Subgroups — #12015

Order 256.

123 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2162	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₈×M₁₆	C₂	1	1	yes	maximal, characteristic
4	#1370	C₂	1	2	yes	maximal, complemented
5	#1367	C₂	1	2	yes	maximal, complemented
6	C₂²×C₁₆	—	2	1	no	—
7	#431	C₄	1	1	yes	characteristic
8	C₄×M₁₆	C₄	1	1	yes	characteristic
9	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
10	C₄⋊C₁₆	—	2	1	no	—
11	C₄⋊C₁₆	C₂²	1	1	yes	characteristic
12	C₄⋊C₁₆	C₂²	1	2	yes	—
13	C₄⋊C₁₆	C₂²	1	1	yes	characteristic
14	C₂²⋊C₁₆	—	2	1	no	—
15	C₄×C₁₆	—	2	1	no	—
16	#322	C₄	1	2	yes	—
17	C₈²	C₂²	1	2	yes	—
18	C₂×M₁₆	C₈	1	1	yes	characteristic
19	C₂×M₁₆	C₈	1	1	yes	characteristic
20	C₂²×C₈	D₄	1	1	yes	characteristic
21	C₂²×C₈	Q₈	1	1	yes	characteristic
22	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
23	C₂×C₁₆	—	2	1	no	—
24	C₂×C₁₆	—	2	1	no	—
25	C₂×C₁₆	—	2	2	no	—
26	C₂×C₁₆	—	4	2	no	—
27	C₄⋊C₈	—	2	1	no	—
28	C₂²⋊C₈	—	2	1	no	—
29	#97	C₈	1	2	yes	—
30	C₄×C₈	C₂×C₄	1	2	yes	—
31	C₄×C₈	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈	C₂×C₄	1	2	yes	—
33	C₄×C₈	D₄	1	1	yes	characteristic
34	C₄×C₈	Q₈	1	1	yes	characteristic
35	C₄×C₈	C₂³	1	1	yes	characteristic
36	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
37	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
38	C₂²×C₄	M₁₆	1	1	yes	characteristic
39	M₁₆	—	2	2	no	—
40	M₁₆	C₁₆	1	4	yes	complemented
41	C₂×C₈	—	2	1	no	—
42	C₂×C₈	—	2	1	no	—
43	C₂×C₈	—	2	2	no	—
44	C₂×C₈	—	4	1	no	—
45	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₈	C₂×C₈	1	2	yes	—
48	C₂×C₈	C₂×C₈	1	2	yes	—
49	C₂×C₈	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
51	C₄²	C₄⋊C₄	1	1	yes	characteristic
52	C₄²	C₂×C₈	1	1	yes	characteristic
53	C₄²	M₁₆	1	1	yes	characteristic
54	C₄²	C₂²×C₄	1	1	yes	characteristic
55	C₁₆	—	4	1	no	—
56	C₁₆	—	4	2	no	—
57	C₁₆	—	4	1	no	—
58	C₂³	C₄⋊C₈	1	1	yes	characteristic
59	C₂×C₄	—	2	1	no	—
60	C₂×C₄	—	2	1	no	—
61	C₂×C₄	—	4	1	no	—
62	C₂×C₄	C₄⋊C₈	1	1	yes	characteristic
63	C₂×C₄	C₄⋊C₈	1	1	yes	characteristic
64	C₂×C₄	C₄⋊C₈	1	1	yes	characteristic
65	C₂×C₄	C₂×C₁₆	1	2	yes	—
66	C₂×C₄	M₃₂	1	2	yes	—
67	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
69	C₂×C₄	C₂×M₁₆	1	1	yes	characteristic
70	C₈	—	2	1	no	—
71	C₈	—	2	1	no	—
72	C₈	—	2	1	no	—
73	C₈	—	2	2	no	—
74	C₈	—	4	2	no	—
75	C₈	C₂×C₁₆	1	4	yes	complemented
76	C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
77	C₈	#137	1	1	yes	characteristic
78	C₂²	—	2	1	no	—
79	C₂²	—	2	1	no	—
80	C₂²	C₄⋊C₁₆	1	2	yes	—
81	C₂²	C₂×C₄⋊C₈	1	1	yes	characteristic
82	C₄	—	2	1	no	—
83	C₄	—	2	1	no	—
84	C₄	—	2	1	no	—
85	C₄	C₄⋊C₁₆	1	2	yes	—
86	C₄	#428	1	1	yes	characteristic
87	C₄	#430	1	1	yes	characteristic
88	C₄	C₂²×C₁₆	1	1	yes	characteristic
89	C₄	C₂×M₃₂	1	1	yes	characteristic
90	C₂	—	2	2	no	—
91	C₂	#1565	1	1	yes	characteristic
92	C₂	C₂×C₄⋊C₁₆	1	1	yes	characteristic
93	C₂	#2153	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₄
commutator	C₄	C₂²×C₁₆
Frattini, cosocle	C₄×C₈	C₂³
socle	C₂²	C₂×C₄⋊C₈
prime radical	C₂³	C₄⋊C₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
4. #1370	C₂	semidirect
5. #1367	C₂	semidirect
40. M₁₆	C₁₆	semidirect
75. C₈	C₂×C₁₆	semidirect

Maximal subgroups

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

subgroup	index
#2162	2
C₂×C₄⋊C₁₆	2
C₈×M₁₆	2
#1370	2
#1367	2