Subgroups — #2551

Order 128.

278 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊Q₈	C₂	1	4	yes	maximal
2	#395	C₂	1	4	yes	maximal
3	#389	C₂	1	4	yes	maximal
4	#384	C₂	1	1	yes	maximal, characteristic
5	C₄×C₄⋊C₄	C₂	1	2	yes	maximal
6	C₂²×Q₈	C₂²	1	4	yes	—
7	C₄⋊Q₈	—	2	16	no	—
8	C₂×C₄⋊C₄	C₂²	1	4	yes	—
9	C₂×C₄⋊C₄	C₂²	1	8	yes	—
10	C₂×C₄⋊C₄	C₂²	1	2	yes	—
11	C₂×C₄⋊C₄	C₂²	1	2	yes	—
12	C₂×C₄⋊C₄	C₂²	1	2	yes	—
13	C₂×C₄²	C₂²	1	4	yes	—
14	C₂×C₄²	C₂²	1	1	yes	characteristic
15	#95	C₂²	1	4	yes	—
16	#95	C₂²	1	4	yes	—
17	C₂×Q₈	—	2	16	no	—
18	C₂×Q₈	—	4	4	no	—
19	C₂²×C₄	C₂³	1	4	yes	—
20	C₂²×C₄	C₂³	1	4	yes	—
21	C₂²×C₄	C₂³	1	1	yes	characteristic
22	C₂²×C₄	C₂³	1	4	yes	—
23	C₂²×C₄	C₂³	1	2	yes	—
24	C₄⋊C₄	—	2	16	no	—
25	C₄⋊C₄	—	2	8	no	—
26	C₄⋊C₄	—	4	2	no	—
27	C₄⋊C₄	D₄	1	8	yes	—
28	C₄⋊C₄	Q₈	1	8	yes	complemented
29	C₄²	—	2	8	no	—
30	C₄²	Q₈	1	4	yes	complemented
31	C₂³	C₂⁴	1	1	yes	characteristic
32	Q₈	—	4	16	no	—
33	C₂×C₄	—	2	4	no	—
34	C₂×C₄	—	2	4	no	—
35	C₂×C₄	—	2	1	no	—
36	C₂×C₄	—	2	4	no	—
37	C₂×C₄	—	2	1	no	—
38	C₂×C₄	—	2	4	no	—
39	C₂×C₄	—	2	8	no	—
40	C₂×C₄	—	2	4	no	—
41	C₂×C₄	C₂×D₄	1	8	yes	—
42	C₂×C₄	C₂×D₄	1	4	yes	—
43	C₂×C₄	C₂×Q₈	1	2	yes	—
44	C₂×C₄	C₂×Q₈	1	8	yes	—
45	C₂×C₄	C₂×Q₈	1	4	yes	—
46	C₂×C₄	C₂×Q₈	1	4	yes	—
47	C₂²	C₂²×D₄	1	2	yes	—
48	C₂²	C₂²×Q₈	1	1	yes	characteristic
49	C₂²	C₂²×Q₈	1	2	yes	—
50	C₂²	H(5,𝔽₂)	1	1	yes	characteristic
51	C₂²	2₋¹⁺⁴	1	1	yes	characteristic
52	C₄	—	2	2	no	—
53	C₄	—	2	8	no	—
54	C₄	—	2	8	no	—
55	C₄	—	4	4	no	—
56	C₄	C₄⋊Q₈	1	8	yes	complemented
57	C₂	C₂×C₄⋊Q₈	1	2	yes	—
58	C₂	#544	1	1	yes	characteristic
59	C₂	D₄²	1	1	yes	characteristic
60	C₂	D₄×Q₈	1	2	yes	—
61	C₂	Q₈²	1	1	yes	characteristic

	subgroup	G/H
center, commutator, Frattini, socle, prime radical, cosocle	C₂³	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

Complements

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

N	H ≅ G/N	type
28. C₄⋊C₄	Q₈	semidirect
30. C₄²	Q₈	semidirect
56. C₄	C₄⋊Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊Q₈	2
#395	2
#389	2
#384	2
C₄×C₄⋊C₄	2