Subgroups — #12512

Order 256.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2241	C₂	1	2	yes	maximal, complemented
2	#2240	C₂	1	2	yes	maximal
3	#1713	C₂	1	1	yes	maximal, characteristic, complemented
4	#1709	C₂	1	1	yes	maximal, characteristic
5	#1311	C₂	1	1	yes	maximal, characteristic, complemented
6	#500	—	2	1	no	—
7	#500	C₂²	1	2	yes	—
8	#494	—	2	1	no	—
9	#494	C₂²	1	1	yes	characteristic
10	#493	C₂²	1	1	yes	characteristic
11	#486	—	2	1	no	—
12	#486	—	2	1	no	—
13	#358	—	2	4	no	—
14	#357	—	2	4	no	—
15	C₄×C₁₆	C₂²	1	2	yes	—
16	C₈²	C₂²	1	1	yes	characteristic
17	C₂×Q₁₆	—	2	2	no	—
18	C₂×Q₁₆	—	4	2	no	—
19	C₂×D₈	—	2	2	no	—
20	C₄⋊Q₈	—	2	1	no	—
21	C₄⋊Q₈	—	2	2	no	—
22	#127	—	2	1	no	—
23	C₂×C₁₆	D₄	1	4	yes	—
24	#107	—	2	2	no	—
25	#107	—	2	2	no	—
26	#107	—	4	2	no	—
27	#102	—	4	2	no	—
28	#102	—	4	2	no	—
29	C₄×C₈	D₄	1	1	yes	characteristic
30	C₄×C₈	D₄	1	1	yes	characteristic
31	C₄×C₈	C₂³	1	1	yes	characteristic
32	C₂×Q₈	—	4	1	no	—
33	C₂×Q₈	—	4	2	no	—
34	C₂×D₄	—	4	1	no	—
35	C₂×D₄	—	4	1	no	—
36	C₂×D₄	—	4	1	no	—
37	Q₁₆	—	4	4	no	—
38	Q₁₆	—	8	4	no	—
39	D₈	—	4	4	no	—
40	C₂×C₈	D₈	1	2	yes	—
41	C₂×C₈	D₈	1	2	yes	—
42	C₂×C₈	C₂×D₄	1	2	yes	—
43	C₄⋊C₄	—	4	2	no	—
44	C₄⋊C₄	—	4	1	no	—
45	C₄⋊C₄	—	4	1	no	—
46	C₄⋊C₄	—	4	2	no	—
47	C₄²	C₂×D₄	1	1	yes	characteristic
48	C₁₆	—	2	4	no	—
49	C₂³	—	8	1	no	—
50	Q₈	—	8	2	no	—
51	Q₈	—	8	4	no	—
52	D₄	—	8	2	no	—
53	D₄	—	8	2	no	—
54	D₄	—	8	2	no	—
55	C₂×C₄	—	8	1	no	—
56	C₂×C₄	—	8	2	no	—
57	C₂×C₄	#127	1	1	yes	characteristic
58	C₂×C₄	C₂×D₈	1	1	yes	characteristic
59	C₂×C₄	C₂×D₈	1	1	yes	characteristic
60	C₈	—	2	2	no	—
61	C₈	—	2	2	no	—
62	C₈	#135	1	4	yes	—
63	C₂²	—	16	1	no	—
64	C₂²	—	16	1	no	—
65	C₂²	—	16	1	no	—
66	C₂²	#493	1	1	yes	characteristic
67	C₄	—	16	1	no	—
68	C₄	—	16	2	no	—
69	C₄	#495	1	2	yes	—
70	C₄	#509	1	2	yes	—
71	C₄	#510	1	2	yes	—
72	C₂	—	32	1	no	—
73	C₂	#1709	1	1	yes	characteristic
74	C₂	#2250	1	1	yes	characteristic
75	C₂	#2251	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#493
commutator, Frattini, cosocle	C₄×C₈	C₂³
prime radical	#493	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ 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
1. #2241	C₂	semidirect
3. #1713	C₂	semidirect
5. #1311	C₂	semidirect

Maximal subgroups

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

subgroup	index
#2241	2
#2240	2
#1713	2
#1709	2
#1311	2