Subgroups — #1673

Order 128.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#500	C₂	1	1	yes	maximal, characteristic, complemented
2	#461	C₂	1	2	yes	maximal
3	C₈×D₄	C₂	1	1	yes	maximal, characteristic
4	#335	C₂	1	1	yes	maximal, characteristic, complemented
5	#332	C₂	1	2	yes	maximal, complemented
6	C₂×SD₁₆	—	2	2	no	—
7	C₂²×C₈	—	2	1	no	—
8	C₄⋊Q₈	C₂²	1	2	yes	—
9	C₄×D₄	C₂²	1	1	yes	characteristic
10	#107	—	2	1	no	—
11	#107	—	2	1	no	—
12	C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₄⋊C₈	C₂²	1	2	yes	—
14	#103	—	2	2	no	—
15	C₂²⋊C₈	—	2	1	no	—
16	C₄×C₈	C₂²	1	1	yes	characteristic
17	C₂×Q₈	—	2	2	no	—
18	C₂×D₄	D₄	1	1	yes	characteristic
19	C₂²×C₄	—	2	1	no	—
20	SD₁₆	—	4	4	no	—
21	C₂×C₈	—	2	1	no	—
22	C₂×C₈	—	2	2	no	—
23	C₂×C₈	—	4	2	no	—
24	C₂×C₈	D₄	1	1	yes	characteristic
25	C₂×C₈	D₄	1	1	yes	characteristic
26	C₄⋊C₄	—	2	2	no	—
27	C₄⋊C₄	—	2	2	no	—
28	C₄⋊C₄	D₄	1	1	yes	characteristic
29	C₂²⋊C₄	—	2	1	no	—
30	C₄²	C₂³	1	1	yes	characteristic
31	C₂³	—	2	1	no	—
32	Q₈	—	4	2	no	—
33	Q₈	—	4	2	no	—
34	D₄	—	2	1	no	—
35	D₄	D₈	1	2	yes	—
36	C₂×C₄	—	2	1	no	—
37	C₂×C₄	—	4	1	no	—
38	C₂×C₄	—	4	1	no	—
39	C₂×C₄	—	4	2	no	—
40	C₂×C₄	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄	#41	1	1	yes	characteristic
43	C₈	—	2	1	no	—
44	C₈	—	4	2	no	—
45	C₈	—	4	1	no	—
46	C₈	SD₁₆	1	2	yes	complemented
47	C₂²	—	2	2	no	—
48	C₂²	—	4	1	no	—
49	C₂²	—	4	1	no	—
50	C₂²	#121	1	1	yes	characteristic
51	C₄	—	2	1	no	—
52	C₄	—	4	1	no	—
53	C₄	—	8	2	no	—
54	C₄	C₂×D₈	1	1	yes	characteristic
55	C₄	C₂×SD₁₆	1	1	yes	characteristic
56	C₄	#135	1	1	yes	characteristic
57	C₄	#137	1	1	yes	characteristic
58	C₂	—	4	2	no	—
59	C₂	#461	1	1	yes	characteristic
60	C₂	#466	1	1	yes	characteristic
61	C₂	#473	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#121
commutator, Frattini, cosocle	C₄²	C₂³
prime radical	C₂×D₄	D₄
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
1. #500	C₂	semidirect
4. #335	C₂	semidirect
5. #332	C₂	semidirect
46. C₈	SD₁₆	semidirect

Maximal subgroups

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

subgroup	index
#500	2
#461	2
C₈×D₄	2
#335	2
#332	2