Subgroups — #13458

Order 256.

162 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#499	C₂	1	1	yes	maximal, characteristic
2	C₂×#365	C₂	1	2	yes	maximal
3	#1945	C₂	1	1	yes	maximal, characteristic
4	#1386	C₂	1	2	yes	maximal
5	#1376	C₂	1	1	yes	maximal, characteristic
6	C₂×#125	—	2	1	no	—
7	C₂×M₃₂	C₂²	1	2	yes	—
8	#499	—	2	4	no	—
9	C₂×#108	C₂²	1	1	yes	characteristic
10	#427	C₂²	1	1	yes	characteristic
11	C₂×#107	—	2	1	no	—
12	C₂×#106	C₂²	1	2	yes	—
13	#424	—	2	1	no	—
14	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
15	#365	—	2	4	no	—
16	#339	—	2	1	no	—
17	C₂×M₁₆	—	2	1	no	—
18	C₂²×C₈	D₄	1	1	yes	characteristic
19	C₂²×C₈	C₂³	1	1	yes	characteristic
20	#125	—	2	4	no	—
21	#125	—	4	2	no	—
22	#117	—	2	1	no	—
23	C₂×C₄⋊C₄	—	2	2	no	—
24	C₂×C₄⋊C₄	—	4	2	no	—
25	C₂×C₄²	D₄	1	1	yes	characteristic
26	M₃₂	—	2	2	no	—
27	M₃₂	Q₈	1	4	yes	—
28	C₂×C₁₆	—	2	2	no	—
29	#108	—	2	2	no	—
30	#107	—	2	4	no	—
31	#107	Q₈	1	2	yes	—
32	#106	—	2	2	no	—
33	#106	—	2	1	no	—
34	#106	D₄	1	4	yes	—
35	C₄⋊C₈	—	4	2	no	—
36	C₄×C₈	—	2	2	no	—
37	C₂²×C₄	—	2	1	no	—
38	C₂²×C₄	—	4	2	no	—
39	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
40	M₁₆	—	4	1	no	—
41	M₁₆	—	4	1	no	—
42	C₂×C₈	—	2	1	no	—
43	C₂×C₈	—	2	1	no	—
44	C₂×C₈	—	4	1	no	—
45	C₂×C₈	SD₁₆	1	2	yes	—
46	C₂×C₈	C₂×D₄	1	2	yes	—
47	C₂×C₈	C₂×Q₈	1	2	yes	—
48	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₈	#41	1	1	yes	characteristic
50	C₄⋊C₄	—	2	4	no	—
51	C₄⋊C₄	—	2	2	no	—
52	C₄⋊C₄	—	4	8	no	—
53	C₄⋊C₄	—	4	2	no	—
54	C₂²⋊C₄	—	4	1	no	—
55	C₄²	—	2	1	no	—
56	C₄²	—	4	1	no	—
57	C₄²	SD₁₆	1	2	yes	—
58	C₁₆	—	2	4	no	—
59	C₂³	AGL(1,ℤ/8)	1	1	yes	characteristic
60	C₂×C₄	—	2	2	no	—
61	C₂×C₄	—	4	2	no	—
62	C₂×C₄	—	4	4	no	—
63	C₂×C₄	—	4	2	no	—
64	C₂×C₄	—	8	4	no	—
65	C₂×C₄	#120	1	1	yes	characteristic
66	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
67	C₂×C₄	C₄⋊Q₈	1	1	yes	characteristic
68	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
69	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
70	C₂×C₄	#137	1	1	yes	characteristic
71	C₈	—	2	2	no	—
72	C₈	—	8	1	no	—
73	C₈	C₂²⋊Q₈	1	2	yes	—
74	C₈	C₄⋊Q₈	1	2	yes	—
75	C₂²	—	2	2	no	—
76	C₂²	#450	1	1	yes	characteristic
77	C₂²	#476	1	1	yes	characteristic
78	C₂²	#501	1	1	yes	characteristic
79	C₄	—	4	2	no	—
80	C₄	—	8	4	no	—
81	C₄	—	8	2	no	—
82	C₄	#395	1	1	yes	characteristic
83	C₄	#448	1	1	yes	characteristic
84	C₄	#475	1	1	yes	characteristic
85	C₄	#498	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	2	1	no	—
88	C₂	#2058	1	1	yes	characteristic
89	C₂	#2231	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
C₂×#499	2
C₂×#365	2
#1945	2
#1386	2
#1376	2