Subgroups — C₂²×#498 #60652

Order 256.

1199 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊Q₈	C₂	1	2	yes	maximal
2	C₂×#498	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#106	C₂	1	4	yes	maximal
4	C₂²×C₄×C₈	C₂	1	1	yes	maximal, characteristic
5	C₂³×Q₈	—	2	2	no	—
6	C₂³×C₈	C₂²	1	2	yes	—
7	C₂×C₄⋊Q₈	—	2	12	no	—
8	C₂×C₄⋊Q₈	C₂²	1	24	yes	—
9	C₂²×C₄⋊C₄	—	2	2	no	—
10	C₂²×C₄⋊C₄	C₂²	1	4	yes	—
11	C₂²×C₄²	C₂²	1	1	yes	characteristic
12	#498	C₂²	1	64	yes	direct product factor
13	C₂×#106	C₂²	1	48	yes	—
14	C₂×C₄×C₈	C₂²	1	12	yes	—
15	C₂²×Q₈	—	2	24	no	—
16	C₂²×Q₈	—	4	16	no	—
17	C₂³×C₄	—	2	4	no	—
18	C₂³×C₄	D₄	1	2	yes	—
19	C₂³×C₄	C₂³	1	1	yes	characteristic
20	C₂²×C₈	Q₈	1	16	yes	complemented
21	C₂²×C₈	C₂³	1	12	yes	—
22	C₄⋊Q₈	—	2	48	no	—
23	C₄⋊Q₈	C₂³	1	32	yes	—
24	C₂×C₄⋊C₄	—	2	24	no	—
25	C₂×C₄⋊C₄	—	2	12	no	—
26	C₂×C₄⋊C₄	C₂³	1	24	yes	—
27	C₂×C₄²	D₄	1	6	yes	—
28	C₂×C₄²	C₂³	1	6	yes	—
29	#106	C₂³	1	64	yes	—
30	C₄×C₈	C₂³	1	16	yes	—
31	C₂⁴	C₂×D₄	1	1	yes	characteristic
32	C₂×Q₈	—	2	32	no	—
33	C₂×Q₈	—	4	96	no	—
34	C₂²×C₄	—	2	24	no	—
35	C₂²×C₄	—	4	16	no	—
36	C₂²×C₄	SD₁₆	1	16	yes	—
37	C₂²×C₄	C₂×D₄	1	3	yes	—
38	C₂²×C₄	C₂×D₄	1	12	yes	—
39	C₂²×C₄	C₂×Q₈	1	8	yes	—
40	C₂²×C₄	C₂⁴	1	3	yes	—
41	C₂×C₈	C₂×Q₈	1	48	yes	complemented
42	C₂×C₈	C₂⁴	1	8	yes	—
43	C₄⋊C₄	—	2	32	no	—
44	C₄⋊C₄	—	2	24	no	—
45	C₄⋊C₄	C₂⁴	1	16	yes	—
46	C₄²	C₂×D₄	1	12	yes	—
47	C₄²	C₂⁴	1	4	yes	—
48	C₂³	C₄⋊Q₈	1	4	yes	—
49	C₂³	C₂×SD₁₆	1	8	yes	—
50	C₂³	C₂²×D₄	1	3	yes	—
51	Q₈	—	4	64	no	—
52	C₂×C₄	—	2	16	no	—
53	C₂×C₄	—	4	48	no	—
54	C₂×C₄	C₄⋊Q₈	1	12	yes	—
55	C₂×C₄	C₂×SD₁₆	1	48	yes	—
56	C₂×C₄	C₂²×D₄	1	3	yes	—
57	C₂×C₄	C₂²×D₄	1	8	yes	—
58	C₂×C₄	C₂²×Q₈	1	12	yes	—
59	C₂×C₄	C₂⁵	1	1	yes	characteristic
60	C₈	C₂²×Q₈	1	16	yes	complemented
61	C₂²	#498	1	16	yes	direct product factor
62	C₂²	C₂×C₄⋊Q₈	1	6	yes	—
63	C₂²	C₂²×SD₁₆	1	12	yes	—
64	C₂²	C₂³×D₄	1	1	yes	characteristic
65	C₄	—	4	16	no	—
66	C₄	C₂×C₄⋊Q₈	1	6	yes	—
67	C₄	C₂²×SD₁₆	1	16	yes	—
68	C₄	C₂³×Q₈	1	2	yes	—
69	C₂	C₂×#498	1	12	yes	direct product factor
70	C₂	C₂²×C₄⋊Q₈	1	1	yes	characteristic
71	C₂	C₂³×SD₁₆	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₂×#498	C₂	direct (N × H)
12. #498	C₂²	direct (N × H)
20. C₂²×C₈	Q₈	semidirect
41. C₂×C₈	C₂×Q₈	semidirect
60. C₈	C₂²×Q₈	semidirect
61. C₂²	#498	direct (N × H)
69. C₂	C₂×#498	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⋊Q₈	2
C₂×#498	2
C₂²×#106	2
C₂²×C₄×C₈	2

Subgroups — C22×#498 #60652

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#498 #60652