Subgroups — #1722

Order 128.

99 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#500	C₂	1	2	yes	maximal, complemented
2	#494	C₂	1	1	yes	maximal, characteristic, complemented
3	#492	C₂	1	1	yes	maximal, characteristic
4	#486	C₂	1	2	yes	maximal
5	#322	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Q₁₆	—	2	2	no	—
7	C₂×SD₁₆	—	2	2	no	—
8	C₄⋊Q₈	C₂²	1	2	yes	—
9	C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	#127	C₂²	1	1	yes	characteristic
11	#107	—	2	4	no	—
12	#102	—	2	4	no	—
13	C₄×C₈	C₂²	1	2	yes	—
14	C₄×C₈	C₂²	1	1	yes	characteristic
15	C₂×Q₈	—	2	1	no	—
16	C₂×Q₈	—	2	2	no	—
17	C₂×D₄	—	2	2	no	—
18	C₂×D₄	—	2	1	no	—
19	Q₁₆	—	4	4	no	—
20	SD₁₆	—	4	4	no	—
21	C₂×C₈	D₄	1	4	yes	—
22	C₂×C₈	D₄	1	2	yes	—
23	C₄⋊C₄	—	2	2	no	—
24	C₄⋊C₄	—	2	2	no	—
25	C₄⋊C₄	—	2	2	no	—
26	C₄²	C₂³	1	1	yes	characteristic
27	C₂³	—	4	1	no	—
28	Q₈	—	4	2	no	—
29	Q₈	—	4	4	no	—
30	D₄	—	4	2	no	—
31	D₄	—	4	4	no	—
32	C₂×C₄	—	4	2	no	—
33	C₂×C₄	—	4	1	no	—
34	C₂×C₄	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄	C₂×D₄	1	2	yes	—
36	C₈	—	2	4	no	—
37	C₈	D₈	1	4	yes	—
38	C₂²	—	8	1	no	—
39	C₂²	—	8	2	no	—
40	C₂²	#127	1	1	yes	characteristic
41	C₄	—	8	1	no	—
42	C₄	—	8	2	no	—
43	C₄	C₂×D₈	1	2	yes	—
44	C₄	#137	1	4	yes	—
45	C₂	—	16	1	no	—
46	C₂	#493	1	1	yes	characteristic
47	C₂	#497	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	#127
commutator, Frattini, cosocle	C₄²	C₂³
prime radical	#127	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ G
LECS	G ▷ C₄² ▷ 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.