Subgroups — #2351

Order 128.

276 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Q₈	C₂	1	2	yes	maximal
2	C₂×C₄×Q₈	C₂	1	1	yes	maximal, characteristic
3	#389	C₂	1	6	yes	maximal
4	#382	C₂	1	3	yes	maximal
5	C₄×C₄⋊C₄	C₂	1	3	yes	maximal
6	C₂²×Q₈	C₂²	1	1	yes	characteristic
7	C₄×Q₈	—	2	8	no	—
8	C₄×Q₈	C₄	1	8	yes	complemented
9	C₂×C₄⋊C₄	C₂²	1	6	yes	—
10	C₂×C₄⋊C₄	C₂²	1	6	yes	—
11	C₂×C₄⋊C₄	C₂²	1	3	yes	—
12	C₂×C₄²	C₂²	1	6	yes	—
13	C₂×C₄²	C₂²	1	3	yes	—
14	#95	C₂²	1	1	yes	characteristic
15	#95	C₂²	1	3	yes	—
16	#95	C₂²	1	6	yes	—
17	C₂×Q₈	C₂×C₄	1	4	yes	—
18	C₂×Q₈	D₄	1	4	yes	—
19	C₂×Q₈	Q₈	1	4	yes	—
20	C₂²×C₄	C₂³	1	6	yes	—
21	C₂²×C₄	C₂³	1	1	yes	characteristic
22	C₂²×C₄	C₂³	1	3	yes	—
23	C₂²×C₄	C₂³	1	2	yes	—
24	C₂²×C₄	C₂³	1	3	yes	—
25	C₄⋊C₄	—	2	12	no	—
26	C₄⋊C₄	—	2	12	no	—
27	C₄⋊C₄	C₂×C₄	1	12	yes	—
28	C₄²	—	2	12	no	—
29	C₄²	C₂×C₄	1	12	yes	—
30	C₂³	C₂⁴	1	1	yes	characteristic
31	Q₈	C₄⋊C₄	1	16	yes	complemented
32	C₂×C₄	—	2	2	no	—
33	C₂×C₄	—	2	6	no	—
34	C₂×C₄	—	2	2	no	—
35	C₂×C₄	—	2	2	no	—
36	C₂×C₄	—	2	6	no	—
37	C₂×C₄	—	2	6	no	—
38	C₂×C₄	—	2	6	no	—
39	C₂×C₄	C₂²×C₄	1	2	yes	—
40	C₂×C₄	C₂²×C₄	1	6	yes	—
41	C₂×C₄	C₂²×C₄	1	6	yes	—
42	C₂×C₄	C₂×D₄	1	6	yes	—
43	C₂×C₄	C₂×Q₈	1	6	yes	—
44	C₂×C₄	#41	1	4	yes	—
45	C₂²	C₂³×C₄	1	1	yes	characteristic
46	C₂²	C₂²×D₄	1	1	yes	characteristic
47	C₂²	C₂²×Q₈	1	1	yes	characteristic
48	C₂²	C₂×#41	1	2	yes	—
49	C₂²	2₋¹⁺⁴	1	2	yes	—
50	C₄	—	2	6	no	—
51	C₄	—	2	2	no	—
52	C₄	—	2	4	no	—
53	C₄	—	4	6	no	—
54	C₄	C₂×C₄⋊C₄	1	12	yes	—
55	C₂	C₂²×C₄⋊C₄	1	1	yes	characteristic
56	C₂	C₄×#41	1	1	yes	characteristic
57	C₂	#519	1	1	yes	characteristic
58	C₂	#548	1	2	yes	—
59	C₂	#557	1	2	yes	—

	subgroup	G/H
center, Frattini, socle, prime radical, cosocle	C₂³	C₂⁴
commutator	C₂²	C₂³×C₄
LCS	G ▷ C₂² ▷ 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
8. C₄×Q₈	C₄	semidirect
31. Q₈	C₄⋊C₄	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
#389	2
#382	2
C₄×C₄⋊C₄	2