Subgroups — #13692

Order 256.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2263	C₂	1	2	yes	maximal, complemented
2	#2246	C₂	1	1	yes	maximal, characteristic, complemented
3	#2178	C₂	1	1	yes	maximal, characteristic, complemented
4	#1422	C₂	1	1	yes	maximal, characteristic, complemented
5	#1419	C₂	1	2	yes	maximal, complemented
6	#508	C₂²	1	1	yes	characteristic
7	C₂×D₁₆	—	2	1	no	—
8	C₂×D₁₆	C₂²	1	2	yes	—
9	#493	—	2	1	no	—
10	#443	—	2	1	no	—
11	SD₆₄	—	2	2	no	—
12	D₃₂	—	2	2	no	—
13	M₆₄	C₂²	1	2	yes	—
14	#368	C₂²	1	1	yes	characteristic
15	#359	—	2	1	no	—
16	C₄×C₁₆	C₂²	1	1	yes	characteristic
17	#135	—	2	1	no	—
18	C₂×D₈	—	2	2	no	—
19	C₂×D₈	—	4	1	no	—
20	#131	—	4	1	no	—
21	#127	—	4	1	no	—
22	Q₃₂	D₄	1	1	yes	characteristic, complemented
23	SD₃₂	—	2	1	no	—
24	D₁₆	—	2	2	no	—
25	D₁₆	—	2	2	no	—
26	D₁₆	—	4	2	no	—
27	D₁₆	D₄	1	1	yes	characteristic, complemented
28	C₂×C₁₆	D₄	1	1	yes	characteristic
29	C₂×C₁₆	C₂³	1	1	yes	characteristic
30	#108	—	2	1	no	—
31	#104	—	4	1	no	—
32	C₄×C₈	D₄	1	1	yes	characteristic
33	C₃₂	—	2	2	no	—
34	#41	—	4	1	no	—
35	C₂×D₄	—	4	2	no	—
36	C₂×D₄	—	8	2	no	—
37	Q₁₆	—	2	1	no	—
38	SD₁₆	—	4	1	no	—
39	D₈	—	2	1	no	—
40	D₈	—	4	2	no	—
41	D₈	—	4	2	no	—
42	D₈	—	8	1	no	—
43	D₈	—	8	1	no	—
44	M₁₆	—	4	1	no	—
45	M₁₆	—	8	1	no	—
46	C₂×C₈	—	8	1	no	—
47	C₂×C₈	D₈	1	1	yes	characteristic
48	C₂×C₈	C₂×D₄	1	1	yes	characteristic
49	C₄²	D₈	1	1	yes	characteristic
50	C₁₆	—	2	1	no	—
51	C₁₆	C₂×D₄	1	1	yes	characteristic
52	C₁₆	#41	1	1	yes	characteristic
53	C₂³	—	8	2	no	—
54	Q₈	—	4	1	no	—
55	D₄	—	4	1	no	—
56	D₄	—	8	1	no	—
57	D₄	—	8	2	no	—
58	D₄	—	8	2	no	—
59	D₄	—	16	4	no	—
60	C₂×C₄	—	2	1	no	—
61	C₂×C₄	—	8	1	no	—
62	C₂×C₄	C₂×D₈	1	1	yes	characteristic
63	C₈	—	8	1	no	—
64	C₈	D₁₆	1	1	yes	characteristic
65	C₈	D₁₆	1	1	yes	characteristic
66	C₈	#121	1	1	yes	characteristic
67	C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
68	C₂²	—	8	1	no	—
69	C₂²	—	16	4	no	—
70	C₂²	—	16	2	no	—
71	C₂²	#509	1	1	yes	characteristic
72	C₄	—	2	2	no	—
73	C₄	—	8	1	no	—
74	C₄	#459	1	1	yes	characteristic
75	C₄	C₂×D₁₆	1	1	yes	characteristic
76	C₂	—	2	1	no	—
77	C₂	—	16	1	no	—
78	C₂	—	32	2	no	—
79	C₂	#2206	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#2206
commutator, Frattini, cosocle	C₂×C₁₆	C₂³
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #2263	C₂	semidirect
2. #2246	C₂	semidirect
3. #2178	C₂	semidirect
4. #1422	C₂	semidirect
5. #1419	C₂	semidirect
22. Q₃₂	D₄	semidirect
27. D₁₆	D₄	semidirect

Maximal subgroups

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

subgroup	index
#2263	2
#2246	2
#2178	2
#1422	2
#1419	2