Subgroups — #13554

Order 256.

149 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#504	C₂	1	1	yes	maximal, characteristic
2	C₂×M₆₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²⋊C₃₂	C₂	1	4	yes	maximal, complemented
5	C₂×#131	C₄	1	1	yes	characteristic
6	#504	—	2	2	no	—
7	#504	—	2	2	no	—
8	C₂×M₃₂	C₄	1	1	yes	characteristic
9	C₂×M₃₂	C₄	1	2	yes	—
10	C₂²×C₁₆	C₂²	1	2	yes	—
11	C₂²×C₁₆	C₂²	1	1	yes	characteristic
12	M₆₄	—	2	2	no	—
13	C₂×C₃₂	—	2	2	no	—
14	C₂×C₃₂	C₂²	1	2	yes	—
15	C₂×C₃₂	C₂²	1	2	yes	—
16	C₂×#41	C₈	1	1	yes	characteristic
17	#131	—	2	2	no	—
18	#131	—	2	2	no	—
19	C₂×M₁₆	C₈	1	1	yes	characteristic
20	C₂×M₁₆	C₈	1	2	yes	—
21	C₂²×C₈	C₂×C₄	1	2	yes	—
22	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
23	M₃₂	—	2	2	no	—
24	M₃₂	—	2	4	no	—
25	C₂×C₁₆	—	2	2	no	—
26	C₂×C₁₆	—	2	2	no	—
27	C₂×C₁₆	C₂×C₄	1	2	yes	—
28	C₂×C₁₆	C₂×C₄	1	1	yes	characteristic
29	C₂×C₁₆	D₄	1	2	yes	—
30	C₂×C₁₆	D₄	1	2	yes	—
31	C₂×C₁₆	C₂³	1	1	yes	characteristic
32	C₃₂	—	2	2	no	—
33	C₃₂	—	2	2	no	—
34	#41	—	2	2	no	—
35	#41	—	2	2	no	—
36	C₂×Q₈	C₁₆	1	1	yes	characteristic
37	C₂×D₄	C₁₆	1	2	yes	—
38	C₂×D₄	C₁₆	1	1	yes	characteristic
39	C₂²×C₄	C₂×C₈	1	2	yes	—
40	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
41	M₁₆	—	2	2	no	—
42	M₁₆	—	2	4	no	—
43	C₂×C₈	—	2	2	no	—
44	C₂×C₈	—	2	2	no	—
45	C₂×C₈	C₂²⋊C₄	1	2	yes	—
46	C₂×C₈	C₂×C₈	1	2	yes	—
47	C₂×C₈	C₂×C₈	1	1	yes	characteristic
48	C₂×C₈	M₁₆	1	2	yes	—
49	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
50	C₁₆	—	2	2	no	—
51	C₁₆	C₂²⋊C₄	1	2	yes	—
52	C₁₆	C₂×D₄	1	1	yes	characteristic
53	C₁₆	C₂×D₄	1	1	yes	characteristic
54	C₂³	C₂×C₁₆	1	2	yes	—
55	C₂³	C₂×C₁₆	1	1	yes	characteristic
56	Q₈	—	2	2	no	—
57	D₄	—	2	4	no	—
58	D₄	—	2	2	no	—
59	C₂×C₄	—	2	2	no	—
60	C₂×C₄	—	2	2	no	—
61	C₂×C₄	C₂²⋊C₈	1	2	yes	—
62	C₂×C₄	C₂×C₁₆	1	2	yes	—
63	C₂×C₄	C₂×C₁₆	1	1	yes	characteristic
64	C₂×C₄	M₃₂	1	2	yes	—
65	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
66	C₈	—	2	2	no	—
67	C₈	C₂²⋊C₈	1	2	yes	—
68	C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₈	C₂×M₁₆	1	1	yes	characteristic
70	C₂²	—	2	2	no	—
71	C₂²	—	2	2	no	—
72	C₂²	—	2	4	no	—
73	C₂²	C₂²⋊C₁₆	1	2	yes	—
74	C₂²	C₂²×C₁₆	1	1	yes	characteristic
75	C₄	—	2	2	no	—
76	C₄	C₂²⋊C₁₆	1	2	yes	—
77	C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
78	C₄	C₂×M₃₂	1	1	yes	characteristic
79	C₂	—	2	2	no	—
80	C₂	—	4	2	no	—
81	C₂	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
82	C₂	#2258	1	2	yes	—

	subgroup	G/H
center, Frattini, cosocle	C₂×C₁₆	C₂³
commutator, socle	C₂²	C₂²×C₁₆
prime radical	C₂×#41	C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₂×M₆₄	C₂	semidirect
3. C₂²×C₃₂	C₂	semidirect
4. C₂²⋊C₃₂	C₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×#504	2
C₂×M₆₄	2
C₂²×C₃₂	2
C₂²⋊C₃₂	2