Subgroups — #8284

Order 256.

215 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#107	C₂	1	1	yes	maximal, characteristic
2	C₈×C₄⋊C₄	C₂	1	2	yes	maximal
3	C₂×#334	C₂	1	1	yes	maximal, characteristic
4	C₂×#322	C₂	1	1	yes	maximal, characteristic
5	#1279	C₂	1	2	yes	maximal
6	C₂×#107	C₄	1	2	yes	—
7	C₂×C₄⋊C₈	C₂²	1	2	yes	—
8	C₂×C₄×C₈	—	2	2	no	—
9	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
12	C₄×C₄⋊C₄	C₂²	1	2	yes	—
13	#340	C₄	1	2	yes	—
14	#336	—	2	2	no	—
15	#334	C₄	1	4	yes	complemented
16	#322	C₄	1	4	yes	complemented
17	C₂²×C₈	—	2	2	no	—
18	C₂²×C₈	—	2	1	no	—
19	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈	D₄	1	2	yes	—
22	C₂×C₄⋊C₄	C₂×C₄	1	4	yes	—
23	C₂×C₄²	—	2	2	no	—
24	C₂×C₄²	C₂³	1	1	yes	characteristic
25	#107	C₈	1	8	yes	complemented
26	C₄⋊C₈	—	2	2	no	—
27	C₄⋊C₈	C₂×C₄	1	4	yes	—
28	C₄×C₈	—	2	1	no	—
29	C₄×C₈	—	4	4	no	—
30	C₄×C₈	C₂×C₄	1	2	yes	—
31	C₄×C₈	C₂×C₄	1	2	yes	—
32	C₄×C₈	C₂×C₄	1	2	yes	—
33	C₄×C₈	C₂×C₄	1	2	yes	—
34	C₄×C₈	Q₈	1	2	yes	—
35	#95	—	2	2	no	—
36	C₂²×C₄	—	2	4	no	—
37	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
39	C₂²×C₄	#41	1	1	yes	characteristic
40	C₂×C₈	—	2	2	no	—
41	C₂×C₈	—	2	2	no	—
42	C₂×C₈	—	2	2	no	—
43	C₂×C₈	—	2	2	no	—
44	C₂×C₈	—	2	4	no	—
45	C₂×C₈	—	4	2	no	—
46	C₂×C₈	—	4	2	no	—
47	C₂×C₈	—	4	2	no	—
48	C₂×C₈	C₄²	1	2	yes	—
49	C₂×C₈	C₄²	1	2	yes	—
50	C₂×C₈	C₄⋊C₄	1	4	yes	—
51	C₂×C₈	C₂×C₈	1	4	yes	—
52	C₄⋊C₄	—	2	4	no	—
53	C₄⋊C₄	C₂×C₈	1	8	yes	—
54	C₄²	—	4	2	no	—
55	C₄²	—	4	2	no	—
56	C₄²	C₂²×C₄	1	1	yes	characteristic
57	C₄²	C₂²×C₄	1	1	yes	characteristic
58	C₄²	C₂×Q₈	1	1	yes	characteristic
59	C₄²	#41	1	1	yes	characteristic
60	C₂³	C₄×D₄	1	1	yes	characteristic
61	C₂×C₄	—	2	2	no	—
62	C₂×C₄	—	2	8	no	—
63	C₂×C₄	—	4	4	no	—
64	C₂×C₄	—	4	4	no	—
65	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
66	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄	#117	1	1	yes	characteristic
68	C₂×C₄	C₄×D₄	1	1	yes	characteristic
69	C₂×C₄	C₄×Q₈	1	1	yes	characteristic
70	C₂×C₄	C₄×Q₈	1	1	yes	characteristic
71	C₂×C₄	C₂²×C₈	1	2	yes	—
72	C₂×C₄	#131	1	2	yes	—
73	C₂×C₄	AGL(1,ℤ/8)	1	2	yes	—
74	C₂×C₄	#137	1	2	yes	—
75	C₈	—	2	2	no	—
76	C₈	—	2	4	no	—
77	C₈	—	4	2	no	—
78	C₈	—	4	4	no	—
79	C₈	C₄×C₈	1	4	yes	complemented
80	C₂²	C₄×C₄⋊C₄	1	1	yes	characteristic
81	C₂²	C₈×D₄	1	2	yes	—
82	C₂²	#441	1	1	yes	characteristic
83	C₂²	#442	1	1	yes	characteristic
84	C₂²	#444	1	2	yes	—
85	C₄	—	2	1	no	—
86	C₄	—	2	1	no	—
87	C₄	—	4	8	no	—
88	C₄	C₂×C₄×C₈	1	1	yes	characteristic
89	C₄	#405	1	1	yes	characteristic
90	C₄	#428	1	2	yes	—
91	C₄	#440	1	2	yes	—
92	C₄	C₈×Q₈	1	2	yes	—
93	C₂	#1579	1	2	yes	—
94	C₂	#1580	1	2	yes	—
95	C₂	C₈×C₄⋊C₄	1	1	yes	characteristic
96	C₂	#1773	1	1	yes	characteristic
97	C₂	#1776	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₄
commutator	C₄	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄²	C₂³
socle, prime radical	C₂³	C₄×D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
15. #334	C₄	semidirect
16. #322	C₄	semidirect
25. #107	C₈	semidirect
79. C₈	C₄×C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×#107	2
C₈×C₄⋊C₄	2
C₂×#334	2
C₂×#322	2
#1279	2