Subgroups — #13556

Order 256.

145 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	2	yes	maximal, complemented
3	#1400	C₂	1	4	yes	maximal, complemented
4	C₂×#131	C₄	1	1	yes	characteristic
5	#504	—	2	2	no	—
6	#504	—	2	2	no	—
7	C₂×M₃₂	C₄	1	1	yes	characteristic
8	C₂×M₃₂	C₂²	1	2	yes	—
9	C₂²×C₁₆	C₄	1	2	yes	—
10	C₂²×C₁₆	C₂²	1	1	yes	characteristic
11	M₆₄	—	2	2	no	—
12	M₆₄	C₂²	1	4	yes	—
13	C₂×C₃₂	—	2	2	no	—
14	C₂×#41	C₈	1	1	yes	characteristic
15	#131	—	2	2	no	—
16	#131	—	2	2	no	—
17	C₂×M₁₆	C₈	1	1	yes	characteristic
18	C₂×M₁₆	C₈	1	2	yes	—
19	C₂²×C₈	C₂×C₄	1	2	yes	—
20	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
21	M₃₂	—	2	2	no	—
22	M₃₂	—	2	4	no	—
23	C₂×C₁₆	—	2	2	no	—
24	C₂×C₁₆	—	2	2	no	—
25	C₂×C₁₆	C₂×C₄	1	2	yes	—
26	C₂×C₁₆	C₂×C₄	1	1	yes	characteristic
27	C₂×C₁₆	D₄	1	2	yes	—
28	C₂×C₁₆	D₄	1	2	yes	—
29	C₂×C₁₆	C₂³	1	1	yes	characteristic
30	C₃₂	—	2	4	no	—
31	#41	—	2	2	no	—
32	#41	—	2	2	no	—
33	C₂×Q₈	C₁₆	1	1	yes	characteristic
34	C₂×D₄	C₁₆	1	2	yes	—
35	C₂×D₄	C₁₆	1	1	yes	characteristic
36	C₂²×C₄	C₂×C₈	1	2	yes	—
37	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
38	M₁₆	—	2	2	no	—
39	M₁₆	—	2	4	no	—
40	C₂×C₈	—	2	2	no	—
41	C₂×C₈	—	2	2	no	—
42	C₂×C₈	C₂²⋊C₄	1	2	yes	—
43	C₂×C₈	C₂×C₈	1	2	yes	—
44	C₂×C₈	C₂×C₈	1	1	yes	characteristic
45	C₂×C₈	M₁₆	1	2	yes	—
46	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
47	C₁₆	—	2	2	no	—
48	C₁₆	C₂²⋊C₄	1	1	yes	characteristic
49	C₁₆	C₂²⋊C₄	1	1	yes	characteristic
50	C₁₆	C₂×D₄	1	2	yes	—
51	C₂³	C₂×C₁₆	1	2	yes	—
52	C₂³	C₂×C₁₆	1	1	yes	characteristic
53	Q₈	—	2	2	no	—
54	D₄	—	2	4	no	—
55	D₄	—	2	2	no	—
56	C₂×C₄	—	2	2	no	—
57	C₂×C₄	—	2	2	no	—
58	C₂×C₄	C₂²⋊C₈	1	2	yes	—
59	C₂×C₄	C₂×C₁₆	1	2	yes	—
60	C₂×C₄	C₂×C₁₆	1	1	yes	characteristic
61	C₂×C₄	M₃₂	1	2	yes	—
62	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
63	C₈	—	2	2	no	—
64	C₈	C₂²⋊C₈	1	2	yes	—
65	C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₈	C₂×M₁₆	1	1	yes	characteristic
67	C₂²	—	2	2	no	—
68	C₂²	—	4	1	no	—
69	C₂²	—	4	2	no	—
70	C₂²	C₂²⋊C₁₆	1	2	yes	—
71	C₂²	C₂²×C₁₆	1	1	yes	characteristic
72	C₄	—	2	2	no	—
73	C₄	C₂²⋊C₁₆	1	2	yes	—
74	C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
75	C₄	C₂×M₃₂	1	1	yes	characteristic
76	C₂	—	2	1	no	—
77	C₂	—	2	2	no	—
78	C₂	—	4	2	no	—
79	C₂	C₂×C₂²⋊C₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	C₂²⋊C₄
commutator	C₂²	C₂²×C₁₆
Frattini, cosocle	C₂×C₁₆	C₂³
socle	C₂	C₂×C₂²⋊C₁₆
prime radical	C₂×#41	C₈
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆ ◁ C₂²×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. #1400	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
#1400	2