Subgroups — C₂×#1372 #12007

Order 256.

131 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#335	C₂	1	2	yes	maximal
3	#1372	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₁₆	C₄	1	1	yes	characteristic
5	C₂²×C₁₆	C₄	1	1	yes	characteristic
6	C₂×C₄⋊C₈	—	2	2	no	—
7	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
8	C₄×C₁₆	C₄	1	2	yes	—
9	C₄×C₁₆	C₂²	1	2	yes	—
10	#335	—	2	2	no	—
11	#335	C₂²	1	4	yes	—
12	C₂²×C₈	—	4	2	no	—
13	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₈	Q₈	1	1	yes	characteristic
15	C₂×C₄²	D₄	1	1	yes	characteristic
16	C₂×C₁₆	—	2	2	no	—
17	C₂×C₁₆	C₈	1	4	yes	complemented
18	C₂×C₁₆	C₂×C₄	1	2	yes	—
19	C₂×C₁₆	C₂×C₄	1	2	yes	—
20	C₄⋊C₈	—	2	4	no	—
21	C₄⋊C₈	—	4	2	no	—
22	C₄×C₈	C₂×C₄	1	1	yes	characteristic
23	C₄×C₈	D₄	1	1	yes	characteristic
24	C₄×C₈	Q₈	1	1	yes	characteristic
25	C₄×C₈	C₂³	1	1	yes	characteristic
26	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
27	C₂²×C₄	D₈	1	1	yes	characteristic
28	C₂²×C₄	Q₁₆	1	1	yes	characteristic
29	C₂×C₈	—	2	2	no	—
30	C₂×C₈	—	4	4	no	—
31	C₂×C₈	—	8	4	no	—
32	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₈	C₂×C₈	1	2	yes	—
35	C₂×C₈	M₁₆	1	2	yes	—
36	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
38	C₄²	C₄⋊C₄	1	1	yes	characteristic
39	C₄²	D₈	1	1	yes	characteristic
40	C₄²	Q₁₆	1	1	yes	characteristic
41	C₄²	C₂×D₄	1	1	yes	characteristic
42	C₁₆	—	2	2	no	—
43	C₁₆	C₂×C₈	1	4	yes	complemented
44	C₂³	#107	1	1	yes	characteristic
45	C₂×C₄	—	2	2	no	—
46	C₂×C₄	C₄⋊C₈	1	2	yes	—
47	C₂×C₄	#107	1	1	yes	characteristic
48	C₂×C₄	#107	1	1	yes	characteristic
49	C₂×C₄	#107	1	1	yes	characteristic
50	C₂×C₄	#108	1	2	yes	—
51	C₂×C₄	SD₃₂	1	4	yes	—
52	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₄	C₂×D₈	1	1	yes	characteristic
54	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
55	C₈	—	2	1	no	—
56	C₈	—	2	1	no	—
57	C₈	—	8	4	no	—
58	C₈	C₄⋊C₈	1	2	yes	—
59	C₈	C₂²×C₈	1	1	yes	characteristic
60	C₈	C₂×M₁₆	1	1	yes	characteristic
61	C₂²	#335	1	2	yes	—
62	C₂²	#367	1	2	yes	—
63	C₂²	#368	1	2	yes	—
64	C₂²	C₂×#107	1	1	yes	characteristic
65	C₄	—	2	1	no	—
66	C₄	—	2	1	no	—
67	C₄	#335	1	2	yes	—
68	C₄	#367	1	2	yes	—
69	C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
70	C₄	C₂×#108	1	1	yes	characteristic
71	C₄	C₂×SD₃₂	1	2	yes	—
72	C₂	#1372	1	4	yes	direct product factor
73	C₂	C₂×#335	1	1	yes	characteristic
74	C₂	C₂×#367	1	1	yes	characteristic
75	C₂	C₂×#368	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₈
commutator	C₈	C₂²×C₈
Frattini, cosocle	C₄×C₈	C₂³
socle, prime radical	C₂³	#107
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #1372	C₂	direct (N × H)
17. C₂×C₁₆	C₈	semidirect
43. C₁₆	C₂×C₈	semidirect
72. C₂	#1372	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×C₄×C₁₆	2
C₂×#335	2
#1372	2

Subgroups — C2×#1372 #12007

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#1372 #12007