Subgroups — #2984

Order 128.

182 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Q₈	C₂	1	1	yes	maximal, characteristic
2	#446	C₂	1	6	yes	maximal, complemented
3	C₈×Q₈	C₂	1	2	yes	maximal, complemented
4	#432	C₂	1	3	yes	maximal
5	#431	C₂	1	3	yes	maximal
6	C₂²×Q₈	C₄	1	1	yes	characteristic
7	C₄×Q₈	C₄	1	3	yes	—
8	C₄×Q₈	C₄	1	1	yes	characteristic
9	C₄×Q₈	C₂²	1	1	yes	characteristic
10	C₄×Q₈	C₂²	1	3	yes	—
11	C₂×C₄⋊C₄	C₄	1	3	yes	—
12	C₂×C₄²	C₂²	1	3	yes	—
13	C₄⋊C₈	C₂²	1	6	yes	—
14	C₄⋊C₈	C₂²	1	6	yes	—
15	C₂²⋊C₈	C₂²	1	1	yes	characteristic
16	C₂²⋊C₈	C₂²	1	3	yes	—
17	#97	C₂²	1	6	yes	—
18	C₄×C₈	C₂²	1	6	yes	—
19	C₂×Q₈	—	2	4	no	—
20	C₂×Q₈	C₂×C₄	1	3	yes	—
21	C₂×Q₈	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄	C₂×C₄	1	3	yes	—
23	C₂²×C₄	C₂×C₄	1	3	yes	—
24	C₂²×C₄	C₂³	1	1	yes	characteristic
25	C₂×C₈	C₂³	1	2	yes	—
26	C₂×C₈	C₂³	1	6	yes	—
27	C₄⋊C₄	C₂×C₄	1	6	yes	—
28	C₄⋊C₄	C₂×C₄	1	3	yes	—
29	C₄⋊C₄	C₂×C₄	1	3	yes	—
30	C₄²	C₂×C₄	1	3	yes	—
31	C₄²	C₂×C₄	1	3	yes	—
32	C₄²	C₂³	1	3	yes	—
33	C₄²	C₂³	1	3	yes	—
34	C₂³	C₂²×C₄	1	1	yes	characteristic
35	Q₈	—	2	6	no	—
36	Q₈	M₁₆	1	4	yes	—
37	C₂×C₄	—	2	6	no	—
38	C₂×C₄	—	2	6	no	—
39	C₂×C₄	—	2	2	no	—
40	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄	C₂²×C₄	1	3	yes	—
42	C₂×C₄	C₂²×C₄	1	3	yes	—
43	C₂×C₄	C₂²×C₄	1	3	yes	—
44	C₂×C₄	C₂²×C₄	1	3	yes	—
45	C₂×C₄	C₂⁴	1	1	yes	characteristic
46	C₈	—	2	2	no	—
47	C₈	—	2	6	no	—
48	C₂²	—	2	1	no	—
49	C₂²	—	2	1	no	—
50	C₂²	#131	1	2	yes	—
51	C₂²	C₂³×C₄	1	1	yes	characteristic
52	C₄	—	2	1	no	—
53	C₄	—	2	3	no	—
54	C₄	—	2	3	no	—
55	C₄	—	2	3	no	—
56	C₄	C₂×M₁₆	1	6	yes	—
57	C₄	2₋¹⁺⁴	1	2	yes	—
58	C₂	—	2	2	no	—
59	C₂	#519	1	1	yes	characteristic
60	C₂	C₂²×M₁₆	1	1	yes	characteristic
61	C₂	C₂×#131	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, cosocle	C₂×C₄	C₂⁴
commutator, socle	C₂²	C₂³×C₄
prime radical	C₂³	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #446	C₂	semidirect
3. C₈×Q₈	C₂	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
#446	2
C₈×Q₈	2
#432	2
#431	2