Subgroups — #2276

Order 128.

314 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Q₈	C₂	1	3	yes	maximal
2	C₄×C₄⋊C₄	C₂	1	9	yes	maximal
3	#376	C₂	1	3	yes	maximal
4	C₂²×Q₈	C₂²	1	1	yes	characteristic
5	C₄×Q₈	C₄	1	24	yes	complemented
6	C₂×C₄⋊C₄	C₂²	1	9	yes	—
7	C₂×C₄²	C₂²	1	6	yes	—
8	C₂×C₄²	C₂²	1	9	yes	—
9	#95	C₂²	1	1	yes	characteristic
10	#95	C₂²	1	9	yes	—
11	C₂×Q₈	C₂×C₄	1	12	yes	—
12	C₂²×C₄	C₂³	1	3	yes	—
13	C₂²×C₄	C₂³	1	9	yes	—
14	C₂²×C₄	C₂³	1	3	yes	—
15	C₄⋊C₄	C₂×C₄	1	36	yes	—
16	C₄²	—	4	6	no	—
17	C₄²	C₂×C₄	1	36	yes	—
18	C₂³	C₂⁴	1	1	yes	characteristic
19	Q₈	C₄²	1	16	yes	complemented
20	C₂×C₄	—	2	9	no	—
21	C₂×C₄	—	2	6	no	—
22	C₂×C₄	—	2	9	no	—
23	C₂×C₄	C₂²×C₄	1	6	yes	—
24	C₂×C₄	C₂²×C₄	1	18	yes	—
25	C₂×C₄	C₂²×C₄	1	18	yes	—
26	C₂²	C₂³×C₄	1	3	yes	—
27	C₂²	H(5,𝔽₂)	1	1	yes	characteristic
28	C₂²	2₋¹⁺⁴	1	3	yes	—
29	C₄	—	2	6	no	—
30	C₄	—	2	18	no	—
31	C₄	C₂×C₄²	1	12	yes	—
32	C₂	C₂²×C₄²	1	1	yes	characteristic
33	C₂	#519	1	3	yes	—
34	C₂	#520	1	3	yes	—

	subgroup	G/H
center, Frattini, socle, prime radical, cosocle	C₂³	C₂⁴
commutator	C₂	C₂²×C₄²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

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

N	H ≅ G/N	type
5. C₄×Q₈	C₄	semidirect
19. Q₈	C₄²	semidirect

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