Subgroups — C₂²×#366 #33405

Order 256.

391 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center, socle, prime radical	C₂⁴	D₈
commutator	C₈	C₂³×C₄
Frattini, cosocle	C₂×C₈	C₂⁴
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×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. C₂×#366	C₂	direct (N × H)
6. C₂²×C₁₆	C₄	semidirect
10. #366	C₂²	direct (N × H)
19. C₂×C₁₆	C₂×C₄	semidirect
37. C₁₆	C₂²×C₄	semidirect
52. C₂²	#366	direct (N × H)
61. C₂	C₂×#366	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂³×C₁₆	2
C₂²×#107	2
C₂×#366	2

Subgroups — C22×#366 #33405

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#366 #33405