Subgroups — #2205

Order 128.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈²	C₂	1	1	yes	maximal, characteristic, complemented
2	#456	C₂	1	3	yes	maximal
3	#356	C₂	1	3	yes	maximal, complemented
4	2₋¹⁺⁴	C₂²	1	1	yes	characteristic
5	#137	—	2	3	no	—
6	C₄⋊Q₈	—	2	3	no	—
7	C₄⋊Q₈	C₂²	1	3	yes	—
8	C₄×Q₈	—	2	3	no	—
9	#104	—	2	3	no	—
10	#101	C₂²	1	3	yes	—
11	#41	—	2	3	no	—
12	#41	—	4	1	no	—
13	C₂×Q₈	—	2	1	no	—
14	C₂×Q₈	—	2	1	no	—
15	C₂×Q₈	—	4	1	no	—
16	C₂×Q₈	D₄	1	3	yes	—
17	C₂×Q₈	C₂³	1	1	yes	characteristic
18	Q₁₆	—	4	3	no	—
19	SD₁₆	—	4	3	no	—
20	M₁₆	—	2	3	no	—
21	C₄⋊C₄	—	2	3	no	—
22	C₄⋊C₄	—	2	6	no	—
23	C₄²	—	2	3	no	—
24	C₄²	D₄	1	3	yes	—
25	Q₈	—	2	4	no	—
26	Q₈	—	2	3	no	—
27	Q₈	—	4	1	no	—
28	Q₈	—	4	3	no	—
29	Q₈	—	8	1	no	—
30	D₄	—	2	3	no	—
31	D₄	—	4	1	no	—
32	C₂×C₄	—	2	3	no	—
33	C₂×C₄	—	2	3	no	—
34	C₂×C₄	—	4	3	no	—
35	C₂×C₄	C₂×D₄	1	3	yes	—
36	C₈	—	4	3	no	—
37	C₂²	—	4	1	no	—
38	C₂²	#120	1	1	yes	characteristic
39	C₄	—	2	6	no	—
40	C₄	—	2	3	no	—
41	C₄	—	4	3	no	—
42	C₄	—	4	1	no	—
43	C₂	—	2	1	no	—
44	C₂	—	8	1	no	—
45	C₂	B(4,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×Q₈	C₂³
prime radical	2₋¹⁺⁴	C₂²
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×Q₈ ◁ G
LECS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

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

N	H ≅ G/N	type
1. Q₈²	C₂	semidirect
3. #356	C₂	semidirect

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