Subgroups — #8814

Order 256.

227 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#1845	C₂	1	2	yes	maximal
2	#1844	C₂	1	2	yes	maximal
3	#1834	C₂	1	2	yes	maximal
4	C₂×C₈²	C₂	1	1	yes	maximal, characteristic
5	C₂×#107	—	2	4	no	—
6	C₂×#106	—	2	4	no	—
7	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
10	#384	C₂²	1	4	yes	—
11	#340	—	2	4	no	—
12	C₈²	C₄	1	4	yes	complemented
13	C₂²×C₈	D₄	1	2	yes	—
14	C₂²×C₈	D₄	1	2	yes	—
15	C₂²×C₈	D₄	1	2	yes	—
16	C₂×C₄⋊C₄	—	2	4	no	—
17	C₂×C₄⋊C₄	—	2	4	no	—
18	C₂×C₄⋊C₄	—	2	4	no	—
19	C₂×C₄²	C₂³	1	1	yes	characteristic
20	#107	—	4	8	no	—
21	#106	—	4	8	no	—
22	C₄×C₈	C₂×C₄	1	2	yes	—
23	C₄×C₈	C₂×C₄	1	2	yes	—
24	C₄×C₈	C₂×C₄	1	2	yes	—
25	C₄×C₈	Q₈	1	2	yes	—
26	C₄×C₈	Q₈	1	2	yes	—
27	C₄×C₈	Q₈	1	2	yes	—
28	C₂²×C₄	—	4	4	no	—
29	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
30	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
31	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
32	C₂×C₈	—	2	4	no	—
33	C₂×C₈	C₄⋊C₄	1	4	yes	—
34	C₂×C₈	C₄⋊C₄	1	4	yes	—
35	C₂×C₈	C₄⋊C₄	1	4	yes	—
36	C₂×C₈	D₈	1	4	yes	—
37	C₂×C₈	SD₁₆	1	8	yes	—
38	C₂×C₈	Q₁₆	1	4	yes	—
39	C₄⋊C₄	—	4	8	no	—
40	C₄⋊C₄	—	4	8	no	—
41	C₄⋊C₄	—	4	8	no	—
42	C₄²	C₂²×C₄	1	1	yes	characteristic
43	C₄²	C₂×Q₈	1	1	yes	characteristic
44	C₄²	C₂×Q₈	1	1	yes	characteristic
45	C₄²	C₂×Q₈	1	1	yes	characteristic
46	C₂³	#127	1	1	yes	characteristic
47	C₂×C₄	—	8	4	no	—
48	C₂×C₄	—	8	4	no	—
49	C₂×C₄	—	8	4	no	—
50	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₄	C₄⋊Q₈	1	1	yes	characteristic
54	C₂×C₄	C₄⋊Q₈	1	1	yes	characteristic
55	C₂×C₄	C₄⋊Q₈	1	1	yes	characteristic
56	C₂×C₄	C₂×D₈	1	2	yes	—
57	C₂×C₄	C₂×SD₁₆	1	4	yes	—
58	C₂×C₄	C₂×Q₁₆	1	2	yes	—
59	C₂×C₄	#135	1	4	yes	—
60	C₈	—	2	4	no	—
61	C₈	#106	1	8	yes	complemented
62	C₈	#107	1	8	yes	complemented
63	C₂²	#384	1	1	yes	characteristic
64	C₂²	#492	1	2	yes	—
65	C₂²	#493	1	1	yes	characteristic
66	C₂²	#494	1	1	yes	characteristic
67	C₂²	#495	1	2	yes	—
68	C₄	—	16	4	no	—
69	C₄	C₂×#106	1	2	yes	—
70	C₄	C₂×#107	1	2	yes	—
71	C₄	#427	1	2	yes	—
72	C₄	#498	1	2	yes	—
73	C₄	#499	1	2	yes	—
74	C₄	#500	1	2	yes	—
75	C₂	#1706	1	2	yes	—
76	C₂	#1707	1	2	yes	—
77	C₂	#1844	1	1	yes	characteristic
78	C₂	#1845	1	1	yes	characteristic
79	C₂	#1846	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	#127
commutator	C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
12. C₈²	C₄	semidirect
61. C₈	#106	semidirect
62. C₈	#107	semidirect

Maximal subgroups

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

subgroup	index
#1845	2
#1844	2
#1834	2
C₂×C₈²	2