Subgroups — #13519

Order 256.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2161	C₂	1	1	yes	maximal, characteristic
2	#2150	C₂	1	1	yes	maximal, characteristic
3	#1945	C₂	1	2	yes	maximal
4	#1389	C₂	1	2	yes	maximal
5	#1384	C₂	1	1	yes	maximal, characteristic
6	C₂×M₃₂	C₂²	1	1	yes	characteristic
7	C₂×M₃₂	C₂²	1	1	yes	characteristic
8	C₂×#108	C₂²	1	2	yes	—
9	#427	C₂²	1	2	yes	—
10	#424	—	2	2	no	—
11	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
12	#368	—	2	2	no	—
13	C₄⋊C₁₆	—	2	2	no	—
14	#339	—	2	2	no	—
15	C₂×M₁₆	—	2	2	no	—
16	C₂²×C₈	D₄	1	1	yes	characteristic
17	C₂²×C₈	C₂³	1	1	yes	characteristic
18	#117	—	2	2	no	—
19	C₂×C₄²	D₄	1	1	yes	characteristic
20	M₃₂	—	2	2	no	—
21	M₃₂	—	2	2	no	—
22	C₂×C₁₆	—	2	1	no	—
23	C₂×C₁₆	D₄	1	2	yes	—
24	#108	—	2	2	no	—
25	#108	—	2	2	no	—
26	#107	—	2	2	no	—
27	#106	Q₈	1	4	yes	—
28	C₄⋊C₈	—	4	4	no	—
29	C₄×C₈	—	2	1	no	—
30	C₄×C₈	—	2	1	no	—
31	C₂²×C₄	—	2	1	no	—
32	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
33	M₁₆	—	4	2	no	—
34	M₁₆	—	4	2	no	—
35	C₂×C₈	—	2	1	no	—
36	C₂×C₈	—	2	2	no	—
37	C₂×C₈	—	4	2	no	—
38	C₂×C₈	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈	C₂×Q₈	1	2	yes	—
40	C₂×C₈	#41	1	2	yes	—
41	C₂×C₈	#41	1	1	yes	characteristic
42	C₄⋊C₄	—	2	4	no	—
43	C₂²⋊C₄	—	4	2	no	—
44	C₄²	—	2	1	no	—
45	C₄²	—	4	2	no	—
46	C₄²	Q₁₆	1	1	yes	characteristic
47	C₄²	Q₁₆	1	1	yes	characteristic
48	C₁₆	—	2	2	no	—
49	C₁₆	—	4	1	no	—
50	C₂³	#135	1	1	yes	characteristic
51	C₂×C₄	—	2	1	no	—
52	C₂×C₄	—	2	1	no	—
53	C₂×C₄	—	4	2	no	—
54	C₂×C₄	—	4	4	no	—
55	C₂×C₄	#121	1	1	yes	characteristic
56	C₂×C₄	C₂²⋊Q₈	1	2	yes	—
57	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
58	C₂×C₄	AGL(1,ℤ/8)	1	2	yes	—
59	C₈	—	4	1	no	—
60	C₈	—	8	2	no	—
61	C₈	#123	1	1	yes	characteristic
62	C₈	#125	1	2	yes	—
63	C₈	C₄⋊Q₈	1	1	yes	characteristic
64	C₂²	—	4	1	no	—
65	C₂²	#467	1	1	yes	characteristic
66	C₂²	#478	1	2	yes	—
67	C₄	—	2	2	no	—
68	C₄	—	4	1	no	—
69	C₄	—	8	4	no	—
70	C₄	#398	1	1	yes	characteristic
71	C₄	#469	1	1	yes	characteristic
72	C₄	#477	1	2	yes	—
73	C₂	—	2	1	no	—
74	C₂	—	2	2	no	—
75	C₂	#2087	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#469
commutator, Frattini, cosocle	C₂²×C₈	C₂³
socle	C₂	#2087
prime radical	C₂³	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Maximal subgroups

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

subgroup	index
#2161	2
#2150	2
#1945	2
#1389	2
#1384	2