Subgroups — #12430

Order 256.

134 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	D₄×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#1667	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	C₂²×C₁₆	—	2	1	no	—
9	#493	C₂²	1	2	yes	—
10	#459	—	2	2	no	—
11	C₈×D₄	C₂²	1	1	yes	characteristic
12	C₄⋊C₁₆	C₂²	1	1	yes	characteristic
13	C₂²⋊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×D₈	—	2	2	no	—
18	C₂×D₈	—	2	2	no	—
19	C₂×D₈	—	4	2	no	—
20	C₂²×C₈	—	2	1	no	—
21	#127	—	2	2	no	—
22	C₄×D₄	D₄	1	1	yes	characteristic
23	D₁₆	—	4	2	no	—
24	D₁₆	—	4	2	no	—
25	C₂×C₁₆	—	2	1	no	—
26	C₂×C₁₆	—	4	2	no	—
27	C₂×C₁₆	D₄	1	1	yes	characteristic
28	C₂×C₁₆	D₄	1	1	yes	characteristic
29	C₄⋊C₈	—	2	2	no	—
30	C₄⋊C₈	D₄	1	1	yes	characteristic
31	#102	—	4	2	no	—
32	C₂²⋊C₈	—	2	1	no	—
33	C₄×C₈	C₂³	1	1	yes	characteristic
34	C₂×D₄	—	4	2	no	—
35	C₂×D₄	—	4	2	no	—
36	C₂×D₄	—	4	2	no	—
37	C₂×D₄	D₈	1	1	yes	characteristic
38	C₂²×C₄	—	2	1	no	—
39	D₈	—	4	2	no	—
40	D₈	—	4	2	no	—
41	D₈	—	8	2	no	—
42	D₈	—	8	4	no	—
43	C₂×C₈	—	2	1	no	—
44	C₂×C₈	—	4	1	no	—
45	C₂×C₈	—	4	1	no	—
46	C₂×C₈	—	4	2	no	—
47	C₂×C₈	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈	#41	1	1	yes	characteristic
49	C₄⋊C₄	D₈	1	1	yes	characteristic
50	C₂²⋊C₄	—	2	1	no	—
51	C₄²	C₂×D₄	1	1	yes	characteristic
52	C₁₆	—	2	1	no	—
53	C₁₆	—	4	1	no	—
54	C₁₆	D₈	1	2	yes	complemented
55	C₂³	—	2	1	no	—
56	C₂³	—	8	2	no	—
57	D₄	—	2	1	no	—
58	D₄	—	8	2	no	—
59	D₄	—	8	2	no	—
60	D₄	—	8	2	no	—
61	D₄	—	8	2	no	—
62	D₄	—	16	2	no	—
63	D₄	D₁₆	1	2	yes	complemented
64	C₂×C₄	—	2	1	no	—
65	C₂×C₄	—	4	1	no	—
66	C₂×C₄	—	4	1	no	—
67	C₂×C₄	#121	1	1	yes	characteristic
68	C₂×C₄	C₂×D₈	1	1	yes	characteristic
69	C₂×C₄	#135	1	1	yes	characteristic
70	C₈	—	2	1	no	—
71	C₈	—	4	1	no	—
72	C₈	—	8	2	no	—
73	C₈	C₂×D₈	1	1	yes	characteristic
74	C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
75	C₂²	—	2	2	no	—
76	C₂²	—	4	1	no	—
77	C₂²	—	4	1	no	—
78	C₂²	—	16	2	no	—
79	C₂²	—	16	2	no	—
80	C₂²	—	16	2	no	—
81	C₂²	#466	1	1	yes	characteristic
82	C₄	—	2	1	no	—
83	C₄	—	4	1	no	—
84	C₄	#459	1	1	yes	characteristic
85	C₄	#472	1	1	yes	characteristic
86	C₄	C₂×D₁₆	1	1	yes	characteristic
87	C₄	#508	1	1	yes	characteristic
88	C₂	—	4	2	no	—
89	C₂	—	32	2	no	—
90	C₂	#1667	1	1	yes	characteristic
91	C₂	#2215	1	1	yes	characteristic
92	C₂	#2221	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#466
commutator, Frattini, cosocle	C₄×C₈	C₂³
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×D₄ ◁ 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. D₄×C₁₆	C₂	semidirect
3. #1667	C₂	semidirect
4. #1371	C₂	semidirect
5. #1361	C₂	semidirect
54. C₁₆	D₈	semidirect
63. D₄	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
D₄×C₁₆	2
#1667	2
#1371	2
#1361	2