Subgroups — #12592

Order 256.

179 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#346	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic
3	#1726	C₂	1	1	yes	maximal, characteristic
4	#1374	C₂	1	4	yes	maximal
5	C₂²×C₁₆	C₂²	1	2	yes	—
6	C₂²×C₁₆	C₂²	1	2	yes	—
7	C₂×#97	C₄	1	2	yes	—
8	C₂×#97	C₄	1	1	yes	characteristic
9	C₂×C₄×C₈	C₂²	1	2	yes	—
10	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
11	#376	C₄	1	1	yes	characteristic
12	#346	C₄	1	4	yes	complemented
13	C₄×C₁₆	C₄	1	4	yes	complemented
14	C₂²×C₈	C₂×C₄	1	2	yes	—
15	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈	C₂³	1	1	yes	characteristic
17	C₂×C₄²	C₂×C₄	1	2	yes	—
18	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
19	C₂×C₁₆	—	2	4	no	—
20	C₂×C₁₆	—	2	4	no	—
21	C₂×C₁₆	C₂×C₄	1	4	yes	—
22	C₂×C₁₆	C₂×C₄	1	4	yes	—
23	#97	—	2	4	no	—
24	#97	C₈	1	4	yes	—
25	C₄×C₈	—	2	4	no	—
26	C₄×C₈	C₂×C₄	1	2	yes	—
27	C₄×C₈	C₂×C₄	1	2	yes	—
28	#95	C₈	1	4	yes	—
29	C₂²×C₄	C₂×C₈	1	4	yes	—
30	C₂²×C₄	C₂×C₈	1	2	yes	—
31	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₈	—	2	2	no	—
33	C₂×C₈	—	2	2	no	—
34	C₂×C₈	C₄²	1	2	yes	—
35	C₂×C₈	C₂×C₈	1	4	yes	—
36	C₂×C₈	M₁₆	1	4	yes	—
37	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₈	#41	1	4	yes	—
40	C₄²	—	2	2	no	—
41	C₄²	—	2	2	no	—
42	C₄²	C₄²	1	2	yes	—
43	C₄²	C₂×C₈	1	2	yes	—
44	C₁₆	—	2	4	no	—
45	C₁₆	—	2	4	no	—
46	C₂³	C₂²×C₈	1	1	yes	characteristic
47	C₂×C₄	—	2	4	no	—
48	C₂×C₄	—	2	4	no	—
49	C₂×C₄	—	2	8	no	—
50	C₂×C₄	C₄×C₈	1	4	yes	—
51	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
52	C₂×C₄	#117	1	2	yes	—
53	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
54	C₂×C₄	C₂×M₁₆	1	2	yes	—
55	C₈	—	2	4	no	—
56	C₈	#97	1	4	yes	—
57	C₈	#117	1	2	yes	—
58	C₈	#117	1	2	yes	—
59	C₂²	C₂×C₄×C₈	1	1	yes	characteristic
60	C₂²	#431	1	2	yes	—
61	C₂²	#504	1	4	yes	—
62	C₄	—	2	4	no	—
63	C₄	—	4	4	no	—
64	C₄	#376	1	1	yes	characteristic
65	C₄	C₂×#97	1	1	yes	characteristic
66	C₄	#431	1	2	yes	—
67	C₂	#1744	1	1	yes	characteristic
68	C₂	#2108	1	2	yes	—
69	C₂	#2164	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
12. #346	C₄	semidirect
13. 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
C₂×#346	2
C₂×C₄×C₁₆	2
#1726	2
#1374	2