Subgroups — #8796

Order 256.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#1568	C₂	1	2	yes	maximal
2	#1450	C₂	1	1	yes	maximal, characteristic
3	#1287	C₂	1	4	yes	maximal
4	#432	C₂²	1	4	yes	—
5	#431	C₂²	1	2	yes	—
6	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
7	#335	—	2	2	no	—
8	#334	—	2	2	no	—
9	#322	C₄	1	2	yes	—
10	C₈²	C₄	1	2	yes	—
11	C₂²×C₈	Q₈	1	1	yes	characteristic
12	C₂²×C₈	Q₈	1	1	yes	characteristic
13	C₂×C₄²	C₂³	1	1	yes	characteristic
14	C₄⋊C₈	—	2	4	no	—
15	C₄⋊C₈	—	2	2	no	—
16	C₂²⋊C₈	—	2	2	no	—
17	C₂²⋊C₈	—	2	4	no	—
18	#97	—	2	4	no	—
19	C₄×C₈	C₂×C₄	1	4	yes	—
20	C₄×C₈	C₂×C₄	1	1	yes	characteristic
21	C₄×C₈	C₂×C₄	1	1	yes	characteristic
22	C₄×C₈	D₄	1	1	yes	characteristic
23	C₄×C₈	D₄	1	1	yes	characteristic
24	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
25	C₂²×C₄	#41	1	2	yes	—
26	C₂×C₈	—	2	2	no	—
27	C₂×C₈	—	2	2	no	—
28	C₂×C₈	—	2	4	no	—
29	C₂×C₈	—	4	4	no	—
30	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
34	C₄²	C₂²×C₄	1	1	yes	characteristic
35	C₄²	C₂×D₄	1	1	yes	characteristic
36	C₄²	#41	1	2	yes	—
37	C₂³	#125	1	1	yes	characteristic
38	C₂×C₄	—	2	1	no	—
39	C₂×C₄	—	2	1	no	—
40	C₂×C₄	—	4	2	no	—
41	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₄	#117	1	2	yes	—
43	C₂×C₄	#124	1	2	yes	—
44	C₂×C₄	#125	1	1	yes	characteristic
45	C₈	—	2	1	no	—
46	C₈	—	2	1	no	—
47	C₈	—	4	4	no	—
48	C₈	—	8	4	no	—
49	C₈	#108	1	2	yes	—
50	C₈	#108	1	2	yes	—
51	C₂²	—	2	2	no	—
52	C₂²	—	2	1	no	—
53	C₂²	#382	1	1	yes	characteristic
54	C₄	—	2	2	no	—
55	C₄	—	2	2	no	—
56	C₄	—	2	1	no	—
57	C₄	C₂×#108	1	1	yes	characteristic
58	C₄	C₂×#108	1	1	yes	characteristic
59	C₂	—	4	1	no	—
60	C₂	#1833	1	2	yes	—
61	C₂	#1837	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#382
commutator	C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
prime radical	C₂³	#125
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

Maximal subgroups

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

subgroup	index
#1568	2
#1450	2
#1287	2