Subgroups — #20329

Order 256.

751 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×M₁₆	C₂	1	6	yes	maximal
2	C₂×#376	C₂	1	1	yes	maximal, characteristic
3	#1726	C₂	1	8	yes	maximal, complemented
4	C₂²×M₁₆	C₂²	1	4	yes	—
5	C₂²×C₄²	C₂²	1	3	yes	—
6	C₄×M₁₆	C₄	1	48	yes	complemented
7	C₂×#97	C₂²	1	12	yes	—
8	C₂×C₄×C₈	C₂²	1	12	yes	—
9	#376	C₄	1	4	yes	—
10	#376	C₂²	1	4	yes	—
11	C₂×#95	C₄	1	4	yes	—
12	C₂³×C₄	C₂×C₄	1	6	yes	—
13	C₂³×C₄	C₂³	1	1	yes	characteristic
14	C₂×M₁₆	C₂×C₄	1	48	yes	—
15	C₂²×C₈	C₂³	1	8	yes	—
16	C₂×C₄²	C₂×C₄	1	6	yes	—
17	C₂×C₄²	C₂×C₄	1	24	yes	—
18	C₂×C₄²	C₂³	1	6	yes	—
19	#97	C₂×C₄	1	48	yes	complemented
20	C₄×C₈	C₂×C₄	1	48	yes	complemented
21	#95	C₂×C₄	1	16	yes	—
22	C₂⁴	C₂²×C₄	1	1	yes	characteristic
23	C₂²×C₄	—	4	12	no	—
24	C₂²×C₄	C₄²	1	24	yes	—
25	C₂²×C₄	C₂²×C₄	1	12	yes	—
26	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
27	C₂²×C₄	C₂²×C₄	1	12	yes	—
28	C₂²×C₄	C₂⁴	1	1	yes	characteristic
29	M₁₆	C₄²	1	64	yes	—
30	C₂×C₈	C₂²×C₄	1	48	yes	—
31	C₄²	C₄²	1	24	yes	—
32	C₄²	C₂²×C₄	1	12	yes	—
33	C₄²	C₂²×C₄	1	12	yes	—
34	C₂³	—	2	4	no	—
35	C₂³	C₂×C₄²	1	6	yes	—
36	C₂³	C₂³×C₄	1	1	yes	characteristic
37	C₂×C₄	—	2	24	no	—
38	C₂×C₄	—	4	24	no	—
39	C₂×C₄	C₂×C₄²	1	24	yes	—
40	C₂×C₄	C₂×C₄²	1	6	yes	—
41	C₂×C₄	C₂×C₄²	1	16	yes	—
42	C₂×C₄	C₂³×C₄	1	6	yes	—
43	C₈	C₂×C₄²	1	32	yes	—
44	C₂²	—	2	12	no	—
45	C₂²	C₄³	1	4	yes	—
46	C₂²	C₂²×C₄²	1	3	yes	—
47	C₂²	#568	1	4	yes	—
48	C₄	—	2	24	no	—
49	C₄	C₄³	1	4	yes	—
50	C₄	C₂²×C₄²	1	4	yes	—
51	C₂	—	2	4	no	—
52	C₂	C₂×C₄³	1	1	yes	characteristic
53	C₂	#2873	1	6	yes	—

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁴
commutator	C₂	C₂×C₄³
socle	C₂³	C₂³×C₄
prime radical	C₂⁴	C₂²×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #1726	C₂	semidirect
6. C₄×M₁₆	C₄	semidirect
19. #97	C₂×C₄	semidirect
20. C₄×C₈	C₂×C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×M₁₆	2
C₂×#376	2
#1726	2