Subgroups — #3483

Order 128.

601 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈²	C₂	1	1	yes	maximal, characteristic, complemented
2	#555	C₂	1	9	yes	maximal, complemented
3	#550	C₂	1	6	yes	maximal, complemented
4	#532	C₂	1	9	yes	maximal, complemented
5	C₄×#41	C₂	1	6	yes	maximal, complemented
6	C₂×#41	C₂²	1	6	yes	—
7	C₂×#41	C₂²	1	6	yes	—
8	C₄⋊Q₈	C₂²	1	9	yes	—
9	#127	C₂²	1	9	yes	—
10	#124	C₂²	1	18	yes	—
11	#121	C₂²	1	36	yes	—
12	C₄×Q₈	C₂²	1	2	yes	—
13	C₄×Q₈	C₂²	1	6	yes	—
14	C₄×D₄	C₂²	1	18	yes	—
15	C₄×D₄	C₂²	1	18	yes	—
16	#117	C₂²	1	18	yes	—
17	C₂×C₄²	C₂²	1	9	yes	—
18	#41	—	2	24	no	—
19	#41	—	4	12	no	—
20	C₂×Q₈	C₂³	1	6	yes	—
21	C₂×Q₈	C₂³	1	2	yes	—
22	C₂×D₄	C₂³	1	18	yes	—
23	C₂×D₄	C₂³	1	18	yes	—
24	C₂²×C₄	C₂³	1	18	yes	—
25	C₂²×C₄	C₂³	1	9	yes	—
26	C₄⋊C₄	C₂³	1	6	yes	—
27	C₄⋊C₄	C₂³	1	18	yes	—
28	C₂²⋊C₄	C₂³	1	36	yes	—
29	C₄²	C₂³	1	9	yes	—
30	C₄²	C₂³	1	9	yes	—
31	C₄²	C₂³	1	6	yes	—
32	C₂³	C₂⁴	1	9	yes	—
33	Q₈	—	4	6	no	—
34	Q₈	#41	1	8	yes	complemented
35	D₄	—	2	36	no	—
36	D₄	—	4	18	no	—
37	C₂×C₄	—	2	36	no	—
38	C₂×C₄	—	2	18	no	—
39	C₂×C₄	C₂⁴	1	6	yes	—
40	C₂×C₄	C₂⁴	1	9	yes	—
41	C₂×C₄	C₂⁴	1	6	yes	—
42	C₂×C₄	C₂⁴	1	1	yes	characteristic
43	C₂²	—	2	18	no	—
44	C₂²	—	2	9	no	—
45	C₂²	C₂⁵	1	1	yes	characteristic
46	C₄	—	2	6	no	—
47	C₄	—	2	9	no	—
48	C₄	C₂×#41	1	12	yes	—
49	C₄	H(5,𝔽₂)	1	2	yes	—
50	C₂	—	4	9	no	—
51	C₂	C₂²×#41	1	2	yes	—
52	C₂	C₂×H(5,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂⁴
commutator, Frattini, socle, cosocle	C₂²	C₂⁵
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ G
LECS	G ▷ C₂² ▷ 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.