Subgroups — #1278

Order 128.

86 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	C₄×M₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₈	C₂	1	1	yes	maximal, characteristic
4	C₂×M₁₆	C₄	1	2	yes	—
5	C₂²×C₈	—	2	1	no	—
6	C₂×C₄²	C₂²	1	1	yes	characteristic
7	C₄⋊C₈	C₄	1	2	yes	—
8	C₂²⋊C₈	—	2	1	no	—
9	#97	—	2	1	no	—
10	C₄×C₈	—	2	1	no	—
11	C₄×C₈	—	2	1	no	—
12	C₄×C₈	—	2	1	no	—
13	C₄×C₈	C₄	1	2	yes	—
14	C₂²×C₄	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄	D₄	1	1	yes	characteristic
16	C₂²×C₄	Q₈	1	1	yes	characteristic
17	M₁₆	—	2	2	no	—
18	M₁₆	C₈	1	4	yes	complemented
19	C₂×C₈	—	2	4	no	—
20	C₂×C₈	—	2	2	no	—
21	C₂×C₈	—	2	2	no	—
22	C₂×C₈	—	2	2	no	—
23	C₄²	C₂×C₄	1	1	yes	characteristic
24	C₄²	C₂×C₄	1	1	yes	characteristic
25	C₄²	D₄	1	1	yes	characteristic
26	C₄²	D₄	1	1	yes	characteristic
27	C₂³	C₄⋊C₄	1	1	yes	characteristic
28	C₂×C₄	—	2	2	no	—
29	C₂×C₄	—	2	2	no	—
30	C₂×C₄	C₄²	1	1	yes	characteristic
31	C₂×C₄	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×C₄	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₄	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₄	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₄	C₂×C₈	1	2	yes	—
37	C₂×C₄	M₁₆	1	2	yes	—
38	C₈	—	2	4	no	—
39	C₈	—	2	4	no	—
40	C₈	—	4	2	no	—
41	C₂²	—	2	1	no	—
42	C₂²	—	2	1	no	—
43	C₂²	#95	1	1	yes	characteristic
44	C₂²	C₄⋊C₈	1	2	yes	—
45	C₄	—	2	1	no	—
46	C₄	—	2	1	no	—
47	C₄	C₄×C₈	1	1	yes	characteristic
48	C₄	#97	1	1	yes	characteristic
49	C₄	C₂²⋊C₈	1	2	yes	—
50	C₄	#104	1	2	yes	—
51	C₄	#108	1	2	yes	—
52	C₂	—	2	2	no	—
53	C₂	#336	1	1	yes	characteristic
54	C₂	#339	1	1	yes	characteristic
55	C₂	#341	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄
commutator	C₄	C₄×C₈
Frattini, cosocle	C₂×C₄²	C₂²
socle	C₂²	#95
prime radical	C₂³	C₄⋊C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ 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
18. M₁₆	C₈	semidirect

Maximal subgroups

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

subgroup	index
#431	2
C₄×M₁₆	2
C₂×C₄×C₈	2