Subgroups — #2213

Order 128.

85 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#509	C₂	1	2	yes	maximal, complemented
2	#493	C₂	1	1	yes	maximal, characteristic, complemented
3	#443	C₂	1	1	yes	maximal, characteristic, complemented
4	#364	C₂	1	1	yes	maximal, characteristic, complemented
5	#361	C₂	1	2	yes	maximal, complemented
6	#135	C₂²	1	1	yes	characteristic
7	C₂×D₈	—	2	1	no	—
8	C₂×D₈	C₂²	1	2	yes	—
9	#131	—	2	1	no	—
10	#127	—	2	1	no	—
11	SD₃₂	—	2	2	no	—
12	D₁₆	—	2	2	no	—
13	M₃₂	C₂²	1	2	yes	—
14	#108	C₂²	1	1	yes	characteristic
15	#104	—	2	1	no	—
16	C₄×C₈	C₂²	1	1	yes	characteristic
17	#41	—	2	1	no	—
18	C₂×D₄	—	2	2	no	—
19	C₂×D₄	—	4	2	no	—
20	Q₁₆	D₄	1	1	yes	characteristic, complemented
21	SD₁₆	—	2	1	no	—
22	D₈	—	2	2	no	—
23	D₈	—	2	2	no	—
24	D₈	—	4	1	no	—
25	D₈	—	4	1	no	—
26	D₈	D₄	1	1	yes	characteristic, complemented
27	M₁₆	—	2	1	no	—
28	M₁₆	—	4	1	no	—
29	C₂×C₈	—	4	1	no	—
30	C₂×C₈	D₄	1	1	yes	characteristic
31	C₂×C₈	C₂³	1	1	yes	characteristic
32	C₄²	D₄	1	1	yes	characteristic
33	C₁₆	—	2	2	no	—
34	C₂³	—	4	2	no	—
35	Q₈	—	2	1	no	—
36	D₄	—	2	1	no	—
37	D₄	—	4	1	no	—
38	D₄	—	4	2	no	—
39	D₄	—	4	2	no	—
40	D₄	—	8	4	no	—
41	C₂×C₄	—	2	1	no	—
42	C₂×C₄	—	4	1	no	—
43	C₂×C₄	C₂×D₄	1	1	yes	characteristic
44	C₈	—	4	1	no	—
45	C₈	D₈	1	1	yes	characteristic
46	C₈	D₈	1	1	yes	characteristic
47	C₈	C₂×D₄	1	1	yes	characteristic
48	C₈	#41	1	1	yes	characteristic
49	C₂²	—	4	1	no	—
50	C₂²	—	8	4	no	—
51	C₂²	—	8	2	no	—
52	C₂²	AGL(1,ℤ/8)	1	1	yes	characteristic
53	C₄	—	2	2	no	—
54	C₄	—	4	1	no	—
55	C₄	#121	1	1	yes	characteristic
56	C₄	C₂×D₈	1	1	yes	characteristic
57	C₂	—	2	1	no	—
58	C₂	—	8	1	no	—
59	C₂	—	16	2	no	—
60	C₂	#459	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#459
commutator, Frattini, cosocle	C₂×C₈	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×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
1. #509	C₂	semidirect
2. #493	C₂	semidirect
3. #443	C₂	semidirect
4. #364	C₂	semidirect
5. #361	C₂	semidirect
20. Q₁₆	D₄	semidirect
26. D₈	D₄	semidirect

Maximal subgroups

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

subgroup	index
#509	2
#493	2
#443	2
#364	2
#361	2