Subgroups — #12455

Order 256.

103 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2252	C₂	1	1	yes	maximal, characteristic, complemented
2	#2169	C₂	1	1	yes	maximal, characteristic
3	#1675	C₂	1	2	yes	maximal
4	#1372	C₂	1	1	yes	maximal, characteristic, complemented
5	#1365	C₂	1	2	yes	maximal, complemented
6	C₂×M₃₂	—	2	1	no	—
7	#500	C₂²	1	2	yes	—
8	#476	—	2	2	no	—
9	C₈×D₄	C₂²	1	1	yes	characteristic
10	#366	—	2	1	no	—
11	#366	—	2	1	no	—
12	C₄⋊C₁₆	C₂²	1	1	yes	characteristic
13	C₂²⋊C₁₆	—	2	1	no	—
14	C₄×C₁₆	C₂²	1	1	yes	characteristic
15	#335	C₂²	1	2	yes	—
16	#333	—	2	2	no	—
17	C₂²×C₈	—	2	1	no	—
18	C₄⋊Q₈	—	2	2	no	—
19	C₄×D₄	D₄	1	1	yes	characteristic
20	M₃₂	—	4	2	no	—
21	C₂×C₁₆	—	2	1	no	—
22	C₂×C₁₆	D₄	1	1	yes	characteristic
23	C₂×C₁₆	D₄	1	1	yes	characteristic
24	#107	—	2	2	no	—
25	#107	—	2	2	no	—
26	#106	—	4	2	no	—
27	C₄⋊C₈	—	2	2	no	—
28	C₄⋊C₈	D₄	1	1	yes	characteristic
29	#102	—	4	2	no	—
30	C₂²⋊C₈	—	2	1	no	—
31	C₄×C₈	C₂³	1	1	yes	characteristic
32	C₂×Q₈	—	4	2	no	—
33	C₂×D₄	D₈	1	1	yes	characteristic
34	C₂²×C₄	—	2	1	no	—
35	C₂×C₈	—	2	1	no	—
36	C₂×C₈	—	4	1	no	—
37	C₂×C₈	—	4	1	no	—
38	C₂×C₈	—	4	2	no	—
39	C₂×C₈	C₂×D₄	1	1	yes	characteristic
40	C₂×C₈	#41	1	1	yes	characteristic
41	C₄⋊C₄	—	4	2	no	—
42	C₄⋊C₄	—	4	2	no	—
43	C₄⋊C₄	D₈	1	1	yes	characteristic
44	C₂²⋊C₄	—	2	1	no	—
45	C₄²	C₂×D₄	1	1	yes	characteristic
46	C₁₆	—	2	1	no	—
47	C₁₆	—	4	1	no	—
48	C₁₆	SD₁₆	1	2	yes	—
49	C₂³	—	2	1	no	—
50	Q₈	—	16	2	no	—
51	D₄	—	2	2	no	—
52	C₂×C₄	—	2	1	no	—
53	C₂×C₄	—	4	1	no	—
54	C₂×C₄	—	4	1	no	—
55	C₂×C₄	—	8	2	no	—
56	C₂×C₄	#121	1	1	yes	characteristic
57	C₂×C₄	C₂×D₈	1	1	yes	characteristic
58	C₂×C₄	#135	1	1	yes	characteristic
59	C₈	—	2	1	no	—
60	C₈	—	4	1	no	—
61	C₈	—	8	2	no	—
62	C₈	C₂×SD₁₆	1	1	yes	characteristic
63	C₈	#137	1	1	yes	characteristic
64	C₂²	—	4	1	no	—
65	C₂²	—	4	1	no	—
66	C₂²	—	4	1	no	—
67	C₂²	#466	1	1	yes	characteristic
68	C₄	—	2	1	no	—
69	C₄	—	4	1	no	—
70	C₄	—	16	2	no	—
71	C₄	#461	1	1	yes	characteristic
72	C₄	#473	1	1	yes	characteristic
73	C₄	#510	1	2	yes	—
74	C₂	—	8	1	no	—
75	C₂	#1673	1	1	yes	characteristic
76	C₂	#2219	1	1	yes	characteristic
77	C₂	#2221	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#466
commutator, Frattini, cosocle	C₄×C₈	C₂³
prime radical	C₂×D₄	D₈
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×D₄ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #2252	C₂	semidirect
4. #1372	C₂	semidirect
5. #1365	C₂	semidirect

Maximal subgroups

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

subgroup	index
#2252	2
#2169	2
#1675	2
#1372	2
#1365	2