Subgroups — #7380

Order 256.

64 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2169	C₂	1	1	yes	maximal, characteristic
2	C₄⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#1397	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×M₃₂	—	2	1	no	—
5	C₈×D₄	C₄	1	1	yes	characteristic
6	C₂×C₃₂	—	2	1	no	—
7	C₂×C₃₂	—	2	1	no	—
8	C₄⋊C₁₆	C₄	1	1	yes	characteristic
9	C₂²⋊C₁₆	—	2	1	no	—
10	C₄×C₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₈	—	2	1	no	—
12	C₄×D₄	C₈	1	1	yes	characteristic
13	M₃₂	—	4	1	no	—
14	M₃₂	—	4	1	no	—
15	C₂×C₁₆	—	2	1	no	—
16	C₂×C₁₆	D₄	1	1	yes	characteristic
17	C₂×C₁₆	D₄	1	1	yes	characteristic
18	C₄⋊C₈	C₈	1	1	yes	characteristic
19	C₂²⋊C₈	—	2	1	no	—
20	C₄×C₈	C₂×C₄	1	1	yes	characteristic
21	C₃₂	—	2	2	no	—
22	C₃₂	—	4	1	no	—
23	C₂×D₄	C₁₆	1	1	yes	characteristic
24	C₂²×C₄	—	2	1	no	—
25	C₂×C₈	—	2	1	no	—
26	C₂×C₈	—	4	1	no	—
27	C₂×C₈	—	4	1	no	—
28	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×C₈	M₁₆	1	1	yes	characteristic
30	C₄⋊C₄	C₁₆	1	1	yes	characteristic
31	C₂²⋊C₄	—	2	1	no	—
32	C₄²	C₂×C₈	1	1	yes	characteristic
33	C₁₆	—	2	1	no	—
34	C₁₆	—	4	1	no	—
35	C₁₆	D₈	1	1	yes	characteristic
36	C₁₆	SD₁₆	1	1	yes	characteristic
37	C₂³	—	2	1	no	—
38	D₄	—	4	1	no	—
39	C₂×C₄	—	2	1	no	—
40	C₂×C₄	—	4	1	no	—
41	C₂×C₄	—	4	1	no	—
42	C₂×C₄	C₂²⋊C₈	1	1	yes	characteristic
43	C₂×C₄	C₂×C₁₆	1	1	yes	characteristic
44	C₂×C₄	M₃₂	1	1	yes	characteristic
45	C₈	—	2	1	no	—
46	C₈	—	4	1	no	—
47	C₈	#102	1	1	yes	characteristic
48	C₈	#104	1	1	yes	characteristic
49	C₂²	—	4	1	no	—
50	C₂²	—	4	1	no	—
51	C₂²	—	4	1	no	—
52	C₂²	C₂²⋊C₁₆	1	1	yes	characteristic
53	C₄	—	2	1	no	—
54	C₄	—	2	1	no	—
55	C₄	—	4	1	no	—
56	C₄	D₄⋊C₈	1	1	yes	characteristic
57	C₄	#350	1	1	yes	characteristic
58	C₂	—	8	1	no	—
59	C₂	D₄⋊C₁₆	1	1	yes	characteristic
60	C₂	#1400	1	1	yes	characteristic
61	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	C₂²	C₂²⋊C₁₆
prime radical	C₂×D₄	C₁₆
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₁₆ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₄⋊C₃₂	C₂	semidirect
3. #1397	C₂	semidirect

Maximal subgroups

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

subgroup	index
#2169	2
C₄⋊C₃₂	2
#1397	2