Subgroups — #12889

Order 256.

155 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic
2	#1846	C₂	1	2	yes	maximal
3	#1380	C₂	1	4	yes	maximal
4	C₂²×C₁₆	C₂²	1	2	yes	—
5	C₂×#107	C₂²	1	4	yes	—
6	C₂×#106	—	2	2	no	—
7	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
8	#382	—	2	2	no	—
9	C₄×C₁₆	C₄	1	4	yes	complemented
10	C₂²×C₈	D₄	1	1	yes	characteristic
11	C₂²×C₈	C₂³	1	1	yes	characteristic
12	C₂×C₄⋊C₄	—	2	4	no	—
13	C₂×C₄²	D₄	1	1	yes	characteristic
14	C₂×C₁₆	—	2	4	no	—
15	C₂×C₁₆	C₂×C₄	1	4	yes	—
16	#107	—	2	8	no	—
17	#106	—	4	4	no	—
18	C₄×C₈	C₂×C₄	1	2	yes	—
19	C₄×C₈	Q₈	1	2	yes	—
20	#95	—	4	4	no	—
21	C₂²×C₄	—	4	4	no	—
22	C₂²×C₄	D₈	1	2	yes	—
23	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
24	C₂×C₈	C₄⋊C₄	1	2	yes	—
25	C₂×C₈	SD₁₆	1	4	yes	—
26	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
27	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
28	C₂×C₈	#41	1	4	yes	—
29	C₄⋊C₄	—	4	4	no	—
30	C₄⋊C₄	—	4	4	no	—
31	C₄²	C₄⋊C₄	1	2	yes	—
32	C₄²	Q₁₆	1	2	yes	—
33	C₁₆	—	2	4	no	—
34	C₂³	C₂×D₈	1	1	yes	characteristic
35	C₂×C₄	—	2	4	no	—
36	C₂×C₄	—	8	4	no	—
37	C₂×C₄	—	8	8	no	—
38	C₂×C₄	#107	1	4	yes	—
39	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₄	#124	1	2	yes	—
41	C₂×C₄	C₂×SD₁₆	1	2	yes	—
42	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
43	C₈	#106	1	4	yes	—
44	C₈	#117	1	2	yes	—
45	C₈	#125	1	2	yes	—
46	C₂²	C₂×#107	1	1	yes	characteristic
47	C₂²	#486	1	2	yes	—
48	C₂²	#508	1	4	yes	—
49	C₄	—	2	4	no	—
50	C₄	—	16	4	no	—
51	C₄	#382	1	1	yes	characteristic
52	C₄	C₂×#106	1	1	yes	characteristic
53	C₄	#487	1	2	yes	—
54	C₂	#1834	1	1	yes	characteristic
55	C₂	#2158	1	2	yes	—
56	C₂	#2242	1	4	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×D₈
commutator	C₂×C₈	C₂²×C₄
Frattini, cosocle	C₂²×C₈	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
9. C₄×C₁₆	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₁₆	2
#1846	2
#1380	2