Subgroups — #2250

Order 128.

99 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×SD₃₂	C₂	1	2	yes	maximal, complemented
2	C₂×D₁₆	C₂	1	2	yes	maximal, complemented
3	#495	C₂	1	1	yes	maximal, characteristic, complemented
4	#493	C₂	1	1	yes	maximal, characteristic, complemented
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	—	2	1	no	—
9	C₂×D₈	C₂²	1	2	yes	—
10	C₂×D₈	C₂²	1	1	yes	characteristic
11	#127	—	2	1	no	—
12	#124	—	2	1	no	—
13	SD₃₂	—	2	4	no	—
14	D₁₆	—	2	4	no	—
15	C₂×C₁₆	C₂²	1	2	yes	—
16	C₄×C₈	C₂²	1	1	yes	characteristic
17	C₂×Q₈	—	2	1	no	—
18	C₂×D₄	—	2	1	no	—
19	C₂×D₄	—	2	2	no	—
20	C₂×D₄	—	4	2	no	—
21	Q₁₆	—	2	2	no	—
22	SD₁₆	—	4	2	no	—
23	D₈	—	2	2	no	—
24	D₈	—	2	2	no	—
25	D₈	—	2	2	no	—
26	D₈	—	4	2	no	—
27	C₂×C₈	D₄	1	1	yes	characteristic
28	C₂×C₈	C₂³	1	1	yes	characteristic
29	C₂²⋊C₄	—	4	2	no	—
30	C₄²	D₄	1	1	yes	characteristic
31	C₁₆	D₄	1	4	yes	complemented
32	C₂³	—	4	1	no	—
33	C₂³	—	4	2	no	—
34	Q₈	—	4	1	no	—
35	Q₈	—	4	1	no	—
36	D₄	—	4	1	no	—
37	D₄	—	4	2	no	—
38	D₄	—	4	2	no	—
39	D₄	—	4	1	no	—
40	D₄	—	8	4	no	—
41	C₂×C₄	—	4	1	no	—
42	C₂×C₄	D₈	1	2	yes	—
43	C₂×C₄	C₂×D₄	1	1	yes	characteristic
44	C₈	D₈	1	2	yes	—
45	C₈	C₂×D₄	1	2	yes	—
46	C₂²	—	8	2	no	—
47	C₂²	—	8	1	no	—
48	C₂²	—	8	4	no	—
49	C₂²	—	8	2	no	—
50	C₂²	C₂×D₈	1	1	yes	characteristic
51	C₄	—	2	2	no	—
52	C₄	—	8	1	no	—
53	C₄	#127	1	1	yes	characteristic
54	C₄	C₂×D₈	1	1	yes	characteristic
55	C₂	—	16	2	no	—
56	C₂	—	16	1	no	—
57	C₂	#493	1	1	yes	characteristic
58	C₂	#509	1	2	yes	—

	subgroup	G/H
center, socle	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

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.