Subgroups — #2844

Order 128.

178 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#400	C₂	1	6	yes	maximal
2	#395	C₂	1	6	yes	maximal
3	#382	C₂	1	3	yes	maximal
4	C₂²×Q₈	C₂²	1	2	yes	—
5	C₂×C₄⋊C₄	C₂²	1	3	yes	—
6	C₂×C₄⋊C₄	C₂²	1	3	yes	—
7	C₂×C₄⋊C₄	C₂²	1	6	yes	—
8	C₂×C₄²	C₂²	1	1	yes	characteristic
9	#95	C₂²	1	12	yes	—
10	#95	C₂²	1	6	yes	—
11	#95	C₂²	1	2	yes	—
12	C₂×Q₈	—	4	6	no	—
13	C₂²×C₄	C₂³	1	6	yes	—
14	C₂²×C₄	C₂³	1	6	yes	—
15	C₂²×C₄	C₂³	1	3	yes	—
16	C₄⋊C₄	—	4	3	no	—
17	C₄⋊C₄	—	4	3	no	—
18	C₄⋊C₄	—	4	6	no	—
19	C₄²	Q₈	1	4	yes	complemented
20	C₂³	C₂⁴	1	1	yes	characteristic
21	Q₈	—	4	8	no	—
22	C₂×C₄	—	2	3	no	—
23	C₂×C₄	—	2	3	no	—
24	C₂×C₄	—	2	6	no	—
25	C₂×C₄	—	2	6	no	—
26	C₂×C₄	—	2	12	no	—
27	C₂×C₄	—	2	12	no	—
28	C₂×C₄	C₂×Q₈	1	6	yes	—
29	C₂²	C₂²×Q₈	1	1	yes	characteristic
30	C₂²	H(5,𝔽₂)	1	3	yes	—
31	C₂²	2₋¹⁺⁴	1	3	yes	—
32	C₄	—	2	6	no	—
33	C₄	—	4	6	no	—
34	C₄	—	4	6	no	—
35	C₂	#544	1	3	yes	—
36	C₂	H(3,𝔽₄)	1	1	yes	characteristic
37	C₂	#563	1	3	yes	—

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

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

N	H ≅ G/N	type
19. C₄²	Q₈	semidirect

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