Subgroups — #2244

Order 128.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#499	C₂	1	1	yes	maximal, characteristic, complemented
2	#493	C₂	1	1	yes	maximal, characteristic
3	#357	C₂	1	4	yes	maximal, complemented
4	#346	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₈	—	2	1	no	—
6	C₂×D₈	C₂²	1	2	yes	—
7	#127	—	2	1	no	—
8	#125	—	2	1	no	—
9	C₂×C₁₆	C₂²	1	2	yes	—
10	#107	C₂²	1	2	yes	—
11	#106	—	2	1	no	—
12	C₄×C₈	C₂²	1	1	yes	characteristic
13	C₂×D₄	—	2	2	no	—
14	C₂×D₄	—	4	2	no	—
15	D₈	—	2	4	no	—
16	D₈	—	4	2	no	—
17	C₂×C₈	D₄	1	1	yes	characteristic
18	C₂×C₈	C₂³	1	1	yes	characteristic
19	C₄⋊C₄	—	2	2	no	—
20	C₄⋊C₄	—	4	2	no	—
21	C₄²	D₄	1	1	yes	characteristic
22	C₁₆	—	2	2	no	—
23	C₂³	—	4	2	no	—
24	D₄	—	4	2	no	—
25	D₄	—	4	2	no	—
26	D₄	—	8	4	no	—
27	C₂×C₄	—	4	2	no	—
28	C₂×C₄	D₈	1	2	yes	—
29	C₂×C₄	C₂×D₄	1	1	yes	characteristic
30	C₈	SD₁₆	1	2	yes	—
31	C₈	#41	1	2	yes	—
32	C₂²	—	8	2	no	—
33	C₂²	—	8	4	no	—
34	C₂²	C₂×D₈	1	1	yes	characteristic
35	C₄	—	2	2	no	—
36	C₄	—	8	2	no	—
37	C₄	#124	1	1	yes	characteristic
38	C₄	C₂×SD₁₆	1	1	yes	characteristic
39	C₂	—	16	2	no	—
40	C₂	#486	1	1	yes	characteristic
41	C₂	#509	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³
prime radical	#493	C₂
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.