Subgroups — #12437

Order 256.

107 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	#2214	C₂	1	2	yes	maximal
3	#2170	C₂	1	1	yes	maximal, characteristic
4	#1373	C₂	1	1	yes	maximal, characteristic, complemented
5	#1358	C₂	1	2	yes	maximal, complemented
6	#510	—	2	2	no	—
7	C₂×Q₃₂	—	2	1	no	—
8	C₂×Q₃₂	—	2	1	no	—
9	#504	—	2	1	no	—
10	#494	C₂²	1	2	yes	—
11	#443	C₂²	1	1	yes	characteristic
12	#364	C₂²	1	1	yes	characteristic
13	#364	C₂²	1	2	yes	—
14	#362	—	2	2	no	—
15	#350	—	2	1	no	—
16	C₄×C₁₆	C₂²	1	1	yes	characteristic
17	#135	D₄	1	1	yes	characteristic
18	C₂×Q₁₆	—	2	2	no	—
19	C₂×Q₁₆	—	2	2	no	—
20	#131	—	2	1	no	—
21	C₄⋊Q₈	—	2	2	no	—
22	Q₃₂	—	4	2	no	—
23	Q₃₂	—	4	2	no	—
24	Q₃₂	—	4	1	no	—
25	Q₃₂	—	4	1	no	—
26	SD₃₂	—	4	2	no	—
27	M₃₂	—	2	1	no	—
28	M₃₂	—	2	2	no	—
29	M₃₂	—	4	1	no	—
30	C₂×C₁₆	—	4	1	no	—
31	C₂×C₁₆	D₄	1	1	yes	characteristic
32	C₂×C₁₆	D₄	1	1	yes	characteristic
33	#108	D₄	1	1	yes	characteristic
34	#104	—	2	1	no	—
35	C₄×C₈	C₂³	1	1	yes	characteristic
36	#41	—	2	1	no	—
37	C₂×Q₈	—	4	2	no	—
38	Q₁₆	—	4	2	no	—
39	Q₁₆	—	4	2	no	—
40	Q₁₆	—	4	2	no	—
41	Q₁₆	—	4	2	no	—
42	Q₁₆	D₈	1	1	yes	characteristic
43	SD₁₆	—	2	1	no	—
44	D₈	D₈	1	1	yes	characteristic
45	M₁₆	—	2	1	no	—
46	M₁₆	—	4	1	no	—
47	C₂×C₈	—	4	1	no	—
48	C₂×C₈	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈	C₂×D₄	1	1	yes	characteristic
50	C₄⋊C₄	—	8	2	no	—
51	C₄²	#41	1	1	yes	characteristic
52	C₁₆	—	2	1	no	—
53	C₁₆	—	4	1	no	—
54	C₁₆	—	4	2	no	—
55	C₁₆	D₈	1	1	yes	characteristic
56	C₁₆	D₈	1	1	yes	characteristic
57	Q₈	—	2	1	no	—
58	Q₈	—	8	2	no	—
59	Q₈	—	8	2	no	—
60	D₄	—	2	1	no	—
61	D₄	—	4	1	no	—
62	C₂×C₄	—	2	1	no	—
63	C₂×C₄	—	4	1	no	—
64	C₂×C₄	—	8	2	no	—
65	C₂×C₄	#121	1	1	yes	characteristic
66	C₈	—	4	1	no	—
67	C₈	C₂×D₈	1	1	yes	characteristic
68	C₈	C₂×D₈	1	1	yes	characteristic
69	C₈	#135	1	1	yes	characteristic
70	C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
71	C₂²	—	4	1	no	—
72	C₂²	#472	1	1	yes	characteristic
73	C₄	—	2	2	no	—
74	C₄	—	4	1	no	—
75	C₄	—	16	2	no	—
76	C₄	#459	1	1	yes	characteristic
77	C₄	#466	1	1	yes	characteristic
78	C₂	—	2	1	no	—
79	C₂	—	8	1	no	—
80	C₂	#1667	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#1667
commutator, Frattini, cosocle	C₄×C₈	C₂³
prime radical	#135	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ 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. #1373	C₂	semidirect
5. #1358	C₂	semidirect

Maximal subgroups

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

subgroup	index
#2247	2
#2214	2
#2170	2
#1373	2
#1358	2