Subgroups — #1831

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	C₄³	C₂	1	1	yes	maximal, characteristic
3	#336	C₂	1	4	yes	maximal
4	C₂²×C₈	C₂²	1	4	yes	—
5	C₂×C₄²	C₄	1	2	yes	—
6	C₂×C₄²	C₄	1	1	yes	characteristic
7	C₂×C₄²	C₄	1	1	yes	characteristic
8	C₂×C₄²	C₂²	1	2	yes	—
9	C₂×C₄²	C₂²	1	1	yes	characteristic
10	C₄⋊C₈	—	2	4	no	—
11	C₂²×C₄	C₂×C₄	1	1	yes	characteristic
12	C₂²×C₄	C₂×C₄	1	2	yes	—
13	C₂²×C₄	C₂×C₄	1	2	yes	—
14	C₂²×C₄	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄	C₂³	1	1	yes	characteristic
16	C₂×C₈	—	2	4	no	—
17	C₂×C₈	—	2	8	no	—
18	C₄²	—	2	4	no	—
19	C₄²	—	2	2	no	—
20	C₄²	—	2	4	no	—
21	C₄²	C₈	1	4	yes	complemented
22	C₄²	D₄	1	2	yes	—
23	C₄²	Q₈	1	2	yes	—
24	C₂³	C₂²×C₄	1	1	yes	characteristic
25	C₂×C₄	—	2	2	no	—
26	C₂×C₄	—	2	2	no	—
27	C₂×C₄	—	2	2	no	—
28	C₂×C₄	—	2	2	no	—
29	C₂×C₄	—	2	2	no	—
30	C₂×C₄	C₄⋊C₄	1	2	yes	—
31	C₂×C₄	C₄⋊C₄	1	2	yes	—
32	C₂×C₄	C₂×C₈	1	4	yes	—
33	C₂×C₄	C₂×C₈	1	2	yes	—
34	C₂×C₄	M₁₆	1	4	yes	—
35	C₂×C₄	M₁₆	1	2	yes	—
36	C₂×C₄	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₄	#41	1	4	yes	—
39	C₈	—	4	4	no	—
40	C₂²	C₂×C₄⋊C₄	1	1	yes	characteristic
41	C₂²	#117	1	2	yes	—
42	C₂²	C₂²×C₈	1	1	yes	characteristic
43	C₂²	C₂×M₁₆	1	1	yes	characteristic
44	C₂²	C₂×M₁₆	1	2	yes	—
45	C₄	—	2	2	no	—
46	C₄	—	2	4	no	—
47	C₄	—	2	4	no	—
48	C₄	C₄⋊C₈	1	4	yes	—
49	C₄	#124	1	2	yes	—
50	C₄	#125	1	2	yes	—
51	C₂	#382	1	1	yes	characteristic
52	C₂	C₂×C₄⋊C₈	1	1	yes	characteristic
53	C₂	#423	1	1	yes	characteristic
54	C₂	#431	1	2	yes	—
55	C₂	#432	1	2	yes	—

	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
21. C₄²	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
C₄³	2
#336	2