Subgroups — #12088

Order 256.

99 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2176	C₂	1	1	yes	maximal, characteristic
2	#1593	C₂	1	2	yes	maximal
3	#1368	C₂	1	1	yes	maximal, characteristic
4	#1334	C₂	1	2	yes	maximal
5	#1313	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₃₂	C₄	1	1	yes	characteristic
7	#446	—	2	2	no	—
8	C₄×Q₁₆	C₂²	1	2	yes	—
9	#367	C₄	1	1	yes	characteristic
10	#358	C₄	1	2	yes	—
11	#346	C₂²	1	1	yes	characteristic
12	#346	C₂²	1	1	yes	characteristic
13	#334	C₂²	1	2	yes	—
14	Q₈⋊C₈	—	2	2	no	—
15	#322	C₂²	1	1	yes	characteristic
16	C₂×Q₁₆	C₂×C₄	1	2	yes	—
17	C₄×Q₈	—	2	2	no	—
18	Q₃₂	—	2	2	no	—
19	C₂×C₁₆	—	2	1	no	—
20	C₂×C₁₆	C₂×C₄	1	1	yes	characteristic
21	C₂×C₁₆	C₂×C₄	1	1	yes	characteristic
22	#107	C₂×C₄	1	2	yes	—
23	C₄⋊C₈	—	2	2	no	—
24	C₄⋊C₈	—	4	2	no	—
25	#103	—	2	2	no	—
26	#97	—	4	2	no	—
27	C₄×C₈	D₄	1	2	yes	—
28	C₄×C₈	C₂³	1	1	yes	characteristic
29	C₂×Q₈	—	2	2	no	—
30	Q₁₆	—	2	2	no	—
31	Q₁₆	—	2	2	no	—
32	C₂×C₈	—	2	2	no	—
33	C₂×C₈	—	4	2	no	—
34	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₈	#41	1	1	yes	characteristic
36	C₄⋊C₄	—	2	2	no	—
37	C₄⋊C₄	—	4	2	no	—
38	C₄²	—	4	2	no	—
39	C₄²	C₂×D₄	1	1	yes	characteristic
40	C₁₆	—	2	1	no	—
41	C₁₆	—	4	1	no	—
42	C₁₆	M₁₆	1	2	yes	—
43	Q₈	—	4	2	no	—
44	Q₈	—	4	2	no	—
45	C₂×C₄	—	4	2	no	—
46	C₂×C₄	—	4	2	no	—
47	C₂×C₄	C₄×D₄	1	1	yes	characteristic
48	C₂×C₄	AGL(1,ℤ/8)	1	2	yes	—
49	C₈	—	2	1	no	—
50	C₈	—	4	2	no	—
51	C₈	—	8	2	no	—
52	C₈	C₂×M₁₆	1	1	yes	characteristic
53	C₈	#131	1	1	yes	characteristic
54	C₂²	#442	1	1	yes	characteristic
55	C₄	—	2	2	no	—
56	C₄	—	8	2	no	—
57	C₄	—	8	2	no	—
58	C₄	#436	1	1	yes	characteristic
59	C₄	#444	1	1	yes	characteristic
60	C₂	#1594	1	1	yes	characteristic
61	C₂	#2180	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂²	#442
commutator	C₂×C₈	C₂²×C₄
Frattini, cosocle	C₄×C₈	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Maximal subgroups

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

subgroup	index
#2176	2
#1593	2
#1368	2
#1334	2
#1313	2