Subgroups — #12045

Order 256.

127 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×D₁₆	C₂	1	1	yes	maximal, characteristic
2	C₈×D₈	C₂	1	2	yes	maximal, complemented
3	#1372	C₂	1	1	yes	maximal, characteristic, complemented
4	#1331	C₂	1	2	yes	maximal, complemented
5	#1311	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₆	C₄	1	1	yes	characteristic
7	C₄×D₈	C₂²	1	2	yes	—
8	C₈×D₄	—	2	2	no	—
9	#366	C₄	1	1	yes	characteristic
10	#357	C₄	1	2	yes	—
11	C₄×C₁₆	C₂²	1	1	yes	characteristic
12	C₄×C₁₆	C₂²	1	1	yes	characteristic
13	#335	C₂²	1	2	yes	—
14	D₄⋊C₈	—	2	2	no	—
15	C₈²	C₂²	1	1	yes	characteristic
16	C₂×D₈	C₂×C₄	1	2	yes	—
17	C₂²×C₈	—	4	2	no	—
18	C₄×D₄	—	2	2	no	—
19	D₁₆	C₈	1	4	yes	complemented
20	C₂×C₁₆	—	2	1	no	—
21	C₂×C₁₆	C₂×C₄	1	1	yes	characteristic
22	C₂×C₁₆	C₂×C₄	1	1	yes	characteristic
23	#107	C₂×C₄	1	2	yes	—
24	C₄⋊C₈	—	2	2	no	—
25	#102	—	2	2	no	—
26	C₂²⋊C₈	—	4	2	no	—
27	C₄×C₈	D₄	1	1	yes	characteristic
28	C₄×C₈	D₄	1	1	yes	characteristic
29	C₄×C₈	C₂³	1	1	yes	characteristic
30	C₂×D₄	—	2	2	no	—
31	C₂²×C₄	—	4	2	no	—
32	D₈	—	2	2	no	—
33	D₈	C₂×C₈	1	4	yes	complemented
34	C₂×C₈	—	2	1	no	—
35	C₂×C₈	—	4	2	no	—
36	C₂×C₈	—	8	4	no	—
37	C₂×C₈	D₈	1	2	yes	—
38	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₈	#41	1	1	yes	characteristic
40	C₄⋊C₄	—	2	2	no	—
41	C₂²⋊C₄	—	4	2	no	—
42	C₄²	C₂×D₄	1	1	yes	characteristic
43	C₁₆	—	2	2	no	—
44	C₁₆	—	2	1	no	—
45	C₁₆	C₂×C₈	1	2	yes	complemented
46	C₂³	—	4	2	no	—
47	D₄	—	2	4	no	—
48	D₄	—	4	2	no	—
49	C₂×C₄	—	4	2	no	—
50	C₂×C₄	—	8	2	no	—
51	C₂×C₄	—	8	2	no	—
52	C₂×C₄	C₄×D₄	1	1	yes	characteristic
53	C₂×C₄	C₂×D₈	1	1	yes	characteristic
54	C₂×C₄	#135	1	1	yes	characteristic
55	C₈	—	2	1	no	—
56	C₈	—	2	2	no	—
57	C₈	—	4	1	no	—
58	C₈	—	8	2	no	—
59	C₈	C₂²×C₈	1	1	yes	characteristic
60	C₈	#131	1	1	yes	characteristic
61	C₂²	—	4	4	no	—
62	C₂²	—	8	2	no	—
63	C₂²	—	8	2	no	—
64	C₂²	C₄×D₈	1	1	yes	characteristic
65	C₄	—	2	1	no	—
66	C₄	—	8	2	no	—
67	C₄	C₈×D₄	1	1	yes	characteristic
68	C₄	#443	1	1	yes	characteristic
69	C₄	#509	1	2	yes	—
70	C₂	—	8	4	no	—
71	C₂	C₈×D₈	1	1	yes	characteristic
72	C₂	#2177	1	1	yes	characteristic
73	C₂	#2179	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₈
commutator	C₈	C₂²×C₈
Frattini, cosocle	C₄×C₈	C₂³
socle	C₂²	C₄×D₈
prime radical	C₂×D₁₆	C₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈² ◁ 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
2. C₈×D₈	C₂	semidirect
3. #1372	C₂	semidirect
4. #1331	C₂	semidirect
5. #1311	C₂	semidirect
19. D₁₆	C₈	semidirect
33. D₈	C₂×C₈	semidirect
45. C₁₆	C₂×C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×D₁₆	2
C₈×D₈	2
#1372	2
#1331	2
#1311	2