Subgroups — #12466

Order 256.

93 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2245	C₂	1	1	yes	maximal, characteristic, complemented
2	#2231	C₂	1	1	yes	maximal, characteristic
3	#2214	C₂	1	1	yes	maximal, characteristic
4	#2171	C₂	1	1	yes	maximal, characteristic
5	#1369	C₂	1	1	yes	maximal, characteristic, complemented
6	#1366	C₂	1	1	yes	maximal, characteristic, complemented
7	#1358	C₂	1	1	yes	maximal, characteristic, complemented
8	#510	—	2	1	no	—
9	#504	—	2	1	no	—
10	#499	C₂²	1	1	yes	characteristic
11	#494	C₂²	1	1	yes	characteristic
12	#443	C₂²	1	1	yes	characteristic
13	#365	—	2	1	no	—
14	#364	C₂²	1	1	yes	characteristic
15	#364	C₂²	1	1	yes	characteristic
16	#364	C₂²	1	1	yes	characteristic
17	#362	—	2	1	no	—
18	#360	—	2	1	no	—
19	#358	—	2	1	no	—
20	#358	—	2	1	no	—
21	#350	—	2	1	no	—
22	#346	C₂²	1	1	yes	characteristic
23	#135	D₄	1	1	yes	characteristic
24	C₂×Q₁₆	—	2	1	no	—
25	C₂×Q₁₆	—	2	1	no	—
26	#131	—	2	1	no	—
27	C₄⋊Q₈	—	2	1	no	—
28	#125	—	2	1	no	—
29	Q₃₂	—	4	1	no	—
30	SD₃₂	—	4	1	no	—
31	M₃₂	—	2	1	no	—
32	M₃₂	—	2	1	no	—
33	M₃₂	—	2	1	no	—
34	M₃₂	—	4	1	no	—
35	C₂×C₁₆	—	4	1	no	—
36	C₂×C₁₆	D₄	1	1	yes	characteristic
37	C₂×C₁₆	D₄	1	1	yes	characteristic
38	#108	D₄	1	1	yes	characteristic
39	#107	—	2	1	no	—
40	#106	—	2	1	no	—
41	#104	—	2	1	no	—
42	C₄×C₈	C₂³	1	1	yes	characteristic
43	#41	—	2	1	no	—
44	C₂×Q₈	—	4	1	no	—
45	Q₁₆	—	4	1	no	—
46	Q₁₆	—	4	1	no	—
47	Q₁₆	—	4	1	no	—
48	Q₁₆	—	4	1	no	—
49	Q₁₆	SD₁₆	1	1	yes	characteristic
50	SD₁₆	—	2	1	no	—
51	D₈	SD₁₆	1	1	yes	characteristic
52	M₁₆	—	2	1	no	—
53	M₁₆	—	4	1	no	—
54	C₂×C₈	—	4	1	no	—
55	C₂×C₈	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈	C₂×D₄	1	1	yes	characteristic
57	C₄⋊C₄	—	4	1	no	—
58	C₄⋊C₄	—	8	1	no	—
59	C₄⋊C₄	—	8	1	no	—
60	C₄²	#41	1	1	yes	characteristic
61	C₁₆	—	2	1	no	—
62	C₁₆	—	2	1	no	—
63	C₁₆	—	4	1	no	—
64	C₁₆	—	4	1	no	—
65	C₁₆	—	4	1	no	—
66	Q₈	—	2	1	no	—
67	Q₈	—	8	1	no	—
68	Q₈	—	8	1	no	—
69	D₄	—	2	1	no	—
70	D₄	—	4	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	4	1	no	—
73	C₂×C₄	—	8	1	no	—
74	C₂×C₄	—	8	1	no	—
75	C₂×C₄	#121	1	1	yes	characteristic
76	C₈	—	4	1	no	—
77	C₈	C₂×SD₁₆	1	1	yes	characteristic
78	C₈	#135	1	1	yes	characteristic
79	C₈	#135	1	1	yes	characteristic
80	C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
81	C₂²	—	4	1	no	—
82	C₂²	#472	1	1	yes	characteristic
83	C₄	—	4	1	no	—
84	C₄	—	4	1	no	—
85	C₄	—	16	1	no	—
86	C₄	—	16	1	no	—
87	C₄	#463	1	1	yes	characteristic
88	C₄	#465	1	1	yes	characteristic
89	C₂	—	2	1	no	—
90	C₂	—	8	1	no	—
91	C₂	#1677	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#1677
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. #2245	C₂	semidirect
5. #1369	C₂	semidirect
6. #1366	C₂	semidirect
7. #1358	C₂	semidirect

Maximal subgroups

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

subgroup	index
#2245	2
#2231	2
#2214	2
#2171	2
#1369	2
#1366	2
#1358	2