Subgroups — #13657

Order 256.

91 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2264	C₂	1	1	yes	maximal, characteristic
2	#2179	C₂	1	1	yes	maximal, characteristic
3	#2176	C₂	1	1	yes	maximal, characteristic, complemented
4	#1426	C₂	1	1	yes	maximal, characteristic, complemented
5	#1420	C₂	1	1	yes	maximal, characteristic, complemented
6	#1418	C₂	1	1	yes	maximal, characteristic
7	#1398	C₂	1	1	yes	maximal, characteristic, complemented
8	#508	C₂²	1	1	yes	characteristic
9	C₂×Q₃₂	C₂²	1	1	yes	characteristic
10	#443	—	2	1	no	—
11	C₄×Q₁₆	—	2	1	no	—
12	Q₆₄	C₄	1	2	yes	complemented
13	SD₆₄	C₄	1	2	yes	complemented
14	M₆₄	C₂²	1	1	yes	characteristic
15	M₆₄	C₂²	1	1	yes	characteristic
16	#368	C₂²	1	1	yes	characteristic
17	#367	C₂²	1	1	yes	characteristic
18	#359	—	2	1	no	—
19	#358	—	2	1	no	—
20	#346	C₂²	1	1	yes	characteristic
21	#135	—	2	1	no	—
22	C₂×Q₁₆	—	2	1	no	—
23	#131	—	4	1	no	—
24	C₄×Q₈	—	4	1	no	—
25	Q₃₂	—	2	1	no	—
26	Q₃₂	C₂×C₄	1	2	yes	—
27	Q₃₂	C₂×C₄	1	1	yes	characteristic
28	SD₃₂	—	2	1	no	—
29	D₁₆	C₂×C₄	1	1	yes	characteristic
30	C₂×C₁₆	D₄	1	1	yes	characteristic
31	C₂×C₁₆	C₂³	1	1	yes	characteristic
32	#108	—	2	1	no	—
33	#107	—	2	1	no	—
34	#104	—	4	1	no	—
35	#103	—	4	1	no	—
36	C₄×C₈	D₄	1	1	yes	characteristic
37	C₃₂	—	2	1	no	—
38	C₃₂	C₂×C₄	1	2	yes	—
39	#41	—	4	1	no	—
40	C₂×Q₈	—	4	1	no	—
41	Q₁₆	—	2	1	no	—
42	Q₁₆	—	2	2	no	—
43	Q₁₆	—	4	1	no	—
44	SD₁₆	—	4	1	no	—
45	D₈	—	2	1	no	—
46	M₁₆	—	4	1	no	—
47	M₁₆	—	8	1	no	—
48	C₂×C₈	—	8	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	C₄²	—	8	1	no	—
54	C₄²	D₈	1	1	yes	characteristic
55	C₁₆	—	2	1	no	—
56	C₁₆	C₂²×C₄	1	1	yes	characteristic
57	C₁₆	#41	1	1	yes	characteristic
58	Q₈	—	4	1	no	—
59	Q₈	—	4	2	no	—
60	Q₈	—	8	1	no	—
61	D₄	—	4	1	no	—
62	D₄	—	8	1	no	—
63	C₂×C₄	—	2	1	no	—
64	C₂×C₄	—	8	1	no	—
65	C₂×C₄	—	8	1	no	—
66	C₂×C₄	—	8	1	no	—
67	C₂×C₄	C₂×D₈	1	1	yes	characteristic
68	C₈	—	2	1	no	—
69	C₈	—	8	1	no	—
70	C₈	C₄×D₄	1	1	yes	characteristic
71	C₈	#135	1	1	yes	characteristic
72	C₂²	—	8	1	no	—
73	C₂²	#509	1	1	yes	characteristic
74	C₄	—	4	1	no	—
75	C₄	—	8	1	no	—
76	C₄	—	8	2	no	—
77	C₄	—	16	1	no	—
78	C₄	C₄×D₈	1	1	yes	characteristic
79	C₄	#509	1	1	yes	characteristic
80	C₂	—	2	1	no	—
81	C₂	—	16	1	no	—
82	C₂	#2177	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#2177
commutator	C₁₆	C₂²×C₄
Frattini, cosocle	C₂×C₁₆	C₂³
prime radical	#508	C₂²
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #346 ◁ 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
3. #2176	C₂	semidirect
4. #1426	C₂	semidirect
5. #1420	C₂	semidirect
7. #1398	C₂	semidirect
12. Q₆₄	C₄	semidirect
13. SD₆₄	C₄	semidirect

Maximal subgroups

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

subgroup	index
#2264	2
#2179	2
#2176	2
#1426	2
#1420	2
#1418	2
#1398	2