Subgroups — #2187

Order 128.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₈	C₂	1	1	yes	maximal, characteristic
2	C₂×SD₃₂	C₂	1	2	yes	maximal, complemented
3	#467	C₂	1	1	yes	maximal, characteristic, complemented
4	#357	C₂	1	2	yes	maximal, complemented
5	C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₄	—	2	1	no	—
7	C₂×Q₁₆	C₂²	1	1	yes	characteristic, complemented
8	C₂×D₈	—	2	4	no	—
9	C₂×D₈	—	2	1	no	—
10	C₂×D₈	C₂²	1	2	yes	—
11	C₂²×C₈	C₂²	1	1	yes	characteristic
12	C₂²⋊Q₈	—	2	1	no	—
13	SD₃₂	—	2	4	no	—
14	C₂×C₁₆	C₂²	1	2	yes	complemented
15	#107	C₂²	1	1	yes	characteristic, complemented
16	#103	—	2	1	no	—
17	C₂⁴	—	4	1	no	—
18	C₂×Q₈	—	2	1	no	—
19	C₂×D₄	—	2	4	no	—
20	C₂×D₄	—	2	2	no	—
21	C₂×D₄	—	4	2	no	—
22	C₂×D₄	—	4	1	no	—
23	C₂²×C₄	D₄	1	1	yes	characteristic
24	Q₁₆	—	2	2	no	—
25	D₈	—	2	4	no	—
26	D₈	—	2	2	no	—
27	D₈	D₄	1	4	yes	—
28	C₂×C₈	—	2	2	no	—
29	C₂×C₈	D₄	1	1	yes	characteristic
30	C₂×C₈	C₂³	1	1	yes	characteristic
31	C₄⋊C₄	—	2	1	no	—
32	C₄⋊C₄	—	4	1	no	—
33	C₂²⋊C₄	—	4	1	no	—
34	C₁₆	—	2	2	no	—
35	C₂³	—	4	4	no	—
36	C₂³	—	4	2	no	—
37	C₂³	—	8	2	no	—
38	C₂³	—	8	2	no	—
39	C₂³	D₈	1	1	yes	characteristic
40	Q₈	—	4	1	no	—
41	Q₈	—	4	1	no	—
42	D₄	—	2	4	no	—
43	D₄	—	4	4	no	—
44	D₄	—	4	2	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	D₈	1	1	yes	characteristic
50	C₂×C₄	C₂×D₄	1	1	yes	characteristic
51	C₈	—	2	1	no	—
52	C₈	C₂×D₄	1	2	yes	—
53	C₂²	—	2	1	no	—
54	C₂²	—	2	1	no	—
55	C₂²	—	4	4	no	—
56	C₂²	—	8	8	no	—
57	C₂²	—	8	2	no	—
58	C₂²	—	8	2	no	—
59	C₂²	SD₃₂	1	2	yes	complemented
60	C₂²	C₂×D₈	1	1	yes	characteristic
61	C₄	—	2	1	no	—
62	C₄	—	8	1	no	—
63	C₄	—	8	1	no	—
64	C₄	#120	1	1	yes	characteristic
65	C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
66	C₂	—	2	2	no	—
67	C₂	—	8	4	no	—
68	C₂	#447	1	1	yes	characteristic
69	C₂	C₂×SD₃₂	1	1	yes	characteristic
70	C₂	#509	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³
prime radical	C₂²×D₈	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₂×SD₃₂	C₂	semidirect
3. #467	C₂	semidirect
4. #357	C₂	semidirect
5. C₂²⋊C₁₆	C₂	semidirect
7. C₂×Q₁₆	C₂²	semidirect
14. C₂×C₁₆	C₂²	semidirect
15. #107	C₂²	semidirect
59. C₂²	SD₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₂²×D₈	2
C₂×SD₃₂	2
#467	2
#357	2
C₂²⋊C₁₆	2