Subgroups — #2180

Order 128.

71 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#510	C₂	1	1	yes	maximal, characteristic
2	#444	C₂	1	1	yes	maximal, characteristic
3	#441	C₂	1	1	yes	maximal, characteristic, complemented
4	#365	C₂	1	1	yes	maximal, characteristic, complemented
5	#362	C₂	1	1	yes	maximal, characteristic, complemented
6	#360	C₂	1	1	yes	maximal, characteristic
7	#347	C₂	1	1	yes	maximal, characteristic, complemented
8	#135	C₂²	1	1	yes	characteristic
9	C₂×Q₁₆	C₂²	1	1	yes	characteristic
10	#131	—	2	1	no	—
11	C₄×Q₈	—	2	1	no	—
12	Q₃₂	C₄	1	2	yes	complemented
13	SD₃₂	C₄	1	2	yes	complemented
14	M₃₂	C₂²	1	1	yes	characteristic
15	M₃₂	C₂²	1	1	yes	characteristic
16	#108	C₂²	1	1	yes	characteristic
17	#106	C₂²	1	1	yes	characteristic
18	#104	—	2	1	no	—
19	#103	—	2	1	no	—
20	#97	C₂²	1	1	yes	characteristic
21	#41	—	2	1	no	—
22	C₂×Q₈	—	2	1	no	—
23	Q₁₆	—	2	1	no	—
24	Q₁₆	C₂×C₄	1	2	yes	—
25	Q₁₆	C₂×C₄	1	1	yes	characteristic
26	SD₁₆	—	2	1	no	—
27	D₈	C₂×C₄	1	1	yes	characteristic
28	M₁₆	—	2	1	no	—
29	M₁₆	—	4	1	no	—
30	C₂×C₈	—	4	1	no	—
31	C₂×C₈	D₄	1	1	yes	characteristic
32	C₂×C₈	C₂³	1	1	yes	characteristic
33	C₄⋊C₄	—	2	1	no	—
34	C₄⋊C₄	—	4	1	no	—
35	C₄²	—	4	1	no	—
36	C₄²	D₄	1	1	yes	characteristic
37	C₁₆	—	2	1	no	—
38	C₁₆	C₂×C₄	1	2	yes	—
39	Q₈	—	2	1	no	—
40	Q₈	—	2	2	no	—
41	Q₈	—	4	1	no	—
42	D₄	—	2	1	no	—
43	D₄	—	4	1	no	—
44	C₂×C₄	—	2	1	no	—
45	C₂×C₄	—	4	1	no	—
46	C₂×C₄	—	4	1	no	—
47	C₂×C₄	—	4	1	no	—
48	C₂×C₄	C₂×D₄	1	1	yes	characteristic
49	C₈	—	2	1	no	—
50	C₈	—	4	1	no	—
51	C₈	C₂²×C₄	1	1	yes	characteristic
52	C₈	#41	1	1	yes	characteristic
53	C₂²	—	4	1	no	—
54	C₂²	AGL(1,ℤ/8)	1	1	yes	characteristic
55	C₄	—	4	1	no	—
56	C₄	—	4	1	no	—
57	C₄	—	4	2	no	—
58	C₄	—	8	1	no	—
59	C₄	C₄×D₄	1	1	yes	characteristic
60	C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
61	C₂	—	2	1	no	—
62	C₂	—	8	1	no	—
63	C₂	#442	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#442
commutator	C₈	C₂²×C₄
Frattini, cosocle	C₂×C₈	C₂³
prime radical	#135	C₂²
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #97 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #441	C₂	semidirect
4. #365	C₂	semidirect
5. #362	C₂	semidirect
7. #347	C₂	semidirect
12. Q₃₂	C₄	semidirect
13. SD₃₂	C₄	semidirect

Maximal subgroups

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

subgroup	index
#510	2
#444	2
#441	2
#365	2
#362	2
#360	2
#347	2