Subgroups — #1589

Order 128.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×D₈	C₂	1	1	yes	maximal, characteristic
2	#436	C₂	1	2	yes	maximal, complemented
3	#335	C₂	1	1	yes	maximal, characteristic, complemented
4	D₄⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₈²	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₈	C₄	1	1	yes	characteristic
7	C₂×M₁₆	—	2	2	no	—
8	C₄×D₄	C₂²	1	2	yes	—
9	#107	C₄	1	1	yes	characteristic
10	C₄⋊C₈	C₂²	1	2	yes	—
11	#102	C₄	1	2	yes	—
12	C₂²⋊C₈	—	2	2	no	—
13	C₄×C₈	C₂²	1	1	yes	characteristic
14	C₄×C₈	C₂²	1	1	yes	characteristic
15	C₄×C₈	C₂²	1	1	yes	characteristic
16	C₂×D₄	C₂×C₄	1	2	yes	—
17	C₂²×C₄	—	2	2	no	—
18	D₈	—	2	2	no	—
19	M₁₆	—	4	4	no	—
20	C₂×C₈	—	2	1	no	—
21	C₂×C₈	—	2	2	no	—
22	C₂×C₈	C₂×C₄	1	1	yes	characteristic
23	C₂×C₈	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈	D₄	1	1	yes	characteristic
25	C₂×C₈	D₄	1	1	yes	characteristic
26	C₄⋊C₄	C₂×C₄	1	2	yes	—
27	C₂²⋊C₄	—	2	2	no	—
28	C₄²	C₂³	1	1	yes	characteristic
29	C₂³	—	2	2	no	—
30	D₄	—	2	2	no	—
31	D₄	—	2	2	no	—
32	C₂×C₄	—	2	2	no	—
33	C₂×C₄	—	4	2	no	—
34	C₂×C₄	—	4	2	no	—
35	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄	#41	1	1	yes	characteristic
38	C₈	—	2	2	no	—
39	C₈	—	2	1	no	—
40	C₈	—	2	1	no	—
41	C₈	—	4	2	no	—
42	C₈	M₁₆	1	2	yes	complemented
43	C₈	D₈	1	2	yes	complemented
44	C₂²	—	4	2	no	—
45	C₂²	—	4	2	no	—
46	C₂²	—	4	2	no	—
47	C₂²	C₄×D₄	1	1	yes	characteristic
48	C₄	—	2	1	no	—
49	C₄	—	4	2	no	—
50	C₄	C₂×M₁₆	1	1	yes	characteristic
51	C₄	#131	1	1	yes	characteristic
52	C₄	C₂×D₈	1	1	yes	characteristic
53	C₄	#135	1	1	yes	characteristic
54	C₂	—	8	2	no	—
55	C₂	#436	1	1	yes	characteristic
56	C₂	C₄×D₈	1	1	yes	characteristic
57	C₂	#443	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄
commutator	C₂×C₄	C₂²×C₄
Frattini, cosocle	C₄²	C₂³
socle	C₂²	C₄×D₄
prime radical	C₂×D₈	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #436	C₂	semidirect
3. #335	C₂	semidirect
4. D₄⋊C₈	C₂	semidirect
5. C₈²	C₂	semidirect
42. C₈	M₁₆	semidirect
43. C₈	D₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×D₈	2
#436	2
#335	2
D₄⋊C₈	2
C₈²	2