Subgroups — #13665

Order 256.

142 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₆₄	C₂	1	2	yes	maximal, complemented
3	#2216	C₂	1	1	yes	maximal, characteristic
4	#1416	C₂	1	2	yes	maximal, complemented
5	C₂²⋊C₃₂	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₁₆	—	2	1	no	—
7	C₂×Q₃₂	—	2	4	no	—
8	C₂×Q₃₂	—	2	1	no	—
9	C₂×Q₃₂	C₂²	1	1	yes	characteristic
10	C₂×Q₃₂	C₂²	1	2	yes	—
11	C₂²×C₁₆	C₂²	1	1	yes	characteristic
12	#467	—	2	1	no	—
13	Q₆₄	—	2	4	no	—
14	C₂×C₃₂	C₂²	1	2	yes	—
15	#366	C₂²	1	1	yes	characteristic
16	#358	—	2	1	no	—
17	C₂²×Q₈	—	4	1	no	—
18	C₂×Q₁₆	—	2	4	no	—
19	C₂×Q₁₆	—	2	2	no	—
20	C₂×Q₁₆	—	2	1	no	—
21	C₂×Q₁₆	—	4	2	no	—
22	C₂×Q₁₆	—	4	1	no	—
23	C₂²×C₈	D₄	1	1	yes	characteristic
24	C₂²⋊Q₈	—	4	1	no	—
25	Q₃₂	—	2	4	no	—
26	Q₃₂	—	2	2	no	—
27	Q₃₂	—	2	2	no	—
28	Q₃₂	D₄	1	4	yes	—
29	C₂×C₁₆	—	2	2	no	—
30	C₂×C₁₆	D₄	1	1	yes	characteristic
31	C₂×C₁₆	C₂³	1	1	yes	characteristic
32	#107	—	2	1	no	—
33	#103	—	4	1	no	—
34	C₃₂	—	2	2	no	—
35	C₂×Q₈	—	4	4	no	—
36	C₂×Q₈	—	4	2	no	—
37	C₂×Q₈	—	4	1	no	—
38	C₂×Q₈	—	8	2	no	—
39	C₂×Q₈	—	8	1	no	—
40	C₂²×C₄	—	8	1	no	—
41	C₂²×C₄	D₈	1	1	yes	characteristic
42	Q₁₆	—	2	4	no	—
43	Q₁₆	—	4	4	no	—
44	Q₁₆	—	4	1	no	—
45	Q₁₆	—	4	2	no	—
46	Q₁₆	—	4	1	no	—
47	C₂×C₈	—	2	1	no	—
48	C₂×C₈	—	2	1	no	—
49	C₂×C₈	D₈	1	1	yes	characteristic
50	C₂×C₈	C₂×D₄	1	1	yes	characteristic
51	C₄⋊C₄	—	4	1	no	—
52	C₄⋊C₄	—	8	1	no	—
53	C₂²⋊C₄	—	8	1	no	—
54	C₁₆	—	2	1	no	—
55	C₁₆	C₂×D₄	1	2	yes	—
56	C₂³	D₁₆	1	1	yes	characteristic
57	Q₈	—	4	4	no	—
58	Q₈	—	8	4	no	—
59	Q₈	—	8	2	no	—
60	Q₈	—	8	1	no	—
61	Q₈	—	8	1	no	—
62	C₂×C₄	—	2	1	no	—
63	C₂×C₄	—	2	1	no	—
64	C₂×C₄	—	8	4	no	—
65	C₂×C₄	—	8	2	no	—
66	C₂×C₄	—	8	1	no	—
67	C₂×C₄	—	8	1	no	—
68	C₂×C₄	D₁₆	1	1	yes	characteristic
69	C₂×C₄	C₂×D₈	1	1	yes	characteristic
70	C₈	—	2	1	no	—
71	C₈	#120	1	1	yes	characteristic
72	C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
73	C₂²	—	2	1	no	—
74	C₂²	—	2	1	no	—
75	C₂²	Q₆₄	1	2	yes	complemented
76	C₂²	C₂×D₁₆	1	1	yes	characteristic
77	C₄	—	2	1	no	—
78	C₄	—	8	4	no	—
79	C₄	—	16	1	no	—
80	C₄	—	16	1	no	—
81	C₄	#447	1	1	yes	characteristic
82	C₄	#509	1	1	yes	characteristic
83	C₂	—	2	2	no	—
84	C₂	#2184	1	1	yes	characteristic
85	C₂	C₂×Q₆₄	1	1	yes	characteristic
86	C₂	#2264	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂³
prime radical	C₂³	D₁₆
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₂×Q₆₄	C₂	semidirect
4. #1416	C₂	semidirect
75. C₂²	Q₆₄	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Q₃₂	2
C₂×Q₆₄	2
#2216	2
#1416	2
C₂²⋊C₃₂	2