Subgroups — #12439

Order 256.

124 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2246	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#1668	C₂	1	2	yes	maximal, complemented
4	#1371	C₂	1	1	yes	maximal, characteristic, complemented
5	#1361	C₂	1	2	yes	maximal, complemented
6	C₂×D₁₆	—	2	1	no	—
7	C₂×D₁₆	—	2	1	no	—
8	#493	C₂²	1	2	yes	—
9	#460	—	2	2	no	—
10	C₈×Q₈	C₂²	1	1	yes	characteristic
11	C₄⋊C₁₆	—	2	1	no	—
12	C₄⋊C₁₆	C₂²	1	1	yes	characteristic
13	C₄×C₁₆	—	2	1	no	—
14	C₄×C₁₆	C₂²	1	1	yes	characteristic
15	#335	C₂²	1	2	yes	—
16	#331	—	2	2	no	—
17	C₂×SD₁₆	—	4	2	no	—
18	C₂×D₈	—	2	2	no	—
19	C₂×D₈	—	2	2	no	—
20	#127	—	2	2	no	—
21	C₄×Q₈	D₄	1	1	yes	characteristic
22	D₁₆	—	4	2	no	—
23	D₁₆	—	4	2	no	—
24	C₂×C₁₆	—	2	1	no	—
25	C₂×C₁₆	D₄	1	1	yes	characteristic
26	C₂×C₁₆	D₄	1	1	yes	characteristic
27	C₄⋊C₈	—	2	1	no	—
28	C₄⋊C₈	—	2	2	no	—
29	C₄⋊C₈	D₄	1	1	yes	characteristic
30	#102	—	4	2	no	—
31	C₄×C₈	—	2	1	no	—
32	C₄×C₈	C₂³	1	1	yes	characteristic
33	C₂×Q₈	D₈	1	1	yes	characteristic
34	C₂×D₄	—	4	2	no	—
35	C₂×D₄	—	4	2	no	—
36	C₂×D₄	—	4	2	no	—
37	SD₁₆	—	8	4	no	—
38	D₈	—	4	2	no	—
39	D₈	—	4	2	no	—
40	D₈	—	8	2	no	—
41	C₂×C₈	—	2	1	no	—
42	C₂×C₈	—	4	2	no	—
43	C₂×C₈	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈	#41	1	1	yes	characteristic
45	C₄⋊C₄	—	2	1	no	—
46	C₄⋊C₄	D₈	1	1	yes	characteristic
47	C₄²	—	2	1	no	—
48	C₄²	C₂×D₄	1	1	yes	characteristic
49	C₁₆	—	2	1	no	—
50	C₁₆	—	4	1	no	—
51	C₁₆	SD₁₆	1	2	yes	complemented
52	C₂³	—	8	2	no	—
53	Q₈	—	2	1	no	—
54	Q₈	D₁₆	1	2	yes	complemented
55	D₄	—	8	2	no	—
56	D₄	—	8	2	no	—
57	D₄	—	8	2	no	—
58	D₄	—	8	2	no	—
59	D₄	—	16	2	no	—
60	C₂×C₄	—	2	1	no	—
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	#121	1	1	yes	characteristic
63	C₂×C₄	C₂×D₈	1	1	yes	characteristic
64	C₂×C₄	#135	1	1	yes	characteristic
65	C₈	—	2	1	no	—
66	C₈	—	4	1	no	—
67	C₈	—	8	2	no	—
68	C₈	C₂×SD₁₆	1	1	yes	characteristic
69	C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
70	C₂²	—	16	2	no	—
71	C₂²	—	16	2	no	—
72	C₂²	—	16	2	no	—
73	C₂²	#466	1	1	yes	characteristic
74	C₄	—	2	2	no	—
75	C₄	—	2	1	no	—
76	C₄	—	4	1	no	—
77	C₄	#460	1	1	yes	characteristic
78	C₄	#472	1	1	yes	characteristic
79	C₄	C₂×D₁₆	1	1	yes	characteristic
80	C₄	#508	1	1	yes	characteristic
81	C₂	—	32	2	no	—
82	C₂	#1668	1	1	yes	characteristic
83	C₂	#2215	1	1	yes	characteristic
84	C₂	#2221	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#466
commutator, Frattini, cosocle	C₄×C₈	C₂³
prime radical	#2246	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈ ◁ 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
2. Q₈×C₁₆	C₂	semidirect
3. #1668	C₂	semidirect
4. #1371	C₂	semidirect
5. #1361	C₂	semidirect
51. C₁₆	SD₁₆	semidirect
54. Q₈	D₁₆	semidirect

Maximal subgroups

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

subgroup	index
#2246	2
Q₈×C₁₆	2
#1668	2
#1371	2
#1361	2