Subgroups — C₈×#97 #7840

Order 256.

303 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#1744	C₂	1	1	yes	maximal, characteristic
2	C₄×#97	C₂	1	1	yes	maximal, characteristic
3	C₄²×C₈	C₂	1	1	yes	maximal, characteristic
4	C₂×#322	C₂	1	2	yes	maximal
5	C₂×C₈²	C₂	1	2	yes	maximal
6	C₂×#97	C₄	1	4	yes	—
7	C₂×C₄×C₈	C₄	1	4	yes	—
8	C₂×C₄×C₈	C₂²	1	2	yes	—
9	C₂×C₄×C₈	C₂²	1	2	yes	—
10	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
12	C₄³	C₂²	1	1	yes	characteristic
13	#336	C₄	1	4	yes	—
14	#322	C₄	1	8	yes	complemented
15	C₈²	C₄	1	8	yes	complemented
16	C₂²×C₈	C₂×C₄	1	4	yes	—
17	C₂²×C₈	C₂×C₄	1	2	yes	—
18	C₂²×C₈	C₂×C₄	1	4	yes	—
19	C₂²×C₈	C₂×C₄	1	2	yes	—
20	C₂×C₄²	C₂×C₄	1	2	yes	—
21	C₂×C₄²	C₂×C₄	1	2	yes	—
22	C₂×C₄²	C₂×C₄	1	2	yes	—
23	C₂×C₄²	C₂³	1	1	yes	characteristic
24	#97	C₈	1	16	yes	direct product factor
25	C₄×C₈	—	2	8	no	—
26	C₄×C₈	C₂×C₄	1	4	yes	—
27	C₄×C₈	C₂×C₄	1	4	yes	—
28	C₄×C₈	C₂×C₄	1	4	yes	—
29	C₄×C₈	C₂×C₄	1	4	yes	—
30	C₄×C₈	C₂×C₄	1	4	yes	—
31	C₄×C₈	C₂×C₄	1	4	yes	—
32	C₂²×C₄	C₄²	1	4	yes	—
33	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
34	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
35	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₈	—	2	4	no	—
37	C₂×C₈	—	2	8	no	—
38	C₂×C₈	C₄²	1	4	yes	—
39	C₂×C₈	C₄²	1	8	yes	—
40	C₂×C₈	C₄²	1	8	yes	—
41	C₂×C₈	C₄²	1	4	yes	—
42	C₂×C₈	C₂×C₈	1	16	yes	—
43	C₂×C₈	M₁₆	1	8	yes	—
44	C₄²	—	2	4	no	—
45	C₄²	—	2	4	no	—
46	C₄²	C₂×C₈	1	8	yes	—
47	C₄²	C₂²×C₄	1	2	yes	—
48	C₄²	C₂²×C₄	1	2	yes	—
49	C₂³	C₂×C₄²	1	1	yes	characteristic
50	C₂×C₄	—	2	4	no	—
51	C₂×C₄	—	2	4	no	—
52	C₂×C₄	C₄×C₈	1	8	yes	—
53	C₂×C₄	C₂×C₄²	1	2	yes	—
54	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
55	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
56	C₂×C₄	C₂×C₄²	1	2	yes	—
57	C₂×C₄	C₂²×C₈	1	4	yes	—
58	C₂×C₄	C₂×M₁₆	1	4	yes	—
59	C₂×C₄	#131	1	4	yes	—
60	C₈	—	2	4	no	—
61	C₈	—	2	8	no	—
62	C₈	C₄×C₈	1	16	yes	complemented
63	C₈	#97	1	8	yes	direct product factor
64	C₂²	C₄³	1	1	yes	characteristic
65	C₂²	C₂×C₄×C₈	1	2	yes	—
66	C₂²	C₄×M₁₆	1	2	yes	—
67	C₂²	#405	1	2	yes	—
68	C₄	—	2	8	no	—
69	C₄	C₂×C₄×C₈	1	4	yes	—
70	C₄	C₂×#97	1	2	yes	—
71	C₄	C₄×M₁₆	1	2	yes	—
72	C₄	#405	1	4	yes	—
73	C₂	C₈×M₁₆	1	4	yes	—
74	C₂	C₄²×C₈	1	1	yes	characteristic
75	C₂	C₄×#97	1	1	yes	characteristic
76	C₂	#1726	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
14. #322	C₄	semidirect
15. C₈²	C₄	semidirect
24. #97	C₈	direct (N × H)
62. C₈	C₄×C₈	semidirect
63. C₈	#97	direct (N × H)

Maximal subgroups

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

subgroup	index
#1744	2
C₄×#97	2
C₄²×C₈	2
C₂×#322	2
C₂×C₈²	2

Subgroups — C8×#97 #7840

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×#97 #7840