Subgroups — #12463

Order 256.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2253	C₂	1	1	yes	maximal, characteristic, complemented
2	#2231	C₂	1	2	yes	maximal
3	#2170	C₂	1	1	yes	maximal, characteristic
4	#1373	C₂	1	1	yes	maximal, characteristic, complemented
5	#1366	C₂	1	2	yes	maximal, complemented
6	#504	—	2	1	no	—
7	#499	C₂²	1	2	yes	—
8	#443	C₂²	1	1	yes	characteristic
9	#367	—	2	1	no	—
10	#366	—	2	1	no	—
11	#365	—	2	2	no	—
12	#364	C₂²	1	1	yes	characteristic
13	#364	C₂²	1	2	yes	—
14	#360	—	2	2	no	—
15	#350	—	2	1	no	—
16	C₄×C₁₆	C₂²	1	1	yes	characteristic
17	#135	D₄	1	1	yes	characteristic
18	#131	—	2	1	no	—
19	#125	—	2	2	no	—
20	M₃₂	—	2	1	no	—
21	M₃₂	—	2	2	no	—
22	M₃₂	—	4	1	no	—
23	C₂×C₁₆	—	4	1	no	—
24	C₂×C₁₆	D₄	1	1	yes	characteristic
25	C₂×C₁₆	D₄	1	1	yes	characteristic
26	#108	D₄	1	1	yes	characteristic
27	#107	—	2	2	no	—
28	#106	—	2	2	no	—
29	#104	—	2	1	no	—
30	C₄×C₈	C₂³	1	1	yes	characteristic
31	#41	—	2	1	no	—
32	Q₁₆	Q₁₆	1	1	yes	characteristic
33	SD₁₆	—	2	1	no	—
34	D₈	Q₁₆	1	1	yes	characteristic
35	M₁₆	—	2	1	no	—
36	M₁₆	—	4	1	no	—
37	C₂×C₈	—	4	1	no	—
38	C₂×C₈	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈	C₂×D₄	1	1	yes	characteristic
40	C₄⋊C₄	—	4	2	no	—
41	C₄⋊C₄	—	8	2	no	—
42	C₄²	#41	1	1	yes	characteristic
43	C₁₆	—	2	1	no	—
44	C₁₆	—	4	1	no	—
45	C₁₆	—	4	2	no	—
46	C₁₆	SD₁₆	1	1	yes	characteristic
47	C₁₆	SD₁₆	1	1	yes	characteristic
48	Q₈	—	2	1	no	—
49	D₄	—	2	1	no	—
50	D₄	—	4	1	no	—
51	C₂×C₄	—	2	1	no	—
52	C₂×C₄	—	4	1	no	—
53	C₂×C₄	—	8	2	no	—
54	C₂×C₄	#121	1	1	yes	characteristic
55	C₈	—	4	1	no	—
56	C₈	C₂×SD₁₆	1	1	yes	characteristic
57	C₈	C₂×Q₁₆	1	1	yes	characteristic
58	C₈	#135	1	1	yes	characteristic
59	C₈	#137	1	1	yes	characteristic
60	C₂²	—	4	1	no	—
61	C₂²	#472	1	1	yes	characteristic
62	C₄	—	4	1	no	—
63	C₄	—	4	1	no	—
64	C₄	—	16	2	no	—
65	C₄	#461	1	1	yes	characteristic
66	C₄	#467	1	1	yes	characteristic
67	C₂	—	2	1	no	—
68	C₂	—	8	1	no	—
69	C₂	#1675	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#1675
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. #2253	C₂	semidirect
4. #1373	C₂	semidirect
5. #1366	C₂	semidirect

Maximal subgroups

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

subgroup	index
#2253	2
#2231	2
#2170	2
#1373	2
#1366	2