Subgroups — #12457

Order 256.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2252	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₁₆	C₂	1	1	yes	maximal, characteristic
3	#1674	C₂	1	2	yes	maximal
4	#1371	C₂	1	1	yes	maximal, characteristic
5	#1364	C₂	1	2	yes	maximal
6	#500	C₂²	1	2	yes	—
7	#462	—	2	2	no	—
8	C₈×Q₈	C₂²	1	1	yes	characteristic
9	#366	—	2	1	no	—
10	#366	—	2	1	no	—
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	#332	—	2	2	no	—
17	C₂×Q₁₆	—	4	2	no	—
18	C₄⋊Q₈	—	2	2	no	—
19	C₄×Q₈	D₄	1	1	yes	characteristic
20	C₂×C₁₆	—	2	1	no	—
21	C₂×C₁₆	D₄	1	1	yes	characteristic
22	C₂×C₁₆	D₄	1	1	yes	characteristic
23	#107	—	2	2	no	—
24	#107	—	2	2	no	—
25	C₄⋊C₈	—	2	1	no	—
26	C₄⋊C₈	—	2	2	no	—
27	C₄⋊C₈	D₄	1	1	yes	characteristic
28	#103	—	4	2	no	—
29	C₄×C₈	—	2	1	no	—
30	C₄×C₈	C₂³	1	1	yes	characteristic
31	C₂×Q₈	—	4	2	no	—
32	C₂×Q₈	D₈	1	1	yes	characteristic
33	Q₁₆	—	8	4	no	—
34	C₂×C₈	—	2	1	no	—
35	C₂×C₈	—	4	2	no	—
36	C₂×C₈	C₂×D₄	1	1	yes	characteristic
37	C₂×C₈	#41	1	1	yes	characteristic
38	C₄⋊C₄	—	2	1	no	—
39	C₄⋊C₄	—	4	2	no	—
40	C₄⋊C₄	—	4	2	no	—
41	C₄⋊C₄	D₈	1	1	yes	characteristic
42	C₄²	—	2	1	no	—
43	C₄²	C₂×D₄	1	1	yes	characteristic
44	C₁₆	—	2	1	no	—
45	C₁₆	—	4	1	no	—
46	C₁₆	Q₁₆	1	2	yes	complemented
47	Q₈	—	2	1	no	—
48	Q₈	—	8	2	no	—
49	Q₈	—	8	2	no	—
50	Q₈	D₁₆	1	2	yes	—
51	C₂×C₄	—	2	1	no	—
52	C₂×C₄	—	2	1	no	—
53	C₂×C₄	—	8	2	no	—
54	C₂×C₄	#121	1	1	yes	characteristic
55	C₂×C₄	C₂×D₈	1	1	yes	characteristic
56	C₂×C₄	#135	1	1	yes	characteristic
57	C₈	—	2	1	no	—
58	C₈	—	4	1	no	—
59	C₈	—	8	2	no	—
60	C₈	C₂×Q₁₆	1	1	yes	characteristic
61	C₈	#137	1	1	yes	characteristic
62	C₂²	#466	1	1	yes	characteristic
63	C₄	—	2	2	no	—
64	C₄	—	2	1	no	—
65	C₄	—	4	1	no	—
66	C₄	—	16	2	no	—
67	C₄	#462	1	1	yes	characteristic
68	C₄	#473	1	1	yes	characteristic
69	C₄	C₂×D₁₆	1	1	yes	characteristic
70	C₄	#508	1	1	yes	characteristic
71	C₂	#1674	1	1	yes	characteristic
72	C₂	#2215	1	1	yes	characteristic
73	C₂	#2222	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²	#466
commutator, Frattini, cosocle	C₄×C₈	C₂³
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
46. C₁₆	Q₁₆	semidirect

Maximal subgroups

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

subgroup	index
#2252	2
Q₈×C₁₆	2
#1674	2
#1371	2
#1364	2