Subgroups — #3383

Order 128.

162 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#557	C₂	1	2	yes	maximal
2	#500	C₂	1	1	yes	maximal, characteristic
3	#499	C₂	1	2	yes	maximal
4	#479	C₂	1	4	yes	maximal
5	#475	C₂	1	2	yes	maximal
6	C₈×Q₈	C₂	1	1	yes	maximal, characteristic
7	C₄×Q₁₆	C₂	1	2	yes	maximal
8	C₄×Q₁₆	C₂	1	1	yes	maximal, characteristic
9	C₂×Q₁₆	C₂²	1	1	yes	characteristic
10	C₄⋊Q₈	—	2	2	no	—
11	C₄⋊Q₈	C₂²	1	2	yes	—
12	#125	—	2	4	no	—
13	#125	C₂²	1	4	yes	—
14	C₄×Q₈	—	2	2	no	—
15	C₄×Q₈	C₂²	1	2	yes	—
16	C₄×Q₈	C₂²	1	4	yes	—
17	C₄×Q₈	C₂²	1	1	yes	characteristic
18	#107	C₂²	1	1	yes	characteristic
19	#107	C₂²	1	2	yes	—
20	#107	C₂²	1	2	yes	—
21	#106	C₂²	1	4	yes	—
22	C₄⋊C₈	C₂²	1	1	yes	characteristic
23	C₄⋊C₈	C₂²	1	2	yes	—
24	#103	C₂²	1	4	yes	—
25	#103	C₂²	1	2	yes	—
26	C₄×C₈	C₂²	1	2	yes	—
27	C₄×C₈	C₂²	1	1	yes	characteristic
28	C₂×Q₈	—	2	2	no	—
29	C₂×Q₈	D₄	1	1	yes	characteristic
30	C₂×Q₈	C₂³	1	2	yes	—
31	Q₁₆	Q₈	1	4	yes	complemented
32	C₂×C₈	C₂³	1	2	yes	—
33	C₂×C₈	C₂³	1	1	yes	characteristic
34	C₂×C₈	C₂³	1	1	yes	characteristic
35	C₄⋊C₄	—	2	4	no	—
36	C₄⋊C₄	—	2	2	no	—
37	C₄⋊C₄	—	2	4	no	—
38	C₄⋊C₄	—	2	2	no	—
39	C₄⋊C₄	—	2	4	no	—
40	C₄⋊C₄	D₄	1	1	yes	characteristic
41	C₄⋊C₄	D₄	1	2	yes	—
42	C₄⋊C₄	C₂³	1	2	yes	—
43	C₄⋊C₄	C₂³	1	4	yes	—
44	C₄²	—	2	4	no	—
45	C₄²	—	2	2	no	—
46	C₄²	C₂³	1	2	yes	—
47	C₄²	C₂³	1	1	yes	characteristic
48	Q₈	—	2	2	no	—
49	Q₈	—	2	2	no	—
50	Q₈	—	4	4	no	—
51	Q₈	C₂×Q₈	1	4	yes	—
52	C₂×C₄	—	2	2	no	—
53	C₂×C₄	—	2	2	no	—
54	C₂×C₄	—	2	4	no	—
55	C₂×C₄	C₂×D₄	1	2	yes	—
56	C₂×C₄	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄	C₂×D₄	1	2	yes	—
58	C₂×C₄	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄	C₂⁴	1	1	yes	characteristic
60	C₈	—	2	2	no	—
61	C₈	—	2	1	no	—
62	C₈	C₂×Q₈	1	2	yes	—
63	C₂²	C₂²×D₄	1	1	yes	characteristic
64	C₄	—	2	4	no	—
65	C₄	—	2	2	no	—
66	C₄	—	2	2	no	—
67	C₄	—	2	1	no	—
68	C₄	—	4	4	no	—
69	C₄	—	4	2	no	—
70	C₄	#135	1	2	yes	—
71	C₄	C₂²×Q₈	1	1	yes	characteristic
72	C₄	2₋¹⁺⁴	1	1	yes	characteristic
73	C₂	D₄×Q₈	1	1	yes	characteristic
74	C₂	C₂×#135	1	1	yes	characteristic
75	C₂	#578	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
31. Q₁₆	Q₈	semidirect

Maximal subgroups

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

subgroup	index
#557	2
#500	2
#499	2
#479	2
#475	2
C₈×Q₈	2
C₄×Q₁₆	2