Subgroups — #12509

Order 256.

153 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2240	C₂	1	4	yes	maximal, complemented
2	#1713	C₂	1	1	yes	maximal, characteristic, complemented
3	#1712	C₂	1	1	yes	maximal, characteristic
4	C₈×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	#500	—	2	1	no	—
6	#500	C₂²	1	2	yes	—
7	#494	—	2	1	no	—
8	#493	—	2	1	no	—
9	#493	—	2	1	no	—
10	#493	C₂²	1	2	yes	—
11	#357	—	2	8	no	—
12	C₄×C₁₆	C₂²	1	2	yes	—
13	C₈²	C₂²	1	1	yes	characteristic
14	C₂×Q₁₆	—	4	2	no	—
15	C₂×D₈	—	2	4	no	—
16	C₂×D₈	—	4	2	no	—
17	C₂×D₈	—	4	2	no	—
18	C₄⋊Q₈	—	2	2	no	—
19	#127	—	2	2	no	—
20	C₂×C₁₆	D₄	1	4	yes	—
21	#107	—	2	4	no	—
22	#107	—	4	2	no	—
23	C₄×C₈	D₄	1	1	yes	characteristic
24	C₄×C₈	D₄	1	1	yes	characteristic
25	C₄×C₈	C₂³	1	1	yes	characteristic
26	C₂×Q₈	—	4	2	no	—
27	C₂×D₄	—	4	2	no	—
28	C₂×D₄	—	4	2	no	—
29	C₂×D₄	—	4	2	no	—
30	Q₁₆	—	8	4	no	—
31	D₈	—	4	8	no	—
32	D₈	—	8	4	no	—
33	D₈	—	8	4	no	—
34	C₂×C₈	D₈	1	2	yes	—
35	C₂×C₈	D₈	1	2	yes	—
36	C₂×C₈	C₂×D₄	1	2	yes	—
37	C₄⋊C₄	—	4	2	no	—
38	C₄⋊C₄	—	4	2	no	—
39	C₄²	C₂×D₄	1	1	yes	characteristic
40	C₁₆	—	2	4	no	—
41	C₂³	—	8	2	no	—
42	Q₈	—	8	4	no	—
43	D₄	—	8	4	no	—
44	D₄	—	8	4	no	—
45	D₄	—	8	4	no	—
46	C₂×C₄	—	8	2	no	—
47	C₂×C₄	#127	1	1	yes	characteristic
48	C₂×C₄	C₂×D₈	1	1	yes	characteristic
49	C₂×C₄	C₂×D₈	1	1	yes	characteristic
50	C₈	D₁₆	1	4	yes	—
51	C₈	SD₃₂	1	4	yes	—
52	C₈	#135	1	4	yes	—
53	C₂²	—	16	2	no	—
54	C₂²	—	16	2	no	—
55	C₂²	—	16	2	no	—
56	C₂²	#493	1	1	yes	characteristic
57	C₄	—	16	2	no	—
58	C₄	#495	1	2	yes	—
59	C₄	C₂×D₁₆	1	2	yes	—
60	C₄	C₂×SD₃₂	1	2	yes	—
61	C₂	—	32	2	no	—
62	C₂	#1709	1	1	yes	characteristic
63	C₂	#2246	1	1	yes	characteristic
64	C₂	#2248	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#493
commutator, Frattini, cosocle	C₄×C₈	C₂³
prime radical	#1712	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #2240	C₂	semidirect
2. #1713	C₂	semidirect
4. C₈×C₁₆	C₂	semidirect

Maximal subgroups

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

subgroup	index
#2240	2
#1713	2
#1712	2
C₈×C₁₆	2