Subgroups — #12909

Order 256.

157 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#368	C₂	1	4	yes	maximal
2	C₂×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic
3	#1848	C₂	1	2	yes	maximal
4	C₂²×C₁₆	C₂²	1	2	yes	—
5	C₂×#108	—	2	2	no	—
6	C₂×#108	C₂²	1	4	yes	—
7	#423	—	2	2	no	—
8	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
9	#368	—	2	8	no	—
10	C₄×C₁₆	C₄	1	2	yes	—
11	C₄×C₁₆	C₄	1	2	yes	—
12	C₂×M₁₆	—	2	4	no	—
13	C₂²×C₈	D₄	1	1	yes	characteristic
14	C₂²×C₈	C₂³	1	1	yes	characteristic
15	C₂×C₄²	D₄	1	1	yes	characteristic
16	C₂×C₁₆	C₂×C₄	1	4	yes	—
17	C₂×C₁₆	D₄	1	4	yes	—
18	C₂×C₁₆	Q₈	1	4	yes	—
19	#108	—	2	8	no	—
20	#108	—	4	4	no	—
21	C₄⋊C₈	—	4	4	no	—
22	C₄×C₈	C₂×C₄	1	1	yes	characteristic
23	C₄×C₈	C₂×C₄	1	1	yes	characteristic
24	C₄×C₈	Q₈	1	1	yes	characteristic
25	C₄×C₈	Q₈	1	1	yes	characteristic
26	C₂²×C₄	Q₁₆	1	2	yes	—
27	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
28	M₁₆	—	4	4	no	—
29	M₁₆	—	4	4	no	—
30	C₂×C₈	—	4	4	no	—
31	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₈	D₈	1	2	yes	—
34	C₂×C₈	Q₁₆	1	2	yes	—
35	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₈	C₂×D₄	1	2	yes	—
37	C₂×C₈	C₂×Q₈	1	2	yes	—
38	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
39	C₄²	C₄⋊C₄	1	1	yes	characteristic
40	C₄²	C₄⋊C₄	1	1	yes	characteristic
41	C₄²	D₈	1	2	yes	—
42	C₁₆	C₄⋊C₄	1	8	yes	—
43	C₂³	C₂×Q₁₆	1	1	yes	characteristic
44	C₂×C₄	—	2	4	no	—
45	C₂×C₄	#107	1	2	yes	—
46	C₂×C₄	#107	1	2	yes	—
47	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₄	#127	1	1	yes	characteristic
49	C₂×C₄	C₄⋊Q₈	1	1	yes	characteristic
50	C₂×C₄	C₂×D₈	1	1	yes	characteristic
51	C₂×C₄	C₂×D₈	1	1	yes	characteristic
52	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
53	C₈	—	8	4	no	—
54	C₈	#107	1	2	yes	—
55	C₈	#107	1	2	yes	—
56	C₈	C₂×C₄⋊C₄	1	2	yes	—
57	C₈	C₄⋊Q₈	1	2	yes	—
58	C₂²	—	2	2	no	—
59	C₂²	C₂×#107	1	1	yes	characteristic
60	C₂²	#494	1	1	yes	characteristic
61	C₂²	#500	1	1	yes	characteristic
62	C₄	—	2	2	no	—
63	C₄	#368	1	4	yes	complemented
64	C₄	#384	1	1	yes	characteristic
65	C₄	C₂×#107	1	1	yes	characteristic
66	C₄	#493	1	1	yes	characteristic
67	C₄	#500	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	2	1	no	—
70	C₂	#1845	1	1	yes	characteristic
71	C₂	C₂×#368	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₈
commutator	C₂×C₈	C₂²×C₄
Frattini, cosocle	C₂²×C₈	C₂³
socle	C₂²	C₂×#107
prime radical	C₂³	C₂×Q₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
63. C₄	#368	semidirect

Maximal subgroups

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

subgroup	index
C₂×#368	2
C₂×C₄×C₁₆	2
#1848	2