Subgroups — #7139

Order 256.

131 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×M₃₂	C₂	1	2	yes	maximal
2	C₂×#322	C₂	1	1	yes	maximal, characteristic
3	C₂×M₃₂	C₄	1	4	yes	—
4	C₂×C₄×C₈	C₄	1	2	yes	—
5	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
6	#346	—	2	2	no	—
7	C₄×C₁₆	—	2	2	no	—
8	#322	—	2	2	no	—
9	C₂²×C₈	C₈	1	4	yes	—
10	C₂²×C₈	C₂×C₄	1	2	yes	—
11	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
12	M₃₂	—	2	4	no	—
13	M₃₂	C₈	1	8	yes	complemented
14	C₂×C₁₆	—	2	4	no	—
15	C₄×C₈	—	2	2	no	—
16	C₄×C₈	—	2	2	no	—
17	C₄×C₈	D₄	1	2	yes	—
18	C₄×C₈	D₄	1	1	yes	characteristic
19	C₄×C₈	Q₈	1	1	yes	characteristic
20	C₂²×C₄	C₄²	1	1	yes	characteristic
21	C₂²×C₄	C₂×C₈	1	2	yes	—
22	C₂×C₈	—	2	4	no	—
23	C₂×C₈	—	4	4	no	—
24	C₂×C₈	C₂²⋊C₄	1	2	yes	—
25	C₂×C₈	C₄⋊C₄	1	2	yes	—
26	C₂×C₈	C₂×C₈	1	4	yes	—
27	C₂×C₈	M₁₆	1	4	yes	—
28	C₄²	C₂²⋊C₄	1	1	yes	characteristic
29	C₄²	C₄⋊C₄	1	1	yes	characteristic
30	C₄²	M₁₆	1	2	yes	—
31	C₁₆	—	2	8	no	—
32	C₂³	C₄×C₈	1	1	yes	characteristic
33	C₂×C₄	—	2	4	no	—
34	C₂×C₄	#95	1	1	yes	characteristic
35	C₂×C₄	C₄×C₈	1	2	yes	—
36	C₂×C₄	#97	1	1	yes	characteristic
37	C₂×C₄	#97	1	2	yes	—
38	C₂×C₄	C₂²⋊C₈	1	2	yes	—
39	C₂×C₄	C₄⋊C₈	1	2	yes	—
40	C₈	—	4	4	no	—
41	C₈	C₂²⋊C₈	1	4	yes	—
42	C₈	C₄⋊C₈	1	4	yes	—
43	C₂²	—	2	2	no	—
44	C₂²	C₈²	1	1	yes	characteristic
45	C₂²	#322	1	1	yes	characteristic
46	C₂²	#336	1	1	yes	characteristic
47	C₄	—	2	2	no	—
48	C₄	#322	1	2	yes	—
49	C₄	#336	1	2	yes	—
50	C₄	#349	1	4	yes	—
51	C₂	—	2	1	no	—
52	C₂	—	2	1	no	—
53	C₂	#1273	1	1	yes	characteristic
54	C₂	#1377	1	2	yes	—

	subgroup	G/H
center	C₄²	C₂²⋊C₄
commutator	C₂²	C₈²
Frattini, cosocle	C₂×C₄×C₈	C₂²
socle	C₂²	#336
prime radical	C₂³	C₄×C₈
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
13. M₃₂	C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×M₃₂	2
C₂×#322	2