Subgroups — #11408

Order 256.

145 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	#1312	C₂	1	2	yes	maximal, complemented
5	#1311	C₂	1	2	yes	maximal, complemented
6	C₂²×C₁₆	C₄	1	2	yes	—
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	C₄	1	2	yes	—
11	C₂²⋊C₁₆	C₄	1	2	yes	—
12	C₄×C₁₆	C₄	1	2	yes	—
13	C₄×C₁₆	C₂²	1	1	yes	characteristic
14	C₄×C₁₆	C₂²	1	2	yes	—
15	C₄×C₁₆	C₂²	1	1	yes	characteristic
16	#322	C₄	1	2	yes	—
17	C₈²	C₂²	1	2	yes	—
18	C₂×M₁₆	C₈	1	2	yes	—
19	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
22	C₂×C₁₆	—	2	4	no	—
23	C₂×C₁₆	C₂×C₄	1	2	yes	—
24	C₂×C₁₆	C₂×C₄	1	4	yes	—
25	C₂×C₁₆	C₂×C₄	1	2	yes	—
26	C₄⋊C₈	C₈	1	2	yes	—
27	C₂²⋊C₈	C₈	1	2	yes	—
28	#97	C₈	1	2	yes	—
29	C₄×C₈	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈	C₂×C₄	1	2	yes	—
33	C₄×C₈	C₂×C₄	1	2	yes	—
34	C₄×C₈	C₂³	1	1	yes	characteristic
35	C₂²×C₄	C₄²	1	1	yes	characteristic
36	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
37	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
38	M₁₆	—	2	4	no	—
39	C₂×C₈	—	2	2	no	—
40	C₂×C₈	—	2	2	no	—
41	C₂×C₈	C₄²	1	1	yes	characteristic
42	C₂×C₈	C₄²	1	1	yes	characteristic
43	C₂×C₈	C₂×C₈	1	4	yes	—
44	C₂×C₈	C₂×C₈	1	4	yes	—
45	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
47	C₄²	C₄²	1	1	yes	characteristic
48	C₄²	C₂×C₈	1	1	yes	characteristic
49	C₄²	C₂×C₈	1	1	yes	characteristic
50	C₄²	C₂²×C₄	1	1	yes	characteristic
51	C₁₆	—	2	2	no	—
52	C₁₆	—	2	4	no	—
53	C₁₆	M₁₆	1	4	yes	complemented
54	C₂³	C₄×C₈	1	1	yes	characteristic
55	C₂×C₄	—	2	1	no	—
56	C₂×C₄	—	2	1	no	—
57	C₂×C₄	—	2	2	no	—
58	C₂×C₄	C₄×C₈	1	1	yes	characteristic
59	C₂×C₄	C₄×C₈	1	1	yes	characteristic
60	C₂×C₄	C₄×C₈	1	1	yes	characteristic
61	C₂×C₄	M₃₂	1	4	yes	—
62	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
63	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
64	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
65	C₈	—	2	1	no	—
66	C₈	—	2	1	no	—
67	C₈	—	2	4	no	—
68	C₈	—	2	4	no	—
69	C₈	C₂×M₁₆	1	2	yes	—
70	C₈	#131	1	2	yes	—
71	C₂²	—	2	1	no	—
72	C₂²	—	2	1	no	—
73	C₂²	#346	1	2	yes	—
74	C₂²	C₂×C₄×C₈	1	1	yes	characteristic
75	C₄	—	2	1	no	—
76	C₄	—	2	1	no	—
77	C₄	#346	1	2	yes	—
78	C₄	C₄×M₁₆	1	1	yes	characteristic
79	C₄	#405	1	1	yes	characteristic
80	C₄	C₂×M₃₂	1	2	yes	—
81	C₄	#504	1	2	yes	—
82	C₂	—	2	2	no	—
83	C₂	C₈×M₁₆	1	1	yes	characteristic
84	C₂	C₂×#346	1	1	yes	characteristic
85	C₂	#2108	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, cosocle	C₄×C₈	C₂³
commutator, socle	C₂²	C₂×C₄×C₈
prime radical	C₂³	C₄×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ 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. #1312	C₂	semidirect
5. #1311	C₂	semidirect
53. C₁₆	M₁₆	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
#1312	2
#1311	2