Subgroups — C₂²×#322 #17311

Order 256.

591 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₈	C₂	1	2	yes	maximal
2	C₂²×C₄×C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×#322	C₂	1	12	yes	maximal, direct product factor
4	C₂³×C₈	C₄	1	4	yes	—
5	C₂³×C₈	C₄	1	2	yes	—
6	C₂²×C₄²	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₈	C₄	1	12	yes	—
8	C₂×C₄×C₈	C₄	1	6	yes	—
9	C₂×C₄×C₈	C₂²	1	12	yes	—
10	C₂×C₄×C₈	C₂²	1	6	yes	—
11	#322	C₂²	1	16	yes	direct product factor
12	C₂³×C₄	C₂×C₄	1	1	yes	characteristic
13	C₂³×C₄	C₂×C₄	1	2	yes	—
14	C₂²×C₈	—	2	16	no	—
15	C₂²×C₈	C₈	1	16	yes	complemented
16	C₂²×C₈	C₂×C₄	1	24	yes	—
17	C₂²×C₈	C₂×C₄	1	12	yes	—
18	C₂×C₄²	C₂×C₄	1	6	yes	—
19	C₂×C₄²	C₂×C₄	1	3	yes	—
20	C₂×C₄²	C₂³	1	3	yes	—
21	C₄×C₈	C₂×C₄	1	24	yes	—
22	C₄×C₈	C₂×C₄	1	12	yes	—
23	C₄×C₈	C₂³	1	8	yes	—
24	C₄×C₈	C₂³	1	4	yes	—
25	C₂⁴	C₄²	1	1	yes	characteristic
26	C₂²×C₄	C₄²	1	3	yes	—
27	C₂²×C₄	C₄²	1	6	yes	—
28	C₂²×C₄	C₂×C₈	1	8	yes	—
29	C₂²×C₄	M₁₆	1	16	yes	—
30	C₂²×C₄	C₂²×C₄	1	3	yes	—
31	C₂²×C₄	C₂²×C₄	1	6	yes	—
32	C₂×C₈	—	2	48	no	—
33	C₂×C₈	C₂×C₈	1	48	yes	complemented
34	C₂×C₈	C₂²×C₄	1	16	yes	—
35	C₂×C₈	C₂²×C₄	1	8	yes	—
36	C₄²	C₄²	1	6	yes	—
37	C₄²	C₂²×C₄	1	6	yes	—
38	C₄²	C₂²×C₄	1	3	yes	—
39	C₄²	C₂⁴	1	1	yes	characteristic
40	C₂³	C₄×C₈	1	4	yes	—
41	C₂³	#97	1	8	yes	—
42	C₂³	C₂×C₄²	1	3	yes	—
43	C₂×C₄	C₄×C₈	1	12	yes	—
44	C₂×C₄	#97	1	24	yes	—
45	C₂×C₄	C₂×C₄²	1	3	yes	—
46	C₂×C₄	C₂×C₄²	1	6	yes	—
47	C₂×C₄	C₂²×C₈	1	12	yes	—
48	C₂×C₄	C₂×M₁₆	1	24	yes	—
49	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
50	C₂×C₄	C₂³×C₄	1	2	yes	—
51	C₈	—	2	16	no	—
52	C₈	C₂²×C₈	1	16	yes	complemented
53	C₂²	#322	1	16	yes	direct product factor
54	C₂²	C₂×C₄×C₈	1	6	yes	—
55	C₂²	C₂×#97	1	12	yes	—
56	C₂²	C₂²×C₄²	1	1	yes	characteristic
57	C₄	C₂×C₄×C₈	1	6	yes	—
58	C₄	C₂×#97	1	12	yes	—
59	C₄	C₂³×C₈	1	2	yes	—
60	C₄	C₂²×M₁₆	1	4	yes	—
61	C₂	C₂×#322	1	12	yes	direct product factor
62	C₂	C₂²×C₄×C₈	1	1	yes	characteristic
63	C₂	C₂²×#97	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄²	C₂²
commutator	C₂	C₂²×C₄×C₈
Frattini, cosocle	C₄²	C₂⁴
socle, prime radical	C₂⁴	C₄²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
3. C₂×#322	C₂	direct (N × H)
11. #322	C₂²	direct (N × H)
15. C₂²×C₈	C₈	semidirect
33. C₂×C₈	C₂×C₈	semidirect
52. C₈	C₂²×C₈	semidirect
53. C₂²	#322	direct (N × H)
61. C₂	C₂×#322	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₈	2
C₂×#322	2

Subgroups — C22×#322 #17311

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#322 #17311