Subgroups — #2241

Order 128.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#500	C₂	1	1	yes	maximal, characteristic
2	#494	C₂	1	1	yes	maximal, characteristic
3	#358	C₂	1	4	yes	maximal
4	C₄×C₁₆	C₂	1	1	yes	maximal, characteristic
5	C₂×Q₁₆	—	2	1	no	—
6	C₂×Q₁₆	C₂²	1	2	yes	—
7	C₄⋊Q₈	—	2	1	no	—
8	C₄⋊Q₈	—	2	1	no	—
9	C₂×C₁₆	C₂²	1	2	yes	—
10	#107	—	2	1	no	—
11	#107	C₂²	1	2	yes	—
12	C₄×C₈	C₂²	1	1	yes	characteristic
13	C₂×Q₈	—	2	2	no	—
14	C₂×Q₈	—	4	1	no	—
15	Q₁₆	—	2	4	no	—
16	Q₁₆	—	4	2	no	—
17	C₂×C₈	D₄	1	1	yes	characteristic
18	C₂×C₈	C₂³	1	1	yes	characteristic
19	C₄⋊C₄	—	2	2	no	—
20	C₄⋊C₄	—	4	1	no	—
21	C₄⋊C₄	—	4	1	no	—
22	C₄⋊C₄	—	4	1	no	—
23	C₄²	D₄	1	1	yes	characteristic
24	C₁₆	—	2	2	no	—
25	Q₈	—	4	2	no	—
26	Q₈	—	4	2	no	—
27	Q₈	—	8	2	no	—
28	C₂×C₄	—	4	2	no	—
29	C₂×C₄	—	4	2	no	—
30	C₂×C₄	D₈	1	1	yes	characteristic
31	C₂×C₄	D₈	1	1	yes	characteristic
32	C₂×C₄	C₂×D₄	1	1	yes	characteristic
33	C₈	SD₁₆	1	2	yes	—
34	C₈	#41	1	2	yes	—
35	C₂²	C₂×D₈	1	1	yes	characteristic
36	C₄	—	8	2	no	—
37	C₄	—	8	2	no	—
38	C₄	SD₃₂	1	2	yes	—
39	C₄	Q₃₂	1	2	yes	—
40	C₄	#124	1	1	yes	characteristic
41	C₄	C₂×SD₁₆	1	1	yes	characteristic
42	C₂	#486	1	1	yes	characteristic
43	C₂	C₂×SD₃₂	1	1	yes	characteristic
44	C₂	C₂×Q₃₂	1	1	yes	characteristic

	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.