Subgroups — #3395

Order 128.

174 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#557	C₂	1	2	yes	maximal
2	#494	C₂	1	3	yes	maximal
3	#462	C₂	1	6	yes	maximal
4	C₈×Q₈	C₂	1	1	yes	maximal, characteristic
5	C₄×Q₁₆	C₂	1	3	yes	maximal
6	C₂×Q₁₆	C₂²	1	3	yes	—
7	C₂×Q₁₆	C₂²	1	6	yes	—
8	C₄⋊Q₈	C₂²	1	6	yes	—
9	#125	—	2	6	no	—
10	C₄×Q₈	—	2	2	no	—
11	C₄×Q₈	C₂²	1	6	yes	—
12	C₄×Q₈	C₂²	1	1	yes	characteristic
13	#107	C₂²	1	1	yes	characteristic
14	C₄⋊C₈	C₂²	1	3	yes	—
15	#103	C₂²	1	6	yes	—
16	C₄×C₈	C₂²	1	3	yes	—
17	C₂×Q₈	D₄	1	1	yes	characteristic
18	C₂×Q₈	C₂³	1	6	yes	—
19	Q₁₆	—	2	6	no	—
20	Q₁₆	—	4	6	no	—
21	C₂×C₈	C₂³	1	3	yes	—
22	C₂×C₈	C₂³	1	1	yes	characteristic
23	C₄⋊C₄	—	2	6	no	—
24	C₄⋊C₄	—	2	6	no	—
25	C₄⋊C₄	—	2	6	no	—
26	C₄⋊C₄	D₄	1	3	yes	—
27	C₄⋊C₄	C₂³	1	2	yes	—
28	C₄²	—	2	6	no	—
29	C₄²	C₂³	1	3	yes	—
30	Q₈	—	2	6	no	—
31	Q₈	—	2	6	no	—
32	Q₈	Q₁₆	1	4	yes	complemented
33	C₂×C₄	—	2	2	no	—
34	C₂×C₄	—	2	6	no	—
35	C₂×C₄	C₂×D₄	1	3	yes	—
36	C₂×C₄	C₂×D₄	1	3	yes	—
37	C₂×C₄	C₂⁴	1	1	yes	characteristic
38	C₈	—	2	3	no	—
39	C₈	#41	1	2	yes	—
40	C₂²	C₂²×D₄	1	1	yes	characteristic
41	C₄	—	2	3	no	—
42	C₄	—	4	6	no	—
43	C₄	—	4	2	no	—
44	C₄	C₂×Q₁₆	1	6	yes	—
45	C₄	C₂×#41	1	1	yes	characteristic
46	C₄	H(5,𝔽₂)	1	1	yes	characteristic
47	C₂	#550	1	1	yes	characteristic
48	C₂	C₂²×Q₁₆	1	1	yes	characteristic
49	C₂	#578	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
32. Q₈	Q₁₆	semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.