Subgroups — #12594

Order 256.

177 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×M₃₂	C₂	1	2	yes	maximal
2	C₄×#97	C₂	1	1	yes	maximal, characteristic
3	#1375	C₂	1	4	yes	maximal
4	C₂×M₃₂	C₂²	1	4	yes	—
5	C₂×#97	C₄	1	2	yes	—
6	C₂×#97	C₂²	1	2	yes	—
7	C₂×C₄×C₈	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
9	C₄³	C₄	1	1	yes	characteristic
10	#346	C₄	1	4	yes	complemented
11	C₄×C₁₆	C₄	1	4	yes	complemented
12	C₂²×C₈	C₂×C₄	1	2	yes	—
13	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₈	C₂³	1	1	yes	characteristic
15	C₂×C₄²	—	2	2	no	—
16	C₂×C₄²	C₂×C₄	1	2	yes	—
17	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
18	M₃₂	—	2	8	no	—
19	C₂×C₁₆	C₂×C₄	1	8	yes	—
20	#97	—	2	4	no	—
21	#97	C₈	1	8	yes	—
22	C₄×C₈	—	2	2	no	—
23	C₄×C₈	C₂×C₄	1	2	yes	—
24	C₄×C₈	C₂×C₄	1	2	yes	—
25	C₂²×C₄	—	2	2	no	—
26	C₂²×C₄	M₁₆	1	2	yes	—
27	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
28	C₂×C₈	—	2	2	no	—
29	C₂×C₈	—	2	2	no	—
30	C₂×C₈	C₄²	1	2	yes	—
31	C₂×C₈	C₂×C₈	1	4	yes	—
32	C₂×C₈	C₂×C₈	1	4	yes	—
33	C₂×C₈	C₂²×C₄	1	2	yes	—
34	C₂×C₈	#41	1	4	yes	—
35	C₄²	—	2	2	no	—
36	C₄²	—	4	4	no	—
37	C₄²	C₄²	1	1	yes	characteristic
38	C₄²	C₄²	1	1	yes	characteristic
39	C₄²	C₂×C₈	1	4	yes	—
40	C₄²	M₁₆	1	2	yes	—
41	C₁₆	—	2	8	no	—
42	C₂³	C₂×M₁₆	1	1	yes	characteristic
43	C₂×C₄	—	2	4	no	—
44	C₂×C₄	—	2	4	no	—
45	C₂×C₄	—	4	4	no	—
46	C₂×C₄	#97	1	2	yes	—
47	C₂×C₄	#97	1	2	yes	—
48	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
49	C₂×C₄	#117	1	2	yes	—
50	C₂×C₄	C₂²×C₈	1	2	yes	—
51	C₂×C₄	C₂×M₁₆	1	1	yes	characteristic
52	C₈	—	2	4	no	—
53	C₈	C₄×C₈	1	4	yes	—
54	C₈	#117	1	4	yes	—
55	C₂²	—	2	2	no	—
56	C₂²	C₂×#97	1	1	yes	characteristic
57	C₂²	#431	1	2	yes	—
58	C₄	—	2	2	no	—
59	C₄	—	4	4	no	—
60	C₄	#347	1	4	yes	—
61	C₄	#376	1	1	yes	characteristic
62	C₄	C₂×C₄×C₈	1	1	yes	characteristic
63	C₄	#431	1	2	yes	—
64	C₂	—	2	2	no	—
65	C₂	#1744	1	1	yes	characteristic
66	C₂	C₂×#347	1	2	yes	—

	subgroup	G/H
center	C₄²	C₄²
commutator	C₄	C₂×C₄×C₈
Frattini, cosocle	C₂²×C₈	C₂³
socle	C₂²	C₂×#97
prime radical	C₂³	C₂×M₁₆
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₄² ◁ 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
11. C₄×C₁₆	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₄×M₃₂	2
C₄×#97	2
#1375	2