Subgroups — #7141

Order 256.

127 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	C₂×M₃₂	C₄	1	2	yes	—
5	C₂²×C₁₆	C₄	1	2	yes	—
6	#431	C₄	1	2	yes	—
7	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
8	#346	—	2	1	no	—
9	C₄×C₁₆	—	2	1	no	—
10	C₄×C₁₆	—	2	1	no	—
11	C₄×C₁₆	—	2	1	no	—
12	#322	—	2	1	no	—
13	C₈²	—	2	1	no	—
14	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
17	M₃₂	—	2	2	no	—
18	M₃₂	C₈	1	4	yes	complemented
19	C₂×C₁₆	—	2	2	no	—
20	C₂×C₁₆	—	2	2	no	—
21	C₂×C₁₆	—	2	2	no	—
22	C₂×C₁₆	C₈	1	4	yes	complemented
23	C₄⋊C₈	C₈	1	4	yes	—
24	C₂²⋊C₈	—	2	2	no	—
25	C₄×C₈	—	2	2	no	—
26	C₄×C₈	D₄	1	1	yes	characteristic
27	C₄×C₈	D₄	1	1	yes	characteristic
28	C₄×C₈	D₄	1	1	yes	characteristic
29	C₄×C₈	Q₈	1	1	yes	characteristic
30	C₂²×C₄	C₄²	1	1	yes	characteristic
31	C₂²×C₄	M₁₆	1	2	yes	—
32	C₂×C₈	—	2	4	no	—
33	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₈	C₂×C₈	1	2	yes	—
38	C₂×C₈	C₂×C₈	1	2	yes	—
39	C₂×C₈	M₁₆	1	2	yes	—
40	C₂×C₈	M₁₆	1	2	yes	—
41	C₄²	C₂²⋊C₄	1	1	yes	characteristic
42	C₄²	C₄⋊C₄	1	1	yes	characteristic
43	C₄²	C₂×C₈	1	2	yes	—
44	C₁₆	—	2	4	no	—
45	C₁₆	—	2	4	no	—
46	C₂³	#97	1	1	yes	characteristic
47	C₂×C₄	—	2	4	no	—
48	C₂×C₄	#95	1	1	yes	characteristic
49	C₂×C₄	C₄×C₈	1	1	yes	characteristic
50	C₂×C₄	C₄×C₈	1	1	yes	characteristic
51	C₂×C₄	C₄×C₈	1	1	yes	characteristic
52	C₂×C₄	#97	1	1	yes	characteristic
53	C₂×C₄	#97	1	1	yes	characteristic
54	C₂×C₄	C₂²⋊C₈	1	2	yes	—
55	C₂×C₄	C₄⋊C₈	1	2	yes	—
56	C₈	—	4	4	no	—
57	C₈	C₂²⋊C₈	1	2	yes	—
58	C₈	C₂²⋊C₈	1	2	yes	—
59	C₈	C₄⋊C₈	1	2	yes	—
60	C₈	C₄⋊C₈	1	2	yes	—
61	C₂²	—	2	2	no	—
62	C₂²	#322	1	1	yes	characteristic
63	C₂²	#322	1	1	yes	characteristic
64	C₂²	#336	1	1	yes	characteristic
65	C₄	—	2	2	no	—
66	C₄	C₈²	1	1	yes	characteristic
67	C₄	#322	1	1	yes	characteristic
68	C₄	#336	1	1	yes	characteristic
69	C₄	#336	1	1	yes	characteristic
70	C₄	#350	1	4	yes	—
71	C₂	—	2	1	no	—
72	C₂	—	2	1	no	—
73	C₂	#1273	1	1	yes	characteristic
74	C₂	#1379	1	2	yes	—

	subgroup	G/H
center	C₄×C₈	D₄
commutator	C₄	C₈²
Frattini, cosocle	C₂×C₄×C₈	C₂²
socle	C₂²	#336
prime radical	C₂³	#97
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
18. M₃₂	C₈	semidirect
22. 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