Subgroups — #2255

Order 128.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#500	C₂	1	1	yes	maximal, characteristic
2	#499	C₂	1	1	yes	maximal, characteristic
3	#367	C₂	1	2	yes	maximal
4	#366	C₂	1	2	yes	maximal
5	#346	C₂	1	1	yes	maximal, characteristic
6	C₄⋊Q₈	—	2	1	no	—
7	#125	—	2	1	no	—
8	C₂×C₁₆	C₂²	1	2	yes	—
9	#107	—	2	1	no	—
10	#107	C₂²	1	1	yes	characteristic
11	#107	C₂²	1	2	yes	—
12	#107	C₂²	1	1	yes	characteristic
13	#106	—	2	1	no	—
14	C₄×C₈	C₂²	1	1	yes	characteristic
15	C₂×Q₈	—	4	1	no	—
16	C₂×C₈	D₄	1	1	yes	characteristic
17	C₂×C₈	C₂³	1	1	yes	characteristic
18	C₄⋊C₄	—	2	1	no	—
19	C₄⋊C₄	—	2	1	no	—
20	C₄⋊C₄	—	2	2	no	—
21	C₄⋊C₄	—	4	1	no	—
22	C₄⋊C₄	—	4	1	no	—
23	C₄⋊C₄	—	4	1	no	—
24	C₄²	D₄	1	1	yes	characteristic
25	C₁₆	Q₈	1	4	yes	complemented
26	Q₈	—	8	2	no	—
27	C₂×C₄	—	4	1	no	—
28	C₂×C₄	—	4	1	no	—
29	C₂×C₄	—	4	2	no	—
30	C₂×C₄	D₈	1	1	yes	characteristic
31	C₂×C₄	D₈	1	1	yes	characteristic
32	C₂×C₄	C₂×D₄	1	1	yes	characteristic
33	C₈	Q₁₆	1	2	yes	—
34	C₈	C₂×Q₈	1	2	yes	—
35	C₂²	C₂×D₈	1	1	yes	characteristic
36	C₄	—	2	1	no	—
37	C₄	—	2	1	no	—
38	C₄	—	8	1	no	—
39	C₄	—	8	2	no	—
40	C₄	—	8	1	no	—
41	C₄	C₄⋊Q₈	1	1	yes	characteristic
42	C₄	C₂×Q₁₆	1	1	yes	characteristic
43	C₂	#500	1	1	yes	characteristic
44	C₂	#509	1	1	yes	characteristic
45	C₂	#510	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	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.

N	H ≅ G/N	type
25. C₁₆	Q₈	semidirect

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