Subgroups — #1485

Order 128.

109 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#533	C₂	1	1	yes	maximal, characteristic
2	#431	C₂	1	1	yes	maximal, characteristic
3	C₄×M₁₆	C₂	1	1	yes	maximal, characteristic
4	Q₈⋊C₈	C₂	1	4	yes	maximal, complemented
5	C₂×M₁₆	—	2	1	no	—
6	C₄⋊Q₈	C₄	1	1	yes	characteristic
7	C₄⋊Q₈	C₄	1	1	yes	characteristic
8	#125	—	2	1	no	—
9	C₂²⋊Q₈	—	2	1	no	—
10	C₂²⋊Q₈	C₄	1	2	yes	—
11	C₄×Q₈	—	2	1	no	—
12	C₄×Q₈	C₂²	1	2	yes	—
13	#117	—	2	1	no	—
14	C₂×C₄²	C₂²	1	1	yes	characteristic
15	C₄⋊C₈	C₂²	1	2	yes	—
16	C₂²⋊C₈	—	2	1	no	—
17	#97	—	2	1	no	—
18	C₄×C₈	—	2	1	no	—
19	C₄×C₈	C₂²	1	2	yes	—
20	C₂×Q₈	—	2	1	no	—
21	C₂×Q₈	C₂×C₄	1	2	yes	—
22	C₂²×C₄	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄	D₄	1	1	yes	characteristic
24	C₂²×C₄	D₄	1	1	yes	characteristic
25	M₁₆	—	2	4	no	—
26	C₂×C₈	—	2	2	no	—
27	C₂×C₈	—	2	2	no	—
28	C₄⋊C₄	—	2	1	no	—
29	C₄⋊C₄	—	2	1	no	—
30	C₄⋊C₄	—	2	1	no	—
31	C₄⋊C₄	—	2	2	no	—
32	C₄⋊C₄	—	2	2	no	—
33	C₄⋊C₄	C₂×C₄	1	2	yes	—
34	C₂²⋊C₄	—	2	2	no	—
35	C₄²	—	2	2	no	—
36	C₄²	C₂×C₄	1	1	yes	characteristic
37	C₄²	D₄	1	1	yes	characteristic
38	C₄²	D₄	1	1	yes	characteristic
39	C₄²	C₂³	1	1	yes	characteristic
40	C₂³	C₂²⋊C₄	1	1	yes	characteristic
41	Q₈	—	2	4	no	—
42	Q₈	—	4	2	no	—
43	C₂×C₄	—	2	2	no	—
44	C₂×C₄	—	2	2	no	—
45	C₂×C₄	—	2	2	no	—
46	C₂×C₄	—	2	2	no	—
47	C₂×C₄	—	2	2	no	—
48	C₂×C₄	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₄	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₄	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄	C₂×D₄	1	1	yes	characteristic
54	C₈	—	2	4	no	—
55	C₈	—	4	2	no	—
56	C₂²	—	2	1	no	—
57	C₂²	—	2	1	no	—
58	C₂²	—	2	1	no	—
59	C₂²	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₄	—	2	1	no	—
61	C₄	—	2	1	no	—
62	C₄	—	2	1	no	—
63	C₄	—	4	2	no	—
64	C₄	—	4	2	no	—
65	C₄	#104	1	2	yes	—
66	C₄	#131	1	2	yes	—
67	C₄	#137	1	2	yes	—
68	C₂	—	4	1	no	—
69	C₂	#408	1	1	yes	characteristic
70	C₂	#419	1	1	yes	characteristic
71	C₂	C₂×#104	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
4. Q₈⋊C₈	C₂	semidirect

Maximal subgroups

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

subgroup	index
#533	2
#431	2
C₄×M₁₆	2
Q₈⋊C₈	2