Subgroups — C₂²×Q₈² #62712

Order 256.

3923 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈²	C₂	1	48	yes	maximal, direct product factor
2	C₂²×C₄⋊Q₈	C₂	1	9	yes	maximal
3	C₂²×C₄×Q₈	C₂	1	6	yes	maximal
4	C₂³×Q₈	C₂²	1	6	yes	—
5	C₂³×Q₈	C₂²	1	2	yes	—
6	Q₈²	C₂²	1	256	yes	direct product factor
7	C₂×C₄⋊Q₈	C₂²	1	216	yes	—
8	C₂×C₄×Q₈	C₂²	1	144	yes	—
9	C₂²×C₄⋊C₄	C₂²	1	18	yes	—
10	C₂²×C₄²	C₂²	1	9	yes	—
11	C₂²×Q₈	—	4	24	no	—
12	C₂²×Q₈	Q₈	1	32	yes	direct product factor
13	C₂²×Q₈	C₂³	1	72	yes	—
14	C₂²×Q₈	C₂³	1	24	yes	—
15	C₂³×C₄	C₂³	1	9	yes	—
16	C₂³×C₄	C₂³	1	6	yes	—
17	C₄⋊Q₈	C₂³	1	576	yes	—
18	C₄×Q₈	C₂³	1	384	yes	—
19	C₂×C₄⋊C₄	C₂³	1	216	yes	—
20	C₂×C₄²	C₂³	1	108	yes	—
21	C₂⁴	C₂⁴	1	1	yes	characteristic
22	C₂×Q₈	—	4	144	no	—
23	C₂×Q₈	C₂×Q₈	1	192	yes	direct product factor
24	C₂×Q₈	C₂⁴	1	96	yes	—
25	C₂×Q₈	C₂⁴	1	32	yes	—
26	C₂²×C₄	—	2	36	no	—
27	C₂²×C₄	C₂×Q₈	1	48	yes	—
28	C₂²×C₄	C₂⁴	1	54	yes	—
29	C₂²×C₄	C₂⁴	1	36	yes	—
30	C₄⋊C₄	C₂⁴	1	288	yes	—
31	C₄²	C₂⁴	1	144	yes	—
32	C₂³	C₂²×Q₈	1	8	yes	—
33	C₂³	H(5,𝔽₂)	1	4	yes	—
34	C₂³	C₂⁵	1	3	yes	—
35	Q₈	—	4	96	no	—
36	Q₈	C₂²×Q₈	1	128	yes	direct product factor
37	C₂×C₄	—	2	108	no	—
38	C₂×C₄	C₂²×Q₈	1	144	yes	—
39	C₂×C₄	C₂⁵	1	36	yes	—
40	C₂×C₄	C₂⁵	1	24	yes	—
41	C₂²	Q₈²	1	16	yes	direct product factor
42	C₂²	C₂³×Q₈	1	12	yes	—
43	C₂²	C₂×H(5,𝔽₂)	1	6	yes	—
44	C₂²	C₂⁶	1	1	yes	characteristic
45	C₄	—	2	36	no	—
46	C₄	C₂³×Q₈	1	48	yes	—
47	C₂	C₂×Q₈²	1	12	yes	direct product factor
48	C₂	C₂⁴×Q₈	1	2	yes	—
49	C₂	C₂²×H(5,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁶
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂×Q₈²	C₂	direct (N × H)
6. Q₈²	C₂²	direct (N × H)
12. C₂²×Q₈	Q₈	direct (N × H)
23. C₂×Q₈	C₂×Q₈	direct (N × H)
36. Q₈	C₂²×Q₈	direct (N × H)
41. C₂²	Q₈²	direct (N × H)
47. C₂	C₂×Q₈²	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×Q₈²	2
C₂²×C₄⋊Q₈	2
C₂²×C₄×Q₈	2

Subgroups — C22×Q82 #62712

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×Q₈² #62712