Subgroups — #1668

Order 128.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#493	C₂	1	1	yes	maximal, characteristic
2	#460	C₂	1	2	yes	maximal, complemented
3	C₈×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#335	C₂	1	1	yes	maximal, characteristic, complemented
5	#331	C₂	1	2	yes	maximal, complemented
6	C₂×SD₁₆	—	2	2	no	—
7	C₂×D₈	—	2	1	no	—
8	C₂×D₈	—	2	1	no	—
9	#127	C₂²	1	2	yes	—
10	C₄×Q₈	C₂²	1	1	yes	characteristic
11	C₄⋊C₈	—	2	1	no	—
12	C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₄⋊C₈	C₂²	1	2	yes	—
14	#102	—	2	2	no	—
15	C₄×C₈	—	2	1	no	—
16	C₄×C₈	C₂²	1	1	yes	characteristic
17	C₂×Q₈	D₄	1	1	yes	characteristic
18	C₂×D₄	—	2	2	no	—
19	C₂×D₄	—	2	2	no	—
20	C₂×D₄	—	2	2	no	—
21	SD₁₆	—	4	4	no	—
22	D₈	—	4	2	no	—
23	D₈	—	4	2	no	—
24	C₂×C₈	—	2	1	no	—
25	C₂×C₈	—	2	2	no	—
26	C₂×C₈	D₄	1	1	yes	characteristic
27	C₂×C₈	D₄	1	1	yes	characteristic
28	C₄⋊C₄	—	2	1	no	—
29	C₄⋊C₄	D₄	1	1	yes	characteristic
30	C₄²	—	2	1	no	—
31	C₄²	C₂³	1	1	yes	characteristic
32	C₂³	—	4	2	no	—
33	Q₈	—	2	1	no	—
34	Q₈	D₈	1	2	yes	complemented
35	D₄	—	4	2	no	—
36	D₄	—	4	2	no	—
37	D₄	—	4	2	no	—
38	D₄	—	4	2	no	—
39	D₄	—	8	2	no	—
40	C₂×C₄	—	2	1	no	—
41	C₂×C₄	—	2	1	no	—
42	C₂×C₄	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄	#41	1	1	yes	characteristic
45	C₈	—	2	1	no	—
46	C₈	—	4	2	no	—
47	C₈	—	4	1	no	—
48	C₈	SD₁₆	1	2	yes	complemented
49	C₂²	—	8	2	no	—
50	C₂²	—	8	2	no	—
51	C₂²	—	8	2	no	—
52	C₂²	#121	1	1	yes	characteristic
53	C₄	—	2	2	no	—
54	C₄	—	2	1	no	—
55	C₄	—	4	1	no	—
56	C₄	C₂×D₈	1	1	yes	characteristic
57	C₄	C₂×SD₁₆	1	1	yes	characteristic
58	C₄	#135	1	1	yes	characteristic
59	C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
60	C₂	—	16	2	no	—
61	C₂	#460	1	1	yes	characteristic
62	C₂	#466	1	1	yes	characteristic
63	C₂	#472	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#121
commutator, Frattini, cosocle	C₄²	C₂³
prime radical	#493	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #460	C₂	semidirect
3. C₈×Q₈	C₂	semidirect
4. #335	C₂	semidirect
5. #331	C₂	semidirect
34. Q₈	D₈	semidirect
48. C₈	SD₁₆	semidirect

Maximal subgroups

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

subgroup	index
#493	2
#460	2
C₈×Q₈	2
#335	2
#331	2