Subgroups — #12906

Order 256.

187 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#367	C₂	1	4	yes	maximal
2	C₂×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic
3	#1845	C₂	1	2	yes	maximal
4	C₂²×C₁₆	C₂²	1	2	yes	—
5	C₂×#107	—	2	2	no	—
6	C₂×#107	C₂²	1	4	yes	—
7	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
8	#384	—	2	2	no	—
9	#367	—	2	8	no	—
10	C₄×C₁₆	C₄	1	4	yes	complemented
11	C₂²×C₈	D₄	1	1	yes	characteristic
12	C₂²×C₈	C₂³	1	1	yes	characteristic
13	C₂×C₄⋊C₄	—	2	4	no	—
14	C₂×C₄⋊C₄	—	4	4	no	—
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	#107	—	2	8	no	—
20	#107	—	4	4	no	—
21	C₄×C₈	C₂×C₄	1	2	yes	—
22	C₄×C₈	Q₈	1	2	yes	—
23	C₂²×C₄	—	4	4	no	—
24	C₂²×C₄	D₈	1	2	yes	—
25	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
26	C₂×C₈	C₄⋊C₄	1	2	yes	—
27	C₂×C₈	D₈	1	2	yes	—
28	C₂×C₈	Q₁₆	1	2	yes	—
29	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
30	C₂×C₈	C₂×D₄	1	2	yes	—
31	C₂×C₈	C₂×Q₈	1	2	yes	—
32	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
33	C₄⋊C₄	—	4	4	no	—
34	C₄⋊C₄	—	4	4	no	—
35	C₄⋊C₄	—	8	8	no	—
36	C₄²	C₄⋊C₄	1	2	yes	—
37	C₄²	Q₁₆	1	2	yes	—
38	C₁₆	C₄⋊C₄	1	8	yes	complemented
39	C₂³	C₂×D₈	1	1	yes	characteristic
40	C₂×C₄	—	8	4	no	—
41	C₂×C₄	—	8	8	no	—
42	C₂×C₄	#107	1	4	yes	—
43	C₂×C₄	SD₃₂	1	8	yes	—
44	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₄	#127	1	1	yes	characteristic
46	C₂×C₄	C₄⋊Q₈	1	1	yes	characteristic
47	C₂×C₄	C₂×D₈	1	1	yes	characteristic
48	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
49	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
50	C₈	#107	1	4	yes	—
51	C₈	C₂×C₄⋊C₄	1	2	yes	—
52	C₈	C₄⋊Q₈	1	2	yes	—
53	C₂²	C₂×#107	1	1	yes	characteristic
54	C₂²	#493	1	1	yes	characteristic
55	C₂²	#500	1	1	yes	characteristic
56	C₂²	C₂×SD₃₂	1	4	yes	—
57	C₄	—	16	4	no	—
58	C₄	#367	1	8	yes	complemented
59	C₄	#384	1	1	yes	characteristic
60	C₄	C₂×#107	1	1	yes	characteristic
61	C₄	#494	1	1	yes	characteristic
62	C₄	#500	1	1	yes	characteristic
63	C₂	#1845	1	1	yes	characteristic
64	C₂	C₂×#367	1	2	yes	—
65	C₂	#2248	1	2	yes	—
66	C₂	#2254	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×D₈
commutator	C₂×C₈	C₂²×C₄
Frattini, cosocle	C₂²×C₈	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ 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
10. C₄×C₁₆	C₄	semidirect
38. C₁₆	C₄⋊C₄	semidirect
58. C₄	#367	semidirect

Maximal subgroups

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

subgroup	index
C₂×#367	2
C₂×C₄×C₁₆	2
#1845	2