Subgroups — #2842

Order 128.

274 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#399	C₂	1	4	yes	maximal, complemented
2	#396	C₂	1	8	yes	maximal, complemented
3	#379	C₂	1	3	yes	maximal
4	C₂⁵	C₂²	1	1	yes	characteristic
5	C₂×C₄⋊C₄	C₂²	1	12	yes	complemented
6	C₂×C₂²⋊C₄	C₂²	1	12	yes	—
7	C₂×C₂²⋊C₄	C₂²	1	6	yes	—
8	#95	C₂²	1	4	yes	complemented
9	C₂⁴	—	4	6	no	—
10	C₂⁴	Q₈	1	4	yes	—
11	C₂⁴	C₂³	1	2	yes	—
12	C₂⁴	C₂³	1	1	yes	characteristic
13	C₂²×C₄	C₂³	1	12	yes	—
14	C₄⋊C₄	—	4	12	no	—
15	C₂²⋊C₄	—	4	12	no	—
16	C₂²⋊C₄	—	4	6	no	—
17	C₂³	—	2	12	no	—
18	C₂³	—	2	6	no	—
19	C₂³	—	4	12	no	—
20	C₂³	—	4	16	no	—
21	C₂³	C₂×Q₈	1	4	yes	—
22	C₂³	C₂×Q₈	1	2	yes	—
23	C₂³	C₂⁴	1	1	yes	characteristic
24	C₂×C₄	—	2	24	no	—
25	C₂×C₄	—	2	12	no	—
26	C₂²	—	2	12	no	—
27	C₂²	—	2	6	no	—
28	C₂²	—	4	16	no	—
29	C₂²	—	4	8	no	—
30	C₂²	—	4	4	no	—
31	C₂²	C₂²×Q₈	1	1	yes	characteristic
32	C₂²	H(5,𝔽₂)	1	6	yes	—
33	C₄	—	4	12	no	—
34	C₂	—	4	4	no	—
35	C₂	—	4	2	no	—
36	C₂	#543	1	3	yes	—
37	C₂	#560	1	4	yes	—

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