Subgroups — #1567

Order 128.

92 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#423	C₂	1	2	yes	maximal
2	C₂×C₄×C₈	C₂	1	1	yes	maximal, characteristic
3	#335	C₂	1	4	yes	maximal, complemented
4	C₂×M₁₆	—	2	2	no	—
5	C₂²×C₈	C₄	1	2	yes	—
6	C₂×C₄²	C₂²	1	1	yes	characteristic
7	C₄⋊C₈	—	2	2	no	—
8	C₄⋊C₈	C₂²	1	4	yes	—
9	C₄×C₈	C₄	1	1	yes	characteristic
10	C₄×C₈	C₄	1	1	yes	characteristic
11	C₄×C₈	C₂²	1	2	yes	—
12	C₂²×C₄	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₄	D₄	1	1	yes	characteristic
14	C₂²×C₄	Q₈	1	1	yes	characteristic
15	M₁₆	—	4	4	no	—
16	C₂×C₈	—	2	4	no	—
17	C₂×C₈	—	2	4	no	—
18	C₂×C₈	C₂×C₄	1	2	yes	—
19	C₂×C₈	C₂×C₄	1	2	yes	—
20	C₄²	C₂×C₄	1	1	yes	characteristic
21	C₄²	D₄	1	1	yes	characteristic
22	C₄²	Q₈	1	1	yes	characteristic
23	C₄²	C₂³	1	1	yes	characteristic
24	C₂³	C₄⋊C₄	1	1	yes	characteristic
25	C₂×C₄	—	2	2	no	—
26	C₂×C₄	—	2	2	no	—
27	C₂×C₄	C₄⋊C₄	1	1	yes	characteristic
28	C₂×C₄	C₄⋊C₄	1	1	yes	characteristic
29	C₂×C₄	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₄	D₈	1	2	yes	—
31	C₂×C₄	Q₁₆	1	2	yes	—
32	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₄	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄	C₂×Q₈	1	1	yes	characteristic
35	C₈	—	2	2	no	—
36	C₈	—	4	4	no	—
37	C₈	M₁₆	1	4	yes	complemented
38	C₂²	—	2	1	no	—
39	C₂²	—	2	1	no	—
40	C₂²	#107	1	2	yes	—
41	C₂²	C₂×C₄⋊C₄	1	1	yes	characteristic
42	C₄	—	2	1	no	—
43	C₄	—	2	1	no	—
44	C₄	#107	1	2	yes	—
45	C₄	#108	1	2	yes	—
46	C₄	C₂×M₁₆	1	2	yes	—
47	C₄	C₂×D₈	1	1	yes	characteristic
48	C₄	C₂×Q₁₆	1	1	yes	characteristic
49	C₂	—	2	2	no	—
50	C₂	#423	1	1	yes	characteristic
51	C₂	C₂×#107	1	1	yes	characteristic
52	C₂	C₂×#108	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄
commutator	C₂×C₄	C₂²×C₄
Frattini, cosocle	C₄²	C₂³
socle	C₂²	C₂×C₄⋊C₄
prime radical	C₂³	C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #335	C₂	semidirect
37. C₈	M₁₆	semidirect

Maximal subgroups

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

subgroup	index
#423	2
C₂×C₄×C₈	2
#335	2