Subgroups — #12597

Order 256.

129 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2110	C₂	1	1	yes	maximal, characteristic
2	#2108	C₂	1	1	yes	maximal, characteristic
3	#1747	C₂	1	1	yes	maximal, characteristic
4	#1379	C₂	1	2	yes	maximal
5	#1378	C₂	1	2	yes	maximal
6	C₂×M₃₂	C₂²	1	2	yes	—
7	C₂×M₃₂	C₂²	1	1	yes	characteristic
8	C₂²×C₁₆	C₂²	1	1	yes	characteristic
9	#405	C₂²	1	2	yes	—
10	#405	C₂²	1	1	yes	characteristic
11	#347	C₄	1	4	yes	complemented
12	#346	C₄	1	2	yes	complemented
13	C₄×C₁₆	C₄	1	2	yes	complemented
14	#338	C₄	1	1	yes	characteristic
15	H(3,ℤ/4)	C₄	1	1	yes	characteristic
16	H(3,ℤ/4)	C₄	1	2	yes	—
17	C₂×M₁₆	C₂×C₄	1	2	yes	—
18	C₂×M₁₆	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈	C₂³	1	1	yes	characteristic
20	#117	C₂×C₄	1	2	yes	—
21	#117	C₂×C₄	1	1	yes	characteristic
22	M₃₂	—	2	2	no	—
23	M₃₂	—	2	2	no	—
24	M₃₂	C₂×C₄	1	4	yes	—
25	C₂×C₁₆	—	2	2	no	—
26	C₂×C₁₆	—	2	2	no	—
27	C₂×C₁₆	C₂×C₄	1	1	yes	characteristic
28	C₂×C₁₆	C₂×C₄	1	2	yes	—
29	C₂×C₁₆	C₂×C₄	1	1	yes	characteristic
30	#97	—	2	2	no	—
31	#97	C₂×C₄	1	2	yes	—
32	C₄×C₈	—	2	2	no	—
33	C₄×C₈	C₂×C₄	1	2	yes	—
34	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
35	M₁₆	—	2	2	no	—
36	M₁₆	—	2	2	no	—
37	M₁₆	—	2	2	no	—
38	C₂×C₈	—	2	2	no	—
39	C₂×C₈	C₄²	1	2	yes	—
40	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₈	#41	1	2	yes	—
43	C₂×C₈	#41	1	2	yes	—
44	C₄⋊C₄	—	2	1	no	—
45	C₄⋊C₄	—	2	2	no	—
46	C₂²⋊C₄	—	2	1	no	—
47	C₂²⋊C₄	—	2	2	no	—
48	C₄²	—	2	2	no	—
49	C₄²	C₄²	1	2	yes	—
50	C₁₆	—	2	1	no	—
51	C₁₆	—	2	1	no	—
52	C₁₆	—	2	4	no	—
53	C₁₆	—	2	2	no	—
54	C₂³	#131	1	1	yes	characteristic
55	C₂×C₄	—	2	2	no	—
56	C₂×C₄	—	4	2	no	—
57	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
58	C₂×C₄	#117	1	2	yes	—
59	C₂×C₄	#131	1	2	yes	—
60	C₂×C₄	#131	1	1	yes	characteristic
61	C₈	—	2	2	no	—
62	C₈	—	4	2	no	—
63	C₈	#117	1	1	yes	characteristic
64	C₈	#117	1	2	yes	—
65	C₈	#117	1	1	yes	characteristic
66	C₂²	—	4	1	no	—
67	C₂²	#405	1	1	yes	characteristic
68	C₂²	#433	1	2	yes	—
69	C₄	—	4	2	no	—
70	C₄	—	8	2	no	—
71	C₄	#376	1	1	yes	characteristic
72	C₄	#405	1	1	yes	characteristic
73	C₄	#433	1	2	yes	—
74	C₂	—	2	1	no	—
75	C₂	—	2	2	no	—
76	C₂	#1746	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#117
commutator	C₂×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₈	C₂³
socle	C₂	#1746
prime radical	C₂³	#131
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ #405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
11. #347	C₄	semidirect
12. #346	C₄	semidirect
13. C₄×C₁₆	C₄	semidirect

Maximal subgroups

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

subgroup	index
#2110	2
#2108	2
#1747	2
#1379	2
#1378	2