Subgroups — #1709

Order 128.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#494	C₂	1	1	yes	maximal, characteristic, complemented
2	#493	C₂	1	1	yes	maximal, characteristic
3	#486	C₂	1	4	yes	maximal, complemented
4	C₈²	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×Q₁₆	—	2	2	no	—
6	C₂×D₈	—	2	2	no	—
7	C₄⋊Q₈	C₂²	1	2	yes	—
8	#127	C₂²	1	2	yes	—
9	#102	—	2	8	no	—
10	C₄×C₈	C₂²	1	2	yes	—
11	C₄×C₈	C₂²	1	1	yes	characteristic
12	C₂×Q₈	—	2	2	no	—
13	C₂×D₄	—	2	4	no	—
14	C₂×D₄	—	2	2	no	—
15	Q₁₆	—	4	4	no	—
16	D₈	—	4	4	no	—
17	C₂×C₈	D₄	1	4	yes	—
18	C₂×C₈	D₄	1	2	yes	—
19	C₄⋊C₄	—	2	4	no	—
20	C₄²	C₂³	1	1	yes	characteristic
21	C₂³	—	4	2	no	—
22	Q₈	—	4	4	no	—
23	D₄	—	4	4	no	—
24	D₄	—	4	8	no	—
25	C₂×C₄	—	4	2	no	—
26	C₂×C₄	C₂×D₄	1	1	yes	characteristic
27	C₂×C₄	C₂×D₄	1	2	yes	—
28	C₈	—	2	4	no	—
29	C₈	D₈	1	4	yes	—
30	C₂²	—	8	2	no	—
31	C₂²	—	8	4	no	—
32	C₂²	#127	1	1	yes	characteristic
33	C₄	—	8	2	no	—
34	C₄	C₂×D₈	1	2	yes	—
35	C₄	#135	1	4	yes	—
36	C₂	—	16	2	no	—
37	C₂	#493	1	1	yes	characteristic
38	C₂	#495	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	#127
commutator, Frattini, cosocle	C₄²	C₂³
prime radical	#493	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ G
LECS	G ▷ C₄² ▷ 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.