Subgroups — #3363

Order 128.

168 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈²	C₂	1	1	yes	maximal, characteristic
2	#557	C₂	1	1	yes	maximal, characteristic
3	#487	C₂	1	3	yes	maximal
4	#477	C₂	1	3	yes	maximal
5	#462	C₂	1	3	yes	maximal
6	C₈×Q₈	C₂	1	1	yes	maximal, characteristic
7	C₄×Q₁₆	C₂	1	3	yes	maximal
8	C₂×Q₁₆	C₂²	1	3	yes	—
9	C₄⋊Q₈	—	2	3	no	—
10	C₄⋊Q₈	C₂²	1	3	yes	—
11	C₄⋊Q₈	C₂²	1	3	yes	—
12	#125	—	2	3	no	—
13	C₄×Q₈	—	2	1	no	—
14	C₄×Q₈	—	2	1	no	—
15	C₄×Q₈	C₂²	1	3	yes	—
16	C₄×Q₈	C₂²	1	3	yes	—
17	C₄×Q₈	C₂²	1	1	yes	characteristic
18	#107	C₂²	1	3	yes	—
19	C₄⋊C₈	C₂²	1	3	yes	—
20	#103	C₂²	1	1	yes	characteristic
21	#103	C₂²	1	3	yes	—
22	#103	C₂²	1	6	yes	—
23	C₄×C₈	C₂²	1	3	yes	—
24	C₂×Q₈	—	2	3	no	—
25	C₂×Q₈	D₄	1	1	yes	characteristic
26	C₂×Q₈	C₂³	1	3	yes	—
27	C₂×Q₈	C₂³	1	1	yes	characteristic
28	Q₁₆	—	2	6	no	—
29	C₂×C₈	C₂³	1	1	yes	characteristic
30	C₂×C₈	C₂³	1	3	yes	—
31	C₄⋊C₄	—	2	3	no	—
32	C₄⋊C₄	—	2	3	no	—
33	C₄⋊C₄	—	2	3	no	—
34	C₄⋊C₄	—	2	3	no	—
35	C₄⋊C₄	—	2	3	no	—
36	C₄⋊C₄	D₄	1	3	yes	—
37	C₄⋊C₄	C₂³	1	3	yes	—
38	C₄⋊C₄	C₂³	1	1	yes	characteristic
39	C₄²	—	2	3	no	—
40	C₄²	—	2	3	no	—
41	C₄²	C₂³	1	3	yes	—
42	Q₈	—	2	1	no	—
43	Q₈	—	2	3	no	—
44	Q₈	—	2	3	no	—
45	Q₈	—	8	3	no	—
46	Q₈	Q₁₆	1	4	yes	complemented
47	Q₈	#41	1	2	yes	—
48	C₂×C₄	—	2	1	no	—
49	C₂×C₄	—	2	1	no	—
50	C₂×C₄	—	2	3	no	—
51	C₂×C₄	—	2	3	no	—
52	C₂×C₄	C₂×D₄	1	3	yes	—
53	C₂×C₄	C₂×D₄	1	3	yes	—
54	C₂×C₄	C₂⁴	1	1	yes	characteristic
55	C₈	—	2	3	no	—
56	C₈	—	2	1	no	—
57	C₂²	C₂²×D₄	1	1	yes	characteristic
58	C₄	—	2	2	no	—
59	C₄	—	2	3	no	—
60	C₄	—	4	3	no	—
61	C₄	—	4	1	no	—
62	C₄	—	4	3	no	—
63	C₄	C₂×Q₁₆	1	6	yes	—
64	C₄	C₂×#41	1	1	yes	characteristic
65	C₄	2₋¹⁺⁴	1	1	yes	characteristic
66	C₂	#548	1	1	yes	characteristic
67	C₂	C₂²×Q₁₆	1	1	yes	characteristic
68	C₂	#577	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
46. Q₈	Q₁₆	semidirect

Maximal subgroups

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

subgroup	index
Q₈²	2
#557	2
#487	2
#477	2
#462	2
C₈×Q₈	2
C₄×Q₁₆	2