Subgroups — #12427

Order 256.

115 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2209	C₂	1	2	yes	maximal
2	#1713	C₂	1	1	yes	maximal, characteristic
3	C₈×Q₁₆	C₂	1	1	yes	maximal, characteristic
4	#1370	C₂	1	1	yes	maximal, characteristic
5	#1362	C₂	1	2	yes	maximal
6	C₂×Q₃₂	—	2	2	no	—
7	#500	—	2	1	no	—
8	#500	—	2	1	no	—
9	#494	C₂²	1	2	yes	—
10	C₈×Q₈	—	2	1	no	—
11	C₄×Q₁₆	C₂²	1	1	yes	characteristic
12	C₄⋊C₁₆	C₂²	1	2	yes	—
13	#358	—	2	2	no	—
14	#335	C₂²	1	1	yes	characteristic
15	Q₈⋊C₈	—	2	1	no	—
16	C₈²	C₂²	1	1	yes	characteristic
17	C₂×Q₁₆	—	2	2	no	—
18	C₂×Q₁₆	—	2	2	no	—
19	C₂×Q₁₆	D₄	1	1	yes	characteristic
20	C₄⋊Q₈	—	2	2	no	—
21	C₄×Q₈	—	2	1	no	—
22	Q₃₂	—	4	4	no	—
23	C₂×C₁₆	—	2	2	no	—
24	#107	—	4	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	C₄⋊C₈	—	4	1	no	—
30	#103	—	2	1	no	—
31	C₄×C₈	—	4	1	no	—
32	C₄×C₈	D₄	1	1	yes	characteristic
33	C₄×C₈	D₄	1	1	yes	characteristic
34	C₄×C₈	C₂³	1	1	yes	characteristic
35	C₂×Q₈	—	2	1	no	—
36	C₂×Q₈	—	4	2	no	—
37	Q₁₆	—	2	1	no	—
38	Q₁₆	—	4	2	no	—
39	Q₁₆	—	4	2	no	—
40	Q₁₆	—	8	2	no	—
41	Q₁₆	D₈	1	2	yes	—
42	C₂×C₈	—	2	1	no	—
43	C₂×C₈	—	4	1	no	—
44	C₂×C₈	D₈	1	1	yes	characteristic
45	C₂×C₈	D₈	1	1	yes	characteristic
46	C₂×C₈	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈	#41	1	1	yes	characteristic
48	C₄⋊C₄	—	2	1	no	—
49	C₄⋊C₄	—	4	1	no	—
50	C₄⋊C₄	—	4	2	no	—
51	C₄⋊C₄	—	4	2	no	—
52	C₄²	—	4	1	no	—
53	C₄²	C₂×D₄	1	1	yes	characteristic
54	C₁₆	—	4	2	no	—
55	Q₈	—	2	2	no	—
56	Q₈	—	4	1	no	—
57	Q₈	—	8	2	no	—
58	Q₈	—	8	2	no	—
59	C₂×C₄	—	4	1	no	—
60	C₂×C₄	—	4	1	no	—
61	C₂×C₄	—	8	2	no	—
62	C₂×C₄	#121	1	1	yes	characteristic
63	C₂×C₄	C₂×D₈	1	1	yes	characteristic
64	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
65	C₈	—	2	2	no	—
66	C₈	—	2	1	no	—
67	C₈	—	2	1	no	—
68	C₈	—	8	1	no	—
69	C₈	Q₃₂	1	2	yes	complemented
70	C₈	C₂×D₈	1	1	yes	characteristic
71	C₈	#135	1	1	yes	characteristic
72	C₂²	#459	1	1	yes	characteristic
73	C₄	—	2	1	no	—
74	C₄	—	4	2	no	—
75	C₄	—	8	1	no	—
76	C₄	—	16	2	no	—
77	C₄	#466	1	1	yes	characteristic
78	C₄	#472	1	1	yes	characteristic
79	C₄	C₂×Q₃₂	1	1	yes	characteristic
80	C₄	#510	1	1	yes	characteristic
81	C₂	#1667	1	1	yes	characteristic
82	C₂	#2209	1	1	yes	characteristic
83	C₂	#2214	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²	#459
commutator, Frattini, cosocle	C₄×C₈	C₂³
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
69. C₈	Q₃₂	semidirect

Maximal subgroups

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

subgroup	index
#2209	2
#1713	2
C₈×Q₁₆	2
#1370	2
#1362	2