Subgroups — #2240

Order 128.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#500	C₂	1	1	yes	maximal, characteristic, complemented
2	#493	C₂	1	1	yes	maximal, characteristic
3	#357	C₂	1	4	yes	maximal, complemented
4	C₄×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₈	—	2	1	no	—
6	C₂×D₈	C₂²	1	2	yes	—
7	C₄⋊Q₈	—	2	1	no	—
8	#127	—	2	1	no	—
9	C₂×C₁₆	C₂²	1	2	yes	—
10	#107	—	2	1	no	—
11	#107	C₂²	1	2	yes	—
12	C₄×C₈	C₂²	1	1	yes	characteristic
13	C₂×Q₈	—	4	1	no	—
14	C₂×D₄	—	2	2	no	—
15	C₂×D₄	—	4	1	no	—
16	C₂×D₄	—	4	1	no	—
17	D₈	—	2	4	no	—
18	D₈	—	4	2	no	—
19	C₂×C₈	D₄	1	1	yes	characteristic
20	C₂×C₈	C₂³	1	1	yes	characteristic
21	C₄⋊C₄	—	2	2	no	—
22	C₄⋊C₄	—	4	1	no	—
23	C₄²	D₄	1	1	yes	characteristic
24	C₁₆	—	2	2	no	—
25	C₂³	—	4	2	no	—
26	Q₈	—	8	2	no	—
27	D₄	—	4	2	no	—
28	D₄	—	4	2	no	—
29	D₄	—	8	2	no	—
30	D₄	—	8	2	no	—
31	C₂×C₄	—	4	2	no	—
32	C₂×C₄	D₈	1	1	yes	characteristic
33	C₂×C₄	D₈	1	1	yes	characteristic
34	C₂×C₄	C₂×D₄	1	1	yes	characteristic
35	C₈	SD₁₆	1	2	yes	—
36	C₈	#41	1	2	yes	—
37	C₂²	—	8	2	no	—
38	C₂²	—	8	2	no	—
39	C₂²	—	8	2	no	—
40	C₂²	C₂×D₈	1	1	yes	characteristic
41	C₄	—	8	2	no	—
42	C₄	D₁₆	1	2	yes	—
43	C₄	SD₃₂	1	2	yes	—
44	C₄	#124	1	1	yes	characteristic
45	C₄	C₂×SD₁₆	1	1	yes	characteristic
46	C₂	—	16	2	no	—
47	C₂	#486	1	1	yes	characteristic
48	C₂	C₂×D₁₆	1	1	yes	characteristic
49	C₂	C₂×SD₃₂	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³
prime radical	#493	C₂
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.

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