Subgroups — #8786

Order 256.

189 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#1842	C₂	1	1	yes	maximal, characteristic
2	#1831	C₂	1	1	yes	maximal, characteristic
3	C₄×#97	C₂	1	1	yes	maximal, characteristic
4	#1277	C₂	1	4	yes	maximal
5	C₂×C₄⋊C₈	—	2	2	no	—
6	C₂×C₄⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄⋊C₈	C₂²	1	2	yes	—
8	C₂×#97	C₄	1	2	yes	—
9	C₂×#97	C₄	1	1	yes	characteristic
10	C₂×#97	C₄	1	1	yes	characteristic
11	C₂×C₄×C₈	C₂²	1	2	yes	—
12	C₄³	C₂²	1	1	yes	characteristic
13	#336	—	2	2	no	—
14	C₂²×C₈	—	2	2	no	—
15	C₂²×C₈	—	2	2	no	—
16	C₂²×C₈	C₂×C₄	1	2	yes	—
17	C₂²×C₈	C₂×C₄	1	2	yes	—
18	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²	D₄	1	2	yes	—
21	C₂×C₄²	Q₈	1	2	yes	—
22	C₂×C₄²	C₂³	1	1	yes	characteristic
23	C₄⋊C₈	—	2	4	no	—
24	C₄⋊C₈	—	2	4	no	—
25	C₄⋊C₈	—	4	4	no	—
26	#97	—	2	4	no	—
27	#97	—	2	2	no	—
28	#97	C₈	1	4	yes	complemented
29	C₄×C₈	—	2	2	no	—
30	C₄×C₈	—	2	2	no	—
31	C₂²×C₄	C₄⋊C₄	1	2	yes	—
32	C₂²×C₄	C₄⋊C₄	1	2	yes	—
33	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
34	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
36	C₂×C₈	—	2	2	no	—
37	C₂×C₈	—	2	2	no	—
38	C₂×C₈	—	2	2	no	—
39	C₂×C₈	—	2	2	no	—
40	C₂×C₈	—	4	2	no	—
41	C₂×C₈	—	4	2	no	—
42	C₂×C₈	—	4	4	no	—
43	C₂×C₈	—	4	2	no	—
44	C₂×C₈	—	4	2	no	—
45	C₂×C₈	C₂×C₈	1	4	yes	—
46	C₂×C₈	M₁₆	1	4	yes	—
47	C₄²	—	2	2	no	—
48	C₄²	—	2	2	no	—
49	C₄²	—	2	2	no	—
50	C₄²	—	4	2	no	—
51	C₄²	C₂×C₈	1	2	yes	—
52	C₄²	M₁₆	1	2	yes	—
53	C₄²	#41	1	2	yes	—
54	C₄²	#41	1	2	yes	—
55	C₂³	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄	—	2	2	no	—
57	C₂×C₄	—	2	2	no	—
58	C₂×C₄	—	2	2	no	—
59	C₂×C₄	—	2	2	no	—
60	C₂×C₄	—	2	2	no	—
61	C₂×C₄	—	2	2	no	—
62	C₂×C₄	C₄⋊C₈	1	4	yes	—
63	C₂×C₄	#117	1	2	yes	—
64	C₂×C₄	#124	1	1	yes	characteristic
65	C₂×C₄	#124	1	1	yes	characteristic
66	C₂×C₄	#125	1	2	yes	—
67	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
68	C₂×C₄	C₂×M₁₆	1	2	yes	—
69	C₂×C₄	C₂×M₁₆	1	1	yes	characteristic
70	C₂×C₄	AGL(1,ℤ/8)	1	2	yes	—
71	C₂×C₄	#137	1	2	yes	—
72	C₈	—	2	4	no	—
73	C₈	—	2	4	no	—
74	C₈	—	8	2	no	—
75	C₈	—	8	2	no	—
76	C₂²	#382	1	1	yes	characteristic
77	C₂²	C₂×C₄⋊C₈	1	1	yes	characteristic
78	C₂²	#423	1	1	yes	characteristic
79	C₂²	#428	1	2	yes	—
80	C₂²	#430	1	2	yes	—
81	C₄	—	2	4	no	—
82	C₄	—	2	2	no	—
83	C₄	—	4	2	no	—
84	C₄	#431	1	2	yes	—
85	C₄	#432	1	2	yes	—
86	C₄	#489	1	2	yes	—
87	C₄	#490	1	1	yes	characteristic
88	C₄	#491	1	1	yes	characteristic
89	C₂	#1565	1	2	yes	—
90	C₂	#1569	1	2	yes	—
91	C₂	#1831	1	1	yes	characteristic
92	C₂	#1836	1	1	yes	characteristic
93	C₂	#1838	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₄
commutator	C₂×C₄	C₂²×C₈
Frattini, cosocle	C₂×C₄²	C₂³
socle, prime radical	C₂³	C₂×C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
28. #97	C₈	semidirect

Maximal subgroups

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

subgroup	index
#1842	2
#1831	2
C₄×#97	2
#1277	2