Subgroups — #2231

Order 128.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#499	C₂	1	1	yes	maximal, characteristic, complemented
2	#443	C₂	1	1	yes	maximal, characteristic
3	#365	C₂	1	2	yes	maximal, complemented
4	#364	C₂	1	1	yes	maximal, characteristic, complemented
5	#360	C₂	1	2	yes	maximal
6	#135	C₂²	1	1	yes	characteristic
7	#131	—	2	1	no	—
8	#125	—	2	1	no	—
9	M₃₂	C₂²	1	2	yes	—
10	#108	C₂²	1	1	yes	characteristic
11	#107	—	2	1	no	—
12	#106	C₂²	1	2	yes	—
13	#104	—	2	1	no	—
14	C₄×C₈	C₂²	1	1	yes	characteristic
15	#41	—	2	1	no	—
16	Q₁₆	Q₈	1	1	yes	characteristic
17	SD₁₆	—	2	1	no	—
18	D₈	Q₈	1	1	yes	characteristic
19	M₁₆	—	2	1	no	—
20	M₁₆	—	4	1	no	—
21	C₂×C₈	—	4	1	no	—
22	C₂×C₈	D₄	1	1	yes	characteristic
23	C₂×C₈	C₂³	1	1	yes	characteristic
24	C₄⋊C₄	—	2	2	no	—
25	C₄⋊C₄	—	4	2	no	—
26	C₄²	D₄	1	1	yes	characteristic
27	C₁₆	—	2	2	no	—
28	Q₈	—	2	1	no	—
29	D₄	—	2	1	no	—
30	D₄	—	4	1	no	—
31	C₂×C₄	—	2	1	no	—
32	C₂×C₄	—	4	1	no	—
33	C₂×C₄	—	4	2	no	—
34	C₂×C₄	C₂×D₄	1	1	yes	characteristic
35	C₈	—	4	1	no	—
36	C₈	SD₁₆	1	1	yes	characteristic
37	C₈	SD₁₆	1	1	yes	characteristic
38	C₈	C₂×Q₈	1	1	yes	characteristic
39	C₈	#41	1	1	yes	characteristic
40	C₂²	—	4	1	no	—
41	C₂²	AGL(1,ℤ/8)	1	1	yes	characteristic
42	C₄	—	4	1	no	—
43	C₄	—	4	1	no	—
44	C₄	—	8	2	no	—
45	C₄	C₂²⋊Q₈	1	1	yes	characteristic
46	C₄	C₂×SD₁₆	1	1	yes	characteristic
47	C₂	—	2	1	no	—
48	C₂	—	8	1	no	—
49	C₂	#476	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#476
commutator, Frattini, cosocle	C₂×C₈	C₂³
prime radical	#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ 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.