Subgroups — #1674

Order 128.

78 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	#462	C₂	1	2	yes	maximal
3	C₈×Q₈	C₂	1	1	yes	maximal, characteristic
4	#335	C₂	1	1	yes	maximal, characteristic
5	#332	C₂	1	2	yes	maximal
6	C₂×Q₁₆	—	2	2	no	—
7	C₄⋊Q₈	C₂²	1	2	yes	—
8	C₄×Q₈	C₂²	1	1	yes	characteristic
9	#107	—	2	1	no	—
10	#107	—	2	1	no	—
11	C₄⋊C₈	—	2	1	no	—
12	C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₄⋊C₈	C₂²	1	2	yes	—
14	#103	—	2	2	no	—
15	C₄×C₈	—	2	1	no	—
16	C₄×C₈	C₂²	1	1	yes	characteristic
17	C₂×Q₈	—	2	2	no	—
18	C₂×Q₈	D₄	1	1	yes	characteristic
19	Q₁₆	—	4	4	no	—
20	C₂×C₈	—	2	1	no	—
21	C₂×C₈	—	2	2	no	—
22	C₂×C₈	D₄	1	1	yes	characteristic
23	C₂×C₈	D₄	1	1	yes	characteristic
24	C₄⋊C₄	—	2	1	no	—
25	C₄⋊C₄	—	2	2	no	—
26	C₄⋊C₄	—	2	2	no	—
27	C₄⋊C₄	D₄	1	1	yes	characteristic
28	C₄²	—	2	1	no	—
29	C₄²	C₂³	1	1	yes	characteristic
30	Q₈	—	2	1	no	—
31	Q₈	—	4	2	no	—
32	Q₈	—	4	2	no	—
33	Q₈	D₈	1	2	yes	—
34	C₂×C₄	—	2	1	no	—
35	C₂×C₄	—	2	1	no	—
36	C₂×C₄	—	4	2	no	—
37	C₂×C₄	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄	#41	1	1	yes	characteristic
40	C₈	—	2	1	no	—
41	C₈	—	4	2	no	—
42	C₈	—	4	1	no	—
43	C₈	Q₁₆	1	2	yes	complemented
44	C₂²	#121	1	1	yes	characteristic
45	C₄	—	2	2	no	—
46	C₄	—	2	1	no	—
47	C₄	—	4	1	no	—
48	C₄	—	8	2	no	—
49	C₄	C₂×D₈	1	1	yes	characteristic
50	C₄	C₂×Q₁₆	1	1	yes	characteristic
51	C₄	#135	1	1	yes	characteristic
52	C₄	#137	1	1	yes	characteristic
53	C₂	#462	1	1	yes	characteristic
54	C₂	#466	1	1	yes	characteristic
55	C₂	#473	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²	#121
commutator, Frattini, cosocle	C₄²	C₂³
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.

N	H ≅ G/N	type
43. C₈	Q₁₆	semidirect

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