Subgroups — C₂³×C₂²⋊C₈ #60059

Order 256.

3823 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×C₄	C₂	1	1	yes	maximal, characteristic
2	C₂⁴×C₈	C₂	1	2	yes	maximal, complemented
3	C₂²×C₂²⋊C₈	C₂	1	28	yes	maximal, direct product factor
4	C₂⁶	C₄	1	1	yes	characteristic
5	C₂⁴×C₄	—	2	16	no	—
6	C₂⁴×C₄	C₄	1	1	yes	characteristic
7	C₂⁴×C₄	C₄	1	14	yes	—
8	C₂⁴×C₄	C₂²	1	14	yes	—
9	C₂⁴×C₄	C₂²	1	1	yes	characteristic
10	C₂³×C₈	—	2	16	no	—
11	C₂³×C₈	C₂²	1	28	yes	complemented
12	C₂×C₂²⋊C₈	C₂²	1	112	yes	direct product factor
13	C₂⁵	—	2	8	no	—
14	C₂⁵	—	2	8	no	—
15	C₂⁵	C₈	1	16	yes	complemented
16	C₂⁵	C₂×C₄	1	14	yes	—
17	C₂⁵	C₂×C₄	1	1	yes	characteristic
18	C₂³×C₄	—	2	8	no	—
19	C₂³×C₄	—	2	112	no	—
20	C₂³×C₄	—	2	112	no	—
21	C₂³×C₄	C₂×C₄	1	14	yes	—
22	C₂³×C₄	C₂×C₄	1	84	yes	—
23	C₂³×C₄	C₂×C₄	1	7	yes	—
24	C₂³×C₄	D₄	1	16	yes	—
25	C₂³×C₄	C₂³	1	28	yes	—
26	C₂³×C₄	C₂³	1	7	yes	—
27	C₂²×C₈	—	2	112	no	—
28	C₂²×C₈	C₂³	1	56	yes	complemented
29	C₂²⋊C₈	C₂³	1	64	yes	direct product factor
30	C₂⁴	—	2	128	no	—
31	C₂⁴	—	2	56	no	—
32	C₂⁴	—	2	56	no	—
33	C₂⁴	C₂²⋊C₄	1	8	yes	—
34	C₂⁴	C₂×C₈	1	8	yes	—
35	C₂⁴	C₂×C₈	1	112	yes	complemented
36	C₂⁴	M₁₆	1	8	yes	—
37	C₂⁴	C₂²×C₄	1	28	yes	—
38	C₂⁴	C₂²×C₄	1	7	yes	—
39	C₂²×C₄	—	2	56	no	—
40	C₂²×C₄	—	2	336	no	—
41	C₂²×C₄	—	2	112	no	—
42	C₂²×C₄	C₂²⋊C₄	1	56	yes	—
43	C₂²×C₄	C₂²×C₄	1	28	yes	—
44	C₂²×C₄	C₂²×C₄	1	56	yes	—
45	C₂²×C₄	C₂²×C₄	1	21	yes	—
46	C₂²×C₄	C₂×D₄	1	56	yes	—
47	C₂²×C₄	C₂⁴	1	8	yes	—
48	C₂²×C₄	C₂⁴	1	7	yes	—
49	C₂×C₈	—	2	112	no	—
50	C₂×C₈	C₂⁴	1	16	yes	complemented
51	C₂³	—	2	56	no	—
52	C₂³	—	2	56	no	—
53	C₂³	—	2	448	no	—
54	C₂³	C₂²⋊C₈	1	64	yes	direct product factor
55	C₂³	C₂×C₂²⋊C₄	1	28	yes	—
56	C₂³	C₂²×C₈	1	28	yes	—
57	C₂³	C₂²×C₈	1	112	yes	complemented
58	C₂³	C₂×M₁₆	1	28	yes	—
59	C₂³	C₂³×C₄	1	8	yes	—
60	C₂³	C₂³×C₄	1	7	yes	—
61	C₂×C₄	—	2	56	no	—
62	C₂×C₄	—	2	112	no	—
63	C₂×C₄	—	2	16	no	—
64	C₂×C₄	C₂×C₂²⋊C₄	1	84	yes	—
65	C₂×C₄	C₂³×C₄	1	8	yes	—
66	C₂×C₄	C₂³×C₄	1	7	yes	—
67	C₂×C₄	C₂²×D₄	1	28	yes	—
68	C₂×C₄	C₂⁵	1	1	yes	characteristic
69	C₈	—	2	16	no	—
70	C₂²	—	2	224	no	—
71	C₂²	—	2	8	no	—
72	C₂²	—	2	8	no	—
73	C₂²	C₂×C₂²⋊C₈	1	112	yes	direct product factor
74	C₂²	C₂²×C₂²⋊C₄	1	14	yes	—
75	C₂²	C₂³×C₈	1	14	yes	—
76	C₂²	C₂³×C₈	1	16	yes	complemented
77	C₂²	C₂²×M₁₆	1	14	yes	—
78	C₂²	C₂⁴×C₄	1	1	yes	characteristic
79	C₄	—	2	8	no	—
80	C₄	C₂²×C₂²⋊C₄	1	14	yes	—
81	C₄	C₂³×D₄	1	2	yes	—
82	C₂	—	2	16	no	—
83	C₂	C₂²×C₂²⋊C₈	1	28	yes	direct product factor
84	C₂	C₂³×C₂²⋊C₄	1	1	yes	characteristic
85	C₂	C₂⁴×C₈	1	1	yes	characteristic
86	C₂	C₂³×M₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴×C₄	C₂²
commutator	C₂	C₂⁴×C₈
Frattini, cosocle	C₂×C₄	C₂⁵
socle	C₂⁵	C₂×C₄
prime radical	C₂⁶	C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₂⁴×C₈	C₂	semidirect
3. C₂²×C₂²⋊C₈	C₂	direct (N × H)
11. C₂³×C₈	C₂²	semidirect
12. C₂×C₂²⋊C₈	C₂²	direct (N × H)
15. C₂⁵	C₈	semidirect
28. C₂²×C₈	C₂³	semidirect
29. C₂²⋊C₈	C₂³	direct (N × H)
35. C₂⁴	C₂×C₈	semidirect
50. C₂×C₈	C₂⁴	semidirect
54. C₂³	C₂²⋊C₈	direct (N × H)
57. C₂³	C₂²×C₈	semidirect
73. C₂²	C₂×C₂²⋊C₈	direct (N × H)
76. C₂²	C₂³×C₈	semidirect
83. C₂	C₂²×C₂²⋊C₈	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁵×C₄	2
C₂⁴×C₈	2
C₂²×C₂²⋊C₈	2

Subgroups — C23×C22⋊C8 #60059

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₂²⋊C₈ #60059