Subgroups — #12465

Order 256.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2244	C₂	1	1	yes	maximal, characteristic, complemented
2	#2231	C₂	1	1	yes	maximal, characteristic, complemented
3	#2213	C₂	1	1	yes	maximal, characteristic
4	#2171	C₂	1	1	yes	maximal, characteristic, complemented
5	#1369	C₂	1	1	yes	maximal, characteristic, complemented
6	#1366	C₂	1	1	yes	maximal, characteristic, complemented
7	#1357	C₂	1	1	yes	maximal, characteristic, complemented
8	#509	—	2	1	no	—
9	#504	—	2	1	no	—
10	#499	C₂²	1	1	yes	characteristic
11	#493	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	#361	—	2	1	no	—
18	#360	—	2	1	no	—
19	#357	—	2	1	no	—
20	#357	—	2	1	no	—
21	#350	—	2	1	no	—
22	#346	C₂²	1	1	yes	characteristic
23	#135	D₄	1	1	yes	characteristic
24	C₂×D₈	—	2	1	no	—
25	C₂×D₈	—	2	1	no	—
26	#131	—	2	1	no	—
27	#127	—	2	1	no	—
28	#125	—	2	1	no	—
29	SD₃₂	—	4	1	no	—
30	D₁₆	—	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×D₄	—	4	1	no	—
45	C₂×D₄	—	8	1	no	—
46	Q₁₆	SD₁₆	1	1	yes	characteristic
47	SD₁₆	—	2	1	no	—
48	D₈	—	4	1	no	—
49	D₈	—	4	1	no	—
50	D₈	—	4	1	no	—
51	D₈	—	4	1	no	—
52	D₈	SD₁₆	1	1	yes	characteristic
53	M₁₆	—	2	1	no	—
54	M₁₆	—	4	1	no	—
55	C₂×C₈	—	4	1	no	—
56	C₂×C₈	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈	C₂×D₄	1	1	yes	characteristic
58	C₄⋊C₄	—	4	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	C₂³	—	8	1	no	—
67	Q₈	—	2	1	no	—
68	D₄	—	2	1	no	—
69	D₄	—	4	1	no	—
70	D₄	—	8	1	no	—
71	D₄	—	8	1	no	—
72	D₄	—	16	2	no	—
73	C₂×C₄	—	2	1	no	—
74	C₂×C₄	—	4	1	no	—
75	C₂×C₄	—	8	1	no	—
76	C₂×C₄	#121	1	1	yes	characteristic
77	C₈	—	4	1	no	—
78	C₈	C₂×SD₁₆	1	1	yes	characteristic
79	C₈	#135	1	1	yes	characteristic
80	C₈	#135	1	1	yes	characteristic
81	C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
82	C₂²	—	4	1	no	—
83	C₂²	—	16	2	no	—
84	C₂²	—	16	1	no	—
85	C₂²	#472	1	1	yes	characteristic
86	C₄	—	4	1	no	—
87	C₄	—	4	1	no	—
88	C₄	—	16	1	no	—
89	C₄	#463	1	1	yes	characteristic
90	C₄	#465	1	1	yes	characteristic
91	C₂	—	2	1	no	—
92	C₂	—	8	1	no	—
93	C₂	—	32	1	no	—
94	C₂	#1677	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#1677
commutator, Frattini, cosocle	C₄×C₈	C₂³
prime radical	#2213	C₂
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. #2244	C₂	semidirect
2. #2231	C₂	semidirect
4. #2171	C₂	semidirect
5. #1369	C₂	semidirect
6. #1366	C₂	semidirect
7. #1357	C₂	semidirect

Maximal subgroups

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

subgroup	index
#2244	2
#2231	2
#2213	2
#2171	2
#1369	2
#1366	2
#1357	2