Subgroups — #1767

Order 128.

146 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×#97	C₂	1	2	yes	maximal
3	C₄³	C₂	1	1	yes	maximal, characteristic
4	#336	C₂	1	2	yes	maximal
5	C₂²×C₈	C₂²	1	4	yes	—
6	C₂×C₄²	C₄	1	2	yes	—
7	C₂×C₄²	C₄	1	2	yes	—
8	C₂×C₄²	C₂²	1	1	yes	characteristic
9	C₂×C₄²	C₂²	1	1	yes	characteristic
10	C₂×C₄²	C₂²	1	1	yes	characteristic
11	C₄⋊C₈	C₄	1	8	yes	complemented
12	#97	—	2	4	no	—
13	C₂²×C₄	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₄	C₂×C₄	1	2	yes	—
15	C₂²×C₄	C₂×C₄	1	2	yes	—
16	C₂²×C₄	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₄	C₂³	1	1	yes	characteristic
18	C₂×C₈	—	2	4	no	—
19	C₂×C₈	—	2	4	no	—
20	C₂×C₈	C₂×C₄	1	8	yes	—
21	C₄²	—	2	4	no	—
22	C₄²	—	2	4	no	—
23	C₄²	—	2	2	no	—
24	C₄²	C₂×C₄	1	4	yes	—
25	C₄²	D₄	1	2	yes	—
26	C₄²	Q₈	1	2	yes	—
27	C₂³	C₂²×C₄	1	1	yes	characteristic
28	C₂×C₄	—	2	2	no	—
29	C₂×C₄	—	2	2	no	—
30	C₂×C₄	—	2	2	no	—
31	C₂×C₄	—	2	2	no	—
32	C₂×C₄	—	2	2	no	—
33	C₂×C₄	C₄²	1	2	yes	—
34	C₂×C₄	C₄²	1	2	yes	—
35	C₂×C₄	C₄⋊C₄	1	4	yes	—
36	C₂×C₄	M₁₆	1	4	yes	—
37	C₂×C₄	M₁₆	1	4	yes	—
38	C₂×C₄	C₂²×C₄	1	2	yes	—
39	C₂×C₄	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄	#41	1	2	yes	—
42	C₈	—	2	8	no	—
43	C₂²	C₂×C₄²	1	1	yes	characteristic
44	C₂²	C₂×C₄⋊C₄	1	1	yes	characteristic
45	C₂²	#117	1	1	yes	characteristic
46	C₂²	C₂×M₁₆	1	2	yes	—
47	C₂²	C₂×M₁₆	1	2	yes	—
48	C₄	—	2	2	no	—
49	C₄	—	2	4	no	—
50	C₄	—	2	4	no	—
51	C₄	#97	1	4	yes	complemented
52	C₄	C₄×D₄	1	2	yes	—
53	C₄	C₄×Q₈	1	2	yes	—
54	C₂	C₄×C₄⋊C₄	1	1	yes	characteristic
55	C₂	C₂×#97	1	1	yes	characteristic
56	C₂	C₄×M₁₆	1	1	yes	characteristic
57	C₂	#423	1	2	yes	—
58	C₂	#432	1	2	yes	—

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂³
commutator	C₂²	C₂×C₄²
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
11. C₄⋊C₈	C₄	semidirect
51. C₄	#97	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×#97	2
C₄³	2
#336	2