Subgroups — #8133

Order 256.

223 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#1744	C₂	1	2	yes	maximal
2	C₄²×C₈	C₂	1	1	yes	maximal, characteristic
3	#1273	C₂	1	4	yes	maximal
4	C₂×C₄×C₈	C₄	1	2	yes	—
5	C₂×C₄×C₈	C₄	1	2	yes	—
6	C₂×C₄×C₈	C₂²	1	2	yes	—
7	C₂×C₄×C₈	C₂²	1	4	yes	—
8	C₄³	C₂²	1	1	yes	characteristic
9	#336	C₄	1	8	yes	—
10	C₂²×C₈	C₂×C₄	1	8	yes	—
11	C₂²×C₈	C₂×C₄	1	4	yes	—
12	C₂×C₄²	C₂×C₄	1	4	yes	—
13	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄²	C₂³	1	1	yes	characteristic
16	C₄×C₈	—	2	4	no	—
17	C₄×C₈	—	2	4	no	—
18	C₄×C₈	—	2	8	no	—
19	C₄×C₈	C₈	1	8	yes	complemented
20	C₂²×C₄	C₄²	1	2	yes	—
21	C₂²×C₄	C₄²	1	2	yes	—
22	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
23	C₂²×C₄	C₂²×C₄	1	2	yes	—
24	C₂×C₈	—	2	4	no	—
25	C₂×C₈	—	2	4	no	—
26	C₂×C₈	—	2	8	no	—
27	C₂×C₈	—	2	16	no	—
28	C₂×C₈	C₂×C₈	1	8	yes	—
29	C₄²	—	2	8	no	—
30	C₄²	C₂×C₈	1	4	yes	—
31	C₄²	M₁₆	1	4	yes	—
32	C₄²	#41	1	4	yes	—
33	C₂³	C₂×C₄²	1	1	yes	characteristic
34	C₂×C₄	—	2	4	no	—
35	C₂×C₄	—	2	4	no	—
36	C₂×C₄	C₄×C₈	1	4	yes	—
37	C₂×C₄	#97	1	4	yes	—
38	C₂×C₄	#117	1	2	yes	—
39	C₂×C₄	#117	1	4	yes	—
40	C₂×C₄	C₂²×C₈	1	2	yes	—
41	C₂×C₄	C₂×M₁₆	1	2	yes	—
42	C₂×C₄	#131	1	8	yes	—
43	C₈	—	2	8	no	—
44	C₈	—	4	8	no	—
45	C₂²	#376	1	1	yes	characteristic
46	C₂²	C₂×C₄×C₈	1	1	yes	characteristic
47	C₂²	C₂×#97	1	1	yes	characteristic
48	C₂²	#405	1	4	yes	—
49	C₄	—	2	4	no	—
50	C₄	—	2	4	no	—
51	C₄	#431	1	4	yes	—
52	C₄	#433	1	8	yes	—
53	C₂	#1450	1	2	yes	—
54	C₂	#1452	1	2	yes	—
55	C₂	#1744	1	1	yes	characteristic
56	C₂	#1746	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
19. C₄×C₈	C₈	semidirect

Maximal subgroups

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

subgroup	index
#1744	2
C₄²×C₈	2
#1273	2