Subgroups — #1279

Order 128.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₈	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₈	—	2	1	no	—
5	C₂²×C₈	—	2	1	no	—
6	C₂×C₄⋊C₄	C₄	1	2	yes	—
7	C₂×C₄²	—	2	1	no	—
8	C₂×C₄²	C₂²	1	1	yes	characteristic
9	C₄⋊C₈	—	2	1	no	—
10	C₄⋊C₈	C₄	1	2	yes	complemented
11	C₄×C₈	—	2	1	no	—
12	C₄×C₈	C₄	1	2	yes	complemented
13	#95	—	2	1	no	—
14	C₂²×C₄	—	2	2	no	—
15	C₂²×C₄	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₄	D₄	1	1	yes	characteristic
17	C₂²×C₄	D₄	1	1	yes	characteristic
18	C₂×C₈	—	2	4	no	—
19	C₂×C₈	—	2	2	no	—
20	C₂×C₈	—	2	2	no	—
21	C₂×C₈	—	4	1	no	—
22	C₂×C₈	—	4	1	no	—
23	C₄⋊C₄	—	2	2	no	—
24	C₄⋊C₄	C₈	1	4	yes	complemented
25	C₄²	—	4	1	no	—
26	C₄²	—	4	1	no	—
27	C₄²	C₂×C₄	1	1	yes	characteristic
28	C₄²	C₂×C₄	1	1	yes	characteristic
29	C₄²	D₄	1	1	yes	characteristic
30	C₄²	Q₈	1	1	yes	characteristic
31	C₂³	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×C₄	—	2	2	no	—
33	C₂×C₄	—	2	4	no	—
34	C₂×C₄	—	4	2	no	—
35	C₂×C₄	—	4	2	no	—
36	C₂×C₄	C₄²	1	1	yes	characteristic
37	C₂×C₄	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₄	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×C₄	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₄	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₄	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₄	C₂×C₈	1	2	yes	—
43	C₂×C₄	M₁₆	1	2	yes	—
44	C₂×C₄	D₈	1	1	yes	characteristic
45	C₂×C₄	SD₁₆	1	1	yes	characteristic
46	C₂×C₄	SD₁₆	1	1	yes	characteristic
47	C₂×C₄	Q₁₆	1	1	yes	characteristic
48	C₈	—	2	4	no	—
49	C₈	—	4	2	no	—
50	C₂²	#95	1	1	yes	characteristic
51	C₂²	C₂²⋊C₈	1	2	yes	—
52	C₂²	#102	1	1	yes	characteristic
53	C₂²	#103	1	1	yes	characteristic
54	C₂²	#104	1	2	yes	—
55	C₄	—	2	1	no	—
56	C₄	—	2	1	no	—
57	C₄	—	4	4	no	—
58	C₄	C₄×C₈	1	1	yes	characteristic
59	C₄	#97	1	1	yes	characteristic
60	C₄	#102	1	1	yes	characteristic
61	C₄	#103	1	1	yes	characteristic
62	C₄	C₄⋊C₈	1	2	yes	—
63	C₄	#106	1	1	yes	characteristic
64	C₄	#107	1	1	yes	characteristic
65	C₂	D₄⋊C₈	1	2	yes	—
66	C₂	Q₈⋊C₈	1	2	yes	—
67	C₂	#336	1	1	yes	characteristic
68	C₂	#339	1	1	yes	characteristic
69	C₂	#340	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄
commutator	C₄	C₄×C₈
Frattini, cosocle	C₂×C₄²	C₂²
socle, prime radical	C₂³	C₂²⋊C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
10. C₄⋊C₈	C₄	semidirect
12. C₄×C₈	C₄	semidirect
24. C₄⋊C₄	C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊C₈	2
C₂×C₄×C₈	2
C₄×C₄⋊C₄	2