Subgroups — #1692

Order 128.

73 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#498	C₂	1	1	yes	maximal, characteristic
2	#477	C₂	1	1	yes	maximal, characteristic
3	#462	C₂	1	1	yes	maximal, characteristic
4	#446	C₂	1	1	yes	maximal, characteristic
5	#335	C₂	1	1	yes	maximal, characteristic
6	#333	C₂	1	1	yes	maximal, characteristic
7	#332	C₂	1	1	yes	maximal, characteristic
8	C₂×Q₁₆	—	2	1	no	—
9	C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₄×Q₈	C₂²	1	1	yes	characteristic
12	#107	—	2	1	no	—
13	#106	—	2	1	no	—
14	#106	—	2	1	no	—
15	C₄⋊C₈	—	2	1	no	—
16	C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₄⋊C₈	C₂²	1	1	yes	characteristic
19	#103	—	2	1	no	—
20	#103	—	2	1	no	—
21	#97	—	2	1	no	—
22	C₄×C₈	C₂²	1	1	yes	characteristic
23	C₂×Q₈	—	2	1	no	—
24	C₂×Q₈	—	2	1	no	—
25	C₂×Q₈	D₄	1	1	yes	characteristic
26	Q₁₆	—	4	2	no	—
27	C₂×C₈	—	2	1	no	—
28	C₂×C₈	—	2	1	no	—
29	C₂×C₈	—	2	1	no	—
30	C₂×C₈	D₄	1	1	yes	characteristic
31	C₂×C₈	D₄	1	1	yes	characteristic
32	C₄⋊C₄	—	2	1	no	—
33	C₄⋊C₄	—	2	1	no	—
34	C₄⋊C₄	—	2	1	no	—
35	C₄⋊C₄	—	2	1	no	—
36	C₄⋊C₄	—	2	1	no	—
37	C₄⋊C₄	D₄	1	1	yes	characteristic
38	C₄²	—	2	1	no	—
39	C₄²	C₂³	1	1	yes	characteristic
40	Q₈	—	2	1	no	—
41	Q₈	—	2	1	no	—
42	Q₈	—	4	1	no	—
43	Q₈	—	4	1	no	—
44	Q₈	—	8	1	no	—
45	C₂×C₄	—	2	1	no	—
46	C₂×C₄	—	2	1	no	—
47	C₂×C₄	—	4	1	no	—
48	C₂×C₄	—	4	1	no	—
49	C₂×C₄	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄	#41	1	1	yes	characteristic
52	C₈	—	2	1	no	—
53	C₈	—	4	1	no	—
54	C₈	—	4	1	no	—
55	C₈	—	4	1	no	—
56	C₈	Q₁₆	1	2	yes	complemented
57	C₂²	#121	1	1	yes	characteristic
58	C₄	—	2	1	no	—
59	C₄	—	4	1	no	—
60	C₄	—	4	1	no	—
61	C₄	—	8	1	no	—
62	C₄	—	8	1	no	—
63	C₄	C₂×Q₁₆	1	1	yes	characteristic
64	C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
65	C₄	#137	1	1	yes	characteristic
66	C₄	#137	1	1	yes	characteristic
67	C₂	#462	1	1	yes	characteristic
68	C₂	#468	1	1	yes	characteristic
69	C₂	#471	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²	#121
commutator, Frattini, cosocle	C₄²	C₂³
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
56. C₈	Q₁₆	semidirect

Maximal subgroups

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

subgroup	index
#498	2
#477	2
#462	2
#446	2
#335	2
#333	2
#332	2