Subgroups — #11399

Order 256.

165 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic
3	#1450	C₂	1	1	yes	maximal, characteristic
4	#1311	C₂	1	2	yes	maximal, complemented
5	C₈×C₁₆	C₂	1	2	yes	maximal, complemented
6	C₂×M₃₂	C₄	1	2	yes	—
7	C₂²×C₁₆	C₄	1	2	yes	—
8	#431	C₄	1	2	yes	—
9	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
10	#346	C₄	1	2	yes	—
11	C₄×C₁₆	C₄	1	2	yes	—
12	C₄×C₁₆	C₂²	1	2	yes	—
13	C₄×C₁₆	C₂²	1	1	yes	characteristic
14	C₄×C₁₆	C₂²	1	1	yes	characteristic
15	#322	C₄	1	2	yes	—
16	C₈²	C₂²	1	2	yes	—
17	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
20	M₃₂	C₈	1	8	yes	complemented
21	C₂×C₁₆	C₈	1	8	yes	complemented
22	C₂×C₁₆	C₂×C₄	1	2	yes	—
23	C₂×C₁₆	C₂×C₄	1	2	yes	—
24	C₂×C₁₆	C₂×C₄	1	4	yes	—
25	C₄⋊C₈	C₈	1	4	yes	—
26	C₂²⋊C₈	C₈	1	4	yes	—
27	C₄×C₈	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈	C₂×C₄	1	4	yes	—
31	C₄×C₈	C₂³	1	1	yes	characteristic
32	C₂²×C₄	C₄²	1	1	yes	characteristic
33	C₂²×C₄	C₂×C₈	1	2	yes	—
34	C₂×C₈	C₄²	1	1	yes	characteristic
35	C₂×C₈	C₄²	1	1	yes	characteristic
36	C₂×C₈	C₂×C₈	1	8	yes	—
37	C₂×C₈	C₂×C₈	1	4	yes	—
38	C₂×C₈	C₂×C₈	1	4	yes	—
39	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
41	C₄²	C₄²	1	1	yes	characteristic
42	C₄²	C₂×C₈	1	2	yes	—
43	C₄²	C₂²×C₄	1	1	yes	characteristic
44	C₁₆	C₂×C₈	1	4	yes	—
45	C₁₆	C₂×C₈	1	4	yes	—
46	C₁₆	C₂×C₈	1	8	yes	—
47	C₂³	C₄×C₈	1	1	yes	characteristic
48	C₂×C₄	—	2	4	no	—
49	C₂×C₄	C₄×C₈	1	2	yes	—
50	C₂×C₄	C₄×C₈	1	1	yes	characteristic
51	C₂×C₄	C₄×C₈	1	2	yes	—
52	C₂×C₄	C₄×C₈	1	2	yes	—
53	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
54	C₂×C₄	C₂²×C₈	1	2	yes	—
55	C₈	—	2	8	no	—
56	C₈	C₄×C₈	1	2	yes	—
57	C₈	C₄×C₈	1	2	yes	—
58	C₈	C₂²×C₈	1	2	yes	—
59	C₈	C₂²×C₈	1	2	yes	—
60	C₂²	—	2	2	no	—
61	C₂²	C₈²	1	2	yes	—
62	C₂²	C₂×C₄×C₈	1	1	yes	characteristic
63	C₄	—	2	2	no	—
64	C₄	C₈²	1	2	yes	—
65	C₄	C₂×C₄×C₈	1	1	yes	characteristic
66	C₄	C₂×C₄×C₈	1	1	yes	characteristic
67	C₄	#504	1	4	yes	—
68	C₂	—	2	2	no	—
69	C₂	C₂×C₈²	1	1	yes	characteristic
70	C₂	#2108	1	2	yes	—

	subgroup	G/H
center	C₄×C₁₆	C₂²
commutator	C₂	C₂×C₈²
Frattini, cosocle	C₄×C₈	C₂³
socle	C₂²	C₂×C₄×C₈
prime radical	C₂³	C₄×C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
4. #1311	C₂	semidirect
5. C₈×C₁₆	C₂	semidirect
20. M₃₂	C₈	semidirect
21. C₂×C₁₆	C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×M₃₂	2
C₂×C₄×C₁₆	2
#1450	2
#1311	2
C₈×C₁₆	2