Subgroups — C₂×#2252 #33723

Order 256.

279 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#500	C₂	1	2	yes	maximal
2	#2252	C₂	1	8	yes	maximal, direct product factor
3	C₂×#366	C₂	1	4	yes	maximal
4	C₂×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Q₈	—	2	2	no	—
6	C₂²×C₁₆	C₂²	1	2	yes	—
7	#500	—	2	4	no	—
8	#500	C₂²	1	8	yes	—
9	C₂×#107	—	2	2	no	—
10	C₂×#107	C₂²	1	4	yes	—
11	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
12	#366	C₂²	1	16	yes	—
13	C₄×C₁₆	C₂²	1	4	yes	—
14	C₂²×Q₈	—	4	2	no	—
15	C₂²×C₈	D₄	1	1	yes	characteristic
16	C₂²×C₈	C₂³	1	1	yes	characteristic
17	C₄⋊Q₈	—	2	8	no	—
18	C₄⋊Q₈	—	4	4	no	—
19	C₂×C₄⋊C₄	—	2	4	no	—
20	C₂×C₄⋊C₄	—	4	2	no	—
21	C₂×C₄²	D₄	1	1	yes	characteristic
22	C₂×C₁₆	Q₈	1	8	yes	complemented
23	C₂×C₁₆	C₂³	1	4	yes	—
24	#107	—	2	8	no	—
25	#107	—	2	4	no	—
26	#107	C₂³	1	8	yes	—
27	C₄×C₈	D₄	1	2	yes	—
28	C₄×C₈	C₂³	1	2	yes	—
29	C₂×Q₈	—	4	8	no	—
30	C₂×Q₈	—	8	8	no	—
31	C₂²×C₄	—	4	4	no	—
32	C₂²×C₄	D₈	1	1	yes	characteristic
33	C₂²×C₄	D₈	1	1	yes	characteristic
34	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
35	C₂×C₈	Q₁₆	1	4	yes	—
36	C₂×C₈	C₂×D₄	1	1	yes	characteristic
37	C₂×C₈	C₂×D₄	1	2	yes	—
38	C₂×C₈	C₂×Q₈	1	4	yes	—
39	C₂×C₈	C₂⁴	1	1	yes	characteristic
40	C₄⋊C₄	—	2	8	no	—
41	C₄⋊C₄	—	4	4	no	—
42	C₄⋊C₄	—	4	8	no	—
43	C₄²	D₈	1	2	yes	—
44	C₄²	C₂×D₄	1	2	yes	—
45	C₁₆	C₂×Q₈	1	8	yes	complemented
46	C₂³	C₂×D₈	1	1	yes	characteristic
47	Q₈	—	8	16	no	—
48	C₂×C₄	—	4	8	no	—
49	C₂×C₄	—	8	8	no	—
50	C₂×C₄	D₁₆	1	4	yes	—
51	C₂×C₄	Q₃₂	1	4	yes	—
52	C₂×C₄	C₄⋊Q₈	1	2	yes	—
53	C₂×C₄	C₂×D₈	1	1	yes	characteristic
54	C₂×C₄	C₂×D₈	1	2	yes	—
55	C₂×C₄	C₂×D₈	1	2	yes	—
56	C₂×C₄	C₂×Q₁₆	1	2	yes	—
57	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
58	C₈	C₄⋊Q₈	1	2	yes	—
59	C₈	C₂×Q₁₆	1	4	yes	—
60	C₈	C₂²×Q₈	1	2	yes	—
61	C₂²	#500	1	2	yes	—
62	C₂²	C₂×D₁₆	1	2	yes	—
63	C₂²	C₂×Q₃₂	1	2	yes	—
64	C₂²	C₂²×D₈	1	1	yes	characteristic
65	C₄	—	8	8	no	—
66	C₄	#500	1	2	yes	—
67	C₄	C₂×D₁₆	1	4	yes	—
68	C₄	C₂×Q₃₂	1	4	yes	—
69	C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
70	C₄	C₂²×Q₁₆	1	1	yes	characteristic
71	C₂	#2252	1	4	yes	direct product factor
72	C₂	C₂×#500	1	1	yes	characteristic
73	C₂	C₂²×D₁₆	1	1	yes	characteristic
74	C₂	C₂²×Q₃₂	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #2252	C₂	direct (N × H)
22. C₂×C₁₆	Q₈	semidirect
45. C₁₆	C₂×Q₈	semidirect
71. C₂	#2252	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×#500	2
#2252	2
C₂×#366	2
C₂×C₄×C₁₆	2

Subgroups — C2×#2252 #33723

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#2252 #33723