Subgroups — #13280

Order 256.

167 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	#2150	C₂	1	1	yes	maximal, characteristic
3	C₄×#108	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	2	no	—
8	#499	C₄	1	4	yes	—
9	C₂×#108	C₂²	1	2	yes	—
10	C₂×#107	C₂²	1	2	yes	—
11	C₂×#106	—	2	1	no	—
12	C₄×M₁₆	—	2	1	no	—
13	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
14	C₄⋊C₁₆	—	2	2	no	—
15	#341	—	2	1	no	—
16	C₂×M₁₆	—	2	2	no	—
17	C₂²×C₈	D₄	1	1	yes	characteristic
18	C₂²×C₈	C₂³	1	1	yes	characteristic
19	#125	—	2	4	no	—
20	#125	—	4	1	no	—
21	#125	—	4	1	no	—
22	C₂×C₄⋊C₄	—	2	2	no	—
23	C₂×C₄⋊C₄	—	4	2	no	—
24	C₂×C₄²	D₄	1	1	yes	characteristic
25	M₃₂	—	2	4	no	—
26	C₂×C₁₆	—	2	2	no	—
27	#108	—	2	2	no	—
28	#108	Q₈	1	4	yes	—
29	#107	—	2	2	no	—
30	#107	C₂×C₄	1	4	yes	—
31	#106	—	2	4	no	—
32	#97	—	4	1	no	—
33	C₄×C₈	—	4	1	no	—
34	C₄×C₈	C₂×C₄	1	2	yes	—
35	C₄×C₈	D₄	1	2	yes	—
36	C₂²×C₄	—	4	2	no	—
37	C₂²×C₄	SD₁₆	1	2	yes	—
38	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
39	M₁₆	—	2	4	no	—
40	M₁₆	—	4	2	no	—
41	C₂×C₈	—	2	2	no	—
42	C₂×C₈	—	4	2	no	—
43	C₂×C₈	C₂²⋊C₄	1	2	yes	—
44	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
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	C₄⋊C₄	—	2	4	no	—
49	C₄⋊C₄	—	4	4	no	—
50	C₄⋊C₄	—	4	4	no	—
51	C₄⋊C₄	—	4	2	no	—
52	C₄²	C₂²⋊C₄	1	2	yes	—
53	C₄²	D₈	1	1	yes	characteristic
54	C₄²	D₈	1	1	yes	characteristic
55	C₁₆	—	4	2	no	—
56	C₂³	C₂×SD₁₆	1	1	yes	characteristic
57	C₂×C₄	—	4	2	no	—
58	C₂×C₄	—	4	4	no	—
59	C₂×C₄	—	8	4	no	—
60	C₂×C₄	#102	1	2	yes	—
61	C₂×C₄	#102	1	2	yes	—
62	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₄	C₂²⋊Q₈	1	2	yes	—
64	C₂×C₄	C₂×D₈	1	1	yes	characteristic
65	C₂×C₄	#137	1	2	yes	—
66	C₈	—	2	2	no	—
67	C₈	—	4	4	no	—
68	C₈	C₄×Q₈	1	2	yes	—
69	C₈	#124	1	1	yes	characteristic
70	C₈	C₄⋊Q₈	1	1	yes	characteristic
71	C₂²	—	2	2	no	—
72	C₂²	C₂×#102	1	1	yes	characteristic
73	C₂²	#475	1	2	yes	—
74	C₄	—	2	2	no	—
75	C₄	—	2	2	no	—
76	C₄	—	8	4	no	—
77	C₄	#391	1	1	yes	characteristic
78	C₄	#419	1	1	yes	characteristic
79	C₄	#474	1	2	yes	—
80	C₂	—	2	2	no	—
81	C₂	#1988	1	1	yes	characteristic
82	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
#2150	2
C₄×#108	2
#1387	2