Subgroups — #1584

Order 128.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#446	C₂	1	1	yes	maximal, characteristic
2	C₈×Q₈	C₂	1	1	yes	maximal, characteristic
3	C₄×Q₁₆	C₂	1	1	yes	maximal, characteristic
4	#335	C₂	1	1	yes	maximal, characteristic
5	Q₈⋊C₈	C₂	1	1	yes	maximal, characteristic
6	Q₈⋊C₈	C₂	1	1	yes	maximal, characteristic
7	#322	C₂	1	1	yes	maximal, characteristic
8	C₂×Q₁₆	C₄	1	1	yes	characteristic
9	C₄×Q₈	C₂²	1	1	yes	characteristic
10	C₄×Q₈	C₂²	1	1	yes	characteristic
11	#107	C₄	1	1	yes	characteristic
12	C₄⋊C₈	—	2	1	no	—
13	C₄⋊C₈	—	2	1	no	—
14	C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₄⋊C₈	C₂²	1	1	yes	characteristic
16	#103	C₄	1	1	yes	characteristic
17	#103	C₄	1	1	yes	characteristic
18	#97	—	2	1	no	—
19	C₄×C₈	—	2	1	no	—
20	C₄×C₈	C₂²	1	1	yes	characteristic
21	C₄×C₈	C₂²	1	1	yes	characteristic
22	C₄×C₈	C₂²	1	1	yes	characteristic
23	C₂×Q₈	C₂×C₄	1	1	yes	characteristic
24	C₂×Q₈	C₂×C₄	1	1	yes	characteristic
25	Q₁₆	—	2	2	no	—
26	C₂×C₈	—	2	1	no	—
27	C₂×C₈	—	2	1	no	—
28	C₂×C₈	—	2	1	no	—
29	C₂×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂×C₈	C₂×C₄	1	1	yes	characteristic
31	C₂×C₈	D₄	1	1	yes	characteristic
32	C₂×C₈	D₄	1	1	yes	characteristic
33	C₄⋊C₄	—	2	1	no	—
34	C₄⋊C₄	—	2	1	no	—
35	C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
36	C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
37	C₄²	—	2	1	no	—
38	C₄²	—	2	1	no	—
39	C₄²	C₂³	1	1	yes	characteristic
40	Q₈	—	2	1	no	—
41	Q₈	—	2	1	no	—
42	Q₈	—	2	1	no	—
43	Q₈	M₁₆	1	2	yes	—
44	C₂×C₄	—	2	1	no	—
45	C₂×C₄	—	2	1	no	—
46	C₂×C₄	—	2	1	no	—
47	C₂×C₄	—	2	1	no	—
48	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄	#41	1	1	yes	characteristic
51	C₈	—	2	1	no	—
52	C₈	—	2	1	no	—
53	C₈	—	2	1	no	—
54	C₈	—	4	1	no	—
55	C₈	—	4	1	no	—
56	C₈	—	4	1	no	—
57	C₈	Q₁₆	1	2	yes	complemented
58	C₂²	C₄×D₄	1	1	yes	characteristic
59	C₄	—	2	2	no	—
60	C₄	—	2	1	no	—
61	C₄	—	4	1	no	—
62	C₄	—	4	1	no	—
63	C₄	—	4	1	no	—
64	C₄	C₂×M₁₆	1	1	yes	characteristic
65	C₄	#131	1	1	yes	characteristic
66	C₄	C₂×Q₁₆	1	1	yes	characteristic
67	C₄	#135	1	1	yes	characteristic
68	C₂	#435	1	1	yes	characteristic
69	C₂	C₄×Q₁₆	1	1	yes	characteristic
70	C₂	#444	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
57. C₈	Q₁₆	semidirect

Maximal subgroups

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

subgroup	index
#446	2
C₈×Q₈	2
C₄×Q₁₆	2
#335	2
Q₈⋊C₈	2
#322	2