Subgroups — #12591

Order 256.

195 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#346	C₂	1	2	yes	maximal
2	C₄²×C₈	C₂	1	1	yes	maximal, characteristic
3	#1374	C₂	1	4	yes	maximal
4	C₂²×C₁₆	C₂²	1	4	yes	—
5	C₂×C₄×C₈	C₄	1	2	yes	—
6	C₂×C₄×C₈	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈	C₂²	1	2	yes	—
8	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
9	C₄³	C₄	1	1	yes	characteristic
10	#346	C₄	1	8	yes	complemented
11	C₂²×C₈	C₂×C₄	1	2	yes	—
12	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₈	C₂³	1	1	yes	characteristic
14	C₂×C₄²	C₈	1	4	yes	—
15	C₂×C₄²	C₂×C₄	1	2	yes	—
16	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
17	C₂×C₁₆	—	2	8	no	—
18	C₂×C₁₆	C₂×C₄	1	8	yes	—
19	C₄×C₈	—	2	4	no	—
20	C₄×C₈	—	2	4	no	—
21	C₄×C₈	C₈	1	4	yes	—
22	C₄×C₈	C₂×C₄	1	4	yes	—
23	C₂²×C₄	C₂×C₈	1	2	yes	—
24	C₂²×C₄	C₂×C₈	1	4	yes	—
25	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
26	C₂×C₈	—	2	2	no	—
27	C₂×C₈	—	2	2	no	—
28	C₂×C₈	C₄²	1	2	yes	—
29	C₂×C₈	C₂×C₈	1	4	yes	—
30	C₂×C₈	M₁₆	1	4	yes	—
31	C₂×C₈	C₂²×C₄	1	2	yes	—
32	C₂×C₈	#41	1	4	yes	—
33	C₄²	—	2	8	no	—
34	C₄²	—	2	2	no	—
35	C₄²	—	2	2	no	—
36	C₄²	C₄²	1	2	yes	—
37	C₄²	C₂×C₈	1	2	yes	—
38	C₁₆	—	2	8	no	—
39	C₂³	C₂²×C₈	1	1	yes	characteristic
40	C₂×C₄	—	2	4	no	—
41	C₂×C₄	—	2	4	no	—
42	C₂×C₄	—	2	4	no	—
43	C₂×C₄	C₄×C₈	1	4	yes	—
44	C₂×C₄	M₃₂	1	8	yes	—
45	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
46	C₂×C₄	#117	1	2	yes	—
47	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
48	C₂×C₄	C₂×M₁₆	1	2	yes	—
49	C₈	—	2	4	no	—
50	C₈	#97	1	4	yes	—
51	C₈	#117	1	4	yes	—
52	C₂²	C₂×C₄×C₈	1	1	yes	characteristic
53	C₂²	#431	1	2	yes	—
54	C₂²	C₂×M₃₂	1	4	yes	—
55	C₄	—	2	4	no	—
56	C₄	—	2	8	no	—
57	C₄	#376	1	1	yes	characteristic
58	C₄	C₂×#97	1	1	yes	characteristic
59	C₄	#431	1	2	yes	—
60	C₂	#1744	1	1	yes	characteristic
61	C₂	C₄×M₃₂	1	2	yes	—
62	C₂	#2163	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
10. #346	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#346	2
C₄²×C₈	2
#1374	2