Subgroups — #13715

Order 256.

88 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	C₂×#366	C₂	1	1	yes	maximal, characteristic
3	#1423	C₂	1	2	yes	maximal, complemented
4	#1416	C₂	1	2	yes	maximal, complemented
5	C₂²⋊C₃₂	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₃₂	C₂²	1	1	yes	characteristic
7	C₂²×C₁₆	C₂²	1	1	yes	characteristic
8	#467	—	2	1	no	—
9	C₂×#107	—	2	1	no	—
10	C₂×C₃₂	C₂²	1	2	yes	—
11	#366	—	2	1	no	—
12	#366	C₂²	1	2	yes	—
13	#366	C₂²	1	1	yes	characteristic
14	#358	—	2	1	no	—
15	C₂×Q₁₆	—	2	1	no	—
16	C₂²×C₈	D₄	1	1	yes	characteristic
17	C₂²⋊Q₈	—	4	1	no	—
18	C₂×C₄⋊C₄	—	4	1	no	—
19	Q₃₂	—	2	2	no	—
20	C₂×C₁₆	—	2	2	no	—
21	C₂×C₁₆	D₄	1	1	yes	characteristic
22	C₂×C₁₆	C₂³	1	1	yes	characteristic
23	#107	—	2	1	no	—
24	#107	—	2	2	no	—
25	#107	—	4	1	no	—
26	#103	—	4	1	no	—
27	C₃₂	—	2	2	no	—
28	C₂×Q₈	—	4	1	no	—
29	C₂²×C₄	—	8	1	no	—
30	C₂²×C₄	D₈	1	1	yes	characteristic
31	Q₁₆	—	4	1	no	—
32	Q₁₆	—	4	1	no	—
33	C₂×C₈	—	2	1	no	—
34	C₂×C₈	—	2	1	no	—
35	C₂×C₈	D₈	1	1	yes	characteristic
36	C₂×C₈	C₂×D₄	1	1	yes	characteristic
37	C₄⋊C₄	—	4	1	no	—
38	C₄⋊C₄	—	4	2	no	—
39	C₄⋊C₄	—	8	1	no	—
40	C₄⋊C₄	—	8	1	no	—
41	C₂²⋊C₄	—	8	1	no	—
42	C₁₆	—	2	1	no	—
43	C₁₆	#41	1	2	yes	—
44	C₂³	D₁₆	1	1	yes	characteristic
45	Q₈	—	8	1	no	—
46	Q₈	—	8	1	no	—
47	C₂×C₄	—	2	1	no	—
48	C₂×C₄	—	2	1	no	—
49	C₂×C₄	—	8	1	no	—
50	C₂×C₄	—	8	1	no	—
51	C₂×C₄	—	8	2	no	—
52	C₂×C₄	—	16	2	no	—
53	C₂×C₄	D₁₆	1	1	yes	characteristic
54	C₂×C₄	C₂×D₈	1	1	yes	characteristic
55	C₈	—	2	1	no	—
56	C₈	#123	1	1	yes	characteristic
57	C₈	#137	1	1	yes	characteristic
58	C₂²	—	2	1	no	—
59	C₂²	—	2	1	no	—
60	C₂²	Q₆₄	1	2	yes	—
61	C₂²	C₂×D₁₆	1	1	yes	characteristic
62	C₄	—	2	1	no	—
63	C₄	—	16	1	no	—
64	C₄	—	16	1	no	—
65	C₄	—	16	2	no	—
66	C₄	#480	1	1	yes	characteristic
67	C₄	#510	1	1	yes	characteristic
68	C₂	—	2	2	no	—
69	C₂	#2232	1	1	yes	characteristic
70	C₂	C₂×Q₆₄	1	1	yes	characteristic
71	C₂	#2263	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. #1423	C₂	semidirect
4. #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
C₂×#366	2
#1423	2
#1416	2
C₂²⋊C₃₂	2