Subgroups — #12008

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	#1564	C₂	1	2	yes	maximal
3	#1372	C₂	1	2	yes	maximal, complemented
4	#1371	C₂	1	2	yes	maximal, complemented
5	C₂²×C₁₆	C₄	1	1	yes	characteristic
6	C₂²×C₁₆	C₄	1	1	yes	characteristic
7	#431	—	2	2	no	—
8	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
9	C₄×C₁₆	C₄	1	1	yes	characteristic
10	C₄×C₁₆	C₄	1	1	yes	characteristic
11	C₄×C₁₆	C₂²	1	1	yes	characteristic
12	C₄×C₁₆	C₂²	1	1	yes	characteristic
13	#335	C₂²	1	4	yes	—
14	#334	—	2	2	no	—
15	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈	Q₈	1	1	yes	characteristic
17	C₂×C₄²	D₄	1	1	yes	characteristic
18	C₂×C₁₆	—	2	2	no	—
19	C₂×C₁₆	C₈	1	4	yes	complemented
20	C₂×C₁₆	C₂×C₄	1	1	yes	characteristic
21	C₂×C₁₆	C₂×C₄	1	1	yes	characteristic
22	C₂×C₁₆	C₂×C₄	1	1	yes	characteristic
23	C₂×C₁₆	C₂×C₄	1	1	yes	characteristic
24	C₄⋊C₈	—	2	4	no	—
25	C₂²⋊C₈	—	4	2	no	—
26	C₄×C₈	—	4	2	no	—
27	C₄×C₈	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈	D₄	1	1	yes	characteristic
29	C₄×C₈	Q₈	1	1	yes	characteristic
30	C₄×C₈	C₂³	1	1	yes	characteristic
31	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
32	C₂²×C₄	D₈	1	1	yes	characteristic
33	C₂²×C₄	Q₁₆	1	1	yes	characteristic
34	C₂×C₈	—	2	2	no	—
35	C₂×C₈	—	4	4	no	—
36	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₈	C₂×C₈	1	2	yes	—
39	C₂×C₈	M₁₆	1	2	yes	—
40	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
42	C₄²	C₄⋊C₄	1	1	yes	characteristic
43	C₄²	D₈	1	1	yes	characteristic
44	C₄²	Q₁₆	1	1	yes	characteristic
45	C₄²	C₂×D₄	1	1	yes	characteristic
46	C₁₆	—	2	1	no	—
47	C₁₆	—	2	1	no	—
48	C₁₆	C₂×C₈	1	2	yes	—
49	C₁₆	C₂×C₈	1	2	yes	—
50	C₂³	#107	1	1	yes	characteristic
51	C₂×C₄	—	2	2	no	—
52	C₂×C₄	—	2	2	no	—
53	C₂×C₄	C₄⋊C₈	1	2	yes	—
54	C₂×C₄	#107	1	1	yes	characteristic
55	C₂×C₄	#107	1	1	yes	characteristic
56	C₂×C₄	#107	1	1	yes	characteristic
57	C₂×C₄	#108	1	2	yes	—
58	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₄	C₂×D₈	1	1	yes	characteristic
60	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
61	C₈	—	2	1	no	—
62	C₈	—	2	1	no	—
63	C₈	—	8	4	no	—
64	C₈	C₄⋊C₈	1	2	yes	—
65	C₈	C₂²×C₈	1	1	yes	characteristic
66	C₈	C₂×M₁₆	1	1	yes	characteristic
67	C₂²	—	2	1	no	—
68	C₂²	—	2	1	no	—
69	C₂²	#335	1	2	yes	—
70	C₂²	C₂×#107	1	1	yes	characteristic
71	C₄	—	2	1	no	—
72	C₄	—	2	1	no	—
73	C₄	#335	1	2	yes	—
74	C₄	#368	1	2	yes	—
75	C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
76	C₄	C₂×#108	1	1	yes	characteristic
77	C₄	#508	1	2	yes	—
78	C₂	—	2	2	no	—
79	C₂	C₂×#335	1	1	yes	characteristic
80	C₂	#2158	1	1	yes	characteristic
81	C₂	C₂×#368	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₈
commutator	C₈	C₂²×C₈
Frattini, cosocle	C₄×C₈	C₂³
socle	C₂²	C₂×#107
prime radical	C₂³	#107
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ 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
4. #1371	C₂	semidirect
19. C₂×C₁₆	C₈	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
#1564	2
#1372	2
#1371	2