Subgroups — #13729

Order 256.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×SD₆₄	C₂	1	2	yes	maximal, complemented
2	C₂×D₃₂	C₂	1	2	yes	maximal, complemented
3	#2249	C₂	1	1	yes	maximal, characteristic, complemented
4	#2246	C₂	1	1	yes	maximal, characteristic, complemented
5	#1397	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Q₃₂	C₂²	1	1	yes	characteristic
7	C₂×SD₃₂	—	2	1	no	—
8	C₂×D₁₆	—	2	1	no	—
9	C₂×D₁₆	C₂²	1	2	yes	—
10	C₂×D₁₆	C₂²	1	1	yes	characteristic
11	#495	—	2	1	no	—
12	#493	—	2	1	no	—
13	SD₆₄	—	2	4	no	—
14	D₃₂	—	2	4	no	—
15	C₂×C₃₂	C₂²	1	2	yes	—
16	C₄×C₁₆	C₂²	1	1	yes	characteristic
17	C₂×Q₁₆	—	2	1	no	—
18	C₂×SD₁₆	—	4	1	no	—
19	C₂×D₈	—	2	2	no	—
20	C₂×D₈	—	2	1	no	—
21	C₂×D₈	—	4	1	no	—
22	#127	—	4	1	no	—
23	#124	—	4	1	no	—
24	Q₃₂	—	2	2	no	—
25	SD₃₂	—	4	2	no	—
26	D₁₆	—	2	2	no	—
27	D₁₆	—	2	2	no	—
28	D₁₆	—	2	2	no	—
29	D₁₆	—	4	2	no	—
30	C₂×C₁₆	D₄	1	1	yes	characteristic
31	C₂×C₁₆	C₂³	1	1	yes	characteristic
32	C₄×C₈	D₄	1	1	yes	characteristic
33	C₃₂	D₄	1	4	yes	complemented
34	C₂×Q₈	—	4	1	no	—
35	C₂×D₄	—	4	1	no	—
36	C₂×D₄	—	4	2	no	—
37	C₂×D₄	—	8	2	no	—
38	Q₁₆	—	4	1	no	—
39	Q₁₆	—	4	1	no	—
40	SD₁₆	—	8	2	no	—
41	D₈	—	4	1	no	—
42	D₈	—	4	2	no	—
43	D₈	—	4	2	no	—
44	D₈	—	4	1	no	—
45	D₈	—	8	2	no	—
46	C₂×C₈	D₈	1	1	yes	characteristic
47	C₂×C₈	C₂×D₄	1	1	yes	characteristic
48	C₂²⋊C₄	—	8	2	no	—
49	C₄²	D₈	1	1	yes	characteristic
50	C₁₆	D₈	1	2	yes	—
51	C₁₆	C₂×D₄	1	2	yes	—
52	C₂³	—	8	1	no	—
53	C₂³	—	8	2	no	—
54	Q₈	—	8	1	no	—
55	Q₈	—	8	1	no	—
56	D₄	—	8	2	no	—
57	D₄	—	8	1	no	—
58	D₄	—	8	2	no	—
59	D₄	—	8	1	no	—
60	D₄	—	16	4	no	—
61	C₂×C₄	—	8	1	no	—
62	C₂×C₄	D₁₆	1	2	yes	—
63	C₂×C₄	C₂×D₈	1	1	yes	characteristic
64	C₈	D₁₆	1	2	yes	—
65	C₈	#127	1	1	yes	characteristic
66	C₈	C₂×D₈	1	1	yes	characteristic
67	C₂²	—	16	2	no	—
68	C₂²	—	16	1	no	—
69	C₂²	—	16	4	no	—
70	C₂²	—	16	2	no	—
71	C₂²	C₂×D₁₆	1	1	yes	characteristic
72	C₄	—	2	2	no	—
73	C₄	—	16	1	no	—
74	C₄	#493	1	1	yes	characteristic
75	C₄	C₂×D₁₆	1	1	yes	characteristic
76	C₂	—	32	2	no	—
77	C₂	—	32	1	no	—
78	C₂	#2246	1	1	yes	characteristic
79	C₂	#2263	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂³
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ 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. C₂×SD₆₄	C₂	semidirect
2. C₂×D₃₂	C₂	semidirect
3. #2249	C₂	semidirect
4. #2246	C₂	semidirect
5. #1397	C₂	semidirect
33. C₃₂	D₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×SD₆₄	2
C₂×D₃₂	2
#2249	2
#2246	2
#1397	2