Subgroups — #13289

Order 256.

171 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	C₄×M₃₂	C₂	1	1	yes	maximal, characteristic
3	#1848	C₂	1	1	yes	maximal, characteristic
4	#1387	C₂	1	4	yes	maximal
5	C₂×#125	—	2	1	no	—
6	C₂×M₃₂	C₂²	1	2	yes	—
7	#499	—	2	1	no	—
8	#499	—	2	1	no	—
9	#499	C₄	1	2	yes	—
10	#499	C₄	1	2	yes	—
11	C₂×#108	—	2	1	no	—
12	C₂×#108	C₂²	1	2	yes	—
13	C₂×#107	C₂²	1	2	yes	—
14	C₂×#106	—	2	1	no	—
15	#423	—	2	1	no	—
16	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
17	#346	—	2	1	no	—
18	C₄×C₁₆	—	2	1	no	—
19	C₂×M₁₆	—	2	2	no	—
20	C₂²×C₈	D₄	1	1	yes	characteristic
21	C₂²×C₈	C₂³	1	1	yes	characteristic
22	#125	—	2	2	no	—
23	#125	—	2	2	no	—
24	#125	—	4	2	no	—
25	C₂×C₄⋊C₄	—	2	2	no	—
26	C₂×C₄⋊C₄	—	4	2	no	—
27	C₂×C₄²	D₄	1	1	yes	characteristic
28	M₃₂	—	2	2	no	—
29	M₃₂	Q₈	1	4	yes	—
30	C₂×C₁₆	—	2	2	no	—
31	#108	—	2	4	no	—
32	#108	—	4	2	no	—
33	#107	—	2	2	no	—
34	#107	C₂×C₄	1	4	yes	—
35	#106	—	2	2	no	—
36	#106	—	2	2	no	—
37	C₄⋊C₈	—	4	2	no	—
38	C₄×C₈	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈	D₄	1	1	yes	characteristic
41	C₄×C₈	D₄	1	1	yes	characteristic
42	C₂²×C₄	—	4	2	no	—
43	C₂²×C₄	SD₁₆	1	2	yes	—
44	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
45	M₁₆	—	4	2	no	—
46	M₁₆	—	4	2	no	—
47	C₂×C₈	—	4	2	no	—
48	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₈	SD₁₆	1	2	yes	—
51	C₂×C₈	Q₁₆	1	2	yes	—
52	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₈	C₂×D₄	1	1	yes	characteristic
54	C₂×C₈	C₂×Q₈	1	2	yes	—
55	C₂×C₈	#41	1	2	yes	—
56	C₄⋊C₄	—	2	4	no	—
57	C₄⋊C₄	—	4	4	no	—
58	C₄⋊C₄	—	4	4	no	—
59	C₄⋊C₄	—	4	2	no	—
60	C₄²	C₂²⋊C₄	1	1	yes	characteristic
61	C₄²	C₂²⋊C₄	1	1	yes	characteristic
62	C₄²	D₈	1	2	yes	—
63	C₁₆	—	2	4	no	—
64	C₂³	C₂×SD₁₆	1	1	yes	characteristic
65	C₂×C₄	—	4	2	no	—
66	C₂×C₄	—	4	4	no	—
67	C₂×C₄	—	8	4	no	—
68	C₂×C₄	#102	1	2	yes	—
69	C₂×C₄	#102	1	2	yes	—
70	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₄	#124	1	1	yes	characteristic
72	C₂×C₄	C₄⋊Q₈	1	1	yes	characteristic
73	C₂×C₄	C₂×D₈	1	1	yes	characteristic
74	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
75	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
76	C₈	—	8	2	no	—
77	C₈	#103	1	2	yes	—
78	C₈	#103	1	2	yes	—
79	C₈	C₄×Q₈	1	2	yes	—
80	C₈	C₂²⋊Q₈	1	2	yes	—
81	C₂²	—	2	2	no	—
82	C₂²	C₂×#102	1	1	yes	characteristic
83	C₂²	#487	1	1	yes	characteristic
84	C₂²	#498	1	1	yes	characteristic
85	C₄	—	2	2	no	—
86	C₄	—	2	2	no	—
87	C₄	—	8	4	no	—
88	C₄	#391	1	1	yes	characteristic
89	C₄	C₂×#103	1	1	yes	characteristic
90	C₄	#486	1	1	yes	characteristic
91	C₄	#500	1	1	yes	characteristic
92	C₂	—	2	1	no	—
93	C₂	—	2	1	no	—
94	C₂	#1990	1	1	yes	characteristic
95	C₂	#2148	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	C₂×#102
commutator	C₂×C₈	C₂²×C₄
Frattini, cosocle	C₂²×C₈	C₂³
prime radical	C₂³	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ 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
C₄×M₃₂	2
#1848	2
#1387	2