Subgroups — #12431

Order 256.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2247	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×C₁₆	C₂	1	1	yes	maximal, characteristic
3	#1669	C₂	1	2	yes	maximal
4	#1371	C₂	1	1	yes	maximal, characteristic, complemented
5	#1362	C₂	1	2	yes	maximal, complemented
6	C₂×Q₃₂	—	2	1	no	—
7	C₂×Q₃₂	—	2	1	no	—
8	C₂²×C₁₆	—	2	1	no	—
9	#494	C₂²	1	2	yes	—
10	#474	—	2	2	no	—
11	C₈×D₄	C₂²	1	1	yes	characteristic
12	C₄⋊C₁₆	C₂²	1	1	yes	characteristic
13	C₂²⋊C₁₆	—	2	1	no	—
14	C₄×C₁₆	C₂²	1	1	yes	characteristic
15	#335	C₂²	1	2	yes	—
16	#332	—	2	2	no	—
17	C₂×Q₁₆	—	2	2	no	—
18	C₂×Q₁₆	—	2	2	no	—
19	C₂²×C₈	—	2	1	no	—
20	C₄⋊Q₈	—	2	2	no	—
21	C₄×D₄	D₄	1	1	yes	characteristic
22	Q₃₂	—	4	2	no	—
23	Q₃₂	—	4	2	no	—
24	C₂×C₁₆	—	2	1	no	—
25	C₂×C₁₆	—	4	2	no	—
26	C₂×C₁₆	D₄	1	1	yes	characteristic
27	C₂×C₁₆	D₄	1	1	yes	characteristic
28	#107	—	4	2	no	—
29	C₄⋊C₈	—	2	2	no	—
30	C₄⋊C₈	D₄	1	1	yes	characteristic
31	#102	—	4	2	no	—
32	C₂²⋊C₈	—	2	1	no	—
33	C₄×C₈	C₂³	1	1	yes	characteristic
34	C₂×Q₈	—	4	2	no	—
35	C₂×D₄	D₈	1	1	yes	characteristic
36	C₂²×C₄	—	2	1	no	—
37	Q₁₆	—	4	2	no	—
38	Q₁₆	—	4	2	no	—
39	Q₁₆	—	8	2	no	—
40	C₂×C₈	—	2	1	no	—
41	C₂×C₈	—	4	1	no	—
42	C₂×C₈	—	4	1	no	—
43	C₂×C₈	—	4	2	no	—
44	C₂×C₈	C₂×D₄	1	1	yes	characteristic
45	C₂×C₈	#41	1	1	yes	characteristic
46	C₄⋊C₄	—	4	2	no	—
47	C₄⋊C₄	—	4	2	no	—
48	C₄⋊C₄	D₈	1	1	yes	characteristic
49	C₂²⋊C₄	—	2	1	no	—
50	C₄²	C₂×D₄	1	1	yes	characteristic
51	C₁₆	—	2	1	no	—
52	C₁₆	—	4	1	no	—
53	C₁₆	D₈	1	2	yes	—
54	C₂³	—	2	1	no	—
55	Q₈	—	8	2	no	—
56	Q₈	—	8	2	no	—
57	D₄	—	2	1	no	—
58	D₄	Q₃₂	1	2	yes	complemented
59	C₂×C₄	—	2	1	no	—
60	C₂×C₄	—	4	1	no	—
61	C₂×C₄	—	4	1	no	—
62	C₂×C₄	—	8	2	no	—
63	C₂×C₄	#121	1	1	yes	characteristic
64	C₂×C₄	C₂×D₈	1	1	yes	characteristic
65	C₂×C₄	#135	1	1	yes	characteristic
66	C₈	—	2	1	no	—
67	C₈	—	4	1	no	—
68	C₈	—	8	2	no	—
69	C₈	C₂×D₈	1	1	yes	characteristic
70	C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
71	C₂²	—	2	2	no	—
72	C₂²	—	4	1	no	—
73	C₂²	—	4	1	no	—
74	C₂²	#466	1	1	yes	characteristic
75	C₄	—	2	1	no	—
76	C₄	—	4	1	no	—
77	C₄	—	16	2	no	—
78	C₄	#459	1	1	yes	characteristic
79	C₄	#472	1	1	yes	characteristic
80	C₄	C₂×Q₃₂	1	1	yes	characteristic
81	C₄	#508	1	1	yes	characteristic
82	C₂	—	4	2	no	—
83	C₂	#1667	1	1	yes	characteristic
84	C₂	#2216	1	1	yes	characteristic
85	C₂	#2222	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#466
commutator, Frattini, cosocle	C₄×C₈	C₂³
prime radical	C₂×D₄	D₈
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×D₄ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #2247	C₂	semidirect
4. #1371	C₂	semidirect
5. #1362	C₂	semidirect
58. D₄	Q₃₂	semidirect

Maximal subgroups

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

subgroup	index
#2247	2
D₄×C₁₆	2
#1669	2
#1371	2
#1362	2