Subgroups — #12499

Order 256.

113 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2226	C₂	1	2	yes	maximal
2	#1713	C₂	1	1	yes	maximal, characteristic, complemented
3	#1589	C₂	1	1	yes	maximal, characteristic
4	#1367	C₂	1	1	yes	maximal, characteristic, complemented
5	#1365	C₂	1	2	yes	maximal, complemented
6	#500	—	2	1	no	—
7	#500	C₂²	1	2	yes	—
8	#494	—	2	1	no	—
9	C₄×D₈	C₂²	1	1	yes	characteristic
10	#436	—	2	1	no	—
11	#367	—	2	2	no	—
12	C₄⋊C₁₆	C₂²	1	2	yes	—
13	#357	—	2	2	no	—
14	#335	C₂²	1	1	yes	characteristic
15	D₄⋊C₈	—	2	1	no	—
16	C₈²	C₂²	1	1	yes	characteristic
17	C₂×Q₁₆	—	4	2	no	—
18	C₂×D₈	D₄	1	1	yes	characteristic
19	C₂×M₁₆	—	4	1	no	—
20	C₄⋊Q₈	—	2	2	no	—
21	C₄×D₄	—	2	1	no	—
22	C₂×C₁₆	—	2	2	no	—
23	#107	—	2	2	no	—
24	#107	—	2	2	no	—
25	#107	—	4	1	no	—
26	#107	—	4	1	no	—
27	#107	D₄	1	1	yes	characteristic
28	C₄⋊C₈	—	2	1	no	—
29	#102	—	2	1	no	—
30	C₂²⋊C₈	—	4	1	no	—
31	C₄×C₈	D₄	1	1	yes	characteristic
32	C₄×C₈	D₄	1	1	yes	characteristic
33	C₄×C₈	C₂³	1	1	yes	characteristic
34	C₂×Q₈	—	4	2	no	—
35	C₂×D₄	—	2	1	no	—
36	C₂²×C₄	—	4	1	no	—
37	Q₁₆	—	8	4	no	—
38	D₈	—	2	2	no	—
39	M₁₆	—	8	2	no	—
40	C₂×C₈	—	2	1	no	—
41	C₂×C₈	—	4	1	no	—
42	C₂×C₈	D₈	1	1	yes	characteristic
43	C₂×C₈	D₈	1	1	yes	characteristic
44	C₂×C₈	C₂×D₄	1	1	yes	characteristic
45	C₂×C₈	#41	1	1	yes	characteristic
46	C₄⋊C₄	—	2	1	no	—
47	C₄⋊C₄	—	4	2	no	—
48	C₄⋊C₄	—	4	2	no	—
49	C₂²⋊C₄	—	4	1	no	—
50	C₄²	C₂×D₄	1	1	yes	characteristic
51	C₁₆	—	4	2	no	—
52	C₂³	—	4	1	no	—
53	Q₈	—	8	4	no	—
54	D₄	—	4	1	no	—
55	D₄	—	4	1	no	—
56	C₂×C₄	—	4	1	no	—
57	C₂×C₄	—	8	1	no	—
58	C₂×C₄	—	8	1	no	—
59	C₂×C₄	—	8	2	no	—
60	C₂×C₄	#121	1	1	yes	characteristic
61	C₂×C₄	C₂×D₈	1	1	yes	characteristic
62	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
63	C₈	—	2	2	no	—
64	C₈	—	2	1	no	—
65	C₈	—	2	1	no	—
66	C₈	—	8	1	no	—
67	C₈	SD₃₂	1	2	yes	—
68	C₈	#137	1	2	yes	—
69	C₂²	—	8	1	no	—
70	C₂²	—	8	1	no	—
71	C₂²	—	8	1	no	—
72	C₂²	#459	1	1	yes	characteristic
73	C₄	—	2	1	no	—
74	C₄	—	8	1	no	—
75	C₄	—	16	2	no	—
76	C₄	#470	1	1	yes	characteristic
77	C₄	#473	1	1	yes	characteristic
78	C₄	C₂×SD₃₂	1	1	yes	characteristic
79	C₄	#510	1	1	yes	characteristic
80	C₂	—	16	1	no	—
81	C₂	#1689	1	1	yes	characteristic
82	C₂	#2208	1	1	yes	characteristic
83	C₂	#2213	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#459
commutator, Frattini, cosocle	C₄×C₈	C₂³
prime radical	C₂×D₈	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ 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
2. #1713	C₂	semidirect
4. #1367	C₂	semidirect
5. #1365	C₂	semidirect

Maximal subgroups

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

subgroup	index
#2226	2
#1713	2
#1589	2
#1367	2
#1365	2