Subgroups — #3249

Order 128.

174 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#533	C₂	1	2	yes	maximal
2	#477	C₂	1	4	yes	maximal, complemented
3	#467	C₂	1	4	yes	maximal
4	#462	C₂	1	4	yes	maximal, complemented
5	C₂×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₁₆	C₂²	1	4	yes	—
7	C₂²×C₈	C₂²	1	2	yes	—
8	C₄⋊Q₈	C₂²	1	2	yes	—
9	C₄⋊Q₈	C₂²	1	2	yes	—
10	#125	—	2	2	no	—
11	C₂²⋊Q₈	—	2	2	no	—
12	C₂²⋊Q₈	C₂²	1	4	yes	—
13	C₄×Q₈	—	2	2	no	—
14	C₄×Q₈	C₂²	1	4	yes	—
15	#117	—	2	2	no	—
16	C₂×C₄²	C₂²	1	1	yes	characteristic
17	#107	C₂²	1	4	yes	—
18	C₄⋊C₈	C₂²	1	4	yes	—
19	#103	C₂²	1	8	yes	—
20	C₂×Q₈	—	2	2	no	—
21	C₂×Q₈	C₂³	1	4	yes	—
22	C₂²×C₄	D₄	1	1	yes	characteristic
23	C₂²×C₄	D₄	1	1	yes	characteristic
24	C₂²×C₄	C₂³	1	1	yes	characteristic
25	Q₁₆	—	4	4	no	—
26	C₂×C₈	—	4	2	no	—
27	C₂×C₈	C₂³	1	4	yes	—
28	C₄⋊C₄	—	2	2	no	—
29	C₄⋊C₄	—	2	2	no	—
30	C₄⋊C₄	—	2	2	no	—
31	C₄⋊C₄	—	2	4	no	—
32	C₄⋊C₄	—	2	4	no	—
33	C₄⋊C₄	C₂³	1	4	yes	—
34	C₂²⋊C₄	—	2	4	no	—
35	C₄²	—	2	4	no	—
36	C₄²	D₄	1	2	yes	—
37	C₄²	C₂³	1	2	yes	—
38	C₂³	C₂×D₄	1	1	yes	characteristic
39	Q₈	—	2	4	no	—
40	Q₈	—	2	4	no	—
41	Q₈	—	4	4	no	—
42	C₂×C₄	—	2	1	no	—
43	C₂×C₄	—	2	1	no	—
44	C₂×C₄	—	2	4	no	—
45	C₂×C₄	—	2	4	no	—
46	C₂×C₄	—	4	1	no	—
47	C₂×C₄	Q₁₆	1	4	yes	complemented
48	C₂×C₄	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄	C₂×D₄	1	2	yes	—
50	C₂×C₄	C₂×D₄	1	2	yes	—
51	C₂×C₄	C₂⁴	1	1	yes	characteristic
52	C₈	—	2	4	no	—
53	C₂²	—	2	1	no	—
54	C₂²	—	2	1	no	—
55	C₂²	C₂×Q₁₆	1	2	yes	—
56	C₂²	C₂²×D₄	1	1	yes	characteristic
57	C₄	—	2	1	no	—
58	C₄	—	2	2	no	—
59	C₄	—	4	4	no	—
60	C₄	—	4	4	no	—
61	C₄	C₂×Q₁₆	1	4	yes	—
62	C₄	H(5,𝔽₂)	1	1	yes	characteristic
63	C₄	2₋¹⁺⁴	1	1	yes	characteristic
64	C₂	—	2	2	no	—
65	C₂	#537	1	1	yes	characteristic
66	C₂	C₂²×Q₁₆	1	1	yes	characteristic
67	C₂	#575	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴
prime radical	C₂³	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #477	C₂	semidirect
4. #462	C₂	semidirect
47. C₂×C₄	Q₁₆	semidirect

Maximal subgroups

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

subgroup	index
#533	2
#477	2
#467	2
#462	2
C₂×C₄⋊C₈	2