Subgroups — #2214

Order 128.

75 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#510	C₂	1	2	yes	maximal
2	#494	C₂	1	1	yes	maximal, characteristic, complemented
3	#443	C₂	1	1	yes	maximal, characteristic
4	#364	C₂	1	1	yes	maximal, characteristic, complemented
5	#362	C₂	1	2	yes	maximal, complemented
6	#135	C₂²	1	1	yes	characteristic
7	C₂×Q₁₆	—	2	1	no	—
8	C₂×Q₁₆	C₂²	1	2	yes	—
9	#131	—	2	1	no	—
10	C₄⋊Q₈	—	2	1	no	—
11	Q₃₂	—	2	2	no	—
12	SD₃₂	—	2	2	no	—
13	M₃₂	C₂²	1	2	yes	—
14	#108	C₂²	1	1	yes	characteristic
15	#104	—	2	1	no	—
16	C₄×C₈	C₂²	1	1	yes	characteristic
17	#41	—	2	1	no	—
18	C₂×Q₈	—	2	2	no	—
19	Q₁₆	—	2	2	no	—
20	Q₁₆	—	2	2	no	—
21	Q₁₆	—	4	1	no	—
22	Q₁₆	—	4	1	no	—
23	Q₁₆	D₄	1	1	yes	characteristic
24	SD₁₆	—	2	1	no	—
25	D₈	D₄	1	1	yes	characteristic
26	M₁₆	—	2	1	no	—
27	M₁₆	—	4	1	no	—
28	C₂×C₈	—	4	1	no	—
29	C₂×C₈	D₄	1	1	yes	characteristic
30	C₂×C₈	C₂³	1	1	yes	characteristic
31	C₄⋊C₄	—	4	2	no	—
32	C₄²	D₄	1	1	yes	characteristic
33	C₁₆	—	2	2	no	—
34	Q₈	—	2	1	no	—
35	Q₈	—	4	2	no	—
36	Q₈	—	4	2	no	—
37	D₄	—	2	1	no	—
38	D₄	—	4	1	no	—
39	C₂×C₄	—	2	1	no	—
40	C₂×C₄	—	4	1	no	—
41	C₂×C₄	—	4	2	no	—
42	C₂×C₄	C₂×D₄	1	1	yes	characteristic
43	C₈	—	4	1	no	—
44	C₈	D₈	1	1	yes	characteristic
45	C₈	D₈	1	1	yes	characteristic
46	C₈	C₂×D₄	1	1	yes	characteristic
47	C₈	#41	1	1	yes	characteristic
48	C₂²	—	4	1	no	—
49	C₂²	AGL(1,ℤ/8)	1	1	yes	characteristic
50	C₄	—	2	2	no	—
51	C₄	—	4	1	no	—
52	C₄	—	8	2	no	—
53	C₄	#121	1	1	yes	characteristic
54	C₄	C₂×D₈	1	1	yes	characteristic
55	C₂	—	2	1	no	—
56	C₂	—	8	1	no	—
57	C₂	#459	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#459
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.