Subgroups — #12451

Order 256.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2252	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×C₁₆	C₂	1	1	yes	maximal, characteristic
3	#1673	C₂	1	2	yes	maximal
4	#1371	C₂	1	1	yes	maximal, characteristic, complemented
5	#1364	C₂	1	2	yes	maximal, complemented
6	C₂²×C₁₆	—	2	1	no	—
7	#500	C₂²	1	2	yes	—
8	#461	—	2	2	no	—
9	C₈×D₄	C₂²	1	1	yes	characteristic
10	#366	—	2	1	no	—
11	#366	—	2	1	no	—
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	#332	—	2	2	no	—
17	C₂×SD₁₆	—	4	2	no	—
18	C₂²×C₈	—	2	1	no	—
19	C₄⋊Q₈	—	2	2	no	—
20	C₄×D₄	D₄	1	1	yes	characteristic
21	C₂×C₁₆	—	2	1	no	—
22	C₂×C₁₆	—	4	2	no	—
23	C₂×C₁₆	D₄	1	1	yes	characteristic
24	C₂×C₁₆	D₄	1	1	yes	characteristic
25	#107	—	2	2	no	—
26	#107	—	2	2	no	—
27	C₄⋊C₈	—	2	2	no	—
28	C₄⋊C₈	D₄	1	1	yes	characteristic
29	#103	—	4	2	no	—
30	C₂²⋊C₈	—	2	1	no	—
31	C₄×C₈	C₂³	1	1	yes	characteristic
32	C₂×Q₈	—	4	2	no	—
33	C₂×D₄	D₈	1	1	yes	characteristic
34	C₂²×C₄	—	2	1	no	—
35	SD₁₆	—	8	4	no	—
36	C₂×C₈	—	2	1	no	—
37	C₂×C₈	—	4	1	no	—
38	C₂×C₈	—	4	1	no	—
39	C₂×C₈	—	4	2	no	—
40	C₂×C₈	C₂×D₄	1	1	yes	characteristic
41	C₂×C₈	#41	1	1	yes	characteristic
42	C₄⋊C₄	—	4	2	no	—
43	C₄⋊C₄	—	4	2	no	—
44	C₄⋊C₄	D₈	1	1	yes	characteristic
45	C₂²⋊C₄	—	2	1	no	—
46	C₄²	C₂×D₄	1	1	yes	characteristic
47	C₁₆	—	2	1	no	—
48	C₁₆	—	4	1	no	—
49	C₁₆	SD₁₆	1	2	yes	complemented
50	C₂³	—	2	1	no	—
51	Q₈	—	8	2	no	—
52	Q₈	—	8	2	no	—
53	D₄	—	2	1	no	—
54	D₄	D₁₆	1	2	yes	—
55	C₂×C₄	—	2	1	no	—
56	C₂×C₄	—	4	1	no	—
57	C₂×C₄	—	4	1	no	—
58	C₂×C₄	—	8	2	no	—
59	C₂×C₄	#121	1	1	yes	characteristic
60	C₂×C₄	C₂×D₈	1	1	yes	characteristic
61	C₂×C₄	#135	1	1	yes	characteristic
62	C₈	—	2	1	no	—
63	C₈	—	4	1	no	—
64	C₈	—	8	2	no	—
65	C₈	C₂×SD₁₆	1	1	yes	characteristic
66	C₈	#137	1	1	yes	characteristic
67	C₂²	—	2	2	no	—
68	C₂²	—	4	1	no	—
69	C₂²	—	4	1	no	—
70	C₂²	#466	1	1	yes	characteristic
71	C₄	—	2	1	no	—
72	C₄	—	4	1	no	—
73	C₄	—	16	2	no	—
74	C₄	#461	1	1	yes	characteristic
75	C₄	#473	1	1	yes	characteristic
76	C₄	C₂×D₁₆	1	1	yes	characteristic
77	C₄	#508	1	1	yes	characteristic
78	C₂	—	4	2	no	—
79	C₂	#1673	1	1	yes	characteristic
80	C₂	#2215	1	1	yes	characteristic
81	C₂	#2222	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#466
commutator, Frattini, cosocle	C₄×C₈	C₂³
prime radical	C₂×D₄	D₈
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. #2252	C₂	semidirect
4. #1371	C₂	semidirect
5. #1364	C₂	semidirect
49. C₁₆	SD₁₆	semidirect

Maximal subgroups

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

subgroup	index
#2252	2
D₄×C₁₆	2
#1673	2
#1371	2
#1364	2