Subgroups — #12436

Order 256.

117 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2246	C₂	1	1	yes	maximal, characteristic, complemented
2	#2213	C₂	1	2	yes	maximal, complemented
3	#2170	C₂	1	1	yes	maximal, characteristic, complemented
4	#1373	C₂	1	1	yes	maximal, characteristic, complemented
5	#1357	C₂	1	2	yes	maximal, complemented
6	#509	—	2	2	no	—
7	C₂×D₁₆	—	2	1	no	—
8	C₂×D₁₆	—	2	1	no	—
9	#504	—	2	1	no	—
10	#493	C₂²	1	2	yes	—
11	#443	C₂²	1	1	yes	characteristic
12	#364	C₂²	1	1	yes	characteristic
13	#364	C₂²	1	2	yes	—
14	#361	—	2	2	no	—
15	#350	—	2	1	no	—
16	C₄×C₁₆	C₂²	1	1	yes	characteristic
17	#135	D₄	1	1	yes	characteristic
18	C₂×D₈	—	2	2	no	—
19	C₂×D₈	—	2	2	no	—
20	#131	—	2	1	no	—
21	#127	—	2	2	no	—
22	SD₃₂	—	4	2	no	—
23	D₁₆	—	4	2	no	—
24	D₁₆	—	4	2	no	—
25	D₁₆	—	4	1	no	—
26	D₁₆	—	4	1	no	—
27	M₃₂	—	2	1	no	—
28	M₃₂	—	2	2	no	—
29	M₃₂	—	4	1	no	—
30	C₂×C₁₆	—	4	1	no	—
31	C₂×C₁₆	D₄	1	1	yes	characteristic
32	C₂×C₁₆	D₄	1	1	yes	characteristic
33	#108	D₄	1	1	yes	characteristic
34	#104	—	2	1	no	—
35	C₄×C₈	C₂³	1	1	yes	characteristic
36	#41	—	2	1	no	—
37	C₂×D₄	—	4	2	no	—
38	C₂×D₄	—	8	2	no	—
39	Q₁₆	D₈	1	1	yes	characteristic
40	SD₁₆	—	2	1	no	—
41	D₈	—	4	2	no	—
42	D₈	—	4	2	no	—
43	D₈	—	4	2	no	—
44	D₈	—	4	2	no	—
45	D₈	D₈	1	1	yes	characteristic
46	M₁₆	—	2	1	no	—
47	M₁₆	—	4	1	no	—
48	C₂×C₈	—	4	1	no	—
49	C₂×C₈	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈	C₂×D₄	1	1	yes	characteristic
51	C₄²	#41	1	1	yes	characteristic
52	C₁₆	—	2	1	no	—
53	C₁₆	—	4	1	no	—
54	C₁₆	—	4	2	no	—
55	C₁₆	D₈	1	1	yes	characteristic
56	C₁₆	D₈	1	1	yes	characteristic
57	C₂³	—	8	2	no	—
58	Q₈	—	2	1	no	—
59	D₄	—	2	1	no	—
60	D₄	—	4	1	no	—
61	D₄	—	8	2	no	—
62	D₄	—	8	2	no	—
63	D₄	—	16	4	no	—
64	C₂×C₄	—	2	1	no	—
65	C₂×C₄	—	4	1	no	—
66	C₂×C₄	#121	1	1	yes	characteristic
67	C₈	—	4	1	no	—
68	C₈	C₂×D₈	1	1	yes	characteristic
69	C₈	C₂×D₈	1	1	yes	characteristic
70	C₈	#135	1	1	yes	characteristic
71	C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
72	C₂²	—	4	1	no	—
73	C₂²	—	16	4	no	—
74	C₂²	—	16	2	no	—
75	C₂²	#472	1	1	yes	characteristic
76	C₄	—	2	2	no	—
77	C₄	—	4	1	no	—
78	C₄	#459	1	1	yes	characteristic
79	C₄	#466	1	1	yes	characteristic
80	C₂	—	2	1	no	—
81	C₂	—	8	1	no	—
82	C₂	—	32	2	no	—
83	C₂	#1667	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#1667
commutator, Frattini, cosocle	C₄×C₈	C₂³
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ 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. #2246	C₂	semidirect
2. #2213	C₂	semidirect
3. #2170	C₂	semidirect
4. #1373	C₂	semidirect
5. #1357	C₂	semidirect

Maximal subgroups

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

subgroup	index
#2246	2
#2213	2
#2170	2
#1373	2
#1357	2