Subgroups — #1714

Order 128.

99 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#500	C₂	1	1	yes	maximal, characteristic, complemented
2	#498	C₂	1	1	yes	maximal, characteristic, complemented
3	#492	C₂	1	1	yes	maximal, characteristic
4	#492	C₂	1	1	yes	maximal, characteristic
5	#487	C₂	1	1	yes	maximal, characteristic, complemented
6	#486	C₂	1	1	yes	maximal, characteristic
7	#322	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×SD₁₆	—	2	2	no	—
9	C₂×SD₁₆	—	2	2	no	—
10	C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₄⋊Q₈	C₂²	1	1	yes	characteristic
12	C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	#127	C₂²	1	1	yes	characteristic
14	#107	—	2	2	no	—
15	#106	—	2	2	no	—
16	#103	—	2	2	no	—
17	#102	—	2	2	no	—
18	C₄×C₈	C₂²	1	1	yes	characteristic
19	C₄×C₈	C₂²	1	1	yes	characteristic
20	C₄×C₈	C₂²	1	1	yes	characteristic
21	C₂×Q₈	—	2	1	no	—
22	C₂×Q₈	—	2	1	no	—
23	C₂×Q₈	—	2	1	no	—
24	C₂×D₄	—	2	1	no	—
25	C₂×D₄	—	2	1	no	—
26	C₂×D₄	—	2	1	no	—
27	SD₁₆	—	4	4	no	—
28	SD₁₆	—	4	4	no	—
29	C₂×C₈	D₄	1	2	yes	—
30	C₂×C₈	D₄	1	2	yes	—
31	C₂×C₈	D₄	1	2	yes	—
32	C₄⋊C₄	—	2	1	no	—
33	C₄⋊C₄	—	2	1	no	—
34	C₄⋊C₄	—	2	1	no	—
35	C₄⋊C₄	—	2	1	no	—
36	C₄⋊C₄	—	2	1	no	—
37	C₄⋊C₄	—	2	1	no	—
38	C₄²	C₂³	1	1	yes	characteristic
39	C₂³	—	4	1	no	—
40	Q₈	—	4	2	no	—
41	Q₈	—	4	2	no	—
42	Q₈	—	4	2	no	—
43	D₄	—	4	2	no	—
44	D₄	—	4	2	no	—
45	D₄	—	4	2	no	—
46	C₂×C₄	—	4	1	no	—
47	C₂×C₄	—	4	1	no	—
48	C₂×C₄	—	4	1	no	—
49	C₂×C₄	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄	C₂×D₄	1	1	yes	characteristic
52	C₈	—	2	2	no	—
53	C₈	—	2	2	no	—
54	C₈	SD₁₆	1	4	yes	complemented
55	C₂²	—	8	1	no	—
56	C₂²	—	8	1	no	—
57	C₂²	—	8	1	no	—
58	C₂²	#127	1	1	yes	characteristic
59	C₄	—	8	1	no	—
60	C₄	—	8	1	no	—
61	C₄	—	8	1	no	—
62	C₄	C₂×SD₁₆	1	2	yes	—
63	C₄	AGL(1,ℤ/8)	1	2	yes	—
64	C₄	#137	1	2	yes	—
65	C₂	—	16	1	no	—
66	C₂	#492	1	1	yes	characteristic
67	C₂	#496	1	1	yes	characteristic
68	C₂	#497	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#127
commutator, Frattini, cosocle	C₄²	C₂³
prime radical	#127	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #500	C₂	semidirect
2. #498	C₂	semidirect
5. #487	C₂	semidirect
7. #322	C₂	semidirect
54. C₈	SD₁₆	semidirect

Maximal subgroups

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

subgroup	index
#500	2
#498	2
#492	2
#487	2
#486	2
#322	2