Subgroups — C₂×#2276 #33998

Order 256.

1827 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×Q₈	C₂	1	3	yes	maximal
2	#2276	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₄×C₄⋊C₄	C₂	1	9	yes	maximal
4	C₂×#376	C₂	1	3	yes	maximal
5	C₂³×Q₈	C₂²	1	1	yes	characteristic
6	C₂×C₄×Q₈	C₄	1	48	yes	complemented
7	C₂×C₄×Q₈	C₂²	1	24	yes	—
8	C₂²×C₄⋊C₄	C₂²	1	9	yes	—
9	C₂²×C₄²	C₂²	1	6	yes	—
10	C₂²×C₄²	C₂²	1	9	yes	—
11	C₄×C₄⋊C₄	C₂²	1	72	yes	—
12	#376	C₂²	1	24	yes	—
13	C₂×#95	C₂²	1	1	yes	characteristic
14	C₂×#95	C₂²	1	9	yes	—
15	C₂²×Q₈	C₂×C₄	1	24	yes	—
16	C₂²×Q₈	C₂³	1	4	yes	—
17	C₂³×C₄	C₂³	1	9	yes	—
18	C₂³×C₄	C₂³	1	3	yes	—
19	C₂³×C₄	C₂³	1	3	yes	—
20	C₄×Q₈	C₂×C₄	1	192	yes	complemented
21	C₂×C₄⋊C₄	C₂×C₄	1	72	yes	—
22	C₂×C₄⋊C₄	C₂³	1	36	yes	—
23	C₂×C₄²	—	4	12	no	—
24	C₂×C₄²	C₂×C₄	1	72	yes	—
25	C₂×C₄²	C₂³	1	24	yes	—
26	C₂×C₄²	C₂³	1	36	yes	—
27	#95	C₂³	1	4	yes	—
28	#95	C₂³	1	36	yes	—
29	C₂⁴	C₂⁴	1	1	yes	characteristic
30	C₂×Q₈	C₄²	1	64	yes	complemented
31	C₂×Q₈	C₂²×C₄	1	48	yes	—
32	C₂²×C₄	—	2	18	no	—
33	C₂²×C₄	—	2	12	no	—
34	C₂²×C₄	—	2	18	no	—
35	C₂²×C₄	C₂²×C₄	1	12	yes	—
36	C₂²×C₄	C₂²×C₄	1	36	yes	—
37	C₂²×C₄	C₂²×C₄	1	36	yes	—
38	C₂²×C₄	C₂⁴	1	6	yes	—
39	C₂²×C₄	C₂⁴	1	18	yes	—
40	C₂²×C₄	C₂⁴	1	6	yes	—
41	C₄⋊C₄	C₂²×C₄	1	144	yes	—
42	C₄²	—	4	24	no	—
43	C₄²	C₂²×C₄	1	144	yes	—
44	C₂³	C₂³×C₄	1	6	yes	—
45	C₂³	H(5,𝔽₂)	1	2	yes	—
46	C₂³	2₋¹⁺⁴	1	6	yes	—
47	C₂³	C₂⁵	1	1	yes	characteristic
48	Q₈	C₂×C₄²	1	64	yes	complemented
49	C₂×C₄	—	2	24	no	—
50	C₂×C₄	—	2	72	no	—
51	C₂×C₄	—	2	18	no	—
52	C₂×C₄	—	2	12	no	—
53	C₂×C₄	—	2	18	no	—
54	C₂×C₄	C₂×C₄²	1	48	yes	—
55	C₂×C₄	C₂³×C₄	1	12	yes	—
56	C₂×C₄	C₂³×C₄	1	36	yes	—
57	C₂×C₄	C₂³×C₄	1	36	yes	—
58	C₂²	C₂²×C₄²	1	4	yes	—
59	C₂²	#519	1	12	yes	—
60	C₂²	#520	1	12	yes	—
61	C₂²	C₂⁴×C₄	1	3	yes	—
62	C₂²	C₂×H(5,𝔽₂)	1	1	yes	characteristic
63	C₂²	C₂×2₋¹⁺⁴	1	3	yes	—
64	C₄	—	2	12	no	—
65	C₄	—	2	36	no	—
66	C₄	C₂²×C₄²	1	24	yes	—
67	C₂	#2276	1	8	yes	direct product factor
68	C₂	C₂³×C₄²	1	1	yes	characteristic
69	C₂	C₂×#519	1	3	yes	—
70	C₂	C₂×#520	1	3	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂⁴
commutator	C₂	C₂³×C₄²
Frattini, cosocle	C₂³	C₂⁵
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂³ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #2276	C₂	direct (N × H)
6. C₂×C₄×Q₈	C₄	semidirect
20. C₄×Q₈	C₂×C₄	semidirect
30. C₂×Q₈	C₄²	semidirect
48. Q₈	C₂×C₄²	semidirect
67. C₂	#2276	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
#2276	2
C₂×C₄×C₄⋊C₄	2
C₂×#376	2

Subgroups — C2×#2276 #33998

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#2276 #33998