Subgroups — #1382

Order 128.

66 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#108	C₂	1	1	yes	maximal, characteristic
3	#427	C₂	1	1	yes	maximal, characteristic
4	C₂×M₁₆	—	2	1	no	—
5	C₂²×C₈	C₂²	1	1	yes	characteristic
6	#117	—	2	1	no	—
7	C₂×C₁₆	—	2	1	no	—
8	C₂×C₁₆	—	2	1	no	—
9	C₂×C₁₆	C₄	1	2	yes	complemented
10	#108	—	2	1	no	—
11	#108	C₄	1	2	yes	complemented
12	#107	C₄	1	2	yes	—
13	#106	—	2	1	no	—
14	C₂²×C₄	D₄	1	1	yes	characteristic
15	M₁₆	—	2	2	no	—
16	M₁₆	—	4	1	no	—
17	C₂×C₈	—	4	1	no	—
18	C₂×C₈	C₂×C₄	1	1	yes	characteristic
19	C₂×C₈	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈	D₄	1	1	yes	characteristic
22	C₂×C₈	D₄	1	1	yes	characteristic
23	C₂×C₈	Q₈	1	1	yes	characteristic
24	C₄⋊C₄	—	2	2	no	—
25	C₂²⋊C₄	—	4	1	no	—
26	C₄²	—	4	1	no	—
27	C₁₆	—	2	2	no	—
28	C₂³	SD₁₆	1	1	yes	characteristic
29	C₂×C₄	—	4	2	no	—
30	C₂×C₄	C₂²⋊C₄	1	1	yes	characteristic
31	C₂×C₄	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×C₄	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₄	D₈	1	1	yes	characteristic
34	C₂×C₄	SD₁₆	1	1	yes	characteristic
35	C₂×C₄	Q₁₆	1	1	yes	characteristic
36	C₈	—	4	2	no	—
37	C₈	C₄²	1	1	yes	characteristic
38	C₈	C₂²⋊C₄	1	1	yes	characteristic
39	C₈	C₄⋊C₄	1	1	yes	characteristic
40	C₈	C₄⋊C₄	1	1	yes	characteristic
41	C₂²	—	2	2	no	—
42	C₂²	#102	1	1	yes	characteristic
43	C₂²	#103	1	1	yes	characteristic
44	C₂²	#106	1	1	yes	characteristic
45	C₄	—	8	2	no	—
46	C₄	#95	1	1	yes	characteristic
47	C₄	#102	1	1	yes	characteristic
48	C₄	#103	1	1	yes	characteristic
49	C₄	#107	1	1	yes	characteristic
50	C₂	—	2	1	no	—
51	C₂	—	2	1	no	—
52	C₂	#340	1	1	yes	characteristic
53	C₂	#359	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₈
commutator	C₈	C₄²
Frattini, cosocle	C₂²×C₈	C₂²
socle	C₂²	#102
prime radical	C₂³	SD₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
9. C₂×C₁₆	C₄	semidirect
11. #108	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₁₆	2
C₂×#108	2
#427	2