₄	C₂×C₈	1	2	yes	—
52	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
53	C₂²×C₄	C₂×C₈	1	4	yes	—
54	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
55	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
56	C₂²×C₄	C₂×C₈	1	4	yes	—
57	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
58	C₂²×C₄	C₂²×C₄	1	2	yes	—
59	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈	—	2	2	no	—
61	C₂×C₈	—	2	2	no	—
62	C₂×C₈	—	2	4	no	—
63	C₂×C₈	—	2	8	no	—
64	C₂×C₈	—	2	4	no	—
65	C₂×C₈	M₁₆	1	4	yes	—
66	C₂×C₈	C₂²×C₄	1	2	yes	—
67	C₂×C₈	C₂²×C₄	1	2	yes	—
68	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈	C₂²×C₄	1	2	yes	—
70	C₂×C₈	#41	1	4	yes	—
71	C₂×C₈	C₂⁴	1	1	yes	characteristic
72	C₄²	C₂×C₈	1	4	yes	—
73	C₄²	C₂×C₈	1	4	yes	—
74	C₄²	C₂×C₈	1	2	yes	—
75	C₄²	C₂×C₈	1	2	yes	—
76	C₄²	C₂²×C₄	1	2	yes	—
77	C₄²	C₂²×C₄	1	2	yes	—
78	C₁₆	—	2	8	no	—
79	C₂³	—	2	2	no	—
80	C₂³	—	2	2	no	—
81	C₂³	C₂×C₁₆	1	4	yes	—
82	C₂³	C₂²×C₈	1	2	yes	—
83	C₂³	C₂²×C₈	1	1	yes	characteristic
84	C₂×C₄	—	2	2	no	—
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	2	2	no	—
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	2	4	no	—
89	C₂×C₄	—	2	8	no	—
90	C₂×C₄	C₂×C₁₆	1	4	yes	—
91	C₂×C₄	C₂×C₁₆	1	4	yes	—
92	C₂×C₄	C₂×C₁₆	1	16	yes	complemented
93	C₂×C₄	M₃₂	1	4	yes	—
94	C₂×C₄	#117	1	2	yes	—
95	C₂×C₄	C₂²×C₈	1	2	yes	—
96	C₂×C₄	C₂²×C₈	1	2	yes	—
97	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
98	C₂×C₄	C₂²×C₈	1	2	yes	—
99	C₂×C₄	C₂²×C₈	1	2	yes	—
100	C₂×C₄	C₂²×C₈	1	2	yes	—
101	C₂×C₄	C₂×M₁₆	1	2	yes	—
102	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
103	C₈	—	2	2	no	—
104	C₈	—	2	2	no	—
105	C₈	#117	1	2	yes	—
106	C₈	C₂×M₁₆	1	4	yes	—
107	C₈	C₂×#41	1	2	yes	—
108	C₂²	—	2	2	no	—
109	C₂²	—	2	2	no	—
110	C₂²	—	2	8	no	—
111	C₂²	#431	1	2	yes	—
112	C₂²	C₂²×C₁₆	1	4	yes	—
113	C₂²	C₂²×C₁₆	1	2	yes	—
114	C₂²	C₂×M₃₂	1	2	yes	—
115	C₂²	C₂³×C₈	1	1	yes	characteristic
116	C₄	—	2	2	no	—
117	C₄	—	2	2	no	—
118	C₄	#431	1	2	yes	—
119	C₄	C₂²×C₁₆	1	4	yes	—
120	C₄	C₂²×C₁₆	1	4	yes	—
121	C₄	C₂×M₃₂	1	4	yes	—
122	C₄	C₂×#117	1	1	yes	characteristic
123	C₄	C₂²×M₁₆	1	1	yes	characteristic
124	C₂	—	2	4	no	—
125	C₂	#2162	1	4	yes	direct product factor
126	C₂	C₂×#431	1	1	yes	characteristic
127	C₂	C₂³×C₁₆	1	1	yes	characteristic
128	C₂	C₂²×M₃₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₈	C₂²
commutator	C₂	C₂³×C₁₆
Frattini, cosocle	C₂×C₈	C₂⁴
socle	C₂³	C₂²×C₈
prime radical	C₂⁴	C₂×C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #2162	C₂	direct (N × H)
3. C₂×C₄⋊C₁₆	C₂	semidirect
5. C₂×C₄×C₁₆	C₂	semidirect
16. C₄⋊C₁₆	C₂²	semidirect
18. C₄×C₁₆	C₂²	semidirect
49. C₂²×C₄	C₁₆	semidirect
92. C₂×C₄	C₂×C₁₆	semidirect
125. C₂	#2162	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₈	2
#2162	2
C₂×C₄⋊C₁₆	2
C₂×C₂²⋊C₁₆	2
C₂×C₄×C₁₆	2

Subgroups — C2×#2162 #33415

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#2162 #33415