Subgroups — #13652

Order 256.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×SD₆₄	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×D₁₆	C₂	1	1	yes	maximal, characteristic
4	#1423	C₂	1	1	yes	maximal, characteristic, complemented
5	#1416	C₂	1	1	yes	maximal, characteristic, complemented
6	#1415	C₂	1	1	yes	maximal, characteristic
7	#1397	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Q₃₂	C₂²	1	1	yes	characteristic
9	C₂×D₁₆	C₂²	1	1	yes	characteristic
10	C₄×Q₁₆	—	2	1	no	—
11	C₄×D₈	—	2	1	no	—
12	SD₆₄	C₄	1	4	yes	complemented
13	C₂×C₃₂	C₂²	1	1	yes	characteristic
14	C₂×C₃₂	C₂²	1	1	yes	characteristic
15	#366	C₂²	1	1	yes	characteristic
16	#366	C₂²	1	1	yes	characteristic
17	#358	—	2	1	no	—
18	#357	—	2	1	no	—
19	C₄×C₁₆	C₂²	1	1	yes	characteristic
20	C₂×Q₁₆	—	2	1	no	—
21	C₂×D₈	—	2	1	no	—
22	C₄×Q₈	—	4	1	no	—
23	C₄×D₄	—	4	1	no	—
24	Q₃₂	—	2	1	no	—
25	Q₃₂	C₂×C₄	1	2	yes	complemented
26	D₁₆	—	2	1	no	—
27	D₁₆	C₂×C₄	1	2	yes	—
28	C₂×C₁₆	D₄	1	1	yes	characteristic
29	C₂×C₁₆	C₂³	1	1	yes	characteristic
30	#107	—	2	1	no	—
31	#107	—	2	1	no	—
32	#103	—	4	1	no	—
33	#102	—	4	1	no	—
34	C₄×C₈	D₄	1	1	yes	characteristic
35	C₃₂	—	2	1	no	—
36	C₃₂	C₂×C₄	1	2	yes	complemented
37	C₂×Q₈	—	4	1	no	—
38	C₂×D₄	—	4	1	no	—
39	C₂²×C₄	—	8	1	no	—
40	Q₁₆	—	2	2	no	—
41	Q₁₆	—	4	1	no	—
42	D₈	—	2	2	no	—
43	D₈	—	4	1	no	—
44	C₂×C₈	D₈	1	1	yes	characteristic
45	C₂×C₈	C₂×D₄	1	1	yes	characteristic
46	C₄⋊C₄	—	4	1	no	—
47	C₄⋊C₄	—	4	1	no	—
48	C₄⋊C₄	—	8	1	no	—
49	C₂²⋊C₄	—	8	1	no	—
50	C₄²	—	8	1	no	—
51	C₄²	D₈	1	1	yes	characteristic
52	C₁₆	—	2	1	no	—
53	C₁₆	C₂²×C₄	1	1	yes	characteristic
54	C₁₆	#41	1	1	yes	characteristic
55	C₂³	—	8	1	no	—
56	Q₈	—	4	2	no	—
57	Q₈	—	8	1	no	—
58	D₄	—	4	2	no	—
59	D₄	—	8	1	no	—
60	C₂×C₄	—	8	1	no	—
61	C₂×C₄	—	8	1	no	—
62	C₂×C₄	—	8	1	no	—
63	C₂×C₄	—	16	2	no	—
64	C₂×C₄	D₁₆	1	1	yes	characteristic
65	C₂×C₄	D₁₆	1	1	yes	characteristic
66	C₂×C₄	C₂×D₈	1	1	yes	characteristic
67	C₈	—	2	1	no	—
68	C₈	C₄×D₄	1	1	yes	characteristic
69	C₈	#135	1	1	yes	characteristic
70	C₂²	—	8	2	no	—
71	C₂²	—	16	1	no	—
72	C₂²	—	16	1	no	—
73	C₂²	C₂×D₁₆	1	1	yes	characteristic
74	C₄	—	2	1	no	—
75	C₄	—	2	1	no	—
76	C₄	—	8	2	no	—
77	C₄	—	16	1	no	—
78	C₄	—	16	1	no	—
79	C₄	C₄×D₈	1	1	yes	characteristic
80	C₄	#508	1	1	yes	characteristic
81	C₂	—	16	2	no	—
82	C₂	C₄×D₁₆	1	1	yes	characteristic
83	C₂	#2263	1	1	yes	characteristic
84	C₂	#2264	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×D₁₆
commutator	C₁₆	C₂²×C₄
Frattini, cosocle	C₂×C₁₆	C₂³
prime radical	C₂×D₁₆	C₂²
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ 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. C₄×Q₃₂	C₂	semidirect
4. #1423	C₂	semidirect
5. #1416	C₂	semidirect
7. #1397	C₂	semidirect
12. SD₆₄	C₄	semidirect
25. Q₃₂	C₂×C₄	semidirect
36. C₃₂	C₂×C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×SD₆₄	2
C₄×Q₃₂	2
C₄×D₁₆	2
#1423	2
#1416	2
#1415	2
#1397	2