Subgroups — #12503

Order 256.

153 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2252	C₂	1	1	yes	maximal, characteristic, complemented
2	#2246	C₂	1	1	yes	maximal, characteristic
3	#2240	C₂	1	2	yes	maximal, complemented
4	#1706	C₂	1	2	yes	maximal, complemented
5	C₈×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₆	—	2	2	no	—
7	#500	C₂²	1	2	yes	—
8	#493	C₂²	1	2	yes	—
9	#492	—	2	2	no	—
10	#486	—	2	2	no	—
11	#366	—	2	2	no	—
12	#357	—	2	4	no	—
13	C₄×C₁₆	C₂²	1	1	yes	characteristic
14	C₄×C₁₆	C₂²	1	1	yes	characteristic
15	C₈²	C₂²	1	1	yes	characteristic
16	C₂×SD₁₆	—	4	4	no	—
17	C₂×D₈	—	2	2	no	—
18	C₂×D₈	—	2	2	no	—
19	C₄⋊Q₈	—	2	2	no	—
20	#127	—	2	2	no	—
21	D₁₆	—	4	4	no	—
22	C₂×C₁₆	D₄	1	2	yes	—
23	C₂×C₁₆	D₄	1	2	yes	—
24	#107	—	2	2	no	—
25	#107	—	2	2	no	—
26	#102	—	4	4	no	—
27	C₄×C₈	D₄	1	1	yes	characteristic
28	C₄×C₈	D₄	1	1	yes	characteristic
29	C₄×C₈	C₂³	1	1	yes	characteristic
30	C₂×Q₈	—	4	2	no	—
31	C₂×D₄	—	4	2	no	—
32	C₂×D₄	—	4	2	no	—
33	C₂×D₄	—	4	2	no	—
34	SD₁₆	—	8	8	no	—
35	D₈	—	4	4	no	—
36	D₈	—	4	4	no	—
37	C₂×C₈	D₈	1	2	yes	—
38	C₂×C₈	D₈	1	2	yes	—
39	C₂×C₈	C₂×D₄	1	1	yes	characteristic
40	C₂×C₈	C₂×D₄	1	1	yes	characteristic
41	C₄⋊C₄	—	4	2	no	—
42	C₄⋊C₄	—	4	2	no	—
43	C₄²	C₂×D₄	1	1	yes	characteristic
44	C₁₆	—	2	2	no	—
45	C₁₆	SD₁₆	1	4	yes	complemented
46	C₂³	—	8	2	no	—
47	Q₈	—	8	4	no	—
48	D₄	—	8	2	no	—
49	D₄	—	8	2	no	—
50	D₄	—	8	4	no	—
51	D₄	—	8	4	no	—
52	C₂×C₄	—	8	2	no	—
53	C₂×C₄	#127	1	1	yes	characteristic
54	C₂×C₄	C₂×D₈	1	1	yes	characteristic
55	C₂×C₄	C₂×D₈	1	1	yes	characteristic
56	C₈	—	2	2	no	—
57	C₈	D₁₆	1	4	yes	complemented
58	C₈	C₂×SD₁₆	1	2	yes	—
59	C₈	#135	1	2	yes	—
60	C₂²	—	16	2	no	—
61	C₂²	—	16	2	no	—
62	C₂²	—	16	2	no	—
63	C₂²	#493	1	1	yes	characteristic
64	C₄	—	16	2	no	—
65	C₄	#492	1	1	yes	characteristic
66	C₄	#495	1	1	yes	characteristic
67	C₄	C₂×D₁₆	1	2	yes	—
68	C₄	#508	1	2	yes	—
69	C₂	—	32	2	no	—
70	C₂	#1706	1	1	yes	characteristic
71	C₂	#2246	1	1	yes	characteristic
72	C₂	#2249	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#493
commutator, Frattini, cosocle	C₄×C₈	C₂³
prime radical	#2246	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. #2252	C₂	semidirect
3. #2240	C₂	semidirect
4. #1706	C₂	semidirect
5. C₈×C₁₆	C₂	semidirect
45. C₁₆	SD₁₆	semidirect
57. C₈	D₁₆	semidirect

Maximal subgroups

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

subgroup	index
#2252	2
#2246	2
#2240	2
#1706	2
C₈×C₁₆	2