Subgroups — #1939

Order 128.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊C₈	C₂	1	2	yes	maximal
2	C₂×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₈	C₂	1	1	yes	maximal, characteristic
4	C₄×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
5	#336	C₂	1	2	yes	maximal
6	C₂²×C₈	C₂²	1	2	yes	—
7	C₂²×C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊C₄	C₄	1	1	yes	characteristic
10	C₂×C₄⋊C₄	C₄	1	1	yes	characteristic
11	C₂×C₄²	C₂²	1	2	yes	—
12	C₂×C₄²	C₂²	1	1	yes	characteristic
13	C₄⋊C₈	—	2	4	no	—
14	C₄⋊C₈	—	2	2	no	—
15	C₄×C₈	—	2	2	no	—
16	#95	C₄	1	2	yes	—
17	C₂²×C₄	C₂×C₄	1	2	yes	—
18	C₂²×C₄	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄	C₂×C₄	1	2	yes	—
20	C₂²×C₄	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄	C₂³	1	1	yes	characteristic
22	C₂×C₈	—	2	1	no	—
23	C₂×C₈	—	2	1	no	—
24	C₂×C₈	—	2	1	no	—
25	C₂×C₈	—	2	1	no	—
26	C₂×C₈	—	2	2	no	—
27	C₂×C₈	—	2	2	no	—
28	C₂×C₈	—	2	2	no	—
29	C₂×C₈	D₄	1	2	yes	—
30	C₂×C₈	Q₈	1	2	yes	—
31	C₄⋊C₄	—	2	2	no	—
32	C₄⋊C₄	C₈	1	4	yes	complemented
33	C₄²	—	2	2	no	—
34	C₄²	—	2	2	no	—
35	C₄²	D₄	1	2	yes	—
36	C₄²	Q₈	1	2	yes	—
37	C₂³	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄	—	2	2	no	—
39	C₂×C₄	—	2	1	no	—
40	C₂×C₄	—	2	1	no	—
41	C₂×C₄	—	2	2	no	—
42	C₂×C₄	—	2	2	no	—
43	C₂×C₄	—	2	2	no	—
44	C₂×C₄	—	2	2	no	—
45	C₂×C₄	C₄⋊C₄	1	2	yes	—
46	C₂×C₄	C₄⋊C₄	1	2	yes	—
47	C₂×C₄	C₂×C₈	1	4	yes	—
48	C₂×C₄	C₂×C₈	1	2	yes	—
49	C₂×C₄	M₁₆	1	2	yes	—
50	C₂×C₄	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₄	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₄	#41	1	1	yes	characteristic
55	C₂×C₄	#41	1	1	yes	characteristic
56	C₈	—	2	2	no	—
57	C₈	—	2	2	no	—
58	C₈	—	4	2	no	—
59	C₂²	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂²	C₄×D₄	1	1	yes	characteristic
61	C₂²	C₄×Q₈	1	1	yes	characteristic
62	C₂²	C₂²×C₈	1	1	yes	characteristic
63	C₂²	C₂×M₁₆	1	1	yes	characteristic
64	C₂²	#131	1	1	yes	characteristic
65	C₂²	#131	1	1	yes	characteristic
66	C₄	—	2	2	no	—
67	C₄	—	2	4	no	—
68	C₄	—	4	2	no	—
69	C₄	C₄⋊C₈	1	4	yes	complemented
70	C₄	#121	1	1	yes	characteristic
71	C₄	C₂²⋊Q₈	1	1	yes	characteristic
72	C₄	#125	1	1	yes	characteristic
73	C₄	C₄⋊Q₈	1	1	yes	characteristic
74	C₂	#389	1	1	yes	characteristic
75	C₂	C₂×C₄⋊C₈	1	1	yes	characteristic
76	C₂	#424	1	1	yes	characteristic
77	C₂	C₈×D₄	1	1	yes	characteristic
78	C₂	#436	1	1	yes	characteristic
79	C₂	C₈×Q₈	1	1	yes	characteristic
80	C₂	#446	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
32. C₄⋊C₄	C₈	semidirect
69. C₄	C₄⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊C₈	2
C₂×C₄×C₈	2
C₄×C₄⋊C₄	2
#336	2