Subgroups — C₄×#366 #12672

Order 256.

183 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#366	C₂	1	2	yes	maximal
2	C₂×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₄×#107	C₂	1	2	yes	maximal
4	#1380	C₂	1	2	yes	maximal
5	C₂²×C₁₆	C₂²	1	1	yes	characteristic
6	C₂²×C₁₆	C₂²	1	1	yes	characteristic
7	C₂×#107	C₂²	1	4	yes	—
8	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
9	C₄×C₄⋊C₄	—	2	2	no	—
10	#366	C₄	1	8	yes	direct product factor
11	C₄×C₁₆	C₄	1	4	yes	complemented
12	#340	—	2	2	no	—
13	C₂²×C₈	D₄	1	1	yes	characteristic
14	C₂²×C₈	C₂³	1	1	yes	characteristic
15	C₂×C₄⋊C₄	—	2	4	no	—
16	C₂×C₄²	—	4	2	no	—
17	C₂×C₄²	D₄	1	1	yes	characteristic
18	C₂×C₁₆	—	2	2	no	—
19	C₂×C₁₆	C₂×C₄	1	2	yes	—
20	C₂×C₁₆	C₂×C₄	1	2	yes	—
21	C₂×C₁₆	C₂×C₄	1	4	yes	—
22	#107	—	2	4	no	—
23	#107	C₂×C₄	1	8	yes	—
24	C₄×C₈	C₂×C₄	1	2	yes	—
25	C₄×C₈	Q₈	1	2	yes	—
26	#95	—	4	2	no	—
27	C₂²×C₄	—	4	4	no	—
28	C₂²×C₄	D₈	1	1	yes	characteristic
29	C₂²×C₄	D₈	1	1	yes	characteristic
30	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
31	C₂×C₈	—	2	2	no	—
32	C₂×C₈	C₄⋊C₄	1	2	yes	—
33	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₈	C₂²×C₄	1	2	yes	—
35	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
36	C₂×C₈	#41	1	2	yes	—
37	C₄⋊C₄	—	2	8	no	—
38	C₄⋊C₄	—	4	4	no	—
39	C₄²	—	8	4	no	—
40	C₄²	C₄⋊C₄	1	2	yes	—
41	C₄²	Q₁₆	1	2	yes	—
42	C₁₆	—	2	2	no	—
43	C₁₆	C₄²	1	4	yes	complemented
44	C₂³	C₂×D₈	1	1	yes	characteristic
45	C₂×C₄	—	2	2	no	—
46	C₂×C₄	—	4	8	no	—
47	C₂×C₄	—	8	4	no	—
48	C₂×C₄	—	8	4	no	—
49	C₂×C₄	#107	1	2	yes	—
50	C₂×C₄	#107	1	2	yes	—
51	C₂×C₄	D₁₆	1	2	yes	—
52	C₂×C₄	Q₃₂	1	2	yes	—
53	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄	C₄×D₄	1	2	yes	—
55	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
56	C₂×C₄	#135	1	2	yes	—
57	C₈	—	2	2	no	—
58	C₈	C₂×C₄²	1	1	yes	characteristic
59	C₈	#117	1	1	yes	characteristic
60	C₈	C₄×Q₈	1	2	yes	—
61	C₂²	C₂×#107	1	1	yes	characteristic
62	C₂²	C₄×D₈	1	2	yes	—
63	C₂²	C₂×D₁₆	1	1	yes	characteristic
64	C₂²	C₂×Q₃₂	1	1	yes	characteristic
65	C₂²	#508	1	2	yes	—
66	C₄	—	2	2	no	—
67	C₄	—	8	8	no	—
68	C₄	#366	1	4	yes	direct product factor
69	C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
70	C₄	#427	1	1	yes	characteristic
71	C₄	C₄×Q₁₆	1	2	yes	—
72	C₂	C₄×#107	1	1	yes	characteristic
73	C₂	C₂×#366	1	1	yes	characteristic
74	C₂	#2158	1	1	yes	characteristic
75	C₂	C₄×D₁₆	1	2	yes	—
76	C₂	C₄×Q₃₂	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	D₈
commutator	C₈	C₂×C₄²
Frattini, cosocle	C₂²×C₈	C₂³
socle, prime radical	C₂³	C₂×D₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
10. #366	C₄	direct (N × H)
11. C₄×C₁₆	C₄	semidirect
43. C₁₆	C₄²	semidirect
68. C₄	#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
C₂×#366	2
C₂×C₄×C₁₆	2
C₄×#107	2
#1380	2

Subgroups — C4×#366 #12672

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#366 #12672