Subgroups — #1288

Order 128.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#432	C₂	1	1	yes	maximal, characteristic
2	C₄×M₁₆	C₂	1	1	yes	maximal, characteristic
3	#383	C₂	1	1	yes	maximal, characteristic
4	C₂×M₁₆	—	2	1	no	—
5	C₂×C₄²	C₂²	1	1	yes	characteristic
6	C₄⋊C₈	—	2	1	no	—
7	C₂²⋊C₈	—	2	1	no	—
8	#97	—	2	1	no	—
9	#97	C₄	1	2	yes	complemented
10	C₄×C₈	C₄	1	2	yes	complemented
11	#95	—	2	1	no	—
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	—	2	4	no	—
19	C₂×C₈	—	2	2	no	—
20	C₂×C₈	—	2	2	no	—
21	C₄²	C₂×C₄	1	1	yes	characteristic
22	C₄²	C₂×C₄	1	1	yes	characteristic
23	C₄²	D₄	1	1	yes	characteristic
24	C₄²	Q₈	1	1	yes	characteristic
25	C₂³	C₂²⋊C₄	1	1	yes	characteristic
26	C₂×C₄	—	2	2	no	—
27	C₂×C₄	—	4	1	no	—
28	C₂×C₄	—	4	1	no	—
29	C₂×C₄	—	4	2	no	—
30	C₂×C₄	—	4	2	no	—
31	C₂×C₄	—	4	2	no	—
32	C₂×C₄	C₄²	1	1	yes	characteristic
33	C₂×C₄	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₄	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×C₄	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₄	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₄	C₄⋊C₄	1	1	yes	characteristic
38	C₈	—	2	4	no	—
39	C₈	—	4	2	no	—
40	C₂²	—	2	1	no	—
41	C₂²	—	2	1	no	—
42	C₂²	#95	1	1	yes	characteristic
43	C₂²	#104	1	2	yes	—
44	C₄	—	2	1	no	—
45	C₄	—	2	1	no	—
46	C₄	—	2	1	no	—
47	C₄	—	2	1	no	—
48	C₄	—	2	1	no	—
49	C₄	—	2	1	no	—
50	C₄	—	8	2	no	—
51	C₂	—	2	2	no	—
52	C₂	H(3,ℤ/4)	1	1	yes	characteristic
53	C₂	#339	1	1	yes	characteristic
54	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. #97	C₄	semidirect
10. 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
#432	2
C₄×M₁₆	2
#383	2