Subgroups — C₂×#2247 #33710

Order 256.

407 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₃₂	C₂	1	4	yes	maximal
2	C₂×#494	C₂	1	2	yes	maximal
3	#2247	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic
5	C₂²×Q₁₆	—	2	2	no	—
6	C₂²×Q₁₆	C₂²	1	4	yes	—
7	C₂×C₄⋊Q₈	—	2	2	no	—
8	C₂×Q₃₂	—	2	16	no	—
9	C₂×Q₃₂	C₂²	1	16	yes	—
10	C₂²×C₁₆	C₂²	1	2	yes	—
11	#494	—	2	4	no	—
12	#494	C₂²	1	8	yes	—
13	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
14	C₄×C₁₆	C₂²	1	4	yes	—
15	C₂²×Q₈	—	2	4	no	—
16	C₂×Q₁₆	—	2	16	no	—
17	C₂×Q₁₆	—	2	8	no	—
18	C₂×Q₁₆	—	2	4	no	—
19	C₂×Q₁₆	—	4	8	no	—
20	C₂×Q₁₆	C₂³	1	8	yes	—
21	C₂²×C₈	D₄	1	1	yes	characteristic
22	C₂²×C₈	C₂³	1	1	yes	characteristic
23	C₄⋊Q₈	—	2	8	no	—
24	C₄⋊Q₈	—	4	4	no	—
25	C₂×C₄⋊C₄	—	4	4	no	—
26	C₂×C₄²	D₄	1	1	yes	characteristic
27	Q₃₂	—	2	32	no	—
28	C₂×C₁₆	D₄	1	8	yes	—
29	C₂×C₁₆	C₂³	1	4	yes	—
30	C₄×C₈	D₄	1	2	yes	—
31	C₄×C₈	C₂³	1	2	yes	—
32	C₂×Q₈	—	2	8	no	—
33	C₂×Q₈	—	4	4	no	—
34	C₂×Q₈	—	4	8	no	—
35	C₂×Q₈	—	4	8	no	—
36	C₂²×C₄	—	4	4	no	—
37	C₂²×C₄	D₈	1	2	yes	—
38	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
39	Q₁₆	—	2	16	no	—
40	Q₁₆	—	4	8	no	—
41	Q₁₆	—	4	16	no	—
42	C₂×C₈	D₈	1	4	yes	—
43	C₂×C₈	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈	C₂×D₄	1	4	yes	—
45	C₂×C₈	C₂×D₄	1	2	yes	—
46	C₂×C₈	C₂⁴	1	1	yes	characteristic
47	C₄⋊C₄	—	4	16	no	—
48	C₄²	D₈	1	2	yes	—
49	C₄²	C₂×D₄	1	2	yes	—
50	C₁₆	C₂×D₄	1	8	yes	—
51	C₂³	C₂×D₈	1	1	yes	characteristic
52	Q₈	—	4	8	no	—
53	Q₈	—	4	8	no	—
54	Q₈	—	8	8	no	—
55	C₂×C₄	—	4	8	no	—
56	C₂×C₄	—	8	8	no	—
57	C₂×C₄	Q₃₂	1	8	yes	complemented
58	C₂×C₄	#127	1	2	yes	—
59	C₂×C₄	C₂×D₈	1	2	yes	—
60	C₂×C₄	C₂×D₈	1	1	yes	characteristic
61	C₂×C₄	C₂×D₈	1	4	yes	—
62	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
63	C₈	#127	1	2	yes	—
64	C₈	C₂×D₈	1	4	yes	—
65	C₈	C₂²×D₄	1	2	yes	—
66	C₂²	#493	1	2	yes	—
67	C₂²	C₂×Q₃₂	1	4	yes	—
68	C₂²	C₂²×D₈	1	1	yes	characteristic
69	C₄	—	8	8	no	—
70	C₄	#493	1	2	yes	—
71	C₄	C₂×Q₃₂	1	8	yes	complemented
72	C₄	C₂×#127	1	1	yes	characteristic
73	C₄	C₂²×D₈	1	1	yes	characteristic
74	C₂	#2247	1	4	yes	direct product factor
75	C₂	C₂×#493	1	1	yes	characteristic
76	C₂	C₂²×Q₃₂	1	2	yes	—

	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
3. #2247	C₂	direct (N × H)
57. C₂×C₄	Q₃₂	semidirect
71. C₄	C₂×Q₃₂	semidirect
74. C₂	#2247	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×Q₃₂	2
C₂×#494	2
#2247	2
C₂×C₄×C₁₆	2

Subgroups — C2×#2247 #33710

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#2247 #33710