Subgroups — #2140

Order 128.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₈	C₂	1	1	yes	maximal, characteristic
2	C₂×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#427	C₂	1	1	yes	maximal, characteristic, complemented
4	#357	C₂	1	4	yes	maximal, complemented
5	C₂²×D₄	—	2	1	no	—
6	C₂×D₈	—	2	1	no	—
7	C₂×D₈	—	2	2	no	—
8	C₂×D₈	C₄	1	4	yes	complemented
9	C₂×D₈	C₂²	1	2	yes	—
10	C₂²×C₈	C₂²	1	1	yes	characteristic
11	#117	—	2	1	no	—
12	M₃₂	—	2	2	no	—
13	C₂×C₁₆	C₂²	1	2	yes	complemented
14	#107	C₂²	1	2	yes	complemented
15	#106	—	2	1	no	—
16	C₂⁴	—	4	1	no	—
17	C₂×D₄	—	2	4	no	—
18	C₂×D₄	—	2	2	no	—
19	C₂×D₄	—	4	1	no	—
20	C₂×D₄	—	4	2	no	—
21	C₂²×C₄	D₄	1	1	yes	characteristic
22	D₈	—	2	4	no	—
23	D₈	—	2	2	no	—
24	D₈	C₂×C₄	1	4	yes	—
25	C₂×C₈	C₂×C₄	1	2	yes	—
26	C₂×C₈	D₄	1	1	yes	characteristic
27	C₂×C₈	D₄	1	2	yes	—
28	C₂×C₈	C₂³	1	1	yes	characteristic
29	C₄⋊C₄	—	2	2	no	—
30	C₂²⋊C₄	—	4	1	no	—
31	C₄²	—	4	1	no	—
32	C₁₆	—	2	2	no	—
33	C₂³	—	4	4	no	—
34	C₂³	—	4	2	no	—
35	C₂³	—	8	4	no	—
36	C₂³	D₈	1	1	yes	characteristic
37	D₄	—	2	4	no	—
38	D₄	—	4	4	no	—
39	D₄	—	4	2	no	—
40	C₂×C₄	—	4	2	no	—
41	C₂×C₄	C₂²⋊C₄	1	2	yes	—
42	C₂×C₄	D₈	1	1	yes	characteristic
43	C₂×C₄	SD₁₆	1	2	yes	—
44	C₂×C₄	C₂×D₄	1	1	yes	characteristic
45	C₈	C₂²⋊C₄	1	2	yes	—
46	C₈	C₂²×C₄	1	1	yes	characteristic
47	C₈	C₂×D₄	1	1	yes	characteristic
48	C₂²	—	2	2	no	—
49	C₂²	—	4	4	no	—
50	C₂²	—	8	8	no	—
51	C₂²	—	8	4	no	—
52	C₂²	#102	1	2	yes	—
53	C₂²	C₂×D₈	1	1	yes	characteristic
54	C₄	—	8	2	no	—
55	C₄	#102	1	2	yes	—
56	C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	C₄	C₂×SD₁₆	1	1	yes	characteristic
58	C₂	—	2	2	no	—
59	C₂	—	8	4	no	—
60	C₂	C₂×#102	1	1	yes	characteristic
61	C₂	#509	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₂×M₃₂	C₂	semidirect
3. #427	C₂	semidirect
4. #357	C₂	semidirect
8. C₂×D₈	C₄	semidirect
13. C₂×C₁₆	C₂²	semidirect
14. #107	C₂²	semidirect

Maximal subgroups

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

subgroup	index
C₂²×D₈	2
C₂×M₃₂	2
#427	2
#357	2