Subgroups — #13543

Order 256.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#1945	C₂	1	1	yes	maximal, characteristic
2	#1844	C₂	1	1	yes	maximal, characteristic
3	#1386	C₂	1	2	yes	maximal
4	#1383	C₂	1	2	yes	maximal
5	#1376	C₂	1	1	yes	maximal, characteristic
6	C₂×M₃₂	C₂²	1	2	yes	—
7	C₂×#108	C₂²	1	1	yes	characteristic
8	#427	C₂²	1	1	yes	characteristic
9	C₂×#106	—	2	1	no	—
10	C₂×#106	C₂²	1	2	yes	—
11	#424	—	2	1	no	—
12	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
13	#384	—	2	1	no	—
14	#339	—	2	1	no	—
15	C₂×M₁₆	—	2	1	no	—
16	C₂²×C₈	D₄	1	1	yes	characteristic
17	C₂²×C₈	C₂³	1	1	yes	characteristic
18	#117	—	2	1	no	—
19	C₂×C₄⋊C₄	—	2	2	no	—
20	C₂×C₄⋊C₄	—	4	2	no	—
21	C₂×C₄²	D₄	1	1	yes	characteristic
22	M₃₂	—	2	2	no	—
23	M₃₂	—	2	2	no	—
24	C₂×C₁₆	—	2	2	no	—
25	#108	—	2	2	no	—
26	#107	Q₈	1	2	yes	—
27	#106	—	2	1	no	—
28	#106	—	2	4	no	—
29	#106	—	4	2	no	—
30	C₄⋊C₈	—	4	2	no	—
31	C₄×C₈	—	2	2	no	—
32	C₂²×C₄	—	2	1	no	—
33	C₂²×C₄	—	4	2	no	—
34	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
35	M₁₆	—	4	1	no	—
36	M₁₆	—	4	1	no	—
37	C₂×C₈	—	2	1	no	—
38	C₂×C₈	—	2	1	no	—
39	C₂×C₈	—	4	1	no	—
40	C₂×C₈	SD₁₆	1	2	yes	—
41	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₈	#41	1	2	yes	—
43	C₂×C₈	#41	1	2	yes	—
44	C₂×C₈	#41	1	1	yes	characteristic
45	C₄⋊C₄	—	2	2	no	—
46	C₄⋊C₄	—	4	2	no	—
47	C₄⋊C₄	—	4	2	no	—
48	C₄⋊C₄	—	8	4	no	—
49	C₂²⋊C₄	—	4	1	no	—
50	C₄²	—	2	1	no	—
51	C₄²	—	4	1	no	—
52	C₄²	Q₁₆	1	2	yes	—
53	C₁₆	—	4	2	no	—
54	C₂³	AGL(1,ℤ/8)	1	1	yes	characteristic
55	C₂×C₄	—	2	4	no	—
56	C₂×C₄	—	2	2	no	—
57	C₂×C₄	—	4	2	no	—
58	C₂×C₄	—	8	4	no	—
59	C₂×C₄	—	8	2	no	—
60	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
61	C₂×C₄	#123	1	1	yes	characteristic
62	C₂×C₄	#124	1	1	yes	characteristic
63	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
64	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
65	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
66	C₈	—	2	2	no	—
67	C₈	—	8	1	no	—
68	C₈	#125	1	2	yes	—
69	C₈	#126	1	2	yes	—
70	C₂²	—	2	2	no	—
71	C₂²	#476	1	1	yes	characteristic
72	C₂²	#484	1	1	yes	characteristic
73	C₂²	#490	1	1	yes	characteristic
74	C₄	—	2	4	no	—
75	C₄	—	8	2	no	—
76	C₄	—	16	2	no	—
77	C₄	#400	1	1	yes	characteristic
78	C₄	#477	1	1	yes	characteristic
79	C₄	#481	1	1	yes	characteristic
80	C₄	#487	1	1	yes	characteristic
81	C₂	—	2	1	no	—
82	C₂	—	2	1	no	—
83	C₂	#2100	1	1	yes	characteristic
84	C₂	#2230	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	#476
commutator, Frattini, cosocle	C₂²×C₈	C₂³
prime radical	C₂³	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ 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
#1945	2
#1844	2
#1386	2
#1383	2
#1376	2