Subgroups — #13716

Order 256.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2216	C₂	1	1	yes	maximal, characteristic
2	#2158	C₂	1	1	yes	maximal, characteristic
3	#1424	C₂	1	1	yes	maximal, characteristic, complemented
4	#1423	C₂	1	1	yes	maximal, characteristic, complemented
5	#1416	C₂	1	1	yes	maximal, characteristic, complemented
6	#1416	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²⋊C₃₂	C₂	1	1	yes	maximal, characteristic
8	C₂×Q₃₂	C₂²	1	1	yes	characteristic
9	C₂²×C₁₆	C₂²	1	1	yes	characteristic
10	#467	—	2	1	no	—
11	#427	—	2	1	no	—
12	C₂×C₃₂	C₂²	1	1	yes	characteristic
13	C₂×C₃₂	C₂²	1	1	yes	characteristic
14	#367	—	2	1	no	—
15	#366	C₂²	1	1	yes	characteristic
16	#366	C₂²	1	1	yes	characteristic
17	#366	C₂²	1	1	yes	characteristic
18	#358	—	2	1	no	—
19	C₂×Q₁₆	—	2	1	no	—
20	C₂²×C₈	D₄	1	1	yes	characteristic
21	C₂²⋊Q₈	—	4	1	no	—
22	#117	—	4	1	no	—
23	Q₃₂	—	2	1	no	—
24	Q₃₂	—	2	1	no	—
25	C₂×C₁₆	—	2	1	no	—
26	C₂×C₁₆	—	2	1	no	—
27	C₂×C₁₆	D₄	1	1	yes	characteristic
28	C₂×C₁₆	C₂³	1	1	yes	characteristic
29	#107	—	2	1	no	—
30	#107	—	2	1	no	—
31	#107	—	2	1	no	—
32	#106	—	4	1	no	—
33	#103	—	4	1	no	—
34	C₃₂	—	2	1	no	—
35	C₃₂	—	2	1	no	—
36	C₂×Q₈	—	4	1	no	—
37	C₂²×C₄	D₈	1	1	yes	characteristic
38	Q₁₆	—	4	1	no	—
39	Q₁₆	—	4	1	no	—
40	C₂×C₈	—	2	1	no	—
41	C₂×C₈	—	2	1	no	—
42	C₂×C₈	D₈	1	1	yes	characteristic
43	C₂×C₈	C₂×D₄	1	1	yes	characteristic
44	C₄⋊C₄	—	4	1	no	—
45	C₄⋊C₄	—	4	1	no	—
46	C₄⋊C₄	—	4	1	no	—
47	C₄⋊C₄	—	8	1	no	—
48	C₂²⋊C₄	—	8	1	no	—
49	C₂²⋊C₄	—	8	1	no	—
50	C₄²	—	8	1	no	—
51	C₁₆	—	2	1	no	—
52	C₁₆	#41	1	1	yes	characteristic
53	C₁₆	#41	1	1	yes	characteristic
54	C₂³	D₁₆	1	1	yes	characteristic
55	Q₈	—	8	1	no	—
56	Q₈	—	8	1	no	—
57	C₂×C₄	—	2	1	no	—
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	—	8	1	no	—
60	C₂×C₄	—	8	1	no	—
61	C₂×C₄	—	8	1	no	—
62	C₂×C₄	—	8	1	no	—
63	C₂×C₄	D₁₆	1	1	yes	characteristic
64	C₂×C₄	C₂×D₈	1	1	yes	characteristic
65	C₈	—	2	1	no	—
66	C₈	#123	1	1	yes	characteristic
67	C₈	#137	1	1	yes	characteristic
68	C₂²	—	2	1	no	—
69	C₂²	—	2	1	no	—
70	C₂²	—	2	1	no	—
71	C₂²	C₂×D₁₆	1	1	yes	characteristic
72	C₄	—	2	1	no	—
73	C₄	—	16	1	no	—
74	C₄	—	16	1	no	—
75	C₄	—	16	1	no	—
76	C₄	—	16	1	no	—
77	C₄	#480	1	1	yes	characteristic
78	C₄	#510	1	1	yes	characteristic
79	C₂	—	4	1	no	—
80	C₂	#2232	1	1	yes	characteristic
81	C₂	#2262	1	1	yes	characteristic
82	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
3. #1424	C₂	semidirect
4. #1423	C₂	semidirect
5. #1416	C₂	semidirect
6. #1416	C₂	semidirect

Maximal subgroups

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

subgroup	index
#2216	2
#2158	2
#1424	2
#1423	2
#1416	2
C₂²⋊C₃₂	2