Subgroups — #12883

Order 256.

171 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2150	C₂	1	2	yes	maximal
2	C₄²×C₈	C₂	1	1	yes	maximal, characteristic
3	#1376	C₂	1	4	yes	maximal
4	C₂×M₃₂	C₂²	1	4	yes	—
5	C₂×C₄×C₈	C₄	1	2	yes	—
6	C₂×C₄×C₈	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈	C₂²	1	2	yes	—
8	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
9	C₄³	C₄	1	1	yes	characteristic
10	C₄⋊C₁₆	—	2	4	no	—
11	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
12	C₂²×C₈	C₂×C₄	1	2	yes	—
13	C₂²×C₈	C₂³	1	1	yes	characteristic
14	C₂×C₄²	—	2	2	no	—
15	C₂×C₄²	C₂×C₄	1	2	yes	—
16	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
17	M₃₂	—	2	8	no	—
18	C₂×C₁₆	—	2	4	no	—
19	C₄×C₈	—	2	4	no	—
20	C₄×C₈	—	2	4	no	—
21	C₄×C₈	C₈	1	2	yes	—
22	C₄×C₈	C₈	1	2	yes	—
23	C₄×C₈	D₄	1	2	yes	—
24	C₄×C₈	Q₈	1	2	yes	—
25	C₂²×C₄	—	2	2	no	—
26	C₂²×C₄	M₁₆	1	2	yes	—
27	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
28	C₂×C₈	—	2	2	no	—
29	C₂×C₈	—	2	2	no	—
30	C₂×C₈	—	2	2	no	—
31	C₂×C₈	C₄⋊C₄	1	2	yes	—
32	C₂×C₈	C₂×C₈	1	4	yes	—
33	C₂×C₈	C₂×D₄	1	1	yes	characteristic
34	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
35	C₂×C₈	#41	1	4	yes	—
36	C₄²	—	2	8	no	—
37	C₄²	—	2	2	no	—
38	C₄²	C₄⋊C₄	1	2	yes	—
39	C₄²	C₂×C₈	1	1	yes	characteristic
40	C₄²	C₂×C₈	1	1	yes	characteristic
41	C₄²	M₁₆	1	4	yes	—
42	C₁₆	—	4	4	no	—
43	C₂³	C₂×M₁₆	1	1	yes	characteristic
44	C₂×C₄	—	2	8	no	—
45	C₂×C₄	—	2	4	no	—
46	C₂×C₄	—	2	4	no	—
47	C₂×C₄	C₄⋊C₈	1	2	yes	—
48	C₂×C₄	C₄⋊C₈	1	2	yes	—
49	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₄	#117	1	2	yes	—
51	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
52	C₂×C₄	C₂×M₁₆	1	2	yes	—
53	C₈	—	2	2	no	—
54	C₈	—	2	4	no	—
55	C₈	#124	1	2	yes	—
56	C₈	#125	1	2	yes	—
57	C₂²	—	2	2	no	—
58	C₂²	C₂×C₄⋊C₈	1	1	yes	characteristic
59	C₂²	#432	1	2	yes	—
60	C₄	—	2	8	no	—
61	C₄	—	2	2	no	—
62	C₄	#364	1	4	yes	—
63	C₄	#382	1	1	yes	characteristic
64	C₄	#423	1	1	yes	characteristic
65	C₄	#431	1	2	yes	—
66	C₂	—	2	2	no	—
67	C₂	#1831	1	1	yes	characteristic
68	C₂	C₂×#364	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂×D₄
commutator	C₂×C₄	C₂²×C₈
Frattini, cosocle	C₂²×C₈	C₂³
socle	C₂²	C₂×C₄⋊C₈
prime radical	C₂³	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Maximal subgroups

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

subgroup	index
#2150	2
C₄²×C₈	2
#1376	2