Subgroups — #12519

Order 256.

180 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2248	C₂	1	2	yes	maximal, complemented
2	#2246	C₂	1	2	yes	maximal, complemented
3	#1712	C₂	1	1	yes	maximal, characteristic, complemented
4	#1709	C₂	1	1	yes	maximal, characteristic, complemented
5	#1311	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×SD₃₂	—	2	4	no	—
7	C₂×D₁₆	—	2	4	no	—
8	#494	C₂²	1	1	yes	characteristic
9	#493	—	2	2	no	—
10	#493	C₂²	1	1	yes	characteristic
11	#493	C₂²	1	2	yes	—
12	#486	—	2	2	no	—
13	C₄×C₁₆	C₂²	1	2	yes	—
14	C₈²	C₂²	1	1	yes	characteristic
15	C₂×Q₁₆	—	2	2	no	—
16	C₂×D₈	—	2	2	no	—
17	C₂×D₈	—	2	2	no	—
18	C₂×D₈	—	2	2	no	—
19	C₂×D₈	—	4	4	no	—
20	C₄⋊Q₈	—	2	1	no	—
21	#127	—	2	1	no	—
22	#127	—	2	2	no	—
23	SD₃₂	—	4	8	no	—
24	D₁₆	—	4	8	no	—
25	C₂×C₁₆	D₄	1	4	yes	—
26	#102	—	4	4	no	—
27	C₄×C₈	D₄	1	2	yes	—
28	C₄×C₈	C₂³	1	1	yes	characteristic
29	C₂×Q₈	—	4	1	no	—
30	C₂×D₄	—	4	1	no	—
31	C₂×D₄	—	4	2	no	—
32	C₂×D₄	—	4	2	no	—
33	C₂×D₄	—	4	4	no	—
34	Q₁₆	—	4	4	no	—
35	D₈	—	4	4	no	—
36	D₈	—	4	4	no	—
37	D₈	—	4	4	no	—
38	D₈	—	8	8	no	—
39	C₂×C₈	D₈	1	4	yes	—
40	C₂×C₈	C₂×D₄	1	2	yes	—
41	C₄⋊C₄	—	4	2	no	—
42	C₄²	C₂×D₄	1	1	yes	characteristic
43	C₁₆	D₈	1	8	yes	complemented
44	C₂³	—	8	1	no	—
45	C₂³	—	8	2	no	—
46	Q₈	—	8	2	no	—
47	D₄	—	8	2	no	—
48	D₄	—	8	2	no	—
49	D₄	—	8	2	no	—
50	D₄	—	8	8	no	—
51	D₄	—	8	4	no	—
52	C₂×C₄	—	8	1	no	—
53	C₂×C₄	#127	1	1	yes	characteristic
54	C₂×C₄	C₂×D₈	1	2	yes	—
55	C₈	—	2	4	no	—
56	C₈	C₂×D₈	1	4	yes	—
57	C₂²	—	16	1	no	—
58	C₂²	—	16	2	no	—
59	C₂²	—	16	4	no	—
60	C₂²	—	16	2	no	—
61	C₂²	#493	1	1	yes	characteristic
62	C₄	—	16	1	no	—
63	C₄	#493	1	2	yes	—
64	C₄	#509	1	4	yes	—
65	C₂	—	32	2	no	—
66	C₂	—	32	1	no	—
67	C₂	#1712	1	1	yes	characteristic
68	C₂	#2250	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	#493
commutator, Frattini, cosocle	C₄×C₈	C₂³
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. #2248	C₂	semidirect
2. #2246	C₂	semidirect
3. #1712	C₂	semidirect
4. #1709	C₂	semidirect
5. #1311	C₂	semidirect
43. C₁₆	D₈	semidirect

Maximal subgroups

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

subgroup	index
#2248	2
#2246	2
#1712	2
#1709	2
#1311	2