Subgroups — #13508

Order 256.

142 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2159	C₂	1	2	yes	maximal
2	#1945	C₂	1	1	yes	maximal, characteristic
3	#1845	C₂	1	1	yes	maximal, characteristic
4	#1387	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×#107	—	2	1	no	—
10	C₂×#107	C₂²	1	2	yes	—
11	#424	—	2	1	no	—
12	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
13	#384	—	2	1	no	—
14	#367	—	2	2	no	—
15	#366	—	2	2	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	#117	—	2	1	no	—
21	C₂×C₄⋊C₄	—	2	2	no	—
22	C₂×C₄⋊C₄	—	4	2	no	—
23	C₂×C₄²	D₄	1	1	yes	characteristic
24	M₃₂	—	2	2	no	—
25	M₃₂	Q₈	1	4	yes	—
26	C₂×C₁₆	—	2	2	no	—
27	#108	—	2	2	no	—
28	#107	—	2	2	no	—
29	#107	—	2	2	no	—
30	#107	—	4	2	no	—
31	#107	D₄	1	1	yes	characteristic
32	#107	D₄	1	1	yes	characteristic
33	#106	—	2	1	no	—
34	C₄⋊C₈	—	4	2	no	—
35	C₄×C₈	—	2	2	no	—
36	C₂²×C₄	—	2	1	no	—
37	C₂²×C₄	—	4	2	no	—
38	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
39	M₁₆	—	4	1	no	—
40	M₁₆	—	4	1	no	—
41	C₂×C₈	—	2	1	no	—
42	C₂×C₈	—	2	1	no	—
43	C₂×C₈	—	4	1	no	—
44	C₂×C₈	D₈	1	2	yes	—
45	C₂×C₈	C₂×D₄	1	1	yes	characteristic
46	C₂×C₈	C₂×Q₈	1	2	yes	—
47	C₂×C₈	#41	1	2	yes	—
48	C₂×C₈	#41	1	1	yes	characteristic
49	C₄⋊C₄	—	2	1	no	—
50	C₄⋊C₄	—	2	1	no	—
51	C₄⋊C₄	—	4	2	no	—
52	C₄⋊C₄	—	4	2	no	—
53	C₄⋊C₄	—	8	4	no	—
54	C₂²⋊C₄	—	4	1	no	—
55	C₄²	—	2	1	no	—
56	C₄²	—	4	1	no	—
57	C₄²	Q₁₆	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	—	2	2	no	—
62	C₂×C₄	—	2	2	no	—
63	C₂×C₄	—	4	1	no	—
64	C₂×C₄	—	4	1	no	—
65	C₂×C₄	—	8	4	no	—
66	C₂×C₄	—	8	2	no	—
67	C₂×C₄	#121	1	1	yes	characteristic
68	C₂×C₄	#123	1	1	yes	characteristic
69	C₂×C₄	C₄⋊Q₈	1	1	yes	characteristic
70	C₂×C₄	C₂×D₈	1	1	yes	characteristic
71	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
72	C₂×C₄	#137	1	1	yes	characteristic
73	C₈	—	2	2	no	—
74	C₈	—	8	1	no	—
75	C₈	C₂²⋊Q₈	1	2	yes	—
76	C₈	#125	1	2	yes	—
77	C₂²	—	2	1	no	—
78	C₂²	—	2	1	no	—
79	C₂²	#459	1	1	yes	characteristic
80	C₂²	#484	1	1	yes	characteristic
81	C₂²	#501	1	1	yes	characteristic
82	C₄	—	2	4	no	—
83	C₄	—	8	1	no	—
84	C₄	—	8	1	no	—
85	C₄	—	16	2	no	—
86	C₄	#398	1	1	yes	characteristic
87	C₄	#462	1	1	yes	characteristic
88	C₄	#480	1	1	yes	characteristic
89	C₄	#500	1	1	yes	characteristic
90	C₂	—	2	1	no	—
91	C₂	—	2	1	no	—
92	C₂	#2083	1	1	yes	characteristic
93	C₂	#2213	1	1	yes	characteristic
94	C₂	#2214	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#459
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
#2159	2
#1945	2
#1845	2
#1387	2
#1376	2