Subgroups — #2209

Order 128.

77 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₃₂	C₂	1	2	yes	maximal
2	#494	C₂	1	1	yes	maximal, characteristic
3	C₄×Q₁₆	C₂	1	1	yes	maximal, characteristic
4	C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	#358	C₂	1	2	yes	maximal
6	C₂×Q₁₆	—	2	1	no	—
7	C₂×Q₁₆	C₂²	1	2	yes	—
8	C₂×Q₁₆	C₂²	1	1	yes	characteristic
9	C₄⋊Q₈	—	2	1	no	—
10	C₄×Q₈	—	2	1	no	—
11	Q₃₂	—	2	4	no	—
12	C₂×C₁₆	C₂²	1	2	yes	—
13	#107	C₂²	1	1	yes	characteristic
14	#103	—	2	1	no	—
15	C₄×C₈	C₂²	1	1	yes	characteristic
16	C₂×Q₈	—	2	1	no	—
17	C₂×Q₈	—	2	2	no	—
18	Q₁₆	—	2	2	no	—
19	Q₁₆	—	2	1	no	—
20	Q₁₆	—	2	2	no	—
21	Q₁₆	—	4	2	no	—
22	Q₁₆	D₄	1	2	yes	—
23	C₂×C₈	D₄	1	1	yes	characteristic
24	C₂×C₈	C₂³	1	1	yes	characteristic
25	C₄⋊C₄	—	2	1	no	—
26	C₄⋊C₄	—	4	1	no	—
27	C₄⋊C₄	—	4	1	no	—
28	C₄⋊C₄	—	4	1	no	—
29	C₄²	—	4	1	no	—
30	C₄²	D₄	1	1	yes	characteristic
31	C₁₆	—	2	2	no	—
32	Q₈	—	2	2	no	—
33	Q₈	—	4	1	no	—
34	Q₈	—	4	2	no	—
35	Q₈	—	4	2	no	—
36	C₂×C₄	—	4	1	no	—
37	C₂×C₄	—	4	1	no	—
38	C₂×C₄	—	4	2	no	—
39	C₂×C₄	D₈	1	1	yes	characteristic
40	C₂×C₄	D₈	1	1	yes	characteristic
41	C₂×C₄	C₂×D₄	1	1	yes	characteristic
42	C₈	—	2	1	no	—
43	C₈	C₂×D₄	1	1	yes	characteristic
44	C₈	#41	1	1	yes	characteristic
45	C₂²	C₂×D₈	1	1	yes	characteristic
46	C₄	—	2	1	no	—
47	C₄	—	4	2	no	—
48	C₄	—	8	2	no	—
49	C₄	—	8	1	no	—
50	C₄	Q₃₂	1	2	yes	complemented
51	C₄	#121	1	1	yes	characteristic
52	C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
53	C₂	#459	1	1	yes	characteristic
54	C₂	C₂×Q₃₂	1	1	yes	characteristic
55	C₂	#510	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

N	H ≅ G/N	type
50. C₄	Q₃₂	semidirect

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