Subgroups — #7393

Order 256.

55 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2183	C₂	1	1	yes	maximal, characteristic
2	C₄⋊C₃₂	C₂	1	1	yes	maximal, characteristic
3	#1397	C₂	1	1	yes	maximal, characteristic
4	C₈×Q₈	C₄	1	1	yes	characteristic
5	C₂×C₃₂	—	2	1	no	—
6	C₂×C₃₂	—	2	1	no	—
7	C₄⋊C₁₆	—	2	1	no	—
8	C₄⋊C₁₆	C₄	1	1	yes	characteristic
9	#346	—	2	1	no	—
10	C₄×C₁₆	C₂²	1	1	yes	characteristic
11	C₄×Q₈	C₈	1	1	yes	characteristic
12	C₂×C₁₆	—	2	1	no	—
13	C₂×C₁₆	D₄	1	1	yes	characteristic
14	C₂×C₁₆	D₄	1	1	yes	characteristic
15	C₄⋊C₈	—	2	1	no	—
16	C₄⋊C₈	C₈	1	1	yes	characteristic
17	C₄×C₈	—	2	1	no	—
18	C₄×C₈	C₂×C₄	1	1	yes	characteristic
19	C₃₂	—	2	2	no	—
20	C₃₂	—	4	1	no	—
21	C₂×Q₈	C₁₆	1	1	yes	characteristic
22	C₂×C₈	—	2	1	no	—
23	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
24	C₂×C₈	M₁₆	1	1	yes	characteristic
25	C₄⋊C₄	—	2	1	no	—
26	C₄⋊C₄	C₁₆	1	1	yes	characteristic
27	C₄²	—	2	1	no	—
28	C₄²	C₂×C₈	1	1	yes	characteristic
29	C₁₆	—	2	1	no	—
30	C₁₆	—	4	1	no	—
31	C₁₆	SD₁₆	1	1	yes	characteristic
32	C₁₆	Q₁₆	1	1	yes	characteristic
33	Q₈	—	4	1	no	—
34	C₂×C₄	—	2	1	no	—
35	C₂×C₄	—	2	1	no	—
36	C₂×C₄	C₂²⋊C₈	1	1	yes	characteristic
37	C₂×C₄	C₂×C₁₆	1	1	yes	characteristic
38	C₂×C₄	M₃₂	1	1	yes	characteristic
39	C₈	—	2	1	no	—
40	C₈	—	4	1	no	—
41	C₈	#103	1	1	yes	characteristic
42	C₈	#104	1	1	yes	characteristic
43	C₂²	C₂²⋊C₁₆	1	1	yes	characteristic
44	C₄	—	2	1	no	—
45	C₄	—	2	1	no	—
46	C₄	—	4	1	no	—
47	C₄	—	4	1	no	—
48	C₄	Q₈⋊C₈	1	1	yes	characteristic
49	C₄	#350	1	1	yes	characteristic
50	C₂	Q₈⋊C₁₆	1	1	yes	characteristic
51	C₂	#1400	1	1	yes	characteristic
52	C₂	#1401	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²⋊C₄
commutator	C₂×C₄	C₂×C₁₆
Frattini, cosocle	C₄×C₁₆	C₂²
socle, prime radical	C₂²	C₂²⋊C₁₆
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₁₆ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×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
#2183	2
C₄⋊C₃₂	2
#1397	2