Subgroups — #2243

Order 128.

71 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#500	C₂	1	1	yes	maximal, characteristic, complemented
2	#495	C₂	1	1	yes	maximal, characteristic
3	#358	C₂	1	2	yes	maximal, complemented
4	#357	C₂	1	2	yes	maximal
5	#346	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Q₁₆	C₂²	1	1	yes	characteristic
7	C₂×SD₁₆	—	2	1	no	—
8	C₂×D₈	C₂²	1	1	yes	characteristic
9	C₄⋊Q₈	—	2	1	no	—
10	#124	—	2	1	no	—
11	C₂×C₁₆	C₂²	1	2	yes	—
12	#107	—	2	1	no	—
13	#107	C₂²	1	2	yes	—
14	C₄×C₈	C₂²	1	1	yes	characteristic
15	C₂×Q₈	—	2	1	no	—
16	C₂×Q₈	—	4	1	no	—
17	C₂×D₄	—	2	1	no	—
18	Q₁₆	—	2	2	no	—
19	SD₁₆	—	4	2	no	—
20	D₈	—	2	2	no	—
21	C₂×C₈	D₄	1	1	yes	characteristic
22	C₂×C₈	C₂³	1	1	yes	characteristic
23	C₄⋊C₄	—	2	2	no	—
24	C₄⋊C₄	—	4	1	no	—
25	C₂²⋊C₄	—	4	1	no	—
26	C₂²⋊C₄	—	4	1	no	—
27	C₄²	D₄	1	1	yes	characteristic
28	C₁₆	—	2	2	no	—
29	C₂³	—	4	1	no	—
30	Q₈	—	4	1	no	—
31	Q₈	—	4	1	no	—
32	Q₈	—	8	2	no	—
33	D₄	—	4	1	no	—
34	D₄	—	4	1	no	—
35	C₂×C₄	—	4	1	no	—
36	C₂×C₄	—	4	2	no	—
37	C₂×C₄	D₈	1	1	yes	characteristic
38	C₂×C₄	D₈	1	1	yes	characteristic
39	C₂×C₄	C₂×D₄	1	1	yes	characteristic
40	C₈	SD₁₆	1	2	yes	—
41	C₈	#41	1	2	yes	—
42	C₂²	—	8	1	no	—
43	C₂²	—	8	1	no	—
44	C₂²	—	8	1	no	—
45	C₂²	C₂×D₈	1	1	yes	characteristic
46	C₄	—	2	1	no	—
47	C₄	—	2	1	no	—
48	C₄	—	8	1	no	—
49	C₄	—	8	2	no	—
50	C₄	#124	1	1	yes	characteristic
51	C₄	C₂×SD₁₆	1	1	yes	characteristic
52	C₂	—	16	1	no	—
53	C₂	#486	1	1	yes	characteristic
54	C₂	#509	1	1	yes	characteristic
55	C₂	#510	1	1	yes	characteristic

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