Subgroups — #1984

Order 128.

146 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	C₂×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₈	C₂	1	1	yes	maximal, characteristic
4	#336	C₂	1	4	yes	maximal
5	C₂²×Q₈	C₄	1	1	yes	characteristic
6	C₂²×C₈	C₂²	1	2	yes	—
7	C₂²×C₈	C₂²	1	2	yes	—
8	C₄×Q₈	—	2	2	no	—
9	C₄×Q₈	—	2	2	no	—
10	C₂×C₄⋊C₄	C₄	1	1	yes	characteristic
11	C₂×C₄⋊C₄	C₄	1	2	yes	—
12	C₂×C₄²	C₂²	1	2	yes	—
13	C₂×C₄²	C₂²	1	1	yes	characteristic
14	C₄⋊C₈	—	2	2	no	—
15	C₄×C₈	—	2	2	no	—
16	C₂×Q₈	—	2	2	no	—
17	C₂×Q₈	—	2	2	no	—
18	C₂×Q₈	C₈	1	4	yes	complemented
19	C₂²×C₄	C₂×C₄	1	2	yes	—
20	C₂²×C₄	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄	C₂×C₄	1	2	yes	—
22	C₂²×C₄	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄	C₂³	1	1	yes	characteristic
24	C₂×C₈	—	2	2	no	—
25	C₂×C₈	—	2	2	no	—
26	C₂×C₈	—	2	2	no	—
27	C₂×C₈	—	2	4	no	—
28	C₂×C₈	Q₈	1	4	yes	complemented
29	C₄⋊C₄	—	2	4	no	—
30	C₄⋊C₄	—	2	2	no	—
31	C₄²	—	2	2	no	—
32	C₄²	—	2	2	no	—
33	C₄²	D₄	1	2	yes	—
34	C₄²	D₄	1	2	yes	—
35	C₂³	C₂²×C₄	1	1	yes	characteristic
36	Q₈	—	2	8	no	—
37	C₂×C₄	—	2	2	no	—
38	C₂×C₄	—	2	2	no	—
39	C₂×C₄	—	2	2	no	—
40	C₂×C₄	—	2	2	no	—
41	C₂×C₄	—	2	4	no	—
42	C₂×C₄	C₂²⋊C₄	1	2	yes	—
43	C₂×C₄	C₂²⋊C₄	1	2	yes	—
44	C₂×C₄	C₂×C₈	1	4	yes	—
45	C₂×C₄	C₂×C₈	1	2	yes	—
46	C₂×C₄	M₁₆	1	2	yes	—
47	C₂×C₄	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄	C₂×Q₈	1	2	yes	—
50	C₂×C₄	#41	1	2	yes	—
51	C₈	—	2	4	no	—
52	C₈	—	4	2	no	—
53	C₂²	C₂×C₂²⋊C₄	1	1	yes	characteristic
54	C₂²	C₄×Q₈	1	2	yes	—
55	C₂²	C₂²×C₈	1	1	yes	characteristic
56	C₂²	C₂×M₁₆	1	1	yes	characteristic
57	C₂²	#131	1	2	yes	—
58	C₄	—	2	4	no	—
59	C₄	—	2	2	no	—
60	C₄	—	4	2	no	—
61	C₄	C₂²⋊C₈	1	4	yes	—
62	C₄	C₂²⋊Q₈	1	2	yes	—
63	C₄	#124	1	1	yes	characteristic
64	C₄	C₄⋊Q₈	1	1	yes	characteristic
65	C₂	#391	1	1	yes	characteristic
66	C₂	C₂×C₂²⋊C₈	1	1	yes	characteristic
67	C₂	#408	1	1	yes	characteristic
68	C₂	C₈×Q₈	1	2	yes	—
69	C₂	#446	1	2	yes	—

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂³
commutator	C₂²	C₂²×C₈
socle, prime radical	C₂³	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
18. C₂×Q₈	C₈	semidirect
28. C₂×C₈	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
C₂×C₄⋊C₈	2
C₂×C₄×C₈	2
#336	2