Subgroups — #12036

Order 256.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2162	C₂	1	1	yes	maximal, characteristic
2	#1567	C₂	1	1	yes	maximal, characteristic
3	#1564	C₂	1	1	yes	maximal, characteristic
4	#1372	C₂	1	2	yes	maximal, complemented
5	#1371	C₂	1	2	yes	maximal, complemented
6	#431	—	2	1	no	—
7	#423	—	2	1	no	—
8	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
9	C₄⋊C₁₆	C₄	1	2	yes	—
10	C₂²⋊C₁₆	C₄	1	2	yes	—
11	C₄×C₁₆	C₂²	1	2	yes	—
12	#335	—	2	1	no	—
13	#335	C₂²	1	1	yes	characteristic
14	#335	C₂²	1	1	yes	characteristic
15	#335	C₂²	1	2	yes	—
16	#334	—	2	1	no	—
17	C₂×M₁₆	—	4	1	no	—
18	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈	Q₈	1	1	yes	characteristic
20	C₂×C₄²	D₄	1	1	yes	characteristic
21	C₂×C₁₆	C₂×C₄	1	4	yes	—
22	C₄⋊C₈	—	2	1	no	—
23	C₄⋊C₈	—	2	1	no	—
24	C₄⋊C₈	—	2	2	no	—
25	C₄⋊C₈	—	4	1	no	—
26	C₂²⋊C₈	—	4	1	no	—
27	C₄×C₈	—	4	1	no	—
28	C₄×C₈	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈	D₄	1	1	yes	characteristic
30	C₄×C₈	Q₈	1	1	yes	characteristic
31	C₄×C₈	C₂³	1	1	yes	characteristic
32	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
33	C₂²×C₄	D₈	1	1	yes	characteristic
34	C₂²×C₄	Q₁₆	1	1	yes	characteristic
35	M₁₆	—	8	2	no	—
36	C₂×C₈	—	2	2	no	—
37	C₂×C₈	—	4	1	no	—
38	C₂×C₈	—	4	1	no	—
39	C₂×C₈	—	4	1	no	—
40	C₂×C₈	—	4	2	no	—
41	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
45	C₄²	C₄⋊C₄	1	1	yes	characteristic
46	C₄²	D₈	1	1	yes	characteristic
47	C₄²	Q₁₆	1	1	yes	characteristic
48	C₄²	C₂×D₄	1	1	yes	characteristic
49	C₁₆	—	2	4	no	—
50	C₂³	#107	1	1	yes	characteristic
51	C₂×C₄	—	2	2	no	—
52	C₂×C₄	—	2	1	no	—
53	C₂×C₄	—	2	1	no	—
54	C₂×C₄	—	2	1	no	—
55	C₂×C₄	—	2	1	no	—
56	C₂×C₄	#107	1	1	yes	characteristic
57	C₂×C₄	#107	1	1	yes	characteristic
58	C₂×C₄	#107	1	1	yes	characteristic
59	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄	C₂×D₈	1	1	yes	characteristic
61	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
62	C₈	—	2	1	no	—
63	C₈	—	2	1	no	—
64	C₈	—	2	1	no	—
65	C₈	—	8	1	no	—
66	C₈	—	8	2	no	—
67	C₈	—	8	1	no	—
68	C₈	#131	1	2	yes	—
69	C₂²	—	2	1	no	—
70	C₂²	—	2	1	no	—
71	C₂²	—	2	1	no	—
72	C₂²	C₂×#107	1	1	yes	characteristic
73	C₄	—	2	1	no	—
74	C₄	—	2	1	no	—
75	C₄	—	2	1	no	—
76	C₄	#368	1	2	yes	—
77	C₄	#424	1	1	yes	characteristic
78	C₄	#430	1	1	yes	characteristic
79	C₄	#509	1	1	yes	characteristic
80	C₄	#510	1	1	yes	characteristic
81	C₂	—	4	1	no	—
82	C₂	#1571	1	1	yes	characteristic
83	C₂	#2159	1	1	yes	characteristic
84	C₂	C₂×#368	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₈
commutator	C₂×C₈	C₂²×C₄
Frattini, cosocle	C₄×C₈	C₂³
socle	C₂²	C₂×#107
prime radical	C₂³	#107
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
4. #1372	C₂	semidirect
5. #1371	C₂	semidirect

Maximal subgroups

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

subgroup	index
#2162	2
#1567	2
#1564	2
#1372	2
#1371	2