Subgroups — C₂²×H(3,𝔽₄) #62688

Order 256.

5123 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×H(3,𝔽₄)	C₂	1	48	yes	maximal, direct product factor
2	C₂²×#124	C₂	1	9	yes	maximal, complemented
3	C₂²×#120	C₂	1	6	yes	maximal, complemented
4	C₂⁶	C₂²	1	2	yes	complemented
5	C₂³×Q₈	C₂²	1	3	yes	complemented
6	C₂³×D₄	C₂²	1	9	yes	—
7	H(3,𝔽₄)	C₂²	1	256	yes	direct product factor
8	C₂×#124	C₂²	1	216	yes	complemented
9	C₂×#120	C₂²	1	144	yes	complemented
10	C₂²×C₂²⋊C₄	C₂²	1	18	yes	complemented
11	C₂²×C₄²	C₂²	1	3	yes	complemented
12	C₂⁵	—	4	24	no	—
13	C₂⁵	C₂³	1	6	yes	—
14	C₂⁵	C₂³	1	24	yes	complemented
15	C₂²×Q₈	—	4	12	no	—
16	C₂²×Q₈	C₂³	1	36	yes	complemented
17	C₂²×D₄	—	4	36	no	—
18	C₂²×D₄	C₂³	1	108	yes	—
19	C₂³×C₄	C₂³	1	9	yes	—
20	#124	C₂³	1	576	yes	complemented
21	#120	C₂³	1	384	yes	complemented
22	C₂×C₂²⋊C₄	C₂³	1	216	yes	complemented
23	C₂×C₄²	C₂³	1	36	yes	complemented
24	C₂⁴	—	2	72	no	—
25	C₂⁴	—	4	144	no	—
26	C₂⁴	—	4	128	no	—
27	C₂⁴	C₂⁴	1	36	yes	—
28	C₂⁴	C₂⁴	1	1	yes	characteristic
29	C₂⁴	C₂⁴	1	32	yes	complemented
30	C₂×Q₈	—	4	72	no	—
31	C₂×Q₈	C₂⁴	1	48	yes	complemented
32	C₂×D₄	—	4	216	no	—
33	C₂×D₄	C₂⁴	1	144	yes	—
34	C₂²×C₄	—	2	36	no	—
35	C₂²×C₄	C₂⁴	1	54	yes	—
36	C₂²⋊C₄	C₂⁴	1	288	yes	complemented
37	C₄²	C₂⁴	1	48	yes	complemented
38	C₂³	—	2	216	no	—
39	C₂³	—	4	96	no	—
40	C₂³	—	4	384	no	—
41	C₂³	—	4	96	no	—
42	C₂³	H(5,𝔽₂)	1	12	yes	—
43	C₂³	C₂⁵	1	24	yes	—
44	C₂³	C₂⁵	1	3	yes	—
45	Q₈	—	4	48	no	—
46	D₄	—	4	144	no	—
47	C₂×C₄	—	2	108	no	—
48	C₂×C₄	C₂⁵	1	36	yes	—
49	C₂²	—	2	72	no	—
50	C₂²	—	4	128	no	—
51	C₂²	—	4	144	no	—
52	C₂²	H(3,𝔽₄)	1	16	yes	direct product factor
53	C₂²	C₂×H(5,𝔽₂)	1	18	yes	—
54	C₂²	C₂⁶	1	1	yes	characteristic
55	C₄	—	2	36	no	—
56	C₂	—	4	24	no	—
57	C₂	C₂×H(3,𝔽₄)	1	12	yes	direct product factor
58	C₂	C₂²×H(5,𝔽₂)	1	3	yes	—

	subgroup	G/H
center, socle	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×H(3,𝔽₄)	C₂	direct (N × H)
2. C₂²×#124	C₂	semidirect
3. C₂²×#120	C₂	semidirect
4. C₂⁶	C₂²	semidirect
5. C₂³×Q₈	C₂²	semidirect
7. H(3,𝔽₄)	C₂²	direct (N × H)
8. C₂×#124	C₂²	semidirect
9. C₂×#120	C₂²	semidirect
10. C₂²×C₂²⋊C₄	C₂²	semidirect
11. C₂²×C₄²	C₂²	semidirect
14. C₂⁵	C₂³	semidirect
16. C₂²×Q₈	C₂³	semidirect
20. #124	C₂³	semidirect
21. #120	C₂³	semidirect
22. C₂×C₂²⋊C₄	C₂³	semidirect
23. C₂×C₄²	C₂³	semidirect
29. C₂⁴	C₂⁴	semidirect
31. C₂×Q₈	C₂⁴	semidirect
36. C₂²⋊C₄	C₂⁴	semidirect
37. C₄²	C₂⁴	semidirect
52. C₂²	H(3,𝔽₄)	direct (N × H)
57. C₂	C₂×H(3,𝔽₄)	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×H(3,𝔽₄)	2
C₂²×#124	2
C₂²×#120	2

Subgroups — C22×H(3,𝔽4) #62688

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×H(3,𝔽₄) #62688