Subgroups — #2249

Order 128.

92 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×SD₃₂	C₂	1	2	yes	maximal, complemented
3	C₂×D₁₆	C₂	1	1	yes	maximal, characteristic
4	#495	C₂	1	2	yes	maximal, complemented
5	C₄×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Q₁₆	C₂²	1	2	yes	—
7	C₂×SD₁₆	—	2	2	no	—
8	C₂×D₈	C₂²	1	2	yes	—
9	#124	—	2	2	no	—
10	Q₃₂	—	2	2	no	—
11	SD₃₂	—	2	4	no	—
12	D₁₆	—	2	2	no	—
13	C₂×C₁₆	C₂²	1	1	yes	characteristic
14	C₂×C₁₆	C₂²	1	1	yes	characteristic
15	C₄×C₈	C₂²	1	1	yes	characteristic
16	C₂×Q₈	—	2	2	no	—
17	C₂×D₄	—	2	2	no	—
18	Q₁₆	—	2	2	no	—
19	Q₁₆	—	2	2	no	—
20	SD₁₆	—	4	4	no	—
21	D₈	—	2	2	no	—
22	D₈	—	2	2	no	—
23	C₂×C₈	D₄	1	1	yes	characteristic
24	C₂×C₈	C₂³	1	1	yes	characteristic
25	C₂²⋊C₄	—	4	4	no	—
26	C₄²	D₄	1	1	yes	characteristic
27	C₁₆	D₄	1	2	yes	—
28	C₁₆	D₄	1	2	yes	—
29	C₂³	—	4	2	no	—
30	Q₈	—	4	2	no	—
31	Q₈	—	4	2	no	—
32	D₄	—	4	2	no	—
33	D₄	—	4	2	no	—
34	C₂×C₄	—	4	2	no	—
35	C₂×C₄	D₈	1	2	yes	—
36	C₂×C₄	C₂×D₄	1	1	yes	characteristic
37	C₈	D₈	1	2	yes	—
38	C₈	C₂×D₄	1	1	yes	characteristic
39	C₈	C₂×D₄	1	1	yes	characteristic
40	C₂²	—	8	2	no	—
41	C₂²	—	8	4	no	—
42	C₂²	C₂×D₈	1	1	yes	characteristic
43	C₄	—	2	2	no	—
44	C₄	—	8	2	no	—
45	C₄	#127	1	1	yes	characteristic
46	C₄	C₂×D₈	1	1	yes	characteristic
47	C₂	—	16	2	no	—
48	C₂	#493	1	1	yes	characteristic
49	C₂	#508	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³
prime radical	C₂×D₁₆	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.