Subgroups — #1947

Order 128.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#107	C₂	1	2	yes	maximal
2	C₂×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#384	C₂	1	1	yes	maximal, characteristic
4	C₄×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
5	#340	C₂	1	2	yes	maximal
6	C₂²×C₈	C₂²	1	2	yes	—
7	C₂×C₄⋊C₄	—	2	1	no	—
8	C₂×C₄⋊C₄	—	2	1	no	—
9	C₂×C₄⋊C₄	C₂²	1	2	yes	—
10	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
12	C₂×C₄²	—	2	1	no	—
13	C₂×C₄²	C₂²	1	1	yes	characteristic
14	#107	—	2	4	no	—
15	C₄⋊C₈	C₄	1	4	yes	complemented
16	#95	—	2	1	no	—
17	C₂²×C₄	—	2	1	no	—
18	C₂²×C₄	—	2	1	no	—
19	C₂²×C₄	—	2	2	no	—
20	C₂²×C₄	D₄	1	1	yes	characteristic
21	C₂²×C₄	D₄	1	1	yes	characteristic
22	C₂²×C₄	C₂³	1	1	yes	characteristic
23	C₂×C₈	—	2	2	no	—
24	C₂×C₈	—	2	2	no	—
25	C₂×C₈	C₂×C₄	1	4	yes	—
26	C₄⋊C₄	—	2	1	no	—
27	C₄⋊C₄	—	2	1	no	—
28	C₄⋊C₄	—	2	2	no	—
29	C₄⋊C₄	—	2	2	no	—
30	C₄⋊C₄	—	4	2	no	—
31	C₄⋊C₄	—	4	2	no	—
32	C₄⋊C₄	D₄	1	2	yes	—
33	C₄⋊C₄	Q₈	1	2	yes	—
34	C₄²	—	4	2	no	—
35	C₄²	C₂×C₄	1	2	yes	—
36	C₄²	Q₈	1	2	yes	—
37	C₂³	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄	—	2	1	no	—
39	C₂×C₄	—	2	1	no	—
40	C₂×C₄	—	2	2	no	—
41	C₂×C₄	—	2	2	no	—
42	C₂×C₄	—	4	1	no	—
43	C₂×C₄	—	4	1	no	—
44	C₂×C₄	—	4	1	no	—
45	C₂×C₄	—	4	1	no	—
46	C₂×C₄	—	4	2	no	—
47	C₂×C₄	—	4	2	no	—
48	C₂×C₄	—	4	2	no	—
49	C₂×C₄	C₄⋊C₄	1	2	yes	—
50	C₂×C₄	C₄⋊C₄	1	2	yes	—
51	C₂×C₄	D₈	1	2	yes	—
52	C₂×C₄	Q₁₆	1	2	yes	—
53	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₄	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₄	#41	1	1	yes	characteristic
58	C₂×C₄	#41	1	1	yes	characteristic
59	C₈	—	2	4	no	—
60	C₂²	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂²	#121	1	1	yes	characteristic
62	C₂²	C₂²⋊Q₈	1	1	yes	characteristic
63	C₂²	C₂×D₈	1	1	yes	characteristic
64	C₂²	C₂×Q₁₆	1	1	yes	characteristic
65	C₂²	AGL(1,ℤ/8)	1	1	yes	characteristic
66	C₂²	#137	1	1	yes	characteristic
67	C₄	—	2	2	no	—
68	C₄	—	4	2	no	—
69	C₄	—	4	2	no	—
70	C₄	—	8	2	no	—
71	C₄	#107	1	4	yes	complemented
72	C₄	C₄×D₄	1	1	yes	characteristic
73	C₄	C₄×Q₈	1	1	yes	characteristic
74	C₄	#125	1	1	yes	characteristic
75	C₄	C₄⋊Q₈	1	1	yes	characteristic
76	C₂	#389	1	1	yes	characteristic
77	C₂	C₂×#107	1	1	yes	characteristic
78	C₂	#428	1	1	yes	characteristic
79	C₂	#459	1	1	yes	characteristic
80	C₂	#462	1	1	yes	characteristic
81	C₂	#474	1	1	yes	characteristic
82	C₂	#477	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
15. C₄⋊C₈	C₄	semidirect
71. C₄	#107	semidirect

Maximal subgroups

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

subgroup	index
C₂×#107	2
C₂×C₄⋊C₈	2
#384	2
C₄×C₄⋊C₄	2
#340	2