Subgroups — #13693

Order 256.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2264	C₂	1	2	yes	maximal
2	#2247	C₂	1	1	yes	maximal, characteristic, complemented
3	#2178	C₂	1	1	yes	maximal, characteristic
4	#1422	C₂	1	1	yes	maximal, characteristic, complemented
5	#1420	C₂	1	2	yes	maximal, complemented
6	#508	C₂²	1	1	yes	characteristic
7	C₂×Q₃₂	—	2	1	no	—
8	C₂×Q₃₂	C₂²	1	2	yes	—
9	#494	—	2	1	no	—
10	#443	—	2	1	no	—
11	Q₆₄	—	2	2	no	—
12	SD₆₄	—	2	2	no	—
13	M₆₄	C₂²	1	2	yes	—
14	#368	C₂²	1	1	yes	characteristic
15	#359	—	2	1	no	—
16	C₄×C₁₆	C₂²	1	1	yes	characteristic
17	#135	—	2	1	no	—
18	C₂×Q₁₆	—	2	2	no	—
19	C₂×Q₁₆	—	4	1	no	—
20	#131	—	4	1	no	—
21	C₄⋊Q₈	—	4	1	no	—
22	Q₃₂	—	2	2	no	—
23	Q₃₂	—	2	2	no	—
24	Q₃₂	—	4	2	no	—
25	Q₃₂	D₄	1	1	yes	characteristic
26	SD₃₂	—	2	1	no	—
27	D₁₆	D₄	1	1	yes	characteristic
28	C₂×C₁₆	D₄	1	1	yes	characteristic
29	C₂×C₁₆	C₂³	1	1	yes	characteristic
30	#108	—	2	1	no	—
31	#104	—	4	1	no	—
32	C₄×C₈	D₄	1	1	yes	characteristic
33	C₃₂	—	2	2	no	—
34	#41	—	4	1	no	—
35	C₂×Q₈	—	4	2	no	—
36	Q₁₆	—	2	1	no	—
37	Q₁₆	—	4	2	no	—
38	Q₁₆	—	4	2	no	—
39	Q₁₆	—	8	1	no	—
40	Q₁₆	—	8	1	no	—
41	SD₁₆	—	4	1	no	—
42	D₈	—	2	1	no	—
43	M₁₆	—	4	1	no	—
44	M₁₆	—	8	1	no	—
45	C₂×C₈	—	8	1	no	—
46	C₂×C₈	D₈	1	1	yes	characteristic
47	C₂×C₈	C₂×D₄	1	1	yes	characteristic
48	C₄⋊C₄	—	8	2	no	—
49	C₄²	D₈	1	1	yes	characteristic
50	C₁₆	—	2	1	no	—
51	C₁₆	C₂×D₄	1	1	yes	characteristic
52	C₁₆	#41	1	1	yes	characteristic
53	Q₈	—	4	1	no	—
54	Q₈	—	8	2	no	—
55	Q₈	—	8	2	no	—
56	D₄	—	4	1	no	—
57	D₄	—	8	1	no	—
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	—	8	1	no	—
60	C₂×C₄	—	8	2	no	—
61	C₂×C₄	C₂×D₈	1	1	yes	characteristic
62	C₈	—	8	1	no	—
63	C₈	D₁₆	1	1	yes	characteristic
64	C₈	D₁₆	1	1	yes	characteristic
65	C₈	#121	1	1	yes	characteristic
66	C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
67	C₂²	—	8	1	no	—
68	C₂²	#509	1	1	yes	characteristic
69	C₄	—	2	2	no	—
70	C₄	—	8	1	no	—
71	C₄	—	16	2	no	—
72	C₄	#459	1	1	yes	characteristic
73	C₄	C₂×D₁₆	1	1	yes	characteristic
74	C₂	—	2	1	no	—
75	C₂	—	16	1	no	—
76	C₂	#2206	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#2206
commutator, Frattini, cosocle	C₂×C₁₆	C₂³
prime radical	#508	C₂²
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₄×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. #2247	C₂	semidirect
4. #1422	C₂	semidirect
5. #1420	C₂	semidirect

Maximal subgroups

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

subgroup	index
#2264	2
#2247	2
#2178	2
#1422	2
#1420	2