Subgroups — C₂×Q₈² #3477

Order 128.

556 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈²	C₂	1	16	yes	maximal, direct product factor
2	C₂×C₄⋊Q₈	C₂	1	9	yes	maximal
3	C₂×C₄×Q₈	C₂	1	6	yes	maximal
4	C₂²×Q₈	C₂²	1	6	yes	—
5	C₂²×Q₈	C₂²	1	2	yes	—
6	C₄⋊Q₈	C₂²	1	72	yes	—
7	C₄×Q₈	C₂²	1	48	yes	—
8	C₂×C₄⋊C₄	C₂²	1	18	yes	—
9	C₂×C₄²	C₂²	1	9	yes	—
10	C₂×Q₈	—	4	12	no	—
11	C₂×Q₈	Q₈	1	16	yes	direct product factor
12	C₂×Q₈	C₂³	1	24	yes	—
13	C₂×Q₈	C₂³	1	8	yes	—
14	C₂²×C₄	C₂³	1	9	yes	—
15	C₂²×C₄	C₂³	1	6	yes	—
16	C₄⋊C₄	C₂³	1	72	yes	—
17	C₄²	C₂³	1	36	yes	—
18	C₂³	C₂⁴	1	1	yes	characteristic
19	Q₈	—	4	24	no	—
20	Q₈	C₂×Q₈	1	32	yes	direct product factor
21	C₂×C₄	—	2	18	no	—
22	C₂×C₄	C₂×Q₈	1	24	yes	—
23	C₂×C₄	C₂⁴	1	18	yes	—
24	C₂×C₄	C₂⁴	1	12	yes	—
25	C₂²	C₂²×Q₈	1	4	yes	—
26	C₂²	H(5,𝔽₂)	1	2	yes	—
27	C₂²	C₂⁵	1	1	yes	characteristic
28	C₄	—	2	18	no	—
29	C₄	C₂²×Q₈	1	24	yes	—
30	C₂	Q₈²	1	4	yes	direct product factor
31	C₂	C₂³×Q₈	1	2	yes	—
32	C₂	C₂×H(5,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁵
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂² ▷ 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.