Subgroups — #12879

Order 256.

183 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊C₁₆	C₂	1	2	yes	maximal
2	C₄²×C₈	C₂	1	1	yes	maximal, characteristic
3	#1374	C₂	1	4	yes	maximal
4	C₂²×C₁₆	C₂²	1	4	yes	—
5	C₂×C₄×C₈	C₄	1	2	yes	—
6	C₂×C₄×C₈	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈	C₂²	1	2	yes	—
8	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
9	C₄³	C₄	1	1	yes	characteristic
10	C₄⋊C₁₆	—	2	4	no	—
11	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
12	C₂²×C₈	C₂×C₄	1	2	yes	—
13	C₂²×C₈	C₂³	1	1	yes	characteristic
14	C₂×C₄²	C₈	1	2	yes	—
15	C₂×C₄²	C₈	1	1	yes	characteristic
16	C₂×C₄²	C₈	1	1	yes	characteristic
17	C₂×C₄²	C₂×C₄	1	2	yes	—
18	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
19	C₂×C₁₆	—	2	4	no	—
20	C₂×C₁₆	—	2	8	no	—
21	C₄×C₈	—	2	4	no	—
22	C₄×C₈	—	2	4	no	—
23	C₄×C₈	—	2	2	no	—
24	C₄×C₈	D₄	1	2	yes	—
25	C₄×C₈	Q₈	1	2	yes	—
26	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
27	C₂²×C₄	C₂×C₈	1	2	yes	—
28	C₂²×C₄	C₂×C₈	1	2	yes	—
29	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
30	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
31	C₂×C₈	—	2	2	no	—
32	C₂×C₈	—	2	2	no	—
33	C₂×C₈	—	2	2	no	—
34	C₂×C₈	C₄⋊C₄	1	2	yes	—
35	C₂×C₈	M₁₆	1	4	yes	—
36	C₂×C₈	C₂×D₄	1	1	yes	characteristic
37	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₈	#41	1	4	yes	—
39	C₄²	—	2	4	no	—
40	C₄²	—	2	2	no	—
41	C₄²	—	2	2	no	—
42	C₄²	—	2	2	no	—
43	C₄²	C₁₆	1	4	yes	complemented
44	C₄²	C₄⋊C₄	1	2	yes	—
45	C₄²	M₁₆	1	2	yes	—
46	C₁₆	—	4	4	no	—
47	C₂³	C₂²×C₈	1	1	yes	characteristic
48	C₂×C₄	—	2	2	no	—
49	C₂×C₄	—	2	2	no	—
50	C₂×C₄	—	2	2	no	—
51	C₂×C₄	—	2	2	no	—
52	C₂×C₄	—	2	2	no	—
53	C₂×C₄	C₄⋊C₈	1	2	yes	—
54	C₂×C₄	C₄⋊C₈	1	2	yes	—
55	C₂×C₄	C₂×C₁₆	1	4	yes	—
56	C₂×C₄	C₂×C₁₆	1	2	yes	—
57	C₂×C₄	M₃₂	1	4	yes	—
58	C₂×C₄	M₃₂	1	2	yes	—
59	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄	#117	1	2	yes	—
61	C₂×C₄	C₂×M₁₆	1	2	yes	—
62	C₂×C₄	C₂×M₁₆	1	1	yes	characteristic
63	C₈	—	2	2	no	—
64	C₈	—	2	4	no	—
65	C₈	#124	1	2	yes	—
66	C₈	#125	1	2	yes	—
67	C₂²	C₂×C₄⋊C₈	1	1	yes	characteristic
68	C₂²	#431	1	2	yes	—
69	C₂²	C₂²×C₁₆	1	1	yes	characteristic
70	C₂²	C₂×M₃₂	1	1	yes	characteristic
71	C₂²	C₂×M₃₂	1	2	yes	—
72	C₄	—	2	2	no	—
73	C₄	—	2	4	no	—
74	C₄	—	2	4	no	—
75	C₄	C₄⋊C₁₆	1	4	yes	—
76	C₄	#382	1	1	yes	characteristic
77	C₄	#423	1	1	yes	characteristic
78	C₄	#432	1	2	yes	—
79	C₂	#1831	1	1	yes	characteristic
80	C₂	C₂×C₄⋊C₁₆	1	1	yes	characteristic
81	C₂	#2150	1	1	yes	characteristic
82	C₂	#2162	1	2	yes	—
83	C₂	#2163	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
43. C₄²	C₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊C₁₆	2
C₄²×C₈	2
#1374	2