Subgroups — C₂³×#124 #62660

Order 256.

7167 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂⁴×D₄	C₂	1	1	yes	maximal, characteristic
3	C₂²×#124	C₂	1	56	yes	maximal, direct product factor
4	C₂³×C₂²⋊C₄	C₂	1	4	yes	maximal, complemented
5	C₂³×C₄²	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂⁶	C₂²	1	2	yes	—
7	C₂³×Q₈	—	2	16	no	—
8	C₂³×Q₈	C₂²	1	28	yes	complemented
9	C₂³×D₄	—	2	16	no	—
10	C₂³×D₄	C₂²	1	28	yes	—
11	C₂⁴×C₄	C₂²	1	2	yes	—
12	C₂⁴×C₄	C₂²	1	2	yes	—
13	C₂⁴×C₄	C₂²	1	1	yes	characteristic
14	C₂×#124	C₂²	1	448	yes	direct product factor
15	C₂²×C₂²⋊C₄	C₂²	1	112	yes	complemented
16	C₂²×C₄²	C₂²	1	28	yes	complemented
17	C₂⁵	—	2	16	no	—
18	C₂⁵	—	2	32	no	—
19	C₂⁵	C₂³	1	28	yes	—
20	C₂⁵	C₂³	1	1	yes	characteristic
21	C₂²×Q₈	—	2	224	no	—
22	C₂²×Q₈	C₂³	1	112	yes	complemented
23	C₂²×D₄	—	2	224	no	—
24	C₂²×D₄	C₂³	1	112	yes	—
25	C₂³×C₄	—	2	16	no	—
26	C₂³×C₄	—	2	16	no	—
27	C₂³×C₄	D₄	1	16	yes	—
28	C₂³×C₄	C₂³	1	28	yes	—
29	C₂³×C₄	C₂³	1	28	yes	—
30	C₂³×C₄	C₂³	1	14	yes	—
31	#124	C₂³	1	512	yes	direct product factor
32	C₂×C₂²⋊C₄	C₂³	1	448	yes	complemented
33	C₂×C₄²	C₂³	1	112	yes	complemented
34	C₂⁴	—	2	112	no	—
35	C₂⁴	—	2	224	no	—
36	C₂⁴	—	4	128	no	—
37	C₂⁴	C₂×D₄	1	8	yes	—
38	C₂⁴	#41	1	16	yes	—
39	C₂⁴	C₂⁴	1	56	yes	—
40	C₂⁴	C₂⁴	1	7	yes	—
41	C₂×Q₈	—	2	448	no	—
42	C₂×Q₈	C₂⁴	1	64	yes	complemented
43	C₂×D₄	—	2	448	no	—
44	C₂×D₄	C₂⁴	1	64	yes	—
45	C₂²×C₄	—	2	112	no	—
46	C₂²×C₄	—	2	112	no	—
47	C₂²×C₄	C₂×D₄	1	112	yes	—
48	C₂²×C₄	C₂⁴	1	56	yes	—
49	C₂²×C₄	C₂⁴	1	56	yes	—
50	C₂²×C₄	C₂⁴	1	28	yes	—
51	C₂²⋊C₄	C₂⁴	1	256	yes	complemented
52	C₄²	C₂⁴	1	64	yes	complemented
53	C₂³	—	2	112	no	—
54	C₂³	—	2	224	no	—
55	C₂³	—	4	448	no	—
56	C₂³	#124	1	64	yes	direct product factor
57	C₂³	C₂²×D₄	1	28	yes	—
58	C₂³	C₂×#41	1	56	yes	—
59	C₂³	C₂⁵	1	16	yes	—
60	C₂³	C₂⁵	1	7	yes	—
61	Q₈	—	2	128	no	—
62	D₄	—	2	128	no	—
63	C₂×C₄	—	2	112	no	—
64	C₂×C₄	—	2	112	no	—
65	C₂×C₄	C₂²×D₄	1	112	yes	—
66	C₂×C₄	C₂⁵	1	16	yes	—
67	C₂×C₄	C₂⁵	1	16	yes	—
68	C₂×C₄	C₂⁵	1	8	yes	—
69	C₂²	—	2	16	no	—
70	C₂²	—	2	32	no	—
71	C₂²	—	4	224	no	—
72	C₂²	C₂×#124	1	112	yes	direct product factor
73	C₂²	C₂³×D₄	1	14	yes	—
74	C₂²	C₂²×#41	1	28	yes	—
75	C₂²	C₂⁶	1	1	yes	characteristic
76	C₄	—	2	16	no	—
77	C₄	—	2	16	no	—
78	C₄	C₂³×D₄	1	16	yes	—
79	C₂	—	4	16	no	—
80	C₂	C₂²×#124	1	28	yes	direct product factor
81	C₂	C₂⁴×D₄	1	1	yes	characteristic
82	C₂	C₂³×#41	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁵	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁶
prime radical	C₂⁴×D₄	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂⁴×Q₈	C₂	semidirect
3. C₂²×#124	C₂	direct (N × H)
4. C₂³×C₂²⋊C₄	C₂	semidirect
5. C₂³×C₄²	C₂	semidirect
8. C₂³×Q₈	C₂²	semidirect
14. C₂×#124	C₂²	direct (N × H)
15. C₂²×C₂²⋊C₄	C₂²	semidirect
16. C₂²×C₄²	C₂²	semidirect
22. C₂²×Q₈	C₂³	semidirect
31. #124	C₂³	direct (N × H)
32. C₂×C₂²⋊C₄	C₂³	semidirect
33. C₂×C₄²	C₂³	semidirect
42. C₂×Q₈	C₂⁴	semidirect
51. C₂²⋊C₄	C₂⁴	semidirect
52. C₄²	C₂⁴	semidirect
56. C₂³	#124	direct (N × H)
72. C₂²	C₂×#124	direct (N × H)
80. C₂	C₂²×#124	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×Q₈	2
C₂⁴×D₄	2
C₂²×#124	2
C₂³×C₂²⋊C₄	2
C₂³×C₄²	2

Subgroups — C23×#124 #62660

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#124 #62660