Subgroups — C₂×H(3,𝔽₄) #3526

Order 128.

704 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	H(3,𝔽₄)	C₂	1	16	yes	maximal, direct product factor
2	C₂×#124	C₂	1	9	yes	maximal, complemented
3	C₂×#120	C₂	1	6	yes	maximal, complemented
4	C₂⁵	C₂²	1	2	yes	complemented
5	C₂²×Q₈	C₂²	1	3	yes	complemented
6	C₂²×D₄	C₂²	1	9	yes	—
7	#124	C₂²	1	72	yes	complemented
8	#120	C₂²	1	48	yes	complemented
9	C₂×C₂²⋊C₄	C₂²	1	18	yes	complemented
10	C₂×C₄²	C₂²	1	3	yes	complemented
11	C₂⁴	—	4	12	no	—
12	C₂⁴	C₂³	1	6	yes	—
13	C₂⁴	C₂³	1	8	yes	complemented
14	C₂×Q₈	—	4	6	no	—
15	C₂×Q₈	C₂³	1	12	yes	complemented
16	C₂×D₄	—	4	18	no	—
17	C₂×D₄	C₂³	1	36	yes	—
18	C₂²×C₄	C₂³	1	9	yes	—
19	C₂²⋊C₄	C₂³	1	72	yes	complemented
20	C₄²	C₂³	1	12	yes	complemented
21	C₂³	—	2	36	no	—
22	C₂³	—	4	24	no	—
23	C₂³	—	4	32	no	—
24	C₂³	C₂⁴	1	12	yes	—
25	C₂³	C₂⁴	1	1	yes	characteristic
26	Q₈	—	4	12	no	—
27	D₄	—	4	36	no	—
28	C₂×C₄	—	2	18	no	—
29	C₂×C₄	C₂⁴	1	18	yes	—
30	C₂²	—	2	36	no	—
31	C₂²	—	4	32	no	—
32	C₂²	—	4	24	no	—
33	C₂²	H(5,𝔽₂)	1	6	yes	—
34	C₂²	C₂⁵	1	1	yes	characteristic
35	C₄	—	2	18	no	—
36	C₂	—	4	12	no	—
37	C₂	H(3,𝔽₄)	1	4	yes	direct product factor
38	C₂	C₂×H(5,𝔽₂)	1	3	yes	—

	subgroup	G/H
center, socle	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.