Subgroups — #7209

Order 256.

111 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	C₂²×C₁₆	C₄	1	2	yes	—
5	#431	C₄	1	1	yes	characteristic
6	C₄×M₁₆	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
8	C₄⋊C₁₆	—	2	1	no	—
9	C₄⋊C₁₆	—	2	1	no	—
10	C₄⋊C₁₆	—	2	1	no	—
11	C₂²⋊C₁₆	C₄	1	2	yes	—
12	C₄×C₁₆	—	2	1	no	—
13	#322	—	2	1	no	—
14	C₈²	—	2	1	no	—
15	C₂×M₁₆	C₈	1	2	yes	—
16	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
19	C₂×C₁₆	—	2	2	no	—
20	C₂×C₁₆	—	2	2	no	—
21	C₂×C₁₆	—	4	2	no	—
22	C₄⋊C₈	—	2	1	no	—
23	C₂²⋊C₈	C₈	1	2	yes	—
24	#97	—	2	1	no	—
25	C₄×C₈	—	2	1	no	—
26	C₄×C₈	—	2	1	no	—
27	C₄×C₈	D₄	1	1	yes	characteristic
28	C₄×C₈	D₄	1	1	yes	characteristic
29	C₄×C₈	D₄	1	1	yes	characteristic
30	C₄×C₈	Q₈	1	1	yes	characteristic
31	C₂²×C₄	C₄²	1	1	yes	characteristic
32	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
33	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
34	M₁₆	—	2	2	no	—
35	M₁₆	C₁₆	1	4	yes	complemented
36	C₂×C₈	—	2	2	no	—
37	C₂×C₈	—	2	1	no	—
38	C₂×C₈	—	2	1	no	—
39	C₂×C₈	—	2	2	no	—
40	C₂×C₈	—	4	1	no	—
41	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
45	C₄²	C₂²⋊C₄	1	1	yes	characteristic
46	C₄²	C₄⋊C₄	1	1	yes	characteristic
47	C₄²	M₁₆	1	1	yes	characteristic
48	C₄²	M₁₆	1	1	yes	characteristic
49	C₁₆	—	4	2	no	—
50	C₁₆	—	4	2	no	—
51	C₂³	C₄×C₈	1	1	yes	characteristic
52	C₂×C₄	—	2	1	no	—
53	C₂×C₄	—	2	1	no	—
54	C₂×C₄	—	4	1	no	—
55	C₂×C₄	#95	1	1	yes	characteristic
56	C₂×C₄	#97	1	1	yes	characteristic
57	C₂×C₄	C₂²⋊C₈	1	1	yes	characteristic
58	C₂×C₄	C₂²⋊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	C₂×C₁₆	1	2	yes	—
62	C₂×C₄	M₃₂	1	2	yes	—
63	C₈	—	2	1	no	—
64	C₈	—	2	1	no	—
65	C₈	—	2	1	no	—
66	C₈	—	2	4	no	—
67	C₈	—	4	2	no	—
68	C₈	#100	1	1	yes	characteristic
69	C₈	#101	1	1	yes	characteristic
70	C₂²	—	2	1	no	—
71	C₂²	—	2	1	no	—
72	C₂²	#336	1	1	yes	characteristic
73	C₂²	C₄×C₁₆	1	1	yes	characteristic
74	C₂²	#346	1	1	yes	characteristic
75	C₄	—	2	1	no	—
76	C₄	—	2	1	no	—
77	C₄	—	2	1	no	—
78	C₄	#343	1	1	yes	characteristic
79	C₄	#344	1	1	yes	characteristic
80	C₄	C₂²⋊C₁₆	1	2	yes	—
81	C₄	C₄⋊C₁₆	1	2	yes	—
82	C₂	—	2	1	no	—
83	C₂	—	2	1	no	—
84	C₂	#1283	1	1	yes	characteristic
85	C₂	#1374	1	1	yes	characteristic
86	C₂	#1378	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²⋊C₄
commutator	C₂²	C₄×C₁₆
Frattini, cosocle	C₂×C₄×C₈	C₂²
socle	C₂²	#336
prime radical	C₂³	C₄×C₈
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
35. M₁₆	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