Subgroups — #12878

Order 256.

181 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2916	C₂	1	1	yes	maximal, characteristic
2	#2113	C₂	1	1	yes	maximal, characteristic
3	#1813	C₂	1	1	yes	maximal, characteristic
4	#1388	C₂	1	2	yes	maximal, complemented
5	#1381	C₂	1	2	yes	maximal, complemented
6	#568	—	2	1	no	—
7	C₂×#131	C₂²	1	1	yes	characteristic
8	C₂×M₃₂	C₂²	1	1	yes	characteristic
9	C₂²×C₁₆	C₂²	1	1	yes	characteristic
10	#430	—	2	1	no	—
11	#430	C₄	1	4	yes	—
12	C₂×#108	—	2	2	no	—
13	C₂×#108	C₂²	1	2	yes	—
14	C₂×#108	C₂²	1	2	yes	—
15	#413	—	2	1	no	—
16	C₂²⋊C₁₆	—	2	2	no	—
17	#338	—	2	1	no	—
18	C₂×#41	D₄	1	1	yes	characteristic
19	#131	—	2	2	no	—
20	#131	—	2	2	no	—
21	#131	—	4	4	no	—
22	C₂×M₁₆	—	2	4	no	—
23	C₂×M₁₆	—	2	2	no	—
24	C₂×M₁₆	—	2	2	no	—
25	C₂×M₁₆	C₂×C₄	1	2	yes	—
26	C₂×M₁₆	D₄	1	1	yes	characteristic
27	C₂²×C₈	D₄	1	2	yes	—
28	C₂²×C₈	C₂³	1	1	yes	characteristic
29	M₃₂	—	2	2	no	—
30	C₂×C₁₆	—	2	1	no	—
31	C₂×C₁₆	—	2	1	no	—
32	C₂×C₁₆	—	2	2	no	—
33	#108	—	2	4	no	—
34	#108	—	2	2	no	—
35	#108	—	2	2	no	—
36	#108	—	2	2	no	—
37	#108	C₂×C₄	1	4	yes	—
38	#101	—	4	1	no	—
39	#100	—	4	1	no	—
40	#41	—	2	2	no	—
41	#41	—	2	2	no	—
42	C₂×Q₈	SD₁₆	1	1	yes	characteristic
43	C₂×D₄	SD₁₆	1	1	yes	characteristic
44	C₂×D₄	Q₁₆	1	2	yes	—
45	C₂²×C₄	C₂²⋊C₄	1	2	yes	—
46	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
47	M₁₆	—	2	2	no	—
48	M₁₆	—	2	4	no	—
49	M₁₆	—	2	4	no	—
50	M₁₆	—	4	4	no	—
51	M₁₆	—	4	2	no	—
52	M₁₆	—	4	2	no	—
53	M₁₆	—	4	2	no	—
54	C₂×C₈	—	2	2	no	—
55	C₂×C₈	—	2	2	no	—
56	C₂×C₈	—	4	4	no	—
57	C₂×C₈	—	4	2	no	—
58	C₂×C₈	—	4	2	no	—
59	C₂×C₈	C₂²⋊C₄	1	2	yes	—
60	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₈	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈	#41	1	2	yes	—
63	C₂×C₈	#41	1	2	yes	—
64	C₁₆	—	4	1	no	—
65	C₁₆	—	4	1	no	—
66	C₂³	#103	1	2	yes	—
67	C₂³	C₂×Q₁₆	1	1	yes	characteristic
68	Q₈	—	2	2	no	—
69	D₄	—	2	4	no	—
70	D₄	—	2	2	no	—
71	C₂×C₄	—	2	2	no	—
72	C₂×C₄	—	2	2	no	—
73	C₂×C₄	#103	1	2	yes	—
74	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄	#123	1	2	yes	—
76	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
77	C₂×C₄	AGL(1,ℤ/8)	1	2	yes	—
78	C₈	—	2	2	no	—
79	C₈	—	4	4	no	—
80	C₈	—	8	2	no	—
81	C₈	#117	1	2	yes	—
82	C₈	#123	1	1	yes	characteristic
83	C₈	#123	1	1	yes	characteristic
84	C₂²	—	2	2	no	—
85	C₂²	—	4	1	no	—
86	C₂²	—	4	2	no	—
87	C₂²	C₂×#103	1	1	yes	characteristic
88	C₂²	#484	1	2	yes	—
89	C₄	—	2	2	no	—
90	C₄	#381	1	1	yes	characteristic
91	C₄	#418	1	1	yes	characteristic
92	C₄	#481	1	2	yes	—
93	C₂	—	2	1	no	—
94	C₂	—	2	2	no	—
95	C₂	—	4	2	no	—
96	C₂	#1824	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#418
commutator	C₂×C₈	C₂²×C₄
Frattini, cosocle	C₂²×C₈	C₂³
socle	C₂	#1824
prime radical	C₂×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
4. #1388	C₂	semidirect
5. #1381	C₂	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#2916	2
#2113	2
#1813	2
#1388	2
#1381	2