Subgroups — #12023

Order 256.

121 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	#1567	C₂	1	2	yes	maximal
3	#1372	C₂	1	4	yes	maximal, complemented
4	C₂²×C₁₆	C₄	1	2	yes	—
5	#423	—	2	2	no	—
6	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
7	C₄×C₁₆	C₄	1	1	yes	characteristic
8	C₄×C₁₆	C₄	1	1	yes	characteristic
9	C₄×C₁₆	C₂²	1	2	yes	—
10	#335	—	2	2	no	—
11	#335	C₂²	1	4	yes	—
12	C₂×M₁₆	—	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	4	no	—
17	C₂×C₁₆	C₂×C₄	1	2	yes	—
18	C₂×C₁₆	C₂×C₄	1	2	yes	—
19	C₄⋊C₈	—	2	4	no	—
20	C₄⋊C₈	—	4	2	no	—
21	C₄×C₈	C₂×C₄	1	1	yes	characteristic
22	C₄×C₈	D₄	1	1	yes	characteristic
23	C₄×C₈	Q₈	1	1	yes	characteristic
24	C₄×C₈	C₂³	1	1	yes	characteristic
25	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
26	C₂²×C₄	D₈	1	1	yes	characteristic
27	C₂²×C₄	Q₁₆	1	1	yes	characteristic
28	M₁₆	—	8	4	no	—
29	C₂×C₈	—	2	2	no	—
30	C₂×C₈	—	2	2	no	—
31	C₂×C₈	—	4	4	no	—
32	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
36	C₄²	C₄⋊C₄	1	1	yes	characteristic
37	C₄²	D₈	1	1	yes	characteristic
38	C₄²	Q₁₆	1	1	yes	characteristic
39	C₄²	C₂×D₄	1	1	yes	characteristic
40	C₁₆	—	2	2	no	—
41	C₁₆	M₁₆	1	4	yes	complemented
42	C₂³	#107	1	1	yes	characteristic
43	C₂×C₄	—	2	2	no	—
44	C₂×C₄	—	2	1	no	—
45	C₂×C₄	—	2	1	no	—
46	C₂×C₄	#107	1	1	yes	characteristic
47	C₂×C₄	#107	1	1	yes	characteristic
48	C₂×C₄	#107	1	1	yes	characteristic
49	C₂×C₄	SD₃₂	1	4	yes	—
50	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₄	C₂×D₈	1	1	yes	characteristic
52	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
53	C₈	—	2	1	no	—
54	C₈	—	2	1	no	—
55	C₈	—	8	4	no	—
56	C₈	#108	1	2	yes	—
57	C₈	C₂×M₁₆	1	2	yes	—
58	C₂²	—	2	1	no	—
59	C₂²	—	2	1	no	—
60	C₂²	#367	1	2	yes	—
61	C₂²	C₂×#107	1	1	yes	characteristic
62	C₄	—	2	1	no	—
63	C₄	—	2	1	no	—
64	C₄	#367	1	2	yes	—
65	C₄	#368	1	2	yes	—
66	C₄	#423	1	1	yes	characteristic
67	C₄	C₂×#108	1	1	yes	characteristic
68	C₄	C₂×SD₃₂	1	2	yes	—
69	C₂	—	2	2	no	—
70	C₂	#1567	1	1	yes	characteristic
71	C₂	C₂×#367	1	1	yes	characteristic
72	C₂	C₂×#368	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₈
commutator	C₂×C₈	C₂²×C₄
Frattini, cosocle	C₄×C₈	C₂³
socle	C₂²	C₂×#107
prime radical	C₂³	#107
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×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	semidirect
41. C₁₆	M₁₆	semidirect

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
#1567	2
#1372	2