Subgroups — #13520

Order 256.

119 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	#2151	C₂	1	1	yes	maximal, characteristic
3	#1953	C₂	1	2	yes	maximal
4	#1385	C₂	1	1	yes	maximal, characteristic
5	#1381	C₂	1	2	yes	maximal
6	C₂×M₃₂	C₂²	1	1	yes	characteristic
7	C₂²×C₁₆	C₂²	1	1	yes	characteristic
8	#430	—	2	2	no	—
9	C₂×#108	C₂²	1	2	yes	—
10	C₂×#108	C₂²	1	2	yes	—
11	#405	C₂²	1	1	yes	characteristic
12	#368	—	2	2	no	—
13	C₄⋊C₁₆	—	2	2	no	—
14	#344	—	2	2	no	—
15	C₂×M₁₆	—	2	2	no	—
16	C₂×M₁₆	—	2	2	no	—
17	C₂×M₁₆	D₄	1	1	yes	characteristic
18	C₂²×C₈	C₂³	1	1	yes	characteristic
19	#117	D₄	1	1	yes	characteristic
20	M₃₂	—	2	2	no	—
21	C₂×C₁₆	—	2	1	no	—
22	C₂×C₁₆	—	2	2	no	—
23	C₂×C₁₆	D₄	1	2	yes	—
24	#108	—	2	2	no	—
25	#108	—	2	2	no	—
26	#108	—	2	2	no	—
27	#108	—	4	4	no	—
28	#108	Q₈	1	4	yes	—
29	#97	—	2	1	no	—
30	C₄×C₈	—	2	1	no	—
31	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
32	M₁₆	—	2	4	no	—
33	M₁₆	—	2	2	no	—
34	M₁₆	—	4	2	no	—
35	M₁₆	—	4	2	no	—
36	M₁₆	—	4	2	no	—
37	C₂×C₈	—	2	1	no	—
38	C₂×C₈	—	4	2	no	—
39	C₂×C₈	—	4	2	no	—
40	C₂×C₈	C₂×D₄	1	1	yes	characteristic
41	C₂×C₈	C₂×Q₈	1	2	yes	—
42	C₂×C₈	#41	1	2	yes	—
43	C₂×C₈	#41	1	1	yes	characteristic
44	C₄⋊C₄	—	2	1	no	—
45	C₂²⋊C₄	Q₁₆	1	2	yes	—
46	C₄²	—	2	1	no	—
47	C₁₆	—	2	2	no	—
48	C₁₆	—	4	1	no	—
49	C₂³	C₂×Q₁₆	1	1	yes	characteristic
50	C₂×C₄	—	2	2	no	—
51	C₂×C₄	#121	1	1	yes	characteristic
52	C₂×C₄	C₂²⋊Q₈	1	2	yes	—
53	C₂×C₄	#135	1	1	yes	characteristic
54	C₂×C₄	AGL(1,ℤ/8)	1	2	yes	—
55	C₈	—	4	1	no	—
56	C₈	—	4	4	no	—
57	C₈	—	8	2	no	—
58	C₈	#123	1	1	yes	characteristic
59	C₈	#125	1	2	yes	—
60	C₈	C₄⋊Q₈	1	1	yes	characteristic
61	C₂²	—	4	1	no	—
62	C₂²	#467	1	1	yes	characteristic
63	C₂²	#477	1	2	yes	—
64	C₄	—	4	2	no	—
65	C₄	#398	1	1	yes	characteristic
66	C₄	#469	1	1	yes	characteristic
67	C₄	#478	1	2	yes	—
68	C₂	—	2	1	no	—
69	C₂	—	2	2	no	—
70	C₂	#2087	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#469
commutator, Frattini, cosocle	C₂²×C₈	C₂³
socle	C₂	#2087
prime radical	C₂³	C₂×Q₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #405 ◁ 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
#2151	2
#1953	2
#1385	2
#1381	2