Subgroups — #13266

Order 256.

171 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2960	C₂	1	1	yes	maximal, characteristic
2	#2150	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic
4	#1379	C₂	1	4	yes	maximal
5	#533	C₄	1	1	yes	characteristic
6	C₂×M₃₂	C₂²	1	2	yes	—
7	C₂²×C₁₆	C₂²	1	2	yes	—
8	#446	—	2	2	no	—
9	C₈×Q₈	—	2	2	no	—
10	#424	C₄	1	2	yes	—
11	#423	C₄	1	1	yes	characteristic
12	#405	C₂²	1	2	yes	—
13	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
14	C₄⋊C₁₆	—	2	2	no	—
15	C₄×C₁₆	—	2	2	no	—
16	C₂×M₁₆	C₂×C₄	1	2	yes	—
17	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈	C₂³	1	1	yes	characteristic
19	C₄⋊Q₈	C₈	1	2	yes	—
20	#125	C₈	1	2	yes	—
21	C₂²⋊Q₈	—	2	2	no	—
22	C₄×Q₈	—	2	2	no	—
23	#117	C₂×C₄	1	2	yes	—
24	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
25	M₃₂	—	2	4	no	—
26	C₂×C₁₆	—	2	2	no	—
27	C₂×C₁₆	—	2	2	no	—
28	C₂×C₁₆	—	2	2	no	—
29	C₂×C₁₆	Q₈	1	4	yes	complemented
30	C₄⋊C₈	—	2	4	no	—
31	C₄⋊C₈	—	2	2	no	—
32	#97	—	2	2	no	—
33	C₄×C₈	—	2	2	no	—
34	C₄×C₈	D₄	1	2	yes	—
35	C₄×C₈	D₄	1	2	yes	—
36	C₂×Q₈	—	2	2	no	—
37	C₂²×C₄	M₁₆	1	2	yes	—
38	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
39	M₁₆	—	2	4	no	—
40	C₂×C₈	—	2	2	no	—
41	C₂×C₈	—	2	2	no	—
42	C₂×C₈	C₂²⋊C₄	1	2	yes	—
43	C₂×C₈	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈	C₂×D₄	1	1	yes	characteristic
45	C₂×C₈	C₂×Q₈	1	2	yes	—
46	C₂×C₈	#41	1	2	yes	—
47	C₄⋊C₄	—	2	4	no	—
48	C₄⋊C₄	—	2	2	no	—
49	C₄⋊C₄	C₂×C₈	1	4	yes	—
50	C₂²⋊C₄	—	2	2	no	—
51	C₄²	—	2	2	no	—
52	C₄²	C₂²⋊C₄	1	2	yes	—
53	C₄²	C₂×C₈	1	1	yes	characteristic
54	C₄²	C₂×C₈	1	1	yes	characteristic
55	C₁₆	—	2	4	no	—
56	C₁₆	—	4	2	no	—
57	C₂³	C₂×M₁₆	1	1	yes	characteristic
58	Q₈	—	4	4	no	—
59	C₂×C₄	—	2	4	no	—
60	C₂×C₄	—	2	4	no	—
61	C₂×C₄	C₂²⋊C₈	1	2	yes	—
62	C₂×C₄	C₂²⋊C₈	1	2	yes	—
63	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₄	C₄×Q₈	1	2	yes	—
65	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
66	C₂×C₄	#131	1	2	yes	—
67	C₈	—	2	2	no	—
68	C₈	—	4	2	no	—
69	C₈	C₂²⋊Q₈	1	2	yes	—
70	C₈	#124	1	1	yes	characteristic
71	C₈	C₄⋊Q₈	1	1	yes	characteristic
72	C₂²	—	2	2	no	—
73	C₂²	C₂×C₂²⋊C₈	1	1	yes	characteristic
74	C₂²	#446	1	2	yes	—
75	C₄	—	2	2	no	—
76	C₄	—	4	4	no	—
77	C₄	#350	1	4	yes	—
78	C₄	#391	1	1	yes	characteristic
79	C₄	#408	1	1	yes	characteristic
80	C₄	C₈×Q₈	1	2	yes	—
81	C₂	—	2	2	no	—
82	C₂	#1984	1	1	yes	characteristic
83	C₂	C₂×#350	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂×D₄
commutator	C₂×C₄	C₂²×C₈
Frattini, cosocle	C₂²×C₈	C₂³
socle	C₂²	C₂×C₂²⋊C₈
prime radical	C₂³	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×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
29. C₂×C₁₆	Q₈	semidirect

Maximal subgroups

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

subgroup	index
#2960	2
#2150	2
C₂×C₄×C₁₆	2
#1379	2