Subgroups — C₂³×#108 #60110

Order 256.

1423 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×M₁₆	C₂	1	2	yes	maximal
2	C₂⁴×C₈	C₂	1	1	yes	maximal, characteristic
3	C₂²×#108	C₂	1	28	yes	maximal, direct product factor
4	C₂⁴×C₄	C₂²	1	1	yes	characteristic
5	C₂²×M₁₆	—	2	14	no	—
6	C₂²×M₁₆	C₂²	1	28	yes	—
7	C₂³×C₈	—	2	2	no	—
8	C₂³×C₈	C₄	1	2	yes	—
9	C₂³×C₈	C₄	1	14	yes	—
10	C₂³×C₈	C₂²	1	14	yes	—
11	C₂×#108	C₂²	1	112	yes	direct product factor
12	C₂⁵	Q₈	1	1	yes	characteristic
13	C₂³×C₄	C₂×C₄	1	1	yes	characteristic
14	C₂³×C₄	C₂×C₄	1	7	yes	—
15	C₂³×C₄	D₄	1	7	yes	—
16	C₂³×C₄	D₄	1	1	yes	characteristic
17	C₂³×C₄	Q₈	1	7	yes	—
18	C₂³×C₄	C₂³	1	7	yes	—
19	C₂×M₁₆	—	2	84	no	—
20	C₂×M₁₆	C₂³	1	56	yes	—
21	C₂²×C₈	—	2	28	no	—
22	C₂²×C₈	C₂×C₄	1	84	yes	—
23	C₂²×C₈	C₂×C₄	1	28	yes	—
24	C₂²×C₈	C₂³	1	28	yes	—
25	#108	C₂³	1	64	yes	direct product factor
26	C₂⁴	—	2	8	no	—
27	C₂⁴	C₄⋊C₄	1	7	yes	—
28	C₂⁴	C₄⋊C₄	1	1	yes	characteristic
29	C₂⁴	C₂×Q₈	1	7	yes	—
30	C₂²×C₄	C₄⋊C₄	1	42	yes	—
31	C₂²×C₄	C₄⋊C₄	1	7	yes	—
32	C₂²×C₄	C₄⋊C₄	1	7	yes	—
33	C₂²×C₄	C₂²×C₄	1	21	yes	—
34	C₂²×C₄	C₂²×C₄	1	7	yes	—
35	C₂²×C₄	C₂×D₄	1	7	yes	—
36	C₂²×C₄	C₂×D₄	1	21	yes	—
37	C₂²×C₄	C₂×Q₈	1	21	yes	—
38	C₂²×C₄	C₂⁴	1	7	yes	—
39	M₁₆	—	2	56	no	—
40	M₁₆	C₂⁴	1	16	yes	—
41	C₂×C₈	—	2	56	no	—
42	C₂×C₈	C₂²×C₄	1	56	yes	—
43	C₂×C₈	C₂²×C₄	1	56	yes	—
44	C₂×C₈	C₂⁴	1	8	yes	—
45	C₂³	—	2	28	no	—
46	C₂³	—	2	28	no	—
47	C₂³	#108	1	8	yes	direct product factor
48	C₂³	C₂×C₄⋊C₄	1	21	yes	—
49	C₂³	C₂×C₄⋊C₄	1	7	yes	—
50	C₂³	C₂²×Q₈	1	7	yes	—
51	C₂×C₄	C₂×C₄⋊C₄	1	42	yes	—
52	C₂×C₄	C₂×C₄⋊C₄	1	21	yes	—
53	C₂×C₄	C₂×C₄⋊C₄	1	21	yes	—
54	C₂×C₄	C₂³×C₄	1	7	yes	—
55	C₂×C₄	C₂³×C₄	1	7	yes	—
56	C₂×C₄	C₂²×D₄	1	7	yes	—
57	C₂×C₄	C₂²×D₄	1	7	yes	—
58	C₂×C₄	C₂²×Q₈	1	7	yes	—
59	C₂×C₄	C₂⁵	1	1	yes	characteristic
60	C₈	—	2	16	no	—
61	C₈	C₂³×C₄	1	16	yes	—
62	C₂²	—	2	42	no	—
63	C₂²	—	2	14	no	—
64	C₂²	C₂×#108	1	28	yes	direct product factor
65	C₂²	C₂²×C₄⋊C₄	1	7	yes	—
66	C₂²	C₂²×C₄⋊C₄	1	7	yes	—
67	C₂²	C₂³×Q₈	1	1	yes	characteristic
68	C₄	C₂²×C₄⋊C₄	1	7	yes	—
69	C₄	C₂²×C₄⋊C₄	1	7	yes	—
70	C₄	C₂⁴×C₄	1	1	yes	characteristic
71	C₄	C₂³×D₄	1	1	yes	characteristic
72	C₂	—	2	7	no	—
73	C₂	—	2	1	no	—
74	C₂	C₂²×#108	1	14	yes	direct product factor
75	C₂	C₂³×C₄⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	D₄
commutator	C₄	C₂⁴×C₄
Frattini, cosocle	C₂×C₄	C₂⁵
socle	C₂⁴	C₄⋊C₄
prime radical	C₂⁵	Q₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ 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
3. C₂²×#108	C₂	direct (N × H)
11. C₂×#108	C₂²	direct (N × H)
25. #108	C₂³	direct (N × H)
47. C₂³	#108	direct (N × H)
64. C₂²	C₂×#108	direct (N × H)
74. C₂	C₂²×#108	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×M₁₆	2
C₂⁴×C₈	2
C₂²×#108	2

Subgroups — C23×#108 #60110

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#108 #60110