Subgroups — #2253

Order 128.

58 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#499	C₂	1	2	yes	maximal
2	#367	C₂	1	2	yes	maximal
3	#366	C₂	1	2	yes	maximal
4	C₄×C₁₆	C₂	1	1	yes	maximal, characteristic
5	#125	—	2	2	no	—
6	C₂×C₁₆	C₂²	1	1	yes	characteristic
7	C₂×C₁₆	C₂²	1	1	yes	characteristic
8	#107	C₂²	1	4	yes	—
9	#106	—	2	2	no	—
10	C₄×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₈	D₄	1	1	yes	characteristic
12	C₂×C₈	C₂³	1	1	yes	characteristic
13	C₄⋊C₄	—	2	4	no	—
14	C₄⋊C₄	—	4	4	no	—
15	C₄²	D₄	1	1	yes	characteristic
16	C₁₆	Q₈	1	2	yes	—
17	C₁₆	Q₈	1	2	yes	—
18	C₂×C₄	—	4	4	no	—
19	C₂×C₄	D₈	1	2	yes	—
20	C₂×C₄	C₂×D₄	1	1	yes	characteristic
21	C₈	Q₁₆	1	2	yes	—
22	C₈	C₂×Q₈	1	1	yes	characteristic
23	C₈	C₂×Q₈	1	1	yes	characteristic
24	C₂²	C₂×D₈	1	1	yes	characteristic
25	C₄	—	2	2	no	—
26	C₄	—	8	4	no	—
27	C₄	C₄⋊Q₈	1	1	yes	characteristic
28	C₄	C₂×Q₁₆	1	1	yes	characteristic
29	C₂	#500	1	1	yes	characteristic
30	C₂	#508	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂²	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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