Subgroups — #1286

Order 128.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#431	C₂	1	1	yes	maximal, characteristic
2	#423	C₂	1	1	yes	maximal, characteristic
3	#382	C₂	1	1	yes	maximal, characteristic
4	C₂×M₁₆	—	2	1	no	—
5	C₂×C₄⋊C₄	—	2	1	no	—
6	C₂×C₄²	C₂²	1	1	yes	characteristic
7	C₄⋊C₈	—	2	1	no	—
8	C₄⋊C₈	—	2	1	no	—
9	C₄⋊C₈	C₄	1	2	yes	complemented
10	C₂²⋊C₈	—	2	1	no	—
11	C₄×C₈	C₄	1	2	yes	complemented
12	#95	—	2	1	no	—
13	#95	C₄	1	2	yes	—
14	C₂²×C₄	—	2	2	no	—
15	C₂²×C₄	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₄	D₄	1	1	yes	characteristic
17	C₂²×C₄	D₄	1	1	yes	characteristic
18	M₁₆	—	4	1	no	—
19	M₁₆	—	4	1	no	—
20	C₂×C₈	—	2	2	no	—
21	C₂×C₈	—	2	2	no	—
22	C₄⋊C₄	—	4	1	no	—
23	C₄⋊C₄	—	4	1	no	—
24	C₄²	C₂×C₄	1	1	yes	characteristic
25	C₄²	C₂×C₄	1	1	yes	characteristic
26	C₄²	D₄	1	1	yes	characteristic
27	C₄²	Q₈	1	1	yes	characteristic
28	C₂³	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×C₄	—	4	1	no	—
30	C₂×C₄	—	4	1	no	—
31	C₂×C₄	—	4	2	no	—
32	C₂×C₄	—	4	2	no	—
33	C₂×C₄	—	4	2	no	—
34	C₂×C₄	C₄²	1	1	yes	characteristic
35	C₂×C₄	C₂²⋊C₄	1	1	yes	characteristic
36	C₂×C₄	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₄	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₄	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₄	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₄	D₈	1	1	yes	characteristic
41	C₂×C₄	SD₁₆	1	1	yes	characteristic
42	C₂×C₄	SD₁₆	1	1	yes	characteristic
43	C₂×C₄	Q₁₆	1	1	yes	characteristic
44	C₈	—	4	2	no	—
45	C₈	—	4	2	no	—
46	C₂²	—	2	1	no	—
47	C₂²	—	2	1	no	—
48	C₂²	#95	1	1	yes	characteristic
49	C₂²	#102	1	1	yes	characteristic
50	C₂²	#103	1	1	yes	characteristic
51	C₄	—	2	1	no	—
52	C₄	—	2	1	no	—
53	C₄	—	2	1	no	—
54	C₄	—	2	1	no	—
55	C₄	—	8	2	no	—
56	C₄	#102	1	1	yes	characteristic
57	C₄	#103	1	1	yes	characteristic
58	C₄	#106	1	1	yes	characteristic
59	C₄	#107	1	1	yes	characteristic
60	C₂	—	2	1	no	—
61	C₂	—	2	1	no	—
62	C₂	H(3,ℤ/4)	1	1	yes	characteristic
63	C₂	#340	1	1	yes	characteristic
64	C₂	#344	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#95
commutator	C₂×C₄	C₄²
Frattini, cosocle	C₂×C₄²	C₂²
prime radical	C₂³	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
9. C₄⋊C₈	C₄	semidirect
11. C₄×C₈	C₄	semidirect

Maximal subgroups

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

subgroup	index
#431	2
#423	2
#382	2