Subgroups — #8266

Order 256.

207 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×C₈	C₂	1	1	yes	maximal, characteristic
2	#1564	C₂	1	2	yes	maximal
3	#1450	C₂	1	2	yes	maximal
4	#1276	C₂	1	2	yes	maximal
5	#431	C₂²	1	4	yes	—
6	C₂×C₄×C₈	C₄	1	2	yes	—
7	C₂×C₄×C₈	C₄	1	2	yes	—
8	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
10	C₄³	C₂²	1	1	yes	characteristic
11	#335	C₄	1	4	yes	complemented
12	#334	C₄	1	4	yes	complemented
13	#322	—	2	2	no	—
14	C₈²	—	2	2	no	—
15	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈	C₂×C₄	1	2	yes	—
18	C₂×C₄²	—	2	2	no	—
19	C₂×C₄²	C₂×C₄	1	2	yes	—
20	C₂×C₄²	C₂³	1	1	yes	characteristic
21	C₄⋊C₈	C₂×C₄	1	8	yes	—
22	C₂²⋊C₈	—	2	4	no	—
23	C₄×C₈	—	2	4	no	—
24	C₄×C₈	—	2	4	no	—
25	C₄×C₈	C₈	1	8	yes	complemented
26	C₄×C₈	C₂×C₄	1	2	yes	—
27	C₄×C₈	C₂×C₄	1	2	yes	—
28	C₄×C₈	D₄	1	1	yes	characteristic
29	C₄×C₈	D₄	1	1	yes	characteristic
30	C₄×C₈	Q₈	1	2	yes	—
31	C₂²×C₄	—	2	2	no	—
32	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
33	C₂²×C₄	#41	1	2	yes	—
34	C₂×C₈	—	2	2	no	—
35	C₂×C₈	—	2	8	no	—
36	C₂×C₈	C₄²	1	2	yes	—
37	C₂×C₈	C₄²	1	1	yes	characteristic
38	C₂×C₈	C₄²	1	1	yes	characteristic
39	C₂×C₈	C₄⋊C₄	1	2	yes	—
40	C₂×C₈	C₄⋊C₄	1	2	yes	—
41	C₂×C₈	C₂×C₈	1	4	yes	—
42	C₂×C₈	C₂×C₈	1	4	yes	—
43	C₂×C₈	M₁₆	1	4	yes	—
44	C₄²	—	2	8	no	—
45	C₄²	—	2	2	no	—
46	C₄²	C₂×C₈	1	4	yes	—
47	C₄²	C₂²×C₄	1	2	yes	—
48	C₄²	C₂×D₄	1	1	yes	characteristic
49	C₄²	C₂×Q₈	1	1	yes	characteristic
50	C₂³	#117	1	1	yes	characteristic
51	C₂×C₄	—	2	8	no	—
52	C₂×C₄	—	2	4	no	—
53	C₂×C₄	—	2	4	no	—
54	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
55	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄	C₄×D₄	1	2	yes	—
57	C₂×C₄	C₄×Q₈	1	2	yes	—
58	C₂×C₄	C₂²×C₈	1	2	yes	—
59	C₂×C₄	C₂×M₁₆	1	2	yes	—
60	C₈	—	2	2	no	—
61	C₈	—	4	8	no	—
62	C₈	C₄×C₈	1	2	yes	—
63	C₈	C₄×C₈	1	2	yes	—
64	C₈	C₄⋊C₈	1	4	yes	—
65	C₈	C₄⋊C₈	1	4	yes	—
66	C₂²	—	2	2	no	—
67	C₂²	C₄×C₄⋊C₄	1	1	yes	characteristic
68	C₂²	#431	1	2	yes	—
69	C₄	—	2	8	no	—
70	C₄	—	2	2	no	—
71	C₄	C₂×C₄×C₈	1	1	yes	characteristic
72	C₄	C₄×M₁₆	1	1	yes	characteristic
73	C₄	C₂×C₄⋊C₈	1	2	yes	—
74	C₄	#443	1	4	yes	—
75	C₂	—	2	2	no	—
76	C₂	C₄×C₄⋊C₈	1	1	yes	characteristic
77	C₂	#1772	1	2	yes	—

	subgroup	G/H
center	C₄×C₈	D₄
commutator	C₄	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄²	C₂³
socle	C₂²	C₄×C₄⋊C₄
prime radical	C₂³	#117
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₈ ◁ 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
11. #335	C₄	semidirect
12. #334	C₄	semidirect
25. C₄×C₈	C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄²×C₈	2
#1564	2
#1450	2
#1276	2