Subgroups — #13634

Order 256.

97 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₆₄	C₂	1	1	yes	maximal, characteristic
2	#2158	C₂	1	1	yes	maximal, characteristic
3	C₂×#366	C₂	1	1	yes	maximal, characteristic
4	#1424	C₂	1	2	yes	maximal, complemented
5	#1423	C₂	1	2	yes	maximal, complemented
6	C₂²×C₁₆	C₂²	1	1	yes	characteristic
7	#427	—	2	1	no	—
8	C₂×#107	—	2	1	no	—
9	M₆₄	C₄	1	4	yes	complemented
10	C₂×C₃₂	C₂²	1	2	yes	—
11	#367	—	2	1	no	—
12	#366	—	2	1	no	—
13	#366	C₂²	1	2	yes	—
14	#366	C₂²	1	1	yes	characteristic
15	#366	C₂²	1	1	yes	characteristic
16	C₂²×C₈	D₄	1	1	yes	characteristic
17	#117	—	4	1	no	—
18	C₂×C₄⋊C₄	—	4	1	no	—
19	C₂×C₁₆	C₂×C₄	1	2	yes	—
20	C₂×C₁₆	D₄	1	1	yes	characteristic
21	C₂×C₁₆	Q₈	1	2	yes	—
22	C₂×C₁₆	C₂³	1	1	yes	characteristic
23	#107	—	2	2	no	—
24	#107	—	2	1	no	—
25	#107	—	2	1	no	—
26	#107	—	4	1	no	—
27	#106	—	4	1	no	—
28	C₃₂	C₂×C₄	1	4	yes	complemented
29	C₂²×C₄	—	8	1	no	—
30	C₂²×C₄	D₈	1	1	yes	characteristic
31	C₂×C₈	C₄⋊C₄	1	2	yes	—
32	C₂×C₈	D₈	1	1	yes	characteristic
33	C₂×C₈	Q₁₆	1	2	yes	—
34	C₂×C₈	C₂×D₄	1	1	yes	characteristic
35	C₄⋊C₄	—	4	1	no	—
36	C₄⋊C₄	—	4	1	no	—
37	C₄⋊C₄	—	4	2	no	—
38	C₄⋊C₄	—	8	1	no	—
39	C₂²⋊C₄	—	8	1	no	—
40	C₄²	—	8	1	no	—
41	C₁₆	C₄⋊C₄	1	2	yes	—
42	C₁₆	C₂²×C₄	1	1	yes	characteristic
43	C₁₆	C₂×Q₈	1	1	yes	characteristic
44	C₂³	D₁₆	1	1	yes	characteristic
45	C₂×C₄	—	8	1	no	—
46	C₂×C₄	—	8	1	no	—
47	C₂×C₄	—	8	2	no	—
48	C₂×C₄	—	16	2	no	—
49	C₂×C₄	#107	1	2	yes	—
50	C₂×C₄	D₁₆	1	1	yes	characteristic
51	C₂×C₄	Q₃₂	1	2	yes	—
52	C₂×C₄	C₂×D₈	1	1	yes	characteristic
53	C₈	#107	1	2	yes	—
54	C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₈	C₂×Q₁₆	1	1	yes	characteristic
56	C₂²	—	2	1	no	—
57	C₂²	—	2	1	no	—
58	C₂²	#366	1	2	yes	—
59	C₂²	C₂×D₁₆	1	1	yes	characteristic
60	C₄	—	16	1	no	—
61	C₄	—	16	2	no	—
62	C₄	—	16	1	no	—
63	C₄	#366	1	2	yes	—
64	C₄	C₂×#107	1	1	yes	characteristic
65	C₄	C₂×Q₃₂	1	1	yes	characteristic
66	C₂	—	2	2	no	—
67	C₂	C₂×#366	1	1	yes	characteristic
68	C₂	#2263	1	1	yes	characteristic
69	C₂	#2264	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×D₁₆
commutator	C₁₆	C₂²×C₄
Frattini, cosocle	C₂×C₁₆	C₂³
prime radical	C₂³	D₁₆
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
4. #1424	C₂	semidirect
5. #1423	C₂	semidirect
9. M₆₄	C₄	semidirect
28. C₃₂	C₂×C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₆₄	2
#2158	2
C₂×#366	2
#1424	2
#1423	2